States of twisted $\mathcal{N}=4$ SYM are not left-right symmetric for general twists, however for the particular case of operator tr$\phi_1^J$ and twists (4.7) indices can still be raised by $\chi$ if one also exchanges $\kappa$ with $\hat{\kappa}$:

$$ \mathbf{Q}^i = \chi^{ij} \mathbf{Q}_j \Big|_{\kappa \leftrightarrow \hat{\kappa}}, \qquad \mathbf{P}^a = \chi^{ab} \mathbf{P}_b \Big|_{\kappa \leftrightarrow \hat{\kappa}}. \qquad (C.2) $$

## C.2 Baxter equation from double scaling limit of QSC

In this section we will derive a finite-difference equation for $\mathbf{Q}_i(u)$ in the double scaling limit. Our starting point will be the Baxter equation (4.13). The computation is similar to that in [23]: we need to construct an ansatz for $\mathbf{P}_a$, expand it in the double scaling limit and plug into (4.13). The ansatz will necessarily contain many unknown coefficients which we will fix by solving (4.13) expanded at large $u$. In the process we will also get a relation between $\Delta$ and the coefficients of the ansatz for $\mathbf{P}_a$.

Let us start with an ansatz for $\mathbf{P}_a$: along the lines of [13], we pull out of $\mathbf{P}_a$ the exponential and power-like prefactors, leaving the part which scales like 1 at infinity:

$$ \begin{aligned} \mathbf{p}_a &= \{A_1 f_1(u), A_2 f_1(-u), A_3 g_1(u), A_4 g_1(-u)\} \\ \mathbf{p}^a &= \{f_2(u), f_2(-u), g_2(u), g_2(-u)\}. \end{aligned} \qquad (C.3) $$

Here we choose $A^a = 1$ and $A_a$ according to (4.9).

Remember that $\mathbf{P}_a$ and $\mathbf{P}^a$ have only one cut - Zhukovsky cut on the real axis. This cut can be resolved by considering $\mathbf{P}_a$ as a function of Zhukovsky variable $x(u)$. In other words, we can represent $f_i$, $g_i$ as series in $x(u)$:

$$ \begin{aligned} f_1 &= 1 + g^{2J} \sum_{n=1}^{\infty} \frac{g^{2n-2} c_{1,n}}{(gx)^n} \\ g_1 &= (gx)^{-J} \left( u^J + \sum_{k=0}^{J-1} c_{2,-k} u^k + \sum_{n=1}^{\infty} \frac{g^{2n} c_{2,n}}{(gx)^n} \right) \\ f_2 &= (gx)^{-J} \left( u^J + \sum_{k=0}^{J-1} c_{3,-k} u^k + \sum_{n=1}^{\infty} \frac{g^{2n} c_{3,n}}{(gx)^n} \right) \\ g_2 &= 1 + g^{2J} \sum_{n=1}^{\infty} \frac{g^{2n-2} c_{4,n}}{(gx)^n} \end{aligned} \qquad (C.4) $$

This ansatz was constructed so as to satisfy the condition that $\mathbf{P}_a$ stays finite as $g \to 0$ and $\tilde{\mathbf{P}}_a$ grows as $O(g^{-J})$. One can see that this is the case if we assume $c_{a,k} \sim 1$. Indeed, the operation tilde (monodromy around a branch point of the Zhukovsky cut) transforms $x(u)$ to $1/x(u)$. Since we want to keep $u = g(x + 1/x)$ finite in the weak coupling regime, when $x \sim 1/g \to \infty$, each power of $x$ should be compensated by at least one power of $g$. Plugging the ansatz (C.4) into (C.3) we get

$$ \begin{aligned} \mathbf{P}_1 &\sim (gx)^{-J/2}(1 + \dots), \\ \mathbf{P}_2 &\sim (gx)^{-J/2}(1 + \dots), \\ \mathbf{P}_3 &\sim (gx)^{-J/2}(\mathrm{Poly}(u) + \dots), \\ \mathbf{P}_4 &\sim (gx)^{-J/2}(\mathrm{Poly}(-u) + \dots), \end{aligned} \qquad (C.5) $$
where Poly($u$) is some polynomial in $u$. The scaling condition for $P_a$ — the finiteness in the weak coupling limit $g \to 0$ — is obviously satisfied. Now consider $\tilde{P}_a$:

$$ 
\begin{aligned}
\tilde{P}_1 & \sim g^{-J}(gx)^{J/2} \left( 1 + g^{2J-2} \sum_{n=1}^{\infty} c_{1,n}(gx)^n \right), \\
\tilde{P}_2 & \sim g^{-J}(gx)^{J/2} \left( 1 + g^{2J-2} \sum_{n=1}^{\infty} c_{1,n}(-gx)^n \right), \\
\tilde{P}_3 & \sim g^{-J}(gx)^{-J/2} \left( \text{Poly}(u) + \sum_{n=1}^{\infty} c_{2,n}(gx)^n \right), \\
\tilde{P}_4 & \sim g^{-J}(gx)^{-J/2} \left( \text{Poly}(-u) + \sum_{n=1}^{\infty} c_{2,n}(-gx)^n \right).
\end{aligned} 
\quad (C.6) $$

The scaling condition for $\tilde{P}_a$ is satisfied as well. To summarize: the scaling of the terms in $f_1, g_2$ is constrained by the scaling of $\tilde{P}$. The scaling of the polynomial terms in $g_1, f_2$ is constrained by finiteness of $P$ as $g \to 0$ and the scaling of the singular terms in $g_1, f_2$ — by $\tilde{P}$. We also want to have a consistency with the weak coupling result [13], that is why the potentially singular parts in $f_1, g_2, \tilde{g}_2, \tilde{f}_1$ have a $g^{2L}$ prefactor for the infinite sums.

In order to proceed we need to constrain the coefficients $c_{a,k}$ of the ansatz above as much as we can. We also want to be able to express $\Delta$ through $c_{a,n}$. To this end, we plug $P_a$ given by the ansatz into (4.13) and expand this equation at $u \to \infty$ assuming that the asymptotics of $Q_i$ are given by (4.6). This yields a system of equations for coefficients $c_{a,n}$ and $\Delta$. The structure of this system of equations depends on $L$. Typically we have to expand the Baxter equation to the fourth subleading order in $1/u$ to get a non-trivial relation for $\Delta$ and a closed system for $c_{a,n}$. We performed the calculations for $J = 2, 3, 4$.

For $J = 2$ and $J = 3$ the resulting relations for $\Delta$ have a form

$$ 
\begin{aligned}
(\Delta - 2)^2 &= f_2(c_{2,0}, c_{3,-1}, c_{4,1}, c_{4,2}, \kappa, \hat{\kappa}), \\
(\Delta - 2)^2 &= f_3(c_{2,-1}, c_{3,-2}, c_{4,1}, c_{4,2}, \kappa, \hat{\kappa}),
\end{aligned} 
\quad (C.7) $$

where $f_2, f_3$ are rational functions of their arguments given in appendix B.2.

Now we need to take the double scaling limit of the relations just derived and of the equation (4.13). Let us define (see (4.8))

$$ s = \sqrt{\kappa \hat{\kappa}}, \quad r = (\kappa/\hat{\kappa})^J, \quad \xi = gs. \quad (C.8) $$

The double scaling limit consists of taking $g \to 0$ and $s \to \infty$ when keeping $\xi$ constant. The parameter $r$ is not entering the Lagrangian (1.1), so we expect it to drop out from final result, although it may be present in the intermediate computations.

Each coefficient $c_{a,k}$ is a regular function of $g, \kappa, \hat{\kappa}$ and so it needs to be expanded in the powers of $g$:

$$ c_{a,k} = \sum_{m=0}^{\infty} c_{a,k,m}(\xi) (g/\xi)^{2m} \quad (C.9) $$
Then we plug these expanded coefficients back into the ansatz (C.4). The resulting expanded $\mathbf{p}_a$ for $J=3$ look as follows:

$$ 
\begin{aligned}
\mathbf{p}_1(u) &= 1 + \frac{4i - 2ir - rc_{3,-1,2}}{ru} (g/\xi)^4 \\
&\quad + \frac{4i\xi^J - 2ir\xi^J - r\xi^J c_{3,-1,2} + rc_{1,2,0}\xi^{3J}u + rc_{1,1,1}\xi^J u^2}{ru^3} (g/\xi)^6 + \dots \\
\mathbf{p}_2(u) &= \mathbf{p}_1(-u) \\
\mathbf{p}_3(u) &= 1 - \frac{2ir}{(r-1)u} - \frac{r(r+1)}{(r-1)^2 u^2} \\
&\quad + \left( \frac{c_{2,-1,1}}{u} + \frac{\sqrt{r}(r(\Gamma_0 - 4) - 4 - \Gamma_0)}{2(r-1)u^2} - \frac{4ir\sqrt{\xi}}{(r-1)u^3} - \frac{2r\sqrt{\xi}(r+1)}{(r-1)^2 u^4} \right) (g/\xi)^2 + \dots \\
\mathbf{p}_4(u) &= \mathbf{p}_3(-u)
\end{aligned}
\quad (C.10) $$

where

$$ (\Delta - 2)^2 = \Gamma_0 + O(g^2) \quad (C.11) $$

and

$$ A_1 = \frac{r}{r-1} - \frac{4\sqrt{r}}{(r-1)}(g/\xi)^2 + \frac{r+7}{(r-1)}(g/\xi)^4 + \dots \quad (C.12) $$

Now we are ready to find the equation for $\mathbf{Q}_i$ in the double-scaling limit. Plug the expanded $\mathbf{P}_a$ back into (4.13). To simplify the equation we define

$$ q_i(u) = \mathbf{Q}_i(u)u^{-J/2}. \quad (C.13) $$

Since the details of computations vary depending on $J$, we performed computations for $J=2, 3, 4$ and present the results below. In all cases the computations are performed with finite $r$ and we confirm that the final equation becomes non-singular as $r \to 0$.<fnref n="23" /> Notice that this was not the case for the intermediate quantities like (C.10). This allows us to set $r$ to 0, which makes more coefficients disappear. Notice that $r \to 0$ limit is exactly the parameter scaling leading to the Lagrangian (1.1), thus the fact that this limit is non-singular in the final equation is a good sign.

## D Details of the quasi-classical expansion

Here we give more orders for the WKB expansion discussed in section 8.6. It is convenient to define the functions via their large $\mathcal{D}$ expansion, instead of hypergeometric functions

$$ \mathcal{N}^{(a)} = \sum_{n=0}^{\infty} c_n^{(a)} \mathcal{D}^{-3n}, \quad (D.1) $$

<footnote n="23">²³We assume that the coefficients $c_{a,k,n}$ stay finite as $r \to 0$, which is confirmed by comparing with the perturbative solution.</footnote>
where

$$
\begin{align*}
c_n^{(0)} &= \frac{3\Gamma(3n)}{\mathfrak{d}(2n+1)\Gamma(n)\Gamma(n+1)^2}, \\
c_n^{(2)} &= \frac{3^{3n+\frac{9}{2}}(n+1)\Gamma(n+\frac{5}{3})\Gamma(n+\frac{7}{3})}{4\pi\mathfrak{d}^5(2n+3)\Gamma(n+1)\Gamma(n+3)}, \\
c_n^{(4)} &= \frac{(7n^3+39n^2+60n+24)\Gamma(3n+6)}{40\mathfrak{d}^6(2n+3)\Gamma(n+1)\Gamma(n+2)\Gamma(n+4)}, \\
c_n^{(6)} &= \frac{(279n^6+3991n^5+22511n^4+63257n^3+92122n^2+64512n+16128)\Gamma(3n+6)}{1680\mathfrak{d}^7(2n+3)\Gamma(n+1)\Gamma(n+2)\Gamma(n+5)}, \\
c_n^{(8)} &= \frac{3^{3n+\frac{11}{2}}p_n\Gamma(n+\frac{8}{3})\Gamma(n+\frac{10}{3})}{44800\pi\mathfrak{d}^8(2n+3)\Gamma(n+1)\Gamma(n+6)}, \tag{D.2}
\end{align*}
$$

where

$$
\begin{equation}
\begin{aligned}
p_n = 3429n^8 &+ 78737n^7 + 762569n^6 + 4040795n^5 + 12701786n^4 + 24004388n^3 \\
& + 26281256n^2 + 14927040n + 3225600.
\end{aligned}
\tag{D.3}
\end{equation}
$$

One can use the above expression to find the log singularity at $\mathfrak{d} = 3$. This singularity is controlled by the large $n$ asymptotics the above coefficients. The coefficient in front of the $\log(\mathfrak{d} - 3)$, computed in this way, and multiplied by $2\pi i$ gives the expansion around $\mathfrak{d} - 3$ for the $\beta$-cycle integral. It produces the following expansion for $\mathfrak{d}$ for fixed $N$ and large $\xi$ as explained in the section 8.6

$$
\begin{equation}
\begin{aligned}
\mathfrak{d} = e^{\pm 2i\pi/3} \left[ \right. & \frac{3 + \frac{\sqrt{3}N}{\bar{\xi}} + \frac{5(3N^2+5)}{36\bar{\xi}^2} + \frac{(11N^2+161)N}{108\sqrt{3}\bar{\xi}^3} + \frac{(129N^4+14754N^2+16781)}{34992\bar{\xi}^4}}{52488\sqrt{3}\bar{\xi}^5} \\
& - \frac{(57N^4-11706N^2-89599)N}{5668704\bar{\xi}^6} - \frac{(831N^6-131805N^4-5307099N^2-4448263)}{306110016\bar{\xi}^7} \\
& \left. + \frac{\sqrt{3}(1017N^6+280749N^4+99326923N^2+501673551)N}{344373768\bar{\xi}^8} \right],
\end{aligned}
\tag{D.4}
\end{equation}
$$

where $\bar{\xi} = \xi e^{\mp i\pi/6}$.

**Open Access.** 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.

References

[1] Ö. Gürdoğan and V. Kazakov, New Integrable 4D Quantum Field Theories from Strongly Deformed Planar N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 117 (2016) 201602 [arXiv:1512.06704] [INSPIRE].

[2] D. Correa, J. Henn, J. Maldacena and A. Sever, The cusp anomalous dimension at three loops and beyond, JHEP 05 (2012) 098 [arXiv:1203.1019] [INSPIRE].
[3] C. Sieg and M. Wilhelm, On a CFT limit of planar $\gamma_i$-deformed $N=4$ SYM theory, Phys. Lett. B 756 (2016) 118 [arXiv:1602.05817] [INSPIRE].

[4] A.B. Zamolodchikov, 'Fishnet' Diagrams as a Completely Integrable System, Phys. Lett. B 97 (1980) 63 [INSPIRE].

[5] N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].

[6] L.N. Lipatov, Asymptotic behavior of multicolor QCD at high energies in connection with exactly solvable spin models, JETP Lett. 59 (1994) 596 [hep-th/9311037] [INSPIRE].

[7] L.D. Faddeev and G.P. Korchemsky, High-energy QCD as a completely integrable model, Phys. Lett. B 342 (1995) 311 [hep-th/9404173] [INSPIRE].

[8] S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Noncompact Heisenberg spin magnets from high-energy QCD: 1. Baxter Q operator and separation of variables, Nucl. Phys. B 617 (2001) 375 [hep-th/0107193] [INSPIRE].

[9] S.E. Derkachov, G.P. Korchemsky, J. Kotanski and A.N. Manashov, Noncompact Heisenberg spin magnets from high-energy QCD. 2. Quantization conditions and energy spectrum, Nucl. Phys. B 645 (2002) 237 [hep-th/0204124] [INSPIRE].

[10] N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum Spectral Curve for Planar $N=4$ super-Yang-Mills Theory, Phys. Rev. Lett. 112 (2014) 011602 [arXiv:1305.1939] [INSPIRE].

[11] N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for arbitrary state/operator in $AdS_5/CFT_4$, JHEP 09 (2015) 187 [arXiv:1405.4857] [INSPIRE].

[12] N. Gromov and F. Levkovich-Maslyuk, Quantum Spectral Curve for a cusped Wilson line in $N=4$ SYM, JHEP 04 (2016) 134 [arXiv:1510.02098] [INSPIRE].

[13] V. Kazakov, S. Leurent and D. Volin, T-system on T-hook: Grassmannian Solution and Twisted Quantum Spectral Curve, JHEP 12 (2016) 044 [arXiv:1510.02100] [INSPIRE].

[14] J. Caetano, O. Gürdoğan and V. Kazakov, Chiral limit of $N=4$ SYM and ABJM and integrable Feynman graphs, arXiv:1612.05895 [INSPIRE].

[15] A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, Quantum Spectral Curve of the $N=6$ Supersymmetric Chern-Simons Theory, Phys. Rev. Lett. 113 (2014) 021601 [arXiv:1403.1859] [INSPIRE].

[16] D. Bombardelli, A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, The full Quantum Spectral Curve for $AdS_4/CFT_3$, JHEP 09 (2017) 140 [arXiv:1701.00473] [INSPIRE].

[17] D. Chicherin, V. Kazakov, F. Loebbert, D. Müller and D.-l. Zhong, Yangian Symmetry for Bi-Scalar Loop Amplitudes, arXiv:1704.01967 [INSPIRE].

[18] O. Mamroud and G. Torrents, RG stability of integrable fishnet models, JHEP 06 (2017) 012 [arXiv:1703.04152] [INSPIRE].

[19] B. Basso and L.J. Dixon, Gluing Ladder Feynman Diagrams into Fishnets, Phys. Rev. Lett. 119 (2017) 071601 [arXiv:1705.03545] [INSPIRE].

[20] C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, TBA, NLO Lüscher correction and double wrapping in twisted AdS/CFT, JHEP 12 (2011) 059 [arXiv:1108.4914] [INSPIRE].

[21] A. Gainutdinov, D. Ridout and I. Runkel, Logarithmic conformal field theory, J. Phys. A 46 (2013) 490301.
[22] J. Caetano, to be published.

[23] N. Gromov and F. Levkovich-Maslyuk, *Quark-anti-quark potential in $N=4$ SYM*, JHEP 12 (2016) 122 [arXiv:1601.05679] [INSPIRE].

[24] M. Alfimov, N. Gromov and V. Kazakov, *QCD Pomeron from AdS/CFT Quantum Spectral Curve*, JHEP 07 (2015) 164 [arXiv:1408.2530] [INSPIRE].

[25] S.E. Derkachov and A.N. Manashov, *Factorization of R-matrix and Baxter Q-operators for generic sl(N) spin chains*, J. Phys. A 42 (2009) 075204 [arXiv:0809.2050] [INSPIRE].

[26] S.E. Derkachov and A.N. Manashov, *Noncompact sl(N) spin chains: BGG-resolution, Q-operators and alternating sum representation for finite dimensional transfer matrices*, Lett. Math. Phys. 97 (2011) 185 [arXiv:1008.4734] [INSPIRE].

[27] N. Gromov, F. Levkovich-Maslyuk and G. Sizov, *Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS/CFT*, JHEP 06 (2016) 036 [arXiv:1504.06640] [INSPIRE].

[28] D.J. Broadhurst, *Evaluation of a Class of Feynman Diagrams for All Numbers of Loops and Dimensions*, Phys. Lett. B 164 (1985) 356 [INSPIRE].

[29] E. Panzer, *On the analytic computation of massless propagators in dimensional regularization*, Nucl. Phys. B 874 (2013) 567 [arXiv:1305.2161] [INSPIRE].

[30] B. Basso and G.P. Korchemsky, *Nonperturbative scales in AdS/CFT*, J. Phys. A 42 (2009) 254005 [arXiv:0901.4945] [INSPIRE].

[31] D. Chicherin, S. Derkachov and A.P. Isaev, *Conformal group: R-matrix and star-triangle relation*, JHEP 04 (2013) 020 [arXiv:1206.4150] [INSPIRE].

[32] N. Gromov, F. Levkovich-Maslyuk and G. Sizov, *Pomeron Eigenvalue at Three Loops in $N=4$ Supersymmetric Yang-Mills Theory*, Phys. Rev. Lett. 115 (2015) 251601 [arXiv:1507.04010] [INSPIRE].

[33] C. Marboe and D. Volin, *Quantum spectral curve as a tool for a perturbative quantum field theory*, Nucl. Phys. B 899 (2015) 810 [arXiv:1411.4758] [INSPIRE].

[34] S. Leurent and D. Volin, *Multiple zeta functions and double wrapping in planar $N=4$ SYM*, Nucl. Phys. B 875 (2013) 757 [arXiv:1302.1135] [INSPIRE].

[35] V. Gurarie, *Logarithmic operators in conformal field theory*, Nucl. Phys. B 410 (1993) 535 [hep-th/9303160] [INSPIRE].

[36] G.P. Korchemsky, *Quasiclassical QCD Pomeron*, Nucl. Phys. B 462 (1996) 333 [hep-th/9508025] [INSPIRE].

[37] G.P. Korchemsky, *Integrable structures and duality in high-energy QCD*, Nucl. Phys. B 498 (1997) 68 [hep-th/9609123] [INSPIRE].

[38] N. Gromov, F. Levkovich-Maslyuk and G. Sizov, *New Construction of Eigenstates and Separation of Variables for SU(N) Quantum Spin Chains*, JHEP 09 (2017) 111 [arXiv:1610.08032] [INSPIRE].

[39] I.M. Gel'fand and M.A. Naimark, *Unitary Representations of the Classical Groups*, Trudy Mat. Inst. Steklov. 36 Izdat. Nauk SSSR, Moscow - Leningrad, (1950), German transl.: Academie-Verlag, Berlin, Germany (1957).

[40] D.P. Zhelobenko, *Compact Lie Groups and Their Representations*, American Mathematical Society, Providence, R.I., U.S.A., (1973).
[41] S.E. Derkachov and A.N. Manashov, *R-Matrix and Baxter Q-Operators for the Noncompact SL(N, C) Invariant Spin Chain*, SIGMA **2** (2006) 084 [nlin/0612003].

[42] J.N. Bernstein, I.M. Gel'fand and S.I. Gel'fand, *Differential Operators on the Base Affine Space and a Study of g-Modules*, Lie Groups and Their Representations, Adam Hilger, London, U.K., (1975).

[43] N. Verma, *Structure of Certain Induced Representations of Complex Semisimple Lie Algebras*, Bull. Amer. Math. Soc. **74** (1968) 160.

[44] S. Winitzki, *Linear Algebra via Exterior Products*, http://www.lulu.com (2010), ISBN 978-1-4092-9496-2.

[45] Y.I. Manin, *Quantum Groups and non-Commutative Geometry*, Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, Canada. (1988).

[46] A. Chervov, G. Falqui and V. Rubtsov, *Algebraic Properties of Manin Matrices 1*, Adv. Appl. Math. **43** (2009) 239 [arXiv:0901.0235].

[47] D. Talalaev, *Quantization of the Gaudin system*, hep-th/0404153 [INSPIRE].

[48] A. Chervov and G. Falqui, *Manin matrices and Talalaev's formula*, J. Phys. A **41** (2008) 194006 [arXiv:0711.2236] [INSPIRE].

[49] F.L. Williams, *Tensor Products of Principal Series Representations*, Springer, (1973), [https://doi.org/10.1007/bfb0067484].