三维四维CFT中张量标量三点函数的重整化分析：新特征与异常

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本文主要探讨了在一般维度的共形场论（CFT）中，标量（scalar）与张量（tensorial）算符混合的三维（d=3）和四维（d=4）三维点函数（3-point functions）的重整化过程。作者Adam Bzowski、Paul McFadden和Kostas Skenderis在他们之前的研究中已经深入研究了纯标量和纯张量算符的关联函数，发现了一些独特性质：标量关联函数的重整化导致了β函数（beta function）的出现，出现了新型的B型共形异常（conformal anomaly），并且在动量空间中表现出意外的解析结构。另一方面，当涉及应力张量（stress tensor）和/或守恒电流（conserved current）时，β函数可能消失，但存在与0/0结构相关的A型异常（A-type anomaly）。混合关联函数则综合了这些特征，包括β函数、B型异常，以及可能出现的新A型异常。通过非扰动方法和一般动量空间分析，作者在文中详细地给出了应力张量、守恒电流和标量（Δ=d和Δ=d-2）在d=3和d=4维度下的重新归一化3点函数的具体结果。他们不仅确定了所有的异常项和β函数，还解析了异常保形Ward变换（anomaly Ward identities）的形式。值得注意的是，在d=3维度中，尽管存在0/0结构，但相关的A型异常并不显著。此外，研究还发现两个电流和一个标量的关联函数，以及两个应力张量和一个标量的关联函数具有通用张量结构，这些结构独立于标量的尺寸，并且在不同螺旋度下会消失。这种普遍性揭示了这些混合关联函数的一些基本特性。本文的研究对于理解共形场论中的对称性和量子效应有着重要的理论价值，尤其是在高能物理和凝聚态物理等领域，它提供了关于标量和张量算符相互作用的深入见解。由于是开放存取（Open Access）资源，这篇文章可供广泛的科研人员免费查阅，促进了知识的传播和进一步的研究。

JHEP11(2018)159

where the constants {u, v

} parametrise our freedom to shift the operator and spacetime

dimensions according to

d →

d = d + 2u, ∆

→

∆

= ∆

+ (u + v

), j = 1, 2, 3. (2.39)

Each choice of the constants {u, v

} thus deﬁnes a diﬀerent regularisation scheme. After the

divergences have been removed, physical dimensions are then restored by sending  → 0.

For the conserved current J

µa

and stress tensor T

µν

, our choice of scheme is restricted

by the necessity for these operators to retain their canonical dimensions, i.e.

∆

d − 1,

∆

d ⇒ v = u. (2.40)

With this scheme, gauge- and diﬀeomorphism invariance are respected and the transverse

Ward identities take the same form in both the regulated and the renormalised theory.

The divergences of triple-K integrals can be directly read oﬀ from a series expansion

of their integrand about the origin [11]. Writing this expansion as

˜α

j=1

x) =

, (2.41)

the divergences arise from terms of the form x

−1+w

(for some ﬁnite nonzero w) that become

poles in the limit as  → 0. Via the Mellin mapping theorem, these divergences are given

by the formula

div

˜α,{

}

−1+w

w

+ O(

). (2.42)

When the singularity condition (2.37) is multiply satisﬁed, the coeﬃcients c

−1+w

them-

selves contain poles in , leading to a higher-order overall divergence.

2.8 Reduction scheme and master integral

For four-dimensional CFTs, all the triple-K integrals we encounter can be derived from a

single ﬁnite master integral I

1{000}

. The corresponding reduction scheme, which we use to

obtain many of our results, is derived in [12]. The master integral I

1{000}

is well-known

in the literature (see e.g., [12, 40–42]) and represents, for example, the 3-point function of

= : ϕ

: for a free conformal scalar ϕ in four dimensions. It can be evaluated as

1{000}

√

−J



− 2 ln

+ ln X ln Y − Li

X − Li



, (2.43)

where J

is given in (2.22) and

X =

−p

+ p

−

√

−J

, Y =

−p

+ p

−

√

−J

. (2.44)

Here, X is eﬀectively a dimensionless complex variable with conjugate

X = 1 − Y .

– 11 –

JHEP11(2018)159

To express our results eﬃciently, we also deﬁne the ﬁnite auxiliary integrals

(ﬁn)

2{111}

= −





2 + p

∂

∂p



+ p



2 + p

∂

∂p



+ p



2 + p

∂

∂p



1{000}

, (2.45)

(ﬁn)

3{222}



2 −p

∂

∂p



2 −p

∂

∂p



2 −p

∂

∂p



1{000}



. (2.46)

Up to scheme-dependent terms, these auxiliary integrals are simply the ﬁnite parts of the

divergent triple-K integrals after which they are named.

Similarly, although we will not

write it this way, the frequently occurring combination (1/4)J

1{000}

is simply the ﬁnite

non-local part of the divergent integral I

0{111}

(see (4.6) of [12]). The derivatives appearing

in these formulae, and in all our later results, can be trivially evaluated using

∂

∂p

1{000}



− p

1{000}

− p

ln p

+ p

− p

) ln p

− p

+ p

) ln p



, (2.47)

with the analogous results for other momenta following by permutation. In this fashion,

for example, we can re-write (2.45) as given in equation (3.48) of [13],

(ﬁn)

2{111}

= −

1{000}

−



+ p

− p

) ln





+ p

− p

) ln





+ p

− p

) ln





. (2.48)

To obtain compact expressions, however, we usually leave such derivatives unevaluated.

2.9 Renormalised 2-point functions

In spacetimes of general dimension d > 2, the momentum-space 2-point functions read

hhO

(p)O

(−p)ii = C

2∆−d

, (2.49)

hhJ

µa

(p)J

νb

(−p)ii = C

µν

(p)p

d−2

, (2.50)

hhT

µν

(p)T

ρσ

(−p)ii = C

T T

µνρσ

(p)p

, (2.51)

while all 2-point functions of diﬀerent operators vanish. These expressions are unsuitable

however when the power of the momenta becomes an even non-negative integer. When

this occurs, the 2-point function is ultralocal in position space (i.e. has support only when

the operator insertions coincide) and can be removed by a local counterterm. The 2-point

function would then vanish implying the operator has zero norm in violation of unitarity.

In reality, in all such cases the corresponding coeﬃcient C

, C

or C

T T

has a UV

divergence, and renormalisation is necessary. For the scalar 2-point function, this occurs

whenever

n = ∆ −

(2.52)

For example, we deﬁne I

2{1+,1+,1+}

= −1/3 + I

(ﬁn)

2{111}

+ O(). From (4.20) of [12], this deﬁnition is

equivalent to (2.48). Alternatively, from (3.16) of [12] with α = 3 and β

= 1 + , we obtain (2.45) after

noting that I

1{,,}

= I

1{000}

+ O(

) and using the dilatation Ward identity for I

1{000}

– 12 –

JHEP11(2018)159

is a non-negative integer, while for 2-point functions of conserved currents and the stress

tensor, renormalisation is necessary in all even spacetime dimensions d = 2N.

To remove these divergences we pass to the dimensionally regulated theory with

d = d + 2u,

∆ = ∆ + (u + v), (2.53)

where u and v are constants parametrising our choice of regularisation scheme. (For stress

tensors and currents, conservation enforces v = u, however for a scalar v is unrestricted.)

Provided v 6= 0, the 2-point functions are now ﬁnite. The overall normalisation constant

is now a divergent function of the regulator,

() =

v

+ C

(0)

+ O(), j ∈ {OO, JJ, T T }. (2.54)

allowing the regulated 2-point functions to be expanded as

hhO

(p)O

(−p)ii

reg

= δ

2∆−d



v

+ C

ln p

+ C

(0)

+ O()



, (2.55)

hhJ

µa

(p)J

νb

(−p)ii

reg

= δ

µν

(p)p

d−2



v

+ C

ln p

+ C

(0)

+ O()



, (2.56)

hhT

µν

(p)T

ρσ

(−p)ii

reg

= Π

µνρσ

(p)p



T T

v

+ C

T T

ln p

+ C

(0)

T T

+ O()



. (2.57)

These expressions can be renormalised through the addition of suitable counterterms.

To quadratic order in the sources, these counterterms are

d+2u

√

g µ

2v



∆−d/2

+ c

µν



(d−4)/2

µνb

+ c

T T

µνρσ



(d−4)/2

µνρσ

, (2.58)

where W

µνρσ

denotes the Weyl tensor and the renormalisation scale µ enters on dimensional

grounds. All the Laplacians are raised to positive integer powers, as required for locality,

since the scalar counterterm exists only when ∆ − d/2 = n while those for the currents

and stress tensors only when d = 2N ≥ 4. Beyond quadratic order in the sources, these

Laplacians should be replaced by their Weyl-covariant generalisations.

Choosing the counterterm coeﬃcients

(−1)

∆−d/2

2v

+ c

(0)

+ O(), (2.59)

(−1)

d/2

4v

+ c

(0)

+ O(), (2.60)

T T

(−1)

d/2

T T

4v

+ c

(0)

T T

+ O(), (2.61)

where the ﬁnite coeﬃcients c

(0)

, c

(0)

and c

(0)

T T

encode a particular choice of renormalisation

scheme, the renormalised correlation functions can now be obtained by subtracting the

– 13 –

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