JHEP Journal of High Energy Physics 1029-8479 JHEP07(2018)174 10.1007/JHEP07(2018)174 Field theoretical derivation of Lüscher’s formula and calculation of finite volume form factors Zoltán Bajnok bajnok.zoltan@wigner.mta.hu János Balog balog.janos@wigner.mta.hu Márton Lájer lajerm@caesar.elte.hu Chao Wu chao.wu@wigner.mta.hu MTA Lendület Holographic QFT Group, Wigner Research Centre for Physics,H-1525 Budapest 114, P.O.B. 49, Hungary Institute for Theoretical Physics, Eötvös Loránd University,H-1117 Budapest Pázmány P. s. 1/A, Hungary 30072018 2018 07 174 13032018 11072018 18072018 OPEN ACCESS, © The Authors 2018 This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 1802.04021

We initiate a systematic method to calculate both the finite volume energy levels and form factors from the momentum space finite volume two-point function. By expanding the two point function in the volume we extracted the leading exponential volume correction both to the energy of a moving particle state and to the simplest non-diagonal form factor. The form factor corrections are given in terms of a regularized infinite volume 3-particle form factor and terms related to the Lüsher correction of the momentum quantization. We tested these results against second order Lagrangian and Hamiltonian perturbation theory in the sinh-Gordon theory and we obtained perfect agreement.

Field Theories in Lower Dimensions Integrable Field Theories Nonperturbative Effects

Article funded by SCOAP3

Introduction

Overview of the method and summary of the results

Finite size energy spectrum

Finite size form factors

Finite volume two-point function

Mirror representation

2$the contribution is subleading to the second L\"uscher order, which is O$({\rm e}^{-2mL})$and which we neglect. This restriction is also necessary for some of our later estimates to be valid. The matrix element$\langle u\vert{\cal O}\vert\beta_1,\beta_2\rangle$can be represented in terms of the S-matrix$S(\theta)$and the 3-particle form factor$F_3(u,\beta_1,\beta_2)$as~\cite{Smirnov-ml-1992vz} \begin{equation} \langle u\vert{\cal O}\vert\beta_1,\beta_2\rangle= \delta(u-\beta_1)F_1+S(\beta_1-\beta_2)\delta(u-\beta_2)F_1 +F_3(u+i\pi-i\epsilon,\beta_1,\beta_2). \label{mat} \end{equation} The integral of its square is divergent and needs to be regularized. ]]> Regularization 0$ is small. We have to pay attention to the following. \begin{itemize} \item[A)] The right hand side of~(\ref{sinhb}) must not cross the cut of the {\tt arcsinh} function (which runs from $i$ to $i\infty$ along the imaginary axis). \item[B)] Avoid points where $C=\cosh u+i\hat q$. \item[C)] Take into account the poles of the regularized matrix elements at $w=\pm(u-b\pm i\epsilon)$. \end{itemize} Problems A) and B) can be easily avoided if $\cosh u<2$ and the parameter $\gamma$ is small enough. The form factor poles can be taken into account explicitly, using the residue theorem. (Only two of the poles lie above the real axis.) After a long computation, we find (up to terms vanishing in the $\epsilon\to0$ limit): \begin{equation} \begin{split} J(u,\psi,q)=\;&\left(\frac{F_1^2}{2\pi\epsilon}-\delta(0)F_1^2\right) \frac{1}{\cosh\psi(\cosh\psi-i\hat q)}+I(u,\psi,q)\\ &+\frac{F_1}{\cosh\psi(\cosh\psi-i\hat q)}[F_3^c(u+i\pi,u,\psi)+F_3^c(u+i\pi,\psi,u)]\\ &+\frac{iF_1^2}{4\pi}\frac{\sinh(u-\psi)[S(\psi-u)-S(u-\psi)]}{\cosh^2\psi(\cosh\psi-i\hat q)^2}\\ &+\frac{iF_1^2}{4\pi}\frac{1}{\cosh\psi(\cosh\psi-i\hat q)}\bigg[ \frac{2[S(u-\psi)-S(\psi-u)]}{u-\psi}\\ &+\frac{\nu[S^\prime(\psi-u)+S^\prime(u-\psi)]}{\cosh\psi}+\frac{\sinh(u-\psi)[S(\psi-u)-S(u-\psi)]}{\nu\cosh\psi}\\ &+\frac{(\sinh\psi+\sinh u)(1+\sinh u\sinh\psi)[S(u-\psi)-S(\psi-u)]}{\nu\cosh^2\psi}\bigg]. \end{split} \label{final} \end{equation} Here the notation \begin{equation} \nu=\cosh\psi+\cosh u \end{equation} is used and $I(u,\psi,q)$ is the shifted integral ($w=v+i\gamma$): \begin{align} I(u,\psi,q)=& \int_{-\infty}^\infty\frac{{\rm d}v}{C(C-\cosh u-i\hat q)}S(-2w) \bigg\{ \frac{iF_1}{2\pi} \left[ \frac{1-S(2w)}{u-b-w} + \frac{S(2w)-1}{u-b+w} \right] \cr & + F_3^c(u+i\pi-b,w,-w) \bigg\}^2. \label{shiftI} \end{align} The (negative) divergent term coming from the denominator is accompanied with a (positive) divergent term coming from the calculation of the numerator. They both multiply the same function. Our main assumption is that the divergences cancel\footnote{Note that putting blindly $x=0$ to the definition of the regularized delta function gives $\delta(0)=1/\pi\varepsilon$.} and the remaining finite terms are correct. Indeed, in appendix~\ref{AppendixB} we show that our heuristic regularization is completely equivalent to the well-defined finite volume regularization. We will make the substitution \begin{equation} \left(\frac{1}{2\pi\epsilon}-\delta(0)\right)\rightarrow\Delta, \end{equation} where $\Delta$ is a finite renormalization constant, which will be fixed later. ]]>

Analytic continuation

L\"uscher's formula

Finite volume form factor

Sinh-Gordon form factors

Hamiltonian perturbation theory

Finite volume form of the Hamiltonian

Time-independent perturbation theory

Corrections to the one-particle energy

\texorpdfstring{\boldmath $\mathcal{O}\left(b^{2}\right)$}{O(b**2)} correction

\texorpdfstring{\boldmath $\mathcal{O}\left(b^{4}\right)$}{O(b**4)} correction

Extracting L\"uscher corrections

Corrections to the form factor \texorpdfstring{\boldmath $\left\langle 0\left(b\right)\left|\varphi\right|q\left(b\right)\right\rangle$}{<0(b)|phi|q(b)>}

\texorpdfstring{\boldmath $\mathcal{O}\left(b^{2}\right)$}{O(b**2)} correction

\texorpdfstring{\boldmath $\mathcal{O}\left(b^{4}\right)$}{O(b**4)} correction

Extracting first L\"uscher correction

Conclusions

Acknowledgments

Perturbative expansion of the sinh-Gordon TBA equations

Details of the Hamiltonian perturbative calculations

The double sum in the energy correction

1$, the upper cut intersects the real axis. The$k_{2}=n_{q}$terms of the double sum in~(\ref{eq:separsum}) were separated for precisely this reason. Now an integral representation can be achieved by writing$S_{\Theta}$as a sum of two contour integrals \begin{equation} S_{\Theta}=\frac{L}{2\pi}\left[\int_{C_{1}}dz\frac{e^{iLz}}{e^{iLz}-1}f_{\Theta}\left(z\right) + \int_{C_{2}}dz\frac{e^{iLz}}{e^{iLz}-1}f_{\Theta}\left(z\right)\right]\label{eq:contour2} \end{equation} with \begin{equation} f_{\Theta}\left(z\right)=\frac{2u\left(\mu^{2}+q^{2}\right)\left(z+i\mu u\right)}{\mu^{2}\left(z-q+i\mu u\right)\left(z+q+i\mu u\right)\left(q^{2}+\mu^{2}u^{2}\right)\sqrt{\left(u^{2}-1\right)\left[\mu^{2}+\left(z+i\mu u\right)^{2}\right]}}. \end{equation} The closed contours$C_{1}$and$C_{2}$are chosen such that$C_{1}$goes from$-\infty-i\epsilon$to$-2\pi L^{-1}-i\epsilon$just below the real axis, then from$-2\pi L^{-1}+i\epsilon$back to$-\infty+i\epsilon$just above the real axis, while$C_{2}$is the mirror image of$C_{1}$with respect to the imaginary axis except that it is also directed counterclockwise. Now both contours can be blown up such that they are tightened to the cuts. As a result of the deformation, the poles of$f_{\Theta}$at$z=\pm q-i\mu u$become encircled in the negative direction which results in additional residual terms. After the variable changes$u\rightarrow\cosh u$,$v\rightarrow\cosh v$, and extending the intagration domain over the real line by symmetrization,\footnote{The region around the branch-overlapped pole needs special treatment.} we get an integral representation of$S_{\Theta}$as \begin{equation} S_{\Theta} = \frac{2L}{i\mu^{2}}\frac{e^{\mu Lu}}{e^{\mu Lu}-1} \frac{\sqrt{\mu^{2}+q^{2}}u}{\left(q^{2}+\mu^{2}u^{2}\right)\sqrt{u^{2}-1}}+\frac{L}{i\pi\mu}\int_{-\infty}^{\infty}dv \left(\lambda\left(u,v\right)s\left(u,v\right)+\mathrm{sing}_{\Theta}\left(u,v\right)\right) \label{eq:sthetint} \end{equation} where \begin{eqnarray} \lambda\left(u,v\right) & = & \frac{e^{\mu L\cosh u}}{e^{\mu L\cosh v}-e^{\mu L\cosh u}}-\frac{e^{\mu L\left(\cosh u+\cosh v\right)}}{e^{\mu L\left(\cosh u+\cosh v\right)}-1}, \\ s\left(u,v\right) & = & \frac{\left(\mu^{2}+q^{2}\right)\cosh u\cosh v}{\left(q^{2}+\mu^{2}\cosh^{2}u\right)\left(q^{2}+\mu^{2}\cosh^{2}v\right)}, \\ \label{Theta singular part} \mathrm{sing}_{\Theta}\left(u,v\right) & = & \frac{2}{\mu L}\frac{1}{u^{2}-v^{2}}\frac{s\left(u,u\right)u}{\sinh u} \end{eqnarray} The term of~(\ref{Theta singular part}) comes from the neighbourhood of the branch-overlapped pole. Note that both$\lambda\left(u,v\right)$and$\mathrm{sing}_{\Theta}\left(u,v\right)$are singular along the lines$u=\pm v$; their sum is, however, finite everywhere. At this stage, we can represent the original double sum as a formula containing the following double integral \begin{eqnarray} \sum_{k_{1},k_{2}\in\mathbb{Z}}D_{1}\left(k_{1},k_{2}\right) & = & L^{2}\int_{-\infty}^{\infty}\frac{du}{2\pi}\int_{-\infty}^{\infty}\frac{dv}{2\pi}\coth\left(\frac{\mu L}{2}\cosh u\right)\left(\lambda\left(u,v\right)s\left(u,v\right)+\mathrm{sing}_{\Theta}\left(u,v\right)\right)\nonumber \\ & & +\left(\text{other terms}\right)\label{eq:DintPV} \end{eqnarray} Since the double integral is absolutely convergent, we can perform a symmetrization of the integrand as \begin{equation} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dudv\:f\left(u,v\right)=\frac{1}{2}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}dudv\:\left(f\left(u,v\right)+f\left(v,u\right)\right) \label{eq:symmetriz} \end{equation} which leaves the value of the integral unchanged. Upon this transformation, the first term of~(\ref{eq:DintPV}) becomes \begin{equation} L^{2}\int_{-\infty}^{\infty}\frac{du}{2\pi}\int_{-\infty}^{\infty}\frac{dv}{2\pi}\frac{1+e^{\mu L\cosh u}+e^{\mu L\cosh v}-3e^{\mu L\left(\cosh u+\cosh v\right)}}{2\left(e^{\mu L\cosh u}-1\right)\left(e^{\mu L\cosh v}-1\right)}s\left(u,v\right)\label{eq:dterm} \end{equation} which can be further simplified as \begin{eqnarray} &&L^{2}\int_{-\infty}^{\infty}\frac{du}{2\pi}\int_{-\infty}^{\infty}\frac{dv}{2\pi}\frac{1+e^{\mu L\cosh u}+e^{\mu L\cosh v}-3e^{\mu L\left(\cosh u+\cosh v\right)}}{2\left(e^{\mu L\cosh u}-1\right)\left(e^{\mu L\cosh v}-1\right)}s\left(u,v\right) \nonumber \\ &&\hspace{4cm}=-\frac{L^{2}}{\mu^{2}}\left(\frac{3}{8}+\int_{-\infty}^{\infty}\frac{du}{2\pi}\frac{1}{e^{\mu L\cosh u}-1}\frac{\mu\sqrt{\mu^{2}+q^{2}}\cosh u}{q^{2}+\mu^{2}\cosh^{2}u}\right)\label{eq:orig_result}\nonumber\\ \end{eqnarray} if we note that \begin{equation} \frac{1+e^{\mu L\cosh u}+e^{\mu L\cosh v}-3e^{\mu L\left(\cosh u+\cosh v\right)}}{2\left(e^{\mu L\cosh u}-1\right)\left(e^{\mu L\cosh v}-1\right)}=-\frac{3}{2}-\frac{1}{e^{\mu L\cosh u}-1}-\frac{1}{e^{\mu L\cosh v}-1} \end{equation} We now calculate the integral of the symmetrized second term of the integrand. For brevity, we introduce the function $\mathrm{sing}\left(u,v\right)=\coth\left(\frac{\mu L}{2}\cosh u\right)\mathrm{sing}_{\Theta}\left(u,v\right).$ In this notation, the symmetrized integral has the following form $\frac{L^{2}}{2}\int_{-\infty}^{\infty}\frac{du}{2\pi}\int_{-\infty}^{\infty}\frac{dv}{2\pi}\left(\mathrm{sing}\left(u,v\right)+\mathrm{sing}\left(v,u\right)\right).$ Both terms of this integrand are divergent by themselves; their sum, however, is finite everywhere. To perform the integrations, we notice that due to the symmetry of \hbox{the~integrand,} \begin{eqnarray} &&\frac{L^{2}}{2}\int_{-\infty}^{\infty}\frac{du}{2\pi}\int_{-\infty}^{\infty}\frac{dv}{2\pi}\left(\mathrm{sing}\left(u,v\right)+\mathrm{sing}\left(v,u\right)\right)\nonumber\\ &&\qquad\qquad=\frac{4L^{2}}{2}\int_{0}^{\infty}\frac{du}{2\pi}\int_{-u}^{u}\frac{dv}{2\pi}\left(\mathrm{sing}\left(u,v\right)+\mathrm{sing}\left(v,u\right)\right) \end{eqnarray} which we regularize\footnote{This regularization comes from regularizing the contour integral around the overlapped pole and then performing the change of variables.} as \begin{eqnarray} &&2L^{2}\lim_{\epsilon\rightarrow0}\left[\int_{\epsilon}^{\infty}\frac{du}{2\pi}\int_{-u+\epsilon}^{u-\epsilon}\frac{dv}{2\pi}\mathrm{sing}\left(u,v\right)+\int_{\epsilon}^{\infty}\frac{du}{2\pi}\int_{-u+\epsilon}^{u-\epsilon}\frac{dv}{2\pi}\mathrm{sing}\left(v,u\right)\right] \nonumber \\ &&\qquad=2L^{2}\lim_{\epsilon\rightarrow0}\left[\int_{\epsilon}^{\infty}\frac{du}{2\pi}\int_{-u+\epsilon}^{u-\epsilon}\frac{dv}{2\pi}\mathrm{sing}\left(u,v\right)+2\int_{0}^{\infty}\frac{du}{2\pi}\int_{u+\epsilon}^{\infty}\frac{dv}{2\pi}\mathrm{sing}\left(u,v\right)\right]\label{eq:singint} \end{eqnarray} In the last step we made use of the identity \begin{equation} \int_{\epsilon}^{\infty}\frac{du}{2\pi}\int_{-u+\epsilon}^{u-\epsilon}\frac{dv}{2\pi}=\int_{-\infty}^{\infty}\frac{dv}{2\pi}\int_{v+\epsilon}^{\infty}\frac{du}{2\pi}, \end{equation} and the fact that$\mathrm{sign}\left(u,v\right)$is symmetric in$v$together with the freedom to switch the labeling of integration variables. Now the integrals over$v$in~(\ref{eq:singint}) can be performed. Combining the remaining$u$integrals, we obtain \begin{equation} \frac{8L}{\mu}\lim_{\epsilon\rightarrow0}\left[-\int_{0}^{\epsilon}\frac{du}{\left(2\pi\right)^{2}}\frac{\tilde{s}\left(u\right)}{\sinh u}\mathrm{arctanh}\left(\frac{u}{u+\epsilon}\right)+\frac{1}{2}\int_{\epsilon}^{\infty}\frac{du}{\left(2\pi\right)^{2}}\frac{\tilde{s}\left(u\right)}{\sinh u}\ln\left(1-\frac{2\epsilon}{\epsilon+2u}\right)\right] \end{equation} with \begin{equation} \tilde{s}\left(u\right)=s\left(u,u\right)\coth\left(\frac{\mu L}{2}\cosh u\right). \end{equation} Both integrands approximate Dirac$\delta$-like peaks centered at$u=0$in the$\epsilon\rightarrow0$limit. Thus, we approximate the regular part$\tilde{s}\left(u\right)$with its value at the top of the peaks, and integrate analytically the singular part. Finally, taking the$\epsilon\rightarrow0limit, we get \begin{align} & \frac{L^{2}}{2}\int_{-\infty}^{\infty}\frac{du}{2\pi}\int_{-\infty}^{\infty}\frac{dv}{2\pi} \left(\mathrm{sing}\left(u,v\right)+\mathrm{sing}\left(v,u\right)\right)\nonumber\\ & \qquad\qquad= \frac{L}{\mu\left(\mu^{2}+q^{2}\right)\pi^{2}}\coth\left(\frac{\mu L}{2}\right) \bigg( \mathrm{Li}_{2}\left(-2\right) +\frac{1}{2}\mathrm{Li}_{2}\left(\frac{1}{4}\right)-\frac{\pi^{2}}{6}+\left(\ln\:2\right)^{2}\bigg) \label{eq:sing_result} \end{align} where\mathrm{Li}_{2}\left(x\right)$is the dilogarithm function \begin{equation} \mathrm{Li}_{2}\left(x\right)=\int_{0}^{\infty}\frac{t}{e^{t}/x-1},\quad x\in\mathbb{C}\setminus\left\{ x\in\mathbb{R}\wedge x\geq1\right\} . \end{equation} Using the above integral representation, the Abel identity \begin{equation} \mathrm{Li}_{2}\left(\frac{x}{1\!-\!y}\right)+\mathrm{Li}_{2}\left(\frac{y}{1\!-\!x}\right)-\mathrm{Li}_{2}\left(\frac{xy}{\left(1\!-\!x\right)\left(1\!-\!y\right)}\right)=\mathrm{Li}_{2}\left(x\right)+\mathrm{Li}_{2}\left(y\right)+\ln\left(1\!-\!x\right)\ln\left(1\!-\!y\right) \end{equation} with$x=-1$and$y=\frac{1}{2}$, and the special value \begin{equation} \mathrm{Li}_{2}\left(-1\right)=-\frac{\pi^{2}}{12}, \end{equation} we obtain \begin{equation} \mathrm{Li}_{2}\left(-2\right)+\frac{1}{2}\mathrm{Li}_{2}\left(\frac{1}{4}\right)-\frac{\pi^{2}}{6}+\left(\ln\:2\right)^{2}=-\frac{\pi^{2}}{4}\label{eq:Li_id} \end{equation} Putting everything together, the double sum of~(\ref{eq:separsum}) can be written as \begin{equation} \sum_{k_{1},k_{2}\in\mathbb{Z}}D_{1}\left(k_{1},k_{2}\right)=C_{1}+C_{2}+C_{\Delta}+C_{r}+C_{\rm ord}+C_{\rm sing}\label{eq:sumc} \end{equation} where: \begin{itemize} \item$C_{1}$and$C_{2}$contains the single sums separated in~(\ref{eq:separsum}). Using~(\ref{eq:singsum1}) and~(\ref{eq:singsum2}), \begin{eqnarray} C_{1} & = & \frac{L}{\omega_{n_{q}}\mu^{2}}\left(\frac{1}{2\pi}+\mu L\int_{-\infty}^{\infty}\frac{du}{2\pi}\frac{e^{\mu L\cosh u}}{\left(e^{\mu L\cosh u}-1\right)^{2}}\cosh u\right)\\ C_{2} & = & \frac{L}{4\omega_{n_{q}}^{2}\mu}\coth\left(\frac{\mu L}{2}\right)-\frac{L}{2\omega_{n_{q}}}\int_{-\infty}^{\infty}\frac{du}{2\pi}\frac{\coth\left(\frac{\mu L}{2}\cosh u\right)}{\omega_{n_{q}}^{2}+\mu^{2}\sinh^{2}u} \end{eqnarray} \item$C_{\Delta}$stands for the term coming from~(\ref{eq:sdelta}) \begin{equation} C_{\Delta}=-\int_{-\infty}^{\infty}\frac{du}{2\pi}\:\frac{L}{\omega_{n_{q}}}\coth\left(\frac{\mu L\cosh u}{2}\right)\frac{q^{2}\sinh^{2}u}{\left(q^{2}+\mu^{2}\cosh^{2}u\right)^{2}} \end{equation} \item$C_{r}$contains the residual terms emerging from the contour deformation of the integral representation of$S_{\Theta}$appearing in~(\ref{eq:sthetint}) \begin{equation} C_{r}=\int_{-\infty}^{\infty}\frac{du}{2\pi}\:\frac{L^{2}}{\mu}\coth\left(\frac{\mu L\cosh u}{2}\right)\frac{e^{\mu L\cosh u}}{e^{\mu L\cosh u}-1}\frac{\sqrt{\mu^{2}+q^{2}}\cosh u}{\left(q^{2}+\mu^{2}\cosh^{2}u\right)} \end{equation} \item$C_{\rm ord}$is the symmetrized double integral contribution~(\ref{eq:orig_result}) \begin{equation} C_{\rm ord}=-\frac{L^{2}}{\mu^{2}}\left(\frac{3}{8}+\int_{-\infty}^{\infty}\frac{du}{2\pi}\frac{1}{e^{\mu L\cosh u}-1}\frac{\mu\sqrt{\mu^{2}+q^{2}}\cosh u}{q^{2}+\mu^{2}\cosh^{2}u}\right) \end{equation} \item Finally,$C_{\rm sing}$is the symmetrized singular contribution~(\ref{eq:sing_result}) \begin{equation} C_{\rm sing}=-\frac{L}{4\mu\left(\mu^{2}+q^{2}\right)}\coth\left(\frac{\mu L}{2}\right). \end{equation} \end{itemize} Combining these terms, significant simplifiactions can be achieved. First of all, notice that$C_{\rm sing}$is cancelled by a similar term appearing in$C_{2}$. As a next step, we combine$C_{\Delta}$and the integral part of$C_{2}$, and perform integration by parts. The resulting boundary term cancels the explicit term appearing in$C_{1}$.$C_{r}$contains an infinite-volume term that can be separated and integrated analytically. Then, the remaining part of the integral in$C_{r}$, the integral part of$C_{\rm ord}$, the integral appearing in$C_{1}$and the result of the previous integration by parts can be combined beautifully together and lead to the following nice representation of the full double sum: \begin{equation} \sum_{k_{1},k_{2}}D_{1}\left(k_{1},k_{2}\right)=\frac{L^{2}}{\mu^{2}}\left(\frac{1}{8}+3\int_{-\infty}^{\infty}\frac{du}{2\pi}\frac{e^{\mu L\cosh u}}{\left(e^{\mu L\cosh u}-1\right)^{2}}\frac{1}{\cosh\left(u-\theta\right)}\right)\label{eq:dsum1-int-1} \end{equation} where we introduced$\theta$as the rapidity variable$q=\mu\sinh\theta$. ]]> Expansion of the form factor Lagrangian perturbation theory Equivalence of finite and infinite volume regularizations Finite volume regularization Infinite volume calculation \epsilon$. This is a different contour deformation, what we used in section 3, but is a completely equivalent regularization. On the shifted contour we can take the $\epsilon\to0$ limit, which basically kills the $\delta$ functions and we arrive at: \begin{equation} \frac{1}{2}\int\frac{du}{2\pi}\frac{d\beta_{1}}{2\pi}\frac{d\beta_{2}}{2\pi}S(\beta_{2}-\beta_{1})F_{3}(u+i\pi-i\eta,\beta_{1},\beta_{2})^{2}g(u-i\eta,\beta_{1},\beta_{2}) \end{equation} which is just the same as the surviving $C_{-}$ contour's contribution $I_{-}$. In the following we compare the remaining terms. In shifting the contour we should pick up the contributions of the poles at $u=\beta_{1}-i\epsilon$ and at $u=\beta_{2}-i\epsilon$. In the following we focus on the integrand only. It is understood that we integrate the expressions for $\beta_{1}$ and $\beta_{2}$. The pole at $u=\beta_{1}-i\epsilon$ has the structure \begin{eqnarray} S(\beta_{2}-\beta_{1})\langle u\vert\phi\vert\beta_{1},\beta_{2}\rangle^{2} & = & -\frac{S_{21}F_{1}^{2}}{(u-\beta_{1}+i\epsilon)^{2}}+\frac{2iS_{21}F_{1}}{u-\beta_{1}+i\epsilon}\\ & & \times\bigg(F_{3}^{c}(u+i\pi-i\epsilon,\beta_{1},\beta_{2})+\frac{iF_{1}S_{12}}{u-\beta_{2}+i\epsilon}\nonumber\\ &&\qquad\qquad\hspace{3cm}-\frac{iF_{1}S_{12}}{u-\beta_{1}-i\epsilon}-\frac{iF_{1}}{u-\beta_{2}-i\epsilon}\bigg)\nonumber \end{eqnarray} The contribution of the double pole is \begin{equation} -\frac{1}{2}iS_{21}F_{1}^{2}\partial_{u}g(u,\beta_{1},\beta_{2})\vert_{u=\beta_{1}-i\epsilon} \end{equation} while the simple pole gives \begin{equation} -F_{1}S_{21}\left(F_{3}^{c}(\beta_{1}+i\pi-2i\epsilon,\beta_{1},\beta_{2})+\frac{iF_{1}S_{12}}{\beta_{1}-\beta_{2}}+\frac{F_{1}S_{12}}{2\epsilon}-\frac{iF_{1}}{\beta_{1}-\beta_{2}-2i\epsilon}\right)g(\beta_{1}-i\epsilon,\beta_{1},\beta_{2}) \end{equation} We also have similar contributions from the pole at $u=\beta_{2}-i\epsilon$, which can be obtained by the $\beta_{1}\leftrightarrow\beta_{2}$ transformation, (where we do not exchange the last two arguments of $g$ ). The divergent term in this formalism appears as \begin{equation} -\frac{F_{1}^{2}}{2\epsilon}\left(g(\beta_{1},\beta_{1},\beta_{2})+g(\beta_{2},\beta_{1},\beta_{2})\right) \end{equation} which is the analogue of $I_{R}$. This term is cancelled by a diagonal one-particle term in the denominator of the two-point function~(\ref{eq:2pt}). By expanding the function $g$ in $\epsilon$ and combining with the double pole terms we get \begin{equation} \frac{iF_{1}^{2}}{2}(1-S_{21})\partial_{u}g(u,\beta_{1},\beta_{2})\vert_{u=\beta_{1}}+\frac{iF_{1}^{2}}{2}(1-S_{12})\partial_{u}g(u,\beta_{1},\beta_{2})\vert_{u=\beta_{2}} \end{equation} The contributions of the connected form factors are \begin{equation} -F_{1}\left(S_{21}F_{3}^{c}(\beta_{1}+i\pi,\beta_{1},\beta_{2})g(\beta_{1},\beta_{1},\beta_{2})+F_{3}^{c}(\beta_{2}+i\pi,\beta_{1},\beta_{2})g(\beta_{2},\beta_{1},\beta_{2})\right) \end{equation} There are two terms where we should be careful with the $\epsilon$ terms. There we use \begin{equation} \frac{1}{\beta_{1}-\beta_{2}\mp2i\epsilon}=P_{\frac{1}{\beta_{1}-\beta_{2}}}\pm i\pi\delta(\beta_{1}-\beta_{2}) \end{equation} The contribution of the $\delta$-function is \begin{equation} 2\pi F_{1}^{2}\delta(\beta_{1}-\beta_{2})g(\beta_{1},\beta_{1},\beta_{1}) \end{equation} which is equivalent to the term $I_{d}$. In the remaining terms the principal value description can be omitted as the full integrand is regular at $\beta_{1}=\beta_{2}$: \begin{equation} iF_{1}^{2}\frac{1}{\beta_{1}-\beta_{2}}\left((1-S_{12})g(\beta_{2},\beta_{1},\beta_{2})-(1-S_{21})g(\beta_{1},\beta_{1},\beta_{2})\right) \end{equation} Clearly, summing up the results we completely agree with the integrand of the finite volume regularization. ]]>

https://doi.org/10.1007/BF01211589 https://doi.org/10.1007/BF01211097 https://doi.org/10.1016/j.nuclphysb.2007.06.027 https://doi.org/10.1016/S0550-3213%2899%2900665-3 https://doi.org/10.1016/j.nuclphysb.2007.07.008 G. Mussardo, Statistical field theory: an introduction to exactly solved models in statistical physics, Oxford Graduate Texts, Oxford U.K. (2009). L. Samaj and Z. Bajnok, Introduction to the statistical physics of integrable many-body systems, Cambridge University Press, Cambridge U.K. (2013). https://doi.org/10.1007/s11005-011-0529-2 https://doi.org/10.1016/j.nuclphysb.2008.08.020 https://doi.org/10.1007/JHEP01%282014%29037 https://doi.org/10.1016/0550-3213%2890%2990333-9 https://doi.org/10.1016/S0550-3213%2896%2900516-0 https://arxiv.org/abs/1707.08027 https://doi.org/10.1016/S0550-3213%2899%2900280-1 https://doi.org/10.1007/JHEP07%282013%29157 https://doi.org/10.1016/j.nuclphysb.2008.04.021 https://doi.org/10.1088/1742-5468/2010/11/P11012 https://doi.org/10.1007/JHEP04%282015%29023 https://doi.org/10.1016/j.nuclphysb.2008.01.021 https://doi.org/10.1007/s11005-011-0512-y https://doi.org/10.1016/0550-3213%2891%2990566-G https://doi.org/10.1103/PhysRevD.76.126008 F.A. Smirnov, Form-factors in completely integrable models of quantum field theory, Adv. Ser. Math. Phys. 14 (1992) 1. https://doi.org/10.1016/0550-3213%2878%2990362-0 https://doi.org/10.1016/0550-3213%2893%2990252-K https://doi.org/10.1103/PhysRevD.91.085011 https://doi.org/10.1007/JHEP04%282015%29042 https://arxiv.org/abs/1710.03853 https://doi.org/10.1007/JHEP06%282017%29058 https://doi.org/10.1007/JHEP05%282017%29124 https://arxiv.org/abs/1505.06745 https://doi.org/10.1016/j.nuclphysb.2016.04.020 https://doi.org/10.1007/JHEP02%282016%29165 https://doi.org/10.1007/JHEP03%282018%29047 https://doi.org/10.1007/JHEP08%282017%29059 https://arxiv.org/abs/1610.09537