欧几里得深度解析：交叉方程与色散关系

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"这篇学术文章是关于解析欧几里得靴，即在欧几里得深层区域解析地求解交叉方程的研究。文章作者是Baur Mukhametzhanova和Alexander Zhiboedova，分别来自哈佛大学物理系和CERN理论物理部门。文章于2019年6月26日提交，10月12日接受，并在10月30日发表在JHEP10(2019)270上，属于Open Access资源，可公开访问。文章的核心内容是探讨如何在欧几里得域中处理复杂的量子场论问题，特别是通过解析方法解决交叉方程。作者关注的是主算子的加权频谱密度在大尺度维度Δ的尾部行为，这是在某一通道中具有特定自旋的算子。他们发现这些尾部可以与另一通道的欧几里得Operator Product Expansion (OPE) 数据相匹配。通过在双通道的欧几里得OPE中加入更多运算符，能够系统地捕捉到1/Δ的次主导尾部效应。为了实现这一目标，作者运用了复数Δ平面上的保形分波的色散关系，这是一种在量子场论中用来处理运算符之间关系的重要工具。同时，他们还利用了洛伦兹反演公式，这是一个在研究散射过程和场论中用于重建物理信息的数学技术，以及复Tauberian定理，这在处理渐近行为和推导幂律行为时非常关键。在一系列示例中，包括对Conformal Field Theories (CFTs)和散射振幅的分析，作者验证了他们的公式，发现两者之间有完美的一致性。这些示例进一步揭示了即使在Δ接近1的较小值时，大的Δ展开也非常有效。特别是在3d Ising模型的预测中，他们的方法表现出了强大的预测能力。最后，作者强调了他们对色散关系的复杂Tauberian定理分析的普遍性，指出这种方法可能适用于许多其他情况，展现出该研究的广泛潜在应用价值。" 这篇研究不仅深入探索了欧几里得域内的量子场论计算，而且还提供了新的分析工具和方法，对于理解和解决相关领域的复杂问题具有重要意义。

JHEP10(2019)270

2.3 Example 2

Consider now a simple example where the complex tauberian theorem (2.13) is applicable.

We consider the spectral density to be

f(E) =

∞

n=0

δ(E − n) . (2.16)

We can explicitly compute the Laplace transform

L(β) =

1 − e

−β



1 +

β +

+ . . .



, (2.17)

and Cauchy moments

(E) = E



m−1

Γ(m + 1)

m−2

2Γ(m)

m−3

12Γ(m − 1)

+ . . . + O





. (2.18)

The error term in this example is a function of the fractional part of E, namely E − [E].

Since 0 ≤ E−[E] < 1, it is indeed O(1) for any E. This is an example of (2.13) with ∆

3 Dispersion relations for conformal partial waves

In this section we derive dispersion relations for conformal partial waves in the complex ∆

plane. These dispersion relations allow us to study separately the contribution of primary

operators of given spin in the s-channel. We then analyze these dispersion relations in

the limit |∆|  1. We ﬁnd that the large ∆ behavior away from the real axis of the

conformal partial waves is controlled by the t-channel OPE data.

In section 5 we will use

these dispersion relations together with complex tauberian theorems to arrive at our ﬁnal

result (5.11).

3.1 Conformal partial waves and Lorentzian inversion formula

Consider a four-point correlator of identical scalar primary operators φ in d ≥ 2 dimensions

hφ(0)φ(z, ¯z)φ(1)φ(∞)i = (z¯z)

−∆

G(z, ¯z), (3.1)

where, as usual, we used conformal invariance to put four points in a plane. The relation

to the conformal cross ratios is

u =

= z¯z, v =

= (1 − z)(1 − ¯z) . (3.2)

The OPE expansions in diﬀerent channels are given by

s − channel : G(z, ¯z) =

∆,J

(z, ¯z)

t − channel : G(z, ¯z) =



z¯z

(1 − z)(1 − ¯z)



∆

∆,J

(1 − z, 1 − ¯z)

(3.3)

In a sense, s-channel operators of dimension ∆ probe t-channel distances

∆

– 8 –

JHEP10(2019)270

(∆) has poles at dimensions of operators appearing in the s-channel OPE with residues

related to OPE coeﬃcients. The precise relation was given in [22]

∆,J

= −Res

∆

→∆







(∆

), ∆ 6= ∆

(∆

) − r

J,∆

J+d−1,∆

+1−d

∆

+1−d

(J + d − 1), ∆ = ∆

(3.10)

where we introduced

∆

= J + d + n, n = 1, 3, 5, . . . (3.11)

and r

J,∆

is deﬁned by (x = ∆ − J − d + 2)

J,∆

Γ(∆ − 1)Γ(∆ + 2 − d)



∆ −





∆ −

d−2





J +

d−2





J +



Γ(J + 1)Γ(J + d − 2)

Γ(2 − x)Γ





Γ(x)Γ



2−x



(3.12)

The reason there is an extra term in (3.10) when ∆ = ∆

is that the conformal blocks

in (3.8) have poles on the real ∆ axis. Their contributions are cancelled by extra poles

of c

(∆), which do not correspond to operators appearing in the OPE. These unphysical

extra poles are explicitly subtracted in the second line of (3.10), as explained in [22].

Since the functions F

∆,J

form an orthogonal basis, one can invert (3.8) and obtain the

Euclidean inversion formula. Further, it is possible to deform the contour of integration in

z, ¯z to Lorentzian kinematics, which entails Caron-Huot’s inversion formula [22].

In the

case of identical external operators it is given by

(∆) =

J,even



∆ −



Γ(∆ − 1)

dzd¯z µ(z, ¯z)G

J+d−1,∆+1−d

(z, ¯z) dDisc G(z, ¯z) (3.13)

where µ(z, ¯z) is the orthogonality measure of partial waves F

∆,J

µ(z, ¯z) =

(z¯z)



z − ¯z

z¯z



d−2

(3.14)

The double-discontinuity is deﬁned [22] by

dDisc G(z, ¯z) = G

eucl

(ρ, ¯ρ) −

G(ρ, ¯ρ − i0) −

G(ρ, ¯ρ + i0) ,

eucl

(ρ, ¯ρ) = G(ρ, 1/¯ρ), z =

4ρ

(1 + ρ)

(3.15)

In a generic CFT the derivation of (3.13) applies only to J > 1.

To recapitulate, c

(∆) are meromorphic shadow symmetric functions. They have poles

at the positions of physical operators appearing in the s-channel OPE as well as a series of

extra (kinematical) poles as dictated by (3.10). For J > 1 c

(∆) could be computed from

the double discontinuity of the correlator using the inversion formula (3.13). In the next

subsection we will show that a direct consequence of (3.13) is polynomial boundedness of

(∆) at large ∆. This allows us to write and study dispersion relations for c

(∆).

For non-identical external operators there will be poles for all integer n.

Recall that we write G

∆,J

for conformal blocks instead of the convention in [22] G

J,∆

– 10 –

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