${\cal N}=4$ SYM Quantum Spectral Curve in BFKL regime

Let us start from briefly describing the relation of the BFKL spectrum in $\mathcal{N}=4$ SYM to those in QCD. The key statement governing this connection is the so-called principle of maximal transcendentality Kotikov:2000pm ; Kotikov:2002ab . One can find a detailed description of this principle in Kotikov:2019xyg . Below we demonstrate this principle in application to the BFKL kernel eigenvalues. In what follows we use the extension of the formula (1) for the case of non-zero conformal spin $n$

	$\displaystyle S$	$\displaystyle=$	$\displaystyle-1+g^{2}\chi_{\textrm{LO}}(\Delta,n)+g^{4}\chi_{\textrm{NLO}}(\Delta,n)+$
		$\displaystyle+$	$\displaystyle g^{6}\chi_{\textrm{NNLO}}(\Delta,n)+g^{8}\chi_{\textrm{NNNLO}}(\Delta,n)+\mathcal{O}(g^{10})\;.$

Consider first the LO BFKL Pomeron kernel eigenvalues for QCD and $\mathcal{N}=4$ SYM  which can be found in Kuraev:1977fs ; Balitsky:1978ic and Kotikov:2000pm respectively. We see that the answers are identical and are given by

$$\chi_{\textrm{LO}}(\Delta,n)=-4\left(\psi\left(\frac{1+n+\Delta}{2}\right)+\psi\left(\frac{1+n-\Delta}{2}\right)-2\psi(1)\right)\;,$$

(3)

where $\Delta$ is the dimension of the state and $n$ is the conformal spin. Moreover, in the LO the BFKL kernel eigenvalues coincide for all 4D gauge theories Kotikov:2000pm .

For the next order in perturbation theory the situation is more subtle. The NLO BFKL kernel eigenvalue⁴⁴4Our identification of the variable $\gamma$ from Kotikov:2000pm with our parameters is: $\gamma=(\Delta-S)/2$, where $\Delta$ is the dimension and $S$ is the spin. for QCD was calculated in Fadin:1998py

			$\displaystyle\chi_{\textrm{NLO}}^{\textrm{QCD}}(\Delta,n)=4\left[-2\Phi\left(n,\frac{1-\Delta}{2}\right)-2\Phi\left(n,\frac{1+\Delta}{2}\right)+6\zeta(3)+\right.$
		$\displaystyle+$	$\displaystyle\left(\frac{67}{9}-2\zeta(2)-\frac{10n_{f}}{9N_{c}}\right)\frac{\chi_{\textrm{LO}}(\Delta,n)}{4}+\psi^{\prime\prime}\left(\frac{1+n-\Delta}{2}\right)+\psi^{\prime\prime}\left(\frac{1+n+\Delta}{2}\right)-$
		$\displaystyle-$	$\displaystyle\frac{1}{2}\left(\frac{11}{3}-\frac{2}{3}\frac{n_{f}}{N_{c}}\right)\left(\frac{\chi_{\textrm{LO}}^{2}(\Delta,n)}{16}-\psi^{\prime}\left(\frac{1+n+\Delta}{2}\right)+\psi^{\prime}\left(\frac{1+n-\Delta}{2}\right)\right)+$
		$\displaystyle+$	$\displaystyle\frac{\pi^{2}\cos\frac{\pi(1+\Delta)}{2}}{\Delta\sin^{2}\frac{\pi(1+\Delta)}{2}}\left.\left(\left(3+\left(1+\frac{n_{f}}{N_{c}^{3}}\right)\frac{3\Delta^{2}-11}{4(\Delta^{2}-4)}\right)\delta^{0}_{n}-\left(1+\frac{n_{f}}{N_{c}^{3}}\right)\frac{\Delta^{2}-1}{8(\Delta^{2}-4)}\delta^{2}_{n}\right)\right]\;,$

where $N_{c}$ is the number of colors and $n_{f}$ is the number of flavours of the fermions in the considered version of QCD. The same quantity in $\mathcal{N}=4$ SYM was first presented in Kotikov:2000pm . We write it down here from Kotikov:2002ab

	$\displaystyle\chi_{\textrm{NLO}}^{\mathcal{N}=4}(\Delta,n)$	$\displaystyle=$	$\displaystyle 4\left[-2\Phi\left(n,\frac{1-\Delta}{2}\right)-2\Phi\left(n,\frac{1+\Delta}{2}\right)+6\zeta(3)+\right.$
		$\displaystyle+$	$\displaystyle\left.\frac{1}{4}\left(\frac{1}{3}-2\zeta(2)\right)\chi_{\textrm{LO}}(\Delta,n)+\psi^{\prime\prime}\left(\frac{1+n-\Delta}{2}\right)+\psi^{\prime\prime}\left(\frac{1+n+\Delta}{2}\right)\right]\;.$

The function $\Phi(n,x)$ in the eigenvalues (2.1) and (2.1) is equal to

	$\displaystyle\Phi(n,x)$	$\displaystyle=$	$\displaystyle\sum\limits_{k=0}^{\infty}\frac{(-1)^{k+1}}{k+x+n/2}\left[\psi^{\prime}(k+n+1)-\psi^{\prime}(k+1)+(-1)^{k+1}(\beta^{\prime}(k+n+1)+\right.$		(6)
		$\displaystyle+$	$\displaystyle\left.\beta^{\prime}(k+1))+\frac{1}{k+x+n/2}\left(\psi(k+n+1)-\psi(k+1)\right)\right]$

and

$$\beta^{\prime}(z)=\frac{1}{4}\left[\psi^{\prime}\left(\frac{z+1}{2}\right)-\psi^{\prime}\left(\frac{z}{2}\right)\right]=\sum_{k=0}^{+\infty}\frac{(-1)^{k+1}}{(z+k)^{2}}\;.$$

(7)

Now we are going to explain the maximal transcendentality principle with the example of NLO BFKL kernel eigenvalue. There is a way to rewrite both $\chi_{\textrm{LO}}(\Delta,n)$ (3) and $\chi_{\textrm{NLO}}(\Delta,n)$ for both QCD (2.1) and $\mathcal{N}=4$ SYM (2.1) in terms of nested harmonic sums. In the LO we get simple result

$$\chi_{\textrm{LO}}(\Delta,n)=-4\left(S_{1}\left(\frac{1+\Delta+n}{2}-1\right)+S_{1}\left(\frac{1+n-\Delta}{2}-1\right)\right)\;,$$

(8)

from which we see that the LO BFKL kernel eigenvalues in QCD and $\mathcal{N}=4$ SYM have transcendentality $1$, where the transcendentality of a harmonic sum $S_{n_{1},\dots,n_{m}}$ is defined to be $|n_{1}|+\dots+|n_{m}|$, transcendentality of a product is given by a sum of the transcendentalities of the multipliers. In the same way one defines the transcendentality of the MZV $\zeta_{n_{1},\dots,n_{m}}$, which is given by the sum $n_{1}+\dots+n_{m}$. In particular, $\pi$ and $\log 2$ have transcedentality $1$.

To express (2.1) and (2.1) through nested harmonic sums we need to introduce some additional formulas. In Costa:2012cb it was shown, that for $n=0$ we can represent the NLO BFKL kernel eigenvalue for $\mathcal{N}=4$ SYM as follows

$$\chi_{\textrm{NLO}}^{\mathcal{N}=4}(\Delta,0)=F_{2}\left(\frac{1+\Delta}{2}\right)+F_{2}\left(\frac{1-\Delta}{2}\right)\;,$$

(9)

where

	$\displaystyle F_{2}(x)$	$\displaystyle=$	$\displaystyle 4\left(-\frac{3}{2}\zeta(3)+\pi^{2}\log 2+\frac{\pi^{2}}{3}S_{1}(x-1)+\right.$
		$\displaystyle+$	$\displaystyle\left.\pi^{2}S_{-1}(x-1)+2S_{3}(x-1)-4S_{-2,1}(x-1)\right)\;.$

Utilizing (9) and (2.1), we are able to rewrite (2.1) in the following way

	$\displaystyle\chi_{\textrm{NLO}}^{\mathcal{N}=4}(\Delta,n)$	$\displaystyle=$	$\displaystyle\frac{1}{2}\left(F_{2}\left(\frac{1+\Delta+n}{2}\right)+F_{2}\left(\frac{1-\Delta-n}{2}\right)+F_{2}\left(\frac{1+\Delta-n}{2}\right)+\right.$		(11)
		$\displaystyle+$	$\displaystyle\left.F_{2}\left(\frac{1-\Delta+n}{2}\right)\right)+4\left(R_{n}\left(\frac{1-\Delta}{2}\right)+R_{n}\left(\frac{1+\Delta}{2}\right)\right)\;,$

where

$$R_{n}(\gamma)=-8\left(S_{-2}\left(\gamma+\frac{n}{2}-1\right)+\frac{\pi^{2}}{12}\right)\left(S_{1}\left(\gamma+\frac{n}{2}-1\right)-S_{1}\left(\gamma-\frac{n}{2}-1\right)\right)\;.$$

(12)

Therefore, from (8), (9), (2.1), (11) and (12) we see that all the contributions to (11) have transcendentality $3$. Furthermore, we see that all the terms of the QCD result with the maximal transcendentality $3$ coincide exactly with the result of $\mathcal{N}=4$ SYM !

In the next part we will explain the relation of the BFKL kernel eigenvalues to the spectrum of length-2 (twist-2 in the case of zero conformal spin) operators in $\mathcal{N}=4$ SYM. Then we will explain how to adapt the QSC for the local operators to describe non-integer spin and conformal spin, which would then lead to the BFKL kernel eigenvalue.

In this part we describe the analytic structure of the length-2 operators with conformal spin in $\mathcal{N}=4$ SYM. Namely, we restrict ourselves to the states with the quantum numbers $J_{1}=2$, $J_{2}=J_{3}=0$. Let us first consider the case of zero conformal spin $S_{2}=0$, which are called the twist-2 $\mathfrak{sl}(2)$ operators

$$\mathcal{O}=\textrm{tr}ZD_{+}^{S}Z+(\textrm{permutations})\;.$$

(13)

In the gauge theory the physical operators should have even number of derivatives to be conformal primaries. The dimensions of these physical operators $\Delta$ were calculated for arbitrary even integer $S$ up to several loops order Lipatov:2001fs ; Kotikov:2002ab ; Kotikov:2003fb . In Gromov:2013pga it was understood how to describe these operators with integer spins within the QSC approach to the leading order in the perturbation theory. This technique was then generalised to the higher orders in Marboe:2014gma ; Marboe:2014sya ; Marboe:2016igj ; Marboe:2017dmb ; Marboe:2018ugv .

As we explain below in order to make a connection with the Regge regime one needs to be able to analytically continue in $S$. In the works GromovKazakovunpublished ; Janik:2013nqa it was understood how to remove the constraint of integer even spin $S$ from the integrable description, leading to the analytic continuation of the anomalous dimension of the twist-2 $\mathfrak{sl}(2)$ operators. At the 1-loop order one gets the following result Kotikov:2002ab ; Kotikov:2005gr

$$\Delta=2+S+8g^{2}\left(\psi(S+1)-\psi(1)\right)+\mathcal{O}(g^{2})\;.$$

(14)

As we can see the formula above (14) is regular for all $S>-1$, but has a simple pole at $S=-1$. In fact as now both $S$ and $\Delta$ are allowed to be non-integer we can invert the function for the range where it is regular and instead consider $S(\Delta)$ which is a more convenient function Kotikov:2002ab ; Brower:2006ea ; Costa:2012cb ; Costa:2013zra ; Brower:2013jga . In particular (14) leads to

$$S_{\Delta>1}(\Delta)=\Delta-2-8g^{2}\left(\psi(\Delta-1)-\psi(1)\right)+\mathcal{O}(g^{4})\;.$$

(15)

Below we describe the method which would allow us to compute this function in $\mathcal{N}=4$ SYM at finite coupling (see the Figure 1 for $n=0$). At the moment we only need to know its main features. We see that $S(\Delta)$ is an even function of $\Delta$. At finite $g$ it has a smooth parabolic shape, however, at weak coupling it becomes piece-wise linear: $S=|\Delta|-2$ for $|\Delta|>1$ and $S=-1$ for $|\Delta|<1$. Thus the $g\to 0$ limit breaks the analyticity and analytic continuation in $S$ does not commute with the $g\to 0$ limit. This implies that even though $S(\Delta)$ is a smooth function at finite $g$, it generates two different series expansions in $g$, depending on the range of $\Delta$. Within this picture the BFKL expansion is the series expansion of $S(\Delta)$ for $|\Delta|<1$ in powers of $g^{2}$. Note that all physical operators belong to the opposite region $\Delta>1$ (the physical operator with the smallest protected dimension $2$ is ${\rm tr}(Z^{2})$), and thus the equation (15) is only valid for $\Delta>1$. At the same time for $|\Delta|<1$ one can use (3) to get

$$S_{|\Delta|<1}(\Delta)=-1+4g^{2}\left(-\psi\left(\frac{1+\Delta}{2}\right)-\psi\left(\frac{1-\Delta}{2}\right)+2\psi(1)\right)+\mathcal{O}(g^{4})\;.$$

(16)

To understand how (15) and (16) are related to each other we have to complexify $\Delta$ and $S$. We explain below how this can be done using QSC at finite $g$, and the resulting function ${\rm Re}\;S(\Delta)$ is presented at the Figure 2. We see that the function $S(\Delta)$ has several sheets, which are connected through a branch cut. When coupling goes to zero the branch points collide forming the singularity in the function $S(\Delta)$. The domains $|\Delta|<1$ and $\Delta>1$ become separated completely by the branch cut dividing the $\Delta$ plane into two parts. Thus one can say that (15) and (16) are the expansions of the two branches of the same function to the left and to the right from the quadratic cut going to infinity along imaginary axis. This observation resolves the seeming paradox that the two functions are not the same. It also allows to make a prediction: the sum $S_{|\Delta|<1}(\Delta)+S_{\Delta>1}(\Delta)$ should be regular around $\Delta=1$ at any order in $g$, as in this combination the branch cut cancels. Indeed, one can check that the simple poles in the both functions around $\Delta=1$ has residues opposite in sign and disappear in the sum at order $g^{2}$. At the higher orders the poles at $\Delta=1$ become more and more severe, but this cancellation also can be verified explicitly to all known orders. This requirement gives non-trivial relations between the functions $S_{\Delta>1}(\Delta)$, obtained perturbatively as an analytic continuation from the dimensions of physical operators, and the BFKL kernel eigenvalue $S_{|\Delta|<1}(\Delta)$.

Now we are ready to turn to non-zero conformal spin. Namely, non-zero conformal spin adds the derivative in the orthogonal direction to the operators (13)

$$\mathcal{O}=\textrm{tr}ZD_{+}^{S_{1}}\partial_{\perp}^{S_{2}}Z+(\textrm{permutations})\;.$$

(17)

The physical states now correspond to non-negative integer $S_{1}$ and $S_{2}$, whose sum is even. We follow the same strategy for (17) as for the case of zero conformal spin. Analogously, having the anomalous dimensions for the physical operators, we can build the analytic continuation in the spins $S_{1}$ and $S_{2}$ and identify them with the spin $S$ and conformal spin $n$ respectively. This analytic continuation is illustrated with the Figure 1. The physical operators are designated with the dots on the operator trajectory. As in the case of zero conformal spin exchange of the roles of $\Delta$ and $S=S_{1}$ allows us to reach the BFKL regime.

Having these analytic properties in mind, we understand how to apply the QSC to the study of the BFKL spectrum: we need to analytically continue to non-integer spin $S$ and conformal spin $n$ not only the anomalous dimensions, but the QSC itself: Q-functions, their asymptotics and analytic structure etc. In the next section we are going to start from the brief description of the QSC basics. Then we will use this setup to consider the calculation of the LO BFKL kernel eigenvalue with non-zero conformal spin by the QSC method.

We are going to present here the formulation of the QSC in terms of the Q-system and gluing conditions, the details of which can be found in Gromov:2014caa ; Gromov:2017blm ; Alfimov:2018cms . Algebraic part of the QSC framework is described as follows. For the $\mathcal{N}=4$ SYM we have the system of $2^{8}$ Q-functions, which are denoted as

$$Q_{a_{1},\ldots,a_{n}|i_{1},\ldots,i_{m}}(u)\;,\quad 1\leq n,m\leq 4.$$

(18)

The Q-functions (18) have two groups of indices: $a$’s are called “bosonic” and $i$’s are called “fermionic”. They are antisymmetric with respect to the exchange of any pair of indices in these two groups. Not all of the 256 Q-functions in question are independent. They are subject to the set of the so-called Plücker’s QQ-relations, which are written in Gromov:2014caa

	$\displaystyle Q_{A|I}Q_{Aab|I}$	$\displaystyle=$	$\displaystyle Q_{Aa|I}^{+}Q_{Ab|I}^{-}-Q_{Aa|I}^{-}Q_{Ab|I}^{+}\;,$		(19)
	$\displaystyle Q_{A|I}Q_{A|Iij}$	$\displaystyle=$	$\displaystyle Q_{A|Ii}^{+}Q_{A|Ij}^{-}-Q_{Aa|Ii}^{-}Q_{Ab|Ij}^{+}\;,$
	$\displaystyle Q_{Aa|I}Q_{A|Ii}$	$\displaystyle=$	$\displaystyle Q_{Aa|Ii}^{+}Q_{Ab|I}^{-}-Q_{Aa|Ii}^{-}Q_{A|I}^{+}\;,$

where $A$ and $I$ are multi-indices from the set $\{1,2,3,4\}$.

The standard normalization is chosen to be

$$Q_{\emptyset|\emptyset}=1\;.$$

(20)

Then the structure of the QQ-relations allows to express the whole Q-system in terms of the “basic” set of $8$ Q-functions $Q_{a|\emptyset}$, $a=1,\ldots,4$ and $Q_{\emptyset|i}$, $i=1,\ldots,4$.

Let us now describe two symmetries of the Q-system, which respect the QQ-relations (19).

• •

  Imposing the so-called unimodularity condition

  $$Q_{1234|1234}=1\;,$$

  (21)

  we are able to introduce the system of Hodge-dual Q-functions

  $$Q^{a_{1}\ldots a_{n}|i_{1}\ldots i_{m}}\equiv(-1)^{nm}\epsilon^{b_{n+1}\ldots b_{4}a_{1}\ldots a_{n}}\epsilon^{j_{m+1}\ldots j_{4}i_{1}\ldots i_{m}}Q_{b_{n+1}\ldots b_{4}|j_{m+1}\ldots j_{4}}\;,$$

  (22)

  where there is no summation over the repeated indices and which satisfy the same QQ-relations (19). The condition (21) allows to write down the following relations between the Q-functions with the lower and upper indices

  	$\displaystyle Q^{a|i}Q_{b|j}=-\delta^{i}_{j}\;,$		$\displaystyle Q^{a|i}Q_{b|i}=-\delta^{a}_{b}\;,$		(23)
  	$\displaystyle Q^{a|\emptyset}=(Q^{a|i})^{+}Q_{\emptyset|i}\;,$		$\displaystyle Q^{\emptyset|i}=(Q^{a|i})^{+}Q_{a|\emptyset}\;,$
  	$\displaystyle Q_{a|\emptyset}Q^{a|\emptyset}=0\;,$		$\displaystyle Q_{\emptyset|i}Q^{\emptyset|i}=0\;.$

• •

  Another symmetry is called the H-symmetry. Its general form is given by

  $$Q_{A|I}\rightarrow\sum\limits_{|B|=|A|,|J|=|I|}(H_{b}^{[|A|-|I|]})_{A}^{B}(H_{f}^{[|A|-|I|]})_{I}^{J}Q_{B|J}\;,$$

  (24)

  where the sum goes over the repeated multi-indices. The definition of $H_{I}^{J}$ is the following: $(H_{f})_{I}^{J}\equiv(H_{F}(u))_{i_{1}}^{j_{1}}(H_{F}(u))_{i_{2}}^{j_{2}}\ldots(H_{F}(u))_{i_{|I|}}^{j_{|I|}}$ and the same for $(H_{b})_{A}^{B}$ and $(H_{B}(u))_{a_{k}}^{b_{k}}$, $k=1,\ldots,|A|$ with $H_{B,F}(u)$ being $4\times 4$ $i$-periodic matrices. The unimodularity condition leads us to the restriction

  $$\det H_{B}(u)\det H_{F}(u)=1\;.$$

  (25)

To proceed with the calculation of the spectrum of $\mathcal{N}=4$ SYM we have to endow the described Q-system with the analytic structure. To start with, we designate the basic Q-functions with this structure as ${\bf P}_{a}$ (same as $Q_{a|\emptyset}$), ${\bf P}^{a}$ ($Q^{a|\emptyset}$), ${\bf Q}_{i}$ ($Q_{\emptyset|i}$) and ${\bf Q}^{i}$ ($Q^{\emptyset|i}$). One can find the asymptotics of the Q-functions in Gromov:2017blm

$${\bf P}_{a}\simeq A_{a}u^{-\tilde{M}_{a}}\;,\quad{\bf P}^{a}\simeq A^{a}u^{\tilde{M}_{a}-1}\;,\quad{\bf Q}_{i}\simeq B_{i}u^{\hat{M}_{i}-1}\;,\quad{\bf Q}^{i}\simeq B^{i}u^{-\hat{M}_{i}}\;,$$

(26)

where $\tilde{M}_{a}$, $a=1,\ldots,4$ and $\hat{M}_{i}$, $i=1,\ldots,4$ are expressed in terms of Cartan charges of ${\rm PSU}(2,2|4)$ (i.e. quantum numbers of the states)

	$\displaystyle\tilde{M}_{a}$	$\displaystyle=$	$\displaystyle\left\{\frac{J_{1}+J_{2}-J_{3}}{2}+1,\frac{J_{1}-J_{2}+J_{3}}{2},\frac{-J_{1}+J_{2}+J_{3}}{2}+1,\frac{-J_{1}-J_{2}-J_{3}}{2}\right\},$
	$\displaystyle\hat{M}_{i}$	$\displaystyle=$	$\displaystyle\left\{\frac{1}{2}\left(\Delta-S_{+}+2\right),\frac{1}{2}\left(\Delta+S_{+}\right),\frac{1}{2}\left(-\Delta-S_{-}+2\right),\frac{1}{2}\left(-\Delta+S_{-}\right)\right\}\;,$		(27)

where $S_{\pm}=S_{1}\pm S_{2}$.

As the Q-system is generated by the set of $8$ basic Q-functions, we first ascribe them the analytic structure dictated by the classical limit of the Q-functions Gromov:2017blm . The minimal choice of the cut structure consistent with the asymptotics (26) is presented on the Figure 3.

Then, according to the (19) we can generate two versions of the Q-system: upper half plane and lower half plane analytic (UHPA and LHPA respectively). In what follows we will define by $\mathcal{Q}_{a|i}$ the UHPA solution of the one of the QQ-relations from (19)

$$\mathcal{Q}_{a|i}^{+}-\mathcal{Q}_{a|i}^{-}={\bf P}_{a}{\bf Q}_{i}$$

(28)

with the large $u$ asymptotic

$$\mathcal{Q}_{a|i}\simeq-i\frac{A_{a}B_{i}}{-\tilde{M}_{a}+\hat{M}_{i}}u^{-\tilde{M}_{a}+\hat{M}_{i}}\;.$$

(29)

Substitution of the formulas ${\bf Q}_{i}=-\mathcal{Q}_{a|i}^{+}{\bf P}^{a}$ or ${\bf P}_{a}=-\mathcal{Q}_{a|i}{\bf Q}^{i}$ into (28) allows to fix the products of the coefficients of the leading asymptotics of the ${\bf P}$- and ${\bf Q}$-functions and for $a_{0},i_{0}=1,\ldots,4$ we get

$$A_{a_{0}}A^{a_{0}}=i\frac{\prod\limits_{j=1}^{4}\left(\tilde{M}_{a_{0}}-\hat{M}_{j}\right)}{\prod\limits_{\begin{subarray}{c}b=1\\ b\neq a_{0}\end{subarray}}\left(\tilde{M}_{a_{0}}-\tilde{M}_{b}\right)}\;,\quad B_{i_{0}}B^{i_{0}}=-i\frac{\prod\limits_{a=1}^{4}\left(\hat{M}_{i_{0}}-\tilde{M}_{a}\right)}{\prod\limits_{\begin{subarray}{c}j=1\\ j\neq i_{0}\end{subarray}}^{4}\left(\hat{M}_{i_{0}}-\hat{M}_{j}\right)}\;,$$

(30)

where there is no summation over the repeated indices $a_{0}$ and $i_{0}$.

An important consequence of the formula ${\bf Q}_{i}=-\mathcal{Q}_{a|i}{\bf P}^{a}$ combined with the formula (28) is the existence of a 4th order Baxter equation for the functions ${\bf Q}_{i}$, $i=1,\ldots,4$ (see Alfimov:2014bwa for the derivation)

	$\displaystyle{\bf Q}_{i}^{[+4]}$	$\displaystyle-$	$\displaystyle{\bf Q}_{i}^{[+2]}\left[D_{1}-{\bf P}_{a}^{[+2]}{\bf P}^{a[+4]}D_{0}\right]+{\bf Q}_{i}\left[D_{2}-{\bf Q}_{a}{\bf P}^{a[+2]}D_{1}+{\bf P}_{a}{\bf P}^{a[+4]}D_{0}\right]-$		(31)
		$\displaystyle-$	$\displaystyle{\bf Q}_{i}^{[-2]}\left[\bar{D}_{1}+{\bf P}_{a}^{[-2]}{\bf P}^{a[-4]}\bar{D}_{0}\right]+{\bf Q}_{i}^{[-4]}=0\;,$

where the functions $D_{0}$, $D_{1}$ and $D_{2}$ are given by

	$\displaystyle D_{0}$	$\displaystyle=$	$\displaystyle\left|\begin{array}[]{llll}{\bf P}^{1[+2]}&{\bf P}^{2[+2]}&{\bf P}^{3[+2]}&{\bf P}^{4[+2]}\\ {\bf P}^{1}&{\bf P}^{2}&{\bf P}^{3}&{\bf P}^{4}\\ {\bf P}^{1[-2]}&{\bf P}^{2[-2]}&{\bf P}^{3[-2]}&{\bf P}^{4[-2]}\\ {\bf P}^{1[-4]}&{\bf P}^{2[-4]}&{\bf P}^{3[-4]}&{\bf P}^{4[-4]}\end{array}\right|\;,\quad D_{1}=\left|(\begin{array}[]{llll}{\bf P}^{1[+4]}&{\bf P}^{2[+4]}&{\bf P}^{3[+4]}&{\bf P}^{4[+4]}\\ {\bf P}^{1}&{\bf P}^{2}&{\bf P}^{3}&{\bf P}^{4}\\ {\bf P}^{1[-2]}&{\bf P}^{2[-2]}&{\bf P}^{3[-2]}&{\bf P}^{4[-2]}\\ {\bf P}^{1[-4]}&{\bf P}^{2[-4]}&{\bf P}^{3[-4]}&{\bf P}^{4[-4]}\end{array}\right|,$		(40)
	$\displaystyle D_{2}$	$\displaystyle=$	$\displaystyle\left|\begin{array}[]{llll}{\bf P}^{1[+4]}&{\bf P}^{2[+4]}&{\bf P}^{3[+4]}&{\bf P}^{4[+4]}\\ {\bf P}^{1[+2]}&{\bf P}^{2[+2]}&{\bf P}^{3[+2]}&{\bf P}^{4[+2]}\\ {\bf P}^{1[-2]}&{\bf P}^{2[-2]}&{\bf P}^{3[-2]}&{\bf P}^{4[-2]}\\ {\bf P}^{1[-4]}&{\bf P}^{2[-4]}&{\bf P}^{3[-4]}&{\bf P}^{4[-4]}\end{array}\right|\;,$		(45)

while the bars over $\bar{D}_{1}$ and $\bar{D}_{2}$ are understood as the complex conjugation $\bar{f}(u)=\overline{f(\bar{u})}$ of the functions defined above. The same equation for ${\bf Q}^{i}$, $i=1,\ldots,4$ is valid, if we turn the functions ${\bf P}^{a}$ into ${\bf P}_{a}$ for $a=1,\ldots,4$.

One can see from (31) that if we are far away (high or low enough) from the real axis in the complex plane, the cut structures (see the Figure 3) of ${\bf P}$- and ${\bf Q}$-functions do not contradict each other. But as we approach the real axis, this is no longer the case and we have to cross the cut. This makes the QQ-relations ambiguous in the vicinity of the cuts, as the shifts by $\pm i/2$ may cross the cut and one may ask which contour to use to reach $u\pm i/2$ from the point $u$ or, in other words, which branch of the multi-valued function to use.

The way to resolve the analytic continuation ambiguity is to interpret the values of the ${\bf Q}$-function in the upper half plane, far enough from the cuts, as a ${\bf Q}$-function with the upper index ${\bf Q}^{i}$ and in the lower half plane the same ${\bf Q}$-function should obey the QQ-relations as if it was a function with lower index ${\bf Q}_{i}^{\uparrow}$. I.e. in the lower half plane this function satisfies (31), whereas in the upper half plane it satisfies the same equation with ${\bf P}_{a}$ interchanged with ${\bf P}^{a}$. The ${\uparrow}$ is added to indicate that ${\bf Q}_{i}^{{\uparrow}}$ does not have cuts in the lower half plane. To close the system of equations we notice that there is yet another way to build the set of functions ${\bf Q}_{i}^{{\uparrow}}$, satisfying (31) and having no cuts in the lower half plane. One can, starting from ${\bf Q}^{i}$, which has no cuts in the upper half plane and ${\bf P}^{a}$ – and build ${\bf Q}_{i}^{\downarrow}$ using QQ-relations. Then the complex conjugate of ${\bf Q}_{i}^{\downarrow}$ will satisfy (31) and will have no cuts in the lower half plane.

Indeed, due to the property, valid for real $S_{1}$, $S_{2}$ and $\Delta$ we can always assume that

$$\bar{{\bf P}}_{a}=C_{a}^{b}{\bf P}_{b}\;,\quad\bar{{\bf P}}^{a}=-C^{a}_{b}{\bf P}^{b}\;,\quad C=\textrm{diag}\{1,1,-1,-1\}$$

(46)

due to the H-symmetry. Thus if we complex conjugate the Baxter equation (31), we get the same equation for $\bar{{\bf Q}}_{j}$. This implies that ${\bf Q}_{i}^{\downarrow}(u)$ is a linear combination of $\bar{{\bf Q}}_{j}$ with regular coefficients

$${\bf Q}^{i}(u)=M^{ij}(u)\bar{{\bf Q}}_{j}(u)\;,\quad{\bf Q}_{i}(u)=\left(M^{-t}\right)_{ij}(u)\bar{{\bf Q}}^{j}(u)\;,$$

(47)

where $-t$ denotes the inversion and transposition of the matrix. Technically it will be more convenient to work with short cuts $[-2g,2g]$. We can connect the branch points of ${\bf Q}$ the way we like, but this would create an infinite ladder of cuts in the lower half plane. It will also modify the notion of the conjugate function, as the conjugation now will also involve the analytic continuation under the cut. So we conclude that in the short cut conventions the gluing condition reads as

$$\tilde{{\bf Q}}^{i}(u)=M^{ij}(u)\bar{{\bf Q}}_{j}(u)\;,\quad\tilde{{\bf Q}}_{i}(u)=\left(M^{-t}\right)_{ij}(u)\bar{{\bf Q}}^{j}(u)\;,$$

(48)

which we will use further. Let us now list the properties of $M^{ij}(u)$, which is called the gluing matrix. The details of the derivation of these properties can be found in Alfimov:2018cms . They are:

• •

  $M^{ij}(u)$ is an $i$-periodic matrix of H-transformation.

• •

  $M^{ij}(u)$ is analytic in the whole complex plane.

• •

  $M^{ij}(u)$ is hermitian as a function

  $$\bar{M}^{ij}(u)=M^{ji}(u)\;.$$

  (49)

There exist another additional symmetry of the ${\bf P}$-functions. The ${\bf P}$-functions of the states with the Cartan charges

	$\displaystyle\tilde{M}_{a}$	$\displaystyle=$	$\displaystyle\left\{2,1,0,-1\right\},$		(50)
	$\displaystyle\hat{M}_{i}$	$\displaystyle=$	$\displaystyle\left\{\frac{1}{2}\left(\Delta-S_{+}+2\right)\;,\frac{1}{2}\left(\Delta+S_{+}\right)\;,\frac{1}{2}\left(-\Delta-S_{-}+2\right)\;,\frac{1}{2}\left(-\Delta+S_{-}\right)\right\}\;,$

where $S_{\pm}\equiv S_{1}\pm S_{2}$, have the certain parity

$${\bf P}_{a}(-u)=(-1)^{a+1}{\bf P}_{a}(u)\;,\quad{\bf P}^{a}(-u)=(-1)^{a}{\bf P}^{a}(u)\;.$$

(51)

The conjugation symmetry and the symmetry above (51) lead to two additional constraints Alfimov:2018cms on the gluing matrix. Let us first concentrate on the physical states, when $S_{1}$ and $S_{2}$ are integer and have the same parity⁵⁵5This is dictated by the cyclicity condition for the states in the $\mathfrak{sl}(2)$ Heisenberg spin chain.. As it follows from the power-like large $u$ asymptotics of the ${\bf Q}$-functions in this case, the only possible ansatz for the gluing matrix is a constant matrix. In Gromov:2014caa ; Alfimov:2018cms it was shown that for the physical length-2 states with the charges (50) from the abovementioned properties of the gluing matrix and two additional constraints mentioned earlier it follows that

$$M^{ij}\left(e^{i\pi\left(\hat{M}_{i}-\hat{M}_{j}\right)}+1\right)=0\;.$$

(52)

From (52), we immediately see that only when the difference between the charges is an odd integer, $M^{ij}$ is non-zero. It is the case only for

	$\displaystyle\hat{M}_{1}-\hat{M}_{2}$	$\displaystyle=$	$\displaystyle-S_{1}-S_{2}+1\;,$		(53)
	$\displaystyle\hat{M}_{3}-\hat{M}_{4}$	$\displaystyle=$	$\displaystyle-S_{1}+S_{2}+1\;.$

Therefore, taking into account the hermiticity of the gluing matrix, for integer $S_{1}$ and $S_{2}$, that have the same parity⁶⁶6At one loop this is dictated by the cyclicity condition for the states in the $\mathfrak{sl}(2)$ Heisenberg spin chain, appearing in the perturbation theory. (i.e. for the physical states) the equations (52) lead to the gluing matrix

$$M^{ij}=\left(\begin{array}[]{cccc}0&M^{12}&0&0\\ \bar{M}^{12}&0&0&0\\ 0&0&0&M^{34}\\ 0&0&\bar{M}^{34}&0\end{array}\right)\;.$$

(54)

In the case of at least one of the spins $S_{1}$ and $S_{2}$ being non-integer, we see, that because all differences $\hat{M}_{i}-\hat{M}_{j}$ are non-integer the equations (52) can have only zero matrix as a solution, if this matrix is assumed to be constant. This leads us to the conclusion that the gluing matrix cannot be constant anymore for non-integer spins. The minimal way to do this keeping it $i$-periodic would be to add exponential contributions and get the following gluing matrix

	$\displaystyle M$	$\displaystyle=$	$\displaystyle\left(\begin{array}[]{cccc}M_{1}^{11}&M_{1}^{12}&M_{1}^{13}&M_{1}^{14}\\ \bar{M}_{1}^{12}&0&0&0\\ \bar{M}_{1}^{13}&0&M_{1}^{33}&M_{1}^{34}\\ \bar{M}_{1}^{14}&0&\bar{M}_{1}^{34}&M_{1}^{44}\\ \end{array}\right)+$		(59)
		$\displaystyle+$	$\displaystyle\left(\begin{array}[]{cccc}0&0&M_{2}^{13}&M_{2}^{14}\\ 0&0&0&0\\ \bar{M}_{2}^{13}&0&0&0\\ \bar{M}_{2}^{14}&0&0&0\\ \end{array}\right)e^{2\pi u}+\left(\begin{array}[]{cccc}0&0&M_{3}^{13}&M_{3}^{14}\\ 0&0&0&0\\ \bar{M}_{3}^{13}&0&0&0\\ \bar{M}_{3}^{14}&0&0&0\\ \end{array}\right)e^{-2\pi u}\;.$		(68)

To summarize, the key observation here is that if we want to consider non-physical states with $S_{1}$ or $S_{2}$ being non-integer, this inevitably requires modification of the gluing conditions. From now on let us use the notations for the spins coming from BFKL physics $S_{1}=S$ and $S_{2}=n$.

In the next Subsection we will briefly describe how the QQ-relations together with the gluing condition (48) can be used to compute numerically the function $S(\Delta,n)$.

The QQ-system together with the gluing conditions allows for the efficient numerical algorithm, developed in Gromov:2015wca . This method was described in details with the Mathematica code attached in a recent review Gromov:2017blm . Here we describe the main steps very briefly.

First, one notices that the ${\bf P}_{a}$ and ${\bf P}^{a}$ functions, $a=1,\ldots,4$, which have only one short cut $[-2g,2g]$ on the main sheet of the Riemann surface, can be parametrized very efficiently as rapidly converging series expansions in powers of $1/x(u)$, where $x(u)=(u+\sqrt{u-2g}\sqrt{u+2g})/(2g)$. After that one can recover the whole Q-system in terms of these expansion coefficients. For that one first uses

$$\mathcal{Q}_{a|i}^{+}-\mathcal{Q}_{a|i}^{-}=-{\mathcal{Q}}_{b|i}^{+}{\bf P}_{a}{{\bf P}}^{b}\;,$$

(69)

which follows directly from (28) and ${\bf Q}_{i}=-\mathcal{Q}_{a|i}{\bf P}^{i}$, to solve for $\mathcal{Q}_{a|i}$. After that one finds ${\bf Q}_{i}$ and $\tilde{\bf Q}_{i}$ from

$${{\bf Q}}_{i}=-{\mathcal{Q}}_{a|i}^{+}{{\bf P}}^{a}\;\;,\;\;\tilde{{\bf Q}}_{i}=-{\mathcal{Q}}_{a|i}^{+}\tilde{{\bf P}}^{a}\;.$$

(70)

Note that this involves $\tilde{{\bf P}}^{a}$, which is the same series as ${{\bf P}}^{a}$, but with $x\to 1/x$ for $u\in[-2g,2g]$. In this way we reconstruct both ${\bf Q}_{i}$ and $\tilde{\bf Q}_{i}$ in terms of the expansion coefficients in ${\bf P}_{a}$ and ${\bf P}^{a}$. Finally, one fixes these coefficients from the gluing condition (48).