共形场论中标量算子的再归一化与异常：三点函数和β函数

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"CFT中的标量三点函数：重新归一化，β函数和异常" 本文深入探讨了共形场理论（CFT）中标量算子的三点函数的再正规化过程，特别是在任意维度下的情况。作者先前的工作证明，保形对称性决定了动量空间内的三点函数，这些函数由三个贝塞尔函数（三倍K积分）的乘积积分来唯一确定。然而，当算子的尺度满足特定关系时，这些三重K积分会出现发散。存在三种类型的发散：超局域、半局域和非局域。超局部发散可以通过局部反条件化来消除，这个过程可能导致新的共形异常出现。半局域发散可以通过源的重新归一化处理，使CFT的相关函数满足带有β函数的Callan-Symanzik方程。非局域发散中，奇异性质在于三元K表示本身，而非三点函数。在这种情况下，CFT的相关函数成为领先的非局部奇异性系数，它们保持对双倍保形变换的不变性，显示出了增强的对称性。文章进一步指出，当异常和β函数同时存在时，相关函数的解析结构会变得复杂，其中包括动量对数的乘积。为了阐明这些概念，作者提供了多个示例，包括自由场实现和AdS/CFT（Anti-de Sitter/Conformal Field Theory）计算。这些示例有助于读者理解在实际情况下如何应用这些理论。这篇论文详细阐述了CFT中标量算子的三点函数在遇到发散情况时的处理方法，包括再正规化、β函数的引入以及可能的共形异常。这些结果对于理解和研究高维量子场论以及其与引力理论的联系，如AdS/CFT对应，具有重要意义。

JHEP03(2016)066

where ∆

= ∆

+ ∆

. This equation shows that the correlation function is a homo-

geneous function of degree ∆

− 2d, which implies that

hhO

)ii = p

∆

−2d





, (4.4)

where F is a general function of two variables.

Let us now discuss the special conformal Ward identity (2.5). As noted in section 2,

it implies two scalar equations. The ﬁrst one reads

0 = K

hhO

)ii



∂

∂p

∂

∂p

∂

∂p

∂

∂p

−

2∆

− d − 1

∂

∂p

−

2∆

+ 2∆

− 3d − 1

∂

∂p



hhO

)ii, (4.5)

while the second equation, K

hhO

)ii = 0, is obtained from this one by

substituting p

↔ p

and ∆

↔ ∆

Let us consider the combinations

= K

−

∂

∂p

D, K

= K

−

∂

∂p

D. (4.6)

The eﬀect of the dilatation terms is to remove the terms with mixed derivatives in (4.5).

In this way we arrive at the particularly simple set of equations discussed in [7],

0 = K

hhO

)ii = K

hhO

)ii, (4.7)

where

= K

−K

, (4.8)

∂

∂p

−

2∆

− d − 1

∂

∂p

, (4.9)

for i, j = 1, 2, 3. Note that K

is the same operator that appeared in our analysis of 2-point

functions, see (3.1).

4.2 General solution

The system of the dilatation and special conformal Ward identities is equivalent to that

deﬁning the generalised hypergeometric function of two variables Appell F

[7, 27] and

from this fact one can infer general properties such as the uniqueness of the solution. An

explicit form of the general solution is given in terms of triple-K integrals [7],

hhO

)ii = c

123

α{β

}

, p

), (4.10)

where c

123

is an integration constant and

α{β

}

, p

) =

∞

dx x

j=1

x). (4.11)

– 12 –

JHEP03(2016)066

is the triple-K integral. Here K

(x) denotes the modiﬁed Bessel function of the second

kind (or the Bessel-K function for short), while the parameters

α =

− 1, β

= ∆

−

, j = 1, 2, 3. (4.12)

Before we proceed to use this result, let us present an elementary derivation of it. We

will start by solving (4.7) using separation of variables,

hhO

)ii = f

). (4.13)

Inserting this ansatz in (4.7), we obtain

= x

, (4.14)

where x

is a constant since the equalities hold for arbitrary p

. The equation K

= x

is equivalent to Bessel’s equation and has the general solution

) = p



x) + a



. (4.15)

The integrand of the triple-K integral is thus itself a solution of the special conformal Ward

identities.

Now, given a solution of the special conformal Ward identities f(p

, p

) =

we can immediately construct a solution of both the special conformal and the dilatation

Ward identities by taking the Mellin transform,

∞

dx x

α−β

f(p

x, p

x). (4.16)

where β

= β

+ β

. To see this, note that

i=1

∂

∂p

f(p

x, p

x) = x

∂

∂x

f(p

x, p

x) (4.17)

and then use integration by parts. In order for this Mellin transform to converge, at least

one of the f

) must be a Bessel-K function, as Bessel-I grows exponentially at large x.

A closer analysis [7, 27] (see also appendix A.3) reveals that in fact all three f

) must

be Bessel-K functions, as otherwise the resulting 3-point function becomes singular for

collinear momentum conﬁgurations (e.g., p

+ p

= p

It remains to discuss convergence at x = 0. As it stands, the triple-K integral converges

only if

α > |β

| + |β

| − 1, p

, p

> 0. (4.18)

However, one can extend the triple-K integral beyond this region by means of analytic

continuation. If one considers the triple-K integral as a function of its parameters with

momenta ﬁxed, then analytic continuation can be used in order to deﬁne the triple-K

everywhere, provided

α + 1 ± β

± β

6= −2k, (4.19)

– 13 –

JHEP03(2016)066

for any choice (of independent) signs and non-negative integer k. When the equality

holds we recover (1.5) and the triple-K integral contains poles (as we will discuss in detail

shortly). In such cases a non-trivial renormalisation of the correlation function (4.10) may

be required.

In summary, when the dimensions are generic, meaning (1.5) is not satisﬁed for any

choice of signs and non-negative integer k, the solution of the dilatation and special confor-

mal Ward identities is (4.10). This is then the analogue of (3.6) for 3-point functions. We

will shortly discuss in detail the special cases but ﬁrst a couple of examples. In these exam-

ples, and those we consider later, it will often be useful to label operators and their sources

according to their (bare) dimensions, as indicated in square brackets. In this notation an

operator of dimension ∆ and its corresponding source are thus O

[∆]

and φ

[d−∆]

Example 1: d = 4 and ∆

= ∆

= 5/2.

This is an example of a ﬁnite correlation function expressible in terms of elementary

functions. The 3-point function is represented by a triple-K integral

hhO

[5/2]

)ii = c (p

)

∞

dx xK

x)K

x), (4.20)

where c is the integration constant. All Bessel K functions with half-integral indices are

elementary. In this case the integral is convergent and evaluates to

hhO

[5/2]

)ii =

cπ

√

+ p

. (4.21)

Example 2: d = 4 and ∆

= ∆

= 2.

In this case the 3-point function is given by

hhO

[2]

)ii = cI

1{000}

(4.22)

It turns that this integral has already been computed in the literature [31, 32] and is given by

1{000}

√

−J



− 2 ln

+ ln



−X





−Y



− Li



−X



− Li



−Y



. (4.23)

where

= (p

+ p

− p

)(p

− p

+ p

)(−p

+ p

)(p

+ p

), (4.24)

X =

− p

√

−J

, Y =

− p

√

−J

. (4.25)

As will be discussed in [33] (see also [7]), triple-K integrals with integral indices can be

obtained from this integral using a recursion method.

– 14 –

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