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The formfactors in 2d sine-Gordon (SG) model were studied using the free field representation in reflectionless points; based on the work of Lukyanov [1], we found the formfactors of the exponential operators $e^{ia\phi}$ and of their descendants $\partial^n \bar{\partial}^m e^{ia\phi}$ in the form of finite sums. We showed that the formfactors are factorized into thr product of universal transcendental part and operator-dependent rational part. We constructed a free field representation for the rational part of the formfactors. In this new representation we checked the merging of breathers, introduced screening operators to construct integrals of motion or derive equations of motion. We expect that this representation will connect the free field approach to formfactors developed by Lukyanov [1], to the other approach by Smirnov [2]
The SG model has the action
$$S[\varphi] = \int d^2x \left\{ \frac{1}{16\pi}(\partial_\mu\varphi)^2-2\mu\cos\beta\varphi \right\}$$
It is the integrable model of QFT as was shown by Zamolodchikov brothers [3]. The integrability of this model means the factorizing of any scattering into two particle scattering. It implies that one has to find the two-particle scattering matrix $S^{\sigma_1'\sigma_2'}_{\sigma_1\sigma_2} (|\theta_1-\theta_2|)$. Here $\sigma$ and $\theta$ respectively denote the topological charge and rapidity of the massive particles of the theory - kinks and antikinks. For kink $\sigma = 1$ and for antikink $\sigma = -1$. This matrix is known exactly [3] from the bootstrap approach and has the form
$$
S(\theta)=\left(\begin{array}{llll}
a(\theta) & & & \\
& b(\theta) & c(\theta) & \\
& c(\theta) & b(\theta) & \\
& & & a(\theta)
\end{array}\right)
$$
where the functions $a,b,c$ satisfy from the bootstrap
$$
b(\theta)=\frac{\operatorname{sh} \frac{\theta}{\xi}}{\operatorname{sh} \frac{i\pi-\theta}{\xi}} a(\theta), \quad
c(\theta)=\frac{\operatorname{sh} \frac{i\pi}{\xi}}{\operatorname{sh} \frac{i\pi-\theta}{\xi}} a(\theta)
$$
Here we introduced a new parameter $\xi = \frac{\beta^2}{1-\beta^2}$. The function $a(\theta)$ is obtained as the only solution by the assumptions of being meromorphic, having no poles on the physical sheet (except the imaginary axis) and satisfying $a(0) = -1$ (classical fact about kinks). This solution is then
$$
a(\theta) = \exp\left(2i\int_0^\infty \frac{dt}{t}\cdot \frac{\operatorname{sh}(\frac{\pi t}{2})\operatorname{sh}(\frac{\pi t}{2}(\xi+1))}{\operatorname{sh}(\pi t)\operatorname{sh}(\frac{\pi t}{2}\xi)} \sin(\theta t)\right)
$$
We see that this matrix is diagonal (i.e. there is no reflection) for the specific values of $\xi$, namely $\xi = \frac{1}{\nu}$, where $\nu \in \mathbb{Z}$. Those values are called reflectionless points. We study them in this work.
The formfactors are closely tied to the S-matrix. They differ from it in definition only by introducing some local operator $\mathcal{O}$. In this work we take $\mathcal{O} = e^{ia\phi}$. The formfactor is defined as
$$
F^{\mathcal{O}}_{\sigma_{2n}...\sigma_1}(\theta_{2n},...\theta_1) \equiv \langle{0}|\mathcal{O}(0)|{A_{\sigma_{2n}}(\theta_{2n})...A_{\sigma_1}(\theta_1)}\rangle_{\mathrm{in}}
$$
where $|{A_{\sigma_{2n}}(\theta_{2n})...A_{\sigma_1}(\theta_1)}\rangle$ is the multi soliton state. Knowing the formfactors allows one to calculate the correlation functions of the operator $\mathcal{O}$ which is of interest.
The S-matrix has first order poles for some values of $\theta$. In QFT a pole of S-matrix often means the presence of the bound state. For the SG model those bound states are called breathers, just like in the classical version of SG model.
Now, the governing function $a(\theta)$ in the S-matrix admits a free field representation by the bosonic field $\phi$ in such a way that $[\phi(\theta);\phi(\gamma)] = \ln a(\theta - \gamma)$. Expressing this field through oscillators as $\phi(\theta)= \int_{-\infty}^{+\infty}\frac{dt}{it}a_te^{i\theta t} $ we get the commutation relations for the generators $a_t$:
\eq$$
[a_t,a_{t'}] = t\cdot \frac{\operatorname{sh} \frac{\pi t}{2}\cdot\operatorname{sh} \frac{\pi t (\xi+1)}{2}}{\operatorname{sh} \pi t \cdot\operatorname{sh} \frac{\pi t \xi}{2}} \delta (t+t')
$$
Lukyanov's approach leads to the formula for the formfactor of exponential operator in terms of the new free bosonic field as
$$
F_{\sigma_{2n}...\sigma_1}(\theta_{2n},...\theta_1)= \mathcal{G}_a \langle\langle \mathcal{Z}_{\sigma_{2n}}(\theta_{2n})...\mathcal{Z}_{\sigma_{1}}(\theta_{1})\rangle \rangle
$$
Here $\mathcal{Z}_\pm (\theta)$ are the operators satisfying Faddeev-Zamolodchikov algebra and being expressed through field $\phi$.
We calculated the formfactors in case $\xi = \frac{1}{\nu}$ to be
$$
F_{+\ldots+-\ldots-}(\theta_{2n},...,\theta_1) =
\frac{\mathcal{G}_aC_+^{2n}}{\mathcal{N}^n}\cdot e^{\frac{a}{\beta}\sum_{k=1}^n \theta_{n+k}-\theta_k}e^{\frac{i\pi n}{2}\nu}
$$
$$
\prod_{k=1}^{n}\sum_{m_k=0}^\nu e^{-\left(\frac{2 a}{\beta}+\nu\right)\delta_{m_k}} w_{m_k} \prod_{l=1}^{n}c_-^{m_l}\prod_{j
$$
Y_+(z')Y^{(m)}_-(z)=\left(z'\over z\right)^{\nu+1\over4}h^{(m)}_{+-}\left(z\over z'\right): Y_+(z')Y^{(m)}_-(z):
$$
$$
Y^{(m')}_-(z')Y^{(m)}_-(z)=\left(z'\over z\right)^{\nu+1\over4}h^{(m'm)}_{--}\left(z\over z'\right): Y^{(m')}_-(z')Y^{(m)}_-(z):
$$
The formfactor in this representation is then
$$
F_\alpha(\theta_1,\ldots,\theta_{2N})_{\sigma_1\ldots\sigma_{2N}}
=\mathcal{G}_\alpha(\mathcal{N} C_+)^{2N}
$$
$$
\prod_{i
We studied breathers in this representation and introduced the screening currents to obtain the integrals of motion. The integral of motion of spin $s$ is
$$ I = _{31} \langle\mathbf{1}|\int du \cdot u^{s-(\nu+1)}\tilde{S}^{-1}(u)Y_+(z_{2n})...Y_+(z_{n+1})Y_-(z_n)...Y_-(z_1)\int dw \cdot w^{-\frac{2\nu+1}{\nu+1}} \cdot S(w) |\mathbf{1}\rangle_{13} $$
[1] S. L. Lukyanov, “Form factors of exponential fields in the sine-Gordon model,” Mod. Phys.
Lett. A12 (1997) 2543–2550, arXiv:hep-th/9703190.
[2] F. A. Smirnov, Form-factors in completely integrable models of quantum field theory, vol. 14.
1992.
[3] A. B. Zamolodchikov and Al. B. Zamolodchikov, “Factorized 𝑆 Matrices in Two-Dimensions
as the Exact Solutions of Certain Relativistic Quantum Field Models,” Annals Phys. 120
(1979) 253–291.
