Operator product expansion coefficients in the exact renormalization group formalism

The exact renormalization group (ERG) provides a framework to study the fundamental aspects of quantum field theories (QFT). For instance, it allows one to give a non-perturbative definition of renormalizable theories Wilson:1973jj and to discuss the realization of symmetries at the quantum level Igarashi:2009tj . On top of being a conceptual framework, ERG offers a framework for practical computations. It has been used as a computational tool for the universal quantities such as critical exponents in statistical field theory (see e.g. Berges:2000ew ; Delamotte:2007pf ), and also as an exploratory tool in a wide range of subjects including quantum gravity R96PaReut2018 .

Besides ERG, the operator product expansion (OPE) offers important insights into non-perturbative aspects of QFTs Wilson:1969zs . Let us denote composite operators as $\left[\mathcal{O}_{a}\right]$ and the product of two composite operators as $\left[\mathcal{O}_{a}\mathcal{O}_{b}\right]$. OPE states the validity of

$\displaystyle\left[\mathcal{O}_{a}\left(x\right)\mathcal{O}_{b}\left(y\right)\right]$

$\displaystyle=\sum_{c}C_{abc}\left(x-y\right)\left[\mathcal{O}_{c}\left(\frac{x+y}{2}\right)\right]$

(1)

inserted into any correlation functions for small $|x-y|$. The existence of OPE has been proved perturbatively by Zimmermann Zimmermann:1972tv and plays a fundamental role in the study of conformal field theories (CFTs). Since both ERG and OPE offer non-perturbative insights into QFT, it is natural to study the relation between the two.

ERG was used to provide a perturbative proof of the existence of OPE Hughes:1988cp ; Keller:1991bz ; Keller:1992by ; Hollands:2011gf ; Holland:2014ifa ; Holland:2014pna , but little effort had been made to explore OPE within ERG beyond perturbation theory. In Pagani:2017tdr a non-perturbative definition of operator products was given, and simple examples of OPE were constructed in the Wilson action framework. Recently, composite operators have been constructed explicitly in the ERG formalism to make contact with various physical observables Becchi:1996an ; Igarashi:2009tj ; Rose:2015bma ; Rose:2016elj ; Daviet:2018lfy ; Pagani:2015hna ; Pagani:2016dof ; Becker:2018quq ; Becker2019Frank2020 .

In the present work we construct explicit examples of OPEs within the effective average action (EAA) framework Wetterich:1992yh . We identify two possible strategies to compute OPEs via ERG. One is based on the construction of operator products and their expansions in composite operators, and the other is based on the computation of three-point functions for theories with conformal invariance. We study explicit solutions of the ERG equations perturbatively with the $\epsilon$-expansions, and compare our results with those obtained via the conformal bootstrap Gaiotto:2013nva ; Hasegawa:2016piv ; Gopakumar:2016wkt ; Gopakumar:2016cpb ; Gopakumar:2018xqi . We also comment on how we may extend our strategies to non-perturbative approximation schemes available within the ERG formalism.

The paper is organized as follows. In section II we define composite operators and their products in the ERG formalism. We then consider the expansion of an operator product in composite operators and outline two possible strategies for the calculation of such an expansion. In section III we give technical remarks to explain how to construct operators at the Wilson-Fisher fixed point. Discussions of a fixed point require fixing a momentum cutoff, and we augment the technical outline given in section II where the momentum cutoff flows. Section IV is a long technical section, where we construct some composite operators explicitly and normalize them appropriately. In section V we apply the results obtained above to extract OPE coefficients and discuss our findings. We summarize our results in section VI and discuss the future perspectives.

In this section we wish to discuss technicalities that are quite important for our calculations. In the main part of this paper we consider a real scalar theory at its criticality in $d=4-\epsilon$ dimensions. In doing so with ERG, we have two alternatives:

C)

We start from a bare theory $S_{B}[\chi]$ with a large but finite cutoff $k_{0}$. We tune the bare parameters to make the theory critical. We then construct the EAA $\Gamma_{k}[\varphi,\varepsilon]$ whose $k$-dependence is determined by ERG. In the limit $k\to 0$, $\Gamma_{k}$ becomes the 1PI generating functional of the correlation functions. A conformal field theory is obtained as the IR limit of the critical theory, i.e., we must look at the correlation functions for the momenta much smaller than the bare cutoff $k_{0}$.

D)

We adopt the dimensionless convention by measuring all dimensionful quantities in appropriate powers of the cutoff $k$. The resulting 1PI EAA $\bar{\Gamma}_{t}[\bar{\varphi},\bar{\varepsilon}]$ has a fixed cutoff of order $1$, and satisfies an ERG differential equation with the Gaussian and Wilson-Fisher fixed-point solutions. $\bar{\Gamma}_{*}[\bar{\varphi},\bar{\varepsilon}]$ at the Wilson-Fisher fixed point gives the correlation functions of a conformal field theory, but only for the momenta much larger than the fixed cutoff.Sonoda:2017rro

We prefer D) because it is easier to construct a fixed point than to tune a bare theory for criticality. For completeness and the convenience of the reader, let us rewrite the relevant ERG differential equations in the dimensionless convention.

We first introduce dimensionless fields by

	$\displaystyle\bar{\varphi}(p)$	$\displaystyle\equiv k^{\frac{d+2}{2}}\varphi(pk)\,,$		(16a)
	$\displaystyle\left[\bar{\mathcal{O}}_{i}\right](p)$	$\displaystyle\equiv k^{d-d_{i}}\left[\mathcal{O}_{i}\right](pk)\,,$		(16b)
	$\displaystyle\bar{\varepsilon}_{i}(p)$	$\displaystyle\equiv k^{d_{i}}\varepsilon_{i}(pk)\,,$		(16c)

where $d_{i}$ is the scale dimension of $\left[\mathcal{O}_{i}\right](x)=\int_{p}e^{ipx}\left[\mathcal{O}_{i}\right](p)$ in the coordinate space at the Gaussian fixed point. It is sometimes called the engineering dimension.

We then define

$$\bar{\Gamma}_{t}[\bar{\varphi},\bar{\varepsilon}]\equiv\Gamma_{k}[\varphi,\varepsilon]\,,$$

(17)

where we have traded the momentum cutoff $k$ for the logarithmic flow time $t$, given by (6). The ERG differential equation for $\bar{\Gamma}_{t}[\bar{\varphi},\bar{\varepsilon}]$ is given by

	$\displaystyle\partial_{t}\bar{\Gamma}_{t}[\bar{\varphi},\bar{\varepsilon}]$	$\displaystyle=\int_{p}\left(\frac{d+2}{2}+p\cdot\partial_{p}\right)\bar{\varphi}(p)\cdot\frac{\delta\bar{\Gamma}_{t}[\bar{\varphi},\bar{\varepsilon}]}{\delta\bar{\varphi}(p)}+\int_{p}\left(d_{i}+p\cdot\partial_{p}\right)\bar{\varepsilon}_{i}(p)\cdot\frac{\delta\bar{\Gamma}_{t}[\bar{\varphi},\bar{\varepsilon}]}{\delta\bar{\varepsilon}_{i}(p)}$
		$\displaystyle\quad-\int_{p}\left(2-p\cdot\partial_{p}\right)R(p)\cdot\frac{1}{2}G_{t;p,-p}[\bar{\varphi},\bar{\varepsilon}]\,,$		(18)

where $G_{t;p,-q}[\bar{\varphi},\bar{\varepsilon}]$ is defined by

$$\int_{q}G_{t;p,-q}[\bar{\varphi},\bar{\varepsilon}]\left(\frac{\delta^{2}\bar{\Gamma}_{t}[\bar{\varphi},\bar{\varepsilon}]}{\delta\bar{\varphi}(q)\delta\bar{\varphi}(-r)}+R(q)\,(2\pi)^{d}\delta(q-r)\right)=(2\pi)^{d}\delta(p-r)\,.$$

(19)

The cutoff function $R(q)$ is a fixed function.

We are interested in the ERG flow from the Gaussian fixed point to the Wilson-Fisher fixed point. We parametrize the flow by $g$ so that the Gaussian fixed point is at $g=0$, and the Wilson-Fisher fixed point is at $g=g_{*}$. We can then replace $\partial_{t}$ by

$$\left(\epsilon g+\beta(g)\right)\partial_{g}$$

where $\epsilon$ is the scale dimension of $g$ at $g=0$. Accordingly, $\bar{\Gamma}(g)[\bar{\varphi}]=\bar{\Gamma}_{t}[\bar{\varphi},0]$ satisfies the ERG equation

$$\left\{\left(\epsilon g+\beta(g)\right)\partial_{g}+\Delta_{\bar{\varphi}}\right\}\bar{\Gamma}(g)[\bar{\varphi}]=-\int_{p}\left(2-p\cdot\partial_{p}\right)R(p)\cdot\frac{1}{2}G(g)_{p,-p}[\bar{\varphi}]\,,$$

(20)

where we define $\Delta_{\bar{\varphi}}$ and $G(g)_{p,-q}[\bar{\varphi}]$ by

$$\Delta_{\bar{\varphi}}\equiv\int_{p}\left(-\frac{d+2}{2}-p\cdot\partial_{p}\right)\bar{\varphi}(p)\cdot\frac{\delta}{\delta\bar{\varphi}(p)}\,,$$

(21)

and

$$\int_{q}G(g)_{p,-q}[\bar{\varphi}]\left(\frac{\delta^{2}\bar{\Gamma}(g)[\bar{\varphi}]}{\delta\bar{\varphi}(q)\delta\bar{\varphi}(-r)}+R(q)\,(2\pi)^{d}\delta(q-r)\right)=(2\pi)^{d}\delta(p-r)\,.$$

(22)

Actually, for the Wilson-Fisher fixed point to exist at $g=g_{*}=\mathrm{O}(\epsilon)$, we need to introduce an anomalous dimension $\frac{1}{2}\eta(g)=\mathrm{O}(g^{2})$ of $\bar{\varphi}$. Since we are only interested in the corrections of order $\epsilon$ to the OPE coefficients, we can ignore it. In Appendix B we obtain $\beta(g)$ to order $g^{2}$ to determine the fixed-point value

$$g_{*}=\frac{(4\pi)^{2}}{3}\epsilon+\mathrm{O}(\epsilon^{2})\,.$$

(23)

Similarly, differentiating (18) with respect to $\bar{\epsilon}_{i}$, we obtain the ERG equation for $\left[\bar{\mathcal{O}}_{i}\right]$ as

	$\displaystyle\left\{\left(\epsilon g+\beta(g)\right)\partial_{g}+d-d_{i}+p\cdot\partial_{p}+\Delta_{\bar{\varphi}}\right\}\left[\bar{\mathcal{O}}_{i}\right](p)-\sum_{j}\gamma_{ij}(g)\left[\bar{\mathcal{O}}_{j}\right](p)$
	$\displaystyle=\int_{q}(2-q\cdot\partial_{q})R(q)\,\frac{1}{2}\int_{r,s}G(g)_{q,-r}[\bar{\varphi}]\frac{\delta^{2}\left[\bar{\mathcal{O}}_{i}\right](p)}{\delta\bar{\varphi}(r)\delta\bar{\varphi}(s)}G(g)_{-s,-q}[\bar{\varphi}]\,.$		(24)

The mixing matrix $\gamma_{ij}(g)$ results from appropriate boundary conditions imposed at $p=0$. Differentiating (18) once more, we obtain the ERG equation for

$$\mathcal{Q}_{ij}(p_{1},p_{2})\equiv-\frac{\delta^{2}\bar{\Gamma}(g)[\bar{\varphi},\bar{\varepsilon}]}{\delta\bar{\varepsilon}_{i}(-p_{1})\delta\bar{\varepsilon}_{j}(-p_{2})}\Bigr{|}_{\varepsilon=0}$$

(25)

as

		$\displaystyle\left\{\left(\epsilon g+\beta(g)\right)\partial_{g}+2d-d_{i}-d_{j}+p_{1}\cdot\partial_{p_{1}}+p_{2}\cdot\partial_{p_{2}}+\Delta_{\bar{\varphi}}\right\}\mathcal{Q}_{ij}(p_{1},p_{2})$
		$\displaystyle\quad-\sum_{k}\left(\gamma_{ik}(g)\mathcal{Q}_{kj}(p_{1},p_{2})+\gamma_{jk}(g)\mathcal{Q}_{ik}(p_{1},p_{2})+\gamma_{ij,k}(g)\left[\bar{\mathcal{O}}_{k}\right](p_{1}+p_{2})\right)$
		$\displaystyle=\int_{q}\left(2-q\cdot\partial_{q}\right)R(q)\,\int_{r,s}G(g)_{q,-r}[\bar{\varphi}]G(g)_{s,-q}[\bar{\varphi}]$
		$\displaystyle\quad\times\left(\frac{1}{2}\frac{\delta^{2}\mathcal{Q}_{ij}(p_{1},p_{2})}{\delta\bar{\varphi}(r)\delta\bar{\varphi}(-s)}+\int_{u,v}\frac{\delta^{2}\left[\bar{\mathcal{O}}_{i}\right](p_{1})}{\delta\bar{\varphi}(r)\delta\bar{\varphi}(-u)}G(g)_{u,-v}[\bar{\varphi}]\frac{\delta^{2}\left[\bar{\mathcal{O}}_{j}\right](p_{2})}{\delta\bar{\varphi}(v)\delta\bar{\varphi}(-s)}\right)\,,$		(26)

where the mixing $\gamma_{ij,k}(g)$ is due to appropriate boundary conditions imposed at $p_{1}=p_{2}=0$.²²2 Eqs. (24) and (III) are derived by employing renormalized sources.

We have thus given the defining ERG differential equations for $\bar{\Gamma}(g)[\bar{\varphi}]$, $\left[\bar{\mathcal{O}}_{i}\right](g;p)$, and $\mathcal{Q}_{ij}(g;p_{1},p_{2})$. We end this section by summarizing how to extract the two- and three-point functions out of these Sonoda:2017rro . We first set $g$ to the fixed-point value $g_{*}$ to go to the Wilson-Fisher fixed point. To extract the two-point function of the elementary scalar field, we expand

$$\bar{\Gamma}(g_{*})[\bar{\varphi}]=\frac{1}{2}\int_{p_{1},p_{2}}\bar{\varphi}(p_{1})\bar{\varphi}(p_{2})\,(2\pi)^{d}\delta(p_{1}+p_{2})\,\bar{\Gamma}^{(2)}(p_{1})+\cdots$$

(27)

in powers of fields. For $p\gg 1$, we obtain the two-point function as

$$\left\langle\bar{\varphi}(p)\bar{\varphi}(q)\right\rangle=\frac{1}{\bar{\Gamma}^{(2)}(p)}\,(2\pi)^{d}\delta(p+q)\,.$$

(28)

Similarly, the field independent part of

$$\left[\bar{\mathcal{O}}_{i}\bar{\mathcal{O}}_{j}\right](g_{*};p_{1},p_{2})=\left[\bar{\mathcal{O}}_{i}\bar{\mathcal{O}}_{j}\right]^{(0)}(p_{1})\,(2\pi)^{d}\delta(p_{1}+p_{2})+\cdots$$

(29)

gives, for $p_{1},p_{2}\gg 1$, the two-point function as

$$\left\langle\bar{\mathcal{O}}_{i}(p_{1})\bar{\mathcal{O}}_{j}(p_{2})\right\rangle=\left[\bar{\mathcal{O}}_{i}\bar{\mathcal{O}}_{j}\right]^{(0)}(p_{1})\,(2\pi)^{d}\delta(p_{1}+p_{2})\,.$$

(30)

Finally, the part quadratic in fields of

$$\left[\bar{\mathcal{O}}_{i}\right](g_{*};p)=\frac{1}{2}\int_{p_{1},p_{2}}\bar{\varphi}(p_{1})\bar{\varphi}(p_{2})\,(2\pi)^{d}\delta(p_{1}+p_{2}-p)\,\left[\bar{\mathcal{O}}_{i}\right]^{(2)}(p_{1},p_{2})+\cdots$$

(31)

gives, for $p_{1},p_{2}\gg 1$, the three-point function as

$$\left\langle\bar{\varphi}(-p_{1})\bar{\varphi}(-p_{2})\bar{\mathcal{O}}_{i}(p)\right\rangle=\left[\bar{\mathcal{O}}_{i}\right]^{(2)}(p_{1},p_{2})\frac{1}{\bar{\Gamma}^{(2)}(p_{1})\bar{\Gamma}^{(2)}(p_{2})}\,(2\pi)^{d}\delta(p_{1}+p_{2}-p)$$

(32)

for $\mathbb{Z}_{2}$-invariant operators $\left[\bar{\mathcal{O}}_{i}\right]$. This equation needs to be modified as (115) if the fixed point has no $\mathbb{Z}_{2}$ invariance.

In the remaining part of the paper we work only in the dimensionless convention. Hence, we omit the bars above the symbols altogether.

In this section we construct explicitly scaling composite operators $\phi_{i}$ at the Wilson-Fisher fixed point in $d=4-\epsilon$ dimensions by solving the ERG. Composite operators are solutions of (24), which we write again as

	$\displaystyle\left(p\cdot\partial_{p}+\Delta_{\varphi}\right)\left[\mathcal{O}_{i}\right](p)+\left(d\,\delta_{ij}-\Delta_{ij}\right)\left[\mathcal{O}_{j}\right](p)$
	$\displaystyle=\frac{1}{2}\int_{q}(2-q\cdot\partial_{q})R(q)\int_{r,s}G_{q,-r}[\Phi]G_{-q,-s}[\Phi]\frac{\delta^{2}\left[\mathcal{O}_{i}\right](p)}{\delta\varphi(r)\delta\varphi(s)}\,,$		(33)

where $G_{p,q}[\Phi]$ is defined by

$$\int_{q}G_{p,q}[\Phi]\left(\frac{\delta^{2}\Gamma_{*}[\varphi]}{\delta\varphi(-q)\delta\varphi(r)}+R(q)(2\pi)^{d}\delta(q-r)\right)=(2\pi)^{d}\delta(p-r)\,,$$

and

$$\Delta_{\varphi}\equiv\int_{q}\left(-\frac{d+2}{2}-q\cdot\partial_{q}\right)\varphi(q)\cdot\frac{\delta}{\delta\varphi(q)}\,.$$

$\Delta_{ij}$ is the matrix of scale dimensions, and it is not diagonal in general:

$$\Delta_{ij}=d_{i}\,\delta_{ij}+\gamma_{ij}\,,$$

where $d_{i}$ is the engineering (mass) dimension of $\mathcal{O}_{i}$ in coordinate space, and $\gamma_{ij}$ is the mixing matrix.

The scaling composite operators $\phi_{i}$ are suitable linear combinations of composite operators that diagonalize the mixing matrix $\Delta_{ij}$. The scaling operators $\left[\Phi_{i}\right]$ for the eigenvalue $\Delta_{i}$ satisfy

	$\displaystyle\left(p\cdot\partial_{p}+d-\Delta_{i}+\Delta_{\varphi}\right)\left[\Phi_{i}\right](p)$
	$\displaystyle=\frac{1}{2}\int_{q}(2-q\cdot\partial_{q})R(q)\int_{r,s}G_{q,-r}[\varphi]G_{-q,-s}[\varphi]\frac{\delta^{2}\left[\Phi_{i}\right](p)}{\delta\varphi(r)\delta\varphi(s)}\,.$		(34)

We can then introduce a normalization constant $\mathcal{N}_{i}$ so that $\phi_{i}\equiv{\cal N}_{i}\left[\Phi_{i}\right]$ satisfies the normalization (12). Note that the coefficient ${\cal N}_{i}$ depends on the space dimension $d$, and therefore on $\epsilon$. Possibly, the simplest example along this line is given by the field $\phi_{1}\equiv{\cal N}_{1}\varphi$, which we describe in section IV.1.

Of course, before solving the ERG for the composite operators and their products, we need to solve the ERG equation to obtain $\Gamma_{*}[\varphi]$ at the fixed point. In this work we use perturbation theory to solve the ERG explicitly. Since solving for $\Gamma_{*}\left[\varphi\right]$ is not the main focus of this paper, we give the first order derivation of $\Gamma_{*}[\varphi]$ in Appendix B. For the present purposes, it suffices to say that the EAA parametrized by $g$ is expressed as

	$\displaystyle\Gamma(g)[\varphi]$	$\displaystyle=\frac{1}{2}\int_{p}\varphi(p)\varphi(-p)v^{(2)}(g;p)$
		$\displaystyle\quad+\frac{1}{4!}\int_{p_{1},\cdots,p_{4}}\prod_{i=1}^{4}\varphi(p_{i})\,(2\pi)^{d}\delta\left(\sum_{i=1}^{4}p_{i}-p\right)\,v^{(4)}(g;p_{1},\cdots,p_{4})\,,$		(35)

where

	$\displaystyle v^{(2)}(g;p)$	$\displaystyle=p^{2}+gv_{1}^{(2)}\,,$		(36a)
	$\displaystyle v^{(4)}(g;p_{1},\cdots,p_{4})$	$\displaystyle=g+g^{2}v_{2}^{(4)}(p_{1},\cdots,p_{4})\,.$		(36b)

$v_{1}^{(2)}$ is a constant given by (140), and the momentum dependent $v_{2}^{(4)}$ is given by (144). The Wilson-Fisher fixed point corresponds to $g=g_{*}=\frac{(4\pi)^{2}}{3}\epsilon$ up to order $\epsilon$. We introduce the high momentum propagator and its derivative by

	$\displaystyle h(p)$	$\displaystyle\equiv\frac{1}{p^{2}+R(p)}\,,$		(37a)
	$\displaystyle f(p)$	$\displaystyle\equiv(p\cdot\partial_{p}+2)h(p)=\frac{(2-p\cdot\partial_{p})R(p)}{\left(p^{2}+R(p)\right)^{2}}\,.$		(37b)

Both $R(p)$ and $f(p)$ decay rapidly at $p\gg 1$. To second order in $g$ the beta function is given by

$$\beta(g)=\beta_{2}\,g^{2}=\left(-3\int_{q}f(q)h(q)\right)g^{2}\,,$$

(38)

where

$$\int_{q}f(q)h(q)=\frac{1}{(4\pi)^{2}}+\mathrm{O}(\epsilon)\,.$$

(39)

Using this result, calculated in Appendix C, we obtain $g_{*}$ given in (23).