大自旋间隔揭示CFT中引力耦合的孤立应力张量OPE

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JHEP07(2018)157

symbol j(ν) inside correlation functions, as in hOOj(ν)i, where it denotes the contribution

of the leading Regge trajectory at some value ν.

In Regge kinematics, recalled below, large N CFT four-point functions are dominated

by the leading Regge trajectory of single trace operators. Its contribution may be resummed

into the exchange of an the eﬀective ﬁeld j(ν); this is achieved by the program of conformal

Regge theory, developed in [21]. A four-point function of conformal, possibly spinning,

primaries is ﬁxed by conformal invariance up to a reduced amplitude A(z, z), where the

conformal cross-ratios are

u = zz =

, v = (1 − z)(1 − z) =

(2.4)

with x

= x

−x

. The Regge limit is reached by analytically continuing z around 1 while

keeping z ﬁxed, and then taking the limit

z, z → 0 ,

ﬁxed. (2.5)

In taking this limit, it is also convenient to use the parameterization

z = σe

, z = σe

−ρ

(2.6)

whereupon the Regge limit is σ → 0 with ρ ﬁxed. For scalars O

, the Regge correlator was

derived in [21]. We will be interested in the analogous result for spinning operators, studied

for speciﬁc cases in [14]. In any d, four-point functions of symmetric traceless tensors can

be written, in a conformal block decomposition, as weighted sums of diﬀerential operators

acting on scalar conformal blocks [37, 38]. The full four-point function is thus expressed

as a sum over a ﬁnite number of distinct such operators, each one representing a unique

tensor structure. In this “diﬀerential basis,”

A(σ, ρ) =

struc

k=1

(k)

12O

(k)

34O

∆,`

(σ, ρ) (2.7)

where G is the scalar conformal block for dimension-∆, spin-` exchange, N

struc

is the

number of independent tensor structures, and d

(k)

12O

(k)

34O

is the product of OPE coeﬃcients

associated to each structure in the exchange of O. We will employ a standard index-free

notation [34] in which Lorentz indices are contracted with null polarization vectors z

O(x, z) ≡ O

...µ

(x)z

. . . z

(2.8)

where z

= 0. In this case, the diﬀerential operators D

are functions of positions and

polarizations.

It is important to distinguish between the exact, leading Regge trajectory at ﬁnite N and the leading

Regge trajectory of single-trace operators in large N CFTs. The latter are not in general required to be

convex, but we will assume the same conditions for the large N trajectory in a neighborhood of the intercept

that includes the stress-tensor point. This has passed some consistency checks [14, 21, 35].

– 8 –

JHEP07(2018)157

2.2 Spinning conformal Regge theory

We will be focused on a four-point function of two scalar primaries φ and two spinning

primaries O

1,2

of spins `

1,2

, z

)φ(x

)i (2.9)

In this case, in the O

→ O → φφ channel, N

struc

is given by the number of independent

three-point tensor structures in hO

Oi. For even parity, generically we have [34],

struc

+ 1)(`

+ 2)(3`

− `

+ 3) (2.10)

Imposing permutation symmetry or conservation reduces this number. Taking the Regge

limit of the reduced amplitude A(σ, ρ) in position space, one ﬁnds

Regge

(σ, ρ) = (Disconnected) +

∞

−∞

dν

struc

k=1

(k)

12j(ν)

1−j(ν)

Ω

iν

(ρ) (2.11)

We leave the (z

, z

)-dependence of A implicit. The ingredients here are as follows. First,

Ω

iν

(ρ) is the H

d−1

harmonic function over a geodesic distance ρ, for scalar dimension

∆ = h −

+ iν. In terms of the H

d−1

bulk-to-bulk propagator Π

iν

(ρ),

Ω

iν

(ρ) =

iν

2π



iν

(ρ) − Π

−iν

(ρ)



, (2.12)

where

iν

(ρ) =

1−h

Γ(h − 1 + iν)

Γ(1 + iν)

(1−h−iν)ρ

(h − 1, h + iν − 1, iν + 1, e

−2ρ

) (2.13)

Next, the {D

} form a basis of Regge diﬀerential operators, and are functions of positions

and polarizations. Finally, α

(k)

12j(ν)

are proportional to the squared OPE coeﬃcients for

Reggeon exchange of the k’th structure in the diﬀerential basis,

(k)

12j(ν)

= X(ν)γ(ν)γ(−ν)d

(k)

12j(ν)

φφj(ν)

h+iν,j(ν)

(2.14)

where γ(ν), X(ν) and K

h+iν,j(ν)

are deﬁned in appendix B. Since we always study hO

φφi

correlators in this paper, we will label squared OPE coeﬃcients only by the operators O

and O

. We will henceforth ignore the disconnected piece of A

Regge

, which corresponds to

identity exchange.

Following [14], we will eventually be imposing unitarity on the Regge limit of the

correlator in impact parameter space. Applied to A

Regge

, the Fourier integral

A(σ, ρ) = (−1)

−

(∆

+∆

)−∆

dp dp e

−2ip·x−2ip·x

B(p, p)

(−p

)

h−

(∆

+∆

)

(−p

)

h−∆

(2.15)

where σ

= x

and σ cosh ρ = −2x · x, deﬁnes the impact parameter representation of

the Regge correlator,

Regge

(S, L) =

∞

−∞

dνS

j(ν)−1

struc

k=1

(k)

12j(ν)

Ω

iν

(L) (2.16)

– 9 –

JHEP07(2018)157

where the impact parameter variables (S, L) are

S = 4|p||p|, cosh L = −

p · p

|p||p|

. (2.17)

The form is essentially identical to (2.11). In these variables, B

Regge

(p, p) is the S → ∞,

ﬁxed L limit of B(p, p). The {

} form a basis of Regge diﬀerential operators in impact

parameter space. The β

(k)

12j(ν)

are proportional to the product of OPE coeﬃcients in this

basis, which we call B:

(k)

12j(ν)

= X(ν)B

(k)

12j(ν)

φφj(ν)

(2.18)

The B

(k)

12j(ν)

basis of OPE coeﬃcients is linearly related to the d

(k)

12j(ν)

basis.

Let us write (2.16) in a compact fashion,

Regge

(S, L) =

∞

−∞

dν ν S

j(ν)−1

X(ν)D(ν) Π

iν

(L) (2.19)

The real operator D is a sum over impact parameter space diﬀerential structures in the

Regge limit,

D(ν) ≡ B

φφj(ν)

struc

k=1

(k)

12j(ν)

(2.20)

We have used the evenness of the integrand to trade Ω for Π using (2.12). We emphasize

that the coeﬃcients B

(k)

12j(ν)

are deﬁned with respect to a basis of tensor structures in the

Regge limit: changing bases can introduce various kinematic factors, as we will see later.

2.2.1 Sliding along the Regge trajectory

Now we evaluate the integral by the saddle point approximation (recall that S → ∞). The

saddle ν

is deﬁned by

− ij

(ν

) =

log S

(2.21)

Note that ν

is a function of L, and that −ij

(ν

) > 0. Expanding B

Regge

around the saddle

point,

− iB

Regge

(S, L) ≈ S

j(ν

)−1

2π

−j

(ν

) log S

X(ν

)

D(ν

) Π

iν

(L) (2.22)

The S

j(ν

)−1

is the hallmark growth, with Reggeon spin j(ν

). By dialing L, we move the

saddle point and access diﬀerent points on the Regge trajectory.

Expanding (2.21) around ν

= 0, using the fact that j

(0) = 0, means keeping L ﬁxed

at large S. The resulting bounds are optimized by taking L  1.

The relation to [14] is [X(ν)D(ν)]

here

= D

there

. Note that in passing from (2.14) to (2.18), we have

absorbed a factor of K

h+iν,j(ν)

, which can have zeroes, in the deﬁnition of B

(k)

12j(ν)

. This will be convenient

when relating our large gap bounds to AdS physics. The B

φφj(ν)

coupling is unique, ﬁxed by a Ward

identity, and is never suppressed at large gap.

– 10 –

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