1 Introduction

Calculation of anomalous dimensions of composite operators belongs to the standard tasks of any quantum field theory calculation. For example, in quantum chromodynamics, anomalous dimensions of leading-twist-two operators govern the scaling behavior of quark and gluon distributions in hadrons and have to be known with high precision. Nowadays the anomalous dimensions are known at three loops; see Refs. [1–3] and references therein. Beyond the two-loop approximation such calculations are feasible only with the help of the advanced methods of computer algebra. Since the calculations are fully automated, finding errors becomes highly nontrivial task and any approach which can provide a check of the final results is very helpful.

Given that the theory depends not only on the coupling constant but also some other parameters such as the dimension of an internal symmetry group, one can organize an expansion over these parameters. The best known example of this kind is the 1 / N expansion; see Ref. [4] for a review. Agreement between the results obtained in the perturbative and 1 / N expansions serves as a powerful test for the validity of calculations. However, the calculations in the 1 / N expansion are much harder than perturbative calculations: only two RG functions—indices of the basic fields in the nonlinear \(\sigma \)-model and Gross–Neveu model—are available at \(1/N^3\) order [5–7]. In QCD the calculations rarely go beyond the leading order in \(1/N_\mathrm{f}\) (where \(N_\mathrm{f}\) is the number of flavors). At leading order in \(1/N_\mathrm{f}\) the anomalous dimensions of twist-two operators were calculated in Refs. [8, 9], but an extension of these results to the next order is hardly possible.

A new approach for calculating the anomalous dimensions of leading-twist operators was proposed in Refs. [10, 11]. It is still a perturbative approach, however, the contributing diagrams are completely different from those in the standard technique. The approach is based on a remarkable property of leading-twist operators: namely, a divergence of such an operator, \(\mathcal {O}_{\mu _1,\ldots \mu _j}\), vanishes in a free theory [12]

$$\begin{aligned} \partial ^{\mu _1}\mathcal {O}_{\mu _1,\ldots \mu _j}(x)=0. \end{aligned}$$
(1)

In the interacting theory the r.h.s. of Eq. (1) is non-zero but proportional to the coupling constant. This identity allows one to extract the \(\ell \)-loop contribution to the anomalous dimension of the operator \(\mathcal {O}_{\mu _1,\ldots \mu _j}\) from \(\ell -1\) loop diagrams only. In particular, the one-loop anomalous dimensions of leading-twist operators do not require calculation of loop integrals at all [11, 13].

The method developed in Ref. [11] is adjusted to local operators and relies heavily upon the so-called “conformal scheme” renormalization [14–16]. In our opinion it is more convenient to stay within the standard \(\overline{\text {MS}}\) scheme and the formalism of non-local (light-ray) operators technique. This technique proves to be more effective and flexible as we will demonstrate on the example of calculation of two-loop anomalous dimensions in the su(n) symmetric \(\varphi ^3\) model [17].

The paper is organized as follows. In Sect. 2 we introduce the model and fix notations. In Sect. 3 we recall the light-ray operator technique. Section 4 is devoted to the calculation of the divergence of the conformal operator. The details of the calculation of two-loop correlators are presented in Sect. 5. Our conclusions are in Sect. 7. In the appendices we explain some technical issues and details of the derivation.

2 Generalities

The su(n) symmetric \(\varphi ^3\) model is a scalar field theory in \(d=6-2\epsilon \equiv 2\mu \) dimensions with an action

$$\begin{aligned} S(\varphi )=&\int \mathrm{d}^dx\left[ \frac{1}{2}(\partial \varphi ^a)^2+\frac{1}{6}gM^\epsilon \,\mathrm{d}^{abc}\varphi ^a\varphi ^b\varphi ^c\right] , \end{aligned}$$
(2)

where \(a=1,\ldots , n^{2}-1\),

$$\begin{aligned} \mathrm{d}^{abc}=2{{\mathrm{tr}}}t^a\{t^b t^c\}, \end{aligned}$$
(3)

\(t^{a}\) are the generators of the su(n) algebra normalized in the conventional way, \({{\mathrm{tr}}}t^a t^b =1/2\). The theory is multiplicatively renormalizable,

$$\begin{aligned} S_R(\varphi ,g)=S(\varphi _0,g_0), \end{aligned}$$
(4)

where

$$\begin{aligned} \varphi _0=Z_\varphi \,\varphi \quad \text {and} \quad g_0=M^\epsilon Z_g g. \end{aligned}$$

Two-loop expressions for the renormalization constants \(Z_1=Z_\varphi ^2\) and \(Z_3= Z_g Z_\varphi ^3\) can be found in Refs. [17, 18]. The \(\beta \)-function of the charge \(u=g^2/(4\pi )^3\) and the field anomalous dimension \(\gamma _\varphi \) are

$$\begin{aligned} \beta (u)&=-2\epsilon u-u^2\frac{n^2-20}{2n} +\mathcal {O}(u^3),\nonumber \\ \gamma _\varphi (u)&= u\frac{n^2-4}{12 n}\left( 1+u\frac{n^2-100}{36 n}\right) +\mathcal {O}(u^3). \end{aligned}$$
(5)

At the critical point \(u_*, \beta (u_*)=0\),

$$\begin{aligned} u_*={4n\epsilon }/(20-n^2)+ O(\epsilon ^2); \end{aligned}$$
(6)

the theory enjoys the scale and conformal invariance.

It is well known that in a conformal theory the form of the two-point correlation function of conformal operators is fixed up to normalization. In particular, the correlator of the conformal traceless symmetric operators has the form

$$\begin{aligned} \langle {\mathcal {O}^{(n)}_{j}(x)\mathcal {O}^{(\bar{n})}_{j'}(y)\rangle }=\delta _{jj'}\delta _{\Delta _j\Delta _{j'}}\frac{C_jI_{n,\bar{n}}^j(x-y)}{((x-y)^2)^{\Delta _j}}. \end{aligned}$$
(7)

Here \(j (j')\) is the spin of the operator. The vectors n and \(\bar{n}\) are two light-like vectors, \(n^2=\bar{n}^2=0,\) and \(\mathcal {O}^{(n)}_j(x)~(\mathcal {O}^{(\bar{n})}_{j'}(y))\) is a contraction of an operator with the vector \(n~(\bar{n})\), for instance,

$$\begin{aligned} \mathcal {O}^{(n)}_j(x)=n^{\mu _1}\ldots n^{\mu _j} \mathcal {O}_{\mu _1\ldots \mu _j}(x). \end{aligned}$$
(8)

\(\Delta _j\) and \(\Delta _{j'}\) are the scaling dimensions of the operators, \(C_j\) is a normalization constant and

$$\begin{aligned} I_{n,\bar{n}}(x)=(n\bar{n})-\frac{2(nx)(\bar{n} x)}{x^2}. \end{aligned}$$
(9)

The scaling dimension of the operator is given by the sum of canonical and anomalous dimensions at a critical point, \(\Delta _j=\Delta _j^{(0)}+\gamma _j\), where \(\gamma _j\equiv \gamma _j(u_*)\). For the leading-twist operators of spin j, \(\Delta _j^{(0)}=2\mu -2+j\).

The anomalous dimension of an operator in MS scheme, \(\gamma _j(u)\), is a function of a coupling constant only. It can be restored from its critical value, \(\gamma _j=\gamma _j(u_*)\), provided that the latter is known as a function of \(\epsilon \).

The correlation function for the divergence of the conformal operator

$$\begin{aligned} \partial \mathcal {O}^{(n)}_j(x)\equiv j\,n^{\mu _2}\ldots n^{\mu _j}\,\partial ^{\mu _1}\mathcal {O}_{\mu _1\mu _2\ldots \mu _j}(x) \end{aligned}$$
(10)

can be obtained from Eq. (7). If x is chosen in the transverse plane, \((xn)=(x\bar{n})=0\), the ratio of the two correlation functions

$$\begin{aligned} \mathcal {T}_j(u_*)&=x^2(n\bar{n}) \frac{\langle {\partial \mathcal {O}_j^{(n)}(x)\partial \mathcal {O}^{(\bar{n})}_j(0)}\rangle }{\langle {\mathcal {O}_j^{(n)}(x)\mathcal {O}_j^{(\bar{n})}(0)}\rangle } \end{aligned}$$
(11)

is a function of the anomalous dimension \(\gamma _j\) only,

$$\begin{aligned} \mathcal {T}_j&= 2j\gamma _j \left[ \frac{(2\mu -3+j)(\mu -1+j)}{\mu -2+j}+2\gamma _j\right] . \end{aligned}$$
(12)

This ratio, in full agreement with the result of Ref. [12], vanishes provided that \(\gamma _j=0\).

The perturbation series for \(\gamma _j\) and \(\mathcal {T}_j\)

$$\begin{aligned} \gamma _j&=u_* \,\gamma _j^{(1)}+u_*^2 \,\gamma _j^{(2)}+\cdots , \nonumber \\ \mathcal {T}_j&= \varkappa _j(u_*\,{T}_j^{(1)}+u_*^2 \, {T}_j^{(2)}+\cdots ), \end{aligned}$$
(13)

are related to each other. For later convenience we choose the normalization factor \(\varkappa _j\) as follows:

$$\begin{aligned} \varkappa _j=\frac{2j(j+2)(j+3)}{j+1}. \end{aligned}$$
(14)

Substituting the series (13) into Eq. (12) one finds the following relations between the expansion coefficients:

$$\begin{aligned} \gamma _j^{(1)}&={T}_j^{(1)},\nonumber \\ \gamma _j^{(2)}&={T}_j^{(2)}-\frac{2(j+1)}{(j+2)(j+3)} (T_j^{(1)})^2 \nonumber \\&\quad +\frac{2j^2+5j+1}{(j+1)(j+2)(j+3)} \epsilon ^{(1)}\,T_j^{(1)}, \end{aligned}$$
(15)

and so on. Here \(\epsilon ^{(1)}=(n^2-20)/4n\).

Since the divergence of the conformal operator is proportional to the coupling constant, \(\partial \mathcal {O}_j^{(n)}\sim O(g)\), the ratio \(\mathcal {T}_j\) contains a “kinematical” factor \(u\sim g^2\). Thus, in order to determine \(\mathcal {T}_j\) and, hence, the anomalous dimension \(\gamma _j\), with \(O(u^\ell )\) accuracy the corresponding correlation functions have to be calculated at one order in u less. In Ref. [11] one-loop anomalous dimensions were reproduced by this method in \(\varphi ^3\) and \(N=4\) SUSY models. Going beyond the leading order requires an effective technique for the calculation of the two-point correlation functions; otherwise one gains nothing in comparison with the standard approach.

3 Light-ray vs. local operators

The first task is to find a convenient description for local operators. As we will argue, the light-ray operator technique [19] is a more suitable one. The light-ray operatorFootnote 1

$$\begin{aligned}{}[\mathcal {O}(x;z_1,z_2)]&=[\varphi ^{a}(x+z_1 n)\varphi ^{a}(x+z_2 n)] \end{aligned}$$
(16)

is defined as the generating function for the renormalized local operators

$$\begin{aligned}{}[\mathcal {O}(x;z_1,z_2)]&\equiv \sum _{k,m} z_1^k z_2^m [\mathcal {O}_{km}](x) \nonumber \\&=\sum _{k,m,k',m'} z_1^k z_2^m\, Z_{km}^{k'm'}\,\mathcal {O}_{k'm'}(x). \end{aligned}$$
(17)

Here \([\mathcal {O}_{km}]\) is the renormalized (in MS scheme) local monomial \(\mathcal {O}_{km}= \partial _+^k\varphi ^a(x)\partial _+^m\varphi ^a(x)/{k!m!}\) and \(\partial _+ =(n \partial _x)\). The sum in Eq. (17) can be replaced by the action of some integral operator on the bare operator

$$\begin{aligned}{}[\mathcal {O}(x;z)]=Z\mathcal {O}(x;z). \end{aligned}$$
(18)

Here we introduced the shorthand notation \(z=\{z_1,z_2\}\). The integral operator Z can be written in the form [18]

$$\begin{aligned} Zf(z)=\int \mathrm{d}\alpha \,\mathrm{d}\beta \, Z(\alpha ,\beta )\, f(z_{12}^\alpha ,z_{21}^\beta ), \end{aligned}$$
(19)

where \(z_{12}^\alpha =\bar{\alpha }z_1+\alpha z_2\) and \(\bar{\alpha }=1-\alpha \). The renormalization kernel \(Z(\alpha ,\beta )\) is given by a series in \(1/\epsilon \) and the coupling u. The light-ray operator (17) satisfies the RG equation

$$\begin{aligned} (M\partial _M +\beta (u)\partial _u+ \mathbb {H}(u))[\mathcal {O}(x;z)]=0, \end{aligned}$$
(20)

where the evolution kernel \(\mathbb {H}\) is an integral operator,

$$\begin{aligned} \mathbb {H}(u)=-\left( M\frac{\mathrm{d}}{\mathrm{d}M} Z\right) \,Z^{-1}+2\gamma _\varphi , \end{aligned}$$
(21)

which encodes all information on the anomalous dimension matrices for local operators.

At the critical point \(u=u_*\) local operators can be classified according to the representations of the conformal group. An operator with the lowest scaling dimension in the representation is called a conformal operator. The leading-twist operator,Footnote 2 \(\mathcal {O}_j\), is uniquely determined by its scaling dimension \(\Delta _j\). The expansion of the light-ray operator (17) over conformal operators and their descendants reads [18]

$$\begin{aligned}{}[\mathcal {O}(x;z)]=\sum _{jk}\Psi _{jk}(z_1,z_2) \,\partial _+^k \mathcal {O}_j(x), \end{aligned}$$
(22)

where the coefficients \(\Psi _{jk}(z_1,z_2)\) are homogeneous polynomials of degree \(j+k\) in \(z_1,z_2\). These polynomials are eigenfunctions of the evolution kernel \(\mathbb {H}(u_*)\),

$$\begin{aligned} \mathbb {H}(u_*)\Psi _{jk}(z)=\gamma _j\,\Psi _{jk}(z), \end{aligned}$$
(23)

with the corresponding eigenvalues being the anomalous dimensions, \(\gamma _j=\gamma _j(u_*)\). Since the theory enjoys conformal invariance at the critical point \(u=u_*\), the evolution kernel commutes with three generators of the collinear subgroup of conformal group,Footnote 3

$$\begin{aligned}{}[S_{\pm ,0},\mathbb {H}(u_*)]=0. \end{aligned}$$
(24)

The generators, however, deviate from their canonical form (see Appendix A),

$$\begin{aligned} S_-^{(0)}&=-\partial _{z_1} - \partial _{z_2}, \nonumber \\ S_0^{(0)}&=z_1\partial _{z_1}+z_2\partial _{z_2}+2, \nonumber \\ S_+^{(0)}&=z_1^2\partial _{z_1}+z_2^2\partial _{z_2}+2(z_1+z_2)\, \end{aligned}$$
(25)

due to quantum corrections,

$$\begin{aligned} S_{\pm ,0}=S_{\pm ,0}^{(0)}+\Delta S_{\pm ,0}. \end{aligned}$$
(26)

Two of the generators are known to all orders,

$$\begin{aligned} \Delta S_-=0, \quad \Delta S_0=-\epsilon +\frac{1}{2} \mathbb {H}(u_*), \end{aligned}$$
(27)

while corrections to the generator of special conformal transformations can be calculated order by order in perturbation theory [18]. The leading correction is

$$\begin{aligned} \Delta S_+=(z_1+z_2)\left( -\epsilon +\frac{1}{2} \mathbb {H}(u_*)\right) +O(\epsilon ^2). \end{aligned}$$
(28)

It follows from Eqs. (23) and (24) that the operators \(S_\pm \) act as raising (lowering) operators on the set of eigenfunctions \(\Psi _{jk}\),

$$\begin{aligned} S_\pm \Psi _{jk}\sim \Psi _{jk\pm 1}. \end{aligned}$$
(29)

In turn \(S_0\) counts conformal spin of the operator \(\mathcal {O}_{jk}\),

$$\begin{aligned} S_0\Psi _{jk}=j_{jk}\Psi _{jk}=\frac{1}{2}(\Delta _{jk}+S_{jk})\Psi _{jk}, \end{aligned}$$
(30)

where \(\Delta _{jk}=\Delta _j+k\) and \(S_{jk}\) are the scaling dimension and spin of the operator, respectively. One derives immediately from (29) that the polynomial \(\Psi _{jk=0}\) accompanying the conformal operator \(\mathcal {O}_j\) in the expansion (22) is a simple power, \(\Psi _{jk=0}(z_1,z_2)\sim (z_1-z_2)^j\) and all other eigenfunctions have the form \(\Psi _{jk}(z_1,z_2)\sim S_+^k (z_1-z_2)^j\).

3.1 Scalar product

Equation (23) can be considered as a standard quantum mechanical problem for the Hamiltonian \(\mathbb {H}\). In order to make this analogy complete one needs to introduce a scalar product on the space of the eigenfunctions. Clearly, such a scalar product has to be adjusted to the symmetries of the problem. At leading order the answer is given by the standard sl(2) invariant scalar product [20, 21]

$$\begin{aligned} (\psi ,\phi )_{0}=\frac{1}{\pi ^2}\iint \limits _{|z_k|<1} \,\mathrm{d}^2z_1 \,\mathrm{d}^2z_2 \,(\psi (z_1,z_2))^\dagger \,\phi (z_1,z_2). \end{aligned}$$
(31)

The integration goes over the unit disks \(|z_{k}|<1, k=1,2\). The generator \(S_0^{(0)}\) is a self-adjoint operator with respect to this scalar product and \((S_+^{(0)})^\dagger =-S_-^{(0)}\). We want to find a deformation of the scalar product (31) that keeps these relations for the complete generators, \(S_0=S_0^\dagger \) and \( S_+^\dagger =-S_-\). Let us look for the solution in the form

$$\begin{aligned} (\psi ,\phi )_{\varpi }=(\psi , \varpi \phi )_0, \quad \varpi =\mathbbm {1}+ u_* \varpi ^{(1)}+\cdots , \end{aligned}$$
(32)

where \(\varpi ^{(1)}\) is a self-adjoint operator with respect to the scalar product (31). The conjugation conditions for the generators imply

$$\begin{aligned} \Delta S_0^{(1)}-(\Delta S_0^{(1)})^\dagger&=[S_0^{(0)},\varpi ^{(1)}], \nonumber \\ \Delta S_+^{(1)}&=[S_+^{(0)},\varpi ^{(1)}]. \end{aligned}$$
(33)

The one-loop corrections to the generators involve the kernel \(\mathcal {H}^{(1)}(\mathbb {H}(u)=\sum _k u^k\mathcal { H}^{(k)})\), which is given by the following expression:

$$\begin{aligned} \mathcal { H}^{(1)}=2(\gamma ^{(1)}_\varphi -\lambda _s \mathcal {H}^+). \end{aligned}$$
(34)

Here \(\lambda _s\) is a color factor, \(\lambda _s=(n^2-4)/n\) and

$$\begin{aligned} \mathcal {H}^+\psi ({z})=\int _0^1\,\mathrm{d}\alpha \int _0^{\bar{\alpha }}\,\mathrm{d}\beta \,\psi (z_{12}^\alpha ,z_{21}^\beta ). \end{aligned}$$
(35)

The explicit expression for \(\varpi ^{(1)}\) and details of the derivation can be found in Appendix B.Footnote 4

Since the eigenfunctions \(\Psi _{jk}\) are mutually orthogonal w.r.t. the scalar product (32), one can represent the conformal operator as the scalar product of the coefficient function with the light-ray operator

$$\begin{aligned} \mathcal {O}_j(x)=(z_{12}^j, [\mathcal {O}(x,z)])_{\varpi }. \end{aligned}$$
(36)

This representation for the conformal operator is the most convenient one for further analysis. We demonstrate it on the following example. The conformal operator is usually defined as an operator which vanishes under special conformal transformations, \(K_{\bar{n}}=K\cdot \bar{n}, \delta _{K_{\bar{n}}}\mathcal {O}_j(0)=0\). This property becomes transparent in the representation (36) if one takes into account that

$$\begin{aligned} \delta _{K_{\bar{n}}}[\mathcal {O}(z)]=2(n\bar{n}) S_+[\mathcal {O}(z)] \end{aligned}$$

(we put here \([\mathcal {O}(z)]=[\mathcal {O}(x=0;z)]\)) and uses the fact that the generators \(S_+\) and \(S_-\) are conjugate to each other w.r.t. the scalar product (32), \(S_+^\dagger =-S_-\).

4 Divergence of conformal operator

In order to construct the divergence of the conformal operator \(\partial \mathcal {O}_j^{(n)}\), Eq. (10), we calculate first the divergence of the light-ray operator (18)

$$\begin{aligned}{}[\partial \mathcal {O}(x;z)]\equiv \frac{\partial }{\partial x^\mu }\frac{\partial }{\partial n_\mu }[\mathcal {O}(x;z)]. \end{aligned}$$
(37)

Taking the n-derivative one cannot, however, keep \(n^2=0\) any longer and has to take into account terms linear in \(n^2\) which account for the trace subtraction in \([\mathcal {O}(x;z)]\). Taking the corresponding modification into account, see e.g. Refs. [11, 19, 22], one gets for (37)

$$\begin{aligned}{}[\partial \mathcal {O}(x;z)]&= \frac{\partial }{\partial x^\mu }\nabla _\mu Z\varphi ^{a}(x+z_1 n)\varphi ^{a}(x+z_2 n) \end{aligned}$$
(38)

where \(\nabla ^\mu \) is a differential operator

$$\begin{aligned} \nabla _\mu =\frac{\partial }{\partial n^\mu }-\frac{1}{2}(\mu -1+n\cdot \partial _n)^{-1} n^\mu \frac{\partial ^2}{\partial n^2}, \end{aligned}$$
(39)

which commutes with the renormalization factor Z and acts on the fields directly. After some simple algebra one gets

$$\begin{aligned}{}[\partial \mathcal {O}(x;z)]&=\frac{1}{2} (S_0^{(\epsilon )}-1)^{-1}Z\{ S_+^{(\epsilon )} \,\partial _x^2\, \mathcal {O}(x;z) \nonumber \\&\quad -L^{(\epsilon )}_{21}\, \partial ^2\varphi ^a(x+z_1n)\varphi ^a(x+z_2n) \nonumber \\&\quad -L_{12}^{(\epsilon )}\,\varphi ^a(x+z_1n)\partial ^2\varphi ^a(x+z_2n)\}, \end{aligned}$$
(40)

where \(S_0^{(\epsilon )}=S_0^{(0)}-\epsilon , S_+^{(\epsilon )}=S_+^{(0)}-\epsilon (z_1+z_2)\), and

$$\begin{aligned} L_{21}^{(\epsilon )}=\partial _{z_2} z_{21}^{2}-\epsilon z_{21}, \quad L_{12}^{(\epsilon )}&=\partial _{z_1} z_{12}^{2}-\epsilon z_{12}. \end{aligned}$$
(41)

Using the equations of motion (EOM) one can replace in this expression

$$\begin{aligned} \partial ^2\varphi ^a(x)\mapsto \frac{1}{2} g M^\epsilon Z_3 Z_1^{-1}d^{abc}\varphi ^b(x)\varphi ^c(x). \end{aligned}$$
(42)

We want to stress here that Eq. (40) holds for arbitrary coupling u but not only at the critical value. Since the l.h.s. of Eq. (40) is a finite (renormalized) operator the r.h.s. can be expressed in terms of renormalized operators with finite coefficients. These operators can be chosen as follows: the two-particle operator \(\mathcal {O}_1=\partial ^{2}\mathcal {O}(x;z)\) and the three-particle operator

$$\begin{aligned} \mathcal {O}_2&=\mathcal {O}^{(d)}(x;w) \nonumber \\&=g\mathrm{d}^{abc}\varphi ^a(x+w_1n)\varphi ^b(x+w_2n)\varphi ^c(x+w_3n), \end{aligned}$$
(43)

where \(w=\{w_1,w_2,w_3\}\). The operators \(\mathcal {O}_1\) and \(\mathcal {O}_2\) mix under renormalization. The mixing matrix (integral operator acting on fields variables) has a lower triangular form,

$$\begin{aligned}{}[\mathcal {O}_k]=Z_{km} \mathcal {O}_m. \end{aligned}$$
(44)

Here \(Z_{11}=Z\) is the renormalization constant of the light-ray operator, Eq. (18), \(Z_{12}=0, Z_{21}=O(u^2)\), and the element \(Z_{22}\) is given, at the one-loop order, by the sum of two-particle kernels

$$\begin{aligned} Z_{11}&=1-\frac{u}{\epsilon }\,\lambda _s\,\mathcal {H}_{12}^{+}+O(u^2), \nonumber \\ Z_{22}&=\mathbbm {1}+\frac{u}{2\epsilon }\sum \limits _{i<k}(\lambda _s\mathcal {H}_{ik}^{d}- \lambda _d\,\mathcal {H}_{ik}^{+})+ O(u^2), \end{aligned}$$
(45)

where \(\lambda _s,\lambda _d\) are color factors

$$\begin{aligned} \lambda _s=(n^2-4)/n, \quad \lambda _d=(n^2-12)/n. \end{aligned}$$
(46)

The kernel \(\mathcal {H}_{ik}^+\) is defined by Eq. (35) and \(\mathcal {H}_{ik}^d\) has the form

$$\begin{aligned} \mathcal {H}_{12}^{d}f(z_1,z_2)=\int _0^1\,\mathrm{d}\alpha \, \alpha \bar{\alpha }\, f(z_{12}^\alpha ,z_{12}^\alpha ). \end{aligned}$$
(47)

The subscripts ik show the arguments the kernel acts on.

Using these results we can rewrite (40) as follows:

$$\begin{aligned}{}[\partial \mathcal {O}(x;z)]&=\frac{1}{2} (S_0^{(\epsilon )}-1)^{-1}\sum \limits _{k=1,2} A_k [\mathcal {O}_k(x;z)]. \end{aligned}$$
(48)

The operators \(A_k\) have the following form:

$$\begin{aligned} A_1&=Z_{11} S_+^{(\epsilon )} Z_{11}^{-1}-M^{-\epsilon } A_2 Z_{21} Z_{11}^{-1}, \nonumber \\ A_2&=-\frac{1}{2}M^\epsilon Z_3 Z_1^{-1}Z_{11}(L_{12}^{(\epsilon )}S_2+L_{21}^{(\epsilon )} S_1)Z_{22}^{-1}, \end{aligned}$$
(49)

where the operators \(S_1,S_2\) map functions of three variables to functions of two variables

$$\begin{aligned}{}[S_1 f](z_1,z_2)&=f(z_1,z_1,z_2), \nonumber \\ [S_2 f](z_1,z_2)&=f(z_1,z_2,z_2). \end{aligned}$$
(50)

At one loop the operators \(A_k\) take the form

$$\begin{aligned} A_1&=S_+(u) + {u}(\lambda _s \mathcal {H}^+-\gamma _\varphi ^{(1)})(z_1+z_2)+O(u^2), \nonumber \\ A_2&=-\frac{1}{2} M^\epsilon ( L_{12}S_2+L_{21} S_1 +(\epsilon -u\lambda _s \mathcal {H}^+)z_{12}S_{12} \nonumber \\&\quad -u z_{12} S_{12} (\lambda _s \mathcal {H}_{13}^d-\lambda _d \mathcal {H}^+_{13})+O(u^2)). \end{aligned}$$
(51)

Here \(S_+(u)\) is given by Eqs. (26), (27) for arbitrary \(u, u_*\rightarrow u\),

$$\begin{aligned} S_{12}=S_1-S_2, \quad L_{km}=L_{km}^{(\epsilon \mapsto 0)} \end{aligned}$$

and deriving (51) we made use of the symmetry of the three-particle operator \(\mathcal {Q}^{(d)}(x;w_1,w_2,w_3)\) under permutation of the w variables.

Let us stress again that the operators \(A_k\) do not contain singular terms for arbitrary u. Using one-loop expressions for Z factors, Eq. (45), it can be checked that all pole terms cancel at order O(u). In particular,

$$\begin{aligned}&Z_3 Z_1^{-1}Z_{11}(L_{12}S_2+L_{21} S_1)Z_{22}^{-1}\nonumber \\&\quad = L_{12}S_2+L_{21} S_1+O(u^2). \end{aligned}$$
(52)

Starting from the representation (36) for the conformal operator, we get for its divergence

$$\begin{aligned} \partial \mathcal {O}_j(x)=(z_{12}^j,[\partial \mathcal {O}(x;z)])_{\varpi }. \end{aligned}$$
(53)

Making use of Eqs. (48)–(51) one finds that the divergence \(\partial \mathcal {O}_j(x)\) is given by the sum of two-particle and three-particle (renormalized) operators

$$\begin{aligned} \partial \mathcal {O}_j(x)=\frac{1}{2(j+1-\epsilon )}(\mathcal {R}^{(2)}_j(x)+\mathcal {R}^{(3)}_j(x)). \end{aligned}$$
(54)

The prefactor on the r.h.s. of Eq. (54) is the eigenvalue of the operator \((S_0^{(\epsilon )}-1)^{-1}\) on the function \(z_{12}^j\). The two-particle term \(\mathcal {R}_j^{(2)}\) has the form

$$\begin{aligned} \mathcal {R}^{(2)}_j(x)&= {\gamma _j}\,(z_{12}^j,\, (z_1+z_2) [\mathcal {O}_1(x;z)])+O(u_*^2). \end{aligned}$$
(55)

We recall that \(\gamma _j=\gamma _j(u_*)\) and \(\mathcal {O}_1(x;z)=\partial ^2 \mathcal {O}(x;z)\). Let us note that the term \(\sim S_+\) in the expression for \(A_1\), Eq. (51), vanishes inside the scalar product since \((z_{12}^j, S_+\cdots )=-(S_-z_{12}^j,\cdots )=0\). In turn, the expression for the three-particle contribution can be written as follows:

$$\begin{aligned} \mathcal {R}^{(3)}_j(x)=\mathcal {R}^{(3, 0)}_j(x)+\mathcal {R}^{(3,1)}_j(x)+O(\epsilon ^2), \end{aligned}$$
(56)

where

$$\begin{aligned} \mathcal {R}^{(3, 0)}_j(x)&=-{M^\epsilon }(z_{12}^j,\,L_{21}S_1 [\mathcal {O}_2(x;z)])_{\varpi }, \nonumber \\ \mathcal {R}^{(3,1)}_j(x)&=-{M^\epsilon }(z_{12}^j,z_{12} S_1 X_j[\mathcal {O}_2(x;z)])_{\varpi } \end{aligned}$$
(57)

and the operator \(X_j\) has the form

$$\begin{aligned} X_j=\epsilon -\gamma _\varphi +\gamma _j/2 -u_*(\lambda _s \mathcal {H}_{13}^d-\lambda _d \mathcal {H}^+_{13}). \end{aligned}$$
(58)

This expression follows immediately from (51) if one takes into account that spin j is even and \(A_2\) is symmetric under interchange \(z_1\leftrightarrow z_2\).

It is clear from (55) that the expansion of the two-particle term \(\mathcal {R}^{(2)}_j\) over conformal operators does not contain the operator of spin j,

$$\begin{aligned} \mathcal {R}_j^{(2)}\sim \partial ^2\left( \sum _{m=0}^{j-2} c_m \partial ^{j-m-2}_+ \mathcal {O}_m(x)\right) . \end{aligned}$$
(59)

Taking into account that \(\langle {\mathcal {O}_j(x)\mathcal {O}_k(0)}\rangle =0\) for \(k<j\) one derives that \(\langle {\mathcal {R}_j^{(2)}(x)\, \partial \mathcal {O}_j(0)}\rangle =0\). This, in virtue of Eq. (54), results in the following relation for the correlators:

$$\begin{aligned} \langle {\mathcal {R}_j^{(2)}(x)\mathcal {R}_j^{(2)}(0)}\rangle =-\langle {\mathcal {R}_j^{(2)}(x)\mathcal {R}_j^{(3)}(0)}\rangle . \end{aligned}$$
(60)

It can be shown that in the correlator \(\langle { \partial \mathcal {O}_j(x) \partial \mathcal {O}_j(0)}\rangle \) one can replace (54) by a simpler expression

$$\begin{aligned} \partial \mathcal {O}_j(x)=\frac{1}{2(j+1)}(\mathcal {R}^{(2)}_j(x)+\mathcal {R}^{(3,0)}_j(x)). \end{aligned}$$
(61)

The omitted terms

$$\begin{aligned} \Delta \mathcal {X}=\frac{1}{2(j+1)}\left( \mathcal {R}^{(3,1)}_j(x)+\frac{\epsilon }{j+1}\mathcal {R}^{(3,0)}_j(x)\right) +O(\epsilon ^2) \end{aligned}$$

give rise to the correction of order \(O(\epsilon ^3)\). In order to verify this it is sufficient to notice that \(\Delta \mathcal {X}\) can be rewritten in the form \((z_{12}^j, F\, S_- z_{12} [\mathcal {O}_2(x;z)])_{\varpi } \), where F is some operator whose explicit expression is not relevant. Inside the correlator the generator \(S_-\) acts, finally, on the function \(z_{12}^j\), nullifying it.

Thus in order to find the anomalous dimension \(\gamma _j\) at order \(O(\epsilon ^2)\) one has to calculate the three correlators

$$\begin{aligned} \langle {\mathcal {O}_j\mathcal {O}_j}\rangle , \quad \langle {\mathcal {R}_j^{(2)}\mathcal {R}_j^{(3,0)}}\rangle , \quad \langle {\mathcal {R}_j^{(3,0)}\mathcal {R}_j^{(3,0)}}\rangle \end{aligned}$$

at one-loop order. We will do it in the next section.

Finally, we note that for \(j=2\) the r.h.s. of Eq. (54) has to vanish identically since the operator \(\mathcal {O}_{\mu \nu }\) is, up to EOM terms, the energy-momentum tensor. The two-particle term \(\mathcal {R}_{j}^{(2)}\) is proportional to the anomalous dimension \(\gamma _j\) and therefore vanishes for \(j=2\). In order to check it for \(\mathcal {R}_j^{(3)}\), it is sufficient to take into account that only the linear term in the expansion of three-particle operator

$$\begin{aligned} \mathcal {O}_2(x; w)&\sim (w_1+w_2+w_3)\cdot \,\mathrm{d}^{abc}\partial _+\varphi ^a(x)\varphi ^b(x)\varphi ^c(x)\nonumber \\&= (S_+^{(1,1,1)}\cdot 1)\,\, \mathrm{d}^{abc}\partial _+\varphi ^a(x)\varphi ^b(x)\varphi ^c(x) \end{aligned}$$

contributes to (56) for \(j=2\). After simple algebra one finds that \(\mathcal {R}_{j=2}^{(3)}=O(\epsilon ^2)\).

5 Correlators

Fig. 1
figure 1

The leading order diagrams for the correlator of two conformal operators, \(\langle {\mathcal {O}^{(n)}_j(x)\mathcal {O}_j^{(\bar{n})}(0)}\rangle \)

Full size image

5.1 LO correlators

In order to give a glimpse of the technique we start with calculation of the necessary correlators at leading order. The correlator \(\langle {\mathcal {O}^{(n)}_j(x)\mathcal {O}_j^{(\bar{n})}(0)}\rangle \) is given by the sum of two diagrams, shown schematically in Fig. 1. They are given by the product of the propagators and give rise to identical contributions to the correlation function. Assuming that x is chosen in a transverse plane, \((x,n)=(x,\bar{n})=0\), one represents the propagator as

$$\begin{aligned} D(x+z_1n-\bar{w}_1 \bar{n})=D(x){(1-z_1\bar{w}_1 r)^{-(\mu -1)}}, \end{aligned}$$
(62)

where \(r=2(n\bar{n})/x^2\) and

$$\begin{aligned} D(x)=\Gamma (\mu -1)\,/(4\pi ^{\mu }{(x^2)^{\mu -1}}). \end{aligned}$$
(63)

At leading order one replaces \(\mu -1\mapsto 2\) so that the second factor in (62) is nothing else than the reproducing kernel, \(\mathcal {K}_{s=1}(z_1, w_1 r)\), corresponding to the spin \(s=1\); see Eq. (A.5). Therefore starting from Eq. (36) one gets for the correlator

$$\begin{aligned} \langle {\mathcal {O}^{(n)}_j(x)\mathcal {O}_j^{(\bar{n})}(0)}\rangle&=2\xi D^2(x)\left( z_{12}^j\left| \prod _{k=1}^2\mathcal {K}_1(z_k, w_k r)\right| w_{12}^j\right) \nonumber \\&=2\xi D^2(x) r^j\, ||w_{12}^j||^2_{11}, \end{aligned}$$
(64)

where \(\xi =n^2-1\) is the isotopic factor, the scalar products correspond to the conformal spin \(s=1\), and we take into account the property of the reproducing kernel (A.6). The norm of \(w_{12}^j\) is given by the following expression:

$$\begin{aligned} ||w_{12}^j||^2_{s_1s_2}=j!\prod _{k=1}^2\frac{\Gamma (2s_k)}{\Gamma (j+2s_k)}\frac{\Gamma (2j+2(s_1+s_2)-1)}{\Gamma (j+2(s_1+s_2)-1)}. \end{aligned}$$
Fig. 2
figure 2

The LO diagrams for the correlator of the divergence of the conformal operators, \(\langle {\partial \mathcal {O}^{(n)}_j(x)\partial \mathcal {O}_j^{(\bar{n})}(0)}\rangle \)

Full size image

The diagrams for the correlator \(\langle {\partial \mathcal {O}^{(n)}_j(x)\partial \mathcal {O}_j^{(\bar{n})}(0)}\rangle \) are shown in Fig. 2. On the leftmost diagram the points \(z_1n\) and \(\bar{w}_1 \bar{n}\) are connected by two propagators,

$$\begin{aligned} D^2(x+z_1n-\bar{w}_1 \bar{n})&=D^2(x){(1-z_1\bar{w}_1 r)^{-2(\mu -1)}} \nonumber \\&=D^2(x)\mathcal {K}_{s=2}(z_1, w_1r). \end{aligned}$$
(65)

Since \(\partial \mathcal {O}_j^{(n)}\sim (z_{12}^j, L_{21} [\mathcal {O}(x,z_1,z_1,z_2)])_{11}\), see Eq. (57), the z-scalar product has the form

$$\begin{aligned} (z_{12}^j, L_{21} \mathcal {K}_2(z_1, w_1 r)\mathcal {K}_1(z_2, w_2 r))_{11}. \end{aligned}$$
(66)

The spins of the reproducing kernels and spins of the scalar product are in discord with each other. However, the operator \(L_{21}=\partial _2 z_{21}^2\) removes this mismatch. It intertwines the representations,

$$\begin{aligned} L_{21} D_2^+\otimes D_1^+= D_1^+\otimes D_1^+ L_{21} \end{aligned}$$

and it can easily be shown, see e.g. Ref. [13], that

$$\begin{aligned} (z_{12}^j, L_{21} \Phi (z_1,z_2))_{11}=-a_j (z_{12}^{j-1}, \Phi (z_1,z_2))_{21}, \end{aligned}$$
(67)

where

$$\begin{aligned} a_j=(j+1){||z_{12}^{j}||_{11}^2}/{||z_{12}^{j-1}||_{21}^2}= \frac{j(j+2)(j+3)}{6}. \end{aligned}$$
(68)

Thus the scalar product (66) takes the form

$$\begin{aligned} -a_j(z_{12}^{j-1}, \mathcal {K}_2(z_1, w_1 r)\mathcal {K}_1(z_2, w_2 r))_{21}=-a_j (r \bar{w}_{12})^{j-1}. \end{aligned}$$
(69)

Restoring all color and symmetry factors one gets for the first diagram

$$\begin{aligned}&-\frac{1}{2} \xi \lambda _s g^2 D^3(x) a_j/(j+1)^2r^{j-1} (L_{21}\bar{w}_{12}^{j-1}, w_{12}^j)_{11} \nonumber \\&\quad = \frac{1}{2}\xi \lambda _s g^2 D^3(x) a_j/(j+1) r^{j-1} ||w_{12}^j||_{11}^2. \end{aligned}$$
(70)

The calculation of the second diagram goes along the same line. One can combine the propagators attached to the point \(z_1\) using Feynman’s trick to get

$$\begin{aligned}&(z_{12}^j, L_{21} \mathcal {K}_2(z_1, w_1 r)\mathcal {K}_2(z_1, w_2 r)\mathcal {K}_1(z_2, w_1 r))_{11} \nonumber \\&\quad = \frac{6(-1)^ja_j (r \bar{w}_{12})^{j-1}}{(j+1)(j+2)}. \end{aligned}$$
(71)

Finally, taking into account that second diagram enters with symmetry factor 2 one gets, in full agreement with (11),

$$\begin{aligned} \mathcal {T}_j(u_*)&=u_*\varkappa _j \gamma _j^{(1)}+O(u_*^2), \end{aligned}$$
(72)

where the one-loop anomalous dimension is, see Eq. (34),

$$\begin{aligned} \gamma _j^{(1)}=\lambda _s\frac{1}{6}\frac{(j-2)(j+5)}{(j+1)(j+2)}. \end{aligned}$$
(73)

Thus the diagrams are easily calculated provided that the spins of the reproducing kernels match that of the scalar product. We will show that this scheme can be extended to loop diagrams as well.

5.2 NLO correlators

Fig. 3
figure 3

NLO correction to the correlator of the conformal operators \(\langle {\mathcal {O}^{(n)}_j(x)\mathcal {O}_j^{(\bar{n})}(0)}\rangle \). The parameter \(\delta =\epsilon /2\)

Full size image

First, it is easy to see that the corrections due to modification of the scalar product cancel out in the ratio of correlators. Indeed, these corrections only influence the norm

$$\begin{aligned} ||w_{12}^j||_{11}^2\mapsto ||w_{12}^j||_{\varpi }^2=(w_{12}^j, (1+\varpi ^{(1)})w_{12}^j)_{11}, \end{aligned}$$

entering the tree-level expressions, Eqs. (64), (70), etc., which cancel out in the ratio of correlators, \(\mathcal {T}_j\), irrespectively of the explicit form of \(\varpi \). This cancelation is expected. Indeed, the problem can be reformulated as a standard quantum mechanical problem for a certain Hamiltonian. A modification of the scalar product produces corrections to the eigenstates (conformal operator). However, the energy shift at the leading order, \(\delta E^{(1)}_\psi =\langle {\psi ^{(0)}|V|\psi ^{(0)}}\rangle \), is not sensitive to such corrections.

The calculation of \(\langle {\mathcal {R}_j^{(2)}\mathcal {R}_j^{(3,0)}}\rangle \) is a bit more involved but straightforward. The corresponding contribution to \(T_j^{(2)}\) reads

$$\begin{aligned} T_j^{(2\rightarrow 3)}=-\frac{1}{2}(\gamma _j^{(1)})^2\, \frac{(j-1)(j^2+4j+9)}{(j+1)(j+2)(j+3)(j+5)}. \end{aligned}$$
(74)

It is convenient to split one-loop corrections to the correlators \(\langle {\mathcal {O}_j \mathcal {O}_j}\rangle \) and \(\langle {\mathcal {R}_j^{(3,0)}\mathcal {R}_j^{(3,0)}}\rangle \) into two groups

  • Self-energy insertions to the propagators.

  • All other loop diagrams.

Taking into account the self-energy correction to the propagators is equivalent to the calculation of the tree-level diagrams with the exact (critical) propagator

$$\begin{aligned} D_c(x)=A(u_*)/(x^2)^{\Delta _\varphi }, \end{aligned}$$
(75)

where \(\Delta _\varphi =\mu -1+\gamma _\varphi \) is the critical dimension of the basic field and the residue \(A(u_*)\) is

$$\begin{aligned} A(u_*)=\frac{\Gamma (\mu -1)}{4 \pi ^\mu \overline{M}^{2\gamma _\varphi }}\left( 1-u_*\lambda _s\frac{5}{36}+O(u_*^2)\right) , \end{aligned}$$
(76)

where \(\overline{M}^2=\pi M^2\mathrm{e}^{\gamma _E}\). For the first diagram in Fig. 1 one gets

$$\begin{aligned} D_c^2(x)\,\, r^j\,\, (z_{12}^j|\mathcal {K}_{s}(z_1,w_1)\mathcal {K}_{s}(z_2,w_2)|w_{12}^j)_{(11),(11)}, \end{aligned}$$
(77)

where \(s=\Delta _\varphi /2=1-(\epsilon -\gamma _\varphi )/2\), the subscripts indicate the conformal spins of the z and w scalar products, respectively. The reproducing kernels in (77) correspond to spin s, which does not match the spins of the scalar product. However, it is easy to see that, due to symmetry, the scalar product with modified spins,

$$\begin{aligned} (z_{12}^j|\mathcal {K}_{s}(z_1,w_1)\mathcal {K}_{s}(z_2,w_2)|w_{12}^j)_{(s_+^\delta ,s_+^\delta ),(s_-^\delta ,s_-^\delta )}, \end{aligned}$$
(78)

where \(s^\delta _\pm =1\pm \delta \) is equal to that in (77) up to terms of order \(\delta ^2\). Therefore, choosing \(s_-^\delta =s\), (\(\delta =(\epsilon -\gamma _\varphi )/2\)), and evaluating the w-product in (78) one gets for (77)

$$\begin{aligned} D_c^2(x)\,\, r^j\,||z_{12}^j||_{s_+^\delta ,s_+^\delta }^2+O(u_*^2). \end{aligned}$$
(79)

The calculation of the diagrams in Fig. 2 goes along the same lines. Finally, the contribution to the ratio of correlators \(\mathcal {T}_j\) from the leading order diagrams and self-energy diagrams can be written in the form (up to \(O(\epsilon ^2)\) terms)

$$\begin{aligned} \mathcal {T}_j^{SE}&=\frac{2u_*\lambda _s}{\varkappa _j}\left( 1+\frac{u_*}{2}\left( \lambda _d-\frac{7}{9}\lambda _s\right) \right) \frac{||z_{12}^{j-1}||^2_{2s_+^\delta ,s_+^\delta }}{||z_{12}^{j}||^2_{s_+^\delta ,s_+^\delta }} \nonumber \\&\quad \times \left( 1-\frac{2\Gamma (4s_-^\delta )(\Gamma (j-1+2s_-^\delta )}{\Gamma (2s_-^\delta )\Gamma (j-1+4s_-^\delta )}\right) . \end{aligned}$$
(80)

Expanding (80) we find for the corresponding contribution to the coefficient \(T^{(2)}_j\)

$$\begin{aligned} {T}^{SE}_j&=\gamma _j^{(1)}\Biggl \{\frac{1}{2}\left( \lambda _d-\frac{7}{9}\lambda _s\right) \nonumber \\&\quad + 2\delta \Biggl [S_{2j+2} -S_{j+3}-S_j +\frac{2}{3}\frac{4j^2+14j+9}{(j+2)(j+3)}\Biggr ]\Biggr \} \nonumber \\&\quad - \frac{4 \lambda _s\,\delta }{(j+1)(j+2)}\left[ S_{j+2}-\frac{(2j+1)(4j+7)}{3(j+1)(j+2)}\right] , \end{aligned}$$
(81)

where \(S_j=\sum _{k=1}^j 1/k\) and

$$\begin{aligned} \delta =\frac{1}{4}\left( -\lambda _d+\frac{1}{3}{\lambda _s}\right) . \end{aligned}$$
(82)

We recall that \(\gamma _j^{(1)}\) is the one-loop anomalous dimension, Eq. (73).

5.3 Loop diagrams

All loop diagrams can be calculated quite easily with the help of several simple tricks. Several examples are given below. Let us start with a correction to the correlator of two conformal operators. The corresponding contribution has the form

$$\begin{aligned} C(\epsilon )=2\,(z_{12}^j| H(x;z,w)|w_{12}^j)_{(11),(11)}, \end{aligned}$$
(83)

where the kernel H(x; z; w) is given by the left diagram shown in Fig. 3 with the parameter \(\delta \rightarrow 0\). We need to find \(C(\epsilon )\) up to terms \(O(\epsilon ^0)\), \(C(\epsilon )=\frac{1}{\epsilon }(c_0+\epsilon c_1+\cdots )\). To this end we proceed as follows. We modify the indices as shown in Fig. 3. This modification does not change the pole structure (see the discussion in Ref. [23]) and, due to the symmetry \(C(\epsilon ,\delta )=C(\epsilon ,-\delta )\), one concludes that

$$\begin{aligned} C(\epsilon ,\delta )=\frac{1}{\epsilon }(c_0+\epsilon c_1+c_2\delta ^2+\cdots ). \end{aligned}$$
(84)

The choice \(\delta =\epsilon /2\) results in the uniqueness of the upper integration vertex and at the same time does not affect first two terms in (84). Using the star–triangle relation for the upper vertex one gets the diagram B in Fig. 3. Using the Feynman formula for the left (right) propagators attached to the integration vertex one can perform the last integral that results in the diagram C in Fig. 3.

This diagram, up to a x-dependent factor, has the form

$$\begin{aligned} \mathcal {K}_{\frac{1}{2}}(z_1,w_1)\int _0^1\,\mathrm{d}\alpha \,\mathrm{d}\beta (\bar{\alpha }\bar{\beta })^{-\delta }(\alpha \beta )^{1-3\delta } \mathcal {K}_{\frac{3}{2}-3\delta }(z_{12}^\alpha ,w_{12}^\beta ). \end{aligned}$$
(85)

In the next step we want to get rid of the parametric integrals. To this end we use the properties of the reproducing kernel (A.6) and represent

$$\begin{aligned}&\mathcal {K}_{s-\delta }(z_{12}^\alpha ,w_{12}^\beta )\nonumber \\&\quad = \int \mathrm{d}^2\xi '\, \mu _{s}(\xi ')\mathcal {K}_{s}(z_{12}^\alpha ,\xi ')\nonumber \\&\qquad \times \int \mathrm{d}^2\xi \, \mu _{s}(\xi )\mathcal {K}_{s-\delta }(\xi ',\xi )\, \mathcal {K}_{s}(\xi ,w_{12}^\beta ) \nonumber \\&\quad = \int \mathrm{d}^2\xi \, \mu _{s+\delta }(\xi )\mathcal {K}_{s}(z_{12}^\alpha ,\xi ) \mathcal {K}_{s}(\xi ,w_{12}^\beta ) +O(\delta ^2), \end{aligned}$$
(86)

where \(s=3/2-2\delta \) and we used Eq. (A.7). Using this expression one can carry out the integrals over \(\alpha ,\beta \) in (85). The resulting expression for H(x; z, w) (up to a prefactor) takes the form of the “sl(2)” diagram shown in Fig. 4.

Fig. 4
figure 4

The “sl(2)” diagram: an arrow line from w to z with index \(\alpha \) stands for the propagator \((1-z\bar{w})^{-\alpha }\). The indices have the following values: \(\alpha =2-3\epsilon /2\), \(\beta =1-\epsilon /2\), and \(\gamma =1\). The black circle denotes an integration vertex with the sl(2) invariant measure \(\mu _{s+\delta }, s+\delta =3/2-\epsilon /4\)

Full size image
Fig. 5
figure 5

One-loop correction to the correlator of the divergence of conformal operators \(\langle {\partial \mathcal {O}^{(n)}_j(x)\partial \mathcal {O}_j^{(\bar{n})}(0)}\rangle \)

Full size image

Now we need to calculate the scalar product (83) with the kernel H(x; z, w) given by the diagram on Fig. 4. The scalar product is a function of indices of left (right) propagators and conformal spins of z (w ) scalar products, \( S(a,b, s_1,s_2|a', b',s'_1,s'_2)\). We need this function up to \(O(\epsilon )\) terms for \(a=a'=\alpha , b=b'=\beta \), and \(s_i=s'_i=1\). As was explained in the previous section the integral with measure \(\mu _s\) can be evaluated easily provided that the sum of the indices of the propagators coming from this vertex is equal to 2s. Taking this into account one finds that the scalar product with shifted indices,

$$\begin{aligned} S(\alpha +\delta ,\beta +\delta , 1+2\delta ,1+4\delta |\alpha -\delta ,\beta -\delta , 1-2\delta ,1-4\delta ), \end{aligned}$$

can be straightforwardly calculated to be

$$\begin{aligned} S(\epsilon )=\frac{j!\Gamma (3-\epsilon )}{\Gamma (j+3-\epsilon )}\, ||z_{12}^{j}||^2_{1+\frac{1}{2}\epsilon ,1+\epsilon } \end{aligned}$$
(87)

and at the same time it differs from the scalar product in question by terms of order \(O(\epsilon ^2)\) only.

Restoring all factors one gets for \(C(\epsilon )\)

$$\begin{aligned} C(\epsilon )&=u_* \, \lambda _s (n^2-1) D^2(x) (x^2\overline{M}^2)^\epsilon \, {r^{j}} \frac{2+\epsilon }{\epsilon }\, S(\epsilon ). \end{aligned}$$
(88)

Since the ratio (11) does not depend on x, it is convenient to put \(x^2\overline{M}^2=1\). Finally, subtracting the counterterms,

$$\begin{aligned} \Delta C(\epsilon )=-\frac{1}{\epsilon }4u_* \, \lambda _s (n^2-1) D^2(x)\, {r^{j}} \frac{||z_{12}^{j}||^2_{1+\frac{1}{2}\epsilon ,1+\frac{1}{2}\epsilon }}{(j+1)(j+2)}, \end{aligned}$$

one obtains

$$\begin{aligned} \frac{C(\epsilon )+\Delta C(\epsilon )}{\langle {\mathcal {O}_j(x)\mathcal {O}_j(0)}\rangle _{0}}={u_*\lambda _s}\frac{2(S_{2j+2}-S_{j+1})}{(j+1)(j+2)}+\cdots \end{aligned}$$
(89)

where the ellipsis stands for higher order terms. The corresponding contribution from (89) to the coefficient \(T_j^{(2)}\) in the ratio of the correlators, see Eq. (11), reads

$$\begin{aligned} T_j^{(o)} = - \gamma _j^{(1)} {\lambda _s}\frac{2(S_{2j+2}-S_{j+1})}{(j+1)(j+2)}. \end{aligned}$$
(90)

The NLO diagrams contributing to the correlator \(\langle {\partial {\mathcal {O}_j}(x)\partial {\mathcal {O}_j}(0)}\rangle \) are shown in Fig. 5 (type A) and Fig. 6 (type B). These diagrams have different color factors: \(\lambda _s\lambda _d\) for the A-diagrams, and \(\lambda _s^2\) for the B-diagrams.

All diagrams of the A-type have the diagram we have just now discussed as a subgraph. The only difference is that in order to kill terms linear in \(\delta \) one has to consider an average of the diagrams, \((D(\delta )+D(-\delta ))/2\). For \(\delta =\epsilon /2\) each of the diagrams, \(D(\pm \delta )\), can be simplified with the help of the star–triangle relation and rewritten in the form of “sl(2)” diagrams. These diagrams in turn can be calculated up to \(O(\epsilon ^2)\) terms in a manner described above. So we skip all details and present the result for each diagram in Appendix C.

Fig. 6
figure 6

NLO diagrams for the correlator of the divergence of conformal operators \(\langle {\partial \mathcal {O}^{(n)}_j(x)\partial \mathcal {O}_j^{(\bar{n})}(0)}\rangle \)

Full size image

All diagrams of B-type shown schematically in Fig. 6 contain two \(2\rightarrow 1\) subgraphs. The diagrams depicted on the right panel are finite, while those on the left panel are divergent. The calculation of these diagrams does not present any problem so that we give only the final results in Appendix C.

Finally, it follows from Eq. (52) that the sum of counterterm diagrams to the diagrams in Figs. 5 and 6 can be written in the form

$$\begin{aligned} 2(Z_3 Z_1^{-1}Z_{j}-1)\,\langle {\partial \mathcal {O}_j^{(n)}(x)\partial \mathcal {O}_j^{(\bar{n})}(0)}\rangle ^{(\epsilon )}_0. \end{aligned}$$
(91)

Here \(Z_j\) is the one-loop renormalization constant for the operator \(\mathcal {O}_j\),

$$\begin{aligned} Z_j=1-\frac{u}{\epsilon }\frac{\lambda _s}{(j+1)(j+2)} \end{aligned}$$
(92)

and

$$\begin{aligned} Z_1=1-\frac{u\lambda _s}{12\epsilon }+O(u^2),&Z_3=1-\frac{u\lambda _d}{4\epsilon }+O(u^2). \end{aligned}$$
(93)

We have put the superscript \((\epsilon )\) to the correlator in order to stress that even the tree-level correlator depends on \(\epsilon \) through a space-time dimension \(d=6-2\epsilon \).

6 Results

The coefficient \(T_j^{(2)}\) in the ratio of the correlators is given by the sum of terms in Eqs. (74), (81), (89), (C.13), and (C.14). In the case of operators with another isotopic symmetry these expressions have to be modified. One can distinguish seven different isotopic projections,

$$\begin{aligned} \mathcal {O}_f^{ab}(x;z_1,z_2)=(P_{f})^{ab}_{a'b'}\varphi ^{a'}(x+z_1n)\varphi ^{b'}(x+z_2n), \end{aligned}$$
(94)

where \(f=1,\ldots ,7\). The projectors \(P_f\) can be found in Ref. [18]. The one-loop anomalous dimension for the operator \(\mathcal {O}_j^{(f)}\) is given by

$$\begin{aligned} \gamma _j^{f(1)}=\frac{1}{6}\left( \lambda _1-\frac{12\lambda _f}{(j+1)(j+2)}\right) , \end{aligned}$$
(95)

where \(\lambda _f\) are the eigenvalues of the operator \(\mathbb {R}^{ab}_{a'b'}=\mathrm{d}^{aa'c}\,\mathrm{d}^{bb'c}\) on the invariant subspaces, \(\mathbb {R}P_f=\lambda _f P_f\). The explicit expressions for \(\lambda _f\) can be found in [18]. We note also that \(\lambda _s=\lambda _1\) and \(\lambda _d=2\lambda _3\).

The modifications of Eqs. (74), (81), (89), (C.13), and (C.14) for the case of arbitrary projections, \(P_f\), are the following:

  • Replace \(\gamma _j^{(1)}\rightarrow \gamma _j^{(1,f)}\) in all expressions.

  • Replace \(\lambda _s\rightarrow \lambda _f\) in the expressions for \(T^{(o)}_j\), \(T^{(5,B)}_j\), \(T^{(5,C)}_j\), \(T^{(6,A)}_j\) and in the last line of \(T^{SE}_j\), Eq. (81).

  • Replace \(\lambda _s\lambda _d\rightarrow 2\nu _f\) in the expression for \(T^{(5,D)}_j\).

Here \(\nu _f\) are the eigenvalues of the invariant operator \(\mathbb {T}^{ab}_{a'b'}=(\mathbb {R}^2)^{a, b'}_{a'b}\); see Ref. [18].

Representing the ratio of the correlators in the form

$$\begin{aligned} \mathcal {T}^{f}_j(u_*)&=\varkappa _j(u_* \,T_j^{f(1)} ~ +~ u_*^2\, T_j^{f(2)} + \cdots ) \end{aligned}$$
(96)

one obtains for the coefficient \(T_j^{f(2)}\)

$$\begin{aligned} T_j^{f(2)}&= \sum _{ab} \lambda _a\,\lambda _b \,T^{(2)}_{j,ab}+\nu _f\, T^{(2)}_{j,f} \end{aligned}$$
(97)

where

$$\begin{aligned} T^{(2)}_{j,ss}&=-\frac{1}{72}\left( \frac{11}{3}+\frac{2j^2+7j-1}{(j+1)(j+2)(j+3)}\right) , \nonumber \\ T^{(2)}_{j,sd}&=\frac{1}{12}\left( \frac{2}{3}+\frac{2j^2+5j+1}{(j+1)(j+2)(j+3)}\right) , \nonumber \\ T^{(2)}_{j,ff}&=2\frac{j^3+8j^2+10j+1}{(j+1)^3(j+2)^3(j+3)}, \nonumber \\ T^{(2)}_{j,fd}&=\frac{1}{(j+1)(j+2)}\left( S_{j+1}-3+\frac{4}{(j+1)(j+3)}\right) , \nonumber \\ T^{(2)}_{j,fs}&=-\frac{1}{3(j+1)(j+2)}\biggl (S_{j+3}-4 \nonumber \\&\quad +\frac{j^2+2j+5}{(j+1)(j+2)(j+3)}\biggr ), \end{aligned}$$
(98)

and

$$\begin{aligned} T^{(2)}_{j,f}&=-\frac{2}{(j+1)^2(j+2)^2}. \end{aligned}$$
(99)

Finally, comparing (96) with the r.h.s. of Eq. (11) we get for the anomalous dimension

$$\begin{aligned} \gamma _j^f(u_*)&=u_*\gamma _j^{f,(1)}+u_*^2\gamma _j^{f,(2)}+\cdots , \end{aligned}$$
(100)

where \(\gamma _j^{f(1)}\) is given by Eq. (95) and \(\gamma _j^{f(2)}\) has the form

$$\begin{aligned} \gamma _j^{f(2)}&=2\gamma _{\varphi }^{(2)}+\frac{2\nu _f}{(j+1)^2(j+2)^2} +2\lambda _f^2\frac{j^2+j-1}{(j+1)^3(j+2)^3} \nonumber \\&\quad -\frac{\lambda _f\lambda _s}{3(j+1)(j+2)}\left[ S_{j+2}-4+\frac{2j+3}{2(j+1)(j+2)}\right] \nonumber \\&\quad +\frac{\lambda _f\lambda _d}{(j+1)(j+2)}\left[ S_{j+2}-3+\frac{1}{j+1}\right] . \end{aligned}$$
(101)

This expression completely agrees with the anomalous dimensions reconstructed from the two-loop evolution kernels [18]. We have also checked that the large j expansion of the anomalous dimensions \(\gamma _j^f(u_*)\) in terms of the quadratic Casimir

$$\begin{aligned} J^2=\left( j+2-\epsilon +\frac{1}{2}\gamma _j^f(u_*)\right) \left( j+1-\epsilon +\frac{1}{2}\gamma _j^f(u_*)\right) \end{aligned}$$

contains only even powers of 1 / J [24, 25].

7 Summary

We have calculated two-loop anomalous dimensions of the leading-twist operators in the \(\varphi ^3\) model using the approach proposed in Refs. [10, 11]. Formally this method allows one to gain one order in the perturbation theory. However, this advantage is illusory since one has to calculate diagrams including finite parts instead of the pole terms in the standard approach. There is also no essential gain as regards the complexity of the calculations.

Nevertheless, both the contributing diagrams and the methods of calculation are quite different in the two approaches. Therefore the calculation of the anomalous dimensions performed in this approach could provide an additional check of the results obtained within the standard approach. Of course, going to the next order is only possible with the advanced methods of computer algebra.