Exact Three-Point Functions of Determinant Operators in Planar N=4 Supersymmetric Yang-Mills Theory

TL;DR

This paper presents a nonperturbative method for calculating three-point correlation functions involving determinant and single-trace operators in planar N=4 SYM, utilizing integrability and gauge/string duality.

Contribution

It introduces a novel approach based on boundary states and integrability to compute these correlators at finite coupling, extending previous methods.

Findings

Derived expressions for correlators at finite coupling.

Validated the approach at weak coupling.

Connected gauge theory correlators with string worldsheet overlaps.

Abstract

We introduce a nonperturbative approach to correlation functions of two determinant operators and one non-protected single-trace operator in planar N=4 supersymmetric Yang-Mills theory. Based on the gauge/string duality, we propose that they correspond to overlaps on the string worldsheet between an integrable boundary state and a state dual to the single-trace operator. We determine the boundary state using symmetry and integrability of the dual superstring sigma model, and write down expressions for the correlators at finite coupling, which we conjecture to be valid for operators of arbitrary size. The proposal is put to test at weak coupling.

Figures

Tables

Table 1. Table 1: The squared structure constants ( 𝔇 𝒪 S ) 2 superscript subscript 𝔇 subscript 𝒪 𝑆 2 \left(\mathfrak{D}_{\mathcal{O}_{S}}\right)^{2} for the spin- S 𝑆 S twist-2 operators.

$S$	${\left({𝔇}_{{𝒪}_S}\right)}^2$
$2$	$\frac{1}{3}-4g^2+g^4\left(56-24{\zeta }_3\right)$
$4$	$\frac{1}{35}-\frac{205g^2}{441}+g^4\left(\frac{70219}{9261}-\frac{20{\zeta }_3}{7}\right)$
$6$	$\frac{1}{462}-\frac{1106g^2}{27225}+g^4\left(\frac{772465873}{1078110000}-\frac{14{\zeta }_3}{55}\right)$
$8$	$\frac{1}{6435}-\frac{14380057g^2}{4509004500}+g^4\left(\frac{5048546158688587}{85305405235050000}-\frac{1522{\zeta }_3}{75075}\right)$

Equations

$$\mathfrak{Z}(a)\equiv {\frac{(1+a^2){\Phi }^1+i(1-a^2){\Phi }^2+2ia{\Phi }^4}{\sqrt{2}}|}_{\begin{array}{c}{x}^{\mu }=\\ (0,a,0,0)\end{array}},$$

$$⟨{\mathcal{D}}_1{\mathcal{D}}_2\mathcal{O}(0)⟩={\left(\frac{a_1-a_2}{a_1{a}_2}\right)}^{\Delta -J}{\mathfrak{D}}_{\mathcal{O}},$$

$$\begin{array}{rl}Z(J,R)& =\sum _{{\psi }_{\mathrm{c}}}⟨\mathcal{G}|{\psi }_{\mathrm{c}}⟩e^{-E_{{\psi }_{\mathrm{c}}}R}⟨{\psi }_{\mathrm{c}}|\mathcal{G}⟩\\ & \stackrel{R\to \infty }{\to }|⟨\mathcal{G}|\Omega ⟩{|}^2{e}^{-E_{\Omega }R}.\end{array}$$

$$F_{\mathbf{A}\mathbf{B}}(u)\equiv {⟨\mathcal{G}|{\mathcal{X}}_{\mathbf{A}}(u){\mathcal{X}}_{\mathbf{B}}(\bar{u})⟩|}_{J\to \infty },$$

$$\begin{array}{rl}& {⟨\mathcal{G}|{\mathbb{S}}_{24}{\mathbb{S}}_{34}|{\mathcal{X}}_1(u){\mathcal{X}}_2(v){\mathcal{X}}_3(\bar{v}){\mathcal{X}}_4(\bar{u})⟩|}_{J\to \infty }\\ =& {⟨\mathcal{G}|{\mathbb{S}}_{13}{\mathbb{S}}_{12}|{\mathcal{X}}_1(u){\mathcal{X}}_2(v){\mathcal{X}}_3(\bar{v}){\mathcal{X}}_4(\bar{u})⟩|}_{J\to \infty },\end{array}$$

$$F_{\mathbf{AB}}(u)={\mathbb{S}}_{\mathbf{AB}}^{\mathbf{CD}}(u,\bar{u})F_{\mathbf{CD}}(\bar{u}).$$

$$F_{\mathbf{A}\mathbf{B}}(u){\mathfrak{C}}^{\mathbf{B}{\mathbf{B}}^{\prime }}{F}_{{\mathbf{B}}^{\prime }{\mathbf{C}}^{\prime }}({\bar{u}}^{2\gamma }){\mathfrak{C}}^{{\mathbf{C}}^{\prime }\mathbf{C}}={\delta }_{\mathbf{A}}^{\mathbf{C}}$$

$$F_{\mathbf{A}\mathbf{B}}(u)=\frac{x^{+}}{x^{-}}\frac{u-\frac{i}{2}}{u}\frac{{\sigma }_B(u)}{\sigma (u,\bar{u})}(-1{)}^{|\dot{A}||B|}{M}_{\mathbf{A},\mathbf{B}},$$

$${\sigma }_B(\bar{u})={\sigma }_B(u),{\sigma }_B(u){\sigma }_B(u^{2\gamma })=\frac{u^2}{u^2+\frac{1}{4}}.$$

$$\begin{array}{rl}\log G(x)=& \oint \frac{dz}{2\pi i}\frac{\log \mathfrak{G}(g(z+z^{-1}))}{x-z},\end{array}$$

$$Z(J,R)=\sum _{{\psi }_o}{e}^{-{\tilde{E}}_{{\psi }_o}J}\stackrel{R\to \infty }{\to }\mathcal{N}\int \mathcal{D}\rho e^{-R{S}_{\mathrm{eff}}[\rho ]}.$$

$$1={\mathbb{R}}_L(u_j)\prod _{k\ne j}\mathbb{S}(u_j,u_k){\mathbb{R}}_R(u_j)\prod _{k\ne j}\mathbb{S}(u_j,{\bar{u}}_k),$$

$$\begin{array}{r}{\left[{\mathbb{R}}_L\right]}_{A\dot{A}}^{B\dot{B}}(u)=\frac{u-\frac{i}{2}}{u}\frac{{\sigma }_B(u^{\gamma })}{\sigma ({\bar{u}}^{\gamma },u^{\gamma })}{\mathcal{S}}_{A\dot{A}}^{B\dot{B}}({\bar{u}}^{\gamma },u^{\gamma }),\end{array}$$

$${\phi }_{a,0}^{+}\equiv -J{\tilde{E}}_a+\log (1+Y_{b,0})\ast ({\mathcal{K}}_{+}{)}_{b,a}^{\bullet \bullet }$$

$$+\log (1+Y_{m,1})\star ({\mathcal{K}}_{+}{)}_{m-1,a}^{▹\bullet }+\log (1+1/Y_{2,2})\hat{\star }({\mathcal{K}}_{+}{)}_{+a}^{y\bullet }$$

$$+\log (1+Y_{1,1})\hat{\star }({\mathcal{K}}_{+}{)}_{-a}^{y\bullet }.$$

$$\begin{array}{rl}⟨\mathcal{G}|\Omega ⟩& =e^{{\sum }_a{\int }_0^{\infty }\frac{du}{2\pi }{\Theta }_a\log (1+Y_{a,0})}\frac{\sqrt{\mathrm{Det}(1-\hat{G})}}{\mathrm{Det}(1-{\hat{G}}_{+})}.\end{array}$$

$${\mathfrak{D}}_{\mathcal{O}}$$

$$\times \sqrt{\prod _{1\le s\le \frac{M}{2}}\frac{u_s^2+\frac{1}{4}}{u_s^2}{\sigma }_B^2(u_s)}\frac{\sqrt{\mathrm{Det}(1-{\hat{G}}^{\bullet })}}{\mathrm{Det}(1-{\hat{G}}_{+}^{\bullet })}.$$

$$\begin{array}{r}{\hat{G}}^{\bullet }\cdot f=\sum _{k=1}^M\frac{i{K}_{1,x}^{\bullet ,X}(u_k,u)}{{\partial }_u\log Y_{1,0}(u_k)}f(u_k)+\hat{G}\cdot f,\\ {\hat{G}}_{+}^{\bullet }\cdot f=\sum _{k=1}^{M/2}\frac{i({\mathcal{K}}_{+}{)}_{1,x}^{\bullet ,X}(u_k,u)}{{\partial }_u\log Y_{1,0}(u_k)}f(u_k)+{\hat{G}}_{+}\cdot f.\end{array}$$

$$u_{\frac{M}{2}+k}={\bar{u}}_k(1\le k\le M/2).$$

$$\varphi (u_j)=2\pi i\left(n_j+\frac{1}{2}\right),n_j\in \mathbb{Z},$$

$$\varphi (u)\equiv$$

$$+\sum _{x,X}\log (1+Y_{X,x})\star K_{x,1}^{X,\bullet }.$$

$$\log Y_{a,0}=$$

$$+\sum _{x,X}\log (1+Y_{X,x})\star K_{x,a}^{X,\bullet },$$

$$\frac{\sqrt{\mathrm{Det}(1-\tilde{\mathsf{S}})}}{\mathrm{Det}(1-{\tilde{\mathsf{S}}}_{+})}=\frac{\sqrt{\det ({\partial }_{u_k}\varphi (u_j))}}{\det ({\partial }_{u_k}{\varphi }^{+}(u_j))},$$

$$\frac{\sqrt{\mathrm{Det}(1-{\hat{G}}^{\bullet })}}{\mathrm{Det}(1-{\hat{G}}_{+}^{\bullet })}=\frac{\sqrt{\det ({\partial }_{u_k}\varphi (u_j))}}{\det ({\partial }_{u_k}{\varphi }^{+}(u_j))}\frac{\sqrt{\mathrm{Det}(1-\tilde{\mathsf{I}})}}{\mathrm{Det}(1-{\tilde{\mathsf{I}}}_{+})}.$$

$$e^{{\sum }_a{\int }_0^{\infty }\frac{du}{2\pi }{\Theta }_a\log (1+Y_{a,0})}\text{and}\frac{\sqrt{\mathrm{Det}(1-\tilde{\mathsf{I}})}}{\mathrm{Det}(1-{\tilde{\mathsf{I}}}_{+})}$$

$${\mathfrak{D}}_{\mathcal{O}}^{\mathrm{asym}}=$$

$$\times \frac{\sqrt{\det ({\partial }_{u_k}\varphi (u_j))}}{\det ({\partial }_{u_k}{\varphi }^{+}(u_j))}.$$

$$\log \left[\frac{{\mathfrak{D}}_{{\mathcal{O}}_S}}{{{\mathfrak{D}}_{{\mathcal{O}}_S}|}_{\text{tree level}}}\right]=$$

$$f_1$$

Full text

I Introduction

To advance our understanding of nonperturbative dynamics in gauge theories, it is useful to study simple models with rich enough structures. $\mathcal{N}=4$ supersymmetric Yang-Mills theory in four dimensions ($\mathcal{N}=4$ SYM) is one of the leading candidates for the following reasons: First it admits the planar large $N_{c}$ limit, which makes it amenable to analytical studies. Second it is a conformal field theory (CFT) and all the correlation functions can be decomposed into two- and three-point functions. Third it can be described alternatively in terms of two-dimensional string worldsheets which we can analyze exactly using integrability. The application of integrability led to a complete determination of two-point functions of local operators Beisert et al. (2012). It was applied also to the three-point function Basso et al. (2015), but the result is still unsatisfactory since it is given by a series expansion which one needs to resum.

In this letter, we present the first fully nonperturbative result for the three-point function valid for a large class of operators 111Some three-point functions are already known at finite coupling. For instance, the structure constant of a Lagrangian operator and two identical operators is given by a derivative of the conformal dimension $\partial_{\lambda}\Delta$ Costa et al. (2010). However, it is determined solely by the conformal dimension and does not provide truly new conformal data.. Specifically, we study the correlator of two determinant operators and one non-protected single-trace operator. By interpreting this correlator as an overlap on the string worldsheet between a boundary state and a state dual to the single-trace operator, we write down nonperturbative expressions using the framework of thermodynamic Bethe ansatz (TBA) Zamolodchikov (1991).

II Set Up and Basic Strategy

The main subject is the three-point function of a non-protected single-trace operator $\mathcal{O}$ and two determinant operators $\mathcal{D}_{1,2}\equiv\det\mathfrak{Z}(a_{1,2})$ 222We chose a configuration suitable for analyzing the symmetry. More general configurations can be obtained by performing the conformal and the R-symmetry transformations. The structure constant $\mathfrak{D}_{\mathcal{O}}$ is not affected by such transformations. with

[TABLE]

where $\Phi^{1,2,4}$ are real scalar fields in $\mathcal{N}=4$ SYM. Owing to the superconformal symmetry, its spacetime dependence is fixed to be Basso et al. (2015); Drukker and Plefka (2009)

[TABLE]

where $\mathfrak{D}_{\mathcal{O}}$ is the structure constant while $\Delta$ and $J$ are the conformal dimension and the R-charge of $\mathcal{O}$.

The goal of this letter is to compute $\mathfrak{D}_{\mathcal{O}}$ nonperturbatively using the gauge/string duality. As discussed in Bissi et al. (2011); Balasubramanian et al. (2002); McGreevy et al. (2000), the duality maps (2) to a closed string in AdS${}_{5}\times$S5 which ends on a geodesic of a D-brane dual to determinant operators (see Figure 1).

On the string worldsheet, this corresponds to an overlap between a boundary state $\langle\mathcal{G}|$ and a state dual to $\mathcal{O}$. To evaluate such an overlap, we first consider a partition function $Z(J,R)$ of a cylinder worldsheet whose ends are capped off by the boundary states (see Figure 2).

In the limit $R\to\infty$, the expansion of $Z(J,R)$ in the closed string channel is dominated by the ground state $|\Omega\rangle$

[TABLE]

By contrast, in the open string channel, $Z(J,R)$ can be viewed as the thermal free energy, and the limit corresponds to the thermodynamic limit in which the volume of the space becomes infinite. This allows us to compute $\langle\mathcal{G}|\Omega\rangle$ using TBA. The result for excited states can be obtained from $\langle\mathcal{G}|\Omega\rangle$ by analytic continuation Dorey and Tateo (1996).

III Constraints on Boundary States

To apply the aforementioned strategy, we first determine the boundary state $\langle\mathcal{G}|$ in the infinite volume ($J\to\infty$) limit. For this, we assume that $\langle\mathcal{G}|$ is an integrable boundary state, namely a state corresponding to a boundary condition which preserves infinite many conserved charges Ghoshal and Zamolodchikov (1994). The assumption is justified a posteriori by agreement with weak-coupling computations as we see later. For integrable boundary states, the overlap in the $J\to\infty$ limit can be factorized into two-particle overlaps

[TABLE]

where $\mathcal{X}$’s are magnons in the $\mathcal{N}=4$ SYM spin chain, and ${\bf A}=A\dot{A}$ and ${\bf B}=B\dot{B}$ are in the bifundamental representation of the $\mathfrak{psu}(2|2)^{2}$ symmetry Beisert (2008). The rapidities $u$ and $\bar{u}$ are parity-conjugate to each other, and satisfy $x^{\pm}(\bar{u})=-x^{\mp}(u)$ where $f^{\pm}(u)\equiv f(u\pm\frac{i}{2})$ and the Zhukovsky variable $x(u)$ is defined by $x(u)\equiv\frac{u+\sqrt{u^{2}-4g^{2}}}{2g}$ with $g\equiv\frac{\sqrt{\lambda}}{4\pi}$ and $\lambda$ being the ’t Hooft coupling constant.

Boundary Yang-Baxter equation

The integrable boundary state satisfies the so-called boundary Yang-Baxter equation (bYBE), which reads (see Figure 3)

[TABLE]

where $\mathbb{S}_{kl}$ is the bulk $S$-matrix Beisert (2008) between $\mathcal{X}_{k}$ and $\mathcal{X}_{l}$.

Watson’s equation

The second constraint is Watson’s equation, which states that an exchange of particles is equivalent to a multiplication of the S-matrix. Explicitly, it reads (see Figure 4)

[TABLE]

Decoupling equation

The last condition is the decoupling condition, which is equivalent to the boundary unitarity in Ghoshal and Zamolodchikov (1994). It states that a pair of particle-antiparticle pairs must decouple from the rest of the overlap (see Figure 5) and reads

[TABLE]

where $\mathfrak{C}$ is the charge conjugation matrix Janik (2006), and $u^{2\gamma}$ is the crossing transformation defined by $x^{\pm}(u^{2\gamma})=\frac{1}{x^{\pm}(u)}$.

Solution

Solving these constraints, the two-particle overlap is fixed to be

[TABLE]

where $|\bullet|$ denotes the grading of the index $\bullet$ and $\sigma(u,v)$ is the bulk dressing phase Beisert et al. (2007). As shown in Appendix A, there are two choices for the matrix part $M_{{\bf A},{\bf B}}$ and we conjecture that the true boundary state is given by a sum of the two. See also discussions below (16). $\sigma_{B}(u)$ is the boundary dressing phase satisfying

[TABLE]

A solution is given by $\sigma_{B}(u)=4^{\frac{1+x^{+}x^{-}}{1-x^{+}x^{-}}}\frac{G(x^{+})}{G(x^{-})}$ with

[TABLE]

and $\mathfrak{G}(u)\equiv\,\frac{\Gamma\left(\frac{1}{2}-iu\right)\Gamma\left(1+iu\right)}{\Gamma\left(\frac{1}{2}+iu\right)\Gamma\left(1-iu\right)}$.

IV $g$-Function for Ground State

We now discuss the ground-state overlap $\langle\mathcal{G}|\Omega\rangle$ for finite $J$. For this we consider $Z(J,R)$ in the open string channel (also known as the mirror channel) and take the limit $R\to\infty$:

[TABLE]

As shown above, in the limit $R\to\infty$ one can replace the sum over $\psi_{o}$ with a path integral of densities $\rho$.

Bethe equation in the mirror channel

The crucial input for writing down $S_{\rm eff}$ is the boundary asymptotic Bethe equation (bABA), which constrains the rapidities of magnons. Schematically, it reads (see Figure 6)

[TABLE]

where $\mathbb{R}_{L,R}$ are the left/right reflection matrices. The reflection matrices are related to the infinite-volume overlap (8) by $[\mathbb{R}_{L}]_{{\bf A}}^{{\bf B}}(u)=[\mathbb{R}_{R}]_{{\bf A}}^{{\bf B}}(\bar{u})=F_{{\bf A}{\bf C}}(u^{\gamma})\mathfrak{C}^{{\bf C}{\bf B}}$ with $u^{\gamma}$ being the mirror transformation defined by $x^{+}(u^{\gamma})=1/x^{+}(u)$ and $x^{-}(u^{\gamma})=x^{-}(u)$. As a result, we find

[TABLE]

where $\mathcal{S}$ is a single copy of the $\mathfrak{psu}(2|2)$ S-matrix Beisert (2008). The structure of $\mathbb{R}_{L,R}$ allows for the unfolding interpretation; the bABA (12) can be mapped to an ABA of a closed string with a single $\mathfrak{psu}(2|2)$ symmetry (see Figure 6).

TBA equation

$S_{\rm eff}$ can be derived from ABA following the standard derivation of TBA. In the $R\to\infty$ limit, (11) can be approximated by the saddle point $\frac{\delta S_{\rm eff}}{\delta\rho}=0$. Due to the unfolding structure, the saddle-point equations coincide with the standard TBA for the spectrum Gromov et al. (2009); Bombardelli et al. (2009); Arutyunov and Frolov (2009) with the identification $Y_{a,s}(\bar{u})=Y_{a,-s}(u)$. They take a form of $\log Y_{a,s}=\varphi^{+}_{a,s}$. For instance, $\varphi^{+}_{a,0}$ reads (the full equations are given in Jiang et al. (2019))

[TABLE]

Here we follow the notations in Bajnok (2012), and $\star$, $\ast$, and $\widehat{\star}$ denote the convolutions along $[-\infty,\infty]$, $[0,\infty]$ and $[-2g,2g]$ respectively. $\mathcal{K}_{+}$ is a symmetrized kernel defined by $K(u,v)+K(u,\bar{v})$ with $K$ being the standard TBA kernel.

g-function

The saddle point value of $S_{\rm eff}$ only gives the exponentially decaying piece in (3), $e^{-E_{\Omega}R}$. To read off the overlap, we need to consider the one-loop fluctuation around the saddle point and the $O(1)$ normalization factor $\mathcal{N}$ in (11). Such analysis was performed in the literature Dorey et al. (2010); Pozsgay (2010); Kostov et al. (2018), and the application to our problem leads to an expression for the ground states, which corresponds to BPS operators in $\mathcal{N}=4$ SYM,

[TABLE]

Here $a$ runs from $1$ to $\infty$ and $\Theta_{a}(u)=i\partial_{u}\log r_{a}(\bar{u})-\pi\delta(u)+\frac{i}{2}\partial_{u}\log S_{aa}^{\bullet\bullet}(u,\bar{u})$ where $S_{ab}^{\bullet\bullet}$ is the bound-state S-matrix and $r_{a}$ is the bound-state reflection factor given in Appendix A. ${\rm Det}$ denotes the Fredholm determinant 333In what follows, ${\rm Det}$ denotes the Fredholm determinant while ${\rm det}$ denotes a standard determinant of a finite-dimensional matrix. and $\hat{G}_{+}$ is an integral kernel defined by $[\hat{G}_{+}]^{(a,s)}_{(b,t)}(u,v)\equiv\frac{\delta\varphi^{+}_{a,s}(u)}{\delta\log Y_{b,t}(v)}$. Similarly, $\hat{G}$ is given by $\frac{\delta\varphi_{a,s}(u)}{\delta\log Y_{b,t}(v)}$ where $\varphi_{a,s}$ are the “right hand sides” of the standard TBA, $\log Y_{a,s}=\varphi_{a,s}$, without identification of $Y$-functions.

V Conjecture for SL(2) Sector

We now generalize (15) to excited states in the SL(2) sector using the analytic continuation trick Dorey and Tateo (1996) following the standard TBA analysis.

$g$-function for excited states

After the analytic continuation, poles of $1/(1+1/Y_{1,0})$ cross the integration contour and modify the overlap (15). As a result, we find that the structure constant $\mathfrak{D}_{\mathcal{O}}$ is given by

[TABLE]

This is the main result of this letter which we conjecture to be valid for any length $J$ and at finite $\lambda$. Here the $J$-dependent prefactor reflects the fact that the true boundary state is a sum of two boundary states as mentioned below (8). $\hat{G}^{\bullet}$ and $\hat{G}_{+}^{\bullet}$ are given by 444The kernels in this letter are related to the ones in Jiang et al. (2019) by the transposition.

[TABLE]

Here and below, $x$ and $X$ take various indices and symbols which represent different bound states. The sum in (17) come from the poles crossing the contours, and $u_{k}$’s are the magnon rapidities satisfying the parity condition,

[TABLE]

They are the solutions to the exact Bethe equations Gromov et al. (2009),

[TABLE]

with

[TABLE]

Exact Gaudin determinants

The result (17) can be rewritten into a form similar to the so-called Gaudin determinants. For this, we first split the kernel $\hat{G}^{\bullet}$ into a sum ${\sf S}$ and an integral ${\sf I}$ (see (17)), and rewrite ${\rm Det}(1-\hat{G}^{\bullet})$ as ${\rm Det}(1-{\sf S}-{\sf I})={\rm Det}(1-\tilde{\sf S})\times{\rm Det}(1-{\sf I})$ with $\tilde{\sf S}\equiv{\sf S}/(1-{\sf I})$. Similarly, ${\rm Det}(1-\hat{G}^{\bullet}_{+})$ can be split into a sum ${\sf S}_{+}$ and an integral ${\sf I}_{+}$, and can be re-expressed as ${\rm Det}(1-\hat{G}^{\bullet}_{+})={\rm Det}(1-\tilde{\sf S}_{+})\times{\rm Det}(1-{\sf I}_{+})$ with $\tilde{\sf S}_{+}\equiv{\sf S}_{+}/(1-{\sf I}_{+})$.

Next we consider $\partial_{u_{k}}\phi(u_{j})$ $(j,k=1,\ldots,M)$. The derivative $\partial_{u_{k}}$ can act either on $p(u_{k})$, $\log S^{\bullet\bullet}_{11}$ or $Y_{X,x}$ in (20). We then eliminate $\partial_{u_{k}}Y_{X,x}$ by considering the excited state TBA (see Gromov et al. (2009) for the full set of equations)

[TABLE]

taking a derivative $\partial_{u_{k}}$ of both sides, and solving for $\partial_{u_{k}}Y_{X,x}$. The parity condition (18) is imposed only at the end of the computation. As a result of these manipulations, we find that $\det(\partial_{u_{k}}\phi(u_{j}))\propto{\rm Det}(1-\tilde{\sf S})$ up to some constant of proportionality. The relation in particular shows that ${\rm Det}(1-\tilde{\sf S})$ is actually a finite-dimensional determinant although it was initially defined as the Fredholm determinant. Details of the rewriting are explained in a toy example in Appendix B.

On the other hand, if we first impose the parity condition (18) and compute the derivatives $\partial_{u_{k}}\phi(u_{j})$, we find that $\det(\partial_{u_{k}}\phi(u_{j}))$ $(j,k=1,\ldots\frac{M}{2})$ is now proportional to ${\rm Det}(1-\tilde{\sf S}_{+})$. Upon taking the ratio, the constants of proportionality cancel out and we obtain

[TABLE]

where $\phi^{+}$ denotes that we are imposing the parity condition before computing derivatives. These determinants can be viewed as the finite-volume version of the Gaudin determinants for the norm of the spin chain. They also resemble the finite-volume one-point functions in sin(h)-Gordon model Jimbo et al. (2011); Negro and Smirnov (2013); Bajnok and Smirnov (2019).

Using this rewriting, we obtain an alternative representation for the Fredholm determinants in (16),

[TABLE]

Asymptotic formula

Using the representation (23), one can take the asymptotic limit of (16), in which the size of the operator becomes large $J\gg 1$. In this limit, the middle-node Y-functions are exponentially suppressed, $Y_{a,0}\to 0$, and one can show that both

[TABLE]

tend to unity. We thus obtain the following expression for the structure constant in the asymptotic limit:

[TABLE]

Note that the determinants on the second line are the standard Gaudin-like determinants since all the finite-size corrections can be dropped. For generalization of (24) to operators outside the SL(2) sector, see Jiang et al. (2019). A similar formula was found at weak coupling for the defect one-point functions de Leeuw et al. (2015).

VI Weak Coupling Test

To test our formula (24), we computed the four-point function of $\mathcal{D}_{1,2}$ and two ${\bf 20}^{\prime}$ operators $\mathcal{O}_{{\bf 20}^{\prime}}$ up to $O(\lambda^{2})$. We then performed the operator product expansion to read off the conformal data of the spin-$S$ twist-2 operators $\mathcal{O}_{S}$. The details are given in Jiang et al. (2019).

The results of the computation are summarized in Table 1. We compared them against the integrability prediction (24) and observed a perfect match. This is quite a nontrivial test of our formalism since the results contain the transcendental number $\zeta_{3}$ and include the contributions from the boundary dressing phase $\sigma_{B}(u)$. Further tests at weak and strong couplings are provided in Jiang et al. (2019).

We also found that the structure constants exhibit a simple large spin behavior up to two loops,

[TABLE]

with $\log S^{\prime}\equiv\log S+\gamma_{E}$, where $\gamma_{E}$ is the Euler-Mascheroni constant, and

[TABLE]

VII Conclusion

In this letter, we applied the TBA formalism to write down a nonperturbative expression for the structure constant of two determinant operators and a single-trace operator in the SL(2) sector of arbitrary size. Our result would provide a foundation for future developments, such as the reformulation in terms of the Quantum Spectral Curve Gromov et al. (2014), as was the case with the TBA for the spectrum. It would also be worth trying to extract various interesting physics from our formula. We also hope that our approach gives useful insights into the three-point functions of single-trace operators Basso et al. (2015).

Acknowledgement

We thank Amit Sever for helpful discussions. SK is supported by DOE grant number DE-SC0009988. EV is funded by the FAPESP grants 2014/18634-9 and 2016/09266-1, and by the STFC grant ST/P000762/1.

Appendix A Overlap and Reflection Matrix

The matrix part of the two-particle overlap is given by

[TABLE]

where

[TABLE]

There are two allowed solutions depending on the value of $z$, $z=\pm i$.

The bound-state reflection factor $r_{a}$ is given by

[TABLE]

[TABLE]

with $f^{[\pm a]}\equiv f(u\pm\frac{ia}{2})$ and

[TABLE]

Appendix B Fredholm Determinant in Toy Model

Here we write down an explicit expression for the Fredholm determinant in a simple toy model which only contains a single species of particles without bound states. We also elucidate the relation between the Fredholm determinants and the exact Gaudin determinants (22).

The excited state TBA is given by $\log Y=\phi$ with

[TABLE]

where $S$ is the S-matrix and $K(u,v)\equiv-i\partial_{u}\log S(u,v)$. The rapidities $u_{k}$’s satisfy the exact Bethe equation

[TABLE]

Following the discussion in the main text, one obtains the deformed Fredholm kernel $\hat{G}^{\bullet}\cdot f={\sf S}\cdot f+{\sf I}\cdot f$, with

[TABLE]

Let us now consider the exact Gaudin norm $\det\left(\partial_{u_{k}}\phi(u_{j})\right)$. Using (32), we obtain

[TABLE]

with $\epsilon(u)\equiv\log Y(u)$. To eliminate $\partial_{u_{k}}\epsilon$, we differentiate the excited state TBA,

[TABLE]

and invert it to get

[TABLE]

Plugging it back to (35), we obtain

[TABLE]

Comparing (34) and (38), we find

[TABLE]

with $\tilde{\sf S}\equiv{\sf S}/(1-{\sf I})$. This relation allows us to relate the Fredholm determinant and the exact Gaudin determinant. The derivation can be readily generalized to $\hat{G}_{+}^{\bullet}$ and to theories with bound states such as $\mathcal{N}=4$ SYM, leading to the identity (22).