# On bosonic Thirring model in Minkowski signature

Mikhail Alfimov<EMAIL_ADDRESS>Andrey Kurakin<EMAIL_ADDRESS>
###### Abstract

We present the way to continue the bosonic Thirring model or
$\beta\gamma$-system with quartic interaction to Minkowski signature, based on
the symmetries of this model. It is shown that the considered Minkowski
version of the model is one-loop renormalizable. Based on this, we find the
amplitudes of scattering of the excitations corresponding to the $\gamma$ and
$\bar{\gamma}$ fields up to the one-loop order. In particular, it was computed
that the $2\rightarrow 2$ amplitudes of these excitations possess property of
reflectionless scattering and thus the corresponding $S$-matrix of such
excitations satisfies the Yang-Baxter equation. The obtained $S$-matrix
elements for $\gamma$ and $\bar{\gamma}$ are shown to coincide with the
corresponding $S$-matrix elements of the solitons in the complex sine-Gordon
model proposed by Dorey and Hollowood.

###### keywords:

Integrability , S-matrix , Yang-Baxter equation , sine-Gordon , Thirring model

††journal: Nuclear Physics B

[first]organization=HSE University,addressline=ul. Usacheva, d. 6,
city=Moscow, postcode=119048, country=Russia

[second]organization=P.N. Lebedev Physical Institute of the Russian Academy of
Sciences, addressline=Leninskiy pr., d. 53, city=Moscow, postcode=119991,
country=Russia

##  1 Introduction

Dual description of deformed $OSp(N|2m)$ sigma models in terms of Toda-like
theories, which are determined by the system of the so-called screening
charges [1], was studied in the recent work [2] by the one of the authors. The
key feature of such description is that the $S$-matrix calculated from the
dual Toda-like Lagrangian order by order in perturbation theory coincides with
the corresponding sigma model $S$-matrix, which represents a deformed solution
of the Yang-Baxter equation. This description was constructed for several
cases, such as $O(N)$ [3, 4] or $\mathbb{C}P^{N}$ [5] deformed sigma
models111For the $O(3)$ case it was first done by Alexey Zamolodchikov
(unpublished).. However, compared to these cases, there are some novel
pecularities in the $OSp(N|2m)$ case. Namely, in the dual Lagrangians of the
$\eta$-deformed $OSp(5|2)$ and $OSp(7|2)$ sigma models because of the presence
of odd coordinates in the target space there appears the so-called bosonic
Thirring model, whose Euclidean action is determined by the Lagrangian of
$\beta\gamma$-system with the quartic interaction

$\mathcal{L}_{\beta\gamma}^{(E)}=\bar{\beta}\partial\bar{\gamma}+\beta\bar{\partial}\gamma-m\bar{\beta}\beta-m\bar{\gamma}\gamma+g^{2}\beta\bar{\beta}\gamma\bar{\gamma}\,.$
(1.1)

As it was said above, one of the ways of confirming the validity of dual
description is to compare the $S$-matrix calculated perturbatively from the
dual Toda-like theory with the exact $S$-matrix originating from the
symmetries of the corresponding sigma model (for the examples see [4]).
However, in contrast to the fermionic version of the Thirring model, the
analogue of (1.1) in Minkowski signature, which allows to correctly reproduce
the $S$-matrix for the corresponding $OSp(N|2m)$ sigma model, is not known.
The objective of the present article is to find such an analogue in the
bosonic case.

Let us first present the information about the tree level. As it was pointed
out in [2], in order to calculate the $S$-matrix for the excitations
corresponding to the fields $\gamma$ and $\bar{\gamma}$, we can take the
Gaussian integral over the fields $\beta$ and $\bar{\beta}$ (see the details
in the A). After this procedure we obtain the classical Lagrangian of complex
sine-Gordon (CSG) theory [6, 7, 8, 9] up to the contribution of some
functional determinant

$\mathcal{L}_{CsG}^{(E)}=\frac{|\partial\psi|^{2}}{1-g^{2}|\psi|^{2}}-m^{2}|\psi|^{2}\,,$
(1.2)

which can be straightforwardly continued to Minkowski signature

$\mathcal{L}_{CsG}^{(M)}=\frac{1}{2}\left(\frac{\partial_{+}\psi_{a}\partial_{-}\psi_{a}}{1-\frac{g^{2}\psi_{a}^{2}}{2}}-m^{2}\psi_{a}^{2}\right)\,,\quad
a=1,2\,,$ (1.3)

where $\psi=(\psi_{1}+i\psi_{2})/\sqrt{2}$. In the article [10] there was
conducted perturbative analysis of the theory (1.3). One-loop $S$-matrix for
this Lagrangian (1.3) was calculated, however, the theory was shown to be not
one-loop renormalizable. Upon supplementing the theory with the proper
counterterm, it becomes one-loop renormalizable and its one-loop $S$-matrix
starts to satisfy the Yang-Baxter equation, which is equivalent to the
condition of reflectionless scattering in this case. Then, in the work [11]
the semiclassical spectrum and the $S$-matrix of the theory (1.3) was studied.
A hypothesis was put forward for the minimal exact $S$-matrix [12] for
specific values of coupling constant of such a theory consistent with the
semiclassical spectrum of the theory.

Our hypothesis is that the theory (1.1), properly continued to the Minkowski
signature, yields the same $S$-matrix for the excitations corresponding to the
fields $\gamma$ and $\bar{\gamma}$ as proposed in [11] order by order in the
coupling constant $g$. The first step to achieve this is to understand how to
interpret the Lagrangian (1.1) in Minkowski spacetime. We offer a way to do
this based on the spacetime and global symmetries of the action (1.1). The
main idea is to look for the $SO(2)$ subgroup of the full symmetry group of
the theory and turn it into the $SO(1,1)$, thus passing to the Lorentz group
symmetry. After doing this the next important question, as we are interested
in loop calculations, is the analysis of divergences. To understand the
renormalization of the fields, mass and coupling constant, we calculate two-
and four-point functions, establishing which counterterms are needed at one-
loop order and partially at two-loop order.

The next step is to calculate the $S$-matrix of the continuation of the theory
(1.1) to the Minkowski signature. This is done by applying the Lehmann-
Symanczik-Zimmermann (LSZ) reduction formula [13] to the four-point
correlation function of the fields $\gamma$ and $\bar{\gamma}$, which yields
the desired amplitudes. We were able to show the coincidence our result with
the amplitudes from [11] at the tree and $1$-loop level, thus confirming our
hypothesis at these orders.

The article is organized as follows. In the Section 2 we remind the reader
facts that are known about the complex sine-Gordon theory regarding its
spectrum and scattering. In the next Section 3 we explain the way we use to
continue the theory (1.1) to the Minkowski signature and setting the
propagators and vertices of such a theory and also show how to obtain the
$\beta\gamma$-system in Minkowski signature from the gauged Wess-Zumino-Witten
model with integrable potential. In the Section 4 we calculate the two-point
functions at one and two loops and also do some analysis of necessary
counterterms. Then, in the Section 5 we turn to the four-point function of the
fields $\gamma$ and $\bar{\gamma}$, from which we extract the $2\rightarrow 2$
amplitude to compare with the $S$-matrix of the CSG theory [11]. In the
Section 6 we make concluding remarks and discuss further open questions.

##  2 Complex sine-Gordon theory

Let us start from refreshing our knowledge about the spectrum and scattering
in the CSG theory [6, 7, 8, 9]. In this section we are going to remember the
mass spectrum of the CSG theory and write down the all-loop $S$-matrix, which
was conjectured earlier in [11]. First we write the weak coupling expansion of
the CSG classical Lagrangian (1.3), which is222Notice that in [14] the authors
use different coupling $\mathbf{g}^{2}$, which is related to ours by
$g^{2}=2\mathbf{g}^{2}$.

$\mathcal{L}_{CsG}=\frac{1}{2}\partial_{+}\psi_{a}\partial_{-}\psi_{a}-\frac{m^{2}}{2}\psi_{a}\psi_{a}+\frac{1}{2}\sum\limits_{k=1}^{+\infty}\left(\frac{g^{2}}{2}\right)^{k}\partial_{+}\psi_{a}\partial_{-}\psi_{a}(\psi_{b}\psi_{b})^{k}\,.$
(2.1)

In this theory we have the solitons as elementary excitations, which are
classified by the charge [15]. Soliton charge is defined modulo $2\pi/g^{2}$

$Q=\pm 1,\ldots,Q_{max}=\pm\left[\frac{2\pi}{g^{2}}\right]\,.$ (2.2)

It was shown in [15], that these solitons have the following mass at the
semiclassical level

$M(Q)=\frac{4m}{g^{2}}\left|\sin\left(\frac{g^{2}Q}{4}\right)\right|\,.$ (2.3)

Also in the same article it was conjectured, that under finite renormalization
of the coupling, which corresponds to adding proper counterterms to the action
(2.1)

$\displaystyle g_{R}^{2}=\frac{g^{2}}{1-\frac{g^{2}}{4\pi}}$ (2.4)

analogous to the sine-Gordon (SG) theory, the semiclassical mass spectrum
(2.3) becomes exact in the coupling constant. Given this information, we can
now turn to the $S$-matrix of such soliton excitations.

In [10] there was computed the one-loop $S$-matrix, which for elementary
excitations of the fields $\psi_{a}$ is given by the following expression

$S_{ij}^{i^{\prime}j^{\prime}}(\theta)=S_{1}(\theta)\delta_{ij}\delta^{i^{\prime}j^{\prime}}+S_{2}(\theta)\delta_{i}^{i^{\prime}}\delta_{j}^{j^{\prime}}+S_{3}(\theta)\delta_{i}^{j^{\prime}}\delta_{j}^{i^{\prime}}\,,$
(2.5)

where all the indices take the values $1$ and $2$ and the functions
$S_{1,2,3}(\theta)$ are

$\displaystyle
S_{1}(\theta)=-\frac{ig^{2}}{2}\coth\theta+i\frac{g^{4}}{8\pi}(\mathop{\mathrm{cosech}}\nolimits\theta+\coth\theta)+\frac{g^{4}}{4}\coth\theta\mathop{\mathrm{cosech}}\nolimits\theta+\mathcal{O}(g^{6})\,,$
(2.6) $\displaystyle
S_{2}(\theta)=1+\frac{ig^{2}}{2}\mathop{\mathrm{cosech}}\nolimits\theta-\frac{g^{4}}{4}\left(\frac{1}{2}+\mathop{\mathrm{cosech}}\nolimits^{2}\theta\right)+\mathcal{O}(g^{6})\,,$
$\displaystyle
S_{3}(\theta)=\frac{ig^{2}}{2}\coth\theta+i\frac{g^{4}}{8\pi}(\mathop{\mathrm{cosech}}\nolimits\theta-\coth\theta)-\frac{g^{4}}{4}\coth\theta\mathop{\mathrm{cosech}}\nolimits\theta+\mathcal{O}(g^{6})\,.$

However, the $S$-matrix determined by (2.5) and (2.6) does not satisfy the
Yang-Baxter equation, which in the case in question is equivalent to the
condition of reflectionless scattering

$S_{1}(\theta)+S_{3}(\theta)=0\,.$ (2.7)

As it was shown in [10] if we supplement the action with the counterterm

$\Delta\mathcal{L}=\frac{m^{2}g^{4}}{16\pi}\psi_{a}\psi_{a}\psi_{b}\psi_{b}\,,$
(2.8)

the one-loop $S$-matrix takes the form

$\displaystyle
S_{1}(\theta)=-\frac{ig^{2}}{2}\coth\theta+i\frac{g^{4}}{8\pi}\coth\theta+\frac{g^{4}}{4}\coth\theta\mathop{\mathrm{cosech}}\nolimits\theta+\mathcal{O}(g^{6})\,,$
(2.9) $\displaystyle
S_{2}(\theta)=1+\frac{ig^{2}}{2}\mathop{\mathrm{cosech}}\nolimits\theta-i\frac{g^{4}}{8\pi}\mathop{\mathrm{cosech}}\nolimits\theta-\frac{g^{4}}{4}\left(\frac{1}{2}+\mathop{\mathrm{cosech}}\nolimits^{2}\theta\right)+\mathcal{O}(g^{6})\,,$
$\displaystyle
S_{3}(\theta)=\frac{ig^{2}}{2}\coth\theta-i\frac{g^{4}}{8\pi}\coth\theta-\frac{g^{4}}{4}\coth\theta\mathop{\mathrm{cosech}}\nolimits\theta+\mathcal{O}(g^{6})\,.$

It is known that the $S$-matrix (2.5) with (2.9) satisfies the Yang-Baxter
equation up to one-loop order thanks to the reflectionless condition

$S_{1}(\theta)+S_{3}(\theta)=\mathcal{O}(g^{6})\,.$ (2.10)

Let us change the basis to the complex one

$\gamma=\frac{1}{\sqrt{2}}(\psi_{1}+i\psi_{2})\,,\quad\bar{\gamma}=\frac{1}{\sqrt{2}}(\psi_{1}-i\psi_{2})\,.$
(2.11)

In the complex basis (2.11) the formulas (2.9) correspond exactly to the
$2\rightarrow 2$ scattering amplitudes of the particles $\gamma$ and
$\bar{\gamma}$, which can be written as

$\displaystyle
S_{\gamma\gamma}^{\gamma\gamma}(\theta)=S_{\bar{\gamma}\bar{\gamma}}^{\bar{\gamma}\bar{\gamma}}(\theta)=S_{2}(\theta)+S_{3}(\theta)\,,$
(2.12) $\displaystyle
S_{\gamma\bar{\gamma}}^{\gamma\bar{\gamma}}(\theta)=S_{1}(\theta)+S_{2}(\theta)\,,$
$\displaystyle
S_{\gamma\bar{\gamma}}^{\bar{\gamma}\gamma}(\theta)=S_{1}(\theta)+S_{3}(\theta)\,.$

Given that at one loop the amplitude is reflectionless (2.10), by using the
crossing symmetry we can express the whole one-loop $S$-matrix in terms of
solely one function

$S(\theta)=S_{1}(\theta)+S_{2}(\theta)=1-\frac{ig^{2}}{2}\tanh\frac{\theta}{2}+\frac{ig^{4}}{8\pi}\tanh\frac{\theta}{2}-\frac{g^{4}}{8}\tanh^{2}\frac{\theta}{2}+\mathcal{O}(g^{6})\,.$
(2.13)

However, introduction of the counterterm (2.8) is not the unique way to
restore the symmetries responsible for the integrability of the model at the
quantum level. As it was explained in [14] by using the gauged Wess-Zumino-
Witten (gWZW) model formulation of the CSG model (1.3) it is possible to
produce the counterterns different from (2.8), which nevertheless lead to the
$S$-matrix, which satisfies the Yang-Baxter equation. The authors of that work
showed that integration of the gWZW action over the fields $A_{\pm}$ produces
the functional determinant contribution, which, under the proper
identification of the fields, yields the $S$-matrix we write down below up to
one-loop order. The advantage of the gWZW or $\beta\gamma$-system approach we
consider below is that the symmetries responsible or integrability are more
manifest in this representation of the CSG model and we do not need to include
additional counterterms if we do not integrate over gauge or $\beta$ fields
respectively.

In relation to the spectrum given by (2.2), (2.3) and (2.4) the particles
$\gamma$ and $\bar{\gamma}$ correspond to the solitons with the charges $Q=1$
and $Q=-1$ respectively. In [11] there was proposed exact $2\rightarrow 2$
$S$-matrix of a pair of solitons with the charges $Q_{1}$ and $Q_{2}$ denoted
as $S_{Q_{1},Q_{2}}(\theta)$. We are interested in the case $Q_{1}=1$,
$Q_{2}=-1$ exact S-matrix

$S_{1,-1}(\theta)=S_{1,1}(i\pi-\theta)=\frac{\sinh\left(\frac{i\pi}{2}-\frac{\theta}{2}+\frac{ig_{R}^{2}}{4}\right)}{\sinh\left(\frac{i\pi}{2}-\frac{\theta}{2}-\frac{ig_{R}^{2}}{4}\right)}\,.$
(2.14)

where the $g_{R}$ is determined by (2.4) and is subject to the condition

$g_{R}^{2}=\frac{4\pi}{\mathrm{k}}\,\quad\mathrm{k}\in\mathbb{N}\,.$ (2.15)

Expanding (2.14) into the series up to the terms of the order $g^{4}$ we
obtain

$S_{1,-1}(\theta)=1-\frac{ig^{2}}{2}\tanh\frac{\theta}{2}-\frac{ig^{4}}{8\pi}\tanh\frac{\theta}{2}-\frac{g^{4}}{8}\tanh^{2}\frac{\theta}{2}+\mathcal{O}(g^{6})\,,$
(2.16)

which in our terms corresponds to the amplitude
$S_{\gamma\bar{\gamma}}^{\gamma\bar{\gamma}}(\theta)$. One can notice the
discrepancy at the order $g^{4}$ in (2.13) and (2.16), which, as commented in
[14], can be explained by the choice of the renormalization scheme. We are to
compare the results (2.13) and (2.16) to the $2\rightarrow 2$ amplitude,
obtained from the version of the theory (1.1) properly analytically continued
to Minkowski signature. We are going to conduct this in the next Section.

##  3 $\beta\gamma$-system in Minkowski space

Our objective now is to find the way to define the bosonic Thirring model
(1.1) in Minkowski signature. Contrary to bosonic case, fermionic Thirring
model in Minkowski signature is known [16]. The main idea is to analyze the
symmetries of the model (1.1) and, based on that find the $SO(2)$ subgroup,
which can be continued to the Lorentz group $SO(1,1)$. Next subsection will
tell us about the symmetries of (1.1).

###  3.1 Symmetries and continuation to Minkowski signature

Analogous to the Coleman-Mandelstam duality [17] in [2] it was shown that at
least at the classical level the CSG model is equivalent to the bosonic
version of the Thirring model, which is determined by the Lagrangian of the
$\beta\gamma$-system with quartic interaction

$\mathcal{L}_{T}=i\mathcal{\bar{B}}\gamma^{\mu}\partial_{\mu}\mathcal{B}-m\mathcal{\bar{B}}\mathcal{B}-\frac{\pi\lambda^{2}}{2}(\mathcal{\bar{B}}\gamma^{\mu}\mathcal{B})^{2},$
(3.1)

where

$\gamma^{1}=\sigma_{1}\,,\quad\gamma^{2}=\sigma_{2}\,,\quad\mathcal{\bar{B}}=\left(\begin{matrix}-i\bar{\beta}&\gamma\end{matrix}\right)\,,\quad\mathcal{B}=\left(\begin{matrix}i\beta\\\
\bar{\gamma}\end{matrix}\right)\,.$ (3.2)

Then in terms of $\beta$ and $\gamma$ fields we get the Lagrangian

$\mathcal{L}_{\beta\gamma}=\beta\bar{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}-m\beta\bar{\beta}-m\gamma\bar{\gamma}-2\pi\lambda^{2}\beta\bar{\beta}\gamma\bar{\gamma}\,.$
(3.3)

If we replace the coupling constant

$-2\pi\lambda^{2}=g^{2}\,,$ (3.4)

the $\beta\gamma$ Lagrangian in Euclidean space takes the same form as in
(1.1)

$\mathcal{L}_{\beta\gamma}=\beta\bar{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}-m\beta\bar{\beta}-m\gamma\bar{\gamma}+g^{2}\beta\bar{\beta}\gamma\bar{\gamma}\,.$
(3.5)

To understand the way of analytic continuation we have to first analyze the
symmetries of the theory (1.1). This theory has the $SO(2)$ symmetry
corresponding to the global rotation of the fields $\beta$ and $\gamma$

$\beta\rightarrow e^{i\alpha}\beta\,,\quad\bar{\beta}\rightarrow
e^{-i\alpha}\bar{\beta}\,,\quad\gamma\rightarrow
e^{-i\alpha}\gamma\,,\quad\bar{\gamma}\rightarrow e^{i\alpha}\bar{\gamma}$
(3.6)

and the symmetry corresponding to the space $SO(2)$ transformation

$\displaystyle\partial\rightarrow e^{i\phi}\partial\,,\quad\beta\rightarrow
e^{\frac{i\phi}{2}}\beta\,,\quad\gamma\rightarrow
e^{\frac{i\phi}{2}}\gamma\,,$ (3.7) $\displaystyle\bar{\partial}\rightarrow
e^{-i\phi}\bar{\partial}\,,\quad\bar{\beta}\rightarrow
e^{-\frac{i\phi}{2}}\bar{\beta}\,,\quad\bar{\gamma}\rightarrow
e^{-\frac{i\phi}{2}}\bar{\gamma}\,.$

Thus, the total symmetry group of the $\beta\gamma$-system in Euclidean space
is $SO(2)\times SO(2)$. We would like to define now this theory in Minkowski
signature and the way to achieve this is to find the $SO(2)$ subgroup inside
this $SO(2)\times SO(2)$ and then to turn it into the $2d$ Lorentz group
$SO(1,1)$.

Let us select subgroup $SO(2)$ in the symmetry group $SO(2)\times SO(2)$ as
follows

$\displaystyle\partial\rightarrow e^{i\phi}\partial\,,\quad\beta\rightarrow
e^{i\phi}\beta\,,\quad\gamma\rightarrow\gamma\,,$ (3.8)
$\displaystyle\bar{\partial}\rightarrow
e^{-i\phi}\bar{\partial}\,,\quad\bar{\beta}\rightarrow
e^{-i\phi}\bar{\beta}\,,\quad\bar{\gamma}\rightarrow\bar{\gamma}\,.$

(we set $\alpha=\phi/2$ when choosing the subgroup). In this case the fields
$\gamma$ and $\bar{\gamma}$ remain invariant under the action of the
considered subgroup $SO(2)$.

Continuation to the Minkowski space with the signature $(+,-)$ is carried out
via the group replacement $SO(2)\rightarrow SO(1,1)$ – Lorentz
transformations, which can be denoted as $x^{2}=ix^{0}$ at the level of
spacetime coordinates (here we follow the same notations as in [2]). Then the
spacetime derivatives are changed as follows

$\partial\rightarrow-\partial_{-}\,,\quad\bar{\partial}\rightarrow\partial_{+}\,,$
(3.9)

whereas the replacement of the fields we define to be

$\displaystyle\beta\rightarrow-\beta_{-}\,,\quad\bar{\beta}\rightarrow\beta_{+}\,,\quad\gamma\bar{\gamma}\rightarrow-\gamma\bar{\gamma}\,.$
(3.10)

Then the Lagrangian in Minkowski space takes in the form

$\displaystyle\mathcal{L}_{\beta\gamma}^{(M)}=\beta_{+}\partial_{-}\bar{\gamma}+\beta_{-}\partial_{+}\gamma-m\beta_{+}\beta_{-}-m\bar{\gamma}\gamma+g^{2}\beta_{+}\beta_{-}\bar{\gamma}\gamma\,.$
(3.11)

Therefore, in the Minkowski signature under the Lorentz boosts from $SO(1,1)$
the fields and derivatives transform as follows

$\partial_{\pm}\rightarrow
e^{\pm\theta}\partial_{\pm}\,,\quad\beta_{\pm}\rightarrow
e^{\mp\theta}\beta_{\pm}\,,\quad\gamma\rightarrow\gamma,\quad\bar{\gamma}\rightarrow\bar{\gamma}\,,$
(3.12)

where $\theta$ is the rapidity of the boost. Thus we have obtained the
formulation of the bosonic Thirring model in the Minkowski signature (3.11).
It is important to notice that the $SO(2)$ subgroup in the symmetry group of
the Euclidean version of the model we consider can be chosen in another way,
what possibly can lead to another analytic continuation of (1.1) to Minkowski
signature.

In the next part we will present another way to derive the action (3.11).

###  3.2 $\beta\gamma$-system from gauged Wess-Zumino-Witten model

It is interesting that the $\beta\gamma$-system (3.11) can be obtained from
the gauged Wess-Zumino-Witten (gWZW) model with integrable potential by
certain field redefinitions. In [14] there was presented the $SU(2)/U(1)$ gWZW
formulation of the CSG model. Namely, its action in the axially gauged case
was obtained in the section 3.3.2 of [14] and reads

$\mathcal{S}_{\textrm{gWZW}}=\frac{k}{4\pi}\int
d^{2}x\left[\partial_{+}\phi\partial_{-}\phi+\sin^{2}\phi\partial_{+}\chi\partial_{-}\chi-\right.\\\
\left.-a_{+}\sin^{2}\phi\partial_{-}\chi-a_{-}\sin^{2}\phi\partial_{+}\chi-
a_{+}a_{-}\cos^{2}\phi-\frac{m^{2}}{2}(\cos 2\phi-1)\right]\,.$ (3.13)

By performing the following shifts of the fields $a_{\pm}$

$a_{\pm}=\tilde{a}_{\pm}-\tan^{2}\phi\partial_{\pm}\chi\,,$ (3.14)

we obtain the transformed action (3.13) in terms of the new fields
$\tilde{a}_{\pm}$

$\mathcal{S}_{\textrm{gWZW}}=\frac{k}{4\pi}\int
d^{2}x\left[\partial_{+}\phi\partial_{-}\phi+\tan^{2}\phi\partial_{+}\chi\partial_{-}\chi-\tilde{a}_{+}\tilde{a}_{-}\cos^{2}\phi-\frac{m^{2}}{2}(\cos
2\phi-1)\right]\,.$ (3.15)

The next shift of the fields $\tilde{a}_{\pm}$ would be

$\tilde{a}_{\pm}=a^{\prime}_{\pm}-\frac{\partial_{\pm}\left(e^{i\chi}\sin\phi\right)}{\cos^{2}\phi}$
(3.16)

and leads us to the action given by the formula (we also made some changes
with the field $\phi$ using trigonometric identities)

$\mathcal{S}_{\textrm{gWZW}}=\frac{k}{4\pi}\int
d^{2}x\left[i(\partial_{-}(\log\cos\phi)\partial_{+}\chi-\partial_{+}(\log\cos\phi)\partial_{-}\chi)+\right.\\\
\left.+a^{\prime}_{+}\partial_{-}\left(e^{-i\chi}\sin\phi\right)+a^{\prime}_{-}\partial_{+}\left(e^{i\chi}\sin\phi\right)-a^{\prime}_{+}a^{\prime}_{-}-m^{2}\sin^{2}\phi+a^{\prime}_{+}a^{\prime}_{-}\sin^{2}\phi\right]\,,$
(3.17)

where the integral in the first line of the Lagrangian equals to $0$ for the
same reason as in the formulas (A.7), (A.8) and (A.9) of the A. And performing
at last the identification of the fields

$e^{i\chi}\sin\phi=\sqrt{\frac{4\pi}{mk}}\gamma\,,\quad
a^{\prime}_{\pm}=\sqrt{\frac{4\pi m}{k}}\beta_{\pm}\,,$ (3.18)

we arrive to the action

$\mathcal{S}_{\textrm{gWZW}}=\int
d^{2}x\left[\beta_{+}\partial_{-}\bar{\gamma}+\beta_{-}\partial_{+}\gamma-m\beta_{+}\beta_{-}-m\gamma\bar{\gamma}+\frac{4\pi}{k}\beta_{+}\beta_{-}\gamma\bar{\gamma}\right]\,,$
(3.19)

whose Lagrangian exactly coincides with (3.11) upon identification of the
coupling constant as

$g^{2}=\frac{4\pi}{k}\,.$ (3.20)

The next important step is to check that integrating out the fields
$\beta_{\pm}$, we are able to reproduce the CSG action with the correct
counterterms. Apart from excluding the $\beta_{\pm}$ fields from the action at
the classical level by imposing the equations of motion for them, we have to
add the functional determinant contribution similar to the one in the A
$-\log\det(1-g^{2}\bar{\gamma}\gamma)$ (up to the $\gamma$ field
normalization).

Following the steps from [14] and utilizing the result of [18], we obtain a
local two-derivative correction coming from the functional determinant

$-\frac{1}{8\pi}\left(\frac{4\pi}{k}\right)^{2}\frac{\partial_{+}(\gamma\bar{\gamma})\partial_{-}(\gamma\bar{\gamma})}{\left(m-\frac{4\pi}{k}\gamma\bar{\gamma}\right)^{2}}\,,$
(3.21)

which leads to the corrected action of the CSG model

$\mathcal{S}_{CsG}^{\textrm{corr}}=\int
d^{2}x\left[\frac{\partial_{+}\gamma\partial_{-}\bar{\gamma}}{m-\frac{4\pi}{k}\gamma\bar{\gamma}}-m\gamma\bar{\gamma}-\frac{1}{8\pi}\left(\frac{4\pi}{k}\right)^{2}\frac{\partial_{+}(\gamma\bar{\gamma})\partial_{-}(\gamma\bar{\gamma})}{\left(m-\frac{4\pi}{k}\gamma\bar{\gamma}\right)^{2}}\right]\,.$
(3.22)

Then, substituting the $\gamma$ field definition from (3.18) into (3.22), we
obtain the following action

$\mathcal{S}_{CsG}^{\textrm{corr}}=\int
d^{2}x\frac{k}{4\pi}\left[\partial_{+}\phi\partial_{-}\phi+\tan^{2}\phi\,\partial_{+}\chi\partial_{-}\chi-m^{2}\sin^{2}\phi-\frac{2}{k}\tan^{2}\phi\,\partial_{+}\phi\partial_{-}\phi\right]\,,$
(3.23)

exactly coinciding with the action (3.38) from [14]. There it was shown that
after field rescaling and redefinition one derives the action, which correctly
reproduces the S-matrix (2.14) up to the one-loop order. Summing up, we
conclude that the $SU(2)/U(1)$ gWZW model and the $\beta\gamma$-system we
consider are related by the field transformation.

In the next section we are going to write the set of Feynman rules for the
theory (3.11), which are subsequently needed for the calculation of the
amplitudes.

###  3.3 Free field propagators and Feynman rules

In this part we are going to derive one of several ingredients necessary to
calculate the amplitudes, namely, the free field propagators. To find the free
field propagators we consider the free classical Lagrangian of
$\beta\gamma$-system

$\mathcal{L}_{\beta\gamma}^{(\textrm{free})}=\beta_{+}\partial_{-}\bar{\gamma}+\beta_{-}\partial_{+}\gamma-m\beta_{+}\beta_{-}-m\bar{\gamma}\gamma\,.$
(3.24)

In the matrix form the free Lagrangian (3.24) can be rewritten as follows

$\mathcal{L}_{\beta\gamma}^{(\textrm{free})}=-\frac{1}{2}\begin{pmatrix}\beta_{+}&\beta_{-}&\bar{\gamma}&\gamma\end{pmatrix}\underbrace{\begin{pmatrix}0&m&-\partial_{-}&0\\\
m&0&0&-\partial_{+}\\\ \partial_{-}&0&0&m\\\
0&\partial_{+}&m&0\end{pmatrix}}_{\hat{K}}\begin{pmatrix}\beta_{+}\\\
\beta_{-}\\\ \bar{\gamma}\\\ \gamma\end{pmatrix}\,.$ (3.25)

Let us apply the Fourier transform to the free action

$S_{\beta\gamma}^{(\textrm{free})}=-\frac{1}{2}\int\frac{d^{2}k}{(2\pi)^{2}}\begin{pmatrix}\tilde{\beta}_{+}(k)\,\tilde{\beta}_{-}(k)\,\tilde{\bar{\gamma}}(k)\,\tilde{\gamma}(k)\end{pmatrix}\underbrace{\begin{pmatrix}0&m&-2ik^{+}&0\\\
m&0&0&-2ik^{-}\\\ 2ik^{+}&0&0&m\\\
0&2ik^{-}&m&0\end{pmatrix}}_{\tilde{\hat{K}}(k)}\begin{pmatrix}\tilde{\beta}_{+}(-k)\\\
\tilde{\beta}_{-}(-k)\\\ \tilde{\bar{\gamma}}(-k)\\\
\tilde{\gamma}(-k)\end{pmatrix}\,,$ (3.26)

where we used the following definitions for the lightcone variables

$x^{\pm}=\frac{x^{0}\pm
x^{1}}{2}\,,\quad\partial_{\pm}=\partial_{0}\pm\partial_{1}\,,\quad
k^{\pm}=\frac{k^{0}\pm
k^{1}}{2}\,,\quad(k,x)=k^{0}x^{0}-k^{1}x^{1}=2(k^{+}x^{-}+k^{-}x^{+})\,.$
(3.27)

In the formula (3.26) we introduced the kinetic operator $\tilde{\hat{K}}(k)$
in Fourier space. In order to determine the set of propagators of the
$\beta\gamma$ theory, we can calculate the inverse matrix of the operator
$\tilde{\hat{K}}(k)$ as

$\tilde{\hat{K}}^{-1}(k)=\begin{pmatrix}0&\frac{-m}{k^{2}-m^{2}}&\frac{-2ik^{-}}{k^{2}-m^{2}}&0\\\
\frac{-m}{k^{2}-m^{2}}&0&0&\frac{-2ik^{+}}{k^{2}-m^{2}}\\\
\frac{2ik^{-}}{k^{2}-m^{2}}&0&0&\frac{-m}{k^{2}-m^{2}}\\\
0&\frac{2ik^{+}}{k^{2}-m^{2}}&\frac{-m}{k^{2}-m^{2}}&0\end{pmatrix}\,.$ (3.28)

Then the propagator matrix is found as $\hat{D}(k)=-i\hat{\tilde{K}}^{-1}(k)$.
Thus, we have the following set of propagators in the Fourier representation
333In the following sections, the prescription $+i\varepsilon$ is omitted in
the expressions in the Green’s function expressions.

$\displaystyle\langle\tilde{\beta}_{+}(k)\tilde{\beta}_{-}(-k)\rangle=\langle\tilde{\beta}_{-}(k)\beta_{+}(-k)\rangle=\langle\tilde{\gamma}(k)\tilde{\bar{\gamma}}(-k)\rangle=\langle\tilde{\bar{\gamma}}(k)\tilde{\gamma}(-k)\rangle=\frac{im}{k^{2}-m^{2}+i\varepsilon}\,,$
(3.29)
$\displaystyle\langle\tilde{\bar{\gamma}}(k)\tilde{\beta}_{+}(-k)\rangle=-\langle\tilde{\beta}_{+}(k)\tilde{\bar{\gamma}}(-k)\rangle=\frac{2k^{-}}{k^{2}-m^{2}+i\varepsilon}\,,\quad\langle\tilde{\gamma}(k)\tilde{\beta}_{-}(-k)\rangle=-\langle\tilde{\beta}_{-}(k)\tilde{\gamma}(-k)\rangle=\frac{2k^{+}}{k^{2}-m^{2}+i\varepsilon}\,.$

Having the free field propagators we are able to sum up the results of the
present subsection by writing down the Feynman rules of the theory, which are
shown in the Figure 1.

$=\frac{2k^{-}}{k^{2}-m^{2}+i\varepsilon}$
---
$\gamma$
---
$\beta_{-}$
---
$k$
---
$=\frac{2k^{+}}{k^{2}-m^{2}+i\varepsilon}$
---
$\bar{\gamma}$
---
$\beta_{+}$
---
$k$
---
$=\frac{im}{k^{2}-m^{2}+i\varepsilon}$
---
$\gamma$
---
$\bar{\gamma}$
---
$k$
---
$=\frac{im}{k^{2}-m^{2}+i\varepsilon}$
---
$\beta_{+}$
---
$\beta_{-}$
---
$k$
---
$=ig^{2}$
---
$\bar{\gamma}$
---
$\gamma$
---
$\beta_{+}$
---
$\beta_{-}$
---
$k_{\bar{\gamma}}$
---
$k_{\gamma}$
---
$k_{\beta_{-}}$
---
$k_{\beta_{+}}$
---
Figure 1: Feynman rules for the Minkowskian $\beta\gamma$-system (3.11).

In the next Section we are going to study the renormalization properties of
our theory by exploring the two-point functions up to two loops. This
information is essential for our main goal of finding the $2\rightarrow 2$
amplitudes in the sector of the particles $\gamma$ and $\bar{\gamma}$.

##  4 Two-point correlation functions

This part is devoted to the calculation of two-point functions of the
Lagrangian fields (3.11) at one and two loops. We have to investigate the
divergences appearing in these functions to understand, which counterterms are
needed to compensate them. Let us start from the one loop order.

###  4.1 One-loop two-point correlation functions

For the correlation function
$\langle\tilde{\gamma}(k)\tilde{\bar{\gamma}}(-k)\rangle$ up to the one loop
order only the diagrams shown in the Figure 2 contribute.

$\gamma$
---
$\bar{\gamma}$
---
$\gamma$
---
$\bar{\gamma}$
---
$\bar{\gamma}$
---
$\gamma$
---
$\gamma$
---
$\bar{\gamma}$
---
$\gamma$
---
$\bar{\gamma}$
---
Figure 2: Contribution of one-loop diagrams into the $\gamma\bar{\gamma}$
propagator.

For the other fields the propagators can vary, however, the form of all such
diagrams is the same. It should be noted that due to the symmetry of loop
integral the diagrams with the half-continuous half-dashed propagator are
equal to $0$. Thus, we obtain the following result

$\displaystyle\langle\tilde{\gamma}(k)\tilde{\bar{\gamma}}(-k)\rangle^{(1)}$
$\displaystyle=g^{2}\frac{m(k^{2}+m^{2})}{(k^{2}-m^{2})^{2}}\int\frac{d^{2}p}{(2\pi)^{2}}\frac{1}{p^{2}-m^{2}}\,,$
(4.1)
$\displaystyle\langle\tilde{\beta}_{+}(k)\tilde{\beta}_{-}(-k)\rangle^{(1)}$
$\displaystyle=g^{2}\frac{m(k^{2}+m^{2})}{(k^{2}-m^{2})^{2}}\int\frac{d^{2}p}{(2\pi)^{2}}\frac{1}{p^{2}-m^{2}}\,,$
$\displaystyle\langle\tilde{\bar{\gamma}}(k)\tilde{\beta}_{+}(-k)\rangle^{(1)}$
$\displaystyle=-ig^{2}\frac{4m^{2}k^{+}}{(k^{2}-m^{2})^{2}}\int\frac{d^{2}p}{(2\pi)^{2}}\frac{1}{p^{2}-m^{2}}\,,$
$\displaystyle\langle\tilde{\gamma}(k)\tilde{\beta}_{-}(-k)\rangle^{(1)}$
$\displaystyle=-ig^{2}\frac{4m^{2}k^{-}}{(k^{2}-m^{2})^{2}}\int\frac{d^{2}p}{(2\pi)^{2}}\frac{1}{p^{2}-m^{2}}\,,$

where the index $(1)$ denotes the one-loop contribution to the correlation
function. We see that all the 1-loop contributions to the two point functions
(4.1) are expressed in terms of the one-loop integral only, which can be
computed by using the dimensional regularization ($d=2-2\epsilon$). The two-
point functions at one-loop level $g^{2}$ are

$\displaystyle\langle\tilde{\gamma}(k)\tilde{\bar{\gamma}}(-k)\rangle^{(1)}$
$\displaystyle=-g^{2}\frac{im(k^{2}+m^{2})}{4\pi(k^{2}-m^{2})^{2}}\left(\frac{1}{\epsilon}-\log\frac{m^{2}}{\mu^{2}}-\gamma_{E}+\log(4\pi)+\mathcal{O}(\epsilon)\right)\,,$
(4.2)
$\displaystyle\langle\tilde{\beta}_{+}(k)\tilde{\beta}_{-}(-k)\rangle^{(1)}$
$\displaystyle=-g^{2}\frac{im(k^{2}+m^{2})}{4\pi(k^{2}-m^{2})^{2}}\left(\frac{1}{\epsilon}-\log\frac{m^{2}}{\mu^{2}}-\gamma_{E}+\log(4\pi)+\mathcal{O}(\epsilon)\right)\,,$
$\displaystyle\langle\tilde{\bar{\gamma}}(k)\tilde{\beta}_{+}(-k)\rangle^{(1)}$
$\displaystyle=-g^{2}\frac{m^{2}k^{+}}{\pi(k^{2}-m^{2})^{2}}\left(\frac{1}{\epsilon}-\log\frac{m^{2}}{\mu^{2}}-\gamma_{E}+\log(4\pi)+\mathcal{O}(\epsilon)\right)\,,$
$\displaystyle\langle\tilde{\gamma}(k)\tilde{\beta}_{-}(-k)\rangle^{(1)}$
$\displaystyle=-g^{2}\frac{m^{2}k^{-}}{\pi(k^{2}-m^{2})^{2}}\left(\frac{1}{\epsilon}-\log\frac{m^{2}}{\mu^{2}}-\gamma_{E}+\log(4\pi)+\mathcal{O}(\epsilon)\right)\,,$

$\gamma$
---
$\bar{\gamma}$
---
$\bar{\gamma}$
---
$\gamma$
---
$=ig^{2}\delta_{m}^{(1)}\frac{mk^{2}}{(k^{2}-m^{2})^{2}}$
---
$=ig^{2}\delta_{m}^{(1)}\frac{m^{3}}{(k^{2}-m^{2})^{2}}$
---
Figure 3: One-loop diagrams corresponding to the contributions of the
counterterms to the $\gamma\bar{\gamma}$ propagators.

where $\gamma_{E}$ is the Euler-Mascheroni constant. Now we are able to find
the counterterms for the Lagrangian, which compensate the divergencies in
(4.2). Our observation is that at one loop it is enough to renormalize the
mass terms only

$m\rightarrow
Z_{m}m=(1+\delta_{m})m=\left(1+g^{2}\delta_{m}^{(1)}+\mathcal{O}\left(g^{4}\right)\right)m\,,$
(4.3)

where $m$ is physical mass (we assume that $m_{\beta}=m_{\gamma}=m$). Then the
Lagrangian, up to terms of order $\sim g^{2}$, takes the following form

$\displaystyle\mathcal{L}=\beta_{+}\partial_{-}\gamma+\beta_{-}\partial_{+}\bar{\gamma}-m\gamma\bar{\gamma}-m\beta_{+}\beta_{-}+g^{2}\beta_{+}\beta_{-}\gamma\bar{\gamma}-g^{2}\delta_{m}^{(1)}m\beta_{+}\beta_{-}-g^{2}\delta_{m}^{(1)}m\bar{\gamma}\gamma\,.$
(4.4)

Taking into account additional vertices, we get the complementary set of
counterterm diagrams, which are drawn in the Figure 3 for the case of
$\gamma\bar{\gamma}$ propagator. For the other propagators the diagrams look
similar.

Therefore, the contribution of the counterterms equals to

$\displaystyle\langle\tilde{\gamma}(k)\tilde{\bar{\gamma}}(-k)\rangle^{(1)}$
$\displaystyle=ig^{2}\frac{m(k^{2}+m^{2})}{(k^{2}-m^{2})^{2}}\delta_{m}^{(1)}\,,$
(4.5)
$\displaystyle\langle\tilde{\beta}_{+}(k)\tilde{\beta}_{-}(-k)\rangle^{(1)}$
$\displaystyle=ig^{2}\frac{m(k^{2}+m^{2})}{(k^{2}-m^{2})^{2}}\delta_{m}^{(1)}\,,$
$\displaystyle\langle\tilde{\bar{\gamma}}(k)\tilde{\beta}_{+}(-k)\rangle^{(1)}$
$\displaystyle=g^{2}\frac{4m^{2}k^{+}}{(k^{2}-m^{2})^{2}}\delta_{m}^{(1)}\,,$
$\displaystyle\langle\tilde{\gamma}(k)\tilde{\beta}_{-}(-k)\rangle^{(1)}$
$\displaystyle=g^{2}\frac{4m^{2}k^{-}}{(k^{2}-m^{2})^{2}}\delta_{m}^{(1)}\,.$

Combining the diagram contributions corresponding to (4.1) together with the
counterterm contributions (4.5), we find the following condition for the
counterterm coefficient $\delta_{m}^{(1)}$

$\delta_{m}^{(1)}=\frac{\Gamma(\epsilon)}{(4\pi)^{1-\epsilon}}\left(\frac{\mu}{m}\right)^{2\epsilon}=\frac{1}{4\pi}\left(\frac{1}{\epsilon}-\log\frac{m^{2}}{\mu^{2}}-\gamma_{E}+\log(4\pi)+\mathcal{O}(\epsilon)\right)\,.$
(4.6)

Now let us turn to the analysis of the two loop contributions to the two-point
functions.

###  4.2 Two-loop two-point functions

At the two-loop level ($\sim g^{4}$) the two-point function is represented by
sunset diagrams, diagrams with two connected loops and two loops separated by
a propagator. The corresponding diagrams for the case of $\gamma\bar{\gamma}$
propagator are shown on the Figure 4. These types of diagrams also contain
divergent terms. Let us denote the contributions coming from these three types
by a, b and c correspondingly. We are going to consider the example of the
$\gamma\bar{\gamma}$ propagator.

$\gamma$
---
$\bar{\gamma}$
---
$\gamma$
---
$\bar{\gamma}$
---
$\bar{\gamma}$
---
$\gamma$
---
Figure 4: Two-loop diagrams for the $\gamma\bar{\gamma}$ propagator.

Let us start from writing down the sunset diagram contributions. We can
classify them by the number of half-dashed half-continuous propagators in the
loops, therefore there are 4 types of of loop integrals in the sunset diagrams

$\displaystyle J_{0}(k^{2})$
$\displaystyle=-\int\frac{d^{2}p_{1}}{(2\pi)^{2}}\frac{d^{2}p_{2}}{(2\pi)^{2}}\,\frac{im^{3}}{(p_{1}^{2}-m^{2}+i\varepsilon)(p_{2}^{2}-m^{2}+i\varepsilon)((k-p_{1}-p_{2})^{2}-m^{2}+i\varepsilon)}\,,$
(4.7) $\displaystyle J_{1}^{\pm}(k)$
$\displaystyle=-\int\frac{d^{2}p_{1}}{(2\pi)^{2}}\frac{d^{2}p_{2}}{(2\pi)^{2}}\,\frac{2m^{2}p_{1}^{\pm}}{(p_{1}^{2}-m^{2}+i\varepsilon)(p_{2}^{2}-m^{2}+i\varepsilon)((k-p_{1}-p_{2})^{2}-m^{2}+i\varepsilon)}\,,$
$\displaystyle J_{2}^{\pm\mp}(k^{2})$
$\displaystyle=\int\frac{d^{2}p_{1}}{(2\pi)^{2}}\frac{d^{2}p_{2}}{(2\pi)^{2}}\,\frac{4imp_{1}^{\pm}p_{2}^{\mp}}{(p_{1}^{2}-m^{2}+i\varepsilon)(p_{2}^{2}-m^{2}+i\varepsilon)((k-p_{1}-p_{2})^{2}-m^{2}+i\varepsilon)}\,,$
$\displaystyle J_{3}^{\pm\mp\pm}(k)$
$\displaystyle=\int\frac{d^{2}p_{1}}{(2\pi)^{2}}\frac{d^{2}p_{2}}{(2\pi)^{2}}\,\frac{8p_{1}^{\pm}p_{2}^{\pm}(k-p_{1}-p_{2})^{\mp}}{(p_{1}^{2}-m^{2}+i\varepsilon)(p_{2}^{2}-m^{2}+i\varepsilon)((k-p_{1}-p_{2})^{2}-m^{2}+i\varepsilon)}\,,$

where the one depicted in the Figure 4 corresponds to the $J_{3}$ type. By
combining all the diagrams of the first type we obtain the total sunset
diagram contribution to the two-point function $\gamma\bar{\gamma}$

$\displaystyle\langle\tilde{\gamma}(k)\tilde{\bar{\gamma}}(-k)\rangle_{a}^{(2)}=g^{4}\frac{k^{2}+m^{2}}{(k^{2}-m^{2})^{2}}J_{0}(k^{2})-\frac{2img^{4}}{(k^{2}-m^{2})^{2}}(k^{+}J^{-}_{1}(k)+k^{-}J_{1}^{+}(k))-$
(4.8)
$\displaystyle-g^{4}\frac{k^{2}+m^{2}}{(k^{2}-m^{2})^{2}}J_{2}^{+-}(k^{2})+\frac{2img^{4}}{(k^{2}-m^{2})^{2}}(k^{+}J_{3}^{-+-}(k)+k^{-}J_{3}^{+-+}(k))=$
$\displaystyle=g^{4}\frac{k^{2}+m^{2}}{(k^{2}-m^{2})^{2}}\left(J_{0}(k^{2})-J_{2}^{+-}(k^{2})\right)-\frac{4img^{4}}{(k^{2}-m^{2})^{2}}k^{+}\left(J^{-}_{1}(k)-k^{+}J_{3}^{-+-}(k)\right)\,.$

The sunset loop integrals can be expressed one in terms of the other, which is
partially done in the B.4.

The next contribution comes from the diagrams with two connected loops of the
type b, which give the result

$\displaystyle\langle\tilde{\gamma}(k)\tilde{\bar{\gamma}}(-k)\rangle_{b}^{(2)}=g^{4}\frac{k^{2}+m^{2}}{(k^{2}-m^{2})^{2}}(I_{1}(0)+I_{2}^{+-}(0))\int\frac{d^{2}p}{(2\pi)^{2}}\frac{im}{p^{2}-m^{2}}=$
(4.9)
$\displaystyle=\frac{ig^{4}m(k^{2}+m^{2})}{16\pi^{2}(k^{2}-m^{2})^{2}}\left(\frac{1}{\epsilon}-\log\frac{m^{2}}{\mu^{2}}-\gamma_{E}+\log(4\pi)+\mathcal{O}(\epsilon)\right)\left(\frac{1}{\epsilon}-2-\gamma_{E}+\log(4\pi)-\log\frac{m^{2}}{\mu^{2}}+\mathcal{O}(\epsilon)\right)\,,$

where the loop integrals $I_{1}(k^{2})$ (5.11), $I_{2}^{+-}(k^{2})$ (5.13) and
$I_{3}^{\pm}(k^{2})$ (5.15) are defined in the Section 5.2 and calculated in
the B.1, B.2 and B.3.

And the last is the contribution from the third group c of diagrams, for which
the loops are separated by a propagator

$\displaystyle\langle\tilde{\gamma}(k)\tilde{\bar{\gamma}}(-k)\rangle_{c}^{(2)}=g^{4}\frac{im(3k^{2}+m^{2})}{(k^{2}-m^{2})^{3}}\int\frac{d^{2}p}{(2\pi)^{2}}\frac{im}{p^{2}-m^{2}}\int\frac{d^{2}p^{\prime}}{(2\pi)^{2}}\frac{im}{p^{\prime
2}-m^{2}}=$ (4.10)
$\displaystyle=\frac{ig^{4}m^{3}(3k^{2}+m^{2})}{16\pi^{2}(k^{2}-m^{2})^{3}}\left(\frac{1}{\epsilon}-\log\frac{m^{2}}{\mu^{2}}-\gamma_{E}+\log(4\pi)+\mathcal{O}(\epsilon)\right)^{2}\,.$

To see which counterterms are needed at the two loop order let us write down
the contribution of diagrams containing the $\delta_{m}^{(1)}$ counterterm
vertices into the two-point function up to the order $\sim g^{4}$. This
contribution is given by

$\displaystyle\langle\tilde{\gamma}(k)\tilde{\bar{\gamma}}(-k)\rangle_{\delta_{m}^{(1)}-\textrm{counterterms}}^{(2)}=-g^{4}\frac{k^{2}+m^{2}}{(k^{2}-m^{2})^{2}}(I_{1}(0)+I_{2}^{+-}(0))\delta_{m}^{(1)}m-$
(4.11)
$\displaystyle-2g^{4}\frac{2im\left(3k^{2}+m^{2}\right)}{(k^{2}-m^{2})^{3}}m\delta_{m}^{(1)}\int\frac{d^{2}p}{(2\pi)^{2}}\frac{im}{p^{2}-m^{2}}+g^{4}\frac{im\left(3k^{2}+m^{2}\right)}{(k^{2}-m^{2})^{3}}(m\delta_{m}^{(1)})^{2}=$
$\displaystyle=-\frac{ig^{4}m^{3}\left(3k^{2}+m^{2}\right)}{16\pi^{2}(k^{2}-m^{2})^{3}}\left(\frac{1}{\epsilon}-\log\frac{m^{2}}{\mu^{2}}-\gamma_{E}+\log(4\pi)+\mathcal{O}(\epsilon)\right)^{2}-$
$\displaystyle-\frac{ig^{4}m\left(m^{2}+k^{2}\right)}{16\pi^{2}(k^{2}-m^{2})^{2}}\left(\frac{1}{\epsilon}-2-\gamma_{E}+\log(4\pi)-\log\frac{m^{2}}{\mu^{2}}+\mathcal{O}(\epsilon)\right)\left(\frac{1}{\epsilon}-\log\frac{m^{2}}{\mu^{2}}-\gamma_{E}+\log(4\pi)+\mathcal{O}(\epsilon)\right)\,.$

If we compare (4.11) with (4.9) and (4.10) we see that the diagrams from these
groups are exactly compensated by the corresponding counterterms from (4.11)
containing $\delta_{m}^{(1)}$. It means that the only remaining divergence
comes from the sunset diagrams contribution to the $\gamma\bar{\gamma}$
correlator

$\displaystyle\langle\tilde{\gamma}(k)\tilde{\bar{\gamma}}(-k)\rangle_{a}^{(2)}=\frac{img^{4}}{16\pi^{2}}\frac{k^{2}+m^{2}}{(k^{2}-m^{2})^{2}}\left(\frac{1}{\epsilon^{2}}+\frac{2}{\epsilon}\bigg{(}-\gamma_{E}+\log
4\pi-\log\left(\frac{m}{\mu}\right)^{2}\right)+\mathcal{O}(\epsilon^{0})\,.$
(4.12)

Then, in the framework of the minimal subtraction scheme, we can write the
counterterm coefficient $\delta_{m}^{(2)}$ in the form

$\delta_{m}^{(2)}=\frac{1}{16\pi^{2}}\left(\frac{1}{\epsilon^{2}}+\frac{2}{\epsilon}\left(-\gamma_{E}+\log
4\pi-\log\left(\frac{m}{\mu}\right)^{2}\right)\right)\,.$ (4.13)

In addition, one can calculate the other two-point functions involving the
fields $\beta$. All the diagrams except for the sunset ones, are exactly
annihilated by the counterterm diagrams with $\delta_{m}^{(1)}$ (4.6). Whereas
the sunset diagram contributions are given by the following result

$\displaystyle\langle\tilde{\beta}_{+}(k)\tilde{\beta}_{-}(-k)\rangle_{a}^{(2)}$
$\displaystyle=g^{4}\frac{k^{2}+m^{2}}{(k^{2}-m^{2})^{2}}\left(J_{0}(k^{2})-J_{2}^{+-}(k^{2})\right)-\frac{4img^{4}}{(k^{2}-m^{2})^{2}}k^{+}\left(J^{-}_{1}(k)-J_{3}^{-+-}(k)\right)\,,$
$\displaystyle\langle\tilde{\gamma}(k)\tilde{\beta}_{-}(-k)\rangle_{a}^{(2)}$
$\displaystyle=\frac{4g^{4}imk^{-}}{(k^{2}-m^{2})^{2}}(J_{2}^{+-}(k^{2})-J_{0}(k^{2}))+\frac{4g^{4}(k^{-})^{2}}{(k^{2}-m^{2})^{2}}(J_{3}^{++-}(k)-J_{1}^{+}(k))+$
(4.14)
$\displaystyle+\frac{g^{4}m^{2}}{(k^{2}-m^{2})^{2}}(J_{3}^{+--}(k)-J_{1}^{-}(k))\,,$
$\displaystyle\langle\tilde{\bar{\gamma}}(k)\tilde{\beta}_{+}(-k)\rangle_{a}^{(2)}$
$\displaystyle=\frac{4g^{4}imk^{+}}{(k^{2}-m^{2})^{2}}(J_{2}^{+-}(k^{2})-J_{0}(k^{2}))+\frac{4g^{4}(k^{+})^{2}}{(k^{2}-m^{2})^{2}}(J_{3}^{+--}(k)-J_{1}^{-}(k))+$
$\displaystyle+\frac{g^{4}m^{2}}{(k^{2}-m^{2})^{2}}(J_{3}^{++-}(k)-J_{1}^{+}(k))\,.$

It is not complicated to check that the counterterm diagrams with
$\delta_{m}^{(2)}$ (4.13) compensate the divergences in (4.2) as well.

In the present section we showed that at one loop the divergences in the two-
point function can be compensated by the renormalization of the mass only and
we do not need any renormalization of the fields at one loop. At two loops we
found that the contributions at this order proportional to $\delta_{m}^{(1)}$
compensate the contributions of all the diagrams into the $\gamma\bar{\gamma}$
correlation function except for the sunset ones. For these diagrams we showed
that the divergencies can be absorbed into $\delta_{m}^{(2)}$.

In the next Section we are going to calculate the $2\rightarrow 2$ amplitudes
in the sector of the particles $\gamma$ and $\bar{\gamma}$ by extracting them
from the four-point correlation function with the usage of the Lehman-
Symanczik-Zimmermann (LSZ) reduction formula in order to compare them with the
$S$-matrix results from [10] and [11].

##  5 Four-point correlation functions

The main objective of our paper is to compare the amplitudes coming from the
Lagrangian (4.4) with the ones calculated in [10] and [11]. We are
concentrated on the $2\rightarrow 2$ amplitudes corresponding to the fields
$\gamma$ and $\bar{\gamma}$. To achieve this goal one can use the correlation
functions

$\langle\gamma(x_{1})\bar{\gamma}(x_{2})\gamma(x_{3})\bar{\gamma}(x_{4})\rangle\,,\quad\langle\gamma(x_{1})\gamma(x_{2})\gamma(x_{3})\gamma(x_{4})\rangle\,,\quad\langle\bar{\gamma}(x_{1})\bar{\gamma}(x_{2})\bar{\gamma}(x_{3})\bar{\gamma}(x_{4})\rangle\,.$
(5.1)

Given the Feynman rules from the Figure 1 it is more convenient to calculate
the corresponding correlation functions in the Fourier representation

$\langle\tilde{\gamma}_{1}\tilde{\bar{\gamma}}_{2}\tilde{\gamma}_{3}\tilde{\bar{\gamma}}_{4}\rangle\,,\quad\langle\tilde{\gamma}_{1}\tilde{\gamma}_{2}\tilde{\gamma}_{3}\tilde{\gamma}_{4}\rangle\,,\quad\langle\tilde{\bar{\gamma}}_{1}\tilde{\bar{\gamma}}_{2}\tilde{\bar{\gamma}}_{3}\tilde{\bar{\gamma}}_{4}\rangle\,,$
(5.2)

where $\tilde{\gamma}_{i}\overset{\textrm{def}}{=}\tilde{\gamma}(k_{i})$ for
brevity. As we already mentioned in the Section 2 the particle $\gamma$ is
associated to the soliton with the charge $Q=1$ and the particle
$\bar{\gamma}$ corresponds to the soliton with the charge $Q=-1$ in the CSG
theory and, in addition, all other $2\rightarrow 2$ amplitudes like
$\gamma\gamma\rightarrow\gamma\gamma$ and
$\bar{\gamma}\bar{\gamma}\rightarrow\bar{\gamma}\bar{\gamma}$ are expressed in
terms of the only one function (2.14). Therefore, it is sufficient to study
the following correlation function

$\langle\tilde{\gamma}_{1}\tilde{\bar{\gamma}}_{2}\tilde{\gamma}_{3}\tilde{\bar{\gamma}}_{4}\rangle=\sum\limits_{n=1}^{+\infty}\tilde{G}^{(n)}(k_{1},k_{2},k_{3},k_{4})$
(5.3)

as the others are related to it by the crossing symmetry transformation
($\tilde{G}^{(k)}(k_{1},k_{2},k_{3},k_{4})$ corresponds to the $k$-th loop
contribution). In the following subsections we will calculate the four point
correlation functions in question at the tree and one-loop level.

###  5.1 Tree-level four-point function

According to our Feynman rules, at the tree level the four-point correlation
function
$\langle\tilde{\gamma}_{1}\tilde{\bar{\gamma}}_{2}\tilde{\gamma}_{3}\tilde{\bar{\gamma}}_{4}\rangle$
is given by following set of diagrams in the Figure 5, which correspond to the
connected diagrams at the order $g^{2}$.

$\gamma_{1}$
---
$\bar{\gamma}_{4}$
---
$\gamma_{3}$
---
$\bar{\gamma}_{2}$
---
$k_{4}$
---
$k_{1}$
---
$k_{3}$
---
$k_{2}$
---
$\gamma_{1}$
---
$\bar{\gamma}_{4}$
---
$\gamma_{3}$
---
$\bar{\gamma}_{2}$
---
$k_{4}$
---
$k_{1}$
---
$k_{3}$
---
$k_{2}$
---
$\gamma_{1}$
---
$\bar{\gamma}_{4}$
---
$\gamma_{3}$
---
$\bar{\gamma}_{2}$
---
$k_{4}$
---
$k_{1}$
---
$k_{3}$
---
$k_{2}$
---
$\gamma_{1}$
---
$\bar{\gamma}_{4}$
---
$\gamma_{3}$
---
$\bar{\gamma}_{2}$
---
$k_{4}$
---
$k_{1}$
---
$k_{3}$
---
$k_{2}$
---
Figure 5: Tree level Feynman diagrams for the correlation function
$\langle\tilde{\gamma}_{1}\tilde{\bar{\gamma}}_{2}\tilde{\gamma}_{3}\tilde{\bar{\gamma}}_{4}\rangle$.

Using the Feynman rules defined in the Section 3, we write the four-point
function in momentum space as the product of vertex term and four propagators
connecting the vertex with the external lines

$\tilde{G}^{(1)}(k_{1},k_{2},k_{3},k_{4})=-g^{2}\frac{4im^{2}(k_{1}^{-}+k_{3}^{-})(k_{2}^{+}+k_{4}^{+})}{(k_{1}^{2}-m^{2})(k_{2}^{2}-m^{2})(k_{3}^{2}-m^{2})(k_{4}^{2}-m^{2})}\,,$
(5.4)

where the external momenta are directed towards the vertex. It should be noted
that for the other two correlation functions from (5.2) on the tree level we
have

$\langle\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}\rangle=\mathcal{O}(g^{4})\,,\quad\langle\bar{\gamma}_{1}\bar{\gamma}_{2}\bar{\gamma}_{3}\bar{\gamma}_{4}\rangle=\mathcal{O}(g^{4})\,.$
(5.5)

The tree level amplitude can be obtained by applying the LSZ reduction formula
to the four-point function. As the result the tree-level amplitude takes the
following form

$\mathcal{M}_{\gamma\bar{\gamma}}^{\gamma\bar{\gamma}(tree)}=-4ig^{2}m^{2}(k_{1}^{-}+k_{3}^{-})(k_{2}^{+}+k_{4}^{+})=\begin{cases}&-4ig^{2}m^{4}\sinh^{2}\frac{\theta_{1}-\theta_{2}}{2}\,,\quad
k_{3}=-k_{2}\,,\;k_{4}=-k_{1}\,,\\\ &0\,,\quad
k_{3}=-k_{1}\,,\;k_{4}=-k_{2}\,,\end{cases}$ (5.6)

where $\theta_{1},\,\theta_{2}$ are rapidities of $\gamma_{1}$ and
$\bar{\gamma}_{2}$ respectively. We also took into account that the energy-
momentum conservation laws for the $(1+1)$-dimensional case has only two
solutions

$k_{3}=-k_{2}\,,\;k_{4}=-k_{1}\;\text{(transition)}\quad\text{or}\quad
k_{3}=-k_{1}\,,\;k_{4}=-k_{2}\;\text{(reflection)}.$ (5.7)

It can be seen that the requirement of the reflectionless of the scattering at
the tree level is satisfied, and, therefore, the Yang-Baxter equation is
fulfilled. If we pass to the dimensionless fields
$\gamma\,,\bar{\gamma}\rightarrow\gamma/\sqrt{m}\,,\bar{\gamma}/\sqrt{m}$ then
the tree-level amplitude of $\gamma\bar{\gamma}\rightarrow\gamma\bar{\gamma}$
scattering has the form

$\displaystyle\mathcal{M}_{\gamma\bar{\gamma}}^{\gamma\bar{\gamma}(tree)}(\theta)=-4ig^{2}m^{2}\sinh^{2}\frac{\theta}{2}\,,$
(5.8)

where we use the notation $\theta=\theta_{1}-\theta_{2}$.

For the special case of $2\rightarrow 2$ scattering, the $S$-matrix element is
related to the amplitude by the following formula

$S(\theta)=1+\frac{\mathcal{M}(\theta)}{4m^{2}\sinh\theta}\,.$ (5.9)

Therefore, combining (5.9) with (5.8), we obtain the
$\gamma\bar{\gamma}\rightarrow\gamma\bar{\gamma}$ scattering amplitude
$S_{\gamma\bar{\gamma}}^{\gamma\bar{\gamma}}(\theta)$ at the tree level takes
the following form

$\displaystyle
S_{\gamma\bar{\gamma}}^{\gamma\bar{\gamma}}(\theta)=1-\frac{ig^{2}}{2}\left(\coth\theta-\mathop{\mathrm{cosech}}\nolimits\theta\right)+\mathcal{O}(g^{4})=1-\frac{ig^{2}}{2}\tanh\frac{\theta}{2}+\mathcal{O}(g^{4})\,,$
(5.10)

which coincides with the CSG results (2.13) and (2.16) at the tree level.

To sum up, in the subsection above we managed show that the tree level
amplitudes for the elementary excitations generated by the fields $\gamma$ and
$\bar{\gamma}$ obtained from the Lagrangian (4.4) exactly coincide with the
same tree level amplitudes from [10] and [11]. In the subsequent subsection we
will explore such a relation at the one loop level.

###  5.2 One-loop four-point function

Let us consider the four-point correlation function
$\langle\tilde{\gamma}_{1}\tilde{\bar{\gamma}}_{2}\tilde{\gamma}_{3}\tilde{\bar{\gamma}}_{4}\rangle$
at the one-loop level. All the connected diagrams giving nonzero contribution
at this order have the structure of a loop consisting of two propagators with
two pairs of external lines connected to it. All one-loop diagrams can be
split into 3 groups according to the set of propagators forming the loops.
They are: with continuous or dashed propagators only, with two half-dashed
half-continuous propagators and with one half-dashed half-continuous
propagator. As an illustration we drew all the diagrams of the second type,
which have the Lorentz indices $+-$ in the loop integral in the Figure 6.

$\gamma_{1}$
---
$\bar{\gamma}_{2}$
---
$\bar{\gamma}_{4}$
---
$\gamma_{3}$
---
$k_{4}$
---
$k_{3}$
---
$k_{1}$
---
$k_{2}$
---
$p$
---
$k-p$
---
$\gamma_{1}$
---
$\bar{\gamma}_{2}$
---
$\bar{\gamma}_{4}$
---
$\gamma_{3}$
---
$k_{4}$
---
$k_{3}$
---
$k_{1}$
---
$k_{2}$
---
$p$
---
$k-p$
---
$k_{1}$
---
$k_{4}$
---
$k_{2}$
---
$k_{3}$
---
$\gamma_{1}$
---
$\bar{\gamma}_{4}$
---
$\bar{\gamma}_{2}$
---
$\gamma_{3}$
---
$p$
---
$k-p$
---
$k_{1}$
---
$k_{4}$
---
$k_{2}$
---
$k_{3}$
---
$\gamma_{1}$
---
$\bar{\gamma}_{4}$
---
$\bar{\gamma}_{2}$
---
$\gamma_{3}$
---
$p$
---
$k-p$
---
$\gamma_{1}$
---
$\bar{\gamma}_{2}$
---
$\gamma_{3}$
---
$\bar{\gamma}_{4}$
---
$k_{1}$
---
$k_{2}$
---
$k_{4}$
---
$k_{3}$
---
$p$
---
$k-p$
---
$\gamma_{1}$
---
$\bar{\gamma}_{2}$
---
$\gamma_{3}$
---
$\bar{\gamma}_{4}$
---
$k_{1}$
---
$k_{2}$
---
$k_{4}$
---
$k_{3}$
---
$p$
---
$k-p$
---
$\gamma_{1}$
---
$\bar{\gamma}_{2}$
---
$\gamma_{3}$
---
$\bar{\gamma}_{4}$
---
$k_{1}$
---
$k_{2}$
---
$k_{4}$
---
$k_{3}$
---
$p$
---
$k-p$
---
$\gamma_{1}$
---
$\bar{\gamma}_{2}$
---
$\gamma_{3}$
---
$\bar{\gamma}_{4}$
---
$k_{1}$
---
$k_{2}$
---
$k_{4}$
---
$k_{3}$
---
$p$
---
$k-p$
---
Figure 6: One-loop contributions with the $+-$ Lorentz structure in the loop
diagrams.

The first group are the diagrams compiled by contractions of the same type
fields, i.e. the contractions of the types
$\wick{\c{1}\beta_{+}(x)\c{1}\beta_{-}(y)}$,
$\wick{\c{1}\gamma(x)\c{1}{\bar{\gamma}}(y)}$. The corresponding loop integral
takes the form

$I_{1}\left(k^{2}\right)=-\int\frac{d^{2}p}{(2\pi)^{2}}\frac{m^{2}}{(p^{2}-m^{2}+i\varepsilon)((k-p)^{2}-m^{2}+i\varepsilon)}\,.$
(5.11)

As we have the $s$-, $t$\- and $u$\- channel diagrams, the loop integral of
the first group $I_{1}$ results in (details can be found in the B)

$\displaystyle
I_{1}\left(s=(k_{1}+k_{2})^{2}\right)=\frac{i}{4\pi}\frac{\theta-i\pi}{\sinh\theta}\,,$
(5.12) $\displaystyle
I_{1}\left(t=(k_{1}+k_{4})^{2}\right)=\begin{cases}-\frac{i}{4\pi}\frac{\theta}{\sinh\theta}\,,\quad
k_{3}=-k_{1}\,,\quad k_{4}=-k_{2}\,,\\\ -\frac{i}{4\pi}\,,\quad
k_{4}=-k_{1},\;k_{3}=-k_{2}\,,\end{cases}$ $\displaystyle
I_{1}\left(u=(k_{1}+k_{3})^{2}\right)=\begin{cases}-\frac{i}{4\pi}\,,\quad
k_{3}=-k_{1}\,,\quad k_{4}=-k_{2}\,,\\\
-\frac{i}{4\pi}\frac{\theta}{\sinh\theta},\;k_{4}=-k_{1},\,k_{3}=-k_{2}\,.\end{cases}$

The second group are the diagrams originating from the contractions of the
different type fields, i.e. from the contractions of the type
$\wick{\c{1}\beta_{+}(x)\c{1}\gamma(y)},\,\wick{\c{1}\beta_{-}(x)\c{1}{\bar{\gamma}}(y)},\,\wick{\c{1}\gamma(x)\c{1}\beta_{+}(y)},\,\wick{\c{1}{\bar{\gamma}}(x)\c{1}\beta_{-}(y)}$.
There are two expressions appearing in the loop integral

$\displaystyle
I_{2}^{\pm\mp}\left(k^{2}\right)=\int\frac{d^{2}p}{(2\pi)^{2}}\frac{4p^{\mp}(k-p)^{\pm}}{(p^{2}-m^{2}+i\varepsilon)((k-p)^{2}-m^{2}+i\varepsilon)}\,,$
(5.13) $\displaystyle
I_{2}^{\pm\pm}\left(k^{2}\right)=\int\frac{d^{2}p}{(2\pi)^{2}}\frac{4p^{\pm}(k-p)^{\pm}}{(p^{2}-m^{2}+i\varepsilon)((k-p)^{2}-m^{2}+i\varepsilon)}\,.$

The difference now is that the integrals (5.13) contain the logarithmic
divergence and therefore have to be regularized ($\mu$ is the regularization
parameter), so the calculation result is (as in the previous case details are
written in the appendix B)

$\displaystyle
I_{2}^{+-}(s)=-\frac{i\theta+\pi}{4\pi}\left(\coth\frac{\theta}{2}-\frac{1}{\sinh\theta}\right)+\frac{i}{4\pi}\left(\frac{1}{\epsilon}-\gamma_{E}+\log(4\pi)-\log\frac{m^{2}}{\mu^{2}}\right)+\mathcal{O}(\epsilon)\,,$
(5.14) $\displaystyle
I_{2}^{+-}(t)=\begin{cases}-\frac{i\theta}{4\pi}\left(\tanh\frac{\theta}{2}+\frac{1}{\sinh\theta}\right)+\frac{i}{4\pi}\left(\frac{1}{\epsilon}-\gamma_{E}+\log(4\pi)-\log\frac{m^{2}}{\mu^{2}}\right)+\mathcal{O}(\epsilon)\,,\,k_{3,4}=-k_{1,2}\,,\\\
\frac{i}{4\pi}\left(\frac{1}{\epsilon}-1-\gamma_{E}+\log(4\pi)-\log\frac{m^{2}}{\mu^{2}}\right)+\mathcal{O}(\epsilon)\,,\,k_{4}=-k_{1}\,,\,k_{3}=-k_{2}\,,\end{cases}$
$\displaystyle
I_{2}^{+-}(u)=\begin{cases}\frac{i}{4\pi}\left(\frac{1}{\epsilon}-1-\gamma_{E}+\log(4\pi)-\log\frac{m^{2}}{\mu^{2}}\right)+\mathcal{O}(\epsilon)\,,\,k_{4}=-k_{1}\,,\,k_{3}=-k_{2}\,,\\\
-\frac{i\theta}{4\pi}\left(\tanh\frac{\theta}{2}+\frac{1}{\sinh\theta}\right)+\frac{i}{4\pi}\left(\frac{1}{\epsilon}-\gamma_{E}+\log(4\pi)-\log\frac{m^{2}}{\mu^{2}}\right)+\mathcal{O}(\epsilon)\,,\,k_{3,4}=-k_{1,2}\,,\end{cases}$
$\displaystyle
I_{2}^{\pm\pm}(s)=-(k_{1}^{\pm}+k_{2}^{\pm})^{2}\left(\frac{i}{4\pi
m^{2}\cosh^{2}\frac{\theta}{2}}+\frac{\pi+i\theta}{8\pi
m^{2}\cosh^{4}\frac{\theta}{2}}\coth\frac{\theta}{2}\right)\,,$ $\displaystyle
I_{2}^{\pm\pm}(t)=\begin{cases}(k_{1}^{\pm}+k_{4}^{\pm})^{2}\left(\frac{i}{4\pi
m^{2}\sinh^{2}\frac{\theta}{2}}-\frac{i\theta}{8\pi
m^{2}\sinh^{4}\frac{\theta}{2}}\tanh\frac{\theta}{2}\right)\,,\,k_{3}=-k_{1},\,k_{4}=-k_{2}\,,\\\
0,\,k_{3}=-k_{2},\,k_{4}=-k_{1}\,.\end{cases}$

The third group are the diagrams consisting of all types of propagators,
dashed, continuous and half-dashed half-continuous. The corresponding loop
integral has the following form

$I_{3}^{\pm}(k)=\int\frac{d^{2}p}{(2\pi)^{2}}\frac{2imp^{\pm}}{(p^{2}-m^{2}+i\varepsilon)((k-p)^{2}-m^{2}+i\varepsilon)}\,.$
(5.15)

The integrals (5.15) are convergent and for the cases we are interested in
they give the result (see the details in the B)

$\displaystyle I_{3}^{\pm}(s)=-\frac{(k_{1}+k_{2})^{\pm}}{4\pi
m\sinh\theta}(i\pi-\theta)\,,$ (5.16) $\displaystyle
I_{3}^{\pm}(t)=\begin{cases}-\frac{(k_{1}-k_{2})^{\pm}\theta}{4\pi
m\sinh\theta}\,,\quad k_{3}=-k_{1}\,,\quad k_{4}=-k_{2}\,,\\\ 0\,,\quad
k_{3}=-k_{2}\,,\quad k_{4}=-k_{1}\,.\end{cases}\,.$

We notice that the u-channel integrals do not exist for the third group of
diagrams.

The total one-loop contribution to the four-point correlation function is too
cumbersome to be written here, but let us comment the disappearance of the
divergence from the final result. It can be noticed that the divergence comes
from the $I_{2}^{+-}$ group of diagrams. Indeed, the divergent contribution
enters to the 4-point function (see Figure 6) with the coefficient

$\displaystyle\frac{1}{4\pi\epsilon}\,4(im)^{2}\left((k_{3}^{-}k_{4}^{+}+k_{1}^{-}k_{2}^{+})+(k_{1}^{-}k_{4}^{+}+k_{3}^{+}k_{4}^{-})-(k_{1}^{-}k_{2}^{+}+k_{1}^{-}k_{4}^{+}+k_{3}^{-}k_{2}^{+}+k_{3}^{-}k_{4}^{+})\right)=0\,,$
(5.17)

which proves the statement that the one-loop correlation function
$\langle\tilde{\gamma}_{1}\tilde{\bar{\gamma}}_{2}\tilde{\gamma}_{3}\tilde{\bar{\gamma}}_{4}\rangle$
is convergent (of course, we take into account the counterterms proportional
to $\delta_{m}^{(1)}$ (4.6)).

In the same way we can answer the question about the one loop renormalization
of the quartic vertex in (4.4). Similarly, one can show that divergence of
four-point function
$\langle\beta_{+1}\beta_{-2}\gamma_{3}\bar{\gamma}_{4}\rangle$ also vanishes.
Logarithmic divergence could appear from the diagrams of the $I_{2}^{+-}$
groups, which have the form as divergent diagrams for
$\langle\gamma_{1}\bar{\gamma}_{2}\gamma_{3}\bar{\gamma}_{4}\rangle$ up to
replacement of input propagators
$(\gamma_{1}\rightarrow\beta_{+1},\,\bar{\gamma}_{2}\rightarrow\beta_{-2})$,
and enters to the correlation function with the coefficient

$\displaystyle\frac{1}{4\pi\epsilon}(16k_{1}^{+}k_{2}^{-}k_{3}^{-}k_{4}^{+}+(im)^{4}+4(im)^{2}(k_{2}^{-}k_{4}^{+}+k_{1}^{+}k_{3}^{-})-$
(5.18)
$\displaystyle-(im)^{4}-4(im)^{2}(k_{2}^{-}k_{4}^{+}+k_{1}^{+}k_{3}^{-})-16k_{1}^{+}k_{2}^{-}k_{3}^{-}k_{4}^{+})=0\,,$

where $k_{1}$, $k_{2}$ are the momenta of $\beta_{+}$, $\beta_{-}$ and
$k_{3}$, $k_{4}$ are the momenta of $\gamma$, $\bar{\gamma}$ respectively.
Thus, we do not need to introduce any $g^{4}$ order counterterms to the
coupling constant $g^{2}$ at the one-loop level.

Applying the LSZ reduction formula and rewriting in terms of the rapidities we
obtain the one-loop contribution to the $2\rightarrow 2$ amplitudes for
$\gamma$ and $\bar{\gamma}$ in the form

$\mathcal{M}_{\gamma\bar{\gamma}}^{\gamma\bar{\gamma}(1-\text{loop})}(\theta)=\begin{cases}-\frac{g^{4}m^{4}}{\pi}\sinh^{2}\frac{\theta}{2}\left(i+\pi\tanh\frac{\theta}{2}\right)\,,\quad
k_{3}=-k_{2}\,,\quad k_{4}=-k_{1}\,,\\\ 0\,,\quad k_{3}=-k_{1}\,,\quad
k_{4}=-k_{2}\,.\end{cases}$ (5.19)

Thus, we observe the reflectionless scattering and, therefore, the Yang-Baxter
equation is satisfied at one loop as well. Now we can extract the contribution
to the element of $S$-matrix by applying the formula (5.9) to (5.19).
Eventually, the transition amplitude from (5.19) gives us exactly the same
one-loop contribution as in (2.16). Therefore, by combining (5.8) with (5.19),
we can write the $S$-matrix element of
$\gamma\bar{\gamma}\rightarrow\gamma\bar{\gamma}$ scattering in our theory

$\displaystyle
S_{\gamma\bar{\gamma}}^{\gamma\bar{\gamma}}(\theta)=1+\frac{ig^{2}}{2}(\mathop{\mathrm{cosech}}\nolimits\theta-\coth\theta)+\frac{ig^{4}}{8\pi}(\mathop{\mathrm{cosech}}\nolimits\theta-\coth\theta)+$
(5.20)
$\displaystyle+\frac{g^{4}}{2}\coth\theta\,\mathop{\mathrm{cosech}}\nolimits\theta-\frac{g^{4}}{2}\left(\frac{1}{2}+\mathop{\mathrm{cosech}}\nolimits^{2}\theta\right)+\mathcal{O}(g^{6})\,.$

which exactly coincides with the one (2.16) from [11], but differs at the one
loop order with (2.13) from [10], which can be attributed to the difference in
the details of renormalization schemes, as it was noticed in [14].

In addition, we recall that in the $\beta\gamma$ action (3.19) originating
from the gWZW model the constant $k$ is related to the coupling as

$g^{2}=\frac{4\pi}{k}\,.$ (5.21)

Taking into account (2.15) we arrive to the relation

$\frac{1}{\mathrm{k}}=\frac{1}{k}+\frac{1}{k^{2}}+\mathcal{O}\left(\frac{1}{k^{3}}\right)\,.$
(5.22)

Given (5.22) it is natural to assume that the formula (2.4) for the relation
between $g^{2}$ and $g_{R}^{2}$ is equivalent to the relation

$k=\mathrm{k}-1\,.$ (5.23)

Checking the relation (5.23) at the higher orders in $k$ will require
calculating the S-matrix at the two-loop order and higher, which is the
subject of future work.

Let us summarize our results and discuss possible open problems in the
conluding section of our work.

##  6 Conclusions

In this article we conducted analysis of the version of bosonic Thirring model
in Minkowski spacetime. We started from remembering the relation of the
Euclidean version of this model to the complex sine-Gordon model via the use
of path integral. Then we offered a way to analytically continue the bosonic
version of Thirring model to the Minkowski signature, putting forward the
hypothesis that such a definition of this theory allows to reproduce the
$S$-matrix of the Minkowski version of complex sine-Gordon model from [11].
Also we managed to transform the gauged Wess-Zumino-Witten model with
integrable potential into this bosonic Thirring model in Minkowski signature
($\beta\gamma$-system) via field redefinitions.

To achieve the goal of reproducing the S-matrix we conducted some perturbative
analysis of the $\beta\gamma$-system (3.11). First of all we calculated the
two-point correlation function of the theory at one loop order in the
Subsection 4.1 and showed that it is enough to have only one loop mass
renormalization terms to absorb the divergences appearing. Then we also found
the two-loop two-point function, showing that there are three types of
diagrams contributing. It appears that the one loop counterterms from the mass
renormalization are sufficient to compensate all the divergences except for
the ones coming from the so-called sunset diagrams. The divergences from the
sunset diagrams can be absorbed into the nex-to-leading order counterterm to
the mass terms in the Lagrangian.

In the next part we concentrated on calculating the amplitudes of the
scattering of $\gamma$ and $\bar{\gamma}$ in order to compare the result with
the $S$-matrix of the complex sine-Gordon theory. By computing the four-point
function
$\langle\tilde{\gamma}_{1}\tilde{\bar{\gamma}}_{2}\tilde{\gamma}_{3}\tilde{\bar{\gamma}}_{4}\rangle$
and applying the LSZ reduction formula we managed to find the corresponding
$\gamma\bar{\gamma}\rightarrow\gamma\bar{\gamma}$ $S$-matrix element up to one
loop order. And indeed it coincides with the same $S$-matrix element from [11]
at this perturbation theory order, which represents the argument in favour of
our initial hypothesis that the theory (3.11) correctly reproduces the
$S$-matrix of the complex sine-Gordon theory. The one-loop result from [10]
differs from ours and the one from [11], however, this could be attributed to
the details of the renormalization scheme used, as it was noticed in [14]. As
a byproduct we checked the the one-loop corrections to the quartic vertex
$\beta_{+}\beta_{-}\bar{\gamma}\gamma$ do not have any divergences, therefore
confirming that there is no need for the counterterms to the coupling constant
at the order $g^{4}$.

An obvious direction for future study is the analysis of the theory (3.11) at
higher loops. It would be important to proceed first of all with the study of
renormalizability properties. The next step would be to find the two-loop
contribution to the 4-point correlation function aiming to extract from it the
two-loop amplitude, which can compared with the all-loop result of [11]. This
can shed some light on which counterterms besides the mass ones are needed at
the two-loop order.

Another interesting problem is to consider several copies of the
$\beta\gamma$-system with the interaction term proportional to
$r_{ijkl}\bar{\beta}^{(i)}\beta^{(j)}\bar{\gamma}^{(k)}\gamma^{(l)}$, where
the indices $(i)$ denote the isotopic index of the field, as it was done in
[19, 20]. It was shown there that a theory with such an interaction is
classically integrable provided that $r_{ijkl}$ satisfies classical Yang-
Baxter equation. One can pose a question whether this condition is sufficient
for the quantum integrability of the Minkowskian version of the model with
such an interaction. The key test for the quantum integrability of such a
coupled $\beta\gamma$-system would be to check the absence of particle
production [21].

To sum up, the result obtained in this article together with the mentioned
directions of future study could improve the understanding of the CSG model
itself and help to progress with the dual description of deformed $OSp(N|2m)$,
as well as other integrable sigma models, as it represents an important
ingredient in the calculation of the $S$-matrix of the dual Toda-like theory.

## Acknowledgements

M.A. and A.K. are especially grateful to Ben Hoare for illuminating ideas and
participating in the work at various stages of the project. M.A. would also
thank Alexey Litvinov, Arkady Tseytlin, Mikhail Vasiliev and Dmitri Bykov for
fruitful and stimulating discussions and Anna Popova for the inspiration and
linguistic assistance during the creation of this text.

##  Appendix A Path integral in the Euclidean $\beta\gamma$-system

In order to demonstrate the correspondence between the $\beta\gamma$-system
and the CSG defined in Euclidean space we are going calculate the path
integral over $\beta$ and $\bar{\beta}$ fields

$\displaystyle\int\mathcal{D}\beta\mathcal{D}\bar{\beta}\exp\left[-\int
dxdy\;\mathcal{L}^{(E)}_{\beta\gamma}\right]=\exp\left[\int
dxdy\;m\gamma\bar{\gamma}\right]\times$ (A.1)
$\displaystyle\times\int\mathcal{D}\beta\mathcal{D}\bar{\beta}\exp\left[-\int
dxdy\left(\beta\bar{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}-m\beta\bar{\beta}+g^{2}\beta\bar{\beta}\gamma\bar{\gamma}\right)\right]\,.$

Let us re-define the fields $\beta$ and $\bar{\beta}$ by making the following
shifts

$\beta\rightarrow\beta^{\prime}=\beta+\frac{\partial\bar{\gamma}}{g^{2}\gamma\bar{\gamma}-m}\,,\quad\bar{\beta}\rightarrow\bar{\beta}^{\prime}=\bar{\beta}+\frac{\bar{\partial}\gamma}{g^{2}\gamma\bar{\gamma}-m}\,.$
(A.2)

Substituting the redefined fields (A.2) into the path integral (A.1) we obtain
the integral of the Gaussian form

$\displaystyle\int\mathcal{D}\beta^{\prime}\mathcal{D}\bar{\beta}^{\prime}\exp\left[-\int
dxdy\;\left(\beta^{\prime}(g^{2}\gamma\bar{\gamma}-m)\bar{\beta}^{\prime}-m\gamma\bar{\gamma}+\frac{\bar{\partial}\gamma\partial\bar{\gamma}}{m-g^{2}\gamma\bar{\gamma}}\right)\right]=$
(A.3)
$\displaystyle=\frac{\pi}{\det(m-g^{2}\bar{\gamma}\gamma)}\exp\left[-\int
dxdy\left(\frac{\bar{\partial}\gamma\partial\bar{\gamma}}{m-g^{2}\gamma\bar{\gamma}}-m\gamma\bar{\gamma}\right)\right]\,.$

If we pass to the massless fields $\gamma^{\prime}$ and
$\bar{\gamma}^{\prime}$

$\gamma\rightarrow\gamma^{\prime}=\frac{\gamma}{\sqrt{m}}\,,\quad\bar{\gamma}\rightarrow\bar{\gamma}^{\prime}=\frac{\bar{\gamma}}{\sqrt{m}}\,,$
(A.4)

then in the exponent we obtain the classical CSG Lagrangian

$\displaystyle\frac{1}{\det(1-g^{2}\bar{\gamma}^{\prime}\gamma^{\prime})}\exp\left[-\int
dxdy\left(\frac{\bar{\partial}\gamma^{\prime}\partial\bar{\gamma}^{\prime}}{1-g^{2}\gamma^{\prime}\bar{\gamma}^{\prime}}-m^{2}\gamma^{\prime}\bar{\gamma}^{\prime}\right)\right]\,.$
(A.5)

Thus, taking path integral (A.1) we get the classical CSG theory, defined in
Euclidean space, modulo the contribution of the functional determinant.

Let us show that the imaginary part of the CSG Lagrangian defined in Minkowski
space is a total derivative and can be ignored and, thus, Lagrangian can be
considered as a real function. Consider the imaginary part of the term

$\displaystyle\frac{\partial_{+}\bar{\gamma}\partial_{-}\gamma}{1-g^{2}\gamma\bar{\gamma}},$
(A.6)

where the fields $\gamma$ and $\bar{\gamma}$ can be written in the form:
$\gamma=re^{i\phi},\;\bar{\gamma}=re^{-i\phi}$. Then, for the imaginary part
of (A.6) we obtain

$\mathop{\mathrm{Im}}\nolimits\left(\frac{\partial_{+}\bar{\gamma}\partial_{-}\gamma}{1-g^{2}\gamma\bar{\gamma}}\right)=\frac{r}{1-g^{2}r^{2}}\left(\partial_{+}r\;\partial_{-}\phi-\partial_{+}\phi\;\partial_{-}r)\right)=\partial_{+}f(r)\;\partial_{-}\phi-\partial_{+}\phi\;\partial_{-}f(r)\,,$
(A.7)

where we introduced the notation

$f(r)=-\frac{1}{2g^{2}}\log(1-g^{2}r^{2})\,.$ (A.8)

Integraing by parts and flipping the derivatives under the integral, for the
imaginary part of the term (A.7) we get

$\displaystyle\int\,dx\,dy\mathop{\mathrm{Im}}\nolimits\left(\frac{\partial_{+}\bar{\gamma}\partial_{-}\gamma}{1-g^{2}\gamma\bar{\gamma}}\right)=\int\,dx\,dy(-f(r)\partial_{+}\partial_{-}\phi+f(r)\partial_{-}\partial_{+}\phi)=0\,,$
(A.9)

where we took into account that the fields vanish at infinity. Thus, the
imaginary part of CSG Lagrangian in Minkowski space is total derivative and we
can consider it as a real Lagrangian.

##  Appendix B Loop integrals calculation

The present Appendix is devoted to the details of the calculation of loop
integrals appearing in the process of deriving the two- and four-point
functions of the theory in question.

###  B.1 $I_{1}$ calculation

Let us calculate loop integral $I_{1}(k^{2})$. This integral is similar to the
well-known loop integral for the theory $\phi^{4}$ and equals to following

$I_{1}(k^{2})=-\int\frac{d^{2}p}{(2\pi)^{2}}\frac{m^{2}}{(p^{2}-m^{2}+i\varepsilon)((k-p)^{2}-m^{2}+i\varepsilon)}=-\frac{m^{2}}{2\pi
k^{2}}\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}\log\frac{1+i\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}}{1-i\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}}\,.$
(B.1)

###  B.2 $I_{3}$ calculation

Let us calculate the loop integral $I_{3}^{\pm}(k)$ corresponding to the loops
formed by the dashed or continuous line and half-dashed half-continuous line

$I_{3}^{\pm}(k)=\int\frac{d^{2}p}{(2\pi)^{2}}\frac{2imp^{\pm}}{(p^{2}-m^{2}+i\varepsilon)((k-p)^{2}-m^{2}+i\varepsilon)}\,.$
(B.2)

On the other hand, the integral can be rewritten as

$I_{3}^{\pm}(k)=\int\frac{d^{2}p}{(2\pi)^{2}}\frac{2im(k-p)^{\pm}}{(p^{2}-m^{2}+i\varepsilon)((k-p)^{2}-m^{2}+i\varepsilon)}\,.$
(B.3)

Comparing both expressions, we obtain the expression for the integral
$I_{3}^{\pm}(k)$ through the integral $I_{1}(k^{2})$

$I_{3}^{\pm}(k)=-\frac{2ik^{\pm}}{m}I_{1}(k^{2})-I_{3}^{\pm}(k)\rightarrow
I_{3}^{\pm}(k)=\frac{ik^{\pm}}{m}I_{1}(k^{2})\,.$ (B.4)

Thus, we derive the following expression for the loop integral
$I_{3}^{\pm}(k)$

$I_{3}^{\pm}(k)=\frac{imk^{\pm}}{2\pi
k^{2}}\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}\log\frac{1+i\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}}{1-i\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}}\,.$
(B.5)

###  B.3 $I_{2}$ calculation

Let us calculate the loop integral $I_{2}^{\pm}(p)$ corresponding to the loops
formed by two mixed lines

$\displaystyle
I^{+-}_{2}(k^{2})=\int\frac{d^{2}p}{(2\pi)^{2}}\frac{4p^{\mp}(k-p)^{\pm}}{(p^{2}-m^{2}+i\varepsilon)((k-p)^{2}-m^{2}+i\varepsilon)}=$
(B.6)
$\displaystyle=\int\frac{d^{2}p}{(2\pi)^{2}}\left(\frac{4p^{\mp}k^{\pm}}{(p^{2}-m^{2}+i\varepsilon)((k-p)^{2}-m^{2}+i\varepsilon)}-\frac{p^{2}}{(p^{2}-m^{2}+i\varepsilon)((k-p)^{2}-m^{2}+i\varepsilon)}\right)=$
$\displaystyle=I_{2a}^{+-}(k^{2})-I_{2b}^{+-}(k^{2})\,,$

where take into account that $p^{2}=4p^{+}p^{-}$.

Consider the contributions of terms to the $I_{2}^{+-}(k^{2})$ separately. Let
us take the second integral (see the calculation of $I_{3}(k)$)

$I^{+-}_{2a}(k^{2})=\frac{1}{4\pi}\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}\log\frac{1+i\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}}{1-i\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}}\,.$
(B.7)

The second contribution has logarithmic divergence. We regularized this
integral using dimensional regularization 444In calculating the integrals we
carried out the Wick rotation defined as
$p^{0}\xrightarrow[]{Euclid}ip_{E}^{0}\,,\,d^{2}p\xrightarrow[]{Euclid}id^{2}p_{E}\,,\,(p_{E},x_{E})=-2(p_{E}^{+}x_{E}^{-}+p_{E}^{-}x_{E}^{+})\,,\;p_{E}^{2}=-4p_{E}^{+}p_{E}^{-}=-p^{2}\,.$

$\displaystyle
I_{2b}^{+-}(k^{2})=\int\frac{d^{2}p}{(2\pi)^{2}}\frac{p^{2}}{(p^{2}-m^{2}+i\varepsilon)((k-p)^{2}-m^{2}+i\varepsilon)}\xrightarrow[]{Euclid}$
(B.8)
$\displaystyle\xrightarrow[]{Euclid}-i\int\frac{\mu^{2\epsilon}d^{2-2\epsilon}p_{E}}{(2\pi)^{2-2\epsilon}}\,\frac{1}{(k-p)_{E}^{2}+m^{2}}-I_{1}(k_{E}^{2})\xrightarrow[]{Mink}\frac{m^{2}}{2\pi
k^{2}}\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}\log\frac{1+i\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}}{1-i\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}}-$
$\displaystyle-\frac{i}{4\pi}\left(\frac{1}{\epsilon}-\gamma_{E}+\log(4\pi)-\log\frac{m^{2}}{\mu^{2}}+\mathcal{O}(\epsilon)\right)\,.$

Therefore, the one-loop integral $I_{2}^{+-}(k^{2})$ takes the form

$\displaystyle
I_{2}^{+-}(k^{2})=\frac{1}{4\pi}\left(1-\frac{2m^{2}}{k^{2}}\right)\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}\log\frac{1+i\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}}{1-i\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}}-$
(B.9)
$\displaystyle+\frac{i}{4\pi}\left(\frac{1}{\epsilon}-\gamma_{E}+\log(4\pi)-\log\frac{m^{2}}{\mu^{2}}+\mathcal{O}(\epsilon)\right)\,.$

Let us calculate the integral $I_{2}^{\pm\pm}(k)$

$\displaystyle
I_{2}^{\pm\pm}(k)=\int\frac{d^{2}p}{(2\pi)^{2}}\frac{4p^{\pm}(k-p)^{\pm}}{(p^{2}-m^{2}+i\varepsilon)((k-p)^{2}-m^{2}+i\varepsilon)}\xrightarrow[]{Euclid}\int\frac{d^{2}p_{E}}{(2\pi)^{2}}\frac{4ip_{E}^{\pm}(k-p)_{E}^{\pm}}{(p_{E}^{2}+m^{2})((k-p)_{E}^{2}+m^{2})}=$
(B.10)
$\displaystyle=\int\limits_{0}^{1}dx\,\int\frac{d^{2}p_{E}}{(2\pi)^{2}}\frac{4ip_{E}^{\pm}(k-p)_{E}^{\pm}}{(p_{E}^{2}-2(p,k)_{E}x+k_{E}^{2}x+m^{2})^{2}}=\int\limits_{0}^{1}dx\,\int\frac{d^{2}p_{E}}{(2\pi)^{2}}\frac{4i(p_{E}^{\pm}+k_{E}^{\pm}x)(k_{E}^{\pm}(1-x)-p_{E}^{\pm})}{(p_{E}^{2}+k_{E}^{2}x(1-x)+m^{2})^{2}}=$
$\displaystyle=\int\limits_{0}^{1}dx\,\int\frac{d^{2}p_{E}}{(2\pi)^{2}}\frac{4i(k_{E}^{\pm})^{2}x(1-x)}{(p_{E}^{2}+k_{E}^{2}x(1-x)+m^{2})^{2}}=\frac{i(k_{E}^{\pm})^{2}}{\pi
k_{E}^{2}}+\frac{2m^{2}(k_{E}^{\pm})^{2}}{\pi
k_{E}^{4}}\sqrt{\frac{k_{E}^{2}}{-4m^{2}-k_{E}^{2}}}\log\frac{1+i\sqrt{\frac{k_{E}^{2}}{-4m^{2}-k_{E}^{2}}}}{1-i\sqrt{\frac{k_{E}^{2}}{-4m^{2}-k_{E}^{2}}}}\,.$

Thus, the integral $I_{2}^{\pm\pm}(k)$ in Minkowski signature has the form

$I_{2}^{\pm\pm}(k)=-\frac{i(k^{\pm})^{2}}{\pi
k^{2}}+\frac{2m^{2}(k^{\pm})^{2}}{\pi
k^{4}}\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}\log\frac{1+i\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}}{1-i\sqrt{\frac{k^{2}}{4m^{2}-k^{2}}}}\,.$
(B.11)

###  B.4 $J_{1,2,3}$ integral calculation

The integral $J_{0}(k^{2})$ is similar to the sunset integral for the
$\phi^{4}$ theory [22]

$\displaystyle
J_{0}(k^{2})=-\int\frac{d^{2}p_{1}}{(2\pi)^{2}}\frac{d^{2}p_{1}}{(2\pi)^{2}}\,\frac{im^{3}}{(p_{1}^{2}-m^{2}+i\varepsilon)(p_{2}^{2}-m^{2}+i\varepsilon)((k-p_{1}-p_{2})^{2}-m^{2}+i\varepsilon)}=$
(B.12) $\displaystyle=-im^{3}(3m^{2}A(k^{2})+B(k^{2}))\,,$

where

$\displaystyle
A(k^{2})=\frac{\Gamma(2)}{(4\pi)^{2}}\int\limits_{0}^{1}dx\,\frac{1}{x(1-x)}\int\limits_{0}^{1}dy\,\frac{y}{\left(-k^{2}y(1-y)+m^{2}\left(y+\frac{1-y}{x(1-x)}\right)\right)^{2}}\,,$
(B.13) $\displaystyle
B(k^{2})=-k^{2}\frac{\Gamma(1)}{(4\pi)^{2}}\int\limits_{0}^{1}dx\,\frac{1}{x(1-x)}\int\limits_{0}^{1}dy\,y(1-y)\frac{1}{\left(-k^{2}y(1-y)+m^{2}\left(1-y+\frac{y}{x(1-x)}\right)\right)^{2}}\,.$

The integral $J_{1}(k)$ can be expressed in terms of the integral
$J_{0}(k^{2})$

$J_{1}^{\pm}(k)=-\frac{2i}{3}\frac{k^{\pm}}{m}J_{0}(k^{2})\,.$ (B.14)

Similar procedure for the integral $J_{2}^{+-}(k^{2})$ yields

$J_{2}^{+-}(k^{2})=-\frac{k^{2}-3m^{2}}{6m^{2}}J_{0}(k^{2})+\frac{i}{2m}\left(\int\frac{d^{2}p}{(2\pi)^{2}}\frac{im}{p^{2}-m^{2}}\right)^{2}\,.$
(B.15)

The third sunset integral can be calculated using the Feynman parameter
technique

$\displaystyle
J_{3}^{\pm\pm\mp}(k)=\int\frac{d^{2}p_{1}}{(2\pi)^{2}}\frac{d^{2}p_{1}}{(2\pi)^{2}}\,\frac{8p_{1}^{\pm}p_{2}^{\pm}(k-p_{1}-p_{2})^{\mp}}{(p_{1}^{2}-m^{2}+i\varepsilon)(p_{2}^{2}-m^{2}+i\varepsilon)((k-p_{1}-p_{2})^{2}-m^{2}+i\varepsilon)}\,.$
(B.16)

Consider individual contributions to the integral $J_{3}^{\pm\pm\mp}(k)$
integral. Let us calculate the following term

$\displaystyle
J_{a}(k)=\int\frac{d^{2}p_{1E}}{(2\pi)^{2}}\frac{d^{2}p_{2E}}{(2\pi)^{2}}\frac{8k^{\mp}p_{1}^{\pm}p_{2}^{\pm}}{(p_{1}^{2}-m^{2}+i\varepsilon)(p_{2}^{2}-m^{2}+i\varepsilon)((k-p_{1}-p_{2})^{2}-m^{2}+i\varepsilon)}\xrightarrow[]{Euclid}$
(B.17)
$\displaystyle\xrightarrow[]{Euclid}\int\frac{d^{2}p_{1E}}{(2\pi)^{2}}\frac{d^{2}p_{2E}}{(2\pi)^{2}}\frac{8k_{E}^{\mp}p_{1E}^{\pm}p_{2E}^{\pm}}{(p_{1E}^{2}+m^{2})(p_{2E}^{2}+m^{2})((k-p_{1}-p_{2})_{E}^{2}+m^{2})}=$
$\displaystyle=8k_{E}^{\mp}\int\frac{d^{2}p_{2E}}{(2\pi)^{2}}\frac{p_{2E}^{\pm}}{p^{2}_{2E}+m^{2}}\int\limits_{0}^{1}dx\int\frac{d^{2}p_{1E}}{(2\pi)^{2}}\frac{p_{1E}^{\pm}}{(p_{1E}^{2}-2(p_{1},q)_{E}x+q^{2}x+m^{2})^{2}}=$
$\displaystyle=8k_{E}^{\mp}\int\limits_{0}^{1}dx\,\frac{1}{4\pi(1-x)}\int\limits_{0}^{1}dy\int\frac{d^{2}p_{2E}}{(2\pi)^{2}}\frac{p_{2E}^{\pm}(k-p_{2})_{E}^{\pm}}{\left(k_{2E}^{2}-2(p_{2},k)_{E}y+k_{E}^{2}y+M^{2}(x,y)\right)^{2}}\,,$

where we defined the function depending only on Feynman parameters

$M^{2}(x,y)=\frac{ym^{2}}{x(1-x)}+m^{2}-m^{2}y\,.$ (B.18)

The integral over the loop momentum $p_{2E}$ is equal to the expression

$\displaystyle\int\frac{d^{2}p_{2E}}{(2\pi)^{2}}\frac{p_{2E}^{\pm}k_{E}^{\pm}-(p_{2E}^{\pm})^{2}}{\left(p_{2E}^{2}-2(p_{2},k)_{E}y+k_{E}^{2}y+M^{2}(x,y)\right)^{2}}=\int\frac{d^{2}p_{2E}}{(2\pi)^{2}}\frac{(p_{2E}^{\pm}+k_{E}^{\pm}y)(k_{E}^{\pm}(1-y)-p_{2E}^{\pm})}{\left(p_{2E}^{2}+k^{2}_{E}y(1-y)+M^{2}(x,y)\right)^{2}}=$
(B.19)
$\displaystyle=\int\frac{d^{2}p_{2E}}{(2\pi)^{2}}\frac{(k_{E}^{+})^{2}\,y(1-y)}{\left(p_{2E}^{2}+k^{2}_{E}y(1-y)+M^{2}(x,y)\right)^{2}}=\frac{(k_{E}^{+})^{2}y(1-y)}{4\pi}\frac{1}{k_{E}^{2}y(1-y)+M^{2}(x,y)}\,.$

Therefore, $J_{a}(k)$ in Minkowski signature has the following form

$\displaystyle\frac{8k^{-}(k^{+})^{2}}{16\pi^{2}}\int\limits_{0}^{1}dx\int\limits_{0}^{1}dy\,\frac{y(1-y)}{1-x}\frac{1}{-k^{2}y(1-y)+M^{2}(x,y)}$
(B.20)

Consider the contribution to $J_{3}^{\pm\pm\mp}(k)$

$\displaystyle
J_{b}(k)=-\int\frac{d^{2}p_{1}}{(2\pi)^{2}}\frac{d^{2}p_{2}}{(2\pi)^{2}}\frac{2p_{1}^{2}p_{2}^{\pm}}{(p_{1}^{2}-m^{2}+i\varepsilon)(p_{2}^{2}-m^{2}+i\varepsilon)((k-p_{1}-p_{2}+i\varepsilon)^{2}-m^{2})}=$
(B.21)
$\displaystyle=-\underbrace{\int\frac{d^{2}p_{1}}{(2\pi)^{2}}\frac{d^{2}p_{2}}{(2\pi)^{2}}\frac{2p_{2}^{\pm}}{(p_{2}^{2}-m^{2})((k-p_{1}-p_{2})^{2}-m^{2})}}_{0}+J_{1}^{+}(k)\,.$

Taking into account the symmetry of $J_{3}^{\pm\pm\mp}(k)$ with respect to the
replacement $p_{1}\leftrightarrow p_{2}$ the given contribution $J_{b}(k)$
enters into $J_{3}^{\pm\pm\mp}(k)$ twice.

Thus we obtain the following expression for integral $J_{3}^{\pm\pm\mp}(k)$ in
Minkowski signature

$\displaystyle
J_{3}^{\pm\pm\mp}(k)=\frac{k^{\pm}(k)^{2}}{8\pi^{2}}\int\limits_{0}^{1}dx\,\frac{1}{1-x}\int\limits_{0}^{1}dy\,\frac{y(1-y)}{-k^{2}y(1-y)+M^{2}(x,y)}+2J_{1}^{\pm}(k)\,.$
(B.22)

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