Two Dimensional Renormalization Group Flows in Next to Leading Order

Zamolodchikov's famous analysis of the RG trajectory connecting successive minimal CFT models $M_p$ and $M_{p-1}$ for $p\gg 1$, is improved by including second order in coupling constant corrections. This allows to compute IR quantities with next to leading order accuracy of the $1/p$ expansion. We compute in particular, the beta function and the anomalous dimensions for certain classes of fields. As a result we are able to identify with a greater accuracy the IR limit of these fields with certain linear combination of the IR theory $M_{p-1}$. We discuss the relation of these results with Gaotto's recent RG domain wall proposal.


Introduction
In his famous paper [1] A. Zamolodchikov has investigated Renormalization Group (RG) flow from the minimal model M p to the M p−1 for large p 1 caused by the relevant operator φ 1,3 . Two main circumstances made it possible to investigate this RG flow using a single coupling constant perturbation theory. First, the conformal dimension of the field φ 1,3 , ∆ 1,3 = 1 − 2 p+1 ≡ 1 − (see Appendix A) is nearly marginal when p 1 and second, the Operator Product Expansion (OPE) of this field with itself produces no relevant field besides the initial one and the unit operator. The method of A. Zamolodchikov not only allowed to identify the IR theory with M p−1 , but also provided detailed description how several classes of local fields behave along the RG trajectory. The analogous RG flow for the N = 1 super minimal models has been investigated in [2].
The main purpose of this paper is a sharpening of Zamolodchikov's analysis, by the inclusion of second order perturbative corrections. It is interesting to note that in all cases we have investigated, the rotation matrix (in the space of fields), that diagonalizes the matrix of anomalous dimensions, does not receive 1/p or 1/p 2 corrections. So an interesting question arises, if any higher order corrections appear at all.
As intermediate results, in this paper we have found several four-point correlation functions in large p limit (see formulae (C.1)).
The initial motivation to carry out these computations came from the recent approach to this RG flow by D. Gaiotto [3]. Using Goddard-Kent-Olive construction, Gaiotto has constructed a non-trivial conformal interface between two successive minimal models and made a striking conjecture, that this interface is the exact RG domain wall which encodes the map between the UV and IR fields. Gaiotto's conjecture survives a strong test: it is fully compatible with the first order parturbative calculations of the mixing amplitudes performed by Zamolodchikov. In this paper we show that this mixing coefficients computed with the help of the perturbation theory up to the second order, unlike those obtained from the Gaiotto's conjecture, do not receive any corrections up to the order 1/p 2 . Nevertheless, this discrepancy might be attributed to the renormalization scheme which is adopted here following Zamolodchikov. Presently the author of this paper does not have any clue how to take into account possible dependencies on the renormalization schemes in order to be able to make any conclusive statement about Gaiotto's conjecture beyond the leading order.
The paper is organized as follows.
In Section 1, we develope some technical tools, necessary to carry out second order in coupling constant calculations.
In Section 2 the β-function and Zamolodchikov's c-function [4] are computed with next to leading order accuracy. The critical value of the renormalized coupling constant, the slope of the β-function as well as the c-function at the critical point are calculated. The results of these computations confirm that also the second order contributions perfectly match with the Zamolodchikoved's conclusion that the IR fixed point corresponds to the CFT M p−1 and that the UV field φ 1,3 flows to the field φ 3,1 of the IR theory. Section 3 is devoted to the renormalization of several series of local fields and to the calculation of their anomalous dimensions. Thus: in Section 3.1 we investigate the renormalization of the fields φ n,n .
In Section 3.2 the renormalization of the fields φ n,n+1 and φ n,n−1 is discussed and the matrix of anomalous dimensions is found. At the fixed point the matrix of anomalous dimensions is diagonalized and its eigenvalues are calculated. In Section 3.3 the same steps are performed for the fields φ n,n+2 , ∂∂φ n,n and φ n,n+2 .
In all cases the predictions of Zamolodchikov successfully withstand the next to leading order test.
In Appendix A some basic facts about the minimal models of 2d CFT are reviewed. The Appendices B and C are devoted to computation of the integrals used in the main text. The Appendix D comments how to calculate the large p limit of those four point correlation functions used in the main text.

Perturbation theory in second order
Suppose the (Euclidean) action density is given by with H 0 being the UV CFT action density, φ a relevant local spinless field and λ the coupling constant. Then for a two-point function up to second order we'll have In this paper we consider a theory, whose UV limit is given by the minimal CFT model M p with p 1 and the perturbing field is φ ≡ φ 1,3 . Leading order corrections in this theory has been investigated by A. Zamolodchikov [1]. Second order computations are more complicated. Indeed, not only the knowledge of four point correlation functions which in general are quite non-trivial in a CFT [5], but also their integrals over two insertion points is required. Fortunately, as we demonstrate below, the conformal invariance allows to perform integration over one of the insertion points explicitly. First let us notice that translational and scale invariance can be exploited to locate the points y 1 , y 2 at y 1 = 1 and y 2 = 0 without loss of generality: (here and below I frequently use the shorthand notation x 12 = x 1 − x 2 , y 12 = y 1 − y 2 et.al.). Any four-point function of primary fields in a CFT essentially depends only on the cross ratio x = x 12 x 34 x 14 x 32 of the insertion points and its conjugate [5] where it is assumed that the fields are spin-less (i.e. ∆ i =∆ i ). Specifying the insertion points as Alternatively specifying x 1 = 1/x, x 2 = ∞, x 3 = 1 and x 4 = 0 and comparing with (1.5) we get the identity which is useful when investigating the correlation functions at large x. After application of (1.4), (1.5) to the four-point function φ(x 1 )φ 2 (0)φ 1 (1)φ(x 2 ) 0 and introduction of the new integration variables two integrations on the r.h.s. of eq. (1.3) become partly disentangled where is the Gaussian hypergeometric function. Above three expressions for I(x) are convenient when exploring the regions x ∼ 0, x ∼ 1 and x ∼ ∞ respectively. Note also that these expressions make explicit the single-valuedness of I(x). Specifying the choice of parameters to (1.9) and applying the identity to the second term of the second equality, the eqs. (1.10) can be rewritten as It is worth noting that in the case when 12 ≡ 1 − 2 = 0 only the third expression is manifestly nonsingular, the first two expressions require a subtle limiting procedure.
Thus for this case it is better to employ the third expression: Let us investigate the behaviour of (1.12) at x ∼ 1. Using standard formulae for the analytic continuation of the hypergeometric function with parameters satisfying the condition a + b − c ∈ Z (see e.g. [7]) one can get convinced that where γ = 0.577216 · · · is the Euler constant and the omitted terms are at most of order |x − 1| 2 log |x − 1| in x → 1 limit. There is no need to investigate the limit x → 0 separately since the obvious symmetry of I(x) with respect to x ↔ 1 − x at x ∼ 0 immediately ensures In this section we calculate the β-function up to 1/p 4 ∼ 4 corrections for the small values of the (renormalized) coupling constant (of order or smaller). As it will become quite clear later for this purpose one should evaluate the integral (1.7) in the special case φ 1 = φ 2 = φ and I(y) given by (1.12) with the accuracy ∼ 1/ . Our strategy will be as follows: separate in the integration region the discs D l,0 = {x ∈ C | |x| < l}, D l,1 = {x ∈ C | |x − 1| < l} and D l,∞ = {x ∈ C | |x| > 1/l} where l is an intermediate length scale such that 0 < l 0 exp(−1/ ) l 1 and l 0 is the ultraviolet scale. For the integral outside these discs we will safely use the small limits of the correlation functions given in the appendix while inside the discs we'll explore (exact in ) OPE. We will see that the trace of the intermediate scale l will be washed out entirely from the final result. In present case the = 0 limit of the four-point function is given by (see appendix C) With required accuracy I(x) ≈ π/ . It is convenient to carry out the integration in radial coordinates x = r exp(iϕ),x = r exp(−iϕ), d 2 x = rdrdϕ. The result of integration over the angular variable ϕ will depend on the region where the radial coordinates takes its value for arbitrary rational function R(x,x) with poles located at x = 0 or x = 1. In particular when R(x,x) is the r.h.s. of the eq. (2.1) we get After performing the remaining elementary integration over r we finally get where (see Fig. 1 ) and the dots stand for negligible terms of order l or l 0 /l. There is a subtlety to be treated carefully here. The fact that the part of the white narrow ring (of width 2l 0 ) outside of the red circle is missing from the integration region Ω l,l 0 is insignificant since its inclusion would produce only negligible terms of order l 0 . Instead we have to subtract the contribution of two lens-like regions of Ω l,l 0 included in the red circle (as already said, the contribution coming from the regions around the singular points will be computed separately exploring OPE). a) Integration over lens-like regions Expanding (2.1) around x ∼ 1 we get where only the singular terms, whose integrals over the region around x = 1 diverge, are presented. The integrals of such terms have been evaluated in Appendix D. As a result the contribution of the lens-like regions, to be subtracted from the r.h.s. of eq. (2.4), is equal to Taking into account (1.14) and that where the first two terms come from the identity and the last two terms from the φ field channel of OPE (2.7) respectively. c) Contribution of x ∼ ∞ At large x we first make use of eq. (1.6) to pass to the inverse variable 1/x ∼ 0 and then we apply OPE. The calculation is similar to the previous case, the main difference being the fact that at this limit I(x) becomes simply π (xx) −2 . The result is Let's pick up all the ingredients: (2.4) times I(x) ≈ 2π , minus (2.6), plus twice (2.9), and plus (2.10). We get As expected, the l dependence disappeared. The presence of the divergent term 3π 2 also is not surprising. In our naive regularization scheme we could cancel this infinity by adding an appropriate, proportional to the area counter-term in action. In fact, if we would have been able to treat the integral (1.7) analytically as continuation from a region of parameters where integral converges, then this kind of non-analytic divergent term couldn't emerge at all. In what follows we'll simply drop out such terms without further ado. Thus for the two point function Following A. Zamolodchikov let us introduce a new coordinate g ("renormalized" coupling constant) in the space of one-parameter family of theories (1.1) instead of the initial coupling λ and introduce the local field φ (g) = ∂ g H (according to (1.1) the initial "bare" perturbing field φ = ∂ λ H). The new coupling g is fixed by the requirement that the two point function Then the β-function can be computed from the identity (see [1]) where Θ is the trace of the energy-momentum tensor. Combining (2.13) and (2.14) one easily finds and The equation (2.15) allows one to express g in terms of λ (the integration constant can be set to zero so that the unperturbed CFT will corresponds to g = 0) or, inversely Inverting and replacing in (2.16) λ in favour of g we get The equation admits a nonzero solution so we have a non-trivial infrared fixed point. In [1] this fixed point has been identified with the minimal model M p−1 and the local field φ (g * ) with the field φ (p−1) 3,1 . Now we are in a position to check this identification more accurately. The anomalous dimension of φ (g * ) is related to the slope of β-function computed from the Kac formula (A.2). Also the shift of the central charge [4] neatly matches to the exact expression

Field renormalization and the UV -IR map
In this section we calculate the matrices of anomalous dimensions for several classes of fields. Diagonalization of these matrices at the IR fixed point provides a detailed map between the UV local fields and their image under RG flow in the IR theory.

Primary fields φ n,n
This is the simplest case to analyze since the fields φ n,n never get mixed with other fields [1]. This follows from the structure of the OPE involving the perturbing field φ 1,3 . The subspace of fields which is generated by the field φ n,n and is closed w.r.t. OPE with φ 1,3 , doesn't contain any other field with a dimension close to ∆ n,n = O( 2 ). We are going to calculate corrections to the anomalous dimension up to the order 4 . That is why for the present purpose the knowledge of the four point function φ(x)φ n,n (0)φ n,n (1)φ(∞) up to 2 correction is required. As in previous case, to find this correlation function we first used AGT relation to find the relevant conformal blocks up to sufficiently large level (actually the computations were performed up to the order x 6 terms). Expanding a conformal block up to 2 and examining first few coefficients of the resulting power series in x it is possible to guess the entire power series and identify it with some elementary function. Having in our disposal the expression for the correlation function we then checked that it satisfies all the nontrivial physical requirements: the single-valuedness and the compatibility with OPE around the points x ∼ 1 and x ∼ ∞. Here is the final expression (see Appendix C) From eq. (1.12), up to order , I(x) is equal to Now we are ready to perform integration over the region Ω l 0 ,l (see Fig. 1). Since the singularities at x ∼ 0 and x ∼ 1 are integrable, we can put l 0 = 0. As in Section 2 the integration over the angular variable should be performed separately for the cases 0 < |x| < 1 and |x| > 1. Integration of rational expressions we have already discussed earlier. As about the logarithmic terms, they can be easily handled first expanding into power series in x if |x| < 1 or in 1/x if |x| > 1. Then we proceed with the radial integration. Both steps are elementary and we present only the final result: Due to the already mentioned mildness of singularities at 0 and 1 the only remaining contribution to be taken into account comes from the neighbourhood of ∞ i.e. from D l,∞ \D l 0 ,∞ .
At large x it is convenient to employ eq. (1.6) and apply the OPE (2.7) with x replaced by 1/x. The correlation function decomposes into a sum of two partial amplitudes one corresponding to the identity and the other to the field φ. a) Contribution of identity The prefactor (xx) −2∆ in (3.4) compensates the factor (xx) 2∆ accompanying the identity operator in OPE and with sufficient accuracy we can replace this partial amplitude by 1. It is straightforward to expand I(x) given by (1.12) at large x keeping only those terms which after integration may produce non-vanishing terms in small l limit Integrating this expression over the region D l,∞ \D l 0 ,∞ , dropping out, as earlier, all singular in l 0 terms and expanding the result up to the linear in terms we get where π (xx) −2 is just the function I(x) with required accuracy, (xx) −2+2 is the prefactor of (3.4), (xx) 1− comes from OPE and the squared structure constant is equal to The integral is converging at the limit l 0 → 0, so we may perform integration over the entire region D l,∞ . The result reads The sum of all contributions (3.3), (3.6), and (3.9) is Combining this with the first order in coupling constant contribution where the value for the structure constant is inserted, we get Let's introduce the renormalized field φ (g) n,n = B(λ)φ n,n by requiring that the two point n,n (0) λ satisfies the normalization condition G n (1, g) = 1 (3.14) so that Then for the anomalous dimension we get (cf. eq. (3.48), derived for a more general situation) In view of (2.18) we find ∆ (g) n,n = ∆ n,n (3.17) So that at the fixed point which completely agrees with the dimension ∆ is robust also against our second order test.
3.2 Renormalization of the fields φ n,n+1 and φ n,n−1 Already in this case one encounters with the phenomenon of mixing. The OPE φ 1,3 φ n,n+1 produces besides φ n,n+1 also the primary field φ n,n−1 , both having dimensions close to 1/4 in large p limit. Thus we have to consider the correlation functions φ(x)φ n,n±1 (0)φ n,n±1 (1)φ(∞) with all four possible choices of signs. The strategy is exactly the same as in previous sections and for each choice we will follow the steps performed in Section 3.1.
Since the structure constant C (1,1)(n,n−1)(n,n+1) = 0 only the field φ which appears in the u-channel OPE gives a nonzero contribution. This contribution is proportional to It remains to collect all the contributions together to get

The matrix of anomalous dimensions
There is no need to calculate the remaining two point functions φ n,n+1 (1)φ n,n−1 (0) λ and φ n,n−1 (1)φ n,n−1 (0) λ since the former is identical with φ n,n−1 (1)φ n,n+1 (0) λ and the latter can be obtained from φ n,n+1 (1)φ n,n+1 (0) λ by simply replacing n → −n. For simplicity of notation let us denote φ n,n+1 ≡ φ 1 and φ n,n−1 ≡ φ 2 , then the two-point functions can be represented as The first order coefficients C α,β are given by From eq. (A.2) for the dimensions we have Explicitly, up to O( ) terms we get and for the second order coefficients we have (see (3.28), (3.38) ) C 1,1 = π 2 (3n 3 + 24n 2 + 64n + 44) 3n(n + 2) 2 2 − 4π 2 (n + 1) (n 3 + 7n 2 + 14n + 5) 3n(n + 2) 2 Obviously the correlation function (3.39) satisfies the Callan-Symanzik equation As in Section 3.1 let us introduce renormalized fields and require that the two point functions G In matrix notations we may write Comparing with (3.44) we see that the renormalized two-point function satisfies the equation where the β function and the renormalized coupling g have been introduced in Section 2 and the matrix of anomalous dimensions Γ is defined as Expanding the matrix B up to second order in λ imposing the normalization condition (3.45) and requiring that the matrix of the anomalous dimensions (3.48) be symmetric, we find Now all the ingredients to calculate the matrix of anomalous dimensions (3.48) are at our disposal. Taking also into account the λ-g relation (2.18), we get Notice that all the matrix elements are regular at = 0, all double and single poles in disappeared. At the fixed point g = g * (see (2.21)) Γ (g * ) It is easy to get the eigenvalues of this matrix n−1,n of the IR CFT M p−1 . We can easily identify also the corresponding normalized eigenvectors and establish the explicit map Remarkably the coefficients in (3.55) did not receive neither nor 2 corrections. Thus it is quite perceivable that under the renormalization scheme (3.45), which we have adopted following A. Zamolodchikov, the relation (3.55) is exact. The same phenomenon we will encounter in the next section where a more involved case of mixing of the three fields φ n,n±2 and ∂∂φ n,n will be considered.
3.3 Renormalization of the fields φ n,n+2 , ∂∂φ n,n and φ n,n−2 The OPE φ 1,3 φ n,n+2 includes fields from the conformal families [φ n,n+4 ], [φ n,n+2 ] and [φ n,n ]. Similarly the product φ 1,3 φ n,n−2 produces fields from the families [φ n,n−4 ], [φ n,n−2 ] and [φ n,n ]. Since the dimensions of the primary fields φ n,n±2 and the descendant field ∂∂φ n,n are close to 1 in large p limit, we have a situation when these three fields effectively get mixed along the RG flow 1 [1]. To find the matrix of anomalous dimensions one has to calculate all the two point correlators of these fields.

Performing integrations we get
where the second and the third lines come from the family [φ n,n ] and the last line, from the φ n,n+2 field of the OPE (3.61).
Also in this case the contribution coming from the region D l,∞ is quite similar to the case discussed in Section 2. We should simply take into account that the contribution of the field φ appearing in the u-channel OPE is proportional to The result (c.f. eq. (2.10)) is I(x) can be replaced by and the result of integration is 16π 2 3(n + 5) The function I(x) will be determined using the first equality in (1.11). For the calculation of the contribution of the field φ n,n+2 it is safe to replace the hypergeometric functions simply by 1. Instead for the contribution of the family [φ n,n ] also the first order in x and inx terms should be taken into account. Below we present expressions for the relevant combinations of structure constants with required accuracy  where the two lines correspond to the [φ n,n ] and φ n,n+2 contributions respectively. ii) D l,1 contribution. The relevant OPE is The relation between the three point functions relevant for this case is Notice the flip of sign compared to (3.62) due to the rearrangement of the points 0 and 1. For the function I(x) the second equality in (1.11) should be used. The hypergeometric functions should be expanded around x = 1. When calculating the contribution of φ n,n+2 it would suffice to keep the constant term only while for the contribution of [φ n,n ] also the terms linear in x − 1 (orx − 1) should be taken into account. During the calculation one encounters the same combinations of the structure constants as in (3.77). Finally we get for the D l,1 contribution a result identical to that of D l,0 given by (3.34). Remember that a similar phenomenon we have encountered earlier in Section 3.2.2. iii) D l,∞ contribution Only the field φ appearing in u-channel OPE gives a nonzero contribution. Since and, from the third equality in (1.11), The large p limit of the four-point function is (see Appendix C) I(x) can be replaced by and the result of the integration is Consequently, from the Appendix D, we see that the contribution of the lens-like regions near x ∼ 1 is 1 + ∆ + ∆ n,n − ∆ n,n−2 2∆ n,n xL −1 1 + ∆ + ∆ n,n − ∆ n,n−2 2∆ n,nxL −1 φ n,n (0) + · · · The impact of L −1 on the three-point function: L −1 φ n,n (0)φ n,n+2 (1)φ(∞) = (∆ n,n + ∆ n,n+2 − ∆) φ n,n (0)φ n,n+2 (1)φ(∞) (3.91) In the first expression of (1.11) for I(x), the hypergeometric functions should be expanded up to the linear order in x (orx) terms. The relevant combination of the structure constants: So, the final result for the D l,0 contribution is ii) D l,1 contribution The relevant OPE: φ(x)φ n,n+2 (1) = |x − 1| 2(∆n,n−∆ n,n+2 −∆ )C (n,n) (1,3)(n,n+2) (3.94) The impact of L −1 on the three-point function: L −1 φ n,n (1)φ n,n−2 (0)φ(∞) = (∆ − ∆ n,n − ∆ n,n−2 ) φ n,n (1)φ n,n−2 (1)φ(∞) (3.95) The combination of structure constants required for this computation coincides with that given by eq. (3.92). The explicit calculation shows that in this case too, the D l,1 contribution is identical to that of D l,0 given by (3.93). Note also that the contribution of D l,∞ is negligible. Combining all the contributions for the case at hand we get

B Computation of I(x)
One way to get the result (1.10) for the integral (1.8) is to notice that I(x) satisfies the hypergeometric differential equation independently with respect to the both variables x andx. The starting point is the identity which shows that as a function of the variable x, I(x) is a linear combination of the hypergeometric functions The same conclusion is true also for the conjugate variablex. The condition that the function I(x) is single valued around the points x = 0 a x = 1 fixes a specific combination of holomorphic and anti-holomorphic parts up to a constant which in its turn can be easily evaluated considering the special case x = 0. The final result is presented in eq. (1.10).

C Four-point functions at large p limit
Since the structure constants of OPE for the minimal models are known (see A.5), to construct the correlation functions it remains to calculate related conformal blocks. According to AGT relation [11] this conformal blocks in a simple fashion are related to the instanton part of the Nekrasov partition function of N = 2 SYM theory with the gauge group SU (2) and with four fundamental hypermultiplets. In the large p limit the minimal models approach to a free theory (the central charge c ≈ 1), so it is not surprising that in this limit conformal blocs of degenerated primary fields become very simple and can be expressed in terms of rational (and also logarithmic in the cases when the leading corrections in 1/p is required to be taken into account) functions of the the cross ratio of the coordinates. It is straightforward to compute Nekrasov partition [12] function up to desired order in instanton expansion using combinatorial formula found in [13] and extended to the case with extra hypermultiplets in [14]. Computing the first few coefficients of the instanton expansion (for more confidence we made calculations up to 6th order ), adjusting appropriately the parameters in order to get the required conformal block and finally taking the large p limit one can easily guess the exact dependence of the conformal block on the cross ratio of the insertion points (which is the same as the instanton counting parameter, from the gauge theory point of view).