Field Theory of Interacting Crumpled Manifolds

g_{R} =  +  +  +  +⋯

Dissertation zur Erlangung des Grades Doktor der Naturwissenschaften (Dr. rer. nat.)

vorgelegt am

Fachbereich Physik der Universität Essen

von

Henryk A. Pinnow

aus Hameln

Essen 2002 2 Interacting Crumpled Manifolds

Abstract

The effect of a δ-interaction on a polymerized membrane of arbitrary internal dimension D has been studied. Depending on the dimensionality of membrane and embedding space, different physical scenarios are observed: The delocalization of a membrane from an attractive defect as well as steric repulsions. The difference of polymers from membranes is emphasized. For the latter, non-trivial contributions appear at the 2-loop level. Furthermore, a “massive scheme” inspired by calculations in fixed dimensions for scalar field theories has been exploited. Despite the fact that these calculations are only amenable numerically, it has been found that in the limit of D → 2 each diagram can be evaluated analytically. This property extends in fact to any order in perturbation theory, allowing for a summation of all orders. An analytically continued expression for the effective coupling of membranes in the scaling limit of large sizes as compared to the microscopic cutoff is obtained. Finally, the construction of an expansion of the effective coupling about D = 2 is presented. Applications to the case of self-avoiding membranes are mentioned.

1. Gutachter: Dr. habil. K.J. Wiese

2. Gutachter: Prof. Dr. L. Schäfer

Tag der Disputation: 19. Juli 2002

Contents

1 Introduction 5 1.1 Physics of fluctuating manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Tethered surfaces and their realizations . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Landau theory of the crumpling transition . . . . . . . . . . . . . . . . . . . . 9 1.4 Properties of the flat phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Properties of the crumpled phase . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Model and physical observables 13 2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Repulsive force exerted by a membrane on a wall . . . . . . . . . . . . . . . . 15 2.3 Monomer density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Unbinding transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Operator product expansion and one-loop result 21

4 Two-loop calculation in a MS-scheme 26 4.1 Operator product expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Numerical calculation in 0 < D < 2 . . . . . . . . . . . . . . . . . . . . . . . 27 4.3 RG-function and extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Calculation in fixed dimension: The massive scheme 33 5.1 Phantom manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.2 Self-avoiding membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Summation of the perturbation series in the limit D = 2 36 6.1 N-loop order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.2 The limit D → 2 and (2−D)-expansion on the torus . . . . . . . . . . . . . . 37 6.3 Asymptotic behavior of the series . . . . . . . . . . . . . . . . . . . . . . . . 43 6.4 The RG-functions in the bare coupling . . . . . . . . . . . . . . . . . . . . . . 46 6.5 Calculation in the renormalized coupling . . . . . . . . . . . . . . . . . . . . . 49 6.6 The limit d = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.7 Laplace-transformed picture . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.8 Discussion of the analytical continuation to D = 2. . . . . . . . . . . . . . . . 52

7 Higher order terms in the (2 − D)- expansion on the torus 55 7.1 Resummed contributions to the expansion in 2 − D up to the fourth order … 57 7.2 Renormalized coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.3 Discussion of possible forms of g̃(D, r) . . . . . . . . . . . . . . . . . . . . . 65

8 Discussion and conclusion 70

A Derivation of the kernel 72 B Mathematica^{◯ R}–programs for the generation of the series contributions in the 2−D-

expansion 73

C Calculation of the diagrams in the 2 − D-expansion 78

D Exact results for polymers 82 D.1 Universal 1/r - repulsion law for polymers . . . . . . . . . . . . . . . . . . . . 82 D.2 Strong coupling expansion for the renormalized coupling g . . . . . . . . . . . 84 D.3 Weak coupling expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 D.4 Semi-flexible polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

E Conformal mapping of the sectors 91

R References 94

1 Introduction

1.1 Physics of fluctuating manifolds

Interacting lines and more generally manifolds play an important role in many areas of modern physics. Examples of lines are self-avoiding polymers [1], vortex-lines in super-conductors [2], directed polymers in a disordered environment, also equivalent to surface growth [3], diffusion of particles and many more. Generalizing results to membranes often poses severe problems, but also new insight into physics. Recently, a lot of work has been devoted to self-avoiding membranes (see [4] for a review). From the point of view of statistical physics one major problem is to study the effect of interactions on the thermodynamical properties of extended fluctuating geometric objects. Generally, there are multi-particle attractive or repulsive interactions involved. One may divide these into two classes: Either (i) one may study the interaction of a single fluctuating object with itself as for instance the well known two-particle excluded volume interaction between any two monomers in a long chain polymer in a good solvent. There, the interaction leads to universal long-distance properties of chains as for example the anomalous scaling of the mean squared end-to-end distance. Or (ii), the interaction may act between different manifolds or between a single manifold and a fixed non-fluctuating object. It is then interesting to study how thermal fluctuations affect the depinning of the manifold from an attractive substrate as well as the steric repulsions from a wall. Both cases have in common that they are fairly well understood as long as the fluctuating objects are one-dimensional [1,5–7]. Referring to the example mentioned above, the long-distance properties of self-avoiding polymers, which are always crumpled on large scales, can be analyzed with renormalization group techniques [8–10], either in the continuous Edwards Hamiltonian [11],

H[r⃗] = \frac{1}{2} ∫_{x∈M} (∇r⃗(x))^{2} + \frac{b_{0}}{2} ∫_{x∈M} ∫_{y∈M} δ^{d}(r⃗(x) − r⃗(y)) ,  (1.1)

or by mapping this model on a local O(N) symmetric φ^{4}-theory in the limit of N = 0 components [12,6,1]. The critical exponents describing the long-distance properties are related to the critical exponents of the corresponding N-vector model at the critical point. What makes (1.1) a non-standard theory is that the interaction is nonlocal. The integration is over the internal volume of the manifold, that is, in the case of polymers along the chain. Obtaining the corresponding results for membranes poses considerable challenges. The generalization of polymers to 2D-surfaces are crystalline fixed-connectivity membranes as they appear for instance in the spectrin network of cell membranes. Considering ”phantom” membranes, which can freely fold into itself, the existence of a bending rigidity induced phase transition separating a high rigidity, low temperature flat phase from a low rigidity, high temperature crumpled phase is well established [13–18]. This is in contrast to polymers, which are always crumpled on large scales. The scaling properties of the crumpled phase of phantom membranes are described by the 2D generalization of the free field part in (1.1). Taking self-avoidance into account, which is modeled in (1.1) through the short range two-body interaction, we expect more open manifolds than those predicted by the free theory, which will be expressed in a nontrivial radius of gyration exponent. The squared radius of gyration R_{g} is defined as the mean squared distance of the manifold constituents from the center of mass. In the case of polymers R_{g} scales like the end-to-end distance. Much effort has been spent on calculating corrections to the radius of gyration exponent within an expansion in the deviation ε from the critical space dimension [19,20]. These calculations can not be performed directly in D = 2, since the naive scaling dimension of the coupling in (1.1) equals

ε(D, d) := [b_{0}] = 2D − \frac{2 − D}{2}d ,  (1.2)

where d denotes the dimension of the embedding space, and is always finite as D → 2. Equivalently, the critical embedding dimension defined through ε(D, d_{c}(D)) = 0 becomes ∞ in this limit. The reason is that the non-self-avoiding membrane densely fills out the embedding space, such that it always ”sees” the interaction. A way to circumvent this problem is to set up the expansion about any point (D < 0, d_{c}(D)) and to extrapolate along an appropriate path in the (D, d)-plane to the physically interesting point (D, d) = (2, 3) [4,21–26]. To second order in ε one then finds a radius of gyration exponent of ν ≈ 0.86 [19,20] , which is a strong correction with respect to the only logarithmic dependence in the non-interacting theory, but nevertheless still indicates existence of a crumpled phase. However, there is no unambiguous evidence for a crumpled phase in experiments [27–30]. Latest Monte-Carlo simulations on plaquette-models [31–34] starting from a discretization of the 2D generalized Hamiltonian (1.1) with system sizes of up to ≈ 17000 plaquettes shows considerable evidence for a vanishing of the above mentioned crumpling transition in the presence of self-avoidance, such that even on large scales fixed-connectivity membranes stay always flat with a radius of gyration exponent of ν ≈ 1. This outcome might be reconciliated with the field theoretic calculation by going to higher loop orders. The aim of this work is to develop techniques which allow to go beyond the two-loop result. We developed such techniques for a simplified model, which replaces the non-local interaction in (1.1) by an interaction with a single point in the embedding space and which is therefore related to case (ii). The corresponding physical situation we think of is the binding and unbinding of a long chain as e.g. a polymer or a membrane from a wall or the wetting of an interface. More precisely, we study the interaction of a single freely fluctuating manifold with another non-fluctuating, fixed object. Depending on whether the interaction is attractive or repulsive, one can distinguish two different scenarios: One may either observe a delocalization transition from an attractive substrate as in wetting phenomena or steric repulsions by a fluctuating manifold. Both cases have in common that excluded volume effects become important. The situation for polymers or a 1-dimensional interface is relatively simple: One knows that in this case polymers interacting with a defect or a wall (excluding or not half of the space) as well as

Figure 1.1: A fluctuating membrane interacting by excluded volume with a surface-like defect.

short-range critical wetting are in the same universality class [35–37]. For two-dimensional interfaces, the situation is more complicated, and a lot of effort has been spent to understand e.g. the delocalization transition [38–45]. Particularly, one is interested in critical wetting for the case of a short-ranged interaction potential. Then, it can not be approximated by a polynomial in the field r⃗ and the conventional field-theoretic approach known from ϕ^{4}−theory fails. This led to a couple of different ansätze [39,40,42–44], among others the functional renormalization group approach of [43,44].

Here we follow a different route: We start by constructing a systematic perturbative renormalization group treatment of the delocalization transition as well as the universal repulsive force exerted by a membrane on a point, line or more generally hyper-plane like defect. We therefore introduce a flexible “phantom” manifold of internal dimension 0 ≤ D ≤ 2. By introducing a δ-potential in a subspace E of dimension d′, part of the embedding space ℝ^{d+d′} (see fig. 1.1) we punish configurations crossing E. Neglecting the effect of self-avoidance between distinct points within the fluctuating manifold, the free energy of a given configuration reads:

H[r⃗] = ∫_{x∈M} d^{D}x \frac{1}{2}(∇r⃗(x))^{2} + g_{0} ∫_{x∈M} d^{D}x ∫_{y⃗∈E} d^{d′}y⃗ δ^{d+d′}(r⃗(x)−y⃗) ,  (1.3)

where r⃗(x) is the position of monomer x ∈ M in embedding space ℝ^{d+d′}, and M denotes the D-dimensional coordinate space of the manifold. In the case of a polymer this is simply the internal chain length. g_{0} is the attractive or repulsive interaction, of dimension (in inverse length-units)

ε(D, d) := [g_{0}] = D − \frac{2 − D}{2}d .  (1.4)

The interaction is naively relevant in the infrared for ε positive, and irrelevant for ε negative. In a perturbative treatment, ε is the natural expansion parameter. The situation is similar to self-avoiding membranes and thus (1.3) has been studied as a “toy-model” for the analysis of the renormalizability of the more complicated interaction in the generalized Edwards model (1.1) for self-avoiding polymerized membranes [46–48]. It was shown to be renormalizable for arbitrary manifold dimensions 0<D<2 [47,48]. This means that the large scale properties are universal, i.e. do neither depend on the regularization scheme used in these calculations, nor on the form of the contact interaction, as long as it is short-ranged. Universal quantities have thus been obtained to one-loop order. They are related, as we will be shown in section 2, to the correction to scaling exponent ω, which as usually in critical phenomena contains deviations from the long-distance scaling behavior.

The aim of this work is two-fold: First, we present the necessary techniques to treat the model (1.3) beyond the leading order. We explicitly perform a two-loop calculation which gives the correction to scaling exponent ω at order ε^{2}. Specializing to membranes one finds that the 2-loop result naively diverges in the limit of D → 2. This is a problem of the ε-expansion, since there, diagrams have to be evaluated at ε(D, d_{c}(D)) = 0, and taking D → 2 implies d_{c}(D) → ∞ causing the result to diverge. Following the above mentioned strategy which has been applied successfully to the calculation of the anomalous dimension of self-avoiding membranes we set up the expansion in ε about some point (D < 0, d_{c}(D)) and extrapolate along some appropriate path to physically interesting points (D, d). It will turn out that the result always depends strongly on the expansion point. Thus, the correction at the two-loop level is ambiguous as long as one is only interested in the limit D → 2. This motivated us to try a “massive scheme” in fixed dimension, i.e. at finite ε. It turns out that the limit D → 2 can then be taken and is regular. Even more, the 2-loop diagram, which for D < 2 can only be calculated numerically, can now be evaluated analytically. This striking property even holds for higher orders, and we are able to give an explicit – quite simple – expression for the perturbation series. In a second step the whole series can be summed and the strong coupling limit analyzed. This is one of the very few cases where one can indeed obtain the exact relation between bare and renormalized coupling in the limit of D → 2, and thus the exact β-function. This result does not depend on the explicit regularization and renormalization prescriptions, and is also obtained for a membrane of spherical or toroidal topology. Furthermore, it turns out that the behavior of the effective coupling, which is logarithmically diverging for large membrane sizes, together with the value of the correction-to-scaling exponent ω, which is therefore forced to be zero, can not be extrapolated from any finite loop expansion. In a final step, we lay the foundations for an expansion about D = 2. In contrast to the leading order, the first order corrections already depend on the cut-off procedure. We studied one specific procedure already in [49], which turns out to reproduce results for polymers at leading order approximately, and even exactly in d = 0. In order to obtain exact results, we decided to set up the expansion in 2 − D on a manifold of toroidal internal topology; this corresponds to periodic boundary conditions. Our aim is to expand the exponent ω about D = 2 and this way to prove the smoothness of the limit D → 2. This will be further discussed in the sections 6 and 7. The following introductory sections are a short exposé on tethered manifolds and the crumpling transition motivating Hamiltonians like (1.1) and (1.3). In section 2 we further discuss the model (1.3) and show, how the correction-to-scaling exponent ω is related to interesting physical observables, which might be also accessible to Monte-Carlo simulations.

1.2 Tethered surfaces and their realizations

Tethered surfaces are systems of particles (atoms or monomers) that are connected to form a regular two-dimensional array embedded in d-dimensional space [50] . Independently from the precise type of the 2D lattice as well as from the form of the bonding potential between the neighboring particles of the network, it is essential that the bonds between the adjacent atoms or monomers of the array can not be broken. This is in contrast to fluid membranes, where molecules making up the membrane can freely circulate within the surface. The distances between adjacent particles are subject to thermal fluctuations which are controlled by the nextnearest-neighbor bonding potentials. This is, too, in contrast to fluid membranes, whose particle density stays constant, while they undergo pronounced shape fluctuations. This property is the reason, why the question is still under debate whether the flat phase of fluid membranes becomes unstable with respect to thermal fluctuations or is rather stabilized, [51–55]. For tethered surfaces the situation is by far clearer: As long as no self-avoidance is taken into account, they are known to undergo a bending rigidity induced continuous phase transition separating a stable flat phase from a crumpled fractal phase. Due to the fixed connectivity tethered surfaces closely resemble linear polymers. The spatial conformation of a linear polymer can be described by a set of position vectors r⃗(x), where x is the internal index and r⃗(x) the location in embedding space of the indexed monomer. Then, a natural generalization to 2D surfaces is made by use of a two-dimensional vector x⃗ as an index denoting the internal position of the

Figure 1.2: Extended view (right) of the crystalline spectrin/actin network which forms the cytoskeleton of the red blood cell membrane (left) .

atom. Such use of a fixed (flat) internal space and its clear separation from the external variable r⃗, on which the Hamiltonian of the system depends, enables an unambiguous definition of the statistical measure for the configurational space, thus avoiding problems regarding a proper measure as they are addressed in [54] in the case of fluid surfaces.

A naturally occurring physical example of tethered membranes are the cytoskeletons of cell membranes, which is essential to their stability as well as functionality. The simplest and most thoroughly studied example is the cytoskeleton of mammalian erythrocytes (red blood cells). The red blood cell cytoskeleton is a tethered network of triangular plaquettes formed primarily by the proteins spectrin and actin (see fig. (1.2) from [4,56,57]). Inorganic realizations of crystalline membranes are graphitic oxide membranes, which are micron size sheets of solid carbon. They are formed by exfoliating carbon with a strong oxidizing agent [28–30]. Furthermore, metal dichalcogenides such as MoS_{2} have also been observed to form rag-like sheets [27].

1.3 Landau theory of the crumpling transition

In this section let us discuss an order parameter theory of the crumpling transition [17]. Any Ddimensional manifold embedded in d- dimensional space can be parametrized by

r⃗ : x⃗ ∈ ℝ^{D} ⟶ r⃗(x⃗) ∈ ℝ^{d} ,  (1.5)

where r⃗(x⃗) points to the position in embedding space of the manifold constituent labeled by x⃗. For D = 1, this represents a polymer, for D = 2 a membrane. Starting from a D- dimensional lattice with nearest neighbor interactions, a statistical description is developed by coarse graining this lattice so that the vector x⃗ becomes a continuous variable, which labels a “bloc” of lattice points. Translational invariance delimits the form of the free energy functional to depend only on the coarse grained tangent vectors t⃗_{α} = ∂r⃗(x⃗)/∂x_{α}, α = 1, …, D and its derivatives. An expansion à la Ginzburg-Landau for small tangents then leads to an effective free energy

H[r⃗] = ∫ d^{D}x [\frac{κ}{2} (∂_{α}∂_{α}r⃗)^{2} + \frac{t}{2} (∂_{α}r⃗)^{2} + u (∂_{α}r⃗ ∂_{β}r⃗)^{2} + ṽ (∂_{α}r⃗ ∂_{α}r⃗)^{2}]
+\frac{b}{2} ∫ d^{D}x ∫ d^{D}y δ^{d}(r⃗(x⃗) − r⃗(y⃗)) .  (1.6)

The last term is a nonlocal excluded volume term, which represents the effects of self-avoidance at large length scales. Identifying the tangents t⃗_{α} with a set of order parameters ϕ_{α}, α = 1, …, D, an analogy with the normal ϕ^{4} theory becomes apparent. However, this analogy is only valid at the mean-field level. At high temperatures one expects a crumpled phase, where t is positive for entropic reasons, while the other terms are irrelevant on large scales. κ denotes the bending rigidity, and the other elastic constants

t, u and ṽ are related to the Lamé-coefficients of Landau elastic theory. At low temperatures, the microscopic surface tangents tend to align in order to minimize the microscopic bending energy, driving the manifold to form a flat phase (t < 0). The manifold is stabilized by the anharmonic terms, provided that u > 0 and v := ṽ + u/D > 0. There is a continuous transition between these phases when t ≡ α(T − T_{c}) = 0. This can be seen within mean-field theory. In the absence of self-avoidance we make the ansatz

r⃗(x⃗) = ∑_{i=1}^{D} ξx_{i}e⃗_{i} ,  (1.7)

where the sum is over two orthogonal cartesian unit vectors spanning a flat surface. Minimizing the corresponding mean-field effective potential,

V (ξ) = Dξ^{2} (\frac{t}{2} + (u + ṽD)ξ^{2}) ,  (1.8)

leads to the following solution:

ξ^{2} = \{ \begin{matrix}0 , t ≥ 0 \\ −\frac{t}{4(u+ṽD)} , t < 0 \end{matrix}

Consequently, there is a flat phase for t < 0 and a crumpled phase for t > 0, separated by a crumpling transition at t = 0. Figure (1.3) shows a “phantom” membrane in typical states below, above and at the crumpling transition (Monte-Carlo samples from [59]).

1.4 Properties of the flat phase

Fluctuational states around the ordered state can be parametrized as

r⃗(x⃗) = ξ [(x_{α} + u_{α})e⃗_{α} + h_{β}e⃗_{β}] ,  (1.9)

where the summation over α is as in (1.7), while the e⃗_{β} are orthogonal to the latter. Specializing to membranes in d = 3 there is a single orthogonal space dimension. Inserting (1.9) into (1.6) leads to lowest non-trivial order to

H[u, h] = ∫ d^{D}x [\frac{κ}{2} (Δh)^{2} + μu_{ij}^{2} + \frac{λ}{2}u_{kk}^{2}] ,  (1.10)

Figure 1.3: Crumpled phase (a), crumpling transition (b) and flat phase (c). No self-avoidance.

where μ = 4uξ^{4}, λ = 8ṽξ^{4} and u_{ij} = \frac{1}{2}[∂_{i}u_{j} + ∂_{i}u_{j} + ∂_{i}h_{β} ∂_{j}h_{β}] denotes the strain matrix. It was shown in [60] that the renormalized bending rigidity grows at large distances and stabilizes the stretched phase against undulations. For small wavevectors q⃗ one finds an effective rigidity κ_{R} with

κ_{R}(q) ∼ \sqrt{k_{B}T K_{0}} q^{−1} ,  K_{0} = \frac{4μ(μ + λ)}{2μ + λ} .  (1.11)

Qualitatively, this is easily understood intuitively: Consider a low energy state of the nearly flat membrane, which is for instance a single low-wavevector undulation ^{1}. It does not necessitate any stretching of bonds between adjacent molecules. However, adding to the long-wavelength deformation with wavevector q⃗_{1} another short-wavelength deformation q⃗_{2} with |q⃗_{1}| ≪ |q⃗_{2}| and q⃗_{1} ⊥ q⃗_{2}, this will induce a space varying gaussian curvature, which inevitably forces to locally stretch and squeeze bonds. Thus, the free energy of the short wavevector undulation is raised up with respect to the same undulation on the flat background. Correspondingly to (1.11), the normal-normal correlation function stays always finite:

lim_{|x|→∞} ⟨n⃗(x)n⃗(0)⟩ < ∞ ,  (1.12)

where n⃗(x) denotes the local unit normal vector on the surface. Therefore, the membrane stays always oriented. The symmetry is broken, and the membrane is flat. This seems to be a violation of the Mermin-Wagner theorem: In fact, the fluctuations in the membrane give rise to long-range interactions, to which the Mermin-Wagner theorem does not apply.

1Out-of-plane fluctuations called undulations are to be distinguished from internal phonon modes.

1.5 Properties of the crumpled phase

In the crumpled phase the free energy is governed by an entropic elasticity [15] such that (1.6) reduces to

H_{0}[r⃗] = \frac{t}{2} ∫ d^{D}x (∇r⃗(x))^{2} + irrel. terms .  (1.13)

The spatial extend of the manifold is characterized by its radius of gyration, which is defined as

R_{g}^{2} = \frac{1}{2N^{2}} ∑_{i,j} ⟨(r⃗_{i} − r⃗_{j})^{2}⟩_{H_{0}} .  (1.14)

In the case of polymers N denotes the number of segments, while for 2D surfaces, correspondingly, it is the number of plaquettes. The radius of gyration is the mean squared distance of the surface constituents from the center of mass of the manifold. It scales with the linear size L of the manifold like

R_{g} ∼ L^{ν} ,  0 ≤ ν ≤ 1  (1.15)

and is related to the plaquette density correlation function, which is easily accessible experimentally by light scattering, small-angle X-ray scattering etc. [61]. For large manifolds R_{g} scales like the two-point function

\frac{1}{2d} ⟨(r⃗(x⃗) − r⃗(y⃗))⟩_{H_{0}} = \frac{|x⃗ − y⃗|^{2−D}}{(2 − D)S_{D}} ,  (1.16)

where a is a microscopic cutoff. One then finds:

R_{g} ∼ \{ \begin{matrix}L^{\frac{2−D}{2}} , D < 0 \\ (ln \frac{L}{a})^{1/2} , D = 2 \end{matrix}

Taking self-avoidance into account gives rise to corrections to the exponent ν. This can be understood within the Flory approximation: It consists in replacing r⃗(x⃗) in (1.6) by the radius of gyration and derivatives with respect to x_{α} by 1/L as well as the integration over the internal volume by L^{D}, leading to

H ≃ κ L^{D−4}R_{g}^{2} + t L^{D−2}R_{g}^{2} + (u + Dṽ) L^{D−4}R_{g}^{4} + b L^{2D}R_{g}^{−d} .  (1.17)

The contribution from the bending rigidity κ can always be neglected with respect to the terms proportional to t and u + Dṽ, respectively. Minimizing (1.17) provides:

R_{g} ∼ \{ \begin{matrix}L , as D ≤ d and t < 0 \\ L^{ν_{Flory}} , ν_{Flory} = \frac{2+D}{2+d} , t > 0 \end{matrix}

For polymers the Flory approximation makes predictions, which are sufficient for experiments. The prediction for membranes in d = 3, ν = \frac{4}{5}, is close to what has been found in field theoretic calculations to two loops (ν ≈ 0.86).

2 Model and physical observables

2.1 The model

We start from the manifold Hamiltonian (1.3). We split the total embedding space ℝ^{d+d′} into E and its orthogonal complement E_{⊥} of dimension d. Each x ∈ V points to a point r⃗(x)=(r⃗_{⊥}(x), r⃗_{∥}(x)), with r⃗_{∥}(x) ∈ E and r⃗_{⊥}(x) ∈ E_{⊥}. The integration over the subspace E is then trivial and gives

∫_{y⃗∈E} δ^{d+d′}(r⃗(x)−y⃗) = δ^{d}(r⃗_{⊥}(x)) .  (2.1)

In the partition function

Z_{total} = ∫ D \left[{ \atop r}\right] exp(−H \left[{ \atop r}\right])  (2.2)

the contributions from the parallel and orthogonal components of r⃗(x) factorize as

Z_{total} = Z_{0} × Z(g_{0})  (2.3)

with

Z_{0} = ∫ D[r⃗_{∥}] exp (−\frac{1}{2}(∇_{∥}r⃗(x))^{2})  (2.4)
Z(g_{0}) = ∫ D[r⃗_{⊥}] exp(−H_{⊥}[r⃗_{⊥}])  (2.5)
H_{⊥}[r⃗_{⊥}] = ∫_{M} d^{D}x (\frac{1}{2}(∇r⃗_{⊥}(x))^{2} + g_{0} δ^{d}(r⃗_{⊥}(x))) .  (2.6)

Since Z_{0} is trivial, we will only consider H_{⊥}[r⃗_{⊥}] and shall drop the subscript _{⊥} for notational simplicity. We keep in mind that cases with d < D make sense, for instance a polymerized (non-selfavoiding) membrane interacting with a wall is described by (2.6) setting D = 2 and d = 1.

Let us discuss (2.6) in more detail: The first term is the elastic energy of the manifold which is entropic in origin. We have scaled elasticity and temperature to unity. The second term models the interaction of the manifold with a single point at the origin in the external d-dimensional space, but we remind that the physical interpretation may well be that of a line or surface. The coupling constant g_{0} may either be positive (repulsive interaction) or negative (attractive interaction). We now give the dimensional analysis. In coordinate-space units, the engineering dimensions are

dim[x] = 1
ν := dim[r⃗] = \frac{2 − D}{2}

ε := dim[∫_{M} d^{D}x δ^{d}(r⃗(x))] = D − νd  (2.7)
g_{0} ∼ μ^{ε}
dim[μ] = −1 ,

Figure 2.1: Critical line defined through ε = 0 ⇔ d_{c}(D) = \frac{2D}{2−D}. The interaction is relevant for points that lie above that line.

where

μ ≡ \frac{1}{L}  (2.8)

is an inverse length scale. The interaction is naively relevant for ε > 0, i.e. d < d_{c} with (see figure 2.1)

d_{c} = \frac{2D}{2−D} ,  (2.9)

irrelevant for ε < 0 and marginal for ε = 0. It has been shown [47,48] that the model is renormalizable for 0 < D < 2 and ε ≥ 0. Results for negative ε are obtained via analytical continuation. One can define the renormalized coupling g as

g := \frac{N}{V_{M}} [Z(0) − Z(g_{0})] L^{ε} ,  (2.10)

Figure 2.2: RG-function and flow for increasing manifold size L for the dimensionless renormalized coupling g:

(a) in the case ε>0, (b) in the case ε<0, (c) in the case ε = 0.

where the normalization N depends on the definition of the path-integral (but not on L) and is chosen such that

g = g_{0}L^{ε} + O(g_{0}^{2}) .  (2.11)

Universal quantities are obtained at fixed-points of the β-function, which is defined as

β(g) := μ \frac{∂g}{∂μ}|_{g_{0}} .  (2.12)

The β-function thus describes, how the effective coupling g changes under scale transformations, while keeping the bare coupling g_{0} fixed. Let us already anticipate the 1-loop result, which we derive later. It reads

β(g) = −εg + \frac{1}{2}g^{2} + O(g^{3}) ,  (2.13)

where g is the dimensionless renormalized coupling. Apart from the trivial solution, g = 0, the flow equation given by (2.12) and (2.13) has a non-trivial fixed point at the zero of the β-function

g^{∗} = 2ε + O(ε^{2}) .  (2.14)

We shall show below that the scaling behavior is described by the slope of the RG-function at the fixed point, which is universal as a consequence of renormalizabilty. The long-distance behavior is then governed by the δ-interaction as considered in our model (2.6), which is the most relevant operator at large scales. Let us now discuss possible physical situations (see fig. 2.2):

(a) ε>0: The RG-flow has an infrared stable fixed point at g^{∗} > 0 and an IR-unstable fixed point at g = 0. The latter describes an unbinding transition whose critical properties are given by the non-interacting system, while the non-trivial IR stable fixed point determines the long-distance properties of the delocalized state, the long-range repulsive force exerted by the fluctuating manifold on the origin – which we remind may be a point, a line or a plane.

(b) ε<0: Now, the long-distance behavior is Gaussian, while the unbinding transition occurs at some finite value of the attractive potential, g^{∗}<0, which corresponds to an infrared unstable fixed point of the β-function. Below g^{∗} the manifold stays always attracted.

(c) ε = 0: This is the marginal situation, where the transition takes place at g^{∗} = 0; we expect logarithmic corrections to scaling.

Note that in the presence of an impenetrable wall constraining the configurational space strictly to half of the embedding space, the above considerations should still apply, when shifting the interaction strength appropriately. We shall discuss that in 2.4.

2.2 Repulsive force exerted by a membrane on a wall

Since we are mainly interested in the long-distance properties of membranes for which always ε > 0 (this is (a) on fig. 2.2), let us try to calculate the repulsive force exerted by the membrane on the origin in the case that this point is strictly forbidden. We will derive a universal expression for this force [48]. We need the (not normalized) membrane density at position r⃗

Z(r⃗, g_{0}) := \frac{1}{V_{M}} ∫ D[r⃗] ∫_{M} d^{D}x δ^{d}(r⃗(x) − r⃗) exp[−H] .  (2.15)

Since the δ-interaction also appears in H, we can relate the density at the origin to the derivative of the partition function with respect to g_{0}:

Z(0, g_{0}) = − \frac{1}{V_{M}} \frac{∂}{∂g_{0}} |_{L} Z(g_{0}) = \frac{1}{N} \frac{∂(gL^{−ε})}{∂g_{0}} |_{L} ,  (2.16)

where g is the renormalized or effective coupling defined in (2.10). Since Z(0, g_{0}) is dimensionless, it depends on the dimensionless combination g_{0}L^{ε} (and ε) only. Using μ = 1/L, we obtain

N Z(0, g_{0}) = L^{−ε} \frac{∂g}{∂g_{0}} |_{L} ≡ −\frac{β(g)}{εg_{0}L^{ε}} .  (2.17)

For the further considerations, it is sufficient to know the behavior of the β-function close to the nontrivial zero. Expanding about g=g^{∗}, we have

β(g) = (g − g^{∗})ω(g^{∗}) + O((g − g^{∗})^{2}) .  (2.18)

Combining (2.16), (2.17) and (2.18), we get

\frac{∂g}{∂(g_{0}L^{ε})}|_{L} = −\frac{β(g)}{εg_{0}L^{ε}} = −\frac{(g − g^{∗})ω(g^{∗})}{εg_{0}L^{ε}} .  (2.19)

The solution of this differential equation is

g − g^{∗} ∼ (g_{0}L^{ε})^{−ω(g^{∗})/ε} + ….  (2.20)

Using (2.17), this implies the scaling law

Z(0, g_{0}) ∼ (g_{0}^{1/ε}L)^{−ω(g^{∗})−ε} .  (2.21)

We now use this result to derive a scaling law for Z(r⃗, g_{0}), with r⃗ ≠ 0. In addition to g_{0}L^{ε} (and ε), Z(r⃗, g_{0}) depends on r⃗; taking into account rotational symmetry and dimensionality, it depends on r⃗ only through the dimensionless combination r/L^{ν}, with r = |r⃗|:

Z(r⃗, g_{0}) = Z(r/L^{ν}, g_{0}L^{ε}) .  (2.22)

For Z(r⃗, g_{0}), the most interesting limit is that of g_{0} → ∞. Physically, this corresponds to strictly forbidding monomers to be at the origin. Therefore it is clear that this limit is well-behaved, and

Z_{∞}(r/L^{ν}) := lim_{g_{0}→∞} Z(r/L^{ν}, g_{0}L^{ε})  (2.23)

is finite. In order to be consistent with (2.21) it has to obey a power law in the scaling regime r ≪ L^{ν}

Z_{∞}(r/L^{ν}) ∼ (r/L^{ν})^{θ} .  (2.24)

Comparing the L-dependence of Z_{∞}(r/L^{ν}) and Z(0, g_{0}), we obtain the exponent identity

θ = \frac{ε + ω(g^{∗})}{ν} .  (2.25)

Finally, from (2.24) we derive the repulsive force between the origin and the manifold

f⃗(r⃗) = ∇r⃗ ln Z_{∞}(|r⃗|/L^{ν}) = θ\frac{r⃗}{r^{2}} .  (2.26)

Note that to derive this result, k_{B}T has been set to 1. Reestablishing the temperature-dependence, we find

f⃗(r⃗) = k_{B}T θ\frac{r⃗}{r^{2}} .  (2.27)

Also note that this argument gives θ = 0 at the Gaussian fixed point, which is necessary since for g_{0} = 0 no force is exerted on the membrane.

2.3 Monomer density

Let us once again turn to the ensemble as defined in (2.16). The expectation value of the monomer density at the potential, ⟨n(0)⟩, where also the manifold is pinned, is obtained from

⟨n(0)⟩ = \frac{1}{Z(0, g_{0})} ∫ D[r⃗] ∫_{V_{M}} δ^{d}(r⃗(x)) δ^{d}(r⃗(s)) e^{−H[r⃗]} = −\frac{∂}{∂g_{0}} ln Z(0, g_{0}) (2.28)

Since g = \frac{N}{V_{M}} [Z(0) − Z(g_{0})]L^{ε} and \frac{∂}{∂g_{0}} = L^{ε} \frac{∂}{∂z},

⟨n(0)⟩ = \frac{1}{g_{0}} (1 + \frac{ω(z)}{ε}) .  (2.29)

where ω(z) = −εz(∂β/∂z)/β(z) and β(z) = −εz∂g/∂z. In particular, lim_{z→∞} ω(z) = ω(g^{∗}) > 0. The monomer density can be measured in a Monte-Carlo simulation and therefore offers a possibility also to check predictions for membranes (D = 2). Let us point out that starting from the natural assumption that as long as ε > 0 the monomer density at the potential stays finite in the scaling limit of large manifolds one may directly deduce the existence of a strong coupling expansion of the effective coupling g(z):

g(z) = g^{∗} − s(ln z)z^{−ω/ε} + O(z^{−w}) ,  (2.30)

where s denotes some function that increases at most subexponentially and w > ω/ε > 0.

2.4 Unbinding transition

Let us discuss the physical situation at the UV-stable fixed point in figure 2.2. The fixed point corresponds to a delocalization transition of the manifold, which is at vanishing coupling g^{∗}=0 for ε>0 and at some finite attractive coupling g^{∗}<0 for ε<0. In the localized phase g<g^{∗}, correlation functions such as ⟨[r⃗(x) − r⃗(y)]^{2}⟩ and the associated correlation length ξ_{∥} (in the D-dimensional internal space) should be finite, as well as the radius of gyration ξ_{⊥}. Approaching the transition point these quantities diverge as [45]

ξ_{∥} ∼ (g^{∗}−g)^{−ν_{∥}} , ξ_{⊥} ∼ (g^{∗}−g)^{−ν_{⊥}} .  (2.31)

Since ξ_{⊥} ∼ ξ_{∥}^{ν}, the exponents ν_{∥} and ν_{⊥} are related through

ν_{⊥} = ν_{∥}ν ,  (2.32)

ν being the dimension of the field (2.7). To relate ν_{∥} to ω, we first observe that ξ_{∥} depends only on g_{0} and the membrane size L, or its inverse μ = 1/L. Writing ξ_{∥} as a function of the renormalized coupling g and the scale μ, we thus have

0 = μ\frac{∂}{∂μ}ξ_{∥}(g, μ)
= μ\frac{∂}{∂μ} (\frac{1}{μ}f(g))
= \frac{1}{μ} (−f(g) + β(g)f′(g)) ,  (2.33)

where the dimension-full factor 1/μ has been factored from the dependence on the dimensionless renormalized coupling g. Using that \frac{∂}{∂g}ξ_{∥}(g) = \frac{1}{μ}f′(g), this can be rewritten as

ξ_{∥}(g) = β(g)\frac{∂}{∂g} ξ_{∥}(g)
≈ ω(g^{∗})(g − g^{∗})\frac{∂}{∂g} ξ_{∥}(g) .  (2.34)

The solution to the above equation is

ξ_{∥} ∼ |g − g^{∗}|^{1/ω(g^{∗})} .  (2.35)

This leads to the identification

ν_{∥} = −\frac{1}{ω(g^{∗})} ,  ν_{⊥} = −\frac{ν}{ω(g^{∗})} .  (2.36)

Note that ω(g^{∗})<0 at the transition. Specializing to (D, d)=(1, 1), we find

ν_{⊥} = 1 , ν_{∥} = 2 .  (2.37)

These exponents are also valid for the delocalization transition of a 1-dimensional interface from an attractive hard wall in 2-dimensional bulk space [39,45,37]. This can be understood as follows: The partition function of a fluctuating polymer interacting with a δ-defect at the boundary of a hard wall can be written as

Z = \frac{1}{2} ∑_{n=0}^{∞} Z_{n} (\frac{1}{2})^{n} e^{−nβE} ,  (2.38)

⟶

Figure 2.3: A polymer (here for simplicity directed) in interaction with a wall (thick dots). Left before and right after flapping up. The grey area is impenetrable.

where the sum runs over the number of contacts with the defect hyper-plane. E denotes the contact energy, β = (k_{B}T)^{−1} and Z_{n} is the constrained partition function of the free interface having exactly n points of contact with the defect. The powers of \frac{1}{2} only appear in the presence of an impenetrable defect and reflect the fact that all configurations in the presence of a hard wall can be obtained by flapping up those parts of the polymer which have penetrated the line (see figure 2.3). This accounts for a factor \frac{1}{2} for each possible flap. Thus, the hard wall constraint translates into a shift in the binding energy according to

E_{no wall} ⟶ E_{wall} = E_{no wall} + k_{B}T ln 2 .  (2.39)

The delocalization transition occurring for (D, d)=(1, 1) at vanishing potential strength is shifted to an attractive interaction dependent on the temperature. Due to the correspondence (2.39) the exponents characterizing the transition remain unchanged.

20 Interacting Crumpled Manifolds

3 Operator product expansion and one-loop result

The partition function of the model has been defined in (2.5). Analogously, we define the expectation value of an observable O(r) as (we denote r for r⃗ from now on)

⟨O(r)⟩_{g_{0}} := \frac{∫ D[r] exp(−H[r])}{Z(g_{0})} .  (3.1)

Most physical observables can be derived from expectation values of general M-point vertex operators like

O(r) = ∏_{j=1}^{M} e^{ip_{j}r(s_{j})} .  (3.2)

Before we go on, let us make some changes in normalizations which will be helpful in the following. First, we rescale the field and the coupling constant according to^{2}

r ⟶ (2−D)^{1/2}r , g_{0} ⟶ (2−D)^{d/2}g_{0} .  (3.3)

Second, we change the integration over internal coordinates to

∫_{x} := \frac{1}{S_{D}} ∫ d^{D}x , S_{D} = 2\frac{π^{D/2}}{Γ(D/2)}  (3.4)

being the volume of the D-dimensional unit-sphere. Third, the normalization of the δ-distribution is changed to

δ̃^{d}(r(x)) := (4π)^{d/2}δ(r(x)) = ∫_{k} e^{ikr(x)}  (3.5)

with

∫_{k} := π^{−d/2} ∫ d^{d}k .  (3.6)

The advantage of these normalizations is that

∫_{k} e^{−k^{2}} = 1 , ∫_{x} Θ(|x| < 1) = 1 .  (3.7)

The model in the new normalizations now reads

H[r] = \frac{1}{2 − D} ∫_{x} \frac{1}{2} (∇r(x))^{2} + g_{0} ∫_{x} δ̃^{d}(r(x)) ,  (3.8)

and (due to the factor of 1/(2 − D) in the above definition) the two-point correlator is

C(x_{i}−x_{j}) := \frac{1}{2d} ⟨[r(x_{i}) − r(x_{j})]^{2}⟩_{0} = |x_{i} − x_{j}|^{2ν} .  (3.9)

2At this stage this is to be undertood only as a formal procedure, that simplifies calculations. All loop integrals are normed with respect to the one loop integral. The precise meaning in the limit D → 2 is discussed in section (6.8).

We now proceed with the calculation of physical observables. As an explicit example, let us consider the perturbation expansion of the 1-point vertex operator

Z^{(p)} := Z(g_{0})⟨e^{ipr(s)}⟩_{g_{0}} = ∑_{N=0}^{∞} \frac{(−g_{0})^{N}}{N!} Z_{N}^{(p)}, ,  (3.10)

where

Z_{N}^{(p)} = ⟨e^{ipr(s)} ∏_{i=1}^{N} ∫_{x_{i}} δ̃^{d}(r(x_{i}))⟩_{0} .  (3.11)

This can be written as

Z_{N}^{(p)} = ∏_{i=1}^{N} ∫_{x_{i}} ∫_{k_{i}} δ̃^{d}(p + ∑ k_{i}) × ⟨exp [i(∑_{i=1}^{N} k_{i}r(x_{i}) + pr(s))]⟩_{0} ,  (3.12)

where we have already integrated out a global translation of the field r. The Gaussian average is:

⟨exp [i(∑_{i=1}^{N} k_{i}r(x_{i}) + pr(s))]⟩_{0} = exp [\frac{1}{2} ∑_{i,j=0}^{N} k_{i}k_{j}C(x_{i}−x_{j})] , k_{0} = p .  (3.13)

The Z_{N}^{(p)} in (3.11) posses short distance singularities for N≥2. In order to analyze these singularities we will use the techniques of normal ordering and operator product expansion in the sequel. In our problem the procedure of normal ordering a vertex operator e^{ik_{j}r(x_{j})} turns out to be quite simple: In any expectation value, we factorize out the contractions between the vertex operators. At one-loop order or equivalently at second order in (3.10) this means that

e^{ik_{1}r(x_{1})}e^{ik_{2}r(x_{2})} = :e^{i(k_{1}r(x_{1})+k_{2}r(x_{2}))}: e^{k_{1}k_{2}C(x_{1}−x_{2})} .  (3.14)

This can be understood as a definition of the normal ordered product :e^{i(k_{1}r(x_{1})+k_{2}r(x_{2}))}:. Note, that equality in the above equation is meant in the sense of operators, that is when inserted into expectation values within the free theory. Let us now turn to an explicit derivation of the OPE of two δ-interactions, i.e. we study (3.14) for small distances x_{1}−x_{2}. Since the short-distance singularities appear only in the internal contractions, the normal ordered product in the r.h.s. of (3.14) is regular for small distances x_{1}−x_{2} and thus can be expanded therein. We then project the resulting terms on the corresponding operators. For that purpose we make a change in the momentum variables according to

( \begin{matrix}k_{1} \\ k_{2} \end{matrix} ) ⟶ ( \begin{matrix}\frac{k_{1}}{2} − k_{2} \\ \frac{k_{1}}{2} + k_{2} \end{matrix} ) .  (3.15)

In internal space we change coordinates to the center of mass system:

x̄ := \frac{1}{2}(x_{1} + x_{2}) , y := x_{2} − x_{1} .  (3.16)

Thus, the r.h.s. of (3.14) takes the form

:e^{i[( \frac{k_{1}}{2} −k_{2})r(x̄−y/2)+( \frac{k_{1}}{2} +k_{2})r(x̄+y/2)]}: e^{(k_{1}^{2}/4−k_{2}^{2})C(y)} =

:e^{ik_{1}r(x̄)} (1 + ik_{2}y∇r(x̄) + ik_{1} \frac{1}{2!} \frac{y^{α}y^{β}}{4} ∇_{α}∇_{β}r(x̄) − \frac{(k_{2}y∇r(x̄))^{2}}{2!} + …): ×

(1 + \frac{k_{1}^{2}}{4} C(y) + …) e^{−k_{2}^{2}C(y)} .  (3.17)

Integration over k_{1}, k_{2} yields the OPE of δ̃^{d}(r(x_{1})) with δ̃^{d}(r(x_{2})):

δ̃^{d}(r(x̄ − \frac{y}{2}))δ̃^{d}(r(x̄ + \frac{y}{2})) = y^{−νd}δ̃^{d}(r(x̄)) − y^{−νd+2ν} \frac{1}{4} ∂_{r}^{2}δ̃^{d}(r(x̄))

−y^{−νd−2ν+2} \frac{1}{4D}δ̃^{d}(r(x̄))(∇r(x̄))^{2}

−y^{−νd+2} \frac{1}{16D}(−Δ_{r})δ̃^{d}(r(x̄))(∇r(x̄))^{2}

+y^{−νd+2} \frac{1}{8D} ∂_{r_{i}} δ̃^{d}(r(x̄))Δr_{i}(x̄) + . . . . .  (3.18)

Restricting ourselves to the most relevant next to leading operators for internal dimension D smaller than, but close to 2, the operator product expansion reads:

δ̃^{d}(r(x̄ − \frac{y}{2}))δ̃^{d}(r(x̄ + \frac{y}{2})) = y^{−νd}δ̃^{d}(r(x̄)) + ∑_{n=1}^{∞} y^{−νd+n(D−2)} \frac{(−∂_{r}^{2})^{n}δ̃^{d}(r(x̄))}{4^{n}n!} . (3.19)

The OPE-coefficient in front of the leading operator, which is the δ-interaction itself, carries the leading short distance singularity. In the following we introduce some diagrammatic notation. Denoting := δ̃^{d}(r(x)) we abbreviate the projection of two nearby δ-interactions on the δinteraction itself as

(  ^{| )} = y^{−νd} .  (3.20)

We now analyze short-distance divergences of the perturbation expansion using the OPE in (3.18). Being interested in the leading scaling behavior, we drop all subdominant terms for 0 < D < 2, such that we only need the leading OPE-coefficient (3.20). The integral over the relative distance in ∫_{y} ( | ) is logarithmically divergent for ε=0. In order to define a counter term we use dimensional regularization, i.e. we set ε>0. The integral is then UV-convergent, but IR-divergent. An IR regulator or equivalently a subtraction scale μ=L^{−1} has to be introduced to define the subtraction operation. Generally, we integrate over all distances bound by the inverse subtraction scale L. The OPE-coefficient in (3.20) then reduces to a number as

⟨  ^{| ⟩}_{L} := ∫_{|y|<L} (  ^{| )} = L^{ε}f(ε, D) .  (3.21)

Let us use the scheme of minimal subtraction (MS). The internal dimension of the membrane is fixed and (3.21) is expanded in a Laurent-series in ε, starting here at order ε^{−1}. Denoting the term of order ε^{p} in ⟨ | ⟩_{L}|_{L=1} by ⟨ | ⟩_{ε^{p}}, the simple pole in (3.21) is

⟨  ^{| ⟩}_{ε^{−1}} = \frac{1}{ε} .  (3.22)

Of course, in this case, due to our normalizations, this is exact. Let us rewrite the Hamiltonian in (2.6) in terms of the renormalized dimensionless coupling g. For perturbative calculations it is custom to write

g_{0} = gZ_{g}μ^{ε} ,  (3.23)

where Z_{g} is the renormalization factor, which is fixed by demanding that observables remain finite in the limit of ε → 0. The Hamiltonian becomes

H[r] = \frac{1}{2−D} ∫_{x} \frac{1}{2}(∇r(x))^{2} + ∫_{x} gZ_{g}μ^{ε} δ̃^{d}(r(x)) .  (3.24)

We find to one-loop order

Z_{g} = 1 + \frac{g}{2}⟨  ^{| ⟩}_{ε^{−1}} + O(g^{2}) .  (3.25)

There is no field-renormalization since the elastic part has also to describe the manifold far away from the origin, for which the interaction term can be neglected. Formally, of course, this follows from the OPE. The renormalization group β-function is defined as

β(g) := μ\frac{d}{dμ}|_{g_{0}} g ,  (3.26)

from what follows together with (3.23) that

β(g) = \frac{−εg}{1 + g \frac{∂}{∂g} ln Z_{g}} .  (3.27)

To one-loop order we obtain from (3.27) and (3.25) as anticipated earlier

β(g) = −εg + \frac{g^{2}}{2} + O(g^{3}) .  (3.28)

The long-distance behavior of the theory is governed by the IR–stable fixed point of the RG– flow. Fixed points are zeros of the β–function:

β(g^{∗}) = 0 ⇒ g^{∗} = 0 or g^{∗} = 2ε .  (3.29)

In section 2.2, we had defined the correction-to-scaling exponent ω. It is obtained from the slope-function ω(g), defined as

ω(g) := \frac{d}{dg} β(g) .  (3.30)

The correction to scaling exponent is ω(g) evaluated at the fixed points, and is found to be

ω(g^{∗}=0) = −ε ,  ω(g^{∗}=2ε) = ε .  (3.31)

Of course, this result is apart from the very existence of a fixed point rather trivial, since it is determined through the leading term of the β-function, which is always −εg. In order to obtain any non-trivial information from the perturbation series it is thus necessary to go beyond the leading order.

Interacting Crumpled Manifolds 25

4 Two-loop calculation in a MS-scheme

4.1 Operator product expansion

Let us now continue with the calculation at the two-loop order. At each order of perturbation theory there is only one new diagram. At two-loop order this comes from three coalescing δinteractions. Again, let us rewrite the product of the three vertex operators as its expectation value times the normal ordered product:

e^{ik_{1}r(x_{1})}e^{ik_{2}r(x_{2})}e^{ik_{3}r(x_{3})}
= :e^{i[k_{1}r(x_{1})+k_{2}r(x_{2})+k_{3}r(x_{3})]}: e^{k_{1}k_{2}C(x_{1}−x_{2})}e^{k_{1}k_{3}C(x_{1}−x_{3})}e^{k_{2}k_{3}C(x_{2}−x_{3})} .  (4.1)

We use the following change of coordinates

\begin{matrix}x̄ := & \frac{1}{3}(x_{1} + x_{2} + x_{3}) \\ a := & x_{2} − x_{1} \\ b := & x_{3} − x_{2} \\ c := & x_{1} − x_{3} \end{matrix} \} ⇔ \{ \begin{matrix}x_{1} = & x̄ + \frac{1}{3}(c − a) \\ x_{2} = & x̄ + \frac{1}{3}(a − b) \\ x_{3} = & x̄ + \frac{1}{3}(b − c) \end{matrix} , .  (4.2)

As at 1-loop order, :e^{i[k_{1}r(x_{1})+k_{2}r(x_{2})+k_{3}r(x_{3})]}: is free of divergences upon approaching the points x_{i}, whereas the UV-divergence comes from the factor e^{k_{1}k_{2}C(x_{1}−x_{2})}e^{k_{1}k_{3}C(x_{1}−x_{3})}e^{k_{2}k_{3}C(x_{2}−x_{3})}. The leading term is thus obtained upon setting x_{1} = x_{2} = x_{3} = x̄ in :e^{i[k_{1}r(x_{1})+k_{2}r(x_{2})+k_{3}r(x_{3})]}:, dropping the gradient terms. Further shifting k_{1} → k_{1} − k_{2} − k_{3}, we obtain

e^{ik_{1}r(x̄)} e^{(k_{1}−k_{2}−k_{3})k_{2}C(a)+k_{2}k_{3}C(b)+(k_{1}−k_{2}−k_{3})k_{3}C(c)} .  (4.3)

Integrating over k_{1}, k_{2} and k_{3} yields the final result

δ̃^{d}(r(x_{1}))δ̃^{d}(r(x_{2}))δ̃^{d}(r(x_{3})) = (  | )δ̃^{d}(r(x̄)) + (  | ′′) (−∂_{r}^{2})δ̃^{d}(r(x̄)) + …,

(4.4)

where the OPE–coefficients are given by

(  | ) = [C(a)C(c) − \frac{1}{4}(C(a) + C(c) − C(b))^{2}]^{−d/2}
= [\frac{1}{4}(a^{ν}+b^{ν}+c^{ν})(b^{ν}+c^{ν}−a^{ν})(a^{ν}+c^{ν}−b^{ν})(a^{ν}+b^{ν}−c^{ν})]^{−d/2} ,  (4.5)

which contributes to the renormalization factor at two-loop order. The first subleading term is (denoting ′′ := (−∂_{r}^{2})δ̃^{d}(r(x̄)))

(  | ′′) = −\frac{1}{4}C(a)C(b)C(c) [\frac{1}{4}(a^{ν}+b^{ν}+c^{ν})(b^{ν}+c^{ν}−a^{ν})(a^{ν}+c^{ν}−b^{ν})(a^{ν}+b^{ν}−c^{ν})]^{−d/2−1}

(4.6)

4.2 Numerical calculation in 0 < D < 2

Let us now turn to the explicit calculation of the second order contribution to the coupling constant renormalization. To two-loop order the renormalization group Z_{g}-factor for the coupling constant reads

Z_{g} = 1 + \frac{g}{2}⟨  ^{| ⟩}_{ε^{−1}} − \frac{g^{2}}{6} [⟨  | ⟩_{ε^{−2},ε^{−1}} − \frac{3}{2}⟨  ^{| ⟩}_{ε^{−1}}^{2}]
+ \frac{g^{2}}{4} ⟨  ^{| ⟩}_{ε^{−1}}^{2} + O(g^{3}) .  (4.7)

Note that this Z_{g}-factor can either be obtained from (2.10) and (3.23) or by expanding the perturbative series and checking that divergent contributions proportional to δ̃(r(x)) cancel.

The formula is arranged such that in the brackets the second order pole in ε cancels. This is, because to leading order in 1/ε, the two-loop diagram factorizes into two one-loop diagrams. The combinatorial factor of 3/2 is composed of a factor of 3 for the number of possible subdivergences, and a factor of 1/2 for the nested integration: The subdivergence in the distance say a only appears in the sector (i.e. part of the domain of integration) in which a is smallest.

Let us define:

I_{2}(D, L) := ⟨  | ⟩_{L}−\frac{3}{2}⟨  ^{| ⟩}_{L}^{2} .  (4.8)

Let us already note that we will later calculate in a “massive” scheme in fixed dimension d. In order to use the results derived here, we keep d arbitrary instead of setting d = d_{c}. We now want to derive an expression which can be integrated numerically. Since the bounds on the integrations in (4.8) are asymmetric (remind that in ⟨ ^{| ⟩}_{L} all 3 distances a, b, and c are bounded by L, whereas ⟨ | ⟩_{L}^{2} is an integral with two bounds only) we treat both terms separately. We start with the first one, which can be written as

I_{2}^{M}(D, L) = ∫_{a,b,c} Θ(a ≤ L)Θ(b ≤ L)Θ(c ≤ L)(  | ) .  (4.9)

The divergence when integrating over the global scale is eliminated by noting that L\frac{∂}{∂L}I_{2}(D, L) = 2ε I_{2}(D, L). There remains of course the sub-divergences, which will be subtracted later by the counter-term. Applying this procedure to (4.9), we obtain

2εI_{2}^{M}(D, L) = ∫_{a,b,c} [δ(a − L)Θ(b ≤ L)Θ(c ≤ L) + δ(c − L)Θ(b ≤ L)Θ(a ≤ L)
+δ(b − L)Θ(a ≤ L)Θ(c ≤ L)](  | ) .  (4.10)

Since the integral is symmetric in a, b, and c, we can replace it by 3 times the first term. Introducing proper normalizations, an explicit representation for the measure and factoring out the explicit L-dependence, we obtain

2εI_{2}^{M}(D, L) = 3L^{2ε} \frac{S_{D−1}}{S_{D}} ∫_{−∞}^{∞} dc_{∥} ∫_{0}^{∞} dc_{⊥}c_{⊥}^{D−2}Θ(b ≤ 1)Θ(c ≤ 1) (  | )|_{a=1,b,c} . (4.11)

We also have to derive an expression for the counter-term. It can be written as

I_{2}^{C}(D, L) = −\frac{3}{2} ∫_{a} ∫_{c} a^{−νd}c^{−νd} Θ(a ≤ L)Θ(c ≤ L) .  (4.12)

Θ(c ≤ a)  B  A  C

Figure 4.1: Area of integration (dashed) of the first integral in (4.13). The intersection with B is to be mapped onto A

Using the same trick to derive w.r.t. the IR-regulator as above, we obtain

2εI_{2}^{C}(D, L) = −\frac{3}{2}L^{2ε} ∫_{A,B,C} a^{−νd}c^{−νd} [δ(a − 1)Θ(c ≤ a) + δ(c − 1)Θ(a ≤ c)] . (4.13)

Using the technique of conformal mapping, which is presented in appendix E, both integrals above can be transformed into one having support in the single sector A defined through b, c ≤ a, (where here a is set to 1). Analogously, we define sectors B and C, as the subset of domains where b respectively c is the longest distance. Note that the first term in (4.13) only has contributions from A and B, while the second can be restricted to A and C after renaming a into b and c into a. Performing explicitly the mapping of both B and C onto A one obtains

∫_{B}(a^{−νd}c^{−νd}Θ(c ≤ a))_{a=1} = ∫_{A} b^{−νd−2ε}c^{−νd}Θ(c ≤ b)
∫_{C}(a^{−νd}b^{−νd}Θ(b ≤ a))_{|a=1} = ∫_{A} c^{−νd−2ε}b^{−νd}Θ(b ≤ c) .  (4.14)

Combining the contributions from all sectors, we arrive at

2εI_{2}^{C}(D, L) = −\frac{3}{2}L^{2ε} ∫_{A} (b^{−νd}+c^{−νd}+b^{−νd}c^{−νd}max(b, c)^{−2ε}) .  (4.15)

Combining (4.11) and (4.15), we obtain an expression for the complete diagram

2εI_{2}(D, L) = 3L^{2ε} \frac{S_{D−1}}{S_{D}} ∫_{−∞}^{∞} dc_{∥} ∫_{0}^{∞} dc_{⊥}c_{⊥}^{D−2}Θ(b ≤ 1)Θ(c ≤ 1)

×\{ [\frac{1}{4}(1+b^{ν}+c^{ν})(b^{ν}+c^{ν}−1)(1+c^{ν}−b^{ν})(1+b^{ν}−c^{ν})]^{−d/2}

−\frac{1}{2} [b^{−νd}+c^{−νd}+b^{−νd}c^{−νd}max(b, c)^{−2ε}] \} .  (4.16)

This expression will also be used later in a massive scheme. Since here we are only interested in the ε-expansion, we take the limit of ε → 0 in the integrand. We have made the (nontrivial) check that the integrand remains a finite integrable function, which can be integrated numerically:

I_{2}(D, L) = 3\frac{L^{2ε}}{2ε} \frac{S_{D−1}}{S_{D}} ∫_{−∞}^{∞} dc_{∥} ∫_{0}^{∞} dc_{⊥}c_{⊥}^{D−2}Θ(b ≤ 1)Θ(c ≤ 1)

×\{ [\frac{1}{4}(1+b^{ν}+c^{ν})(b^{ν}+c^{ν}−1)(1+c^{ν}−b^{ν})(1+b^{ν}−c^{ν})]^{−d_{c}/2}

−\frac{1}{2} [b^{−νd_{c}}+c^{−νd_{c}}+b^{−νd_{c}}c^{−νd_{c}}] \}

+O(ε^{0}) .  (4.17)

Let us now proceed with the explicit numeric evaluation of (4.17). There are integrable singularities both in c_{∥} and c_{⊥}. To disentangle them [26,20] we make a change of variables from Cartesian coordinates (c_{∥}, c_{⊥}) to radial and angular coordinates (c, α) according to

c_{∥} = c cos(α) ,  c_{⊥} = c sin(α) .  (4.18)

such that

∫_{−∞}^{∞} dc_{∥} ∫_{0}^{∞} dc_{⊥}c_{⊥}^{D−2} ⟶ ∫_{0}^{1} dc c^{D−1} ∫_{0}^{π/2} dα (sin(α))^{D−2} ,  (4.19)

where the upper bounds of π/2 on α and 1 on c is due to the bound of 1 on b and c. We can further restrict integration to the half-sector with c < b. In this sector, there remain singularities for small α and small c. The reader is invited to verify that they are eliminated by a second change of variables

α = \frac{π}{2} β^{\frac{1}{D−1}}
c = c̃^{\frac{1}{2−D}} .

D 0.1 6.71 0.2 2.11 0.3 0.847 0.4 0.325 0.5 0.0807 0.6 -0.0415 0.7 -0.0949 D 0.8 -0.102 0.9 -0.0712 1.0  0 1.1 0.123 1.2 0.321 1.3 0.644 1.4 1.20 \begin{matrix}D \\ 1.5 & 2.24 \\ 1.6 & 4.45 \\ 1.7 & 10.4 \\ 1.8 & 36.7 \\ 1.9 & 658 \\ 2.0 & ∞ \end{matrix}

Figure 4.2: Numerically obtained results for the two-loop diagram in the MS-scheme. Some values are tabulated.

Figure 4.3: Correction to scaling exponent ω for (D=1, d=1) with different extrapolation parameters (D, d), (D, ε(D, d)), (D, D_{c}(d)) (from left to right). The abscissa labels the corresponding internal dimension D_{0} of the departure point on the critical line ε(D_{0}, d_{0})=0.

The above method works for D > 1 only. For D < 1, one has to analytically continue the integration over c_{⊥} in (4.11). This can be done by partial integration, since the integrand does not explicitly depend on c_{⊥} but on c = \sqrt{c_{⊥}^{2} + c_{∥}^{2}}. To have no boundary terms, one should partially integrate before mapping everything onto the sector A. We have explicitly checked consistency with the earlier used measure for 1<D<2. Note finally, that in the special case of D = 1 in (4.17) the integration measure becomes a distribution reducing the integration measure to a line along c_{∥}.

The diagram is shown on figure (4.2). It grows as D^{−1} for D → 0 and exponentially for D → 2:

I_{2}(D) ∼ (\frac{3}{4})^{−d_{c}/2} ∼ (\frac{3}{4})^{−D/(2−D)} .  (4.20)

In the case of polymers D=1 it vanishes. The reason is that the diagrams factorize, and thus the 1-loop result is correct to all orders when working in an appropriate scheme. (See e.g. [4].)

4.3 RG-function and extrapolation

From (3.27) the renormalization β-function in second order of perturbation theory reads

β(g) = −εg + \frac{g^{2}}{2} − \frac{g^{3}}{3} I_{2}(D) + O(g^{4}) .  (4.21)

Interacting Crumpled Manifolds 31

(x, y) ; (D, d) (1,1) (2,1) (2,2) (2,3)
(D, Dc) , max 0.55 2.80 2.43 2.10
(D, d) , max 0.55 2.80 2.43 2.20
(D, ε) , max 0.55

Figure 4.4: Correction to scaling exponent ω for different extrapolation points (D, d): Extrapolation parameters are (D, D_{c}(d)) (top) and (D, d) (bottom). The abscissa labels the corresponding internal dimension D_{0} of the departure point on the critical line ε(D_{0}, d_{0})=0. Furthermore, the two-loop values with minimal sensitivity with respect to the departure point are tabulated.

The long-distance behavior of the theory is governed by the IR–stable fixed point of the RG– flow. Fixed points are given by zeros of the β–function:

β(g^{∗}) = 0  ⇒ g^{∗} = 0 ∨ −ε + \frac{g^{∗}}{2} − \frac{g^{∗2}}{3} I_{2}(D) + O(g^{∗3}) = 0 .  (4.22)

The physically interesting nontrivial fixed point is the one that is next to zero. In an ε–expansion it reads:

g^{∗} = 2ε + \frac{8}{3}I_{2}(D)ε^{2} + O(ε^{3}) .  (4.23)

The correction to scaling exponent ω at this fixed point is found in an ε-expansion to be

ω(g^{∗}) = ε − \frac{4}{3}I_{2}(D)ε^{2} + O(ε^{3}) .  (4.24)

The question now arises of how to calculate ω for a given physical situation. The general strategy that we follow in the MS-scheme is that we make use of the freedom to choose any point on the critical line (2.2) around which to start our expansion [25,26]. To expand towards some physically interesting point (D, d) we furthermore dispose of the freedom to follow any path starting on the curve ε=0. Since our expansion (4.24) is exact in D but only of second order in ε, we can equally well expand it to second order in D around any point on the critical curve. Now we can change our extrapolation path through an invertible transformation (x, y) = (x(D, d), y(D, d)). One expresses D and ε as functions of x and y and reexpands ω to second order in x and y around the point (x_{0}, y_{0}) = (x_{0}(D_{0}, d_{c}(D_{0})), y_{0}(D_{0}, d_{c}(D_{0}))), where we recall that d_{c}(D) is defined such that ε(D, d_{c}(D)) = 0. The aim is to find an optimal choice of variables (x, y). The guidelines for such a choice have been discussed in [20], where this procedure has been successfully applied to extrapolate results for the anomalous dimension of self-avoiding membranes. As a general rule, we “trust” most a result that is the least sensitive to the starting point of the extrapolation. Therefore, we would like to find a plateau for some suitable chosen variables (x, y). This procedure works well for polymers, or more generally for membranes with inner dimension close to 1. However it turns out that for membranes it works much less well than in the self-avoiding case: we could not find a plateau, but at best an extremum.

Some extrapolations are shown in the following figures (4.3, 4.4). We start with polymers interacting with a δ-wall corresponding to the point (D, d) = (1, 1). (Note that for d < 2 the interaction becomes relevant for polymers.) This can be seen on fig. 4.3. The values obtained from the plateaus appearing at the two-loop level (ω≈0.55) are quite close to the exact value ω = 1/2, known from the fact that the result in D = 1 at one-loop is exact. From the latter it follows that it is best to use (D, ε) as extrapolation parameters and to stay in D = 1.

Let us now turn to the results obtained for ω extrapolated to points D = 2 (fig. 4.13): The extrapolated value depends strongly on the starting-point on the critical line. We can identify a least sensitive value, which lies below D = 1, reflecting the D-dependence of the two-loop diagram. We obtain identical results for different extrapolation paths. Note, that if we fix d = 1 and perform an ε-expansion in D as is commonly done [62], then, this corresponds to starting from D_{c}(d=1) = 2/3 < 1. The d-dependence of the extrapolated values is recovered in the massive scheme, which we discuss in the next section.

5 Calculation in fixed dimension: The massive scheme

5.1 Phantom manifolds

An alternative scheme to an ε-expansion is a calculation in fixed dimension. This way we hope to circumvent the difficulty that we faced when calculating the two-loop diagram in the previous section in the limit of D → 2: Since the diagram behaves like (\frac{4}{3})^{d/2}, it becomes large when the dimension of the embedding space is large. When working in an ε-expansion, d and D are related by ε = D − d(2 − D)/2 and sending ε → 0 before D → 2 imposes that d → ∞. Thus one would like to work at finite d, thus finite ε. The diagrams are UV-convergent for d < d_{c}. They are cut-off in the infrared by a finite membrane size. Alternatively, we can use a “soft” IR cut-off, that is we introduce a chemical potential τ into the Hamiltonian and sum over all manifold sizes. For polymers (D = 1) this scheme corresponds to a “massive scheme” in an O(N)-symmetric, scalar ϕ^{4}-theory as can be seen via a de Gennes Laplace transformation on the N = 0 component limit. (The correlator in momentum-space reads \frac{1}{k^{2}+t}) In the following we keep a hard IR regulator and calculate the two-loop diagram in fixed dimension. We use the analytical expression given in (4.16). For the numerical integration in 1 < D < 2 we proceed as before. The result for different embedding dimensions d is shown graphically on figure 5.1, where also some explicit values in the limit of D = 2 are given. Note, that the exact factorization-property in the limit of D = 1 is only valid in the MS-scheme or in a massive scheme with suitable soft cut-off as mentioned above. However, even the method is not tailored to capture that feature, the obtained value ω = 0.6 in the case of (D, d) = (1, 1) is not too far from the exact value 1/2. We expect that higher order terms would again reproduce the exact value of ω = 1/2. Also note that results for membranes (D = 2) are close to those from the ε-expansion, especially for small d.

\begin{matrix}d & I_{2|_{D=2}} \\ 0.5 & -0.437 \\ 1 & -0.410 \\ 1.5 & -0.386 \\ 2 & -0.358 \\ 2.5 & -0.331 \\ 3 & -0.299 \\ 3.5 & -0.265 \\ 4 & -0.229 \end{matrix}

Figure 5.1: 2-loop diagram in fixed dimension for embedding dimensions d=0.5 to d=4 (from bottom to top). The curves are only shown for values of D such that ε≥0.

5.2 Self-avoiding membranes

Up to now we were considering a D-dimensional polymerized “phantom” membrane interacting with a point-like impurity in d-dimensional embedding space modeled through a δ-potential.

\begin{matrix}d & ω_{|_{D=2}} \\ 0.5 & 2.82 \\ 1 & 2.80 \\ 1.5 & 2.77 \\ 2 & 2.75 \\ 2.5 & 2.72 \\ 3 & 2.69 \\ 3.5 & 2.65 \\ 4 & 2.60 \end{matrix}

Figure 5.2: Correction to scaling exponent ω for embedding dimension d=0.5 to d=4 (from top to bottom). ε is restricted to ε≥0; otherwise one has to work with an explicit UV-cutoff.

The infinite fractal dimension of the free manifold was causing problems, and since all operators of the type (−Δ_{r})^{n}δ(r), n≥0, attain the same relevance in D = 2, it is not clear whether the model will break down. Physically, the problem is much better defined for self-avoiding membranes. From field-theoretical calculations, which have recently been refined up to twoloop [19,20], we know that two-dimensional self-avoiding manifolds embedded in three dimensional space have an anomalous dimension of ν^{∗} ≈ 0.85. We now try to overcome the problem of an infinite Hausdorff-dimension by a crude simplification: We approximate the self-avoiding manifold by a Gaussian theory with the same scaling dimension, that is a Hamiltonian of the type [47]

H[r] = \frac{1}{k−D} ∫_{x} \frac{1}{2} r(x)(−Δ_{x})^{k/2}r(x) ,  (5.1)

where

r : x ∈ ℝ^{D} ⟶ r(x) ∈ ℝ^{d}

as before providing the two-point correlator in our normalizations:

C(x, y) = |x−y|^{k−D} .  (5.2)

The scaling dimension is

ν = \frac{k−D}{2} .  (5.3)

We recover (3.9) by setting k = 2, but the model can be continued analytically to any real value of k, with k ≥ 2. Setting

k^{∗} = 2ν^{∗} + D  (5.4)

we get a Gaussian manifold with identical scaling dimension as obtained for self-avoiding crumpled membranes at two-loop. The critical embedding dimension of the interaction then reads

d_{c} = \frac{2D}{k^{∗}−D} = \frac{D}{ν^{∗}} ,  (5.5)

that is the interacting is relevant in d<d_{c}. For membranes d_{c}=\frac{2}{0.85} ≈ 2.4. All operators of the type (−Δ_{r}^{n}δ^{d}(r)) are at least naively irrelevant for n > 0, as

[(−Δ_{r}^{n}δ^{d}(r))] = −ν^{∗}(d + 2n)  (5.6)

and the corresponding coupling has in inverse length units the dimension

[g_{0}^{(n)}] = D − ν^{∗}(d + 2n) < 0 ,  n > 0 .  (5.7)

We may repeat the calculation in the massive scheme setting the free scaling dimension ν^{∗}=0.85 and (D, d)=(2, 1). The result is shown in table (5.2). Let us stress that this method can not be turned into a systematic expansion, since perturbation theory neglects all effects of the non-Gaussian nature of the nontrivial crumpled state. We only know that for d → ∞ the Gaussian variational approximation for the Hamiltonian, i.e. (5.1) becomes exact [63–65].

I  ω(g^{∗}) θ(g^{∗})
1-loop 1.000 1.15 2.71
2-loop -0.379 1.49 3.10

exponent ω and the contact exponent θ.

Table 5.1: Two-loop results for self-avoiding membranes (D=2) approximated through a Gaussian theory having identical scaling dimension ν^{∗}=0.85. Results are shown for the corresponding diagram, the correction to scaling

6 Summation of the perturbation series in the limit D = 2

In the last section we introduced a “massive” scheme allowing to perform the renormalization procedure in fixed dimension. As can be seen in figures 5.1 and 5.2, the limit D → 2 can be taken in the 2-loop diagram and is smooth. Surprisingly, this limit can be calculated analytically: Setting ν = 0 in (4.16), the integrand becomes constant, and thus trivial to calculate. This is a striking property. Even more, if we slightly modify the regularization prescription, this property holds to all loop orders in perturbation theory. Below, we will give an analytic expression for the complete perturbation series, which allows for analyzing the strong coupling limit. Furthermore, we show how corrections in (2 − D) can be incorporated. We will see that only the latter depend on the explicit cut-off procedure, a point which will be further clarified below. In the following– since we perform calculations on a torus – we slightly modify the rescaling of the field and the bare coupling:

r → r(S_{D}(2 − D))^{1/2} , g_{0} → g_{0}(S_{D}(2 − D))^{d/2} .  (6.1)

Our aim is to evaluate the perturbation series of the effective coupling, which reads:

g = \frac{g_{0}L^{ε}}{V_{M}} ∑_{N=0}^{∞} \frac{(−g_{0})^{N}}{(N+1)!} ⟨ ∏_{i=1}^{N+1} ∫_{x_{i}} δ^{d}(r(x_{i}))⟩_{0} .  (6.2)

The two-point function gets a factor (S_{D}(2 − D))^{d/2} due to the rescaling (6.1). All other factors (S_{D}(2 − D))^{d/2} will be dropped. We discuss this at the end of this section in 6.8.

6.1 N-loop order

In order to calculate the N–loop contribution to the perturbation theory we need the SDE of N + 1 contact interactions. The normal ordered product of the corresponding vertex operators reads

e^{ik_{1}r(x_{1})}e^{ik_{2}r(x_{2})} · . . . · e^{ik_{N+1}r(x_{N+1})} = :e^{i(∑_{n=1}^{N+1} k_{n}r(x_{n}))}: ∏_{i,j=1}^{N+1} e^{\frac{1}{2} k_{i}k_{j}C(x_{i}−x_{j})} .  (6.3)

We choose x_{N+1} as root in the renormalization procedure (for the precise definition, see below) and therefore k_{N+1} to integrate over in order to obtain the δ-distribution (or its derivative), onto which to project. We thus shift

k_{N+1} ⟶ k_{N+1} − ∑_{j=1}^{N} k_{j}

and rewrite the quadratic form in (6.3) as

\frac{1}{2} ∑_{i,j=1}^{N+1} k_{i}k_{j}C(x_{i}−x_{j}) = ∑_{j=1}^{N} k_{N+1}k_{j}C(x_{N+1}−x_{j}) + \frac{1}{2} ∑_{i,j=1}^{N} k_{i}k_{j}C(x_{i}−x_{j})

⟶ ∑_{j=1}^{N} k_{N+1}k_{j}C(x_{N+1}−x_{j}) − ∑_{i,j=1}^{N} k_{i}k_{j}C(x_{N+1}−x_{j}) + \frac{1}{2} ∑_{i,j=1}^{N} k_{i}k_{j}C(x_{i}−x_{j})

= ∑_{j=1}^{N} k_{N+1}k_{j}C(x_{N+1}−x_{j}) − ∑_{i,j=1}^{N} k_{i}k_{j} \frac{C(x_{N+1}−x_{i}) + C(x_{N+1}−x_{j}) − C(x_{i}−x_{j})}{2} .

(6.4)

(6.3) then becomes (up to subdominant terms^{3} involving spatial derivatives of r)

:exp [ik_{N+1}r(x_{N+1})]:

× exp [∑_{j=1}^{N} k_{N+1}k_{j}C(x_{N+1}−x_{j}) − ∑_{i,j=1}^{N} k_{i}k_{j} \frac{C(x_{N+1}−x_{i}) + C(x_{N+1}−x_{j}) − C(x_{i}−x_{j})}{2}]

(6.5)

and the integration over k_{N+1} produces the δ-interaction and its derivatives; the latter come from the first term in (6.4). They are subdominant and thus will be dropped. We abbreviate the quadratic form in (6.4) as (D_{ij}), where the matrix elements are

D_{ij} = \frac{1}{2} [C(x_{N+1}−x_{i}) + C(x_{N+1}−x_{j}) − C(x_{i}−x_{j})] , i ≠ j ,
D_{ii} = C(x_{N+1}−x_{i}) .  (6.6)

Integrating out the momenta k_{1}, … k_{N} yields the operator product expansion of N + 1 contact interactions as

δ̃^{d}(r(x_{1})) δ̃^{d}(r(x_{2})) … δ̃^{d}(r(x_{N+1})) = [det(D_{ij})]^{−d/2} δ̃^{d}(r(x_{N+1}))
+ subdominant operators .  (6.7)

6.2 The limit D → 2 and (2−D)-expansion on the torus

Whereas (6.7) is still completely general, we now want to study the limit D → 2 as well as perturbations above it. We start by remarking that the propagator C(x) may be rewritten as

C(x) = ℂ^{(0)}(x) + (2 − D)ℂ(x) ,  (6.8)

where ℂ^{(0)}(x) denotes the limit D → 2 of C(x). Accordingly, we define matrices 𝔇^{(0)} and 𝔇, such that

D_{ij} = 𝔇_{ij}^{(0)} + (2 − D)𝔇_{ij} ,  (6.9)

where

𝔇_{ij}^{(0)} = \frac{1}{2} ,  i ≠ j
𝔇_{ii}^{(0)} = 1  (6.10)

3Here we argue in terms of an OPE. Note that in order to make these arguments rigorous, one should ask of whether these terms affect the effective action at constant background-field. This is not the case, since then ∇r = 0.

and

𝔇_{ij} = \frac{μ^{−2ν}}{2} (ℂ|(x_{N+1}−x_{i})μ| + ℂ|(x_{N+1}−x_{j})μ| − ℂ|(x_{i}−x_{j})μ|) ,  i ≠ j
𝔇_{ii} = μ^{−2ν}ℂ|(x_{N+1}−x_{i})μ| ,  (6.11)

μ = 1/L. Note, that we obtained (6.10) from the limit D → 2 in (6.6), and we used for C(x) the infinite D-space propagator C(x) = |x|^{2−D}. However, the constance in the limit D → 2 for noncoinciding points stays conserved, even if we are considering closed manifolds. Thus, (6.9) together with (6.10) always stay valid as well as the following considerations: Due to (6.9),

det(D) = exp[Trln(D_{ij})]
= det(𝔇^{(0)}) · exp[Trln[1 + (2 − D)[𝔇^{(0)}]^{−1}D̃]] ,  (6.12)

where [𝔇^{(0)}]^{−1} denotes the inverse matrix to 𝔇^{(0)}. Denoting

M := [𝔇^{(0)}]^{−1}𝔇 ,  (6.13)

let us expand the OPE-coefficient in (6.7) up to fourth order in 2−D. Expanding first the logarithm,

ln[1 + (2−D)M] = (2−D)M − \frac{(2−D)^{2}}{2} M^{2} + \frac{(2−D)^{3}}{3} M^{3} − \frac{(2−D)^{4}}{4} M^{4} + O((2−D)^{5})

and inserting into (6.12) one arrives at

[det(D)]^{−d/2} = [det(𝔇^{(0)})]^{−d/2} [1 − \frac{d}{2} [(2 − D)TrM

−\frac{(2 − D)^{2}}{2} TrM^{2} + \frac{(2 − D)^{3}}{2} TrM^{3} − \frac{(2 − D)^{4}}{4} TrM^{4}]

+\frac{d^{2}}{8} [(2 − D)^{2}Tr^{2}M − (2 − D)^{3}TrM TrM^{2} + (2 − D)^{4} [\frac{1}{4}Tr^{2}M^{2} + \frac{2}{3}TrM TrM^{3}]]

−\frac{d^{3}}{48} [(2 − D)^{3}Tr^{3}M − (2 − D)^{4} \frac{3}{2}Tr^{2}M TrM^{2}] + \frac{d^{4}}{384}(2 − D)^{4}Tr^{4}M] + O((2 − D)^{5})

(6.14)

To check the above result consider that

ln(1 + x) = x − \frac{x^{2}}{2} + \frac{x^{3}}{3} − \frac{x^{4}}{4} + ⋯

and

(ax + bx^{2} + cx^{3} + cx^{4})^{2} = a^{2}x^{2} + 2abx^{3} + (b^{2} + 2ac)x^{4} + ⋯

as well as

(ax + bx^{2} + cx^{3} + cx^{4})^{3} = a^{3}x^{3} + 3a^{2}bx^{4} .

The zeroth order is [det(𝔇^{(0)})]^{−d/2}. In order to proceed we need the latter and the inverse matrix [𝔇^{(0)}]^{−1}. Let us first calculate the determinant. From (6.9) we have

𝔇^{(0)} = μ^{−2ν} ( \begin{matrix}1 & \frac{1}{2} & … & \frac{1}{2} \\ \frac{1}{2} & 1 & ... & ⋮ \\ ⋮ & ... & ... & \frac{1}{2} \\ \frac{1}{2} & … & \frac{1}{2} & 1 \end{matrix} ) = \frac{μ^{−2ν}}{2} (( \begin{matrix}1 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋮ & 1 \end{matrix} ) + ( \begin{matrix}1 & … & 1 \\ ⋮ & ... & ⋮ \\ 1 & … & 1 \end{matrix} )) = \frac{μ^{−2ν}}{2} (1 + NP) ,

P = \frac{1}{N} ( \begin{matrix}1 & … & 1 \\ ⋮ & ... & ⋮ \\ 1 & … & 1 \end{matrix} )  (6.15)

being the projector onto (1, 1, …, 1). Denoting by dim Im(P) the dimension of the space onto which P projects one has the general formula:

det(1 + aP) = (1 + a)dim Im(P) .

Since here dim Im(P) = 1, we find

det(𝔇^{(0)}) = μ^{−2νN} \frac{1 + N}{2^{N}} .  (6.16)

The inverse matrix of (1 + NP) can be inferred from the ansatz

(1 + NP)(1 + bP) = 1 .

Since P is a projector P^{2} = P and this implies

(N + b + Nb) P = 0 .

Therefore, b = −N/(N + 1) and

[𝔇^{(0)}]^{−1} = 2 (1 − \frac{N}{N + 1}P) μ^{2ν} .  (6.17)

To obtain the first order contribution in (2−D) from eq. (6.12) we only need M = [𝔇^{(0)}]^{−1}𝔇, which reads

(M_{ij}) = ([𝔇^{(0)}]^{−1}𝔇)_{ij} = (2𝔇_{ij} − \frac{2}{1 + N} ∑_{k=1}^{N} 𝔇_{ik}) μ^{2ν} .  (6.18)

The trace of (6.18) can easily be performed, with the result

TrM = (\frac{2N}{1 + N} ∑_{i=1}^{N} 𝔇_{ii} − \frac{2}{1 + N} ∑_{i=1}^{N} ∑_{k=1}^{N}(1 − δ_{ik})𝔇_{ik}) μ^{2ν} .  (6.19)

In each order of perturbation theory we have to integrate the expression (6.7) over internal distances. These integrals have to be regularized in the infrared through an appropriate IR

Figure 6.1: Regularization scheme for the N-loop diagrams on manifolds with toroidal topology (periodic boundary conditions). Here: D = 2

cut-off. In the preceding sections we have cut off the integrals such that all relative distances between internal points were smaller than L = 1/μ. Here, we are considering a finite manifold of toroidal topology. In order to be consistent we need modify the regularization prescription (3.21): Specializing to a squared piece of membrane all components of internal D-dimensional vectors x⃗_{i} are running from 0 to L. Accordingly, all components of the difference vectors x⃗_{i} − x⃗_{N+1}, where x⃗_{N+1} is a fixed, but arbitrarily chosen point, the root of the subtraction prescription, are bounded by L. Implicitly, all other distances are restricted through L as well (see fig. 6.1). Of course, on the torus the correlator has to be modified.

To simplify the calculations, we further introduce the following notation:

\overline{f(x_{i_{1}}, …, x_{i_{k}})} := ∫_{x_{1}} ⋯ ∫_{x_{N}} f(x_{i_{1}}, …, x_{i_{k}})  (6.20)

with normalization

∫_{x} := ∫ d^{D}x ∏_{j=1}^{D} θ(x^{j} < 1) ,  (6.21)

which is chosen such that the overbar can be thought of as an averaging procedure, and especially

\overline{1} = 1  (6.22)

Thanks to our regularization prescription the integral of (6.19) over internal points can be replaced by μ^{−ND} (for the integration measure L^{ND}) times

\overline{TrM} = \frac{2N^{2}}{1 + N} \overline{ℂ(x_{N+1} − x_{i})}

− (\frac{2N(N − 1)}{1 + N}) (\overline{ℂ(x_{N+1} − x_{i})} − \frac{1}{2} \overline{ℂ(x_{i} − x_{j})})

= \frac{2N}{1 + N} \overline{ℂ(x_{N+1} − x_{i})} + \frac{N(N − 1)}{1 + N} \overline{ℂ(x_{i} − x_{j})} , .  (6.23)

Due to the internal symmetry of closed manifolds the expression above can be further simplified, since

\overline{ℂ(x_{N+1} − x_{i})} = \overline{ℂ(x_{i} − x_{j})} ≡ \overline{ℂ(x)} .  (6.24)

Introducing a diagrammatic notation

:= \overline{ℂ(x)} ,  (6.25)

the N-loop integral reads up to first order in 2−D

∫ (det D)^{−d/2} = μ^{−Nε} (\frac{1 + N}{2^{N}} )^{−d/2} × [1 − \frac{d}{2}(2−D) (N  ) + O((2−D)^{2})] . (6.26)

For the further analysis we will not only need (6.23), but also the terms appearing to second order in 2 − D in (6.14). We derived expressions like (6.23) for Tr^{2}M and TrM^{2} with Mathematica^{◯ R}. The Mathematica^{◯ R}–program will be presented and discussed in the appendix. It is based on the fact that all terms to appear in the expansion (6.14) are of the form Tr^{n}M^{m} or products of the latter and therefore can be written as P(N)/(N+1)^{k}, where n, m, k ∈ N and P(N) is some polynomial in N. It will turn out soon that it is convenient to expand the polynomial P(N) in terms of the following base:

{1, N, N(N−1), N(N−1)(N−2), …, ∏_{j=0}^{k}(N−j), …} .  (6.27)

We obtain:

\overline{TrM^{2}} = \frac{−2N(N−1) − N(N−1)(N−2)}{1+N} \overline{ℂ(x)}^{2} + \frac{2N + 3N(N−1) + N(N−1)(N−2)}{1+N} \overline{ℂ^{2}(x)} ,

(6.28)

and

\overline{Tr^{2}M} = \frac{4N(N−1) + N(N−1)(N−2)}{1+N} \overline{ℂ(x)}^{2} + \frac{2N}{1+N} \overline{ℂ^{2}(x)} .  (6.29)

Diagrammatically, the averages can be rewritten as

:= \overline{ℂ(x)}^{2} ,  (6.30)

and

:= \overline{ℂ^{2}(x)} .  (6.31)

Like in the case of the first order diagram (6.28) and (6.29) are highly simplified as compared to an open manifold. The averages over both any open chain of two propagators like in (6.30) and any closed loop of two propagators like in (6.31) are independent from the position of the root x_{N+1}. Diagrammatically, this symmetry can be written as

≡  =  =  =  =  ,  (6.32)

and

≡  =  .  (6.33)

Combining (6.14), (6.23), (6.28) and (6.29) the N–loop integral reads up to the second order in 2 − D:

∫ (det D)^{−d/2} = μ^{−Nε} (\frac{1 + N}{2^{N}} )^{−d/2} × [1 − \frac{d}{2}(2−D) (N )

+ \frac{d}{4}(2−D)^{2} (\frac{−2N(N−1) − N(N−1)(N−2)}{1+N}  + \frac{2N + 3N(N−1) + N(N−1)(N−2)}{1+N}  )

+ \frac{d^{2}}{8} (2−D)^{2} (\frac{4N(N−1) + N(N−1)(N−2)}{1+N}  + \frac{2N}{1+N}  )

+O((2−D)^{3})] .  (6.34)

Then, we finally obtain for the renormalized coupling g up to the second order in^{4} 2 − D:

g = g_{0}μ^{−ε} (1 + ∑_{N=1}^{∞} \frac{(−g_{0} 2^{d/2}μ^{−ε})^{N}}{(N+1)!(1+N)^{d/2}} [1 − (2−D)\frac{d}{2} (N  ) +

+ \frac{d}{4}(2−D)^{2} (\frac{−2N(N−1) − N(N−1)(N−2)}{1+N}  + \frac{2N + 3N(N−1) + N(N−1)(N−2)}{1+N}  )

+ \frac{d^{2}}{8} (2−D)^{2} (\frac{4N(N−1) + N(N−1)(N−2)}{1+N}  + \frac{2N}{1+N}  )

+O(2 − D)^{3}])  (6.35)

This can also be written as

g2^{d/2} = − ∑_{N=0}^{∞} \frac{(−g_{0}2^{d/2}μ^{−ε})^{N+1}}{N! (N+1)^{d/2+1}} + (2−D)\frac{d}{2} ∑_{N=0}^{∞} \frac{(−g_{0}2^{d/2}μ^{−ε})^{N+1}N}{N! (N+1)^{d/2+1}}

−(2−D)^{2} \frac{d}{4} ∑_{N=0}^{∞} \frac{(−g_{0}2^{d/2}μ^{−ε})^{N+1})(−2N(N−1) − N(N−1)(N−2))}{N! (N+1)^{d/2+2}}

4The expression in parenthesis is the inverse renormalization Z_{g}-factor which relates the bare coupling g_{0} to the renormalized coupling g according to gZ_{g}μ^{ε} = g_{0}.

−(2−D)^{2} \frac{d}{4} ∑_{N=0}^{∞} \frac{(−g_{0}2^{d/2}μ^{−ε})^{N+1})(2N + 3N(N−1) + N(N−1)(N−2))}{N! (N+1)^{d/2+2}}

−(2−D)^{2} \frac{d^{2}}{8} ∑_{N=0}^{∞} \frac{(−g_{0}2^{d/2}μ^{−ε})^{N+1})(4N(N−1) + N(N−1)(N−2))}{N! (N+1)^{d/2+2}}

−(2−D)^{2} \frac{d^{2}}{8} ∑_{N=0}^{∞} \frac{(−g_{0}2^{d/2}μ^{−ε})^{N+1})(2N)}{N! (N+1)^{d/2+2}}  + O(2 − D)^{3}  (6.36)

Absorbing a factor of 2^{d/2} both in g and g_{0} and defining the dimensionless coupling z := g_{0}μ^{−ε}, we obtain the final result

g = z ∑_{N=0}^{∞} \frac{(−z)^{N}}{N! (N+1)^{d/2+1}} + (2−D)\frac{d}{2} z^{2} ∑_{N=0}^{∞} \frac{(−z)^{N}}{N! (N+2)^{d/2+1}}

−(2−D)^{2} \frac{d}{4} (2z^{3} ∑_{N=0}^{∞} \frac{(−z)^{N}}{N! (N+3)^{d/2+2}} − z^{4} ∑_{N=0}^{∞} \frac{(−z)^{N}}{N! (N+4)^{d/2+2}} )

−(2−D)^{2} \frac{d}{4} (2z^{2} ∑_{N=0}^{∞} \frac{(−z)^{N}}{N! (N+2)^{d/2+2}} −3z^{3} ∑_{N=0}^{∞} \frac{(−z)^{N}}{N! (N+3)^{d/2+2}} +z^{4} ∑_{N=0}^{∞} \frac{(−z)^{N}}{N! (N+4)^{d/2+2}} )

−(2−D)^{2} \frac{d^{2}}{8} (−4z^{3} ∑_{N=0}^{∞} \frac{(−z)^{N}}{N! (N+3)^{d/2+2}} +z^{4} ∑_{N=0}^{∞} \frac{(−z)^{N}}{N! (N+4)^{d/2+2}} )

−(2−D)^{2} \frac{d^{2}}{8} (2z^{2} ∑_{N=0}^{∞} \frac{(−z)^{N}}{N! (N+2)^{d/2+2}} )  + O(2 − D)^{3} .  (6.37)

This formula is the starting point for further analysis in the subsequent sections.

6.3 Asymptotic behavior of the series

In the following we are interested in the limit of large z (strong repulsion), which also is the scaling behavior of infinitely large membranes. We therefore need an analytical expression for sums like (6.37) in the limit of large z. An example is the leading order expression

g = z ∑_{N=0}^{∞} \frac{(−z)^{N}}{N! (N+1)^{d/2+1}} .  (6.38)

For the following analysis we need an integral representation of the above series. We claim that for all k, d > 0

∑_{N=0}^{∞} \frac{(−z)^{N}}{N!(k + N)^{d/2}} = \frac{1}{Γ(\frac{d}{2})} ∫_{0}^{∞} dr r^{d/2−1} exp[−z e^{−r} − kr] .  (6.39)

This can be proven as follows:

\frac{1}{Γ(\frac{d}{2})} ∫_{0}^{∞} dr r^{d/2−1} exp[−z e^{−r} − kr] = \frac{1}{Γ(\frac{d}{2})} ∑_{N=0}^{∞} \frac{(−z)^{N}}{N!} ∫_{0}^{∞} dr r^{d/2−1}e^{−(N+k)r}

= \frac{1}{Γ(\frac{d}{2})} ∑_{N=0}^{∞} \frac{(−z)^{N}}{N!} \frac{Γ(\frac{d}{2})}{(N + k)^{d/2}} .

This integral-representation is not the most practical for our purpose. It is better to set r → s := e^{−r} which yields

∑_{N=0}^{∞} \frac{(−z)^{N}}{N!(k + N)^{d/2}} = \frac{1}{Γ(\frac{d}{2})} ∫_{0}^{1} ds s^{k−1}(− ln s)^{d/2−1}e^{−sz} .  (6.40)

This formula is already very useful for some purposes. It is still advantageous to make a second variable-transformation s → y := sz, yielding

∑_{N=0}^{∞} \frac{(−z)^{N}}{N!(k + N)^{d/2}} = \frac{(ln z)^{d/2−1}}{Γ(\frac{d}{2})z^{k}} ∫_{0}^{z} dy y^{k−1} (1 − \frac{ln y}{ln z} )^{d/2−1} e^{−y} .  (6.41)

Finally we remark that we usually have the following combination

f_{k}^{d}(z) := z^{k} ∑_{N=0}^{∞} \frac{(−z)^{N}}{N!(k + N)^{d/2}} = \frac{(ln z)^{d/2−1}}{Γ(\frac{d}{2})} ∫_{0}^{z} dy y^{k−1} (1 − \frac{ln y}{ln z} )^{d/2−1} e^{−y} . (6.42)

It satisfies the following simple recursion relation, which is helpful to calculate the β-function:

z \frac{d}{dz} f_{k}^{d}(z) = f_{k}^{d−2}(z) .  (6.43)

The derivative above can also be rewritten as

z \frac{d}{dz} f_{k}^{d}(z) = kf_{k}^{d}(z) − f_{k+1}^{d}(z) ,  (6.44)

such that one obtains a useful formula in order to isolate the dominant behavior for large z:

f_{k+1}^{d}(z) = kf_{k}^{d}(z) − f_{k}^{d−2}(z)  (6.45)

From (6.39) f_{k}(z) > 0 for all k, d > 0 and the behavior for large z is obtained by expanding (1 − \frac{ln y}{ln z})^{d/2−1} for small \frac{1}{ln z}

f_{k}^{d}(z) = \frac{(ln z)^{d/2−1}}{Γ(\frac{d}{2})} [ ∫_{0}^{∞} dy y^{k−1}e^{−y}

−\frac{1}{ln z} (\frac{d}{2} − 1) ∫_{0}^{∞} dy y^{k−1} ln y e^{−y}

+ O (\frac{1}{(ln z)^{2}} ) ]

+ O(e^{−z}) .  (6.46)

The result is

f_{k}^{d}(z) = \frac{(ln z)^{d/2−1}Γ(k)}{Γ(\frac{d}{2})} (1 − \frac{1}{ln z} \frac{d−2}{2} \frac{Γ′(k)}{Γ(k)} + …) .  (6.47)

For later reference, we note a list of useful formulas (γ = 0.577216 … is the Euler-constant)

f_{1}^{d}(z) = \frac{(ln z)^{d/2−1}}{Γ(\frac{d}{2})} + \frac{γ(ln z)^{d/2−2}}{Γ(\frac{d−2}{2})} + \frac{(6γ^{2} + π^{2})(ln z)^{d/2−3}}{12Γ(\frac{d−4}{2})} + …  (6.48)
f_{2}^{d}(z) = \frac{(ln z)^{d/2−1}}{Γ(\frac{d}{2})} + \frac{(γ − 1)(ln z)^{d/2−2}}{Γ(\frac{d−2}{2})} + \frac{(6γ(γ − 2) + π^{2})(ln z)^{d/2−3}}{12Γ(\frac{d−4}{2})} + …  (6.49)
f_{3}^{d}(z) = \frac{2(ln z)^{d/2−1}}{Γ(\frac{d}{2})} + \frac{(2γ − 3)(ln z)^{d/2−2}}{Γ(\frac{d−2}{2})} + \frac{(6γ^{2} − 18γ + 6 + π^{2})(ln z)^{d/2−3}}{6Γ(\frac{d−4}{2})} + ….

(6.50)

Note that with the above notations, the relation (6.37) expressing g as a function of z becomes

g(z) = f_{1}^{d+2}(z) + (2−D)\frac{d}{2}f_{2}^{d+2}(z)

−(2−D)^{2} \frac{d}{4} [(2f_{3}^{d+4}(z) − f_{4}^{d+4}(z))  + (2f_{2}^{d+4}(z) − 3f_{3}^{d+4}(z))  ]

−(2−D)\frac{d^{2}}{8} [(−4f_{3}^{d+4}(z) + f_{4}^{d+4}(z))  + (2f_{2}^{d+4}(z))  ] + O(2 − D)^{3} .

(6.51)

Making several times use of the relation (6.45), which can be rationalized with the help of an algebraic manipulation program (see Mathematica^{◯ R}–program in the appendix), the last equation can be cast into a form, that makes the leading behavior for large z obvious at each order of the expansion in 2 − D. Let us return to the notation of (6.24) and let us define a ”connected” correlation function ℂ_{c}(x) as

ℂ_{c}(x) := ℂ(x) − \overline{ℂ(x)} .  (6.52)

Skipping the argument of the correlation function to be averaged, the notation simplifies to

\overline{ℂ(x)} ≡ \overline{ℂ}  a.s.f.,  (6.53)

and we have

\overline{ℂ_{c}^{2}} = \overline{ℂ^{2} − 2\overline{ℂ}ℂ + \overline{ℂ}^{2}} = \overline{ℂ^{2}} − \overline{ℂ}^{2} ,  (6.54)

such that (6.51) becomes

g(z) = f_{1}^{d+2}(z) + (2−D)\frac{d}{2}\overline{ℂ} f_{2}^{d+2}(z)

−(2−D)^{2} \frac{d}{4} [2\overline{ℂ_{c}^{2}} f_{1}^{d+4}(z) + (\overline{ℂ}^{2}−4\overline{ℂ_{c}^{2}}) f_{1}^{d+2}(z) + (2\overline{ℂ_{c}^{2}}−\overline{ℂ}^{2}) f_{1}^{d}(z) − \overline{ℂ_{c}^{2}} f_{1}^{d−2}(z)]

−(2−D)^{2} \frac{d^{2}}{8} [2\overline{ℂ_{c}^{2}} f_{1}^{d+4}(z) − (2\overline{ℂ_{c}^{2}}+\overline{ℂ}^{2}) f_{1}^{d+2}(z) + 2\overline{ℂ}^{2} f_{1}^{d}(z) − \overline{ℂ}^{2} f_{1}^{d−2}(z)]

+ O(2 − D)^{3} .  (6.55)

Note, that the leading behavior for large values of z has been absorbed into the terms ahead, respectively. That is, at second order in 2 − D, the leading behavior is isolated in f_{1}^{d+4}(z). In (6.55) it turns out that the diagrams to be calculated are \overline{ℂ} and \overline{ℂ_{c}^{2}}. On the torus this corresponds to the evaluation of two discrete sums as will be reported in the appendix. We find:

\overline{ℂ} = S_{D} ∑_{k≠0} \frac{1}{k⃗^{2}} − \frac{1}{2π(2 − D)} = −1.4967(6) + S_{D} · 0.3422(1)(2 − D) + O(2 − D)^{2} (6.56)

and

\overline{ℂ_{c}^{2}} = S_{D}^{2} ∑_{k≠0} \frac{1}{k⃗^{4}} = 0.043(1) + O(2 − D) .  (6.57)

k⃗ is D-dimensional with components k_{i} = 2π/L n_{i}, and the indices n_{i} are running from −∞ to ∞, n_{i} = 0 being excluded from the summation.

In the next two sections we shall present two formalisms which give results for ω at second order. It will turn out that the results catch qualitatively correctly the cross-over to D = 1; however they differ in the numerical values. This problem has already been reported in [49], where the same procedure has been applied to open manifolds. There, we suspected this to be a reflection of the fact that the (2−D)-expansion were not systematic due to an inconsistent regularization prescription. Now working with closed manifolds of finite size, regularization problems do not arise. Although the expansion on a torus considerably simplifies at second order as compared to an open manifold ((6.32), (6.33)), however there remain series problems with these schemes. They can be attributed to the fact that at second and higher order in 2 − D the β–function does not show up a finite limit for z → ∞, as it should at the fixed point (lim_{z→∞} β(z) = 0). This problem, however, is no surprise. We expect the β-function to decay for large values of z up to some scaling function like a powerlaw:

β(z) ∼ z^{−ω/ε} .  (6.58)

The dominant exponent containing the correction-to-scaling exponent ω will have some expansion in 2 − D. Therefore, an expansion of the β-function in powers of 2 − D will provide some alternating series involving at each order powers of logarithms of z. Since the expansion of β(z) is alternating in 2 − D as we consider the leading behavior, there arises a zero at some finite ln z^{∗} ∼ (2 − D)^{−2}. The scheme to be presented first is trying to circumvent the problem of a diverging (truncated) β–function by treating the zero as a fixed point. Within the second scheme to be presented thereafter, we first invert the series for the renormalized coupling providing the bare coupling as a function of the latter. This also allows for expressing the β-function in terms of the renormalized coupling. Unfortunately, it turns out that within the 2 − D–expansion both schemes do not provide a systematic way to gain information about the correction-to-scaling exponent ω at the fixed point.

6.4 The RG-functions in the bare coupling

The renormalization group β-function is defined as

β(g) := μ\frac{d}{dμ} _{0}g .  (6.59)

Since it is hard to invert the function g(z), it is advantageous to study β as function of the bare coupling z

β(g) ≡ β(z) = −εz \frac{d}{dz} g .  (6.60)

Taking to each order only those terms into account, which carry the dominant behavior for large values of z, up to second order in 2 − D, we find from (6.51) using (6.43)

β(z) = −ε [f_{1}^{d}(z) + (2 − D)\frac{d}{2} \overline{ℂ} f_{2}^{d}(z) − (2 − D)^{2} \frac{d}{2} (1+\frac{d}{2}) \overline{ℂ_{c}^{2}} f_{1}^{d+2} + …] . (6.61)

Inserting the asymptotic expansions from the last section, this becomes

β(z) = −\frac{ε}{Γ(\frac{d}{2})} [(ln z)^{d/2−1} (1 + (2 − D)\frac{d}{2} \overline{ℂ}) − (2 − D)^{2} (1+\frac{d}{2}) \overline{ℂ_{c}^{2}} (ln z)^{d/2} + …] ,

(6.62)

where omitted terms have either one more power of (2 − D) or 1/ ln z. As shown in section 2, universal properties are given by the correction to scaling exponent ω, reading

ω(z) := \frac{dβ(g)}{dg} = \frac{dβ(z)}{dz} \frac{1}{\frac{dg}{dz}} = −\frac{εz}{β(z)} \frac{dβ(z)}{dz} .  (6.63)

Let us first study D = 2. β(z) and ω(z) are plotted on figure 6.2. We see that for d < 2 the β-function becomes 0 for z → ∞. For d > 2, β(z) has no zero, but the flow of g is still to infinity. In both cases ω is given by ω = lim_{z→∞} ω(z). Taking only the leading term in the (2 − D)-expansion of β(z), ω(z) is given by

ω(z) = −ε\frac{z \frac{d}{dz}f_{1}^{d}(z)}{f_{1}^{d}(z)} → \frac{d − 2}{2 ln z} → 0 .  (6.64)

From (6.62) it is clear, that to first order in 2 − D we do not obtain any correction to the above result. So, let us also take into account the second order in 2 − D. Then the ”fixed point” (a zero) for small enough 2 − D of (6.62) is at the finite value

ln z^{∗} = \overline{ℂ_{c}^{2}} \frac{2}{2 + d} \frac{1}{(2 − D)^{2}} + · · · .  (6.65)

Next, when trying to calculate ω(z), we face the following problem: Since we truncated the series for g at order two in 2 − D, β′(z) does not vanish at the fixed point, i.e. the zero of β(z). This might lead to the conclusion that ω(z^{∗}) is always ∞, an absurd result. However since this is a consequence of the truncation of the series, we follow the strategy also to expand the denominator of ω(z) in powers of 2 − D. From (6.61) and (6.63) we obtain:

ω(z) = −ε [\frac{f_{1}^{d−2}(z)}{f_{1}^{d}(z)} − (2 − D)^{2} \frac{d}{2} (1+\frac{d}{2}) \overline{ℂ_{c}^{2}} \frac{f_{1}^{d}(z)}{f_{1}^{d}(z)}
+ (2 − D)^{2} \frac{d}{2} (1 + \frac{d}{2}) \overline{ℂ_{c}^{2}} \frac{f_{1}^{d+2}(z)f_{1}^{d−2}(z)}{(f_{1}^{d}(z))^{2}} + ⋯] . (6.66)

Figure 6.2: β-function (left) and ω (right) as functions of the dimensionless bare coupling z for different dimensions d: d = 0.25, 0.5, 0.75, …, 3 (from top to bottom, respectively). Note, that β always has a fixed point at z=0 with ω(0)= − ε. Furthermore, there is a fixed point for 0<d<2 at z=∞ with ω(∞)=0, the latter remaining true for d≥2. In d=2 β tends to −ε as z → ∞. Above, β diverges.

Inserting the asymptotic series we find

ω(z) = −ε [\frac{d − 2}{2} \frac{1}{ln z} − (2 − D)^{2} \frac{d}{2} (1+\frac{d}{2}) \overline{ℂ_{c}^{2}} + (2 − D)^{2} \frac{d}{2} (1 + \frac{d}{2}) \overline{ℂ_{c}^{2}} \frac{d − 2}{d} + ⋯] .

(6.67)

Inserting the fixed point (6.65), we arrive at

ω(z^{∗}) = ε [2 + \frac{d}{2} − \frac{d^{2}}{4} ] \overline{ℂ_{c}^{2}} (2 − D)^{2} .  (6.68)

This should be checked against the exact result in D = 1, which reads

ω(D=1) = ε .  (6.69)

Due to the smallness of \overline{ℂ_{c}^{2}} our resummation-procedure thus only reproduces qualitatively the behavior for finite 2 − D. Let us point out that the limit D = 2 has been analyzed in a completely consistent way within this scheme. Setting D = 2 in (6.67) the correction-to-scaling exponent is obtained from the limit z → ∞ in (6.64). Therefore, we always find ω = 0 (independently from d > 0) in D = 2. On the other hand, starting from an ansatz like (6.58) and expanding it in powers of 2 − D, the lowest order coefficient of ω/ε is obtained from the constant lowest order term in 2 − D in the corresponding equation to (6.67). However, the ansatz (6.58) is oversimplified as will be discussed in section 7. In particular it does not cover the limit D = 2. Before further discussing the validity of the procedure, let us turn to the second scheme, namely the calculation in the renormalized coupling.

[figure: g; β(g)]

Figure 6.3: β−function in terms of the renormalized coupling g truncated at order 160, Pade-resummed, and plotting only that part for which the truncated series converges. (This can e.g. be tested by taking away the last few terms of the series.) This is compared to the asymptotic behavior (6.72) (proportional to 1/g for large g). d is set to 1, and we used the diagonal (80,80)-Pade approximant, which was find to converge best. (The non-resummed expression starts to diverge already at g ≈ 1.8 at this order.)

6.5 Calculation in the renormalized coupling

We start from g(z) given in (6.55). Taking to each order in 2 − D only the dominant term for large z into account this can be approximated by

g(z) = \frac{(ln z)^{d/2}}{Γ(\frac{d}{2}+1)} [1 + (2 − D)\frac{d}{2} \overline{ℂ} − (2 − D)^{2} \frac{d}{2} \overline{ℂ_{c}^{2}} ln z] .  (6.70)

Using (6.70), we can write ln z as a function of g. To second order in 2 − D, this reads

ln z = g̃^{2/d} (1 − (2−D)\overline{ℂ} + (2−D)^{2} \frac{d}{2}\overline{ℂ}^{2}) + g̃^{4/d}(2−D)^{2} \overline{ℂ_{c}^{2}} ,
g̃ := g Γ(\frac{d}{2} + 1) .  (6.71)

We now write the β-function in terms of g̃. Starting from (6.62) we obtain

β(g̃) = −\frac{ε}{Γ(\frac{d}{2})} [g̃^{1−2/d} (1 + (2−D) \overline{ℂ} + (2−D)^{2} \overline{ℂ}^{2} \frac{(d−4)(d−2)}{8}) − (2−D)^{2} 2\overline{ℂ_{c}^{2}} g̃] .

(6.72)

Fixed points are at the zeros of the β-function. The non-trivial one is at

g̃^{∗−2/d} = (2 − D)^{2} 2\overline{ℂ_{c}^{2}} + · · · .  (6.73)

The correction to scaling exponent ω is simply obtained by evaluating the derivative of β with respect to g at the fixed point. In terms of g̃ this is

ω(g̃) = \frac{dβ(g)}{dg} = \frac{dβ(g̃)}{dg̃} \frac{dg̃}{dg} = Γ(\frac{d}{2} + 1) \frac{dβ(g̃)}{dg̃}

= −ε\frac{Γ(\frac{d}{2}+1)}{Γ(\frac{d}{2})} [(1−\frac{2}{d}) g̃^{−2/d} (1 + (2−D) \overline{ℂ} + (2−D)^{2} \overline{ℂ}^{2} \frac{(d−4)(d−2)}{8}) − (2−D)^{2} 2\overline{ℂ_{c}^{2}}] .

(6.74)

Figure 6.4: Qualitative behavior for the β-function in D = 1, D = 2 and result anticipated for D ≈ 1.5.

Inserting the fixed point g̃^{∗} from (6.73) we find

ω(g̃^{∗}) = ε 2\overline{ℂ_{c}^{2}} (2 − D)^{2} .  (6.75)

Again, this is only qualitatively correct as compared to the exact result ω(D=1) = ε in D = 1, and coincides with the result obtained from the previous scheme only in d = 0 and d = 2. There is a series problem that has to be mentioned and that applies to the ”bare scheme” from the previous subsection as well as to the calculation in renormalized coupling: In the ”bare scheme” ω(z) is evaluated at the fixed point ln z^{∗} ∼ (2 − D)^{−2}. However, each higher order power in ln z in (6.67) is coming up only with an extra power in 2 − D, therefore providing arbitrarily diverging contributions in 2 − D. The same applies to the second scheme, where ω(g) has an expansion in powers of (2−D)g^{2/d} and the fixed point lies at g^{∗−2/d} ∼ (2 − D)^{2}.

Let us finally point out that in the limit D = 2 the true asymptotic behavior of the β−function in terms of the renormalized coupling g is obtained from the completely summed series (6.38) leading to (6.72) for large g. Conversely, if one tries to invert (6.38) and truncates it taking only a finite number of orders into account, it is at least possible to reach the asymptotic regime – however, for large enough g the truncated β−function wildly oscillates and thus strongly deviates from the true behavior. In figure (6.3) the Pade-resummed truncated β−function up to order g^{160} in d = 1 is compared with the exact, asymptotic flow-function. One notices that the truncated β-function even though improved through a Pade-Resummation hardly gets into touch with the asymptotic regime. The same applies to the slope-function ω(g), which is not shown in fig. (6.3).

Note that the above arguments suggest quite intuitively the behavior of the exact β-function in 1 < D < 2. Whereas for D = 1, the β-function is a parabola, and for D = 2 it decays like a power-law for large g at least as long as d < 2, the β-function for values of D between these two extremes should cut the axes β(g) = 0 at a finite value of g, which for D → 2 wanders off to infinity, thus by continuity forcing the exponent ω to go to 0 for D → 2.

6.6 The limit d = 0

Let us shortly consider the limit of vanishing bulk space dimension d, which is completely solvable for arbitrary internal dimensions D. However, the strong coupling limit of g(z) is not analytic around d = 0. Starting from (6.38) and setting d = 0 one almost immediately obtains:

g(z) = 1 − e^{−z} ,  (6.76)

which can be inverted providing the bare coupling in terms of the renormalized coupling as

z = ln(1 − g)^{−1} .  (6.77)

Recalling the definition of the β−function (2.12) we find:

β(g) = ε(1 − g) ln(1 − g)  (6.78)

with fixed points at

g^{∗} = 0  ∨  g^{∗} = 1 .  (6.79)

Then, the correction-to-scaling exponent ω as well can be evaluated in terms of the renormalized coupling as

ω(g) = \frac{dβ(g)}{dg}|_{g^{∗}} = −ε (1 + ln(1 − g))|_{g^{∗}} ,  (6.80)

such that one finds

ω(g^{∗}=0) = −ε  and ω(g^{∗}=1) = +∞ .  (6.81)

Note that (6.76) is obtained from the leading order contribution in (6.55). All other terms do not contribute. The same result is found from the exact solution of the limit D = 1 setting d = 0 as is shown in the appendix (D.27).

6.7 Laplace-transformed picture

Let us finally point out that the correction to scaling exponent ω found at leading order remains unchanged in the Laplace-transformed picture introduced in section 5, that is we sum over all membrane-sizes weighted by a chemical potential τ. The sum can easily be done, leading to an additional factor of Γ(1+\frac{Nε}{D}), which is N! in the limit of D → 2. (6.37) then has to be replaced by

g = − ∑_{N=1}^{∞} \frac{(−z)^{N}}{N^{d/2}} ,  (6.82)

which is a special function called polylogarithmic function

g = −Li_{\frac{d}{2}} (−z) ,  (6.83)

and accordingly from (6.60) the RG β-function reads

β(z) = ε · Li_{\frac{d}{2} −1}(−z)  (6.84)

while the correction to scaling exponent becomes (6.63)

ω(z) = −ε\frac{Li_{\frac{d−4}{2}} (−z)}{Li_{\frac{d−2}{2}} (−z)} .  (6.85)

Indeed, we find the same fixed point behavior as before and lim_{z→∞} ω(z) = 0 for arbitrary d>0. This follows from the logarithmic divergence of g(z) in the limit of strong coupling.

6.8 Discussion of the analytical continuation to D = 2.

Before we go on with the analysis of the expansion in powers of 2 − D let us discuss the analytical continuation to D = 2 in more detail. In particular we would like to discuss the proper treatment of the factor (2 − D)^{d/2}, which becomes singular in D = 2 and which appears to the Nth power in the Nth loop integral in the perturbation expansion of the effective coupling g(z). It is due to the form of the two-point difference correlator, which reads in dimensions 0 < D < 2:

C(x − y) := \frac{1}{2d} ⟨|r⃗(x⃗) − r⃗(y⃗)|^{2}⟩_{H_{0}} = \frac{|x − y|^{2−D}}{(2 − D)S_{D}} ,  (6.86)

where S_{D} denotes the D-dimensional volume of the unitsphere. Let us recall the expression for the perturbation series for the effective coupling in powers of the scaling variable z := g_{0}L^{ε}:

g(z) = z ∑_{N=0}^{∞} \frac{(−z)^{N}}{(N+1)!} ∫_{x_{1}} ⋯ ∫_{x_{N}} (det D_{ij})^{−d/2} ,  (6.87)

where the internal linear size of the manifold has been scaled out of the integrations, which are left with the measure (6.21) on the torus. The quadratic form D_{ij} is

D_{ij} = \frac{1}{2} [C(x_{N+1}−x_{i}) + C(x_{N+1}−x_{j}) − C(x_{i}−x_{j})] .  (6.88)

The arising of N powers of (2 − D)^{d/2} in the Nth loop order is then clear. Let us show how this factor can be absorbed into a rescaling of the bare coupling z. Substituting r′ = r(S_{D}(2 − D))^{1/2} in the functional integration, the free part of the Hamiltonian (2.6) gets a prefactor (S_{D}(2 − D))^{−1} and, accordingly, the bare coupling is replaced with

z → z(S_{D}(2 − D))^{d/2}  (6.89)

in order to leave the physics invariant. Then, the two-point function in the rescaled variables reads

C(x − y) = |x − y|^{2−D} .  (6.90)

Expressing the perturbation series (6.87) still in powers of the old coupling one has:

g(z) = (S_{D}(2−D))^{−d/2} (z(S_{D}(2−D))^{d/2} ∑_{N=0}^{∞} \frac{(−z(S_{D}(2−D))^{d/2})^{N}}{(N+1)!} ∫_{x_{1}} ⋯ ∫_{x_{N}} (det D_{ij})^{−d/2})

(6.91)

where, thanks to (6.90) the remaining integrand, (det D_{ij})^{−d/2}, now has an expansion in powers of 2 − D as we discussed in the preceding subsections. The overall factor (S_{D}(2−D))^{−d/2} is contributed from the Jacobian in the integration over the position of the center of mass. Actually, as compared to (6.87) in (6.91) nothing has changed. Summing up all loops at each order in 2 − D of the integrand, we arrive at (6.55):

(S_{D}(2−D))^{d/2}g(z) = f_{1}^{d+2}(z(S_{D}(2−D))^{d/2}) + (2−D)\frac{d}{2}\overline{ℂ} f_{2}^{d+2}(z(S_{D}(2−D))^{d/2}) + ….

(6.92)

Let us consider a reference membrane of linear internal size L in D = 2: The expansion of the (dimensionfull) effective coupling of a manifold of linear internal size L/(S_{D}(2−D))^{d/(2ε)} in D < 2 then reads:

L^{ε}g(z) = L^{ε}(f_{1}^{d+2}(z) + (2−D)\frac{d}{2}\overline{ℂ} f_{2}^{d+2}(z) + O(2 − D)^{2}) ,  (6.93)

where the singular scaling factor (S_{D}(2−D))^{d/(2ε)} has canceled out. In the limit D = 2 we are left with

g(z) = f_{1}^{d+2}(z) .  (6.94)

It is important to point out that performing the limit D → 2 in this way corresponds to studying the scaling limit a/L ≪ 1, where a denotes a microscopic cutoff. Therefore, in addition to (6.94) we expect corrections to scaling for small membrane sizes. Equivalently, our statement is that after summing up the complete perturbation series in the limit D = 2 calculating each diagram with an explicit ultraviolet cutoff a and the exact two point function, then, the longdistance behavior is expected to be independent from the cutoff a in the scaling limit a/L ≪ 1. It will coincide with the behavior that we obtained from the resummed, analytically continued diagrams after having sent a to zero. Let us finally emphasize that there is no alternative to forcing the membrane size to ∞ when approaching the limit D = 2. This is in contrast to what is done in the rescaling procedures in (3.3) and (6.1), which imply an infinite bare coupling g_{0} in the limit D = 2. As discussed above the rescaling of the field contributes a Jacobian in the perturbation series of g(z) from the integration over the position of the center of mass and can be only eliminated by forcing the size to ∞. The Jacobian has been dropped in the preceding sections, which is completely safe as long as one is only interested in the exponent ω. There is a simple way to anticipate the divergence of the fixed point in the limit D → 2 and to justify that the analytical continuation of the exponent ω to D = 2 must become zero as we found it: The β-function at two loops reads (4.21)

β(g) = −εg + \frac{g^{2}}{2} I_{1}(D) − \frac{g^{3}}{3} I_{2}(D) + O(g^{4}) ,  (6.95)

where in contrast to (4.21) we have not made the value of the one loop integral, I_{1}(D), explicit. Within the ε-expansion one finds a nontrivial fixed point at g^{∗} (4.23):

g^{∗} = \frac{2ε}{I_{1}(D)} + \frac{8}{3} \frac{I_{2}(D)}{I_{1}^{3}(D)}ε^{2} + O(ε^{3}) .  (6.96)

As we already discussed above each loop integral I_{N} is coming up with N powers of (2−D)^{d/2}, as long as we do not perform the rescaling procedure, such that the shift to infinity in the limit D = 2 of the fixed point g^{∗} can already be anticipated from (6.96). However calculating the correction -to-scaling exponent ω, the ε-expansion reads (4.24):

ω(g^{∗}) = ε − \frac{4}{3} \frac{I_{2}(D)}{I_{1}^{2}(D)}ε^{2} + O(ε^{3}) .  (6.97)

It turns out that all loop integrals appear only in a form, where they are normed with respect to the one loop integral. This way any singular factor from the one loop integral drops out. The reasoning above is not restricted to the ε-expansion. Even more, it has turned out that only within the calculations in fixed dimensions (D, d) the remaining combinations I_{2}(D)/I_{1}^{2}(D) have smooth expansions in 2 − D, from what we may conclude that as long as one is interested in the analytical continuation of the exponent ω to D = 2 it is completely justified to norm all loop integrals with respect to the one loop integral. This is what the rescaling procedure does.

7 Higher order terms in the (2 − D)- expansion on the torus

In this section we will render the analysis of the strong coupling behavior of the theory in D slightly below 2 more systematic. For this purpose we will carry out the expansion in powers of 2 − D up to the fourth order. This will allow for an insight into the general structure of the 2 − D-expansion. Next, we will give two different ansätze for the exact renormalized coupling g(D, z) as a function of the internal dimension D and show, how their free parameters get matched with the obtained expansion in 2 − D. From the second order on, it is quite a tedious work to collect all terms to appear in the expansion. We accomplished this with the help of a Mathematica^{◯ R}-program, which we discuss in the appendix. However, we have only calculated the diagrams to appear up to the second order. To get an idea of the general form of the expansion we also needed the resummed terms to the third and fourth order, even if we do not have the explicit values of the corresponding diagrams available. Of course, our main purpose is to get information about the strong coupling behavior for finite 2 − D close to zero. First, let us list the diagrams to appear up to the fourth order:

= \overline{ℂ(x_{i}−x_{j})} ,  (7.1)

which contributes to first order in 2 − D. To second order one needs in addition

= \overline{ℂ^{2}(x_{i}−x_{j})} .  (7.2)

To third order diagrams with new topology are

= \overline{ℂ^{3}(x_{i}−x_{j})} ,

= \overline{ℂ(x_{i}−x_{j})ℂ(x_{j}−x_{k})ℂ(x_{i}−x_{k})} .  (7.3)

Finally, to fourth order we need:

= \overline{ℂ^{4}(x_{i}−x_{j})} ,

= \overline{ℂ^{2}(x_{i}−x_{j})ℂ(x_{k}−x_{j})ℂ(x_{k}−x_{i})} ,

= \overline{ℂ(x_{i}−x_{j})ℂ(x_{k}−x_{j})ℂ(x_{l}−x_{k})ℂ(x_{i}−x_{l})} .  (7.4)

We will express the resummed series in terms of averages over both the correlator ℂ(x) (6.8) and the connected correlator ℂ_{c}(x) (6.52). One can switch to one or the other representation depending on which might be the more convenient. Before stating the results let us remind the definition of the correlator ℂ(x), which was

C(x) = ℂ^{(0)}(x) + (2 − D)ℂ(x) ,  (7.5)

where ℂ^{(0)}(x) is the limit D → 2 of the two-point correlator C(x). In (6.52) we have already introduced a connected correlation function

ℂ_{c}(x) = ℂ(x) − \overline{ℂ} ,  (7.6)

such that

\overline{ℂ_{c}^{2}} = \overline{ℂ^{2}} − \overline{ℂ}^{2} ,  (7.7)

and the notation for the leading behavior of the renormalized coupling g could be shortened (6.55). From the definition (7.6), we obtain other useful averages, which we will need at higher orders in 2 − D. To third order we will need:

\overline{ℂ_{c}^{3}} = \overline{ℂ^{3}} − 3\overline{ℂ}\overline{ℂ^{2}} + 2\overline{ℂ}^{3}  (7.8)

and

\overline{ℂ_{c}^{△}} = \overline{ℂ_{c}(x_{i}−x_{j})ℂ_{c}(x_{j}−x_{k})ℂ_{c}(x_{k}−x_{i})}
= \overline{ℂ(x_{i}−x_{j})ℂ(x_{j}−x_{k})ℂ(x_{k}−x_{i})} + 3\overline{ℂ}^{2}\overline{ℂ(x_{i}−x_{j})} − 3\overline{ℂ}\overline{ℂ(x_{i}−x_{j})ℂ(x_{j}−x_{k})} − \overline{ℂ}^{3}
= \overline{ℂ^{△}} − \overline{ℂ}^{3} ,  (7.9)

where x_{i}, x_{j}, x_{k} are distinct points, and the average is over their positions. In (7.9) we exploited the symmetry of the closed manifold as it is diagrammatically expressed in (6.32), and the definition of \overline{ℂ^{△}} is self-evident. Furthermore, we will need to fourth order in 2 − D:

\overline{ℂ_{c}^{4}} = \overline{ℂ^{4}} + 12\overline{ℂ^{2}}\overline{ℂ}^{2} − 4\overline{ℂ^{3}}\overline{ℂ} − 3\overline{ℂ}^{4} ,  (7.10)

\overline{ℂ_{c}^{⋄}} = \overline{ℂ_{c}(x_{i}−x_{j})ℂ_{c}(x_{j}−x_{k})ℂ_{c}(x_{k}−x_{l})ℂ_{c}(x_{l}−x_{i})}
= \overline{ℂ^{⋄}} + 5\overline{ℂ}^{4}  (7.11)

and

\overline{ℂ_{c}} = \overline{ℂ_{c}^{2}(x_{i}−x_{j})ℂ_{c}(x_{i}−x_{k})ℂ_{c}(x_{k}−x_{j})}
= \overline{ℂ} − 2\overline{ℂ^{△}}\overline{ℂ} − \overline{ℂ^{2}}\overline{ℂ}^{2} + 2\overline{ℂ}^{4} .  (7.12)

7.1 Resummed contributions to the expansion in 2 − D up to the fourth order

In the following we will state all terms, which appear in the expansion of the renormalized coupling g(z) up to the fourth order in 2 − D according to (6.14). We have to calculate at order N of perturbation theory:

\overline{TrM} = N \overline{ℂ}  (7.13)

Inserting this into the perturbation series and summing up the resulting terms to all orders in N generates the following contributions in the 2 − D-expansion of the renormalized coupling:

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{TrM}(−z)^{N+1}}{N!} = \overline{ℂ}f_{1}^{d+2}(z) − \overline{ℂ}f_{1}^{d}(z) ,  (7.14)

which contributes to first order in 2 − D. To second order in 2 − D, we have:

\overline{TrM^{2}} = \frac{−2N(N−1) − N(N−1)(N−2)}{1+N} \overline{ℂ}^{2} + \frac{2N + 3N(N−1) + N(N−1)(N−2)}{1+N} \overline{ℂ^{2}} ,

(7.15)

providing

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{TrM^{2}}(−z)^{N+1}}{N!} = (−2\overline{ℂ}^{2} + 2\overline{ℂ^{2}})f_{1}^{d+4}(z) + (5\overline{ℂ}^{2} − 4\overline{ℂ^{2}})f_{1}^{d+2} + (−4\overline{ℂ}^{2} + 3\overline{ℂ^{2}})f_{1}^{d}(z)
−(−\overline{ℂ}^{2} + \overline{ℂ^{2}})f_{1}^{d−2}(z) ,  (7.16)

and

\overline{Tr^{2}M} = \frac{4N(N−1) + N(N−1)(N−2)}{1+N} \overline{ℂ}^{2} + \frac{2N}{1+N} \overline{ℂ^{2}} ,

(7.17)

providing

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{Tr^{2}M}(−z)^{N+1}}{N!} = (−2\overline{ℂ}^{2} + 2\overline{ℂ^{2}})f_{1}^{d+4}(z) + (\overline{ℂ}^{2} − 2\overline{ℂ^{2}})f_{1}^{d+2}(z)
+2\overline{ℂ}^{2}f_{1}^{d}(z) − \overline{ℂ}^{2}f_{1}^{d−2}(z) .  (7.18)

Let us rewrite this in terms of the connected correlator (7.7):

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{TrM^{2}}(−z)^{N+1}}{N!} = 2\overline{ℂ_{c}^{2}} f_{1}^{d+4}(z) + (−4\overline{ℂ_{c}^{2}} + \overline{ℂ}^{2})f_{1}^{d+2}(z) + (−\overline{ℂ}^{2} + 3\overline{ℂ_{c}^{2}}) f_{1}^{d}(z)
−\overline{ℂ_{c}^{2}} f_{1}^{d−2}(z) ,  (7.19)

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{Tr^{2}M}(−z)^{N+1}}{N!} = 2\overline{ℂ_{c}^{2}} f_{1}^{d+4}(z) − (2\overline{ℂ_{c}^{2}} + \overline{ℂ}^{2})f_{1}^{d+2}(z)
+2\overline{ℂ}^{2} f_{1}^{d}(z) − \overline{ℂ}^{2} f_{1}^{d−2}(z)  (7.20)

58 Interacting Crumpled Manifolds

Let us now state the terms to appear at third order in 2 − D, which we derived with the help of a Mathematica^{◯ R}- program (N denotes the loop order.):

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{TrM^{3}}(−z)^{N+1}}{N!} = 4(6\overline{ℂ}^{3} − 3\overline{ℂ}\overline{ℂ^{2}} + \overline{ℂ^{3}} − 4\overline{ℂ^{△}})f_{1}^{d+4}(z)
+ (−62\overline{ℂ}^{3} + 36\overline{ℂ}\overline{ℂ^{2}} − 10\overline{ℂ^{3}} + 36\overline{ℂ^{△}})f_{1}^{d+2}(z)
+ (63\overline{ℂ}^{3} − 39\overline{ℂ}\overline{ℂ^{2}} + 9\overline{ℂ^{3}} − 32\overline{ℂ^{△}})f_{1}^{d}(z)
+ (−33\overline{ℂ}^{3} + 18\overline{ℂ}\overline{ℂ^{2}} − 3\overline{ℂ^{3}} + 17\overline{ℂ^{△}})f_{1}^{d−2}(z)
+ (9\overline{ℂ}^{3} − 3\overline{ℂ}\overline{ℂ^{2}} − 6\overline{ℂ^{△}})f_{1}^{d−4}(z) + (−\overline{ℂ}^{3} + \overline{ℂ^{△}})f_{1}^{d−6}(z)

(7.21)

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{TrM TrM^{2}}(−z)^{N+1}}{N!} = 4(6\overline{ℂ}^{3} − 3\overline{ℂ}\overline{ℂ^{2}} + \overline{ℂ^{3}} − 4\overline{ℂ^{△}})f_{1}^{d+6}(z)
+ (−50\overline{ℂ}^{3} + 26\overline{ℂ}\overline{ℂ^{2}} − 8\overline{ℂ^{3}} + 32\overline{ℂ^{△}})f_{1}^{d+4}(z)
+ (29\overline{ℂ}^{3} − 16\overline{ℂ}\overline{ℂ^{2}} + 6\overline{ℂ^{3}} − 20\overline{ℂ^{△}})f_{1}^{d+2}(z)
+ (\overline{ℂ}^{3} − \overline{ℂ}\overline{ℂ^{2}} − 2\overline{ℂ^{3}} + 4\overline{ℂ^{△}})f_{1}^{d}(z)
+ (−5\overline{ℂ}^{3} + 4\overline{ℂ}\overline{ℂ^{2}})f_{1}^{d−2}(z) + (\overline{ℂ}^{3} − \overline{ℂ}\overline{ℂ^{2}})f_{1}^{d−4}(z)

(7.22)

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{Tr^{3}M}(−z)^{N+1}}{N!} = 4(6\overline{ℂ}^{3} − 3\overline{ℂ}\overline{ℂ^{2}} + \overline{ℂ^{3}} − 4\overline{ℂ^{△}})f_{1}^{d+6}(z)
+ (−26\overline{ℂ}^{3} + 6\overline{ℂ}\overline{ℂ^{2}} − 4\overline{ℂ^{3}} + 24\overline{ℂ^{△}})f_{1}^{d+4}(z)
+ (−3\overline{ℂ}^{3} + 12\overline{ℂ}\overline{ℂ^{2}} − 8\overline{ℂ^{△}})f_{1}^{d+2}(z)
+ 3(\overline{ℂ}^{3} − 2\overline{ℂ}\overline{ℂ^{2}})f_{1}^{d}(z)
+ 3\overline{ℂ}^{3}f_{1}^{d−2}(z) − \overline{ℂ}^{3}f_{1}^{d−4}(z)

(7.23)

Again, let us rewrite this in terms of connected correlators:

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{TrM^{3}}(−z)^{N+1}}{N!} = 4(\overline{ℂ_{c}^{3}} − 4\overline{ℂ_{c}^{△}})f_{1}^{d+4}(z)
+ (−10\overline{ℂ_{c}^{3}} + 36\overline{ℂ_{c}^{△}} + 6\overline{ℂ}\overline{ℂ_{c}^{2}})f_{1}^{d+2}(z)
+ (9\overline{ℂ_{c}^{3}} − 32\overline{ℂ_{c}^{△}} − 12\overline{ℂ}\overline{ℂ_{c}^{2}} + \overline{ℂ}^{3})f_{1}^{d}(z)
+ (3\overline{ℂ_{c}^{3}} − 17\overline{ℂ_{c}^{△}} − 9\overline{ℂ}\overline{ℂ_{c}^{2}} + \overline{ℂ}^{3})f_{1}^{d−2}(z)
− 3(2\overline{ℂ_{c}^{△}} + \overline{ℂ}\overline{ℂ_{c}^{2}})f_{1}^{d−4}(z) + \overline{ℂ_{c}^{△}} f_{1}^{d−6}(z) ,

(7.24)

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{TrM TrM^{2}}(−z)^{N+1}}{N!} = 4(\overline{ℂ_{c}^{3}} − 4\overline{ℂ_{c}^{△}})f_{1}^{d+6}(z)
+ (−8\overline{ℂ_{c}^{3}} + 32\overline{ℂ_{c}^{△}} + 2\overline{ℂ}\overline{ℂ_{c}^{2}})f_{1}^{d+4}(z)
+ (6\overline{ℂ_{c}^{3}} − 20\overline{ℂ_{c}^{△}} + 2\overline{ℂ}\overline{ℂ_{c}^{2}} − 6\overline{ℂ}^{3})f_{1}^{d+2}(z)
+ (−2\overline{ℂ_{c}^{3}} + 4\overline{ℂ_{c}^{△}} − 7\overline{ℂ}\overline{ℂ_{c}^{2}} + 2\overline{ℂ}^{3})f_{1}^{d}(z)
− (\overline{ℂ}^{3} − 4\overline{ℂ}\overline{ℂ_{c}^{2}})f_{1}^{d−2}(z) − \overline{ℂ}\overline{ℂ_{c}^{2}} f_{1}^{d−4}(z) ,

(7.25)

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{Tr^{3}M}(−z)^{N+1}}{N!} = 4(\overline{ℂ_{c}^{3}} − 4\overline{ℂ_{c}^{△}})f_{1}^{d+6}(z)
+ (−4\overline{ℂ_{c}^{3}} + 24\overline{ℂ_{c}^{△}} − 6\overline{ℂ}\overline{ℂ_{c}^{2}})f_{1}^{d+4}(z)
+ (−8\overline{ℂ_{c}^{△}} + 12\overline{ℂ}\overline{ℂ_{c}^{2}} + \overline{ℂ}^{3})f_{1}^{d+2}(z)
− 3(\overline{ℂ}^{3} + 2\overline{ℂ}\overline{ℂ_{c}^{2}})f_{1}^{d}(z)
+ 3\overline{ℂ}^{3}f_{1}^{d−2}(z) − \overline{ℂ}^{3}f_{1}^{d−4}(z) .

(7.26)

To fourth order in 2 − D we obtain:

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{TrM^{4}}(−z)^{N+1}}{N!}

= 8(90\overline{ℂ}^{4} − 36\overline{ℂ}^{2}\overline{ℂ^{2}} + 3\overline{ℂ^{2}}^{2} + 4\overline{ℂ}\overline{ℂ^{3}} − \overline{ℂ^{4}} + 24\overline{ℂ} − 36\overline{ℂ^{⋄}} − 48\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+8}(z)
+ 4(501\overline{ℂ}^{4} − 222\overline{ℂ}^{2}\overline{ℂ^{2}} + 23\overline{ℂ^{2}}^{2} + 28\overline{ℂ}\overline{ℂ^{3}} − 6\overline{ℂ^{4}} + 128\overline{ℂ} − 180\overline{ℂ^{⋄}} − 272\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+6}(z)
− 4(561\overline{ℂ}^{4} − 273\overline{ℂ}^{2}\overline{ℂ^{2}} + 31\overline{ℂ^{2}}^{2} + 38\overline{ℂ}\overline{ℂ^{3}} − 7\overline{ℂ^{4}} + 136\overline{ℂ} − 178\overline{ℂ^{⋄}} − 308\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+4}(z)
+ (1359\overline{ℂ}^{4} − 696\overline{ℂ}^{2}\overline{ℂ^{2}} + 82\overline{ℂ^{2}}^{2} + 100\overline{ℂ}\overline{ℂ^{3}} − 16\overline{ℂ^{4}} + 304\overline{ℂ} − 396\overline{ℂ^{⋄}} − 736\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+2}(z)
+ (−500\overline{ℂ}^{4} + 248\overline{ℂ}^{2}\overline{ℂ^{2}} − 36\overline{ℂ^{2}}^{2} − 32\overline{ℂ}\overline{ℂ^{3}} + 5\overline{ℂ^{4}} − 92\overline{ℂ} + 154\overline{ℂ^{⋄}} + 252\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d}(z)
+ (116\overline{ℂ}^{4} − 48\overline{ℂ}^{2}\overline{ℂ^{2}} + 12\overline{ℂ^{2}}^{2} + 4\overline{ℂ}\overline{ℂ^{3}} − \overline{ℂ^{4}} + 12\overline{ℂ} − 47\overline{ℂ^{⋄}} − 48\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d−2}(z)
+ (−16\overline{ℂ}^{4} + 4\overline{ℂ}^{2}\overline{ℂ^{2}} − 2\overline{ℂ^{2}}^{2} + 10\overline{ℂ} + 4\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d−4}(z) + (\overline{ℂ}^{4} − \overline{ℂ^{⋄}})f_{1}^{d−6}(z)

(7.27)

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{Tr^{2}M^{2}}(−z)^{N+1}}{N!}

= 8(90\overline{ℂ}^{4} − 36\overline{ℂ}^{2}\overline{ℂ^{2}} + 3\overline{ℂ^{2}}^{2} + 4\overline{ℂ}\overline{ℂ^{3}} − \overline{ℂ^{4}} + 24\overline{ℂ} − 36\overline{ℂ^{⋄}} − 48\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+8}(z)
+ 4(501\overline{ℂ}^{4} − 222\overline{ℂ}^{2}\overline{ℂ^{2}} + 23\overline{ℂ^{2}}^{2} + 28\overline{ℂ}\overline{ℂ^{3}} − 6\overline{ℂ^{4}} + 128\overline{ℂ} − 180\overline{ℂ^{⋄}} − 272\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+6}(z)
− 4(537\overline{ℂ}^{4} − 261\overline{ℂ}^{2}\overline{ℂ^{2}} + 28\overline{ℂ^{2}}^{2} + 40\overline{ℂ}\overline{ℂ^{3}} − 8\overline{ℂ^{4}} + 132\overline{ℂ} − 172\overline{ℂ^{⋄}} − 296\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+4}(z)
+ (1111\overline{ℂ}^{4} − 572\overline{ℂ}^{2}\overline{ℂ^{2}} + 44\overline{ℂ^{2}}^{2} + 120\overline{ℂ}\overline{ℂ^{3}} − 24\overline{ℂ^{4}} + 272\overline{ℂ} − 624\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+2}(z)
+ (260\overline{ℂ}^{4} − 122\overline{ℂ}^{2}\overline{ℂ^{2}} − 10\overline{ℂ^{2}}^{2} + 48\overline{ℂ}\overline{ℂ^{3}} − 10\overline{ℂ^{4}} + 72\overline{ℂ} − 80\overline{ℂ^{⋄}} − 160\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d}(z)
+ (6\overline{ℂ}^{4} + 18\overline{ℂ}^{2}\overline{ℂ^{2}} − 15\overline{ℂ^{2}}^{2} + 8\overline{ℂ}\overline{ℂ^{3}} − 2\overline{ℂ^{4}} + 8\overline{ℂ} − 8\overline{ℂ^{⋄}} − 16\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d−2}(z)
+ (8\overline{ℂ}^{4} − 14\overline{ℂ}^{2}\overline{ℂ^{2}} + 6\overline{ℂ^{2}}^{2})f_{1}^{d−4}(z)
+ (−\overline{ℂ}^{4} + 2\overline{ℂ}^{2}\overline{ℂ^{2}} − \overline{ℂ^{2}}^{2})f_{1}^{d−6}(z)

(7.28)

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{TrM TrM^{3}}(−z)^{N+1}}{N!}

= 8(90\overline{ℂ}^{4} − 36\overline{ℂ}^{2}\overline{ℂ^{2}} + 3\overline{ℂ^{2}}^{2} + 4\overline{ℂ}\overline{ℂ^{3}} − \overline{ℂ^{4}} + 24\overline{ℂ} − 36\overline{ℂ^{⋄}} − 48\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+8}(z)
+ 4(441\overline{ℂ}^{4} − 186\overline{ℂ}^{2}\overline{ℂ^{2}} + 18\overline{ℂ^{2}}^{2} + 22\overline{ℂ}\overline{ℂ^{3}} − 5\overline{ℂ^{4}} + 114\overline{ℂ} − 168\overline{ℂ^{⋄}} − 236\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+6}(z)
− 2(800\overline{ℂ}^{4} − 351\overline{ℂ}^{2}\overline{ℂ^{2}} + 39\overline{ℂ^{2}}^{2} + 41\overline{ℂ}\overline{ℂ^{3}} − 9\overline{ℂ^{4}} + 204\overline{ℂ} − 294\overline{ℂ^{⋄}} − 430\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+4}(z)
+ (661\overline{ℂ}^{4} − 279\overline{ℂ}^{2}\overline{ℂ^{2}} + 36\overline{ℂ^{2}}^{2} + 23\overline{ℂ}\overline{ℂ^{3}} − 6\overline{ℂ^{4}} + 186\overline{ℂ} − 258\overline{ℂ^{⋄}} − 364\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+2}(z)
+ (−90\overline{ℂ}^{4} + 21\overline{ℂ}^{2}\overline{ℂ^{2}} − 6\overline{ℂ^{2}}^{2} + 6\overline{ℂ}\overline{ℂ^{3}} − 48\overline{ℂ} + 60\overline{ℂ^{⋄}} + 59\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d}(z)
+ (−24\overline{ℂ}^{4} + 15\overline{ℂ}^{2}\overline{ℂ^{2}} − 3\overline{ℂ}\overline{ℂ^{3}} + 6\overline{ℂ} − 6\overline{ℂ^{⋄}} + 11\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d−2}(z)
+ (10\overline{ℂ}^{4} − 3\overline{ℂ}^{2}\overline{ℂ^{2}} − 7\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d−4}(z) + (−\overline{ℂ}^{4} + \overline{ℂ}\overline{ℂ^{△}})f_{1}^{d−6}(z)

(7.29)

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{Tr^{2}M TrM^{2}}(−z)^{N+1}}{N!}

= 8(90\overline{ℂ}^{4} − 36\overline{ℂ}^{2}\overline{ℂ^{2}} + 3\overline{ℂ^{2}}^{2} + 4\overline{ℂ}\overline{ℂ^{3}} − \overline{ℂ^{4}} + 24\overline{ℂ} − 36\overline{ℂ^{⋄}} − 48\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+8}(z)
+ 4(381\overline{ℂ}^{4} − 150\overline{ℂ}^{2}\overline{ℂ^{2}} + 13\overline{ℂ^{2}}^{2} + 16\overline{ℂ}\overline{ℂ^{3}} + 4\overline{ℂ^{4}} − 100\overline{ℂ} + 156\overline{ℂ^{⋄}} − 200\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+6}(z)
+ 4(251\overline{ℂ}^{4} − 90\overline{ℂ}^{2}\overline{ℂ^{2}} + 8\overline{ℂ^{2}}^{2} + 8\overline{ℂ}\overline{ℂ^{3}} − 3\overline{ℂ^{4}} + 70\overline{ℂ} − 116\overline{ℂ^{⋄}} − 128\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+4}(z)
+ (159\overline{ℂ}^{4} − 4\overline{ℂ}^{2}\overline{ℂ^{2}} − 4\overline{ℂ^{2}}^{2} − 12\overline{ℂ}\overline{ℂ^{3}} + 2\overline{ℂ^{4}} − 40\overline{ℂ} + 72\overline{ℂ^{⋄}} − 72\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+2}(z)
+ (50\overline{ℂ}^{4} − 53\overline{ℂ}^{2}\overline{ℂ^{2}} + 8\overline{ℂ^{2}}^{2} + 16\overline{ℂ}\overline{ℂ^{3}} − 8\overline{ℂ} + 16\overline{ℂ^{⋄}} − 32\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d}(z)
+ (−4\overline{ℂ}^{4} + 5\overline{ℂ}^{2}\overline{ℂ^{2}} − 2\overline{ℂ^{2}}^{2} − 4\overline{ℂ}\overline{ℂ^{3}} + 8\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d−2}(z)
+ (−6\overline{ℂ}^{4} + 5\overline{ℂ}^{2}\overline{ℂ^{2}})f_{1}^{d−4}(z) + (\overline{ℂ}^{4} − \overline{ℂ}^{2}\overline{ℂ^{2}})f_{1}^{d−6}(z)

(7.30)

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{Tr^{4}M}(−z)^{N+1}}{N!}

= 8(90\overline{ℂ}^{4} − 36\overline{ℂ}^{2}\overline{ℂ^{2}} + 3\overline{ℂ^{2}}^{2} + 4\overline{ℂ}\overline{ℂ^{3}} − \overline{ℂ^{4}} + 24\overline{ℂ} − 36\overline{ℂ^{⋄}} − 48\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+8}(z)
+ 4(261\overline{ℂ}^{4} − 78\overline{ℂ}^{2}\overline{ℂ^{2}} + 3\overline{ℂ^{2}}^{2} + 4\overline{ℂ}\overline{ℂ^{3}} − 2\overline{ℂ^{4}} + 72\overline{ℂ} − 132\overline{ℂ^{⋄}} − 128\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+6}(z)
+ 4(61\overline{ℂ}^{4} + 9\overline{ℂ}^{2}\overline{ℂ^{2}} − 6\overline{ℂ^{2}}^{2} − 8\overline{ℂ}\overline{ℂ^{3}} + 24\overline{ℂ} − 72\overline{ℂ^{⋄}} − 8\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+4}(z)
+ (−89\overline{ℂ}^{4} + 36\overline{ℂ}^{2}\overline{ℂ^{2}} − 12\overline{ℂ^{2}}^{2} − 16\overline{ℂ}\overline{ℂ^{3}} − 48\overline{ℂ^{⋄}} + 128\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d+2}(z)
+ (36\overline{ℂ}^{2}\overline{ℂ^{2}} − 32\overline{ℂ}\overline{ℂ^{△}})f_{1}^{d}(z) + (6\overline{ℂ}^{4} − 12\overline{ℂ}^{2}\overline{ℂ^{2}})f_{1}^{d−2}(z)
+ 4\overline{ℂ}^{4}f_{1}^{d−4}(z) − \overline{ℂ}^{4}f_{1}^{d−6}(z)

(7.31)

Rewriting everything in terms of connected correlators we find:

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{Tr^{4}M}(−z)^{N+1}}{N!}

= 8(222\overline{ℂ}^{4} + 6\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 3\overline{ℂ_{c}^{2}}^{2} − \overline{ℂ_{c}^{4}} + 24\overline{ℂ_{c}} − 36\overline{ℂ_{c}^{⋄}})f_{1}^{d+8}(z)
+ 4(804\overline{ℂ}^{4} + 12\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 3\overline{ℂ_{c}^{2}}^{2} − 2\overline{ℂ_{c}^{4}} + 72\overline{ℂ_{c}} − 132\overline{ℂ_{c}^{⋄}} − 4\overline{ℂ}\overline{ℂ_{c}^{3}} + 16\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d+6}(z)
− 4(432\overline{ℂ}^{4} − 3\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} − 6\overline{ℂ_{c}^{2}}^{2} + 24\overline{ℂ_{c}} − 72\overline{ℂ_{c}^{⋄}} − 8\overline{ℂ}\overline{ℂ_{c}^{3}} + 40\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d+4}(z)
+ (287\overline{ℂ}^{4} − 36\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} − 12\overline{ℂ_{c}^{2}}^{2} − 48\overline{ℂ_{c}^{⋄}} − 16\overline{ℂ}\overline{ℂ_{c}^{3}} + 128\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d+2}(z)
+ (4\overline{ℂ}^{4} + 36\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} − 32\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d}(z)
+ (−6\overline{ℂ}^{4} − 2\overline{ℂ}^{2}\overline{ℂ_{c}^{2}})f_{1}^{d−2}(z) + 4\overline{ℂ}^{4}f_{1}^{d−4}(z) − \overline{ℂ}^{4}f_{1}^{d−6}(z)

(7.32)

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{TrM^{2} Tr^{2}M}(−z)^{N+1}}{N!}

= 8(222\overline{ℂ}^{4} + 6\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 3\overline{ℂ_{c}^{2}}^{2} − \overline{ℂ_{c}^{4}} + 24\overline{ℂ_{c}} − 36\overline{ℂ_{c}^{⋄}})f_{1}^{d+8}(z)
+ 4(960\overline{ℂ}^{4} + 24\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + \overline{ℂ_{c}^{2}}^{2} − 4\overline{ℂ_{c}^{4}} + 100\overline{ℂ_{c}} − 156\overline{ℂ_{c}^{⋄}})f_{1}^{d+6}(z)
− 4(714\overline{ℂ}^{4} + 20\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 8\overline{ℂ_{c}^{2}}^{2} − 3\overline{ℂ_{c}^{4}} + 70\overline{ℂ_{c}} − 116\overline{ℂ_{c}^{⋄}} − 4\overline{ℂ}\overline{ℂ_{c}^{3}} + 12\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d+4}(z)
+ (889\overline{ℂ}^{4} + 36\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} − 2\overline{ℂ_{c}^{2}}^{2} − 28\overline{ℂ}\overline{ℂ_{c}^{3}} − 4\overline{ℂ_{c}^{4}} + 80\overline{ℂ_{c}} − 144\overline{ℂ_{c}^{⋄}} + 88\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d+2}(z)
+ (−99\overline{ℂ}^{4} + 3\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 8\overline{ℂ_{c}^{2}}^{2} + 16\overline{ℂ}\overline{ℂ_{c}^{3}} − 8\overline{ℂ_{c}} + 16\overline{ℂ_{c}^{⋄}} − 48\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d}(z)
+ (3\overline{ℂ}^{4} − 11\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} − 2\overline{ℂ_{c}^{2}}^{2} − 4\overline{ℂ}\overline{ℂ_{c}^{3}} + 8\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d−2}(z)
+ (−\overline{ℂ}^{4} − 5\overline{ℂ}^{2}\overline{ℂ_{c}^{2}})f_{1}^{d−4}(z) − \overline{ℂ}^{2}\overline{ℂ_{c}^{2}}f_{1}^{d−6}(z)

(7.33)

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{Tr^{2}M^{2}}(−z)^{N+1}}{N!}

= 8(222\overline{ℂ}^{4} + 6\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 3\overline{ℂ_{c}^{2}}^{2} − \overline{ℂ_{c}^{4}} + 24\overline{ℂ_{c}} − 36\overline{ℂ_{c}^{⋄}})f_{1}^{d+8}(z)
+ 4(1116\overline{ℂ}^{4} + 36\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 23\overline{ℂ_{c}^{2}}^{2} − 6\overline{ℂ_{c}^{4}} + 128\overline{ℂ_{c}} − 180\overline{ℂ_{c}^{⋄}} + 4\overline{ℂ}\overline{ℂ_{c}^{3}} − 16\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d+6}(z)
− 4(1080\overline{ℂ}^{4} + 47\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 28\overline{ℂ_{c}^{2}}^{2} − 8\overline{ℂ_{c}^{4}} + 132\overline{ℂ_{c}} − 172\overline{ℂ_{c}^{⋄}} + 8\overline{ℂ}\overline{ℂ_{c}^{3}} − 32\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d+4}(z)
+ (2111\overline{ℂ}^{4} + 148\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 44\overline{ℂ_{c}^{2}}^{2} + 24\overline{ℂ}\overline{ℂ_{c}^{3}} − 24\overline{ℂ_{c}^{4}} + 272\overline{ℂ_{c}} − 328\overline{ℂ_{c}^{⋄}} − 80\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d+2}(z)
+ (−538\overline{ℂ}^{4} − 74\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 10\overline{ℂ_{c}^{2}}^{2} − 8\overline{ℂ}\overline{ℂ_{c}^{3}} + 10\overline{ℂ_{c}^{4}} − 72\overline{ℂ_{c}} + 80\overline{ℂ_{c}^{⋄}} + 8\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d}(z)
+ (59\overline{ℂ}^{4} + 20\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} − 15\overline{ℂ_{c}^{2}}^{2} − 2\overline{ℂ_{c}^{4}} + 8\overline{ℂ_{c}} − 8\overline{ℂ_{c}^{⋄}})f_{1}^{d−2}(z)
+ (−2\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 6\overline{ℂ_{c}^{2}}^{2})f_{1}^{d−4}(z) − \overline{ℂ_{c}^{2}}^{2}f_{1}^{d−6}(z)

(7.34)

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{TrM TrM^{3}}(−z)^{N+1}}{N!}

= 8(222\overline{ℂ}^{4} + 6\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 3\overline{ℂ_{c}^{2}}^{2} − \overline{ℂ_{c}^{4}} + 24\overline{ℂ_{c}} − 36\overline{ℂ_{c}^{⋄}})f_{1}^{d+8}(z)
+ 4(1038\overline{ℂ}^{4} + 30\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 18\overline{ℂ_{c}^{2}}^{2} − 5\overline{ℂ_{c}^{4}} + 114\overline{ℂ_{c}} − 168\overline{ℂ_{c}^{⋄}} + 2\overline{ℂ}\overline{ℂ_{c}^{3}} − 8\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d+6}(z)
− 2(1818\overline{ℂ}^{4} + 54\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 39\overline{ℂ_{c}^{2}}^{2} + \overline{ℂ}\overline{ℂ_{c}^{3}} − 9\overline{ℂ_{c}^{4}} + 204\overline{ℂ_{c}} − 294\overline{ℂ_{c}^{⋄}} − 22\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d+4}(z)
+ (1583\overline{ℂ}^{4} + 48\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 36\overline{ℂ_{c}^{2}}^{2} − \overline{ℂ}\overline{ℂ_{c}^{3}} − 6\overline{ℂ_{c}^{4}} + 186\overline{ℂ_{c}} − 258\overline{ℂ_{c}^{⋄}} + 8\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d+2}(z)
+ (−358\overline{ℂ}^{4} − 21\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} − 6\overline{ℂ_{c}^{2}}^{2} + 6\overline{ℂ}\overline{ℂ_{c}^{3}} − 48\overline{ℂ_{c}} + 60\overline{ℂ_{c}^{⋄}} − 37\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d}(z)
+ (35\overline{ℂ}^{4} + 12\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} − 3\overline{ℂ}\overline{ℂ_{c}^{3}} + 6\overline{ℂ_{c}} − 6\overline{ℂ_{c}^{⋄}} + 23\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d−2}(z)
+ (−3\overline{ℂ}^{2} − 7\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d−4}(z) + \overline{ℂ}\overline{ℂ_{c}^{△}}f_{1}^{d−6}(z)

(7.35)

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{TrM^{4}}(−z)^{N+1}}{N!}

= 8(222\overline{ℂ}^{4} + 6\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 3\overline{ℂ_{c}^{2}}^{2} − \overline{ℂ_{c}^{4}} + 24\overline{ℂ_{c}} − 36\overline{ℂ_{c}^{⋄}})f_{1}^{d+8}(z)
+ 4(1116\overline{ℂ}^{4} + 36\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 23\overline{ℂ_{c}^{2}}^{2} − 6\overline{ℂ_{c}^{4}} + 128\overline{ℂ_{c}} − 180\overline{ℂ_{c}^{⋄}} + 4\overline{ℂ}\overline{ℂ_{c}^{3}} − 16\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d+6}(z)
− 4(1110\overline{ℂ}^{4} + 39\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 31\overline{ℂ_{c}^{2}}^{2} + 10\overline{ℂ}\overline{ℂ_{c}^{3}} − 7\overline{ℂ_{c}^{4}} + 136\overline{ℂ_{c}} − 178\overline{ℂ_{c}^{⋄}} − 36\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d+4}(z)
+ (2473\overline{ℂ}^{4} + 72\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 82\overline{ℂ_{c}^{2}}^{2} + 36\overline{ℂ}\overline{ℂ_{c}^{3}} − 16\overline{ℂ_{c}^{4}} + 304\overline{ℂ_{c}} − 396\overline{ℂ_{c}^{⋄}} − 128\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d+2}(z)
+ (−955\overline{ℂ}^{4} − 12\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} − 36\overline{ℂ_{c}^{2}}^{2} − 12\overline{ℂ}\overline{ℂ_{c}^{3}} + 5\overline{ℂ_{c}^{4}} − 92\overline{ℂ_{c}} + 154\overline{ℂ_{c}^{⋄}} + 68\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d}(z)
+ (288\overline{ℂ}^{4} + 12\overline{ℂ_{c}^{2}}^{2} − \overline{ℂ_{c}^{4}} + 12\overline{ℂ_{c}} − 47\overline{ℂ_{c}^{⋄}} − 24\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d−2}(z)
+ (−60\overline{ℂ}^{4} − 2\overline{ℂ_{c}^{2}}^{2} + 10\overline{ℂ_{c}^{⋄}} + 4\overline{ℂ}\overline{ℂ_{c}^{△}})f_{1}^{d−4}(z) + (6\overline{ℂ}^{4} − \overline{ℂ_{c}^{⋄}})f_{1}^{d−6}(z)

(7.36)

7.2 Renormalized coupling

Our purpose is to guess the exact form of the renormalized coupling g(z) as a function of 2 − D on a manifold of toroidal topology. The information that is of our disposal are the exact results in both D = 2 (6.38) and D = 1 (D.24) as well as the exact resummed series up to the fourth order in 2 − D. From the structure of the expansion of g(z) in powers of 2 − D it is clear (see also discussion of the Mathematica^{◯ R}-program in the appendix) that to each order in 2 − D g(z) can be expressed in terms of f_{1}^{d}s (6.45). Since

f_{1}^{d}(z) = \frac{z}{Γ(\frac{d}{2})} ∫_{0}^{∞} dr r^{d/2−1} exp[−ze^{−r} − r] ,  (7.37)

the exact g(z) as a function of 2 − D will be of the form:

g(z) ≡ g(D, z) = z ∫_{0}^{∞} dr g̃(D, r) exp[−ze^{−r} − r] ,  (7.38)

where g̃(D, r) has some expansion in powers of 2−D. To zeroth order corresponding to D = 2 one has:

g̃(2, r) = \frac{r^{d/2}}{Γ(\frac{d}{2}+1)} .  (7.39)

Analyzing the series data from the preceding subsection reveals the following structure of the expansion of g̃(D, z):

g̃(D, r) = r^{d/2} (\frac{1}{Γ(\frac{d}{2}+1)} + (2 − D)P(2−D, r)) ,  (7.40)

where P(2−D, r) is analytic in D = 2, such that

P(2−D, r) = ∑_{n=0}^{∞} P_{n}(r)(2 − D)^{n} ,  (7.41)

and the nth order coefficient P_{n}(r) has a Laurent series in r:

P_{n}(r) = ∑_{j=−n_{max}}^{n} p_{n_{j}}r^{j} .  (7.42)

Note that powers α of r on the level of g̃ (7.40) translate after integrating over r (7.38) into powers α of log z (clarify 6.46). Therefore, positive powers of r dominate the behavior for large z (strong coupling). Let us write down the terms of the expansion of g̃(D, r) up to the third order in 2 − D as one obtains them from the generated series contributions of the preceding subsection:

g̃(D, r) = r^{d/2} [\frac{1}{Γ(\frac{d}{2}+1)} + (2 − D)\frac{d}{2}\overline{ℂ} (\frac{1}{Γ(\frac{d}{2}+1)} − \frac{r^{−1}}{Γ(\frac{d}{2})})

−(2 − D)^{2} (\frac{d}{2} (1+\frac{d}{2}) \overline{ℂ_{c}^{2}} \frac{r}{Γ(\frac{d}{2}+2)} + \frac{\frac{d}{4} (\overline{ℂ}^{2} − 4\overline{ℂ_{c}^{2}}) − \frac{d^{2}}{8} (2\overline{ℂ_{c}^{2}} + \overline{ℂ}^{2})}{Γ(\frac{d}{2}+1)}

+ \frac{(\frac{d}{4} (−\overline{ℂ}^{2} + 3\overline{ℂ_{c}^{2}}) − \frac{d^{2}}{8} (2\overline{ℂ}^{2})) r^{−1}}{Γ(\frac{d}{2})} + \frac{(−\frac{d}{4} \overline{ℂ_{c}^{2}} + \frac{d^{2}}{8} \overline{ℂ}^{2}) r^{−2}}{Γ(\frac{d}{2}−1)} )

+(2 − D)^{3} (\frac{(\frac{d^{2}}{2} + \frac{d^{3}}{12}) (\overline{ℂ_{c}^{3}} − 4 \overline{ℂ_{c}^{△}}) r^{2}}{Γ(\frac{d}{2}+3)}

+ \frac{(d (\overline{ℂ_{c}^{3}} − 4 \overline{ℂ_{c}^{△}}) + \frac{d^{2}}{8} (−8 \overline{ℂ_{c}^{3}} + 32 \overline{ℂ_{c}^{△}} + 2 \overline{ℂ}\overline{ℂ_{c}^{2}}) + \frac{d^{3}}{48} (−4 \overline{ℂ_{c}^{3}} + 24 \overline{ℂ_{c}^{△}} − 6 \overline{ℂ}\overline{ℂ_{c}^{2}})) r}{Γ(\frac{d}{2}+2)}

+\frac{(\frac{d}{4} (−10 \overline{ℂ_{c}^{3}} + 36 \overline{ℂ_{c}^{△}}+6\overline{ℂ}\overline{ℂ_{c}^{2}}) +\frac{d^{2}}{8} (6 \overline{ℂ_{c}^{3}}−20 \overline{ℂ_{c}^{△}} + 2 \overline{ℂ}\overline{ℂ_{c}^{2}}−6 \overline{ℂ}^{3}) + \frac{d^{3}}{48} (−8 \overline{ℂ_{c}^{△}}+12 \overline{ℂ}\overline{ℂ_{c}^{2}}+\overline{ℂ}^{3}))}{Γ(\frac{d}{2}+1)}

+\frac{(\frac{d}{4} (9\overline{ℂ_{c}^{3}} − 32\overline{ℂ_{c}^{△}} − 12\overline{ℂ}\overline{ℂ_{c}^{2}} + \overline{ℂ}^{3}) +\frac{d^{2}}{8} (−2\overline{ℂ_{c}^{3}} + 4\overline{ℂ_{c}^{△}} − 7\overline{ℂ}\overline{ℂ_{c}^{2}} + 2\overline{ℂ}^{3}) − \frac{d^{3}}{16} (\overline{ℂ}^{3} + 2\overline{ℂ}\overline{ℂ_{c}^{2}})) r^{−1}}{Γ(\frac{d}{2})}

+\frac{(\frac{d}{4} (3\overline{ℂ_{c}^{3}} − 17\overline{ℂ_{c}^{△}} − 9\overline{ℂ}\overline{ℂ_{c}^{2}} + \overline{ℂ}^{3}) −\frac{d^{2}}{8} (\overline{ℂ}^{3} − 4\overline{ℂ}\overline{ℂ_{c}^{2}}) + \frac{d^{3}}{16} \overline{ℂ}^{3}) r^{−2}}{Γ(\frac{d}{2}−1)}

+ \frac{(−\frac{3d}{4} (2\overline{ℂ_{c}^{△}} + \overline{ℂ}\overline{ℂ_{c}^{2}}) −\frac{d^{2}}{8} \overline{ℂ}\overline{ℂ_{c}^{2}} − \frac{d^{3}}{48} \overline{ℂ}^{3}) r^{−3}}{Γ(\frac{d}{2}−2)} + \frac{\frac{d}{4} \overline{ℂ_{c}^{△}} r^{−4}}{Γ(\frac{d}{2}−3)} )]

(7.43)

Clearly, one discovers the form (7.40). Let us also state g̃(D, r) up to the fourth order, but taking to each order only the dominant contribution into account:

g̃(D, r) = r^{d/2} [\frac{1}{Γ(\frac{d}{2}+1)} + (2 − D)\frac{(\frac{d}{2})}{Γ(\frac{d}{2}+1)}\overline{ℂ} − (2 − D)^{2} \frac{\frac{d}{2} (1 + \frac{d}{2})}{Γ(\frac{d}{2}+2)} \overline{ℂ_{c}^{2}} r

+ (2 − D)^{3} \frac{(\frac{d^{2}}{2} + \frac{d^{3}}{12}) (\overline{ℂ_{c}^{3}} − 4\overline{ℂ_{c}^{△}})}{Γ(\frac{d}{2}+3)}r^{2}

− (2 − D)^{4} \frac{(d + \frac{3}{4}d^{2} + \frac{d^{3}}{4} + \frac{d^{4}}{48}) (222\overline{ℂ}^{4} + 6\overline{ℂ}^{2}\overline{ℂ_{c}^{2}} + 3\overline{ℂ_{c}^{2}}^{2} − \overline{ℂ_{c}^{4}} + 24\overline{ℂ_{c}} − 36\overline{ℂ_{c}^{⋄}})}{Γ(\frac{d}{2}+4)}r^{3} + ⋯]

(7.44)

For completeness let us state the result of the 2 − D-expansion of the renormalized coupling up to the fourth order. This time performing the integration in (7.38), such that we obtain g as a function of the bare coupling in terms of the functions f_{k}^{d}(z), whose asymptotic behavior has been analyzed in (6.46). For this purpose and for the sake of more compactness we need introduce a new notation: Since all series contributions are of the form as stated in (B.1), we introduce vectors 𝕄 such that

∑_{loops}(det 𝔇^{(0)})^{−d/2} \frac{\overline{∏_{i=1}^{l} (TrM^{n_{i}})^{m_{i}}}(−z)^{N}}{N!} ≡ ∑_{j=min}^{max}𝕄_{(\begin{matrix}m_{1} & m_{2} & ⋯ & m_{l} \\ n_{1} & n_{2} & ⋯ & n_{l} \end{matrix})}f_{1}^{d+2j}(z)
≡ 𝕄_{(\begin{matrix}m_{1} & m_{2} & ⋯ & m_{l} \\ n_{1} & n_{2} & ⋯ & n_{l} \end{matrix}|\begin{matrix}max \\ min \end{matrix} )}^{j}f_{1}^{d+2j}(z) ,  (7.45)

where max and min are some integers, and summation over the double index j is implicit. Inserting the results for the resummed series contributions into (6.14) we find for the renormalized coupling to the fourth order in 2 − D:

g(z) = f_{1}^{d+2}(z) − (2 − D) \frac{d}{2}𝕄_{( \begin{matrix}1 \\ 1 \end{matrix}| \begin{matrix}1 \\ 0 \end{matrix} )}^{j}f_{1}^{d+2j}(z) + (2 − D)^{2} (\frac{d}{4}𝕄j ; \begin{aligned}( & {1 \atop 2} & | \\ & & \begin{matrix}2 \\ −1 \end{matrix} & ) \end{aligned}f_{1}^{d+2j}(z) + \frac{d^{2}}{8} 𝕄j ; \begin{aligned}( & {2 \atop 1} & | \\ & & \begin{matrix}2 \\ −1 \end{matrix} & ) \end{aligned}f_{1}^{d+2j}(z))
− (2 − D)^{3} (\frac{d}{4}𝕄j ; \begin{aligned}( & {1 \atop 3} & | \\ & & \begin{matrix}2 \\ −3 \end{matrix} & ) \end{aligned}f_{1}^{d+2j}(z) + \frac{d^{2}}{8} 𝕄j ; \begin{aligned}( & 1 1 1 2 & | \\ & & \begin{matrix}3 \\ −2 \end{matrix} & ) \end{aligned}f_{1}^{d+2j}(z) + \frac{d^{3}}{48}𝕄j ; \begin{aligned}( & {3 \atop 1} & | \\ & & \begin{matrix}3 \\ −2 \end{matrix} & ) \end{aligned}f_{1}^{d+2j}(z))
+ (2 − D)^{4} (\frac{d}{8}𝕄j ; \begin{aligned}( & {4 \atop 1} & | \\ & & \begin{matrix}4 \\ −3 \end{matrix} & ) \end{aligned}f_{1}^{d+2j}(z) + \frac{d^{2}}{8} (\frac{1}{4}𝕄j ; \begin{aligned}( & {2 \atop 2} & | \\ & & \begin{matrix}4 \\ −3 \end{matrix} & ) \end{aligned}f_{1}^{d+2j}(z) + \frac{2}{3}𝕄j ; \begin{aligned}( & 1 1 1 3 & | \\ & & \begin{matrix}4 \\ −3 \end{matrix} & ) \end{aligned}f_{1}^{d+2j}(z))
+\frac{d^{3}}{32}𝕄j ; \begin{aligned}( & 2 1 1 2 & | \\ & & \begin{matrix}4 \\ −3 \end{matrix} & ) \end{aligned}f_{1}^{d+2j}(z) + \frac{d^{4}}{384}𝕄j ; \begin{aligned}( & {4 \atop 1} & | \\ & & \begin{matrix}4 \\ −3 \end{matrix} & ) \end{aligned}f_{1}^{d+2j}(z)) + O(2 − D)^{5}  (7.46)

The vector entries 𝕄^{j} are to be be taken from subsection 7.1.

7.3 Discussion of possible forms of g̃(D, r)

Before making a guess for the exact g̃(r) and thus implicitly for g(z), let us summarize the properties, to which g(z) as a function of the internal dimension D has to obey.

(I) In the limit D = 2 g(z) has to tend to the exact form as it is given by (6.38). Note, that in this limit g(z) diverges logarithmically. There is no strong coupling expansion in D = 2.

(II) A phantom polymer closed to form a ring corresponds to the limit D = 1 of the toroidal manifold. This case has also been solved exactly. There are both weak- and strong coupling expansions available (D.23,D.34). In particular, g(z) tends to a finite fixed point value as z tends to ∞. In view of the logarithmic divergence in D = 2 and the renormalizability of the theory for internal dimensions 0 < D < 2 [47,48], we expect finiteness of lim_{z→∞} = g^{∗} as soon as 2 − D becomes positive. Then, we expect the existence of a strong coupling expansion starting with the fixed point value and a term, which encodes the correction to scaling behavior of the theory:

g(D, z) = g^{∗}(D) + C_{1}(D, ln z) z^{−ω_{1}/ε} + C_{2}(D, ln z) z^{−ω_{2}/ε} + ⋯,  (7.47)

ω_{1}, ω_{2}, ω_{2}−ω_{1} > 0 a.s.f.., and the scaling functions C_{i} diverge at most subexponentially for large arguments. Powerlaw behavior in the strong coupling limit necessitates the resummation of the series expansion (7.43) to turn into some exponentially growing function g̃(D, r). This can be seen as follows:

f_{1+ \frac{ω}{ε}}^{2} (z) = z^{1+ \frac{ω}{ε}} ∫_{0}^{∞} dr exp[−z e^{−r} − (1+ω/ε)r] = Γ(1+\frac{ω}{ε}) + O(e^{−z})
⇒ z^{−ω/ε} = \frac{z}{Γ(1+\frac{ω}{ε})} ∫_{0}^{∞} dr exp[−z e^{−r} − (1+ω/ε)r] + O(e^{−z})
= \frac{z}{Γ(1+\frac{ω}{ε})} ∑_{n=0}^{∞} ∫_{0}^{∞} dr \frac{(−ω/ε r)^{n}}{n!} exp[−z e^{−r} − r] + O(e^{−z})

= \frac{1}{Γ(1+\frac{ω}{ε})} ∑_{n=0}^{∞} \frac{(−ω/ε)^{n}}{n!} f_{1}^{2(n+1)}(z) + O(e^{−z}) ,  (7.48)

where it is to be understood that ω is expanded in powers of 2 − D. Let us now postulate a possible form of g̃(D, r) obeying to the demanded properties (I) and (II):

g̃(D, r) = C (\frac{1 − e^{− \frac{ω}{ε} rS(2−D,r)}}{ω/ε} )^{d/2} ,  (7.49)

where S is analytic in D = 2 such that

S(2−D, r) = ∑_{n=0}^{∞} S_{n}(r)(2 − D)^{n}  (7.50)

and to each order in 2 − D, there is a Laurent series of the S_{n}(r) of the form

S_{n}(r) = ∑_{j=−∞}^{0} s_{n_{j}} r^{j} .  (7.51)

Furthermore, ω/ε has some expansion in powers of 2 − D:

ω/ε = ω_{n}(2 − D)^{n} + O(2 − D)^{n+1} .  (7.52)

Let us once again stress that powers in r at the level of g̃(r) translate into powers in log z, while exponential growth in r is providing powerlaw behavior for large z. Therefore, (7.49) obeys to the demanded properties (I) and (II). Note in particular, that setting S = 1 and integrating over r according to (7.37) the expression in parentheses resembles to

log z = lim_{D→2} \frac{1 − z^{−(2−D)}}{2 − D} .  (7.53)

The r.h.s. expression above grows logarithmically in the limit D = 2 and shows up a fixed point together with a strong coupling expansion as soon as 2 − D > 0. Let us expand (7.34) in powers of 2 − D taking only the leading term of ω/ε into account:

g̃(D, r) = C (r S(2−D, r) − \frac{ω_{n}(2−D)^{n}r^{2}}{2!} S^{2}(2−D, r) + ⋯)^{d/2}  (7.54)

Inserting the expansion of S(2−D, r),

S(2−D, r) = S_{0}(r) + (2 − D)S_{1}(r) + O(2 − D)^{2} ,  (7.55)

and expanding everything provides:

g̃(D, r) = C r^{d/2} S_{0}(r) ×
× (1 + (2−D)\frac{d}{2} \frac{S_{1}(r)}{S_{0}(r)} + (2−D)^{2} \frac{d}{2} \frac{S_{2}(r)}{S_{0}(r)} − \frac{ω_{n}(2−D)^{n}d}{4}r

+\frac{(2−D)^{2}}{2!} \frac{d}{2} (\frac{d}{2}−1) (\frac{S_{1}(r)}{S_{0}(r)})^{2} + O(2 − D)^{3})  (7.56)

Matching the coefficients of the expansion above with the result in (7.43) we set:

C = \frac{1}{Γ(\frac{d}{2}+1)}
S_{0}(r) = 1
S_{1}(r) = \overline{ℂ} (1 − \frac{d}{2} \frac{1}{r} + O(1/r^{2}))
ω_{n} ≡ ω_{2} = 2 \overline{ℂ_{c}^{2}} ,  (7.57)

where we have skipped all higher order poles in the Laurent series of S_{1}(r) From the above considerations it follows that the correction to scaling exponent ω reads to lowest order in 2 − D:

ω = ε 2\overline{ℂ_{c}^{2}} (2 − D)^{2} + O(2 − D)^{3} .  (7.58)

The result coincides with what we have found in the preceding section from the calculation within the scheme in the renormalized coupling. Let us point out, however, that the scheme above is not unambiguous. Going to the next order it will turn out that the sum in (7.51) actually has to be extended to j = 1 in order to provide consistency with the expansion (7.43). But then, it is not unambiguous, whether the term proportional to r^{d/2+1} has to be attributed to ω_{2} or rather to S_{2}(r). From some simple general considerations one can easily see that it is in principle impossible to extract any information about the correction to scaling exponent from an expansion like (7.43) as long as one has no further information about the behavior of scaling functions like S(r) available. The only thing that we can conclude for sure is that the correction to scaling exponent in an ensemble of a manifold of toroidal topology and internal dimension close to D = 2, which interacts with a single short-range defect, is proportional to some power of 2 − D and that this power is larger than 1:

ω ∼ (2 − D)^{n} ,  n ≥ 2 .  (7.59)

Actually, higher than linear powers of r in the exponent (7.49) are fatal, because they destroy the scaling behavior for z → ∞, which was intended to be encoded in the exponent ω/ε. Therefore, let us discuss another ansatz for g̃(D, r) that does not show higher than linear powers of r in the exponent as in (7.49):

g̃(D, r) = C (\frac{1 − S(2−D, r) e^{− \frac{ω}{ε} r}}{ω/ε} )^{d/2} ,  (7.60)

where S can be written as

S(2−D, r) = 1 + \frac{ω}{ε} r S̃(2−D, r)  (7.61)

S̃ being analytic in D = 2 such that

S̃(2−D, r) = ∑_{n=1}^{∞} S̃_{n}(r)(2 − D)^{n}  (7.62)

and to each order in 2 − D, there is a Laurent series of the S̃_{n}(r) of the form

S̃_{n}(r) = ∑_{j=−n_{min}}^{n_{max}} s̃_{n_{j}} r^{j} .  (7.63)

In contrast to (7.51) there is no restriction for positive powers in r, but we have to set

S_{0}(r) ≡ 1  (7.64)

from the beginning in order to obey to the properties (I) and (II). Furthermore, ω/ε has some expansion in powers of 2 − D:

ω/ε = ω_{n}(2 − D)^{n} + O(2 − D)^{n+1} .  (7.65)

Expanding first the expression in parenthesis of (7.60) taking only the lowest order of ω/ε into account leads to

\frac{1 − S(2−D, r) e^{− \frac{ω}{ε} r}}{ω/ε} = −r (S̃_{1}(r)(2 − D) + S̃_{2}(r)(2 − D)^{2} + S̃_{3}(r)(2 − D)^{3})
+ r (1 + \frac{ω}{ε} r S̃_{1}(r)(2 − D)) − \frac{ω}{ε} \frac{r^{2}}{2!} + ⋯,  (7.66)

where we have expanded S up to the third order in 2 − D. ω is at least of second order in 2 − D. Skipping all higher than third order terms and expanding everything in (7.54), we obtain:

g̃(D, r) = C r^{d/2} (1 − S̃_{1}(r)(2 − D) − (\frac{ω_{2}}{2!} r + S̃_{2}(r)) (2 − D)^{2}
− (−\frac{ω_{2}}{2!} r S̃_{1}(r) + \frac{ω_{3}}{2!} r + S̃_{3}(r)) (2 − D)^{3})^{d/2}
= C r^{d/2} (1 − \frac{d}{2} (S̃_{1}(r)(2 − D) + (\frac{ω_{2}}{2!} r + S̃_{2}(r)) (2 − D)^{2}

+ (−\frac{ω_{2}}{2!} r S̃_{1}(r) + \frac{ω_{3}}{2!} r + S̃_{3}(r)) (2 − D)^{3})

+\frac{d}{4} (\frac{d}{2}−1) (S̃_{1}^{2}(r)(2 − D)^{2} + 2S̃_{1}(r) (S̃_{2}(r) + \frac{ω_{2}}{2!} ) (2 − D)^{3})

+ \frac{d}{12} (\frac{d}{2}−1) (\frac{d}{2}−2) (S̃_{1}^{3}(r)(2 − D)^{3}))  (7.67)

This time one may set

C = \frac{1}{Γ(\frac{d}{2}+1)}
S_{0}(r) = 1
S̃_{1}(r) = −\overline{ℂ} (1 − \frac{d}{2} \frac{1}{r} + O(1/r^{2}))
ω_{2} = 2 \overline{ℂ_{c}^{2}} ,  (7.68)

in order to match with the expansion in (7.43). Again, we recover the same result for ω as from the previous ansatz and the calculation within the renormalized coupling:

ω = ε 2\overline{ℂ_{c}^{2}} (2 − D)^{2} + O(2 − D)^{3} .  (7.69)

However, there still remains the ambiguity in attributing the second order terms either to ω_{2} or to S̃_{2} as it is obvious from (7.67). Nevertheless, making any choice for ω_{2} it is clear, that the third order term S̃_{3}(r) has sufficiently many parameters free to get the third order reconciliated with the second order a.s.f.. Of course, the behavior of S(r) for large values of r is supposed to stay subexponential. This corresponds to the same demand on the scaling functions C_{i} in (7.47). There is some additional information about the scaling function S(r) necessary, in order to enable an unambiguous analysis of the scaling behavior for dimensions D close to 2 . From the structure of the expansion (7.46), however, it is not possible to make a guess about the precise properties of S(r). A way out might be a complete solution of the problem in D = 1 within the 2 − D expansion.

8 Discussion and conclusion

In this work, we have discussed the perturbation expansion of a D-dimensional elastic manifold interacting via a δ-interaction with a fixed point in embedding space. This calculation can be done exactly for D = 1, but becomes non-trivial for D ≠ 1. Interestingly and quite surprisingly, it simplifies considerably in the limit of D → 2, if one decides to work at finite ε. In that limit we were able to obtain an explicit expression for the renormalized coupling as function of the bare one, in terms of a non-trivial series. Analysis of this series shows that in the limit of large values of the bare coupling z, corresponding to strong repulsion or equivalently to large membrane sizes, the renormalized coupling g diverges logarithmically as a function of z. This yields a vanishing correction-to-scaling exponent ω in the limit D = 2. Furthermore, dependent on the dimensionality d of the embedding space the β-function shows up a zero at infinite bare coupling as 0 ≤ d < 2. Then, the renormalization flow tends to a fixed point, and the theory becomes scale invariant in this limit. Due to the logarithmic divergence of the renormalized coupling, however, the corresponding zero of the β-function in terms of the latter is, too, shifted to infinity. This is quite a remarkable result, because it reveals that the strong coupling behavior of a ”phantom”, polymerized membrane in a crumpled phase interacting with a δ-like defect in embedding space is only accessible through a non-perturbative treatment and always largely deviates qualitatively from any finite loop expansion, be it within a minimal subtraction scheme or at finite ε. Also, it is important to note that this result is completely independent of the regularization procedure. This does no longer hold true beyond the leading order, which should be accessible to an expansion in 2 − D. We constructed its first order in a specific regularization scheme in [49]. While this reproduces qualitatively correctly the known result in D = 1 (and even exactly for d = 0), it suffers from its inconsistent renormalization scheme, which neglects the boundaries of a finite manifold. We used a hard cutoff in position space, while working with the infinite D-space correlator. It seems that only in an ε-expansion this procedure is systematic. Now, we have overcome this problem by constructing the 2 − D-expansion on a manifold of toroidal shape of finite size, thus imposing periodic boundary conditions on the field. There is no further infrared cutoff necessary. We have carried out the expansion of the renormalized coupling up to the fourth order in 2 − D. This reveals part of the general structure of the expansion. It is important to point out that in considering g as a function of the bare coupling the limits D → 2 and strong coupling (z → ∞) can not be interchanged. While g tends to infinity as z does in D = 2, we expect finiteness of this limit as soon as 2 − D > 0 and the existence of a strong coupling expansion as found for polymers (D = 1). We were able to guess an exact g(D, z) as a function of z and the internal dimension D, which satisfies these properties and which can be reconciliated with the available expansion in 2 − D by an appropriate matching of its free parameters. However, this matching is not unambiguous and therefore does not provide a decent determination of the prefactor in the relation

ω ∼ (2 − D)^{2} .  (8.1)

The correction-to-scaling exponent ω is closely related to observables, which are accessible to Monte-Carlo simulations. This is for instance the monomer density at the potential of a membrane, which is pinned by one of its points close to the defect.

While results for the pinning problem are interesting in its own, the main motivation is certainly to obtain a better understanding of self-avoiding polymerized membranes. This model is well-behaved physically, since its fractal dimension d_{f} is bounded by the dimension of the embedding space d_{f} < d. Preliminary studies [66] indicate that this problem can also be attacked by the methods developed in this work. This would be very welcome to check the 2-loop calculations [20,19] and the large-order behavior [67] on one side and numerical results (e.g. [34]) on the other.

A Derivation of the kernel

Starting from the free theory with Hamiltonian

H_{0} = \frac{1}{2 − D} ∫_{x} \frac{1}{2}(∇r(x))^{2}  (A.1)

yields the difference correlator

C^{kl}(x_{1}−x_{2}) := ⟨\frac{1}{2}[r_{k}(x_{1})−r_{l}(x_{2})]^{2}⟩_{0} = δ^{kl}C(x_{1}−x_{2})  (A.2)

with

C(x_{1}−x_{2}) = |x_{1}−x_{2}|^{2−D} .  (A.3)

It can be shown as in [4] to be a solution in the sense of distributions of the Laplacian equation

ΔC(x) = (2 − D)S_{D}δ^{D}(x) .  (A.4)

The difference correlator may be considered as the analytical continuation to values of D < 2 of the usual two-point correlator. Let us shortly derive (A.3) starting from the well known form of the kernel in Fourier space

G(k⃗_{1}, k⃗_{2}) = \frac{δ^{D}(k⃗_{1}+k⃗_{2})}{(k⃗_{1})^{2}} ,  (A.5)

such that

G(x) = \frac{1}{(2π)^{D}} ∫ d^{D}k \frac{exp[ik⃗x⃗]}{k^{2}}  (A.6)

Introducing a Schwinger parametrization and splitting the integration over the momenta into a part integrating over the component, which is parallel to x⃗, and its complementary part leads to

G(x) = \frac{1}{(2π)^{D}} ∫_{0}^{∞}ds ∫_{−∞}^{∞} dk_{∥} ∫ d^{D−1}k_{⊥} e^{ik_{∥}x−s(k_{∥}^{2}+k_{⊥}^{2})}

= \frac{1}{(2π)^{D}} ∫_{0}^{∞}ds ∫_{−∞}^{∞} dk_{∥} e^{−sk_{∥}^{2}+ik_{∥}x} (∫_{−∞}^{∞} dk_{⊥} e^{−sk_{⊥}^{2}} )^{D−1}

= \frac{1}{(2π)^{D}} ∫_{0}^{∞}ds (\frac{π}{s} )^{D/2} e^{− \frac{x^{2}}{4s}} = \frac{1}{(2\sqrt{π})^{D}} ∫_{0}^{∞}ds′ s^{′D/2−2}e^{−s′ \frac{x^{2}}{4}}

= \frac{1}{(2\sqrt{π})^{D}} Γ (\frac{D}{2} − 1) (\frac{x^{2}}{4} )^{−(D/2−1)} = \frac{1}{S_{D}(2 − D)}|x|^{2−D}  (A.7)

≡ \overline{ℂ^{2}(x_{i}−x_{j})ℂ(x_{i}−x_{k})ℂ(x_{j}−x_{k})ℂ(x_{l}−x_{k})}

Figure B.1: Example of a fifth order diagram in the 2 − D-expansion.

B Mathematica^{◯ R}–programs for the generation of the series contributions in the 2 − D-expansion

All terms to appear in the expansion in 2 − D are of the form

∏_{i=1}^{k}(TrM^{m_{i}})^{n_{i}} , m_{i}, n_{i}, k ∈ N  (B.1)

as follows from (6.14). To N-loop order, M is an N × N-matrix as defined in (6.18). The diagrams to be evaluated at nth order in 2−D are positional averages of an ensemble of internal points on the manifold, which are connected through n correlators as shown in the example of figure (B.1). The numerical value of the diagrams is determined through the topology of the graph, whose knots are given by the internal points and whose links correspond to the correlators joining them. It furthermore depends on the number of knots making up closed loops in the graph, but it does not depend on the loop order. Then, the N-loop contribution at nth order in 2 − D depends on the numerical value of the diagrams each coming up with a combinatorial factor involving the number of ways to form the corresponding graph from N+1 independent internal points. The combinatorial factor is a polynomial in N. Together with some power of the factor (N + 1)^{−1} from (6.17) the general form of the term (B.1) can be written as

∏_{i=1}^{k}(TrM^{m_{i}})^{n_{i}} = \frac{P (N)}{(N + 1)^{n_{max}}} , n_{max} ≤ ∑_{i=1}^{k} m_{i}n_{i} ,  (B.2)

and P (N) is some polynomial in N. The degree of the polynomial is less than 2n, if n denotes the order in 2 − D. In order to write down the term (B.1) as a function of N we need know P (N). Since its degree is bounded through 2n, one can determine its coefficients through a fit-procedure to be applied on the values of (B.1) at the first 2n+1 loop orders. This is done by the Mathematica^{◯ R}– modules, which are presented on the following pages. The module ”Dreset” initializes the matrices 𝔇^{(0)−1} and 𝔇_{ik} (6.18) to obtain M. The ”For” loop, which is individual for each term (B.1) to be analyzed, recognizes the diagrams to appear and performs the evaluation to N = 1, …, N_{max} ≤ 2n + 1 loops. Of course, the ”values” are expressions involving finitely many diagrams. The module ”AnalyseListfkd” fits the N_{max} points through a polynomial being expanded in terms of the base given in (6.27). Summing up all loop orders each term of the form (B.1) translates into some expression, which can be written in terms of the functions f_{k}^{d}(z) as introduced in (6.39). This procedure is done by the module ”Recognizefkd”. Making several times use of the relation (6.45) all f_{k}^{d′}s can be transformed into some f_{1}^{d′}s making the leading behavior in the strong coupling limit obvious, which is then completely encoded in the upper index. This is done by ”Reducefkd”. Finally, it might be convenient to express the diagrams in terms of positional averages of internal points, which are rather joint by the connected correlator as introduced in (7.6). The transcription of all expressions is performed by ”Trans2Conn”. The application of the discussed modules on the second order terms in the 2 − D-expansion is also included in the sequel.

Interacting Crumpled Manifolds 75 76 Interacting Crumpled Manifolds Interacting Crumpled Manifolds 77

C Calculation of the diagrams in the 2 − D-expansion

In this section we will calculate the diagrams to appear in the 2 − D expansion on the torus. It turns out that to obtain \overline{ℂ} and \overline{ℂ_{c}^{2}} we need evaluate two sums over discrete wavevectors obeying to periodic boundary conditions on the torus. Let us first derive the latter before turning to the explicit evaluation. Starting from the definition of the difference correlator C(x),

C(x) := G(x) − G(0) ,  (C.1)

where G(x) is the usual two–point correlator, we obtain C(x) through an inverse discrete Fourier–transformation from G(k) = 1/k⃗^{2}, which reads:

C(x) = \frac{1}{L^{D}} ∑_{k⃗≠0} \frac{1}{k⃗^{2}} (1 − e^{ik⃗x⃗}) ,  k⃗ = \frac{2π}{L} n⃗ , n⃗ ∈ Z × Z\\{\{0\} × N\} .  (C.2)

Performing the averaging procedure

\overline{C(x)} = ∫_{x} C(x) ,  (C.3)

where 1/L^{D} ∫_{x} e^{ik⃗x⃗} = δD ; k⃗ is to be taken into account, the calculation of \overline{C(x)} reduces to

\overline{C(x)} = I_{1} := ∑_{k≠0} \frac{1}{k⃗^{2}} ,  k⃗ = \frac{2π}{L} n⃗ ,  (C.4)

where k⃗ is D-dimensional, and the indices n_{i} are running from −∞ to ∞, n_{i} = 0 being excluded from the summation. Of course, in the expansion in powers of 2 − D we need an analytic continuation to real values of D. Finally, to obtain \overline{ℂ(x)} we have to subtract \overline{ℂ^{(0)}(x)} from \overline{C(x)}. Due to our normalizations:

\overline{ℂ(x)} = S_{D} (\overline{C(x)} − \frac{\overline{ℂ^{(0)}(x)}}{2π(2 − D)}) ,  (C.5)

where S_{D} denotes the volume of the unit sphere and \overline{ℂ^{(0)}(x)} = 1. Turning to \overline{ℂ_{c}^{2}(x)}, we first note that within our normalizations we have

S_{D}^{−2}\overline{ℂ^{2}(x)} = \overline{(C(x) − ℂ^{(0)}(x)/(2π(2−D)))^{2}} = \overline{C(x)^{2}} − 2\frac{\overline{C(x)}}{2π(2−D)} + \frac{1}{(2π(2−D))^{2}} (C.6)

and

S_{D}^{−2}\overline{ℂ(x)}^{2} = \overline{C(x)}^{2} − 2\frac{\overline{C(x)}}{2π(2−D)} + \frac{1}{(2π(2−D))^{2}}  (C.7)

according to (C.5), such that

\overline{ℂ_{c}^{2}(x)} ≡ \overline{ℂ^{2}(x)} − \overline{ℂ(x)}^{2} = S_{D}^{2} (\overline{C^{2}(x)} − \overline{C(x)}^{2}) .  (C.8)

Knowing already the sum to be evaluated to obtain \overline{C}, (C.4), what is left to obtain is:

\overline{C^{2}(x)} = ∫_{y} ∫_{x} C^{2}(x) = \frac{1}{L^{2D}} ∫_{y} ∫_{x} ∑_{k_{i}≠0} ∑_{p_{j}≠0} \frac{1}{k⃗^{2}} \frac{1}{p^{2}} (e^{ik⃗x⃗} − 1)(e^{ip⃗x⃗} − 1)

= ∑_{k_{i}≠0} ∑_{p_{j}≠0} \frac{1}{k⃗^{2}} \frac{1}{p^{2}} (δ_{k⃗+p⃗}^{D} − δD ; k⃗ − δD ; p⃗ + 1) = ∑_{k_{1}≠0} \frac{1}{k⃗^{4}} + [∑_{k_{1}≠0} \frac{1}{k⃗^{2}} ]^{2} . (C.9)

Therefore,

S_{D}^{−2}\overline{ℂ_{c}^{2}(x)} = I_{2} := ∑_{k≠0} \frac{1}{k⃗^{4}} ,  k_{i} = \frac{2π}{L} n_{i} .  (C.10)

Let us first calculate I_{1}: Setting L = 1, renaming the indices n_{i} into k_{i} and introducing a Schwinger parametrization we have:

I_{1} = \frac{1}{(2π)^{2}} ∑_{{k_{i}=−∞ \atop k_{i}≠0}}^{∞} \frac{1}{k⃗^{2}} = \frac{1}{(2π)^{2}} = ∑_{{k_{i}=−∞ \atop k_{i}≠0}}^{∞} ∫_{0}^{∞}ds e^{−sk⃗^{2}}

= \frac{1}{(2π)^{2}} ∫_{0}^{∞}ds ( ∑_{{k=−∞ \atop k≠0}}^{∞} e^{−sk^{2}} )^{D} = \frac{1}{(2π)^{2}} ∫_{0}^{∞} \frac{ds}{s^{2}} ( ∑_{{k=−∞ \atop k≠0}}^{∞} e^{−k^{2}/s})^{D} ,  (C.11)

where it is to be noted that the sum in the last line is only one-dimensional. Furthermore, from now on it is clear, how I_{1} is analytically continued to real values of D. In order to evaluate this sum, we will make use of a Poisson-transformation, which reads:

∑_{n=−∞}^{∞} e^{−A(n−z/2)^{2}} = \sqrt{\frac{π}{A}} ∑_{l=−∞}^{∞} e^{− \frac{π^{2}l^{2}}{A} +iπlz} .  (C.12)

The contribution from l = 0 is the approximation of the l.h.s. through a gaussian integral. Our aim is to calculate the coefficients of the 2 − D expansion of I_{1} numerically using some algebraic manipulation program. Then, the integration interval in (C.11) has to be made finite. This is done as follows: For any s_{0} > 0 we have

I_{1} = \frac{1}{(2π)^{2}} ∫_{0}^{s_{0}} \frac{ds}{s^{2}} ( ∑_{{k=−∞ \atop k≠0}}^{∞} e^{−k^{2}/s})^{D} + \frac{1}{(2π)^{2}} ∫_{s_{0}}^{∞} \frac{ds}{s^{2}} ( ∑_{{k=−∞ \atop k≠0}}^{∞} e^{−k^{2}/s})^{D} .  (C.13)

For any finite s_{0} > 0, the sum in the first integral can be truncated at some finite k_{max} for all s ∈ [0, s_{0}]. For the second integral (corresponding to large values of s) we make use of the poissonian formula (C.12) with z = 0 and choose s_{0} in a way that terms l ≠ 0 can be neglected:

( ∑_{{k=−∞ \atop k≠0}}^{∞} e^{−k^{2}/s})^{D} = (\sqrt{πs} − 1 + 2 ∑_{l=1}^{∞} e^{−π^{2}l^{2}s})^{D} ≈ (\sqrt{πs})^{D}(1 − (\sqrt{πs})^{−1})^{D} . (C.14)

Then, the second integral can be evaluated approximately through

\frac{1}{(2π)^{2}} ∫_{s_{0}}^{∞} \frac{ds}{s^{2}} ( ∑_{{k=−∞ \atop k≠0}}^{∞} e^{−k^{2}/s})^{D} ≈ \frac{1}{(2π)^{2}} ∑_{n=0}^{∞} \frac{Γ(n − D)π^{(D−n)/2}}{Γ(n + 1)Γ(−D)} ∫_{s_{0}}^{∞}ds s^{\frac{D−n−4}{2}}

= \frac{1}{(2π)^{2}} ∑_{n=1}^{∞} \frac{2Γ(n − D)π^{(D−n)/2}}{Γ(n + 1)Γ(−D)(2 − D + n)}s0^{\frac{2−D+n}{2}} + 2\frac{s0^{\frac{2−D}{2}}}{2π^{(2−D/2)}(2 − D)} . (C.15)

From (C.13) and (C.15) we then have:

I_{1} ≈ \frac{1}{(2π)^{2}} [∫_{0}^{1} \frac{ds s_{0}}{s^{2}} (2 ∑_{k=1}^{∞} e^{−k^{2} \frac{s_{0}}{s}} )^{D} + ∑_{n=1}^{∞} \frac{2Γ(n − D)π^{(D−n)/2}}{Γ(n + 1)Γ(−D)(2 − D + n)}s0^{\frac{2−D+n}{2}} + \frac{2π^{D/2}}{2 − D}s0^{\frac{2−D}{2}} ] .

(C.16)

The third term above shows up a pole in 2 − D, which can be easily subtracted expanding the expression in powers of 2 − D. The pole is

I_{1} = \frac{1}{2π(2 − D)} + O((2 − D)^{0}).  (C.17)

The precision of the machine that we used to evaluate (C.16) was sufficient in a way that we could select s_{0} from an interval, such that the sum appearing in the integrand could be truncated at some finite k_{max} and the result was independent from the precise value of s_{0} within the desired order of accuracy, therefore, justifying the approximation in (C.15). Of course, the other series can be safely truncated as well. Setting for instance s_{0} = 10 and k_{max} = n_{max} = 20 we obtain with Mathematica^{◯ R}:

I_{1} = \frac{1}{2π(2 − D)} − 0.2382(1) + 0.3422(1)(2 − D) + O((2 − D)^{2}) .  (C.18)

On the torus we scaled the square root of the volume of the D-dimensional unitsphere into the field. Accordingly, comparing with (C.4) and (C.5) we then find:

\overline{ℂ} = −1.4967(6) + S_{D} · 0.3422(1)(2 − D) + O((2 − D)^{2}) .  (C.19)

Let us turn to the evaluation of I_{2} following the same strategy as above. Again, setting L = 1 and introducing a Schwinger parametrization leads to:

I_{2} = \frac{1}{(2π)^{4}} ∫_{0}^{∞}ds s^{−3} ( ∑_{{k=−∞ \atop k≠0}}^{∞} e^{−k^{2}/s})^{D}

= \frac{1}{(2π)^{4}} [∫_{0}^{1} \frac{ds}{s_{0}^{2}s^{3}} (2 ∑_{k=1}^{∞} e^{−k^{2}/(s_{0}s)})^{D} + ∫_{s_{0}}^{∞}ds s^{−3} ( ∑_{{k=−∞ \atop k≠0}}^{∞} e^{−k^{2}/s})^{D}]  (C.20)

Once again applying the Poisson-transformation (C.12) with z = 0 on the second integral involving large values of s, we approximate

∫_{s_{0}}^{∞}ds s^{−3} ( ∑_{{k=−∞ \atop k≠0}}^{∞} e^{−k^{2}/s})^{D} ≈ π^{D/2} ∫_{s_{0}}^{∞}ds s^{D/2−3} (1 − (πs)^{(−1/2)})

∑_{n=0}^{∞} \frac{Γ(n − D)π^{(D−n)/2}}{Γ(n + 1)Γ(−D)} ∫_{s_{0}}^{∞}ds s^{\frac{D−n}{2} −3} = ∑_{n=0}^{∞} \frac{2Γ(n − D)π^{(D−n)/2}}{Γ(n + 1)Γ(−D)(4 + n − D)}s0^{\frac{D−n−4}{2}} ,

(C.21)

such that we arrive at

I_{2} ≈ \frac{1}{(2π)^{4}} [∫_{0}^{1} \frac{ds}{s_{0}^{2}s^{3}} (2 ∑_{k=1}^{∞} e^{−k^{2}/(s_{0}s)})^{D} + ∑_{n=0}^{∞} \frac{2Γ(n − D)π^{(D−n)/2}}{Γ(n + 1)Γ(−D)(4 + n − D)}s0^{\frac{D−n−4}{2}} ] .

(C.22)

There is no pole in 2 − D. Since I_{2} appears at second order in 2 − D we only need its value at D = 2. Again, it is safe to truncate both series appearing in (C.22) at some finite values k_{max} and n_{max}, and s_{0} has to be chosen from an appropriate interval. Setting k_{max} = n_{max} = 20 and s_{0} = 10 we obtain with Mathematica^{◯ R}:

I_{2} = 0.00109(1) + O((2 − D)) ,  (C.23)

or– due to the rescaling of the field by S_{D}^{1/2}–

\overline{ℂ_{c}^{2}} = 0.043(1) + O((2 − D)) .  (C.24)

The implementation of the summations with Mathematica^{◯ R}is shown in the sequel.

D Exact results for polymers

In this section we would like to derive some results for gaussian polymer chains interacting with a δ-like defect. The path integrals then turn out to be factorizable and thus exactly solvable. Let us start from the (bulk) polymer propagator being defined as the conditional probability of finding the internal point s′ at the position r_{f} when starting in s at r_{i}. The propagator can be written as a functional integral for a restricted partition function of the free chain according to

Z_{s,s′}(r_{i}|r_{f}) = ∫ D[r] δ̃(r(s)−r_{i})δ̃(r(s′)−r_{f}) e^{−H[r]}
= ∫_{k} e^{−k^{2}|s′−s|+ik(r_{f} −r_{i})} = |s′−s|^{−d/2} e^{− \frac{(r_{f} −r_{i})^{2}}{4|s′−s|}} .  (D.1)

The propagator (D.1) possesses the Markov property

Z_{s,s′′}(r_{i}|r_{f}) = ∫ d^{d}r_{m} Z_{s,s′}(r_{i}|r_{m}) Z_{s′,s′′}(r_{m}|r_{f}) .  (D.2)

D.1 Universal 1/r - repulsion law for polymers

We have shown in section 2.2 by a scaling argument that as long as r ≪ L^{ν}, the restricted partition function of a manifold pinned at one of its internal points scales as Z_{∞}(r/L^{ν}) ∼ (r/L^{ν})^{θ}, with the contact-exponent given by (2.25). It is instructive to prove this for the special case of polymers in d = 1. According to (2.1) and (2.6) this corresponds to a polymer in 3dspace interacting with a δ-potential on a plane. For this purpose we need the restricted partition function (D.1) in presence of a short-range interaction modeled through a δ-potential, which is situated at the origin of the embedding space. The restricted partition function in the presence of the interaction will be denoted by Z_{s,s′}^{g_{0}}(r_{i}|r_{f}). We consider the case, where the chain is fixed at its ends, such that s=0 and s′=L. We furthermore switch to a grand-canonical ensemble and denote

Z_{τ}^{g_{0}}(r_{i}|r_{f}) := ∫_{0}^{∞} dL Z_{0,L}^{g_{0}}(r_{i}|r_{f}) e^{−τL} ,  (D.3)

τ being some chemical potential. Z_{τ}^{g_{0}}(r_{i}|r_{f}) can be expanded in a perturbation series in the coupling g_{0} according to

Z_{τ}^{g_{0}}(r_{i}|r_{f}) = ∑_{n=0}^{∞}(−g_{0})^{n}Z_{τ}^{g_{0},n}(r_{i}|r_{f}) .  (D.4)

The Z_{τ}^{g_{0},n}(r_{i}|r_{f}) are

Z_{τ}^{g_{0},n}(r_{i}|r_{f}) := ∫_{0}^{∞}dL ∫_{0<x_{1}<...<x_{n}<L} Z_{0,x_{1}}(r_{i}|0) Z_{x_{1},x_{2}}(0|0) × …
… × Z_{x_{n−1},x_{n}}(0|0) Z_{x_{n},L}(0|r_{f}) e^{−τL}
= [∫_{0}^{∞} dx e^{−xτ}Z_{0,x}(r_{i}, 0)][∫_{0}^{∞} dx e^{−xτ}Z_{0,x}(0, 0)]^{n−1}[∫_{0}^{∞} dx e^{−xτ}Z_{0,x}(0, r_{f})]
= Z_{τ}^{0}(r_{i}|0) Z_{τ}^{0}(0|0)^{n−1} Z_{τ}^{0}(0|r_{f}) ,  (D.5)

where we have used that after integration over L, the integrals factorize; we further have defined

Z_{τ}(r_{i}|r_{f}) = ∫_{0}^{∞} dsZ_{0,s}(r_{i}|r_{f})e^{−τs}
= ∫_{0}^{∞} ds ∫_{k} e^{−(k^{2}+τ)s+ik(r_{f} −r_{i})} = ∫_{k} \frac{e^{ik(r_{f} −r_{i})}}{k^{2} + τ} .  (D.6)

In d=1 the integration over the momenta k in (D.6) leads to

Z_{τ}(r_{i}|r_{f})_{|d=1} = \sqrt{\frac{π}{τ}} e^{−\sqrt{τ}|r_{f} −r_{i}|} .  (D.7)

Let us now lock at what terms appear in the n-th order coefficient in (D.5): From the initial point of the chain a factor Z_{τ}(r_{i}|0) is contributed, while the final point is coming up with Z_{τ}(0|r_{f}). Furthermore, the internal points give n−1 powers of Z_{τ}(0|0), where

Z_{τ}(0|0) = \sqrt{\frac{π}{τ}} .  (D.8)

The (n=0)-coefficient is special: Clearly, one has

Z_{τ}^{g_{0},0}(r_{i}|r_{f}) = Z_{τ}(r_{i}|r_{f}) .  (D.9)

Inserting (D.8)-(D.9) into (D.5) we obtain for (D.4):

Z_{τ}^{g_{0}}(r_{i}|r_{f}) = \sqrt{\frac{π}{τ}} [e^{−\sqrt{τ}|r_{f} −r_{i}|} + e^{−\sqrt{τ}(|r_{f} |+|r_{i}|)} ∑_{n=1}^{∞} (−g_{0}\sqrt{\frac{π}{τ}} )^{n}]

= \sqrt{\frac{π}{τ}} [e^{−\sqrt{τ}|r_{f} −r_{i}|} − e^{−\sqrt{τ}(|r_{f} |+|r_{i}|)} \frac{1}{1 + \frac{1}{g_{0}} \sqrt{\frac{τ}{π}}} ] .  (D.10)

The limit of g_{0} → ∞ can be taken and leaves us with the beautiful result

Z_{τ}^{∞}(r_{i}|r_{f}) = \sqrt{\frac{π}{τ}} (e^{−\sqrt{τ}|r_{f} −r_{i}|} − e^{−\sqrt{τ}(|r_{f} |+|r_{i}|)}) .  (D.11)

This is the propagator of a scalar field with Dirichlet boundary conditions. It has applications when studying critical phenomena of e.g. an Ising magnet in half-space [68,69]. Performing the inverse Laplace-transformation of (D.11) we arrive at

Z_{L}^{∞}(r_{i}|r_{f}) = \frac{1}{L^{1/2}} (e^{− \frac{(r_{f} −r_{i})^{2}}{4|L|}} − e^{− \frac{(|r_{f} |+|r_{i}|)^{2}}{4|L|}} ) .  (D.12)

We conclude that a δ- potential acting in a hyper-plane suppresses all configurations which penetrate or touch it, when taking its amplitude to infinity. Finally, in order to prove the universal repulsion law, we start from (D.12) and integrate over all final positions r_{f}:

Z_{L}^{∞}(r_{i}) = ∫_{−∞}^{∞} dr_{f} Z_{L}^{∞}(r_{i}|r_{f}) = 2\sqrt{π} (1 − erf\frac{|r_{i}|}{2\sqrt{L}}) ,  (D.13)

erf denoting the error function. For small arguments we have

1 − erf(x) ∼ x .  (D.14)

Furthermore, Z_{L}^{∞}(r_{i}) denotes the restricted partition function of a chain pinned at one of its ends in r_{i}. To obtain Z_{∞}(r/L^{ν}) in (2.23) we have to evaluate:

Z_{∞}(r/L^{ν}) = \frac{1}{L} ∫_{0}^{L} ds Z_{s}^{∞}(r)Z_{L−s}^{∞}(r) ∼ (\frac{r}{\sqrt{L}})^{2}  (D.15)

according to (D.14) in the scaling regime r≪L^{1/2}. We thus find that θ = 2, which confirms (2.25) for D = 1 and d = 1. The above proof can be extended to arbitrary dimensions 0<d<2, where ε>0 for polymers. Then, θ = 4 − 2d.

It would be nice to make the same arguments for membranes. However, the proof is based on a drastic simplification, which only occurs in D=1 and shows up in the factorizability of loop diagrams as in (D.5). This has no extension to manifolds of internal dimension D>1.

D.2 Strong coupling expansion for the renormalized coupling g

In the following we want to derive an expression for the renormalized coupling g as defined in (2.10) in terms of a strong coupling expansion in powers of the bare coupling. This is needed in the fixed membrane size ensemble as it is also considered in the 2 − D-expansion. In order to make matching our strong coupling series about D = 2 with the known exact result for polymers, we shortly derive the latter both for an open as well as a closed chain. Starting from the definition of the polymer-propagator let us point out that the full partition function then is defined as

Z^{O} = ∫_{r_{f}} ∫_{r_{i}} Z_{s,s′}(r_{i}|r_{f}) , open end polymer  (D.16)
Z^{C} = ∫_{r} Z_{s,s′}(r|r) , closed polymer  (D.17)

In both cases the integrand is expanded in powers of the bare coupling according to (D.4). Generally, the nth order coefficient in (D.5) is coming up with n − 1 powers of Z_{τ}(0|0) to be here calculated for arbitrary d < 2. Accordingly,

Z_{τ}(0|0) = ∫_{k} \frac{1}{k^{2} + τ} = π^{−d/2} \frac{S_{d} \frac{π}{2}τ^{−ε}}{sin(\frac{dπ}{2})} = Γ(ε)τ^{−ε} ,  (D.18)

where we have used the normalization of the integral according to (3.6) and

Γ(ε)Γ(1−ε) = \frac{π}{sin(πε)} and S_{d} = 2\frac{π^{d/2}}{Γ(\frac{d}{2})} .

Note that here ε = 1 − d/2. Furthermore, there is a Z_{τ}(r_{i}|0)Z_{τ}(0|r_{f}) from the end-points contributing according to (D.16) in the case of an open end chain

∫_{r_{i}} ∫_{r_{f}} Z_{τ}(r_{i}|0)Z_{τ}(0|r_{f}) = (∫_{r} ∫_{k} \frac{e^{−ikr}}{k^{2} + τ} )^{2} = τ^{−2} ,  (D.19)

Interacting Crumpled Manifolds 85

Figure D.1: ”Daisy”-diagrams contributing to Z^{O} (left) and Z^{C} (right), (D.16), each leaf representing the integration over an internal distance.

while for a closed chain one is obtaining:

∫_{r} Z_{τ}(r|0)Z_{τ}(0|r) = ∫_{r} ∫_{k_{i}} ∫_{k_{f}} \frac{e^{ir(k_{f} −k_{i})}}{(k_{i}^{2} + τ)(k_{f}^{2} + τ)} = ∫_{k} \frac{1}{(k^{2} + τ)^{2}} = εΓ(ε)τ^{−ε−1} (D.20)

The case n=0 is special. For an open polymer we have to evaluate

∫_{r_{i}} ∫_{r_{f}} Z_{τ}^{g_{0},0}(r_{i}|r_{f}) = ∫_{r_{i}} ∫_{r_{f}} ∫_{k} \frac{e^{ik(r_{f} −r_{i})}}{k^{2} + τ} = V_{ℝ^{d}}τ^{−1} ,  (D.21)

while a closed chain is yielding

∫_{r} Z_{τ}^{g_{0},0}(r|r) = V_{ℝ^{d}} ∫_{k} \frac{1}{k^{2} + τ} = V_{ℝ^{d}}Γ(ε)τ^{−ε}  (D.22)

Diagrammatically, the nth order contribution to the partition function (D.16) can be written according to (D.18-D.22) as shown in fig. (D.1).

Let us now calculate Z^{0} − Z^{g_{0}} for a closed chain:

Z^{0} − Z^{g_{0}} = −ετ^{−1} ∑_{n=1}^{∞}(−Γ(ε)(g_{0}τ^{−ε})^{n} = ετ^{−1} \frac{Γ(ε)g_{0}τ^{−ε}}{1 + Γ(ε)g_{0}τ^{−ε}}

= ετ^{−1} \frac{1}{1 + τ^{ε}(Γ(ε)g_{0})^{−1}} = ετ^{−1} ∑_{n=0}^{∞} (−\frac{τ^{ε}}{Γ(ε)g_{0}} )^{n} = ε ∑_{n=0}^{∞} (−\frac{1}{Γ(ε)g_{0}} )^{n} τ^{nε−1} . (D.23)

Performing an inverse Laplace-transformation, we arrive at

g_{C} := (Z^{0} − Z^{g_{0}})(L) = \frac{1}{2πi} ∫_{γ−i∞}^{γ+i∞}dτ e^{τL}(Z^{0} − Z^{g_{0}})

= ε [1 + ∑_{n=1}^{∞} (−\frac{1}{Γ(ε)g_{0}L^{ε}} )^{n} \frac{Γ(nε) sin(πnε)}{π}] = ε [1 + ∑_{n=1}^{∞} (−\frac{1}{Γ(ε)g_{0}L^{ε}} )^{n} \frac{1}{Γ(1−nε)}] ,

(D.24) where we have used that for any μ > 0

\frac{1}{2πi} ∫_{γ−i∞}^{γ+i∞}dτ τ^{μ−1}e^{τL} = \frac{Γ(μ) sin(πμ)}{π}L^{−μ} .  (D.25)

Z^{0} −Z^{g_{0}} as given in (D.24) is the dimensionless renormalized coupling g in the case of a closed chain. Note that in the calculation of the partition function of an open end chain there has to be performed one more integration over the field r, such that only (Z^{0} − Z^{g_{0}})L^{(ε−1)} =: g, 1−ε being the dimension of the field, becomes dimensionless. Before we proceed let us shortly consider how to treat the limit of vanishing bulk space dimension d in this framework: Setting d = 0 and equivalently ε = 1 for polymers, then, the renormalized coupling is obtained from (D.23) and (D.24) as

g_{C}(d=0) = \frac{1}{2πi} ∫_{γ−i∞}^{γ+i∞}\frac{dτ}{τ} e^{τL} \frac{g_{0}}{g_{0} + τ} .  (D.26)

This time, the inverse Laplace-transformation can be immediately evaluated in terms of residues. First order poles of the integrand are located at τ = 0 and τ = −g_{0}, therefore providing

g_{C}(d=0) = 1 − e^{−g_{0}L} ,  (D.27)

which is consistent with what we have found within the expansion in 2 − D (6.76). Let us calculate the inverse Laplace-transformation of Z^{0} − Z^{g_{0}} for an open end chain:

Z^{0} − Z^{g_{0}} = τ^{−2}g_{0} ∑_{n=1}^{∞}(−(g_{0}Γ(ε)τ^{−ε})^{n−1} = \frac{g_{0}τ^{−2}}{1 + (g_{0}Γ(ε)τ^{−ε})}

= \frac{τ^{ε−2}}{Γ(ε)} \frac{g_{0}Γ(ε)τ^{−ε}}{1 + g_{0}Γ(ε)τ^{−ε}} = \frac{τ^{ε−2}}{Γ(ε)} \frac{1}{1 + (g_{0}Γ(ε))^{−1}τ^{ε}} = \frac{τ^{ε−2}}{Γ(ε)} ∑_{n=0}^{∞} (−\frac{1}{g_{0}Γ(ε)})^{n} τ^{nε} , (D.28)

leading to

(Z^{0} − Z^{g_{0}})(L) = \frac{1}{2πi} ∫_{γ−i∞}^{γ+i∞}dτ e^{τL}(Z^{0} − Z^{g_{0}})

= −\frac{1}{Γ(ε)} ∑_{n=0}^{∞} (−\frac{1}{g_{0}Γ(ε)})^{n} ∫_{γ−i∞}^{γ+i∞}dτ \frac{L τ^{(n+1)ε−1}}{(n + 1)ε − 1} e^{τL}

= −\frac{1}{Γ(ε)} ∑_{n=0}^{∞} (−\frac{1}{g_{0}Γ(ε)})^{n} L^{1−(n+1)ε} \frac{Γ((n + 1)ε)}{(n + 1)ε − 1} \frac{sin(π(n + 1)ε)}{π}

= L^{1−ε} [\frac{sin(πε)}{(1 − ε)π} + ∑_{n=1}^{∞} (−\frac{1}{L^{ε}g_{0}Γ(ε)})^{n} \frac{1}{Γ(2−nε)}] ,  (D.29)

where we have used that for 0 < μ < 1

∫_{γ−i∞}^{γ+i∞}dτ τ^{μ−2} e^{τL} = \frac{τ^{μ−1}}{μ − 1} e^{τL}|_{γ−i∞}^{γ+i∞} − L ∫_{γ−i∞}^{γ+i∞}dτ \frac{τ^{μ−1}}{μ − 1} e^{τL} ,  (D.30)

and the boundary term vanishes. The dimensionless renormalized coupling then is defined as in (2.10), such that

g_{O} := (Z^{0} − Z^{g_{0}})(L)L^{ε−1} = \frac{sin(πε)}{(1 − ε)π} + ∑_{n=1}^{∞} (−\frac{1}{L^{ε}g_{0}Γ(ε)})^{n} \frac{1}{Γ(2−nε)} . (D.31)

From the strong coupling expansions (D.24) and (D.31) one easily obtains the fixed point values of g. For the closed chain we find

g_{C}^{∗} = ε ,  (D.32)

while for the open end chain we have

g_{O}^{∗} = \frac{sin(πε)}{(1 − ε)π} .  (D.33)

D.3 Weak coupling expansion

In order to be complete let us turn the expansions (D.24) and (D.31) into weak coupling expansions. This can be easily done starting for instance from the expansion in the Laplacetransformed picture for the closed chain as given in the top line in (D.23). Then,

g_{C} := (Z^{0} − Z^{g_{0}})(L) = \frac{1}{2πi} ∫_{γ−i∞}^{γ+i∞}dτ e^{τL}(Z^{0} − Z^{g_{0}}) = \frac{−ε}{2πi} ∑_{n=1}^{∞}(−g_{0}Γ(ε))^{n} ∫_{γ−i∞}^{γ+i∞} dτ τ^{−nε−1} e^{τL}

= −ε ∑_{n=0}^{∞} \frac{(−g_{0}Γ(ε)L^{ε})^{n}}{Γ(1+nε)} ,  (D.34)

where we have simply used the analytical continuation of (D.25) to arbitrary values of μ. (D.25) can be obtained from integrating along a branch cut, which makes it necessary to have μ > 0. But μ can be always rendered positive through a sufficient number m of partial integrations according to (D.30). Having chosen m appropriately, it drops out after the inverse Laplace transformation has been performed. Let us apply the same procedure to the weak coupling expansion of the renormalized coupling of the open chain in the top line in (D.28):

(Z^{0} − Z^{g_{0}})(L) = \frac{1}{2πi} ∫_{γ−i∞}^{γ+i∞}dτ e^{τL}(Z^{0} − Z^{g_{0}}) = \frac{g_{0}}{2πi} ∑_{n=1}^{∞}(−(g_{0}Γ(ε))^{n−1} ∫_{γ−i∞}^{γ+i∞} dτ τ^{−(n−1)ε−2} e^{τL}

= Lg_{0} ∑_{n=1}^{∞} \frac{(−g_{0}Γ(ε)L^{ε})^{n−1}}{Γ(2+(n−1)ε)} ,

(D.35) from what follows that

g_{O} := L^{ε−1}(Z^{0} − Z^{g_{0}}) = g_{0}L^{ε} ∑_{n=0}^{∞} \frac{(−g_{0}Γ(ε)L^{ε})^{n}}{Γ(2+nε)} .  (D.36)

D.4 Semi-flexible polymers

Theories consisting in some one-dimensional manifold interacting with a single defect turn out to be always one-loop exact. This is due to the factorizability of the path-integrals like (D.5) to appear in the perturbation theory. Let us now discuss another model taking once again advantage of this property. Consider so-called semi-flexible polymers, whose configurations are controlled by bending elasticity. For low temperatures or equivalently on short scales their configurational space can be parametrized by some d-dimensional field r⃗, being defined through the displacement of the individual mass points with respect to some flat background, therefore allowing for no overlaps or even self-intersections of the polymer. Then, the curvature along the manifold can be approximated through the usual Laplacian ∂_{tt}r⃗(t) and the free Hamiltonian reads:

H_{0}[r⃗] = ∫_{0}^{L} dt \frac{1}{2}(Δr⃗(t))^{2} .  (D.37)

Let us now consider the interaction with some short-range, δ-like defect, acting in the displacement space and being located at the origin: A short distance analysis shows that any two nearby intersections of the polymer with the defect appear like a single, parallel intersection [70]. This is due to the bending stiffness of the manifold suppressing short-wavelength fluctuations. Taking the most relevant, orientational dependent interaction into account, which is

H_{int}[r⃗] = v_{0} ∫_{0}^{L} dt δ^{d}(r⃗(t))δ^{d}(∇r⃗(t)) ,  (D.38)

let us shortly sketch, how to completely sum up the perturbation theory in this case following the same lines as in the preceding sections. Once again, we need to find the propagator of the corresponding Schrödinger-type equation, which is defined through the path-integral:

G_{t,t′}(r_{i}, v_{i}|r_{f}, v_{f}) = ∫ D[r⃗] δ^{d}(r(t)−r_{i})δ^{d}(∇r(t)−v_{i})δ^{d}(r(t′)−r_{f})δ^{d}(∇r(t′)−v_{f}) e^{−H_{0}[r⃗]} ,

(D.39)

where r_{i}, v_{i} and r_{f}, v_{f} denote position and slope of the initial and final point, respectively. Be G_{t,t′}^{v_{0}} the corresponding propagator in presence of the interaction (D.38). Switching to a grandcanonical ensemble,

G_{τ}^{v_{0}}(r_{i}, v_{i}|r_{f}, v_{f}) = ∫_{0}^{∞}dL G_{0,L}(r_{i}, v_{i}|r_{f}, v_{f}) e^{−τL} ,  (D.40)

the perturbation series of G_{τ}^{v_{0}}(r_{i}, v_{i}|r_{f}, v_{f}) can then be written as

G_{τ}^{v_{0}}(r_{i}, v_{i}|r_{f}, v_{f}) = ∑_{n=0}^{∞}(−v_{0})^{n}G_{τ}^{v_{0},n}(r_{i}, v_{i}|r_{f}, v_{f}) .  (D.41)

However, solving the path-integral of the free problem (D.39) poses considerable problems, when treating the energyless tilt mode. The solution has been obtained from the equivalent problem of the corresponding Schrödinger-type equation [71–73]. It reads:

G_{t,t′}(r_{i}, v_{i}|r_{f}, v_{f}) = (\frac{\sqrt{3}}{π(t′−t)^{2}} )^{d} e^{− \frac{6}{(t′−t)^{3}} S_{t,t′}(r_{i},v_{i}|r_{f} ,v_{f})} ,  (D.42)

where

S_{t,t′}(r_{i}, v_{i}|r_{f}, v_{f}) = (r_{f}−r_{i} − v_{i}(t′−t))^{2}+\frac{(t′−t)^{2}}{3} (v_{f}−v_{i})^{2}−(t′−t)(r_{f}−r_{i}−v_{i}(t′−t))(v_{f}−v_{i})

(D.43)

is the action of the minimal energy path. In analogy to (D.5) the path-integral to be calculated to obtain G_{τ}^{v_{0},n}(r_{i}, v_{i}|r_{f}, v_{f}) factorizes after ”time-ordering” the distances:

G_{τ}^{v_{0},n}(r_{i}, v_{i}|r_{f}, v_{f}) = [∫_{0}^{L} dL e^{−τL}G_{0,L}(r_{i}, v_{i}|0, 0)] [∫_{0}^{L} dL e^{−τL}G_{0,L}(0, 0|0, 0)]^{n−1} ×
× [∫_{0}^{L} dL e^{−τL}G_{0,L}(0, 0|r_{f}, v_{f})] ,  n ≥ 1 .  (D.44)

The n = 0 contribution is obtained from

G_{τ}^{v_{0},0}(r_{i}, v_{i}|r_{f}, v_{f}) = ∫_{0}^{L} dL e^{−τL}G_{0,L}(r_{i}, v_{i}|r_{f}, v_{f}) .  (D.45)

It is hard to calculate the Laplace-transform of the propagator (D.42), but the renormalized coupling v can be easily obtained from Z^{0} − Z^{v_{0}}, Z^{v_{0}} being the full partition function given by

Z^{v_{0}}(τ) = ∫ d^{d}r_{i} d^{d}v_{i} d^{d}r_{f} d^{d}v_{f} G_{τ}^{v_{0}}(r_{i}, v_{i}|r_{f}, v_{f})  (D.46)

in the Laplace-transformed picture. Since

∫_{0}^{∞}dL e^{−τL} ∫ d^{d}r_{i} d^{d}v_{i} G_{0,L}(r_{i}, v_{i}|0, 0) = 3^{−d/2} \frac{Γ(1−2d)}{τ^{1−2d}}  (D.47)

and

∫_{0}^{∞}dL e^{−τL}G_{0,L}(0, 0|0, 0) = (\frac{\sqrt{3}}{π})^{d} \frac{Γ(1−2d)}{τ^{1−2d}} ,  (D.48)

we find:

(Z^{0} − Z^{v_{0}})(τ) = −Γ(1−2d) τ^{−(1−2d)} (\frac{3\sqrt{3}}{π})^{−d} ∑_{n=1}^{∞}(−v_{0})^{n} (\frac{3^{d/2}Γ(1−2d)}{π^{d}}τ^{−(1−2d)})^{n}

= \frac{3^{−d}Γ^{2}(1−2d) v_{0}τ^{−2(1−2d)}}{1 + v_{0}τ^{−(1−2d)} \frac{3^{d/2}Γ(1−2d)}{π^{d}}} = (\frac{3\sqrt{3}}{π})^{−d} Γ(1−2d)τ^{−(1−2d)} ∑_{n=0}^{∞} (−v_{0}τ^{−(1−2d)} \frac{3^{d/2}Γ(1−2d)}{π^{d}})^{−n} ,

(D.49) where we have turned the series into a strong coupling expansion. Performing an inverse Laplace transformation,

\frac{1}{2πi} ∫_{γ−i∞}^{γ+i∞}dτ e^{τL}(Z^{0} − Z^{v_{0}})(τ)

= (\frac{3\sqrt{3}}{π})^{−d} Γ(1−2d)L^{1−2d} ∑_{n=0}^{∞} (\frac{3^{d/2}Γ(1−2d)}{π^{d}})^{−n} \frac{(−v_{0}L^{1−2d})^{−n}}{Γ(1 + (n−1)(1−2d))} , (D.50)

the dimensionless renormalized coupling v reads:

v := (Z^{0} − Z^{v_{0}})(L)L^{−(1−2d)} = (\frac{3\sqrt{3}}{π})^{−d} Γ(1−2d) ∑_{n=0}^{∞} (\frac{3^{d/2}Γ(1−2d)}{π^{d}})^{−n} \frac{(−v_{0}L^{1−2d})^{−n}}{Γ(1 + (n−1)(1−2d))} .

(D.51)

The fixed point is obtained sending the dimensionless bare coupling z := v_{0}L^{1−2d} to ∞:

v^{∗} := lim_{z→∞} v(z) = (\frac{3\sqrt{3}}{π})^{−d} \frac{Γ(1−2d)}{Γ(2d)} .  (D.52)

From the strong coupling expansion (D.51) one may immediately read off the value of the correction to scaling exponent ω at the fixed point:

ω(v^{∗}) = 1 − 2d .  (D.53)

B  A  C

Figure E.1: The sectors A, B and C

E Conformal mapping of the sectors

In order to calculate the two-loop diagram efficiently, one wants to write it as an integral over a finite domain only. To do so we need the technique of conformal mapping of the sectors, which also serves for analytically continuing the measure of integration to internal dimensions D < 1. This technique has been extensively used and well documented in the context of self-avoiding tethered membranes [26,20,4], but we repeat the presentation here for completeness. Generally, in evaluating the three-point divergences we need to integrate over some domain in the upper half-plane (see 4.11). The measure of integration reads

∫_{y} = \frac{S_{D−1}}{S_{D}} ∫_{−∞}^{∞} dy_{1} ∫_{0}^{∞}dy_{2} (y_{2})^{D−2}  (E.1)

and the integrand is a function f of the three distances a, b and c between the points O = (0, 0), H = (−L, 0) and Y = (y_{1}, y_{2}), given explicitly by (see fig. E.1)

a = L = fixed, b = \sqrt{(y_{1})^{2} + (y_{2})^{2}} , c = \sqrt{(L−y_{1})^{2} + (y_{2})^{2}} .  (E.2)

In our problem f is homogeneous, of degree λ, but not necessarily symmetric:

f(κa, κb, κc) = κ^{−λ}f(a, b, c) .  (E.3)

Let us now explicitly show, how sectors can be mapped onto each other. Specializing to the mapping B → A we choose a coordinate system as given in figure (E.1). The mapping B → A is mediated by a special conformal transformation, the inversion with respect to the circle around O and radius L. In complex coordinates this is (Ȳ is the complex conjugate of Y)

Y ⟶ Ỹ = \frac{L^{2}}{Ȳ} = Y \frac{L^{2}}{b^{2}} ⇔ Y = Ỹ \frac{L^{2}}{b̃^{2}}  (E.4)

such that

( \begin{matrix}y_{1} \\ y_{2} \end{matrix} ) ⟶ ( ỹ_{1} ; ỹ_{2} ) = ( \frac{y_{1}L^{2}}{b^{2}} ; \frac{y_{2}L^{2}}{b^{2}} ) ⇔ ( \begin{matrix}y_{1} \\ y_{2} \end{matrix} ) = ( \frac{ỹ_{1}L^{2}}{b̃^{2}} ; \frac{ỹ_{2}L^{2}}{b̃^{2}} ) .  (E.5)

This change of coordinates gives a Jacobian for the measure (E.1)

(\frac{L^{2}}{b̃^{2}} )^{D} .  (E.6)

It can easily be seen that the mapping B → A is one to one: First, be Y =(y_{1}, y_{2}) ∈ B, that is b > L and b > c. Then,

c̃^{2} = ((ỹ_{1}−L)^{2} + (ỹ_{2})^{2}) = \frac{L^{4} − 2y_{1}L^{3} + b^{2}L^{2}}{b^{2}} = L^{2} \frac{c^{2}}{b^{2}} < L^{2}  (E.7)

and

b̃^{2} = L^{2} \frac{L^{2}}{b^{2}} < L^{2} .  (E.8)

Since b̃^{2}, c̃^{2} < L^{2}, Ỹ ∈ A. Second, for the inverse mapping, Y (Ỹ ∈ A) ∈ B is checked as follows:

c^{2} = b^{2} + L^{2} − 2y_{1}L = b^{2} + L^{2} − 2ỹ_{1} \frac{L^{3}}{b̃^{2}} < b^{2} ,  (E.9)

since

c̃^{2} = b̃^{2} − 2ỹ_{1}L + L^{2} < L^{2} ⇔ L^{2} − 2ỹ_{1} \frac{L^{3}}{b̃^{2}} < 0

and

b^{2} = L^{2} \frac{L^{2}}{b̃^{2}} > L^{2} .  (E.10)

Since b > L and b > c, Y ∈ B. Finally, let us look at how the whole integral transforms. First inserting the transformed variables including the Jacobian (E.6) and second using the homogeneity (E.3), we arrive at

∫_{Y ∈B} f(a = L, b, c) = ∫_{Ỹ ∈A} (\frac{L}{b̃})^{2D} f (L, \frac{L^{2}}{b̃} , \frac{c̃L}{b̃}) = ∫_{Ỹ ∈A} (\frac{L}{b̃})^{2D−λ} f(b̃, ã = L, c̃) . (E.11)

Let us restate the above calculation in a completely geometric interpretation as sketched in the figure. The mapping consists of two steps:

• The rescaling with respect to O by a factor L/b which maps Y onto Y′ and H onto H′. (E.3) implies that f is changed by a factor (\frac{L}{b})^{−λ}.

• A mirror operation, which maps Y′ onto H and H′ onto Ỹ, leaving invariant the origin O. This operation is a permutation of the first two arguments of f.

The mapping Y → Ỹ corresponds to the special conformal transformation (E.4). Analogously, we find that

∫_{Y ∈C} f(a = L, b, c) = ∫_{Ỹ ∈A} (\frac{L}{c̃})^{2D−λ} f(c̃, b̃, ã = L) .  (E.12)

Acknowledgements

Allen, die zum Gelingen dieser Arbeit beigetragen haben, möchte ich herzlich danken: Hans Werner Diehl, der mir die Mitarbeit in seiner Arbeitsgruppe ermöglicht hat und mit zahlreichen Anregungen und Kritik beigetragen hat. Kay J. Wiese, der auch nach seinem Fortgang an die Strände Kaliforniens (ITP, Santa Barbara) die Arbeit stets engagiert betreut hat und bei dem meine zahlreichen Fragen fast immer Gehör fanden. Mein zweiwöchiger Aufenthalt in Santa Barbara bleibt in guter Erinnerung. Insbesondere auch eine lange Diskussion mit Andreas Ludwig, der sich eingehend für den Limes D → 2 und die Entwicklung in 2 − D interessiert hat. Für anregende Diskussionen oder Tips danke ich weiterhin besonders Lothar Schäfer und Ralf Blossey. Darüber hinaus danke ich allen weiteren Mitarbeitern der Arbeitsgruppe und des Fachbereichs Physik, die den Aufenthalt in Essen interessant oder angenehm gemacht haben, sei es wissenschaftlich gesehen oder sei es aufgrund anderer Ablenkungen. Insbesondere Andrea Ostendorff, Anja Gerwinski, Alexandra Apel, Boris Shalaev, Frau Thälker, Florian Jasch, Hannes Hager, Hans Werner Diehl, Harald Karl, Ira Terwyen, Jens Eggers, Jouni Kallounki, Kay Wiese, Knut Elsner, Lothar Schäfer, Martin Smock, Mykola Sphot, Philipp Kuhn, Ralf Blossey, Reinhard Leidl, Rona Rangsch, Rüdiger Oberhage, Sutapa Mukherij und Ulla Lensen. Diese Arbeit ist von der DFG im Rahmen des Leibniz Programms Di 378/2-1 unterstützt worden.

94 Interacting Crumpled Manifolds

R References

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