CFT data in the Gross-Neveu model

Some of the most interesting phenomena in nature are associated with the regime of the strong interaction and therefore are not accessible to the standard perturbative treatment. Particularly, strongly-coupled physics occurs in the critical regime and is described by conformal field theories (CFTs) Polyakov:1970xd . From the standpoint of fundamental physics, such a regime is reached at the end of the renormalization group (RG) flow of a quantum field theory (QFT), either in the deep infra-red (IR) or in the asymptotically ultra-violet (UV) regime. Moreover, different systems can have their RG flows terminate at the same CFT, in a manifestation of the renowned phenomenon of critical universality.

The universality principle served as one of the inspirations to classify CFTs based on the algebra of primary operators and their observable properties, such as scaling dimensions and operator product expansion (OPE) coefficients, without necessarily specifying the Lagrangian of the underlying theory. In particular, the microscopic specifics of the underlying theory are ignored while the symmetries of the system and general consistency conditions (such as unitarity) constraining the available space of parameters come forward. The corresponding program is known as the conformal bootstrap, and it has been under active development over recent years Parisi:1972zm ; Polyakov:1974gs ; Ferrara:1973yt ; ElShowk:2012ht ; Simmons-Duffin:2016gjk .

This paper is motivated by the desire to expand our understanding of critical dynamics in $d=3$ dimensions. The well-known example of a three-dimensional critical system is furnished by the IR fixed point of the $O(n)$ vector model. This model can be viewed as a continuum limit description of the critical $n$-vector model on a lattice Stanley:1968ef ; Guida:1998bx ; ElShowk:2012ht . The latter in turn generalizes the three-dimensional Ising model, describing the second-order phase transition of a ferromagnet. The fermionic counterpart of the $O(n)$ vector model is given by the $U(n)$ models with quartic fermion couplings, which we choose as the main focus of this paper.

The fixed point of the $O(n)$ vector model exists in a perturbative Wilson-Fisher regime when the model is considered in $4-\epsilon$ dimensions for small values of $\epsilon$. The three-dimensional physics (after a proper re-summation) is rather well approximated by setting $\epsilon=1$ Wilson:1971dc . Similarly, the $U(n)$ fermionic model with the four-fermion Gross-Neveu interaction is asymptotically free in two dimensions¹¹1In fact, it possesses a number of remarkable properties in two dimensions, such as the dynamical breaking of chiral symmetry and generation of the IR scale via the dimensional transmutation, making it a 2d toy-model of quantum chromodynamics Gross:1974jv . but possesses the Wilson-Fisher type of fixed point in the UV limit in $2+\epsilon$ dimensions Gross:1974jv ; ZinnJustin:1991yn .

While the critical vector models are non-perturbative in general $d$, a popular approach to study them is given by the $1/N$ expansion, around an infinitely large number $N\rightarrow\infty$ of the degrees of freedom of the system (see Parisi:1975im and references therein). The $1/N$ expansion and conformal bootstrap techniques are well suited to study strongly-coupled critical regime and have proven to be remarkably successful methods for extracting the CFT data such as the scaling dimensions and the OPE coefficients, see Vasiliev:1981yc ; Vasiliev:1981dg ; Vasiliev:1975mq ; Vasiliev:1982dc ; Gracey:1990wi ; ZinnJustin:1991yn ; Lang:1991kp ; Gracey:1992cp ; Vasiliev:1992wr ; Vasiliev:1993pi ; Gracey:1993kb ; Gracey:1993kc ; Lang:1993ct ; Petkou:1995vu ; Petkou:1994ad ; Derkachov:1993uw ; Derkachov:1997ch ; Leonhardt:2003du ; Fei:2014yja ; Fei:2014xta ; Gracey:2015tta ; Manashov:2016uam ; Diab:2016spb ; Giombi:2016fct ; Fei:2016sgs ; Gracey:2016mio ; Manashov:2017rrx ; Gracey:2018ame ; Alday:2019clp ; Giombi:2019upv ; Goykhman:2019kcj ; Goykhman:2020ffn for an incomplete list of related references.²²2 While the quartic coupling in the $O(n)$ vector model in $4<d<6$ dimensions, and the Gross-Neveu coupling in $2<d<4$ dimensions are non-renormalizable by power counting, these theories are renormalizable at each order in the $1/N$ expansion Parisi:1975im ; Rosenstein:1988pt . See also Goykhman:2019kcj where consistency check for the existence of a fixed point in the $O(n)$ vector model in $2<d<6$ dimensions was carried out by examining the Callan-Symanzik equations.

Recently the power of the background field method was emphasized in the context of large-$N$ vector models Goykhman:2020ffn . It was shown that formally fixing some of the degrees of freedom to their non-dynamical background values provides a simple short-cut to the calculation of effective vertices at sub-leading orders in the $1/N$ expansion. In particular, this procedure allows one to easily extract finite parts of the three-point correlation functions (OPE coefficients). This is contrasted with the typical focus of the research in large $N$ critical vector models on the calculation of the critical exponents. In the literature, the background field method has been used effectively for calculations in cases where the field acquires a vacuum expectation value (v.e.v.) as a result of spontaneous symmetry breaking Goldstone:1962es . However, in the case of Goykhman:2020ffn , as well as in this paper, the considered fields do not necessarily acquire a v.e.v., and therefore the method should be viewed only as a calculation tool.

In this paper, we further establish the power of the background field method for the study of critical vector models. To this end, we will scrutinize the UV critical regime of the $U(n)$ fermionic model with the Gross-Neveu interaction, and compute the full $s^{2}ss$ and $s^{2}\bar{\psi}\psi$ effective vertices for the fermion $\psi$, the Hubbard-Stratonovich field $s$, and the composite operator $s^{2}$ in terms of the corresponding conformal triangles.³³3See Vasiliev:1993pi for the earlier calculation of the $s^{2}ss$ conformal triangle, which focused on its singular part for the purpose of calculating the anomalous dimension $\gamma_{s^{2}}$. We extend their result by adding the finite part of this conformal triangle, essential for the calculation of the OPE coefficients. This is the first time that the background field method is used to calculate correlation functions involving a composite operator. In a simple way, this method provides a very powerful technique to calculate the finite part of non-planar diagrams contributing to these conformal triangles. We further use these conformal triangles to calculate the OPE coefficients appearing in the correlation functions involving the operator $s^{2}$, including the two-point function $\langle s^{2}s^{2}\rangle$, and the three-point functions $\langle s^{2}\bar{\psi}\psi\rangle$ and $\langle s^{2}ss\rangle$.

It is well known that the UV fixed point of the GN model in $2<d<4$ dimensions is described by the same CFT as the IR fixed point of the Gross-Neveu-Yukawa (GNY) model ZinnJustin:1991yn . Such an example of critical universality can also be interpreted in terms of the GNY model being a UV completion of the GN model. Moreover, the GNY model can be studied perturbatively in $d=4-\epsilon$ dimensions, and the corresponding CFT data should match the CFT data of the GN model ZinnJustin:1991yn . We perform such a consistency check on all of our results obtained in this paper.⁴⁴4In a similar spirit, the $O(N)$ vector model in $4<d<6$ dimensions has been conjectured to have a UV completion in terms of the vector model coupled to a dynamical scalar field with cubic interaction. This conjecture has received numerous verifications up to the fourth order in perturbation theory Fei:2014yja ; Fei:2014xta ; Gracey:2015tta ; Diab:2016spb ; Giombi:2019upv ; Goykhman:2019kcj .

The rest of this paper is organized as follows. In section 2 we set up our conventions and review the known results in the literature that will be useful for the purposes of this paper. In section 3 we use the background field method to calculate the new conformal triangles up to the next-to-leading order in the $1/N$ expansion. We derive the $s^{2}\bar{\psi}\psi$ conformal triangle in subsection 3.1, and the $s^{2}ss$ conformal triangle in subsection 3.2. In section 4 we carry out the calculation of the $\langle s^{2}ss\rangle$ three-point function at the next-to-leading order in the $1/N$ expansion. In the process, we derive various OPE coefficients. In section 5 we calculate the $\langle s^{2}\bar{\psi}\psi\rangle$ three-point function at the next-to-leading order in the $1/N$ expansion. In section 6 we demonstrate how the $s^{2}ss$ conformal triangle can be calculated from the $\langle s^{2}s^{2}\rangle$ two-point function, which in particular serves as a non-trivial consistency check for our results. We conclude with the discussion in section 7, where we also outline possible future directions.

Consider the $U(n)$-invariant fermionic model in $2\leq d\leq 4$ dimensions with the quartic Gross-Neveu interaction,

$$S=\int d^{d}x\,\left(\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi+\frac{g}{N}\,\left(\bar{\psi}\psi\right)^{2}\right)\,,$$

(1)

where $\psi$ is the $n$-component multiplet of Dirac fermions. According to the standard conventions $N=n\,\textrm{tr}\mathbb{I}$, where $\mathbb{I}$ is the unit matrix in the $2^{[d/2]}$-dimensional space of Dirac spinors. We will be working in the Euclidean signature, with Hermitian gamma-matrices, $(\gamma^{\mu})^{\dagger}=\gamma^{\mu}$, such that $\{\gamma^{\mu},\gamma^{\nu}\}=2\delta^{\mu\nu}\,\mathbb{I}$. Below in this paper we will skip keeping explicit track of the $U(n)$ indices where it does not cause a confusion.⁵⁵5In particular, when writing down the Feynman rules, we will omit the Kronecker delta-symbols for the $U(n)$ indices.

Using the standard trick, known as the Hubbard-Stratonovich transformation, we can rewrite the action (1) in terms of the original fermions $\psi$ as well as an auxiliary scalar field $s$,

$$S=\int d^{d}x\,\left(\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-\frac{1}{4g}\,s^{2}+\frac{1}{\sqrt{N}}\,s\bar{\psi}\psi\right)\,.$$

(2)

The Hubbard-Stratonovich field $s$ becomes dynamical due to the fermion loop diagrams. The model (2) is believed to reach a non-trivial UV fixed point in $2<d<4$ dimensions ZinnJustin:1991yn ,⁶⁶6Such a fixed point can be studied perturbatively in $2+\epsilon$ dimensions. and we will be studying the corresponding CFT.

We will use the following Feynman rules for the bare propagators of the Dirac fermion and the Hubbard-Stratonovich field $s$, and the leading order interaction vertex:

A fermionic loop generates the factor of $-n\,\textrm{tr}(\mathbb{I})=-N$, where the minus sign appears as a consequence of the Wick contractions due to the anti-commuting nature of the fermion field. We will also use the following notation for the regularized Hubbard-Stratonovich propagator:

where $\delta$ is the regularization parameter which is taken to zero at the end of calculation Vasiliev:1975mq , $\mu$ is the renormalization mass scale, and $\Delta_{s}=1+\gamma_{s}$ is dimension of the Hubbard-Stratonovich field, which acquires the anomalous term $\gamma_{s}$. Our notations for the scalar and fermion lines with a general exponent and a unit amplitude will be as follows:

According to this notation, we will typically skip explicitly labeling exponents of the bare propagators of the fundamental fermion and the Hubbard-Stratonovich field.

The full fermion $\psi$ and Hubbard-Stratonovich $s$ propagators, including the anomalous dimensions $\gamma_{\psi,s}$ as well as corrections to the amplitudes $A_{\psi,s}$, will be denoted with solid blobs, and are given by

The full propagator of the composite field $s^{2}$ will be denoted as

where we defined

$$C_{s^{2}}=2C_{s}^{2}$$

(3)

to be the leading order amplitude of the $\langle s^{2}s^{2}\rangle$ propagator. The leading order propagator amplitudes are given by ZinnJustin:1991yn

$\displaystyle C_{\psi}=\frac{\Gamma\left(\frac{d}{2}\right)}{2\pi^{\frac{d}{2}}}\,,\qquad C_{s}=-\frac{2^{d}\sin\left(\frac{\pi d}{2}\right)\Gamma\left(\frac{d-1}{2}\right)}{\pi^{\frac{3}{2}}\,\Gamma\left(\frac{d}{2}-1\right)}\,,$

(4)

The anomalous dimensions at the next-to-leading order are ZinnJustin:1991yn

$$\gamma_{\psi}=-\frac{1}{N}\,\frac{2^{d-1}\sin\left(\frac{\pi d}{2}\right)\Gamma\left(\frac{d-1}{2}\right)}{\pi^{\frac{3}{2}}\,d\,\Gamma\left(\frac{d}{2}-1\right)}+{\cal O}\left(1/N^{2}\right)\,,\qquad\gamma_{s}=-4\frac{d-1}{d-2}\,\gamma_{\psi}+{\cal O}\left(1/N^{2}\right)\,,$$

(5)

and $1/N$ corrections to the propagators amplitudes are Manashov:2017rrx

$$A_{\psi}=-\frac{2}{d}\,\gamma_{\psi}+{\cal O}\left(1/N^{2}\right)\,,\quad A_{s}=-\left(H_{d-2}+\frac{2}{d}+\pi\,\cot\left(\frac{\pi d}{2}\right)\right)\,\gamma_{s}+{\cal O}\left(1/N^{2}\right)\,,$$

(6)

where $H_{n}$ is $n$th harmonic number.

We will also need the $s\bar{\psi}\psi$ effective vertex, represented by the corresponding conformal triangle Polyakov:1970xd , and denoted with a solid blob

where the exponents

$$\alpha=\frac{d-1-\gamma_{s}}{2}\,,\qquad\beta=1-\gamma_{\psi}+\frac{\gamma_{s}}{2}$$

(7)

are such that the integration vertices $x_{1,2,3}$ become unique when the propagators are attached to them. The $\bar{\psi}\psi s$ conformal triangle diagram is a sum of the leading order tree-level diagram, the sub-leading loop corrections, and the $\bar{\psi}\psi s$ vertex counter-term. In particular, the $\bar{\psi}\psi s$ vertex counter-term cancels the divergences from the sub-leading order loop diagram, see Goykhman:2020ffn for the detailed explanation.

To the next-to-leading order in $1/N$ we can expand

$$Z_{s\bar{\psi}\psi}=Z_{s\bar{\psi}\psi}^{(0)}\left(1+\delta Z_{s\bar{\psi}\psi}+{\cal O}(1/N)\right)\,,$$

(8)

where Manashov:2017rrx ⁷⁷7See also Goykhman:2020ffn for a recent derivation.

	$\displaystyle Z_{s\bar{\psi}\psi}^{(0)}$	$\displaystyle=-\frac{(d-2)\Gamma\left(\frac{d}{2}\right)^{2}}{4\pi^{d}}\,(2\gamma_{\psi}+\gamma_{s})\,,$		(9)
	$\displaystyle\delta Z_{s\bar{\psi}\psi}$	$\displaystyle=-\frac{2}{d-2}\,\gamma_{s}\,.$		(10)

For the analysis of loop diagrams we we will be using the following propagator merging relations Gracey:1990wi

where we defined

	$\displaystyle A(\Delta)$	$\displaystyle=\frac{\Gamma\left(\frac{d}{2}-\Delta\right)}{\Gamma(\Delta)}\,,\qquad V(\Delta_{1},\Delta_{2})=\frac{\Gamma\left(\frac{d+1}{2}-\Delta_{1}\right)}{\Gamma(\Delta_{1}+\frac{1}{2})}\frac{\Gamma\left(\frac{d+1}{2}-\Delta_{2}\right)}{\Gamma(\Delta_{2}+\frac{1}{2})}\,,$		(11)
	$\displaystyle U(\Delta_{1},\Delta_{2},\Delta_{2})$	$\displaystyle=\pi^{\frac{d}{2}}\,A(\Delta_{1})A(\Delta_{2})A(\Delta_{3})\,,$		(12)

and the uniqueness relations for $\Delta_{1}+\Delta_{2}+\Delta_{3}=d$ DEramo:1971hnd ; Symanzik:1972wj ; Gracey:1990wi ,

	$\displaystyle\int d^{d}x_{4}\,\frac{1}{|x_{14}|^{2\Delta_{1}}|x_{24}|^{2\Delta_{2}}|x_{34}|^{2\Delta_{3}}}$	$\displaystyle=\frac{U(\Delta_{1},\Delta_{2},\Delta_{3})}{|x_{12}|^{d-2\Delta_{3}}|x_{13}|^{d-2\Delta_{2}}|x_{23}|^{d-2\Delta_{1}}}\,,$		(13)
	$\displaystyle\int d^{d}x_{4}\,\frac{x_{24}^{\mu}\gamma_{\mu}x_{41}^{\nu}\gamma_{\nu}}{|x_{14}|^{2\Delta_{1}+1}|x_{24}|^{2\Delta_{2}+1}|x_{34}|^{2\Delta_{3}}}$	$\displaystyle=\frac{\pi^{\frac{d}{2}}A(\Delta_{3})V(\Delta_{1},\Delta_{2})x_{31}^{\mu}\gamma_{\mu}x_{23}^{\nu}\gamma_{\nu}}{|x_{12}|^{d-2\Delta_{3}}|x_{13}|^{d-2\Delta_{2}+1}|x_{23}|^{d-2\Delta_{1}+1}}\,.$		(14)

Loops in position space are simply additive:⁸⁸8Notice that the minus sign in the r.h.s. of the last loop is independent of the minus sign in the factor of $-N$ appearing in the Feynman rule for the fermionic loop.

The structure of three-point functions in CFTs is completely fixed by the conformal symmetry. It is usually convenient to decompose contributions to the three-point functions (and the corresponding OPE coefficients) into the terms originating from the corresponding conformal triangle Polyakov:1970xd , and the terms due to the amplitudes of the propagators attached to the conformal triangle Goykhman:2019kcj . In this section we will make use of the background field method to calculate the $s^{2}\bar{\psi}\psi$ and $s^{2}ss$ conformal triangles.

The results obtained in this section will be used below in section 4 to derive the $1/N$ correction $A_{s^{2}}$ to the amplitude of the $\langle s^{2}s^{2}\rangle$ propagator, and the $1/N$ corrections to the three-point function $\langle s^{2}ss\rangle$. Finally, in section 5 we will use these results to obtain the next-to-leading order corrections to the three-point function $\langle s^{2}\bar{\psi}\psi\rangle$.