N=4 SYM的BFKL谱：非零共形自旋的微扰框架

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"N $$ \mathcal{N} $$ = 4的BFKL谱：非零保形自旋" 这篇论文探讨的是N $$ \mathcal{N} $$ = 4超对称 Yang-Mills (SYM) 理论中的BFKL谱，特别是在非零保形自旋情况下的研究。BFKL谱是高能粒子散射理论中的一个重要概念，由Balitsky、Fadin、Kuraev和Lipatov提出，用于描述胶子交换过程中的渐近行为。在N $$ \mathcal{N} $$ = 4 SYM理论中，由于其超对称性，该模型具有许多理想化的特性，使得它成为理论物理学家研究强相互作用的一个重要工具。论文基于量子光谱曲线（Quantum Spectral Curve, QSC）建立了一个非微扰框架，这允许研究者对BFKL谱进行全局分析，扩展了之前仅限于特定共形自旋值的方法。QSC是一种强大的工具，用于处理强耦合领域的量子场论问题，尤其在计算渐近自由度方面表现突出。作者们利用这个框架重新推导了已知的BFKL Pomeron特征值的微扰结果，并且得到了新的预测。Pomeron是高能量散射中一个重要的概念，它代表了散射截面随着能量增加而增长的效应。他们特别关注了Lipatov自旋链的Faddeev-Korchemsky Baxter方程，这些方程与非零保形自旋下的BFKL核特征值相关联。此外，他们在保形自旋为| n | = 1且Δ= 0的点进行了非微扰分析，得到了BFKL截断函数的小耦合扩展以及至3环阶的4环阶部分结果。这些结果对于理解高能量过程中的强相互作用至关重要。论文还包括了数值算法的实现，这些算法作为arXiv提交的辅助文件提供。通过数值计算，作者们能够得到任意保形自旋的截断函数在强耦合展开下的解析公式，这是对BFKL理论的进一步拓展和深化。这篇论文为N $$ \mathcal{N} $$ = 4 SYM理论中的BFKL谱提供了新的理解和计算工具，特别是在非零保形自旋的情况下，这对于高能物理和量子场论的研究具有重要意义。

JHEP07(2018)181

(u), (2.26) and remembering the QQ-relation for Q-function with 4 “fermionic” indices

∅|1234

= det

1≤k,l≤4

[5−2l]

∅|k

one can show that

∅|1234

= det







[+3]

[+1]

[−1]

[−3]

···

i[+3]

i[+1]

i[−1]

i[−3]

···

[+3]

[+1]

[−1]

[−3]







←j-th row . (2.27)

First of all let us show that the matrix elements M

do not have any branch points. To

see this let us notice that for Im u < −3/2 the Q-function on the l.h.s. and the determinant

on the r.h.s. of (2.27) are analytic and do not have branch points. Therefore in the same

region and due to the i-periodicity M

has to be free from the branch points in the whole

complex plane.

Second, in principle, M

can have poles. As M

is i-periodic, the existence of a

pole, for example, in the point u

automatically leads to the inﬁnite number of poles in

the points u

+ ik, where k ∈ Z. However, at least in the region Im u < −3/2 the r.h.s.

of (2.27) is analytic, thus the poles of M

have to be compensated by the zeroes of

∅|1234

in the same points or, in other words, there exists k

such that

∅|1234

+ ik) = 0 for

k ≤ k

. In its turn this means that the number of zeroes of

∅|1234

is inﬁnite and these

zeroes accumulate at inﬁnity. We know that

∅|1234

has power-like asymptotic, then there

exist such a and b that

∅|1234

(u) − au

= O



b−1



, u → ∞. (2.28)

However, evaluating the l.h.s. of (2.28) at u = u

+ik for k ≤ k

leads to a contradiction with

the r.h.s. of (2.28). From this contradiction we conclude that M

cannot have any poles.

Summarizing what was said we see that the gluing matrix M

is analytic in the whole

complex plain. For the physical state, which means that the spins S

and S

are integer

and the asymptotics of the functions

are the power-like, analyticity and i-periodicity of

the gluing matrix M

leads us to the conclusion that it is constant in this case.

Now let us return back to the equations (2.26). As we know that the matrix M

free of any singularities, then analytically continuing both sides of the equations (2.26) to

the sheet with the short cuts we obtain

= M



−t



. (2.29)

Analytic properties of the Q-functions allow us to establish one important property

of the gluing matrix. Analytically continuing both sides of the ﬁrst equation from (2.29),

using the fact that due to the quadratic nature of the branch points analytic continuation

and complex conjugation commute with each other, and then applying the second equation

from (2.29) we derive

= M



−t



. (2.30)

– 11 –

JHEP07(2018)181

Noticing that (2.30) is true for any point u and applying the same trick as above and using

the analyticity properties of the gluing matrix, we arrive to the conclusion that



−t



= δ

(2.31)

and the gluing matrix is hermitian

(u) = M

(u) . (2.32)

In what follows we are going mainly to deal with the Q-functions on the physical

sheet therefore from now on we omit the hats and checks above the designations of the

Q-functions implying that all the Q-functions are considered on the sheet with the short

cuts if the opposite is not mentioned speciﬁcally.

Now we are ready to ﬁnd the other constraints on the gluing matrix which follow from

the conjugation and parity symmetries of the Q-system. To do this we will need the 4th or-

der Baxter equation for the functions Q

(u) to see if the certain properties of the P

(u) and

(u) can allow us to relate Q

(u),

(u) and Q

(−u). In the two subsequent subsections

we analyze the implications of the conjugation and parity properties respectively.

2.3 Complex conjugation symmetry

Let us now concentrate on the conjugation properties of the P- and Q-functions assuming

the charges ∆, S

and S

to be real. In [17] from the reality of the energy, Y- and T-

functions and the fact that the complex conjugation supplemented by the certain sign

factor is the symmetry of the Q-system it is shown that complex conjugation is equivalent

to some H-symmetry transformation already mentioned in the subsection 2.1. The matrix

of this transformation was proven in [17] to be constant due to analytic properties and

power-like asymptotic of the P-functions. Then there was found a transformation which

allows to make all P-functions with lower indices purely real and thus the P-functions

with upper indices pure imaginary. However, in our calculations we use the diﬀerent

normalization and make the other H-rotation by multiplying P

and P

by i and thus P

and P

also by i and obtain

1,2

= P

1,2

3,4

= −P

3,4

1,2

= −P

1,2

3,4

= P

3,4

or, in other words

= C

= −C

, C = diag{1, 1, −1, −1}. (2.33)

Given the conjugation properties (2.33), we see that the Baxter equation (2.11), written

for the Q-functions with prescribed analytic properties

[+4]

− Q

[+2]

− P

[+2]

a[+4]

+ Q

− P

a[+2]

+ P

a[+4]

− (2.34)

−Q

[−2]

+ P

[−2]

a[−4]

+ Q

[−4]

= 0

with D

and

given by (2.12) with Q

a|∅

= P

, remains the same, but for

(u) now.

Thus, the functions

satisfy the equation (2.34) too. As the functions

(u) constitute

– 12 –

JHEP07(2018)181

where ω

are the i-periodic functions on the sheet with the short cuts. Remembering (2.41)

and applying (2.44) on the sheet with the short cuts we obtain the equation

ˆω

− ˆω



− δ



ˆω

. (2.45)

On the other hand, the functions ω

are i-periodic on the sheet with the short cuts, thus

on the sheet with the long cuts their analytic continuation under the cut on the real axis

is given by the simple formula

ˇω



ˇω



. (2.46)

Rewriting (2.45) on the sheet with the long cuts gives us



ˇω



− ˇω



− δ



ˇω

. (2.47)

If we return several steps back, we can derive from the QQ-relations the equation for the

function Q

ab|ij

, which looks almost exactly like (2.47)

ab|ij+

−

ab|ij−



− δ



ab|kl−

. (2.48)

Recalling the notion of µ-functions introduced in the QSC framework in [16, 17], which are

i-periodic on the sheet with the long cuts, we multiply both sides by ˇµ

, which leads us to



ˇµ

ab|ij−



−



ˇµ

ab|ij−





− δ



ˇµ

ab|kl−



. (2.49)

Therefore, the functions ˇω

and ˇµ

ab|ij−

satisfy the same equation. As the functions

ˇµ

ab|ij−

are antisymmetric in i and j due to the antisymmetry of the Q-function Q

ab|ij

it is natural to impose the constraint that ω

is also antisymmetric and

= M

Ω

= −M

Ω

= −ω

. (2.50)

In the following subsections we are going to exploit (2.50) to constrain the gluing matrix

for diﬀerent spins S

and S

2.4 Parity symmetry

Now we are going to describe the parity properties of the Q-system. For a large class of

states the P-functions possess the certain parity. Such states include the states with the

charges J

= 2, J

= J

= 0, which we consider in the rest of the paper and also the ground

state with the charges J

= 3 and J

= J

= 0, which is relevant for the BFKL Odderon

eigenvalue (see [33, 34]). Thus, for the case J

= 2, J

= J

= 0 we have

= {2,1,0, −1}, (2.51)



(∆−S

−S

+2) ,

(∆+ S

(−∆−S

+2) ,

(−∆+ S

−S

)



As we understood the analytic structure of P- and Q-functions, taking into account

the asymptotics of these functions expressed in terms of the charges (2.51), it is natural

– 14 –

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