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Nuclear Physics B 820 [FS] (2009) 707-752

www.elsevier.com/locate/nuclphysb

Disordered O(n) loop model and coupled conformal field theories

Hirohiko Shimada

Department of Basic Sciences, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan Received 17 April 2009; accepted 18 May 2009 Available online 22 May 2009

Abstract

A family of models for fluctuating loops in a two-dimensional random background is analyzed. The models are formulated as O(n) spin models with quenched inhomogeneous interactions. Using the replica method, the models are mapped to the M ^ 0 limits of M-layered O(n) models coupled each other via $1,3 primary fields. The renormalization group flow is calculated in the vicinity of the decoupled critical point, by an epsilon expansion around the Ising point (n = 1), varying n as a continuous parameter. The one-loop beta function suggests the existence of a strongly coupled phase (0 < n < n*) near the self-avoiding walk point (n = 0) and a line of infrared fixed points (n* < n < 1) near the Ising point. For the fixed points, the effective central charges are calculated. The scaling dimensions of the energy operator and the spin operator are obtained up to two-loop order. The relation to the random-bond q-state Potts model is briefly discussed. © 2009 Elsevier B.V. All rights reserved.

PACS: 05.10.Cc; 05.50.+q; 11.25.Hf; 75.10.Nr

Keywords: Disordered system; Conformal perturbation theory; Replica method; Scaling dimension; Renormalization group; Complex Selberg integral

1. Introduction

A variety of random, or disordered systems are known to have non-trivial universalities. These include the universality consisting of the well-known three symmetry classes which appear in the weak localization effects to the conductance [1] or the level correlations [2] in disordered electronic systems, and the universality in the wave function multifractality at the integer quantum Hall transition in which disorder plays an essential role [3]; even the universality in the distri-

E-mail address: shimada@dice.c.u-tokyo.ac.jp.

0550-3213/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2009.05.013

bution of the zeros of the Riemann zeta function is expected to have its origin in some chaotic system [4], and it may possibly be closely related to some disordered system. It is amazing that some of these systems in appropriate limits are phenomenologically well described by such simple ansatz as the one in the random matrix theory. It is, however, by no means certain that these phenomena are well understood theoretically. The most fundamental problem to be understood is how the universalities in disordered systems emerge from a microscopic structure of each system.

A lot of effort has been made to investigate this direction, and in some cases we have a partial answer; we can first formulate a system microscopically, then by using a very crude approximation (such as 'dimensional reduction' used in disordered electronic systems [2]), deduce universality directly in some carefully chosen limit. Even under such fortunate circumstances, it is usually the case that we do not have much knowledge on the way along which the system deviates from the universality away from the limit. This is, roughly speaking, because a disordered system formulated on a microscopic ground typically has highly nonlinear interactions induced by disorder and becomes almost intractable without any crude approximations. Thus, it is important to study a simple model which is tractable with a sensible approximation; analysis of such a model and ideas used there may lead to a practical way to obtain qualitative results on the deviation from the universality in more complicated disordered systems and to an insight on the emergence of the universality itself.

With this broad motivation in mind, among the diverse disordered systems, we take a problem in statistical physics, where relations between microscopic properties and macroscopic behavior of models are often quite intelligible. More specifically, we study a model with quenched disorder defined on a lattice focusing on its possible critical behavior. Understanding the behavior of such a system governed by the Hamiltonian which contains quenched inhomogeneous interactions is especially important, both from the practical perspective that there are no translationally invariant perfect crystals in the real systems and from theoretical interests inspired by the other disorder-induced phenomena.

As is well known, the solid notion of the universality class has been established in the field of critical phenomena without disorder. Nonlinear interactions inherent in a model are essential to understand the existence of the non-trivial universality classes [5]; one should confront with the nonlinearities in a more powerful framework than the mean field approximation. The ideas of scaling and renormalization group (RG) give such a framework. In this framework, one introduces a theory space and a vector field on it, which generates a flow that indicates the scale dependence of a theory considered. Assuming critical points in the second order phase transitions are described by scale invariant field theories, they correspond to the fixed points of the flow. The flow can be determined, step by step, by a path integration on the fluctuations belonging to a certain scale; the nonlinear interactions induce the mutual coupling among degrees of freedom at different scales. As a result, the global configuration of the flow can become non-trivial. Then we can read off the universality classes from the sets of RG eigenvalues which characterize the flow linearized around the fixed points. Furthermore, it is known that the ideas of RG are not only powerful to resolve the universality classes in the critical phenomena of pure systems but also flexible enough to deal with some disordered systems.

For the target of this paper, a system governed by an inhomogeneous Hamiltonian, it is natural to consider a configuration of the interactions as one realization from an ensemble that respects given probability distribution. Basic observables to be discussed can be related to the free energy in a large enough system, which is expected to be self-averaging over the disorder distribution.

We should therefore study the quenched average: the average over the disorder distribution, of the free energy obtained by tracing over the statistical mechanical degrees of freedom.

Actually, evaluating the quenched average of the free energy is notoriously difficult task, since we should evaluate the average of the logarithm of the partition function in a non-translational invariant realization of the disorder. It is practical to use either one of the two methods: replica [6] or supersymmetry [2,7]. When these methods are applicable, we are left with an effective theory with the translational invariance, but this time, with the additional nonlinear interactions induced by the disorder. It should be noted that the system acquires enhanced symmetry, namely, the replica permutation symmetry or the supergroup symmetry, respectively.

For a weakly disordered system, one can consider the disorder-induced couplings in the effective theory as a perturbation to the corresponding pure theory. A natural question is whether the induced coupling destructs the critical phenomena of the pure system, or not. A basic and general result on this direction is known as the "Harris criterion" that tells when the disorder can be neglected at large scales; if the couplings are irrelevant in the RG sense, the disorder cannot change the universality class of the system from that of the pure system [8]. However when they are relevant or marginal, one should work harder to analyze the flow, namely, proceed to calculation of loop diagrams formed by the disorder-induced couplings. This procedure corresponds to the calculation of the RG beta function up to the second or higher order in the couping. In the case that the beta function has a zero in addition to the one corresponding to the pure fixed point, the flow can (depending on the sign of the beta function) transfer, at large scale, the theory to another fixed point; one recognizes that the disordered system belongs to a non-trivial universality class. The RG eigenvalues belonging to the universality class can be predicted perturbatively. In general, without a special reason to be integrable, one cannot solve given interacting statistical model. Therefore, the importance of such a crossover from one fixed point to the other cannot be overemphasized.

We shall study a one-parameter family of disordered models and discuss their universality class in two dimensions. Thus, let us briefly note the special role of two dimensions in the study of the universality class of pure systems first, and then comment on the current status of corresponding study in disordered systems. Working in two dimensions provides us with an ideal circumstance to study the crossover in depth. First, in two dimensions the conformal symmetry is infinite dimensional, and assuming the conformal symmetry (instead of the scale invariance only) enables us to classify the possible fixed points of unitary theories with minimal symmetry [9]. Second, it can be shown that the RG flow is irreversible for unitary theories ("c-theorem") [10]. Third, there are known examples of the crossover in which, at the non-trivial fixed point reachable via the flow from other fixed points, the realized enhanced-symmetries are known and the non-perturbative results can be obtained [11-13].

In the disordered systems in two dimensions, on the other hand, fixed points created by disorder are expected to be described by CFT's, but by more general, non-unitary ones. The lack of the unitarity gives rise to the challenging problems of determining the universality class in disordered systems. Major current examples concerned with the restrictive cases where the supersymmetry method can be used. They typically lead to logarithmic CFT's where the disorder-induced coupling is marginally irrelevant [7,14,15]. In these cases, the coupling goes to zero under the RG flow, and the models do not cause any non-trivial crossover. There is also a consideration on the running of the effective central charge defined on the basis of the supersymmetry [14], in analogy with the c-theorem in pure (unitary) systems [10]. At present, however, not much is known about the crossover cases, and hence our understanding of the disordered critical points is quite limited.

One of a few exceptional examples showing the crossover is the random-bond q-state Potts model in two dimension1 [6,18,19]. The model at q = 2 is the random-bond Ising model and does not show the crossover; it is equivalent to the random-mass fermion model, or using the replica method, to the multi-color Gross-Neveu model, in all of which the interaction is marginally irrelevant [7]. Now in the Z2-invariant scalar field theory (without disorder), a nontrivial fixed point emerges when the dimension d is considered as a continuous parameter in the range d < 4 [5]. In the random-bond q-state Potts model, it is also fundamental to consider the parameter q as a continuous number. One has then a non-trivial fixed point in the region q > 2. The perturbative calculation of the RG eigenvalues belonging to this non-trivial universality class is well under control. In particular, the theoretical prediction for the exponent of spin-spin correlation function is in good agreement with the numerical simulation [20].

In this paper, we introduce the disordered version of the O(n) model2 and study the crossover in it. Our model has its own physical importance at certain integral values of n; it corresponds to the random-bond XY (n = 2), the random-bond Ising (n = 1) and the polymers (or, self-avoiding walks) in random environment (n = 0). But more importantly, we consider the models for continuous values of n; this leads to another example of the crossover in disordered system.

For continuous values of n, as explained below, we have a family of models which describe fluctuating loops in a two-dimensional random background. As is well known, elementary excitations in a spin system like the O(n) model can be considered as non-local geometrical objects, namely, loops in the high-temperature expansion. It has been repeatedly emphasized that these loops are analogous to the closed trajectories of some particles [21,22]. Now the parameter n controls a statistics of particles in the random potential; we expect distinct, non-trivial behavior according to the value of n. In order to study these, we use the replica method as in the studies of the random-bond Potts model [6,18,19]; the method leads us to consider several, say M, layers of two-dimensional O(n) models coupled each other via the disorder-induced coupling, and to take the M ^ 0 limit in the end.

The loops in the pure O(n) model become scale invariant at critical temperature for |n| < 2. We investigate critical behavior in the disordered O(n) model using conformal perturbation theory around the one-parameter family of CFT's corresponding to a line of the pure O(n) critical points [23]. The existence of the crossover in our model suggests that there is a one-parameter family of non-unitary CFT's. In this respect, we mention recent development of the stochastic Loewner evolution (SLE) [24], and its application to non-unitary critical points [25,26]. The line of the fixed points in our model may serves as a natural target of such study in disordered system.

The organization of the paper is the following. In Section 2, we formulate the model on a lattice, and explain the types of the quenched disorder considered. Then we use the replica method and take the disorder average. As a result, we reach an intriguing picture of particles going up and down across the two-dimensional layers, thus forming a whole connected diagrams. Then we discuss the relation between the observables on a lattice and the scaling fields in a continuum limit. We see that the nontrivial cases in which the disorder is relevant occur for n < 1. In Section 3, we perform the one-loop calculation and discuss the existence of a non-trivial fixed point. The one-loop beta function suggests the existence of a threshold n*; the fixed point exists for n* < n < 1, while the strongly coupled phase exists in 0 < n < n* .In Section 4, we perform a two-loop calculation for n* <n < 1, using the full information of the four-point function

1 Other exception includes the disordered Dirac fermion problems [16,17].

2 The O(n) model is another natural extension of the Ising model other than the q-state Potts model.

provided by the O(n) CFT. The next-leading order correction for the thermal exponent and the lowest order correction for the spin exponent are then found. In Section 5, we calculate the effective central charge defined in the replica formalism. We find that this increases, along the flow, against the c-theorem which is responsible for unitary theories. We conclude in Section 6, and comment on few further directions. Appendix A provides the formulas on the critical Liouville field theory used in the paper. In Appendices B and C, we describe the calculation of the integral in the two-loop calculation of the beta function and in the spin scaling dimensions, respectively. Finally, Appendix D is devoted to the derivation of the integral formula and the expansion techniques. The main formula takes the form of the scattering amplitude reflecting the picture that the particle forming the loop can propagate via intermediate states while going across the replica layers.

2. Formulation of the model

In this section, we formulate disordered O(n) loop models on a lattice and discuss their continuum limit. In Section 2.1, three types of the disordered lattice models are mapped to homogeneous coupled lattice models by the replica method. We introduce the known CFT for the homogeneous O(n) model and discuss the relation between lattice and continuum in Section 2.2. In Section 2.3, the effective action for the continuum disordered model is derived making use of the operator product expansion (OPE).

2.1. Disordered loop and coupled loop model on a lattice

We start with the partition function of pure O(n) model on a two-dimensional lattice:

where si is a «-component spin on a site i, and s) represents a measure on the isotropic internal space; using the notation Trsi for the tracing operation f \\i^(si)dnsi■, it satisfies Trsi 1 = 1, TrSis¡- = 0 and Trsi (si ■ si) = n. The interaction is short-ranged, and the notation (i, j) refers to a link between the nearest-neighbor sites of the lattice.

A basic idea in this paper is to continue the parameter n to non-integral values and to take advantage of known continuum properties of the corresponding O(n) model [23]. On a lattice, the partition function (1) of the spin system with an arbitrary value of n can be interpreted as a model for fluctuating loops. It becomes especially simple on a honeycomb lattice, and then tracing over the spin degrees of freedom yields,

where the summation is taken over the configuration of the closed loops [27]. Now the weight per bond of loops is t, while the weight per closed loop is n, which is called a fugacity and can take non-integral values. The model in (1) shows universal critical behavior when |n| < 2, and we may use the same continuum description regardless of specific lattice structures.3

(i,j )

config. of loops

3 This is valid in the dilute phase but not in the dense phase. For instance, the intersections which may occur in the model on a square lattice become relevant in the dense phase and discriminate it from the model on the honeycomb lattice. For an explanation of the dilute and the dense phase, see below.

Another interpretation to make sense out of the non-integral values of n is possible for |n| < 2. By orienting the loops, one can assign a complex Boltzmann weight e'x (e—'x) to each clockwise (anti-clockwise) loop with a phase angle x determined from the relation n = e'x + e—'x = 2 cos x .This notion of the oriented loops here is standard in the Coulomb gas (CG) methods [28], where the loops are mapped to the level lines of a height model [28-30]. One can also consider the loop as a closed trajectory of some particle. The local phase factor exp(i&x/2n) is then associated, at a site, with each turn of the particle through an angle 0. This is a lattice version of the spin factor discussed in [22]. In a sense, the value of n controls statistics of the particles.

The qualitative behavior of the O(n) model can be summarized as follows. When |n| < 2, the model has three different phases separated by a critical point t = tc .4 The region t < tc corresponds to the high temperature phase of the spin model, and the length of loop measured by the unit of the lattice spacing is finite. On the other hand, the average length of the loop is divergent either at t = tc or in t > tc. The O(n) model at the critical point t = tc is called in the "dilute" phase, since the fraction of the number of sites visited by some loop is zero. In t > tc, this fraction becomes non-zero and the corresponding phase is called "low temperature" or "dense" phase. It is known that the shapes of the loops in the dilute and the dense phase are described by the SLEk with the Brownian motion amplitude k < 4 and k > 4, respectively [24].

We now give the partition function of the random model as

Z[{t},n] = Trs¡]~[(1 + tijsi • sj), (3)

where the local interactions tij between spins are position dependent and the notation {t} refers to some definite configuration of the interaction. We consider the configuration {t} as a realization taken from some ensemble with a probability distribution functional P [{t}]. One might assume a non-local probability distribution functional, but here we restrict ourselves to study the case of short range correlation. This means that the interaction t = tij on each link independently respects single distribution function P(t). Later, we shall let P(t) a Gaussian-like distribution.

Given an inhomogeneous realization {t}, the tracing over the spin degrees of freedom is still possible:

#loops

Z[{t},n] = m n#loops n tai(1)ai(2)tai(2)ai(3) ••• tal(Ll)al(1), (4)

config. of loops l=1

where al(i) (i e{1, 2,...,Ll}) denotes a site which belongs to the ith loop of a length Ll. Now the path of the particles forming loops should avoid the links with higher cost; we expect its behavior to change, according to the value of n. The situation is, to some extent, analogous to the problem of the electrons in a random potential which has been studied in connection with the Anderson localization [31]. As the weight t is different from link to link (Fig. 1(a)), the exact methods applicable in the pure O(n) model, such as the CG method, cannot be applied here.

Assuming the short-range correlation between the disorder, we can study the self-averaging quantities such as the free energy and the translationally averaged correlation functions by calculating the quenched average [32]. For example, neglecting the surface effects, the free energy of a large enough system A can be considered as a sum of the free energies of many macroscopically large subsystems of A each with a different realization. Since by the short-range assumption these

4 On a hexagonallattice, to be concrete, it is given by tc(n) = (2 + -J 2 — n) 1/2.

Fig. 1. (a) Loops on a inhomogeneous lattice. The grayscale on each link represents the magnitude of the weight tij. (b) Loops on several sheets of homogeneous lattices are coupled each other. An example of a term with the first order in the coupling t2.

realizations are independent each other, the total free energy per site of A in the thermodynamical limit takes, with the probability one, the quenched averaged value defined by

f = Nmf II dtijPj-ln Z[{t},n])/N.

^ (ij)

Here, we have used the fact that a free energy per site of a subsystem with N sites is given by f = (- ln Z)/N. The overline is used to indicate the averaging over the distribution. It should be noted that, in this argument, we need a macroscopic number of macroscopically large subsystems.

In this paper, we use the replica method to evaluate the various quantities related to the quenched average of the free energy (5). This method is based on the identity:

ZM - 1 ln Z = lim -.

Using this, the problem of calculating the quenched average ln Z is reduced to the task of obtaining another quenched average ZM correctly for M & 0. To evaluate the latter, we first prepare the M e N layers of replicated inhomogeneous O(n) models with the same realization {t}, and then take an average over the distribution. Since, in general, the moments of the interaction tij are non-zero, the resulting theory is a coupled M layers of O(n) models. We calculate the quantity ZM for finite M e N, take the limit M ^ 0 in the end. This procedure gives, at least formally, gives the desired average.

Although we perform the detailed analysis in continuum theory, let us proceed, for a while, within the discrete lattice formulation. The purpose is two-fold: (i) to get some insight on underlying physics, as we can concretely see the elementary processes existing in the model and (ii) to see that there are possibly many lattice models described by the same continuum field theory. Now, we consider the M copies of the model (3) and take the disorder average. This gives

(Z [{t},n])M = j Y\dtl}P(t^Y\ Tr^n (1 + tijs(a) ■ sf)

(i,j) a = 1 (ij)

= n T^n 1 + tl£ s(a) ■ sja) + t2^ s(

a = 1 (ij )\ a = 1 a<b

+ (higher order terms),

s(a) ■ s(a)s(b) . s(b)

where in the last line, the first two moments are denoted as t1 = tij and t2 = t2-, and we omit the terms with the higher moments for simplicity. The term t1 (s(a) ■ s(a)) represents a walk on

the same replica layer; the generic case with t1 = 0 is discussed in the following. In addition to this, the theory acquires nonlinear couplings. For instance, when we look at a particular path of a loop, the coupling t2 (s(a) ■ s(a)s(b) ■ sf^) for a < b is considered as a branching process across

' J 1 J

the replica direction as shown in Fig. 1(b). On a lattice, evaluating directly the contribution of the diagrams with the second or higher orders in such processes should involve some sophisticated enumerative combinatorics. Notice, however, that the translational invariance of the individual O(n) model is now restored. Thus, in the continuum limit, we can use the knowledge of the homogeneous O(n) field theory.

The restriction on the sum to the off-diagonal sector (a < b) in the t2 term follows from the fact that the original partition function (3) is linear in the bonds (tijsi ■ sj). When we talk about the universality, however, this formal linearity and resulting off-diagonal property should be reconsidered; we should rather consider a larger parameter space including more general couplings generated under block spin transformations.

In order to see this, firstly, it is instructive to remind that the pure model (1) corresponds to the Hamiltonian H = ln(1 + tsi ■ sj), and originates from the high temperature expansion of another spin system which has the same O(n) symmetry but has the different Hamiltonian H = -ßj> ■ sj. From the RG perspective, these two models are regarded as two different points in the same theory space determined from the O(n) symmetry. Actually, both systems are believed to belong to the same universality class. In fact, with the knowledge of the scaling dimensions of fields obtained by the CG method [28], we may argue that these two Hamiltonians are equivalent modulo irrelevant operators. In particular, the field (si ■ sj)2 represents the double occupancy of the bonds in the partition function, and the presence of it enables the loops to overlap. Although such a field is not contained in the partition function (1), it is natural to think that the term (si ■ sj)2 is generated, under block spin transformation, from the single occupation (si ■ sj). But, as we will see in the end of Section 2.2, the term (si ■ sj)2 is irrelevant in the dilute phase, or equivalently, near the critical point. Because of this, we can say that the linear expression (1) nicely summarize the characteristics of the dilute phase in generic models with the O(n) symmetry.

Taking these considerations into account, we now briefly discuss the theory space of the coupled models derived from the disordered ones. Our concern is near the decoupled critical point; applying the knowledge of the pure O(n) model, we note the inessential differences that arise from different formulations of disordered models on a lattice. For concreteness, let us consider another disordered model:

Z[{ß},n] = Trsi []exp(ßijsi ■ sj), (8)

with a bond distribution P(ß). The formal operations lead to, again, a coupled model:

_ m / m m \

Z[V= n n exP A Es(a) • 4fl)+fc £ • sfsb • j

a = l (i,j) \ a = l a = \,b=\ /

+ (higher order terms), (9)

where ¡3\ = j3ij and = 2 (fifj — j3ij2) are the first and the second cumulants obtained from the distribution P(jij), respectively. Comparing this model (9) with the previous one (7), we notice

that the differences between these two are simply the presence of the exponential function and

Sa),(b) (b Sj Si Sj

that of the diagonal coupling: f)2 (s

(a) . s^)s(b) . s(b) ) with a = b in (9). Both features lead to the

term (s(a) ■ s(a))2 i.e. the process of the double occupancy of the bonds on the same replica plane.

Such a term can also be generated in the model (7) under block spin transformation. However, when the corresponding pure model is in the dilute phase, the process is irrelevant. For this reason, we expect both of the coupled models (7) and (9) are described by a same field theory.

A related model, which is more suitable for taking continuum limit, can be formulated with a probability distribution Ps (/) for disorders / on each site:

Z[{/},n] = i*«]!exp[(/i + xj)si ■ j (10)

In this model, the bond strength ¡3ij = /i + /j respects a distribution P obtained by the convolution of two Ps i.e. Ps * Ps = P .If this distribution P (¡3) is identical to the distribution P(¡) in (8), then this model becomes similar to that of (8). Still, the model (10) is different from the model (8), since there exists the correlation between the nearest-neighbor bonds (say ¡¡ij and ¡ik) connected at a site. Nevertheless, the correlation is still short range; we expect a similar behavior at large scales.

We can argue that this correlation existing in the model (10) is not essential to distinguish it from the model (8) by examining how these two models are mapped into the theory space of the coupled models. Averaging over the disorder / in (10) yields, at a site i, a term Y,a b j k(sia) ■

) as a leading nonlinear coupling. In particular, the j = k part of this coupling is induced by the correlation between the nearest-neighbor bonds in the model (10). If this coupling were, with some special reason, never generated in the model (9) mapped from (8), we would seriously distinguish (10) from (8) in the first place. But, it has already been contained in (9), implicitly. Indeed, although the expression (9) is written with respect to links, this can be recasted in terms of sites thanks to the discrete rotational and the discrete translational invariance of the coupled model on a lattice. Then we recognize, by expanding the exponential, this coupling is already present in the model (9).

To summarize, in the dilute phase, there are two important processes on a lattice: the walk on the same layer (s(a) ■ s j)) and the branching to another layer (s(a) ■ s<j,)s(b) ■ s(b)) for a = b. It is plausible to think that these coupled models discussed here are, at large scales, described by a simple continuum field theory.

2.2. Critical point of the homogeneous O(n) model

When we approach a critical point of some lattice model defined in the dimension d, the correlation length f of the model, which is usually the order of the lattice cut-off a, becomes arbitrary large. The continuum limit (or, scaling limit) of the lattice model can be reached by taking a ^ 0, while keeping f fixed. There is nice class of local lattice observables {i^} which have the finite limit

hm a-2(M +-+A*n)(lat(x1) ■ ■■ini(x„)) = [h(x1) ■ ■■in(xn)), (11)

for an appropriate choice of scaling dimensions {2A$k}. With the idea of the block spin transformation, the scaling fields lk(xk) may be considered as coarse grained versions of fi^1 over

Table 1

Correspondence between lattice observables and continuum fields. The value of p is given by p = 1/(2 - 2a2_).

Lattice Continuum Primary fields Scaling dimension

energy T.J si • sj £ (x) 01,3 -2 + 4a-

spin sa i „ a(x) 0p-1,p 1 - a-/2 - 3/(8a-)

polarization sa s- aii(x) 01,0 1 - a-/2

encounter Hj(si • Sj)2 aiv(x) 02,0 1 - a-/2 + 3/(2a-)

some region of size L, centered at xk, with a ^ L ^ f. These scaling dimensions are determined dynamically from the interactions in the system, and related to the RG eigenvalues y^k via = d — 2Aj,k. When y^k is positive (negative), the RG flow near the critical point is unstable (stable), and the corresponding observable and fields are relevant (irrelevant).

Around the critical point in the theory space of the O(n) model, there are, at least, two directions which are unstable (i.e. relevant) under the RG flow to the infrared. These directions are spanned by the coupling constants associated with the two most important pairs {fi^^h} of local lattice observables and the corresponding scaling fields. On the lattice side, one is the energy density Y, j si ■ sj and the other is the spin vector sf. In the scaling limit, they renormal-ize to become the scaling fields called the energy operator E(x) and the spin operator a(x). The scaling dimension of E(r) and that of a(r) is related to the leading thermal and the leading magnetic eigenvalue of the spin system, respectively. These two positive eigenvalues correspond to the linearized RG flow near the critical point, and determine the thermodynamic exponents [32].

So far, we have described the qualitative structure of the theory space focusing on the correspondence between the lattice observables and the continuum scaling fields. It is tantalizing that there is no versatile scheme for a given lattice model to establish this type of correspondence and to proceed to quantitative analysis using a scaling theory at hand. However, the O(n) model was particularly successful in this respect; the geometric nature such as the loop representation of the partition function made it possible to access some of the exact results on the scaling dimensions by the CG method [28] before the emergence of the conformal field theory [9]. These exact results by the CG method, or by the other means [33] then led to the conjecture claiming that a certain one-parameter family of CFT's describe the critical points of O(n) model for the continuous value of n (|n| < 2) [23]. This claim was studied numerically for n = 0 [34] and further checked for |n| < 2 by the Bethe ansatz [35].

We assume this one-parameter family of CFT's [23] in the rest of the paper. The correlation functions of primary fields are represented in terms of the vertex operators {e'ak0(x)} with an appropriate choice of charges {ak}, where 0(x) is a bosonic scalar field [23,29,30]. The necessary formulae on the vertex operator representation are given in Appendix A. In this construction, we have two marginal operators (the screening vertex operators). In the O(n) model, the screening charge a— is related to n as

n = — 2 cos (nf a—). (12)

The energy operator and the spin operator is identified with the primary fields as follows:

E(x) ^ 01,3, a(x) ^ <fip—i,p, (13)

where the value of p is given by p = 1f(2 — 2a—). These identifications are based on the dimensions originally obtained by the CG methods given in Table 1. Using the formulas (A.5) and

(A.6), the dimensions of these operators are checked as

2A£ = -2a1i3ar-3 = -2(-a-)(a+ + 2a-) = -2 + 4a-, (14)

2Aa = -2ap-i,paj-^ = -2((2 - p)a+ + (1 - p)a-)(pa+ + (1 + p)a-)

= 1 - a-/2 - 3/(8a-), (15)

where a+a- = -1 in (A.3) is used.

The relation to the two-dimensional critical statistical models are summarized below. The O(n) model at n = 2 (a- = 1) and n = 1 (a- = |) are the XY model and the Ising model, respectively. The Ising model is the only minimal unitary model [9] that belongs to the O(n) family. Note that the Ising model is also regarded as the q-state Potts model at q = 2 [6]. The model at n = 0 (a- = 3) corresponds to the polymer, or self avoiding walk (SAW), as the qualitative result was obtained by e = 4 - d expansion [36]. The model in two dimensions is related, under the SLE duality [24], to the percolation (Potts model at q = 1). Both models are non-unitary, and various scaling dimensions in the models are numerically studied [34]. Finally, the model at n =-2 (a- = 2) is the loop-erased random walk (LERW). The one-parameter family of O(n) CFT's are related to the SLE by

k = 4a-, (16)

where k is the strength of the Brownian motion which drives the evolution SLE. These critical O(n) models covers 2 < k < 4.

In the rest of the section, we discuss more on the correspondence between the lattice and the continuum. The relation between the loop configuration and the correlation function in the continuum theory is given; the latter is the object of our study henceforth. The reader who is not interested in the lattice may skip the following.

In this paper, we restrict our consideration to the calculation of the RG flow in the subspace spanned by the coupling constant associated with the energy operator E and the spin operator a. There is, however, another important RG eigenvalue apart from the ones associated with these two; the eigenvalue is responsible for the geometric property of the loops. In other words, the thermal and the magnetic eigenvalues are not sufficient to characterize the local shape of the loops even at a primitive level. To see this, let us first remind that, in the SAW (n = 0), the fractal dimension Dp of the loop is given by the thermal eigenvalue ye, and thus Dp = 1/v [36]; it is then given by ye = 2 - 2Ae = | in two dimensions. The generalization of this result to arbitrary n (|n| < 2) was discussed by relating the O(n) loops to the clusters in the percolation [37]. The fractal dimension for n = 0 should then be written as

Dp =-, (17)

with ap = 1 (in the percolation theory, ap is called the Fisher exponent, and v the correlation length exponent). Further, the fractal dimension Dp is expected to be the RG eigenvalue from the polarization operator (or, the two-leg operator): (0lat,0| = [s"sf ,aII(x)} with the vector indexes a = f, for which the scaling dimension can be obtained by the CG method (see Table 1). The fractal dimension of the O(n) loop in the dilute phase is then given by Dp = 2 - 2AaI = 1 + a-/2.5

5 This fractal dimension is also derived, in a rigorous way, from the SLE [24].

Diagrammatically, the polarization operator an(x), when inserted into a correlation function, creates two curves emanating from a point x, while the spin operator a(x) creates only one curve. The curves created by an (x) repel each other; they have different colors a = j and hence cannot connect by themselves. They need to be annihilated by the other operator an(y). By contrast, the insertion of the energy operator E(x) require that a point x be passed by some loop; the curve passing the point will connect each end to form the loop.

However, forming a loop becomes more and more difficult as n tends to zero; the loop segments repel each other in the SAW. In this sense, the role of the energy operator E approaches that of the polarization operator aII. In fact, at the SAW, these two operators has the same scaling dimension 2A = and this explain aF = 1 in (17). In the CFT context, by using the scaling dimensions obtained by the CG method, aII(x) can be identified with the <^,0 primary field [30]. Then, aF = 1 can be confirmed by the well-known equivalence between the primary field:

$r,s = 0m—r,m+1—s (0 < r < m, 0 < S < m + 1), (18)

in the mth minimal model with the central charge c = 1 — 6/m(m + 1). Indeed, in the SAW, which is the m = 2 minimal model, $1,3 (the energy operator E) should have the same dimension as the field 01,o (the polarization operator aII), since (1, 3) + (1, 0) = (2, 3).

There are, of course, many other composite fields constructed as products of the several local lattice observables. A quantitative discussion on such fields is given, which is based on the OPE and the correlation inequalities [38]. Their result suggests, in a generic situation, that the scaling dimension of a composite field exceeds the sum of those of the elementary fields due to the repulsion between them. This implies higher-order composite fields are more likely to be irrelevant in the RG. In our context, we mention the four-leg operator j(si ■ sj)2,aIV(x)} as an important example [28,30]. The insertion of the field aIV(x) into a correlation function corresponds to requiring two of the loop segments to overlap; in the particle picture, the trajectories should encounter at the point x. Again, the dimension of the field aIV(x) is obtained by the CG method, and identified with the primary field 02,0 in the CFT. Now, we see that it is relevant in the dense phase (k > 4) and irrelevant in the dilute phase (k < 4) as shown in Fig. 2. As used in Section 2.1, the latter observation is a key ingredient in understanding the universality of the pure O(n) model in the dilute phase. Further, at the level of coupled models, this is crucial in our expectation that (7) and (9) are, near the decoupled critical points, described by a same field theory.

2.3. Scaling limit of the disordered O(n ) models

We shall go into the continuum formulation of the disordered models by using the CFT description of the critical O(n) model. In Section 2.1, we have considered, on the lattice, three disordered models whose partition functions are given by (3), (8) and (10). As we have seen, after the disorder averaging, they have much in common when mapped to the coupled models. When the couplings between replicas are sufficiently small, one could, by looking at (7) for instance, guess the action of the coupled models in the continuum limit. But we shall take the other way. Instead, we first consider the continuum limit of the disordered model (10), and then take the disorder average in the continuum theory. The advantage is that we can use the OPE to deal with the composite operators.

In the following, we will use the relation:

n(1 + tsi sj) ~ exp jY.si ■ sj ^ exp —S* + m 1 d 2x E (x) ■ (19)

(i,j) L (i,j) J j

The first equivalence symbol means that these two pure model belong to the same universality class, as discussed in the paragraph above (8). In the right-hand side, we formally write the action of the O(n) CFT as S.t.. The coupling constant m is proportional to the reduced temperature, which is given by (t - tc)/tc, or (fi - fic)/fic.

Using the relation (19), it is natural to assume the continuum limit of the model (10) is described by

r M M / M

z[ {ri,n]M — YlV®{a)(x) exp -J2 S(a) + d2xm(x)lj2 E(a)(x)

~ —1 L a = 1 \a = 1

where S^ is the action of the O(n) CFT on the replica plane (a). Here, we formally write elementary fields (in the sense of [39]) as @(a) (x) and replace the tracing Tya in (10) to / Vi>a(x),

which denotes the path integration over the fluctuation of @(a)(x). The realization [m} of the local weights im on lattice sites in (10) are now replaced to its continuum counterpart, that is, a configuration of a scalar function m(x). Physically, this m(x) can be considered as a locally fluctuating reduced temperature. We assume m(x) respects a single distribution function P(m(x)) which is independent of the position x. Hereafter, we suppress the argument of the partition function

Z[[M},n]. Then the averaging operation is written as

ZM -f Vm(x)P {m(x)) z(21)

where Vm(x) denotes a path integral measure Wx dm(x). Let ¡k be a kth cumulant of the distribution function P(m(x)). In the important case of a Gaussian distribution P(m(x)) — exp[-(m(x) - m0)2/2g0], we have ¡i — m0, ¡¡2 — g0 and ¿¡t — 0 for k > 3. Then, by the averaging over the disorder, we obtain

ZM — '

a — 1

J f\V0a(x) exp -jP Sia) + j d2x(l1j2 E(a)(x) + Sj(x)

a—1 a—1 a—1

Si(x) — ¡2 £ E(a) (x)E(b) (x) + £(a) (x)£(b) (x)E(c) (x) + •••■ (23)

a,b a,b,c

In this formal expression, we should note that the nonlinear part Sj(x) contains the composite operators i.e. the products of the several fields at the same point on the same replica plane. We deal with them by using the fusion rules in the O(n) model. According to the identification E ^ 01,3 in (13), the fusion rule in the thermal sector reads

01,3 • 01,3 ~ 01,1 + 01,3 + (24)

or, equivalently,

E • E - J + E + E'. (25)

Here, E' — 015 is the next-leading energy operator. Since this operator is irrelevant as shown in Fig. 2, we neglect the E • E ^ E' part of the OPE (25). The identity field J — 011 contributes as a trivial shift of the free energy. The emergence of the energy operator E in the right-hand side of (25) is a characteristic of O(n) model with n — 1, which will be discussed in the paragraph below (35). Because of this contribution (E • E ^ E), qualitatively speaking, we should have

Fig. 2. The scaling dimensions 2A of the primary fields 01,3, 01,5 and 02,0 plotted as functions of the SLE parameter k = 4a-. The horizontal line at 2A = 1 shows the marginality of the most relevant couping E(a) (x)E(b (x).

a hierarchical flow of the coupling constants: • •• ^ f3 ^ f2 ^ f1. For example, the diagonal (a = b) part of f2J2 E(a\x)E(b\x) mixes with f1 J] E(a) and hence the flow f2 ^ f1 occurs.6 Thus, by redefining the coupling constants fk, we get

M „ / M

ZM = j f\V0a(x) exp - J2 sia) + f d2 x imof^ E (a)(x) + Si(x)

a=1 L a=1 \ a=1 /

Si(x) = goJ2 E(a\x)E(b)(x) + f3 E(a\x)E(b\x)E(c\x) + ..., (27)

a=b a=b,b=c,c=a

where we use the symbol m0, g0 and fk (k > 3) for the new coupling constants. Now, the effective interaction SI(x) is defined without composite operators. It should be noted that we restrict our considerations on the theory space which is replica symmetric; we assume that the coupling constant for, say E(a\x)E(b\x), is independent of the pair (a,b).

These coupling constants are determined by the distribution function P(m(x)) in (21), and are non-zero in general. The massless limit (m0 ^ 0) of the decoupled model (go = 0, fk = 0) remains obviously as a fixed point. We now change the distribution function P(m(x)), and gradually turn on the effective interaction SI in (27) while keeping m0 = 0. The large scale behavior is then dominated by the most relevant field: E(a)(x)E(b)(x). The dimension of this field is given by 4Ae , which is the twice the dimension of the energy operator.

As we can infer from (14), this field becomes marginal at n = 1 (a- = 4). Since 0(1) model is the Ising model, this serves as a simple check of the known marginality of the disorder in the random-bond Ising model [6,7]. Hence, we use the parametrization:

a- = 4 - €, (28)

to perform the epsilon expansion in the next section.

6 In the coupled model at M = 0, this flow may cause shift in the critical temperature.The magnitude of this effect is proportional to the number of diagonal elements M, the structure constant (35) and £2- However, since the disordered model corresponds to M ^ 0, we assume it is negligible.

3. Renormalization group flow in the one-loop calculation

In this section, we discuss the scale dependence of the theory (27) in the one-loop RG calculation. It will turn out that under certain circumstances, the disordered O(n) model has a pair of the ultraviolet (UV) and the infrared (IR) fixed points, as in the random-bond q-state Potts model [6,19]. We will investigate the properties of the IR fixed point by which the large scale behavior of the theory is dominated.

3.1. The epsilon expansion and procedure ofRG

We shall calculate the RG flow of the coupled model (27) perturbatively in the epsilon expansion using the parameter in (28). Generally, a flow is determined only by the most fundamental properties of a theory such as the symmetries and the dimension of the space. In order to grasp the topology of a theory space, it is often useful to think that these symmetries are dependent on some continuous parameters [6,19,39,40], or that the dimension itself is a continuous parameter [5]. Although the change is gradual when we look locally at a generic point as varying these parameters, the global structure of the flow, on the other hand, may change drastically at certain critical values of the parameters. A typical topological transition is caused by a branching of one merged fixed point into a pair of the UV fixed point and the IR fixed point. The idea of epsilon expansion provides us with a firm ground to perform a perturbation calculation in the non-trivial region after the branching.

In the parameter space, we can infer the location of the branching by the presence of a marginal field: the hallmark of the merged fixed points. In the renowned example of the Z2-invariant scalar field theory [5], the most relevant interaction 04 becomes marginal at the dimension d — 4. In d — 4 - e, the Gaussian fixed point and the Wilson-Fisher fixed point emerge; we can do a solid perturbative calculation by setting an appropriate coordinate [40] in which the coupling constant remains O(e).

In the disordered O(n) model, as mentioned in the previous section, the most relevant part of the effective interaction SJ in (27) is the term E(a)(x)E(b)(x), which is marginal at n — 1 and is relevant for n < 1 (see Fig. 2). The next-leading relevant term E(a) (x)E(b) (x)E(c) (x) is irrelevant for n > 0 and is marginal at n — 0 (SAW).7 We restrict our analysis on n > 0, keep the most relevant part

and discard the other higher-order terms in (27).

The implementation of the RG is as follows. We introduce a cut-off length scale r, which serves as an effective lattice spacing, and determine a renormalized coupling g(r) by integrating out the short-distance degrees of freedom:

7 It may be noted that the role of this field is analogous to that of the 06 term in the Z2-invariant scalar field theory [44]. This term is irrelevant if d > 3 and becomes marginal at d — 3. In 3 < d < 4, we have only the two fixed points mentioned above.

r-&"g(rMSI(x)SI(œ)\

= g0(S,<x№<»»o +1 f SyfrMMMl

\y-x\<r

+ 3- f d 2yd 2z (SI<x>SI<y>SI<z>SI<^>)0 + •••, (30)

\y-x\<r, \z-x\<r

where the insertions of the interaction fields SI are restricted onto a disk of radius r, and the coupling is measured by projecting perturbative contributions on SI<<x>>. The symbol (■ ■ -}0 represents the unperturbed correlation function. A trivial scaling factor r-8e is introduced in order to make the renormalized coupling dimensionless. At this stage in the RG, as usual, a finite redefinition of the coupling is possible; physical quantities (scaling dimensions, for instance) is invariant under a finite coordinate reparametrization of the theory space [40]. To fix this ambiguity, we choose a minimal subtraction scheme. Then we get the renormalized coupling in the form:

g<r> = r8e (go + G2<e, r>g0 + G3 <e, rg + ■■■), (31)

where G2<e, r> and G3<e, r> have only poles in e and no regular part.8 The beta function is then obtained by differentiating the renormalized coupling (31) with respect to <lnr>. Using the first two terms in the fusion rule (25), we make contractions and list, in Fig. 3, the possible diagrams for the beta function.

3.2. One-loop beta function, a line of random fixed points and a strong coupling region

We here calculate the one-loop beta function. The process represented by the diagram in Fig. 3(a) involves three layers, and the number of the ways for this contraction is 4<M - 2>. Then we have

G2,i(r,e) = 4(M - 2) J d2y [E(x)E(y)lo

\ y - x \ <r

= 4n<M - 2>(8-)' (32)

where we have used (£<x>E<y» =\x - y\-AAe and 4Ag = 2 - 8e.

Besides this, up to the one-loop order, there is the other contribution represented by Fig. 3(b) which is obtained by using twice the sub-leading part of the OPE (E ■ E ^ E):

1 i ,2..i Cee<«-> V_„_r^:(,2^(r"

G2,2(r,t) = 1.2/ = 2, [Ç^)]2^). «33,

\y-x\<r

Here, C^g (a—) is the structure constant of the CFT which appear in the OPE:

CEE i . CEE c . CEE

(zz )2Ae (zz)Ae (Zz )2AE

CI cE C E E (0, 0) ■ E (z, z) = , -EA I + E +-„EE ,, E + ■■■, (34)

8 More precisely, we keep the combination r8e/8e = 1/8e + lnr + O<e> and omit the other parts.

Fig. 3. The diagrams for the beta function. A gray disk and a nearby pair of the disks represent an energy operator E(a) and the interaction Si = J]a=b E(a)E(b), respectively. An arrow corresponds to the sub-leading part of the OPE in (34) and a shaded-square represents four-point function. The external lines to the point infinity in (30) are amputated. The second order: (a) G2 0 and (b) G2 2(r,e); the third order: (c) G3 i(r, e), (d) G3 2(r,e), (e) G3 3(r,e) and (f) G3 , 4(r,e).

where we write just the primary operators to represent their conformal families including the descendants. The structure constants are, in general, determined by the requirement that the operator algebra be associative. In practice, the crossing symmetry of the four-point function is strong enough to fix them [9]; the actual values are obtained by using either the connection matrix of the hypergeometric function [41], or the vertex operator representation of the four-point function [42]. This reads,

r£ , , ! o ,3 [-Y3(P)Y(2 - 3p)]2

C££(fi) = -4(1 - 2P) —2,0 Wo—-rr-' (35)

Y2(2p)Y(3 - 4p)

where we have used temporary notations p = a- and y(x) = T(x)/T(1 - x).

The structure constant C^g (a-) has information about the important selection rules in the pure O(n) model. First, observe that the square of the structure constant (35) has a zero at a- = | (the Ising model, e = 0) and a pole at a- = 3 (SAW, e = -¡i;). The former zero is due to the self-duality of the Ising model [43]. In the vicinity of the critical point, the duality transformation changes the sign of the reduced temperature t. Since the energy operator E couples to t, if the model is invariant under the duality, it should be odd under the duality. Then E is not allowed to appear in the right-hand side of the fusion rule (25).

The pole at a- = 3 emerges because of the strong repulsion between the loop segments in the limit n ^ 0. Since the normalization of operators are fixed such that C^ = 1, what really happens is the divergence of the ratio C/C^£. When the two loop-segments approach, they will either (i) form a complete loop (to contribute the free energy) or (ii) form together a joined loop segment. Since, in the limit n ^ 0 the process (i) is strongly suppressed by the repulsion between the loop segments, the ratio, indeed, diverges.

Taking these interpretations into the account, more intuitive representations of the second-order contributions are possible. The term G2, 1(r, e) is represented by the diagram in which two open segments lie on two layers and one closed loop on another layer (Fig. 4(a)), while the term G2 2(r e) is represented by the diagram in which two open segments lie on two layers (Fig. 4(b)).

Fig. 4. The second order diagrams; see also Fig. 3. (a) G;, 1 (r,e): the three-layer process with a closed loop (b) G;,2(r,e): the two-layer process.

Now we sum up (32) and (33) to get

g(r) = r8e jg0 + 2^[2(M — 2) + [cEe(a—)]2] (^^ + Og)

By introducing g = r 8eg0, we solve (36) with respect to g :

g = g(r) - ie \-2(M - 2) + [Cee {a2-)f] g(r)2 + o {g(r)3).

Using this, the beta function up to the one-loop order is then obtained as9

P(g) =

= 8eg(r) + 2(M — 2) + \CESS (a—)]2}g(r)2 + O(g(r)3).

dln(r)

This beta function of the coupled model has a non-trivial zero when the coefficient of g(r)2 is negative. Since the disordered model corresponds to the limit M ^ 0, there is a competition between the negative term from G24 (r, e) and the positive one from G22(r, e). For the term from G2,2, we use the exact expression (35) of the structure constant. Then we have two regions in 0 <n < 1 divided by a threshold n*, where the RG flows are qualitatively distinct from each other (Fig. 5). In the first region (0 < n < n*), the domination of the term G2,2 leads to a monotonic increasing beta function; the theory flows to strong coupling. In the second region (n* <n < 1), on the other hand, the theory flows to an IR fixed point. Though the perturbative calculation is reliable only in the region g < O(e), from a formal point of view we have the line of the IR fixed points which extends from (1, 0) to (n*, ro) in the (n, g)-plane. The threshold obtained from the beta function (38) is given by n* = 0.233636 • • •, which corresponds to [C^(a-)]2 = 4 or, equivalently, e = 12 • 0.796164 • ••. It should be noted, however, that the divergence of the coupling g at n = n* is an artifact of the one-loop calculation; if there is a positive g3 term,10 the line of the IR fixed points terminates at (nc,gc) = (n* ,gfc*) with a finite coupling g'c*.

The strong coupling phase of the disordered model (M ^ 0) near n ^ 0 is formed by the dominance of the two-layer process G2,2(r, e) over the three-layer process with a closed loop G2,1 (r, e). The qualitative behavior of the absolute ratio R(n) = |G2,2/G2,11 is independent of M except a special case with M = 2, where the three-layer process is prohibited. Diagrammat-ically, a large R(n) implies that the ratio of (the number of inter-layer hopping) to (the average

9 A similar form of beta function appears in e = d — 2 expansion of the random-bond Ising model [32].

10 Although the e dependence of the third order coefficient is not calculated in this paper, we anticipate a positive g3 term from a continuation of the result at e ^ in Section 4.

Fig. 5. A schematic flow diagram of the couping constant g with respect to n of the disordered O(n) model suggested by the one-loop beta function (38). The bold curve represents the positions of the IR fixed points.

number of complete loops per layer) is also large. Near n ~ 0 region, the particles, which are destined to form a whole connected diagram, favor to sew layers together rather than to stay on one layer and to walk around avoiding their traces.

We shall henceforth proceed to the two-loop calculation to investigate the IR fixed points in the region near n = 1 (e ^ y^), where regarding the structure constant Cgg in (35) as O(e) is justifiable. In this region, the calculation turns out to be parallel to that in the study of the random-bond Potts model by Dotsenko et al. [19]. At one-loop level, Cgg = O(e) implies G22 = O(e); in the minimal subtraction scheme, we drop this term. The beta function is then

P(g) = = 8eg + 4n(M - 2)g2 + O(g3). (39)

dln(r) v 7

We left the two-loop calculation of the intermediate region where Cg g ~ 1 as a future problem. 4. Scaling dimensions up to the two-loop calculation

4.1. The beta function at two-loop order

There are four types of third order diagrams for the beta function as listed in Fig. 3(c), (d), (e) and (f). As the diagram G3 3 (r, e) in Fig. 3(e) has extra e2 factor from the structure constants and the diagram G34(r, e) in Fig. 3(f) has no pole in e, we omit these terms in the following. The G3,1 (r, e) in Fig. 3(c) can be calculated as

G3,1 (r, e) = - • 12(M - 2)(M - 3) j d2yd2z(£(x)E(y)}0(£(y)E(z))0

|y—x\<r,\z-x |<r

-,2 , i-2+8e i_ _ n-2+8e

= 8n(M - 2)(M - 3) J dy y-1+16e j d2z |z|-2+8e |z - 1|

y<r |z|< TyT

= 16n2(M - 2)(M - 3)(8e)2 + O(e^, (40)

where we have used the asymptotic behavior of the integral for 1 ^ R:

d2z |z|-2+8e |z - 1|-2+8e = 4n — + O(R-2+16e). (41)

It should be emphasized that we get no simple pole in (40).

Next, we have G3,2(r, e) in Fig. 3(d) containing four-point functions:

G3,2(r,t) = 3 ' 24(M - 2) J d2yd2z {£(x)E(y)E(z)E(w))0{£(y)E(z))0

^-x^r, |z-x|<r

-1+16^

= 8n(M - 2) J dyy-1+16e1{r/\y\,e), (43)

where we define

I(R,e) = j d2z[S(0)E(1)E(z)£(<x))n[E(1)E(z})0. (44)

Actually, instead of I(R, e ), we shall calculate I(œ,e ) by analytic continuation in Appendix B, and see that I(œ,e) itself has no poles in e. Although, in the appendix, we prove this fact by applying the contiguity relation between generalized hypergeometric series, the fact can also be interpreted as a result of a cancellation between two poles with opposite sign, as explained in the following.11

In the limit \z\ ^ œ, the integrand in (44) behaves as \z\-4^£ (4Ag = 2 - 8e), since the four-point function, using the identity operator I in (34) as the leading intermediate channel, decouples into two two-point functions. Hence, I(œ,e) has the contributions from both regions \z\ < R and \z\ > R, each of which contains a pole +2nR8e/8e and -2nR8e/8e, respectively; they are canceled out each other. Observing that I(R, e) does not have the latter pole, and estimating the corrections, we obtain

I(R, e) = 2n-+1(œ,e) + O(R-1+8e). (45)

As the first term corresponds to the decoupling limit z ^ x or y ^ x of the four-point function in (42), it leads to the same contribution as (40), only if we replace (M - 2) in (43) by (M -2)(M - 3). Now, using the vertex operator representation, we write I(œ,e) in (45) as

2_ j2 ,2.

I(œ,e) = rfj d2zd2ud2v [Vai3(1)VoF3(z)

■{Vai3 (0)Vai3 (l)Vm (z)VaV3 (œ)Va- (u)Va- (v)} = N j d2zd2ud2v \z\4ai3a13\1 - z\8ai3aT3|(z - u)(z - v)\4ma-

. \uv(u - 1)(v - 1)|4ai3a-\u - v\4a-, (46)

11 A similar cancellation of poles has already been observed in the random-bond Potts model [19].

where N is the normalization factor with the four-point function given in (A.7). Substituting (45) and the result (B.31) for I(ro,e) into (43), we obtain

G3,2(r, e) = 16n2(M - 2)

+ O (e

We get, by collecting the third order terms (40) and (47), the renormalized coupling constant in (31) as

g(r) = r8e | g0 + 4n (Mn 2) r8eg2 + 16n2

(M - 2)2 (M - 2)

r 16eg0

+ O(g4).

Then (48) is solved with respect to g = r8eg0 as

(M - 2) 2 2

g = g(r) - 4n( )g(r)2 + 16n2 8e

(M - 2)2 (M - 2)

g(r)3 + O (g(r)4). (49)

Consequently, we obtain the RG beta function

P(g) = ^^ = 8eg + 4n(M - 2)g2 - 16n2(M - 2)g3 + O(g4). d ln r y '

As is expected from the known equivalence of the random-bond Ising model to the random-mass fermion model and the Gross-Neveu model [6,7], at the Ising point e = 0 (n = 1), the expression (50) reduces to the two-loop beta function for the Gross-Neveu model [46]. Solving P(gc) = 0 for the disordered model (M ^ 0), we obtain the coupling constant at the random fixed point as

gc = - + — + O(e3).

4.2. Correction to the scaling dimensions

In our perturbation theory, the two-point correlation function between fields O(0) and O(ro) is calculated as

p(0)O(ro)) = O(0)O(ro))0 + g0 I d2x {SI(y)O(0)O(ro)]0

d 2y d 2z Si (y)Si (z)O(0)O(ro) 0 +

|yl<r, |z|<r

Again, as in (30), we restrict the insertion of the interaction SI around the field O(0) within the disk of radius r. From this, we can determine a renormalization constant Zq for the field O perturbatively as

O(0)O(ro)) = Zq O(0)O(ro))0. (53)

The two-point functions in the theories with different cut-offs r and sr are related as

Fig. 6. The contributions for the scaling dimensions. The diagrams (a), (b), (c) and (d) are for Z£, while (e) Sx(r, e) and (f) S3(r,e) are for Za. The white circle represents the spin operator o(a) ; for symbols, see also the caption in Fig. 3.

=ZzOgr)s—AAo o(°)o(R)ig(r)

r g(sr)

YO (g)

s—4Ao l0(0)0(R)

r,g(r)'

L g(r)

where, in the second line, we have introduced the anomalous dimension

Y (g) =

dZo (g) d ln r

Now consider the large s behavior of (54) and let the upper limit g(rs) of the integral tend to gc of the random fixed point; the beta function j(g) tends to zero, while the anomalous dimensions Yq (g), as we will see, remain finite both for the energy operator E and the spin operator a. This means the integral is dominated by the contribution from the region g & gc, and we obtain

2( Ao — AV0Y) = —yo (gc),

where we mean, by 2A|R and 2AUV, the scaling dimension of the field O at the IR (the random) fixed point and the UV fixed point, respectively.

4.3. Scaling dimension of the energy operator

The first and second order diagrams for the renormalization constant of the energy operator Z£ are listed in Fig. 6(a), (b), (c) and (d). As the integral in the diagram in Fig. 6(c) does not have pole in e, we omit this term from the calculation. Since the integrals are same as those for the two-loop beta function, by adapting combinatorial factors, we have

Ze = 1 + g0

2(M — 1) 2(M — 2)

G2,1 +

8(M — 1)(M — 2) 2(M — 2)(M — 3)

G3,1 +

4(M — 1) 4(M — 2)

(M — 1) 2 2 1 + 4^g + 8^ 2g2

(M — 1)(2M — 3) (M — 1)

where the results (32), (40) and (47) are used in the second line. Again, by using the expression for g in (49), we get

Y£(g) = ^^ = 4n(M - 1)g(r) - 8n2(M - 1)g(r)2 + O(g(r)3). (58)

dlnr v '

Consequently, by using (51) in (56), we obtain

2(4R - ¿UV) = -YE(gc) = 4e + 8e2 + O(e3). (59)

4.4. Scaling dimension of the spin operator

We shall here calculate the renormalization constant for spin operator Za up to the third order in g0, which gives the lowest order correction to the scaling dimension 2Aa at the IR fixed point. According to the operator algebra (see the diagrams in Fig. 6), there are no contribution at the first order, and one contribution at the second order S2(r, e) and the other one at the third order

S3(r,c):

Za = 1 + S2 (r,e) + S3(r,e) + O(g4). (60)

The second order diagram in Fig. 6(e) is given by

S2(r,e) = g2 ■ 4(M - 1) J d2yd2z{a(x)£(y)£(z)a(^))0{£(y)E(z))0, (61)

\y-x\, |z-x|<r

while the third order diagram in Fig. 6(f) is given by S3(r,e) = g3 ■ 24(M - 1)(M - 2)

d 2yd 2zd2wla(x)£ (y)E (z)a(œ)) J £ (y)E (w)\J E (w)E (z)) 0.

\y-x\<r, |z-x|<r

\W-X\<r (62)

In evaluating the integrals (61) and (62), we keep only the terms with the lowest powers in e. For this reason, it turns out that we can add certain extra regions to these integrals. First, by adding the region \z - x\ > r to (61), we get

S2(r,€) = 4ng2(M - 1) y dyy-1+1&K2(œ,e),

with K2(œ,e) defined as

K2(œ,e) = j d2z(a(O)E(1)E(z)a(œ))0{E(1)E(z))Q. (64)

Second, if we add the region \w - x| > r and |z - x| > r to (62), we see that the third order contribution S3(r, e) has the similar structure as the second order one S2(r, e) in (61). In fact, under a trivial change of variables, we obtain the factorized form:

S3(r,e) = 8ng0(M - 1)(M - 2) J dyy-1+24^d2w |w(w - 1)|-4^£K3(œ,e), (65)

with the definition

K3(œ,e) = j d2z |1 — z|2—A[o(0)£(1)£(z)a(œ))0.

Then, we write the integrals (64) and (66) using the vertex operators: K2(ro,e) = nJ d2zd2ud2v |1 - z|-A

• (%-Tp(0)Va13 (1)Va13 (z)Vap-hp(ro)Va- (u)Va- (v))

= nJ d2zd2ud2v |z|4a^a1311 - z|4a13a13-4A£ |uv|4ap-1pa-

• |u - v|4a22|(- - u)(1 - v)(z - u)(z - v) |4a13a-, (67)

K3(œ,e) = N J d2zd2ud2v |z|4ap—^p»1311 — z|4a13a13+2—A iuvi4aP—1,Pa

■ |u — v|4a2 |(1 — u)(1 — v)(z — u)(z — v) |4a13a— (68)

and give the results in Appendix C.

Now we obtain, by collecting the contributions (63) and (65), the renormalization constant for the spin operator as

ln Zff = 4ng2(M — 1)—K2(œ,e) 16e

+ 8ng0(M — 1)(M — 2)

r24e n Tfe 2e

K3(œ,e) + O (g4), (69)

where we have used (41) to obtain the third order term. Differentiating this with respect to (ln r) and using the result (C.10) for the integrals K2(ro,e) and K3(ro,e), we obtain the anomalous dimension for the spin operator as

Ya(g) = 4n(M - 1)jg - (M - 2) 2Lg2} 2Nne\U2 + W2 - 8YZne]

+ 8n(M - 1)(M - 2)ng3 • 2Nne[U2 + W2 - UYZne] + O(g4) = 8Nn2(M - 1){eg2[U2 + W2 - 8YZ] - 4g3(M - 2)YZ} + O(g4), (70)

where the expression for the bare coupling constant (49) is used. For the definition of the constants N, U, W, Y and Z, see (A.7) and (C.5). The limit M ^ 0 for the disordered O(n) model yields,

Ya(g) = -8Nn 2eg(r)2[U2 + W2] + O(g(r)4). (71)

Now, from (56), we obtain

2(A? - AUV) = -Ya(gc) T4( 1) = 8 '-Y e3 + O(e4)

K2 (sin n ) 3 ( 4) = 128-e 3 + O (e4), (72)

where, in the last line, we have used the complete elliptic integral of the first kind defined by

d0 - k2 sin2 0

K(k) = I =, (73)

for the purpose of the comparison with the result in the random-bond Potts model [19]; this will be discussed in Section 6.

5. Effective central charge

The central charge, in general, characterize a system by counting the number of the critical fluctuations in it. If the replica method is used, however, the central charge becomes rather trivial; we are always left with the vanishing central charge c = 0 because of the formal limit M ^ 0 in the end. Nevertheless, we can define a so-called effective central charge characterizing a random fixed point as follows [18]. Consider the M layers of the coupled model and the relation

cir(M) = Mcuv + Ac(M), (74)

where cuv and cIR(M) are the central charge of the pure system and that of the non-trivial fixed point, respectively. The effective central charge is then defined as

dciR(M)

ceff =

. (75)

In order to obtain the cIR(M), it is useful to recall the definition of the C-function and the differential equation satisfied by it [10,32]. To this end, we consider the vicinity of some critical theory S* in the theory space spanned by the coupling constants {g'}:

S = S* - J d2xg'&i(x), (76)

and then specialize to our disordered O(n) model. The response of the action under the transformation z' ^ z' + a' (z) can be expressed by the stress tensor Tlv as

SS =-2-j d2rT'J,vd'lav. (77)

As usual, a particular component of the stress tensor Tzz = 4(Tn - T22 - 2iT\2) and the trace of it 4Tzz = Tn + T22 are denoted as T and &, respectively. As the RG flow dg' = j' dt occurs under the dilatation z' ^ (1 + dt)z', definitions (76) and (77) leads to the relation:

&(x) = 2nj'$i(x). (78)

From T and &, one can consider three-types of the two-point functions:

{T(z,~z)T(0,0)) = F(T)/z4, (79)

[T(z,z)&(0, 0)) = H(r)/z3z, (80)

&(z,z)&(0, 0)) = G(r)/z2z2, (81)

which are measured at the scale t = ln(zz). The C-function is then defined as 3

C = 2F - H -- G. (82)

= -(2n)2 G'JP'P'. (85)

The conservation of the stress tensor = 0 yields dzT(z, z) + 1 dz&(z, z) = 0, and one can easily show

—C = -- G. (83)

After fixing the reference scale t to t = 0, the C-function depends only on the coupling constants gi. We introduce the Zamolodchikov metric on the theory space as

Gij = (zz)2(@i(z,z)@j(0,0))|zz=1. (84)

From (78) and (83), we have

1 i d 3

-P'—C = —(

r dgi 4

For unitary theories, the positivity condition implies that the metric should be positive definite. The C-function is stable at the point with P' = 0 i.e. a fixed point and take the same value as the central charge as we can see from the definitions (79)—(82).

In our case, the perturbing field is quadratic in energy operator as in (29):

0 = Si = J2 E {a)(x)E {b)(x), (86)

and hence the metric (84) is

G = 2M(M - 1). (87)

This metric becomes negative in the limit M ^ 0, thereby violating the c-theorem. Note also that our normalization for the energy operator E is such that Cr££ = 1. Substituting the beta function (39) into (85), we obtain

(8e)3 ( )

AC(M) = -6n2 • 2M(M - 1)--—-t + O(e4)

( ) ( ) 6[4n(M - 2)]^ 1 '

3 M(M - 1) ( 4)

= -64e3 -Mh2r2 + O ^. (88)

From the definition of the effective central charge (75), we have

cff - cUfV = 16e3 + O(e4). (89)

Since we have the IR fixed points in e > 0, this always increase under the RG flow as expected.

6. Discussion and conclusions

Before concluding this paper, we discuss the relation to the known results and then comment on possible future directions.

The RG flow in our disordered O(n) model near the disordered Ising point, which is inferred from the results (50), (58) and (70), is qualitatively similar to that in the random-bond q-state Potts model [6,19]. The crossover to an IR fixed point occurs only when the most relevant interaction, which is quadratic in energy operators as in (86), becomes relevant; the region correspond to n < 1 for the disordered O(n) model and q > 2 for the random-bond q-state Potts model. This can be understood by recalling how the two families of the pure models, namely, the O(n) models and the q-state Potts models coincide at the Ising point (n = 1, q = 2) [23,43].

Fig. 7. The scaling dimension of the energy operators for the O(n) model (2 A13) and for the q-state Potts model (2^2,1) plotted as functions of the SLE parameter k = \a2_. The horizontal line at 2A^ = 1 shows the marginality of the disorder-induced coupling. The arrows indicate the directions of the expansion parameters e and ep.

The energy operator in the O(n) model is identified with the primary field 013 as we have stated in (13), while the energy operator in the Potts model is identified with the field 02,1. These primary fields are, in a general minimal CFT, different objects as one expects from the fact that a four-point function of primary fields §rs is determined from an ordinary differential equation of order rs. Accordingly, in the vertex operator representation of the four-point functions, we should include two of the screening operators V— in the O(n) model and one of the other screening operator V+ in the Potts model, respectively. However, the field 013 reduces to the other field 02,1 at the Ising point (the m = 3 minimal model) because of the equivalence (18). At the level of the scaling dimensions, the reduction can be summarized in Fig. 7, in which the line of 2A 1,3 and the curve of 2A21 intersect at the Ising point (k = 3).

Up to one-loop order, when the deviation parameter e from the Ising point defined in (28) is small and neglecting the structure constant (35) in the O(n) model can be justified, the beta function is determined essentially from the scaling dimension of the energy operator as in (39); thus similarity with the random-bond q-state Potts model is natural from Fig. 7. By contrast, the reduction of the two-loop beta function (50), which involves four-point functions, to that of the q = 2 random-bond Potts model in the limit n ^ 1 serves as a non-trivial check of the vertex operator representations.

Remarkably, two lines of the IR fixed points lie the opposite sides of the Ising point; if we introduce a parameter

eK = 3 - k, (90)

the random fixed points of the disordered O(n) models and those of the random-bond q-state Potts models exist in the region €k > 0 and €k < 0, respectively. Another natural parameter ep for the Potts models can be taken as a+ = 3 — eP, where the IR fixed points emerges in the region eP > 0.12 The definition (28), the relation (16) and a+a— = 1 from (A.3) lead to relations:

e = 4 eK = — 9 eP — g eP + O (eP). (91)

12 It should be noted that yet another parameter e defined as a+ = 4 + e is used in [19].

Then the results for the scaling dimension of the energy operator in the random fixed point of the disordered O(n) model (59) and that of the random-bond q-state Potts models [6,19] can be summarized in one expression as

TR 1 2 , h + ^ + O(4) (eK> 0; O(n)),

2Af = 1 + -(1 - 2Ag) + O e3 = 2 2 3 (92)

E 2 [1 + f el + O(e3K) (eK< 0; Potts),

where both results are measured from the Tsing value 2Ag = 1. Naturally, the scaling dimensions at these TR fixed points imply faster decays in the correlation functions: Ag < A® as one expects with a general RG flow [45]. The ratio between the coefficients in e^ is which is the square of the ratio of the derivative of the line 2 A 1,3 at the Tsing point to that of the curve 2A2,i in Fig. 7.

Contrastingly, the coefficients in the spin scaling dimensions are transcendental numbers. Using the relation (91), our result (72) for the disordered O(n) model is written as

2(- KV0(„) = 2^^^ + °(C k > 0). (93)

Then we quote, from Ref. [19], the result for the random-bond q-state Potts model:

'KR - 4uv)potts=27 r(-1 ^\)2$+°& «p> 0)

(-Ck)3 + O(e4) (€K< 0), (94)

where, in the second line, we have used the elliptic integral defined in (73). As a first observation, notice that both coefficients are consisting of the numbers K(cos0) known as "singular values" of the elliptic integrals for which the modular ratios K(cos 0)/K(sin 0) belong to quadratic irrational numbers. The ratios here are v^I and V3 for the disordered O(n) model and the random-bond q-state Potts model, respectively.13 These modular ratios are exceptional in that the modular angles 0 are commensurate to n, namely, 0 = n/4 and 0 = n/12.

The coefficient in (93) and that in (94) characterize the universality classes of the disordered models as well as of the corresponding coupled two-dimensional CFT's in the zero-layer limits. Now, it is noteworthy that they are simply related to the quantities originating from the specific structures of three-dimensional lattices; the coefficient in (93) for the disordered O(n) model and that in (94) for the random-bond q-state Potts model correspond to the Watson's triple integrals [53,54] which represent the special values of Green's functions on the body centered cubic (bcc) lattice and on the face centered cubic (fcc) lattice, respectively. For our disordered O(n) model, to be concrete, the coefficient in (93) can be written as

, n n n ,

K2(sin f) 1 1 r r f dudvdw \

2 ( 3 4) =H — 1- , (95)

n3 2n \ n3J J J 1 - cos u cos v cos w 1

x 0 0 0 '

where the quantity in the parenthesis is the Watson integral for the bcc lattice. Tt would be interesting to see whether this coefficient can be derived, asymptotically at large scales, by a

13 The two singular values have number theoretic origins and belong to the sequence of the values derived by Chowla-Selberg formula [52]; the relevant part of the formula is obtained from the functional equations of the Epstein's zeta functions related to the quadratic fields Q(V —1) and Q(V—3).

combinatoric analysis of coupled loop model on a lattice such as the one defined from the partition function (7).

These remarks are also useful to give representations for the two coefficients in terms of rational integrals in order to see they are "periods" proposed in Ref. [50]. Looking back the arguments, both coefficients are obtained not as products but as ratios between the complex Selberg integrals [48] and the non-symmetric extensions of them such as the one obtained in Appendix D, and thus it is not a priory clear that they are periods even if each of these Selberg-type integrals independently turns out to be a period. But if we start from the Watson integrals, they can be transformed into rational integrals [53], and hence the coefficients in (93) and that in (94), characterizing the two universality classes interconnected at the disordered Ising CFT, belong to, in fact, periods.14

Now, we list future directions in the following:

(i) The fractal dimensions and SLE with the replica symmetry. We have seen that there is a line of IR fixed points for n* < n < 1. An immediate issue to be discussed is the fractal dimension DF of the loops in the IR fixed points. One may figure out the corrections for the Fisher exponent in (17) by perturbative calculation like the one performed in this paper. Related result has recently appeared for the random-bond q-state Potts model [62]. Further, the effective scale invariance on the line of the IR fixed points suggests the possibility of constructing one-parameter family of new SLE's. In order to describe the shape of the disordered loops, stochastic motion over the effective space enhanced by the replica permutation symmetry may be considered in analogy with the SLE driven by a composition of the stochastic motion in the real space and that in the internal space such as an SU(2) [25] or a ZN [26].

(ii) The strong coupling phase. We have seen that the coupled O(n) model has moved to strongly-coupled theory by the two-layer process in Fig. 4(b) in the region 0 < n < n*. The physical picture of the strong coupling region is important for the polymers in disordered environment [59]. Especially for this region, consideration on the theory space with the replica symmetry breaking (RSB) is desirable. Although the RSB-RG flow has been proposed in the random-bond q-state Potts model [60], numerical studies there supports the result of the replica symmetric fixed point rather than that of the RSB fixed point. We anticipate our model has more reason to favor the RSB situation due to formations of replica bound states boosted by the presence of the two-layer process. We should add that a large class of disordered models which includes strongly-coupled layered O(n) models has recently been discussed from the view point of the AdS/CFT duality [61]. Actually, the focus of their analysis is on large-n, while our strong coupling region lies at n « 0; although a direct application seems to be difficult, this direction may be valuable in order to grasp the nature of strong coupling phases induced by disorder.

(iii) Non-local observables. We have studied the scaling dimensions of the fields corresponding to the local segments of loops. In the pure O(n) model, some results are known on non-local observables such as the area of loops [57] or the linking numbers around multiple points [58]. A challenging problem is the study of the off-diagonal pair-correlation between the global shapes of loops in the disordered model. The whole shape of a loop itself is, of course, difficult to handle. But extracting an essential part of such a correlation between non-local objects in a simple disordered system may possibly provide us with the basis of other important problems. In chaotic systems, for instance, the off-diagonal correlation between periodic orbits, which are analogous

14 More precisely, as suggested in the expression (95), the coefficient in (93) for the disordered O(n) model belongs to the 1/n-extended period ring P[1/n] defined also in Ref. [50].

to the disordered loops studied in this paper, is already a central issue in understanding the quantum level statistics [55,56]. It would be interesting if the possible relation between the disordered loops and periodic orbits in a chaotic system could be clarified.

In conclusion, we have studied the disordered O(n) models in two dimensions. We formulated the inhomogeneous loop model on a lattice. After adopting the replica method in order to take the disorder average, the models were mapped to the coupled layers of the homogeneous loop models. Some of the possible differences in the lattice formulation of the disordered models were argued to be irrelevant using the properties of the pure model in the dilute phase. Then we discussed the continuum limit of the disordered model assuming the identification between the energy operator E and the primary field 01,3 in the vicinity of the pure critical O(n) model. We considered the RG flow in the replica symmetric theory space. Since the most relevant interaction E(a)E(b) becomes marginal at the disordered Ising model n = 1, we used the epsilon expansion methods near n = 1 to perform a perturbative calculation. At one-loop level, the RG flow suggests that there exist a critical value n*, the strong disorder region in 0 < n < n* and a line of the random fixed points in n* < n < 1. These distinct bahaviors are controlled by the structure constant C££ which encodes the selection rules in the pure O(n) model. We interpreted the OPE intuitively and explained the qualitative picture of the strongly-coupled phase. Then we perform the two-loop calculation near n = 1, where C££ is O(e). The beta function is equivalent to that of the Gross-Neveu model and thus checked the previously known result in the random-bond Ising model. The corrections to the scaling dimensions of the energy operator and the spin operator were calculated up to two-loop order. Finally, we calculated the effective central charge and see this increases under the RG flow.

Acknowledgements

The author would like to thank S. Hikami for his continuous encouragement, stimulating discussion on replica methods, and reading of the manuscript. The author has greatly benefited from enlightening discussions with S. Ryu, T. Takebe and D. Bernard about disordered systems and conformal invariance. It is pleasure to thank A. Kuniba, K. Fukushima, T. & H. Shimada, H. Seki, Y. Kondo, H. Nakanishi and K. Nomura for various comments and encouragement. The author would also like to thank T. Yoshimoto, T. Kimura, H. Katsura and K. Ino for useful discussions and their interest in the work.

Appendix A. The vertex operators in the critical Liouville field theory

The correlation functions in the O(n) model can be described by the vertex operators in the critical Liouville theory.15 The basic idea is to deform the Gaussian free field theory by putting the background charge (—2a0) at the infinity. This is accomplished by coupling the scalar field P to the scalar curvature R. The action of the theory is formally given by

J d2x [(Vp)2 + 4ia0Ry + k+eia+v + X—eia— P].

15 Although it is often referred as the "Coulomb gas representation" [19,43], we call it the "vertex operator representation" in the critical Liouville theory [29] in distinction with the "Coulomb gas method" [28].

The corresponding central charge is determined from the OPE of the stress-energy tensor and turns out to be

c = 1 — _4a_. (A.2)

The correlation function of the vertex operators Vak = e'akp in this theory is non-zero only if the overall charge neutrality ^k ak = 2a0 is satisfied. The screening vertex operators V± = eia±p are incorporated into the correlation functions in order to ensure the neutrality. The charges of the screening operators are chosen such that they are marginal, because otherwise the conformal invariance of the Gaussian theory would be violated. This condition determine the charges:

a± = a0 + 1. (A.3)

The charge a— in the O(n) model and in the mth minimal model are given by

n = —_cos(п/a2_) and a2_ =—m—, (A.4)

respectively. Tn this paper, we fix our convention concerning the sign of the charges such that a0 > 0 and a0 < 0 describe the dilute (critical) and the dense (low-temperature) phase of O(n) model, respectively. The charge ar,s and its conjugate a^ is defined as a linear combination of a+ and a—:

ar,s = 2 (1 — r)a+ + 2 (1 — s)a—,

ars = 2 (1 + r)a+ + 2 (1 + s)a—= 2a0 — av. (A.5)

The scaling dimensions of the primary operators 0r,s is given by

2Ar,s = —2ar,sar,s = 1 [(ra+ + sa—)2 — (a+ + a—)2]. (A.6)

Tn the two-loop calculation in Section 4, we use the vertex operator representations of the four-point functions {E(0)E(1)E(z)E(<x>))0 for (46) and {a(0)E(1)E(z)o(<x>))0 for (67) and (68). Tn order to satisfy the neutrality, we should include two of the screening operator V- in the vertex correlation functions. The normalization factors in the four-point functions are determined in a decoupling limit, such that the structure constant is fixed to unity. The value can be calculated by the complex Selberg integral [48] as well as by the method described in Appendix D; the results are the same for both types of the four-point functions:

T( 1 )8

N—1 = + O(e) = 8m4 + O(e), (A.7)

where m = F( 1 )2/(23/2n1/2) = _.6__05755 ••• is the lemniscate constant.

Appendix B. Some details of the integral for the two-loop beta function and the scaling dimension of the energy operator

Tn this appendix, we thoroughly use the result of Appendix D. We read off the exponents in (D.1) from the vertex operator representation (46). 2a = 4a13a^, 2b = 8a13a^, 2a' = 2b' = 4a13a—, 2f = 4a^a—, 2g = 4a—. For convenience, we interchange the values of a and b, and

those of a' and b' (the latter is trivial, since here a' = b'). These interchanges are possible by an obvious symmetry between 0 and 1 in the integral (D.1).

As a result, we have a = 2b = —2f = 4a13aj"3 = — 2 + 8e, a' = b' = — g = 2a13a- = - 2 + 2e. From these parameters, we determine the integral by the scattering amplitude formula (D.7). The scattering matrix M are given by (D.8) and read,

—4ne 4ne —16nV

—4ne

,—4ne + 16n 2e2 4ne — 16n 2e2 —4ne,

+ O (e

From (D.20), the initial and the final state basis in (D.7) are given by

2 + 2e — 2 + 6e

— 2 + 2e

5 — 2e

— 1 + 4e 1 — 4e

— 1 + 8e 2 — 4e

— 2e

2 + 2e —5 + 2e —2 + 2e

— 2 + 6e 2 — 4e

— 2 + 2e

2 + 2e

2 — 4e 4e

2 + 2e 2 — 4e

2 — 2e 1 — 4e — 1 + 4e

J1— =

2 — 10e 4e —1 + 4e

— 6e 2 — 4e

— 2e

— 2 + 2e 2 — 2e 3 — 2e

§ — 10e — 1 + 2e — 2 + 2e

— 6e 2 — 4e

— 2 + 2e

2 — 4e

2 — 2e 1 — 4e

— 2e

— 1 + 2e

2 — 2e

2 — 2e

J3—=

5 — 10e — 1 + 2e —1 + 4e

2 — 6e —4e

2 — 2e 2-4e

1 — 4e

2 — 2e

where J± (l = 1, 2, 3) are real triple integrals defined in (D.9)-(D.14), and the symbol ^ - j is

explained in (D.19). We should discuss the order of these integrals J± in e; for this purpose, we can use the series representation derived in Appendix D.3. From (D.22), the prefactors y± are calculated as,

(K+,K2+,K3+) = (— 3ne, —16n, — £) + O(e°),

, ) / n 16n 9n2 9n2 \

(Kf,K2—,Y3—) = (- + —, 8n — + — (—1 + 8log2)J + O(e). (B.4)

Naively, we expect the leading terms in Jl = ylSl is the same order in e as the yl, since all the triple series Sl contains the term with (i, j,k) = (0, 0,0), which is unity and hence O(1). Actually, it is not the case for J+; the series S+ is O(e) because of the non-trivial cancellation between the constant terms.

Since this observation makes the crucial point in the calculation, we describe the calculation of J+ in detail. First, from (D.23), we have

to to to

s+ = EEE

i=0 j=0 k=0

(—1 + 4e)i (1 — 4e)j( 3 — 2e)k

i!j !k!

(2 + 2e)j+k( —■ + 6e)i+j+k(—1 + 8e)i+j (1 + 4e)j+k(2 + 4e)i+j +k(1 + 4e)i+j '

Denoting the summand in (B.5) as Si,j,k, we observe, from the definition of the Pochhammer symbol (x)k = x(x + 1) ■■■ (x + k - 1), that Si,j,k gets a factor of O(e) whenever each of two conditions {i > 2,i + j > 2} is satisfied. Thus, we let

«1 = £ S0,0,k, a2 = ^2 S1,0,k, a3 = ^2 S0,1,k, k^O k^O k^O

«4 = Y So,j,k, «5 = £ Si,j,k, (B.6)

where the leading order of a1, a2, a3 and a4, a5 are O(e0) and O(e1), respectively. To be specific, at the accuracy of O(e), we are left with the following:

ai = 3F2 2 2 , „., 2 , , ; 1 = 1--4 ^ ,-4--+ O(e), (B.7)

1 + 4e 2 + 4e ' J n„3

a2 = (—1 + 15 A 3F2 ( 3 — 2e, f + M + * ) =—1 ^ + + o

2 \ 4 2 J\ 1 + 4e, 3 + 4e J 4 3n3 ( ),

a3 = (3 — 49e) 3F2 ( 2 — 54 +4 *+ 6e ) = 3 . 4£^ + O(e), (B.9)

3 \8 4 J3 ^ 2 + 4e, 3 + 4e / 8 9n3 ( ), y ;

where the values of generalized hypergeometric functions 3 F2 at unity are used. The arguments, which is always taken at unity, are suppressed in the first equalities in (B.8), (B.9) and henceforth. Now, we see the aforementioned cancellation at O(e0), and get

a1 + a2 + a3 = O(e). (B.10)

This is an example of identities known as the "contiguity relation" between hyper-geometric functions. Note that this type of the leading order cancellation occurs only for S+. We note that the cancellations at this order guarantees a consistency in the RG scheme. Resulting O(e) term can be evaluated numerically,16 using

(x + e)k = (x)k■ {1 + e[f(x + k) — f(x)] + O(e2)}, (B.11)

where ty(x) is the di-gamma function. The other terms contributing O(e) in S+ are

a4 + a5 = -Y (2)i(1)j(2h+j(5h+J + O(e2). (B.12)

4 5 16 jj (1)i(3)j(4)i+j(3)i+j y ' y J

To evaluate this double series, it is useful to note the following identity:

y (a),(b)j(c),+j(d),+j _ F ( a + b,c,d ) (B13)

(1)i(1)j(e)i+j(f)i+j 3 2\ e,f J. ' • '

16 Actually, there have recently been extensive studies on the closed form evaluation of the expansion of the hyperge-ometric series in algorithmic approach, for example [51]. However, to our knowledge, the desired expansions here have not been published.

This holds because (a + b)k has the same binomial expansion as (a + b)k. By comparing the order S terms of the following expression,

(3)K-1 + S)j()l+](2)+ / 1 + S, -1, 1 \ (B14)

(1)i(1)j(2)i+j(1)i+j 3 2\ 2, 1 y ■ ;

one can express the right-hand side of (B.12) in terms of a single series:

a4 + as = Qx ■ e + O(e2),

/ 133 \ / 1 _i_ s-11 \ Qx = 3F2Î 23222 j + 8F«i 2 +2 7 , 2) , (B.15)

where FS(- ■ ■) denotes the coefficients of the order S terms in the 3F2 function at unity. In this case,

3F2 ( 1 + S , x1, 1 ) - sF^ 1, ^ 2 ) = S ■ F^ 1 + S , ^ 1 ) + O(S2). (B.16)

We use this notation hereafter. Now, collecting the O(e) parts of (B.7)-(B.9) and (B.15), we obtain

(3n 2 \

Z_6L + O(e°) J (0 ■ e0 + Ae + O(e2)) = A + O(e), (B.17)

where we have used (B.4) for y+. The constant A is given by —3n

A = ■

t3 2 xi13 2 )—^ ( 2 é2 )+f4 2 A—2+4S

_1 F ( 3, 3, 1 + 6S V 3 F ( 3 — 2S, 2 + 2S, 1 + 6S\ '

4 1 + 4S, 3 + 4S / + 8 SV 2 + 4S, 3 + 4S / + Qx This completes the calculation of /+. Since the linear relations (D.24) are solved as

(B.18)

J+ = 2 (Jx— — 2 J2— + J3—) — 2ne( Jf + J3—) + O(e), (B.19)

J2+ = —J2— — 4ne( J— + Jf) + O(e), (B.20)

J3+ = J— — Jf + 4ne(—J2— + J3—) + O(e), (B.21)

the knowledge of the other two bases are sufficient in the formula (D.7). Actually, we can calculate Jf and J3— in similar manners. Let us define constants D and F as follows:

~ (—1 + 4e)i(3 — 2e)](2 — 2e)k (5 — 10e)j+k( 1 — 6e)i+j +k(—2 + 2e)i+j

i,£k=0 i]!k! (2 — 6e)j+k (5 — 2e)i+]+k (2)i+j

+ D e + O (e2), (B.22)

1 )4 4)

J— = Y— ■ = + D + O(e), (B.23)

^ (—1 + 4e)i(1 — 4e)j(3 — 2e)k (§ — 10ej+k( 1 — 6e)i+j+k(—4e)i+j 3 = .jfc^ i'! J! k! (2 — 8ej+k (3 — 8e)i+j+k (2 — 8e)i+j

4T( 4 )4

+ F e + O (e2), (B.24)

£( 4 )4

J— = y— ■ S— = + F + O(e), (B.25)

333 16ne

where, again, (B.4) has been used for y- and y3—. Then, D and F are given by 2£(4)4 n2 (133) 3n2 n

d = —+ —(1 + 3log2) 3F21 2;2:2--Q1 — Q2

3n 8 B ) V 3,2 J 16 10

F (3—25,2+25,2+25 V — fx (2—25,2+25, +25

64 5 \ 2 + 45,3 8 H 1 + 45,2

( ) (B.26)

p £(1)\ d — 65, 3 — 25, 2 — 105 \ 3n2

F = ^T(—1 + 8log2) + frfM 2 2—V 3 — 285 ) — (B.27)

where Q2 denotes well-converging double series

Q2 = V —--(1 )j(3)j(3)k(7)j +k. (B.28)

jk 5 + j + k (3)j(1)j(1)k(2)j+k

Now, by substituting (B.23), (B.25) and the matrix elements (B.1) into the formula (D.7 ), we obtain

n1)4 ( r(1 )4 )2

(—2N )—1l (cx,e) = —^(_A + D + F — J+ — J— + 16n 2e2( —■4— 4 v 2 2 ' \ 16ne)

+ O(e) 1\4

(2 A + D + F) + O(e), (B.29)

£( 4 )4

where, in the first and the second line, we have used (B.20) and (B.21), respectively. Our concern is to evaluate this, from (B.18), (B.26) and (B.27), within high numerical accuracy. To this end, we have used asymptotic behavior of the di-gamma function for the evaluation of Fs. We obtained the value

2A + D + F = 27.50074327 (21), (B.30)

which is in good agreement with another value 4m2 = _7.500743_7_08 •••. Assuming the latter value in (B.29) and using (A.7), we obtain

£( 1 )8

I (<x>,e) = — 2N —— + O(e) = —+ O(e). (B.31)

Appendix C. Integrals for the correction coefficient of the spin field dimension

We here calculate the integrals K2(r, e) in (64) and K3(r, e) in (66). From (67) and (68), we read off the exponents defined in (D.1). After interchanging the values of the parameters a ^ b and a' ^ b' (as in Appendix B), we have a = 2a23 — 2Ag = 2 + 2e, b = 2ap_x pa13 = — 4 + e,

a' = f = 2a13a- = —2 + 2e, b' = 2a t = 4 - e, and g = 2a- = 1 - 2e for K2; we

have almost the same set of the parameters as K but a = 2a23 + 1 - 4A£ = 2 + 6e for K3. We substitute these into (D.8) and get, both for K2 and K3, the same matrix:

1 / 2ne —2ne

^ = — I —1 + 3ne 2ne

— 1 + 3ne —1 + 3ne

+ O (e

It is useful to adopt a temporary notation [x] which takes the value x and x + 4 for K2 and K3, respectively. From (D.20), the basis in (D.7) is then determined as

J3—=

— 1 + [8]e 4 — e

3 — 2e

— 1 + [6]e 5 — 2e 4 — e 2 + [2]e — 2 + 2e — 4 + e

4 — [7]e 2 — 4e 1

— 2 + 2e

5 — 2e

2 — 2e

— 4 + e

— 2 + 2e

— 1 + [8]e

— 2 + [6]e

— 1 + 2e —I + 2e 4 — [7]e 2 — [6]e — 2 + 2e — 4 + e

— 3 + 2e

— 4 + e

5 — e —2 + 2e

— 1 + [8]e

— 2 + 2e +e

+ 2e 2 + 2e

4 — [7]e 2 — [6]e

— 2 + 2e

2 — 2e

4 — e

5 — 2e

2 — 2e

— 4 + e

— 4 + e

2 — 2e 1 — e

4 — [7]e — 2 + 2e — 4 + e

For convenience of the presentation, we use the following notation both for K2 and K3 :

(X, Y, Z) = (J+,J2+,J3+), (U,V,W) = (Jf, J2—, J3— ). (C.5)

From (C.2), (D.22) and (D.23), we first observe that X is O(e°); this is guaranteed by the same contiguity relation as in (B.10). As a result, we realize that all of X, Y and Z are regular in e. By the combination of this observation and the linear relations (D.24), we infer V = O(e), which suggests a non-trivial cancellation at O(e—x) occurs in the triple series S— determined from (C.3) and (D.23). Now, by noting V = O(e), the linear relations (D.24) can be casted as

X — Z = W/V2 + O(e), Y + Z = V2ne(U + W) + O(e2), tineY = 42ne(U + W) — V/V2 + O(e2

(C.6) (C.7) (C.8)

— 4 + e

where we have introduced the variable Q, which takes the values:

(8 for K2, (C9) Q = ) 12 for K3. ( )

It should be noted that the leading order part of both U and W are O(1), and are common for K2 and K3. Using the formula (D.7) for (C.1) and (C.5), with the help of (C.6)-(C.8), we obtain

K* = 2ne(U2 + W2 - QYZ) + O(e2), (C.10)

where the notation Q in (C.9) is used to express both K2 and K3 in parallel. Actually, only the combination U2 + W2 is necessary for the disordered model (M ^ 0). Substituting the parameters in (C.2) and (C.4) into (D.22) and (D.23), we obtain

32V2n ^ ^ ^ (3)i(-4)j(-4)k (4)j+k(2)i+j+k(4)i+j (C11)

U = 7 Z^Z^Z^ i\;\k\ (3s ,3n /1K ' (C.11)

7 i=0 j=0 k=0 i j (2 )j+k( 2 )i+j+k(T)!'+j

W = 9n7/2 (I)i(4)j(-4)k(I)j+k(2)i+j+k(4)i+j (C12)

8r( 4 )2 j=0 k= i !j!k! (2)j+k(3)i+j+k( 4 )■+;' .

We obtain numerically

U2 + W2 = 671'0 ± 0'3, (C.13)

which is compatible, within error bar (±0'05%), with the value

T(4)12

% = 64 ^ = 670'78 •••' (C.14)

8n 6 n3

Unfortunately, the convergence of the triple series U is slow, and thus our numerical accuracy is not good here. Nevertheless, we assume, for U2 + W2, the value in (C.14).

Appendix D. Integrals

In our calculation of the RG functions, we should deal with a multiple integral over C3: I = JJI d2zd2ud2v |z|2fl|1 - zl2b|v - z|2/\u - z|2/

■ |u|2fl'|1 - u^^u - v|2g|v|2fl'|1 - v|2b'' (D.1)

Since this form of the integral comes from the correlation functions of the vertex operators (or, more plainly, from the interaction between charged particles in a two-dimensional plane), it seems to be ubiquitous in physics and mathematics. For example, the integral can be interpreted as a six-particle closed string amplitude [47]. The integral in a special, symmetric case of parameters (a = a', b = b' and / = g) is well studied in the context of twisted cohomology, and known as the "complex Selberg integral" [48]. What we need, however, is the formula in a non-symmetric case, when doing perturbation theory around a conformal fixed point. In this respect, a formula for two variables and that for three variables in a special (degenerate) case were used in the study of random-bond Potts model by Dotsenko et al. [19,60]. We extend their results and derive a formula for (D.1) in a systematic way. The formula, obtained in (D.7), takes form of a scattering amplitude.

Fig. D.1. (a) Two circles Sa and SB in the definition (D.3) of the regularization of a interval [A, B]. (b) The same regularization but in a simpler representation used in Appendix D.4.

D.1. Regularization of the one-dimensional intervals

We encounter with strong algebraic singularities that make the multiple integrals divergent. For this reason, we consider an analytic continuation in the parameter of the integrals. In order to keep the discussion clear, it is helpful to make the way of the analytic continuation explicit. The analytic continuation is achieved by the use of a "regularization" of the intervals [48,49], which we now describe using the following one-dimensional simple example.

Consider an integral on a real interval

dx (x — A)p(B — x)qf(x), (D.2)

where f(x) is an analytic function on a neighborhood of the interval [A, B]. If f(A) = 0 and f(B) = 0, the condition Rep < —1 or Req < —1 makes the integral divergent. The regularization of the interval [A, B] in the integral (D.2) is given by a replacement

reg: [A,B] —-—-+ [A + S,B — S] +-—-. (D.3)

1 — exp(2nip) 1 — exp(2niq)

Here, as in Fig. D.1, SA and SB are positively oriented circles of radius S which have centers A and B, and start at A + S and B — S, respectively. By replacing the interval [A, B] by the regularized one "reg[A, B]" and taking the limit S ^ 0, the value of the integral (D.2) remains same for Re p > — 1 and Re q > — 1 since the contributions from two additional circles vanish, and is now finite also for Re p < —1 or Re q < —1 unless p e Z or q e Z. In the latter case, adding two circles corresponds to the subtraction of infinite quantities, and the resulting finite value is what is known as the "Hadamard finite part" of the integral.17

The definition here is natural in the following sense. Starting with the multiple integral (D.1) and introducing an infinitesimal imaginary part, we shall reach an iterated integral in which each integral is on the regularization of the interval rather than on the usual real interval. This regularization comes from the pairing of the paths on the upper and the lower half planes both of which detour the branch points. All the one-dimensional integrals in this appendix should be regarded as the integral over the regularization of the interval.

D.2. Decomposition of the multiple integral

We shall consider the integrand in (D.1) as a function defined on C3 x C3 rather than on C3; writing z = Z1 + iz2 in the variable z, to be specific, the first factor in the integrand in (D.1):

17 Actually, the definition (D.3) of the regularization of a interval [A, B] is proportional to well-known "Pochhammer contour" which is used to define the analytic continuation of the hypergeometric functions.

\z\2a = (z1 + zf )a is now defined on (z1,z2) e C x C. Then the path of z2 is rotated by the angle n/2 - 2n, without hitting any singularity, for a positive infinitesimal number n. By introducing a new variable zo, we rewrite \z\2a as follows:

\z\2a = (z2 + z2)a -^(z2 + (ie-2'nzof)°

= (z+ - in[z+ - z-])a(z- + in[z+ - z-])a, (D.4)

where the notation z± = zi ± z0 is introduced. In the following, we call {z+,u+,v+} and {z-,u-,v-}, respectively, holomorphic and antiholomorphic variables. Using a notation Xn = n[X+ - X-] for a quantity X, we decompose the integral (D.i) as

0 'HUH-

I =(-) ¡I J J J J dz+ du+ dv+ dz- du-dv- J+ ■ J-, (D.5)

-to,to]6

J+ = (z+ - izn)a(z+ - 1 - izn)b{z+ - u+ i[z - u]v)f(z+ - v+ i[z - v]n)J ■ (u+ - iun)a (u+ - 1 - iun)b (u+ - v+ - i[u - v]nY(v+ - ivn)a

(v+ - 1 - ivn)b'

J- = (z- + izn)a(z- - 1 + izn)b(z- - u- + i[z - u]nf (z- - v- + i[z - v]n) ■ (u- + iun)a (u- - 1 + iun)b (u- - v- + i[u - v]^g(v- + ivn)a

■ (v-- 1 + iv-nf. (D.6)

Here, J+ and J- are weakly dependent, through an infinitesimal number n, on the antiholomorphic (-) variables and the holomorphic (+) variables, respectively. If we fix the holomorphic variables first, the dependence of J- on the holomorphic variables determines relative positions of the integration paths and the two branch points 0 and 1 on the complex planes of the antiholomorphic variables. For this relative positions, we can observe there are two very distinct cases.

The first case is trivial, and corresponds to a choice of the holomorphic variables such that at least one of the z+,u+ and v+ lies outside the interval [0, 1]. Then, according to (D.6), one can find at least one integration path on the antiholomorphic plane such that all of the points 0, 1 and the other paths projected are located on the same side of the plane. Therefore, the path can be contracted to a point by adding a semicircle of infinite radius and the corresponding integral should vanish.

There is, however, the second case in which all of the z+,u+ and v+ belong to the interval (0, 1). On the antiholomorphic planes, this means all the paths of z-,u- and v- intersect the segment [0, 1] in the same sequence as z+,u+ and v+ lie on the interval (0, 1). Take a sequence: z+ < v+ < u+ as in Fig. D.2(a), for instance. Now, we deform the paths so that the resulting paths encircle the point 1, the real axis from the upper and the lower side (Fig. D.2(b)). Since the path nearest to the point 1 is that of u-, we first deform it, then that of v-, and finally that of z-. Note, in deforming the path of, say v-, the fixed variable u- is projected on the v--plane as a branch point.

As a result of the deformation, we have a pairing of the paths in the upper and the lower half planes for each variable z-,u- and v-. Each paring of the path can be decomposed into a sum of the integral over the regularizations of intervals (see Appendix D.1 for the definition of the regularizations and Ref. [48] for the summation of them).

Fig. D.2. Deformation of the paths on the antiholomorphic planes. Both figures correspond to the sequence of the holo-morphic variables < v+ < u+. (a) Before the deformations. (b) After the deformations. The five symbols A, C, D, E and F are assigned to each regularization of the interval. The case with v- <u- is omitted here, and thus the regularization B in Fig. D.3 does not appear.

Taking the presence of the branch cuts into account, we now attach the appropriate factors on these regularizations. Since each factor comes as a difference of two phase factors on the upper and the lower half plane, it takes the form of a sin-function. In the following, we assign the numbers {1, 2, 3} for the sequences of variables {(z± < v± < u±), (v± < z± < u±), (v± < u± < z±)}, respectively.

Consequently, from (D.6), we obtain a scattering-type formula

I = (-2) • J J+ J+) M J-

/s(b)s(bf ) s(b + f)s(b' ) s(b + 2f)s (bf ) \

• [s(bf ) + s(g + b' )] • [s(bf ) + s(g + b' )] • [s(bf ) + s(g + b' )]

s(b)s(b' ) s(b)s(bf )2 s(b + f)s(bf )

•[s(f + b' ) + s(f + g + b' )] + s(b + f)s(f + g + b' ) • [s(bf ) + s(g + b' )]

s(b)s(f + b' ) s (bf )s (f + b' ) s (b)s (bf ) V •[sf + b') + s(f + g + b')] + s(b)s(b')s(f + g + b') • [s(bf) + s(g + b')] )

where a notation s(x) = sin(^x) is used.18 The dimension of the basis is (3!)/2 = 3, where two in the denominator comes from the symmetry between u and v. The factor two in (D.7) is necessary because of the same symmetry in J + basis. Each matrix element of M is a sum of two term; each term is a product of three sin-functions attached onto the regularization of the intervals on z-, u- and v--planes. Further, we have defined the initial and the final state basis as

J+ = f du I dv I dzza(1 - z)b(v - z)f(u - z)f H+ (u, v), 0 0 0 1 u z

= J du j dz j dvza(1 - z)b(z - v)f(u - z)f H+ (u,v),

(D.10)

18 It would be interesting if a possible relation between the scattering matrix M in (D.7) and M± in (D.24) in this paper, and the intersection matrix and the monodromy invariant Hermitian form in [49] were elaborated.

Fig. D.3. Pairings of the paths (the regularizations of the intervals) that contribute the matrix elements of M. Each branch correspond to the regularization (see also Fig. D.2 for the branches A, C, D, E and F). The branches B and C, for instance, correspond to the regularization in the v--plane with the sequence v- < u- and u- < v-, respectively. Each regularization has a certain factor on it due to the presence of the branch cuts.

= f dzjduj dvza(1 - z)b(z - v)f(z - u)f H+ (u,v),

to TO TO

J- = f dz j dv j duza(z - 1)b(v - z)f(u - z)fH_(u,v),

1 z v to TO TO

= j dv j dz j duza(z - 1)b(z - v)f(u - z)f H-(u, v),

1 v z TO TO TO

j dv j du j dzza(z - 1)b(z - v)f (z - u)f H_(u, v),

where we have used notations

H+ (u, v) = ua(1 - u)b (u - v)gva (1 - v) H-(u, v) = ua'(u - 1)b/(u - v)gva'(v - 1)

(D.11)

(D.12)

(D.13)

(D.14)

(D.15) (D.16)

The matrix elements of M are conveniently understood if we draw the tree-diagrams which show the sequences of the variables under the process of the deformation (Fig. D.3).

D.3. Representation of the basis through triple hypergeometric series

By a suitable change of variables, the integral in (D.9) is transformed into an integral over the unit cube:

j+ = j du j dv j dz u2+a+2a!+2f+g(1 - u)b'v 1+a+a/+f(1 -v)gza(1 - z)f 0 0 0

■ (1 - vz)f (1 - uvz)b(1 - uv)b. (D.17)

In general, all of J± (l = 1, 2, 3) in (D.9)-(D.14) can be cast into the same form, namely,

J dû J dv J i

dz û k-1(1 -û)k-1v f-1(1 - v)/x -1Zv-1(1 -Z)v

(1 - vZ)-p(1 - ûvZ)-q(1 - ûv)-

(D.18)

with each different set of exponents (X, v, k',i',v',p,q,r}. In this paper, the value of the triple integral in the right-hand side of (D.18) is denoted as a compact symbol:

k k' / fx'

(D.19)

Then each of the base J± (I = 1,2, 3) in (D.9)-(D.14) looks like,

£ 1 + b' -f 2 + a + a' + f 1 + g -b 1 + a 1 + f -b'

- b - 2b' 1 + b -f

1 - £ + a + b

1 + f -b'

-1 - a'-b'-f - g 1 + g -b'

£ 1 + b' -g 2 + a + a' + f 1 + f -b' 1 + a' 1 + f -b

-£ - b - 2b' 1 + b' -g 1 - £ + a' + b' 1 + f -b' -1 - a'-b'-f - g 1 + f -b

£ 1 + b -f 2 + 2a' + g 1 + f -b' 1 + a' 1 + g -b'

-£ - b - 2b' 1 + b' -f 1 - £ + a' + b' 1 + g -b -1 - a - b - 2f 1 + f -b'

(D.20)

where, for brevity, a notation £ = 3 + a + 2a' + f + 2g is used.

Performing the binomial expansion for the last three factors in (D.18), we get the following triple hypergeometric series:

J± = Y±- S±, ±= r(k)r(kQ Y(f)Y(f') r^r^) V(k + k') r(f + /') r(v + v') ,

œ œ œ

i=0 j=0 k=0

(p)i(q)j(r)k

(k)j+k(/x)i+j+k(v)i+j

i ! j !k! (k + k')j+k(f + f')i+j+k(v + v ' )i+j '

(D.21) (D.22)

(D.23)

where the Pochhammer symbol (x)k = x(x + 1)■■■(x + k — 1) is used. The parameters (k, v, k',i',v',p, q,r} is related to the exponents {a, b, a',b',f, g} in (D.1) as indicated in

(D.20). This series representation is particularly useful when some of {p, q,r, X, /i,v} are nonpositive integer plus O(e). In that case, a separation of the order in e occurs.

D.4. Linear relations between the basis

For generic values of the parameters {a, b,af,b,g,f}, there exist three independent linear relations between the triple integral basis {J + ,J-}. Although in principle we can evaluate the coefficients in the epsilon expansion of these integrals using the series expressions, but in practice some of the bases happened to be difficult to expand in e, while the others to be more straightforward. Hence, these relations are necessary in our epsilon expansion calculation of the RG functions.

The explicit form of the relations are,

= -M- I J-

(D.24)

/ s(a)s(ar)

• 2s(a' + g/2)c(g/2)

s(a)s(a')

• 2s(a' + f + g/2)c(g/2)

s(a)s(a' + f)

• 2s(a' + f + g/2)c(g/2)

/ s(a + b + 2f)s(a'+ b' + g + f)

• 2s(a' + b' + f + g/2)c(g/2)

s(a + b + f)s(a' + b' + g + f)

• 2s(a' + b' + f + g/2)c(g/2)

s (a + f)s(a' )

• 2s(ar + g/2)c(g/2)

s (a! )

• [s(a)s(a' ) + s(a + f)s(a' + g + f)]

s(a)s(a')

• 2s(af + f + g/2)c(g/2)

s (a + b + 2f)s(a' + b' + g + f) • 2s(a' + b' + g/2)c(g/2)

s(af+ b' + g + f)

•[s(a + b + 2f)s(a! + b' + g + f) + s(a + b + f)s(a' + b' )]

(D.25)

s(a + b)s(a' + b' + g + f) s (a + b + f)s(a' + b' + g + f)

\ • 2s(a' + b' + f + g/2)c(g/2) • 2s(ar + bf + g/2)c(g/2)

s (a + 2f)s(a' )

• 2s(a' + g/2)c(g/2)

s (a + f)s(a')

• 2s(ar + g/2)c(g/2)

s(a)s(a')

• 2s(a' + g/2)c(g/2)/

s (a + b + 2f)s(a' + b' + g)

• 2s(a' + b'+ g/2)c(g/2)

s (a + b + 2 f)s(a' + b' + g + f)

• 2s(a'+ b'+ g/2)c(g/2)

s (a + b + 2 f)s(a' + b' + g + f)

• 2s(a' + b' + f + g/2)c(g/2) /

(D.26)

where we have used the notation c(x) = cos(nx). We now derive the first column of the relation (D.24) for an illustration. Since it is the analyticity of the integrand on the region except the branch cuts that makes the relation valid, the basic strategy is to deform successively each integration path defined on each complex plane. To keep track of the successive deformations of the integration paths on the three complex planes, we use a simple semicircular diagram to represent the regularization of the interval (see Fig. D. 1 ).

Fig. D.5. The successive deformations of the paths on the three complex planes. The notations e(x) = exp(nx) and s(x) = sin(nx) are used. Each semicircular shape represents the regularization of interval (see Fig. D.1). The dotted curves indicate the integrations over the variable z, while the solid curves indicate that of u or v. The vertical lines indicate the relative positions of the branch cuts induced for the other variables.

Consider the z-plane in which all the branch cuts are taken along the real axis from each branch point to positive infinity. Since the integrand has no branch points except the non-negative real axis, the integral along the contour C in Fig. D.4 is zero.

Then the z-integral on a regularized interval reg[0,w] can be expressed by a certain linear combination of the integral on the intervals reg[w, v], reg[v, 1] and reg[1, to] as shown in the second equality in Fig. D.5. Each coefficient comes from the pairing of the two segments which lie on opposite sides of the branch cuts. In this way, we can shift the integration path for each variables to the intervals in the positive real direction.

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