A Lightcone Commutator and Stress-Tensor Exchange in d=4 CFTs

Motivated by developing a field-theoretic algebraic approach to the universal part of multi-stress tensor conformal blocks of a scalar four-point function in a class of higher-dimensional CFTs, we construct a mode operator, ${\cal L}_m$, near the lightcone in d=4 CFTs and show that it leads to a Virasoro-like commutator, including a regularized central-term. As an example, we describe how to reproduce the d=4 single-stress tensor exchange contribution in the lightcone limit by a mode summation. We comment on possible generalizations.

In d > 2, the conformal group is finite-dimensional and stress tensors generally do not form an algebra. One should not expect to find a model-independent way that universally captures the full stress-tensor contribution. Is it therefore completely hopeless if one desires to generalize a similar story to d > 2 in a certain way?
The motivation of the present work comes from recent growing evidence [23][24][25][26][27][28][29][30] indicating that a certain universality of multi-stress tensors in a large class of d > 2 CFTs (including holographic CFTs) appears in the limit where operators in a correlator approach each others lightcone or, equivalently, in the lowest-twist limit. We will mainly focus on CFTs with a large central charge and a scalar four-point function with two heavy and two light scalars. The universality, more precisely, means that the operator product expansion (OPE) coefficients of the lowest-twist multi-stress tensors are "protected", in the sense that they are fixed by dimensions of scalars and the central charge C T . The higher-twist OPE coefficients, on the other hand, can be contaminated by other model-dependent parameters. From the gravity side's viewpoint, the universality implies that these coefficients are insensitive to higher-curvature terms in the bulk gravitational action, i.e. they can be determined by Einstein gravity. In d = 2, such universality can be explained by the Virasoro symmetry. The recent d > 2 results share intriguing similarities with d = 2 CFTs and we are motivated to search for a possible Virasoro-like derivation in the lightcone limit in d > 2 CFTs; we focus on d = 4 in this note.
A recent effort towards this direction was made in [24]. Based on the most general stress-tensor commutators consistent with the Poincaré algebra in local QFT [31], it was shown that, under an assumption on the Schwinger term, a Virasoro-like stress-tensor commutator emerges near the lightcone in d = 4 CFTs. Here, we would like to start to build a bridge between the stress-tensor commutator and the conformal block decomposition of a scalar four-point function. We shall also comment on a potential relationship between the Schwinger-term assumption and the validity of the lightcone universality in d = 4 CFTs.
To build a bridge between the d = 4 stress-tensor commutator and the scalar four-point correlator, it is desirable to construct an effective mode operator, similar to the generator L m in d = 2. An immediate obstacle, however, is that the d = 4 stress-tensor commutator has a UV cut-off, Λ, dependent central-term. (Λ has mass dimension one.) We will propose an L m , defined near the lightcone, and show that, using the d = 4 stress-tensor commutator, it results in a Virasoro-like [L m , L n ]. The basic picture of this construction is that we treat the additional two-dimensional transverse space as a thin layer with a thickness defined by a short-distance cut-off . The product 2 Λ 2 gives a dimensionless constant whose value will be fixed by the conventional OPE coefficient of the single-stress tensor exchange. Introducing a thin region, instead of infinite transverse space, may be interpreted as a lightcone limit, where we arrange scalars to live on a d = 2 plane and the stress tensors contribute only near the plane.
We will describe how to use this d = 4 lightcone commutator [L m , L n ] to compute the single-stress tensor exchange in the lightcone limit by a direct mode summation. This computation generalizes the Virasoro-algebra derivation of the one-graviton contribution to the identity block in d = 2 CFTs described in [15].
The more general case, beyond single-stress tensor, becomes more involved partially because the stress-tensor-scalar, T O, OPE in d = 4 has a delicate structure. While the general story is left to future work, we will make some preliminary remarks on a possible multi-stress tensor generalization.

Stress-tensor commutator near the lightcone
We start with the stress-tensor commutation relation in d = 4 CFTs in Minkowski space- Using the tracelessness condition, one can write the relevant component of the stress tensor in the lightcone limit as An important point is that the purely-spatial components of the stress tensor generally do not admit a modelindependent commutator [31]. However, in the case where stress tensors are inserted in a scalar correlator, the transverse components are suppressed in the lightcone limit. (By lightcone limit, we mean that we consider 4 scalars to lie on an x + − x − plane with config- , and then take x − → 0. We also send stress tensor's The dominating contribution in the lightcone limit is [24] − where , and ∆ is a Laplacian. Note that the central-term contains a UV cut-off Λ-dependent piece. We have set x − → 0 in the above commutator. More formally, one can write x − = and then focus on the leading small-contribution. The result (1) is valid only when the Schwinger-term in the stress-tensor commutator is a c-number. That is, the central-term in (1) is assumed to be the same as the expectation value of the stress-tensor commutator. A priori, however, there might be an additional operator Schwinger-term. It remains an interesting question to ask in what class of d = 4 CFTs the Schwinger-term is effectively a c-number (in the lightcone limit) as it may be related to the validity of the universality in d = 4 CFTs.
In what follows, we shall simply assume that we focus on the class of d = 4 CFTs where the Schwinger-term is effectively a c-number and adopt (1).

A mode operator and
To develop a Virasoro-like effective representation theory for the class of d = 4 CFTs whose lowest-twist subsector has a universal meaning, one would like to explore possible constructions of an effective mode generator, denoted as L m , which is defined via integrating the coordinates of a stress tensor out. Our goal here is to find an L m such that, when combined with the stress-tensor commutator near the lightcone in d = 4 CFTs, it can lead to a commutator [L m , L n ] which (i) satisfies the Jacobi identity and (ii) has a regularized central-term.
Since the difference between T ++ and T ++ is suppressed in the lightcone limit, as mentioned, we simply adopt T ++ in the following to have simpler expressions.
Let us Wick rotate to a Euclidean plane, ds 2 (The subscript will be dropped.) We keep the extra twodimensional transverse directions uncompactified. 1 Consider the following ansatz: Changing the power of x + corresponds to shifting m; we adopt m + 1 for later convenience.
The smear function f (y, z; j, k) generally can depend on new mode numbers, j, k, associated with transverse coordinates. The integrals along the transverse directions are necessary as the stress-tensor commutator contains Dirac delta-functions; just sending y, z to zero in the stress tensor does not give a sensible commutator.
The Jacobi identity severely constrains the form of f (y, z; j, k). We propose where a short-distance scale is introduced for the transverse directions. The stress-tensor contribution therefore comes only from a very thin region near a d = 2 plane. 2 The stress-tensor commutator (1) and (3) give, in the lightcone limit, where the Cauchy integral theorem was used. 3 In this notation, δ m+n,0 has mass dimension −m − n, and δ m+n,2 has dimension −m − n + 2, both with magnitude unit. Keeping an explicit for the limit of x − is irrelevant in deriving (4), but it will be useful when we later consider a scalar correlator with a stress tensor inserted.
Here we consider that the large UV cut-off term suppresses the last piece of (4), which causes tension with the Jacobi identity. The product of the UV cut-off Λ and the shortdistance regulator provides a dimensionless parameter. This constant will be fixed (see below) by the conventional single-stress tensor OPE coefficient that contains C T defined via the stress-tensor two-point function, T µν (x)T λρ (0) = C T I µν,λρ x 8 . (See [32] for related notations.) We arrive at an effective lightcone commutator: with α = 56 405π 2 in our convention. 4 2 One may adopt asymmetric limits, x − → , L → b , but b can be absorbed into the overall undetermined coefficient of the central-term discussed below. We use the conventional single-stress tensor OPE coefficient to fix the undetermined constant. 3 The radial ordering is implicit. In the lightcone limit, ∂ + T ++ is also suppressed. In d = 2, the centralterm of the stress-tensor commutator is finite. In terms of the generator L m , the d = 2 stress-tensor commutator leads to the well-known Virasoro algebra of the form [L m , L n ]. 4 If one first redefines Λ → Λ in (1) to absorb C T , one needs to reintroduce C T via Λ 2 2 ∼ C T . This process looks ad hoc and we do not adopt here. However, it is interesting to note that a similar identification appears in the soft-theorem related literature: see (147) in [33]. There, it is argued that the central charge can be related to internal soft exchanges. I thank L. Fitzpatrick for a discussion.
It should be emphasized that unlike in d = 2 CFTs where the Virasoro algebra represents an exact symmetry, (5) is an effective description. In (5), we have ignored contributions suppressed in the lightcone limit with a large UV cut-off, and we have assumed a class of d = 4 CFTs with a c-number Schwinger-term.

A Virasoro-like derivation of d = 4 single-stress tensor exchange
The effective lightcone algebra (5) looks formally the same as the d = 2 Virasoro algebra.
We may assume that, in a universal class of d = 4 CFTs, there exists a lowest-twist subsector where the associated intermediate states, |α s (s denotes a subspace), can be effectively organized into a Virasoro-like representation theory. In some sense, the lightcone limit acts like picking the holomorphic sector out and we do not need to introduce "L m ".
Focusing on such a subspace, we may try to follow the terminology of the highest-weight representation in d = 2 CFTs: L 0 |h = h|h and L m |h = 0 for m ≥ 1. The modes L m with m < 0 generate descendants. The vacuum |0 , preserving the maximal numbers of symmetries, is the associated state of the identity operator that has h = 0. One important difference, however, is that T O OPE structure in d = 4 is more delicate. 5 Using the T O OPE to express L m as a general differential operator will not be included in the present note. Here, we focus on a Virasoro-like derivation of the single-stress tensor exchange in the lightcone limit in d = 4. This derivation does not require knowing L m as a general differential operator because we can use the three-point function T OO , together with (5) and (3). The three-point function of the stress tensor with two scalar primaries in d = 4 with c T OO = − 2∆ 3π 2 . We will focus on the identity block in d = 4 CFTs at large C T with the heavy-light limit: ∆ H ∼ C T , ∆ L ∼ O(1). The single-stress tensor exchange contribution, discussed below, may be computed without explicitly imposing these limits, but we will still formally adopt ∆ H and ∆ L in what follows, having in mind a potential generalization involving multi-stress tensors.
In the lightcone limit, we assume that the corresponding intermediate states can be 5 For explicit expressions at the first few orders in the OPE, see, for instance, Appendix B of [34]. effectively generated by the operator L m acting on the vacuum. Introduce a basis which formally represents a normalized one-graviton state. Assume m > 1 here. We may relate the single-stress tensor contribution to the conformal block in the lightcone limit to the following object: We have switched to the conventional variables z,z defined by where u, v are conformal cross-ratios. In the configuration (8) Let us first compute the numerator of (8) using (9), (6), and (3). The computation is short but can be thorny as it involves a certain order of limits.
Denote y T , z T as the transverse coordinates for T ++ and r 2 = y 2 T + z 2 T . We have in a smallz expansion. In performing the contour integral, we have picked the pole due to O L (z,z). We should consider the stress tensor's lightcone limit as x − → , instead of directly setting x − = 0 from the start in this correlator computation. After performing the remaining integrals, A similar procedure, using (9), gives the following leading non-zero contribution: From the above explicit computations (11) and (12), we see that the leading contributions vanish at m = 2 and m = 3. We interpret that the vacuum is annihilated effectively by the operators L † 2 and L † 3 . By effective, we mean L m is in a correlator with the lightcone limit imposed. Note we are interested in developing an effective representation theory in d = 4 CFTs where stress tensors are in a scalar correlator; conditions L † 2 |0 = L † 3 |0 = 0 without referring to the lightcone limit can be too strong. In d = 2, the corresponding (11) and (12) both give a factor of (m − 1), but the lightcone limit is not necessary.
To define a normalizable basis with the normalization factor N m , we shall use the commutator (5). We may first assume, in the lightcone limit, a formal relation L † m = f (m)L −m with m > 1 where L −m does not annihilate the vacuum. One can compute The vanishing results obtained using L † m at m = 2, 3 then require f (m) ∼ (m − 2)(m − 3). However, as this ratio formally appears in both the numerator and the denominator of (8) it is irrelevant in computing the d = 4 scalar correlator. Thus, effectively, we write where To have a non-vanishing final result in the limit → 0, we perform an overall rescaling The 2 factor can be related to the volume of the transverse space. We obtain, in the which is the single-stress tensor block in the lightcone limit in d = 4 CFTs [35].

Concluding remarks and outlook
We have described an alternative derivation of the single-stress tensor block in d = 4 CFTs using a Virasoro-like approach. The effective mode generator L m is defined by integrating the stress tensor near a d = 2 plane where scalars live. This picture suggests that one may deal with the UV divergence in the stress-tensor commutator via the finite product Λ 2 2 .
It would be interesting to see if there is another way of renormalizing the central-term of the stress-tensor commutator.
In a recent wok [28], an ansatz has been proposed for the multi-stress tensor sector of the heavy-light scalar correlator in the lightcone limit. Assuming such an ansatz, the resulting OPE coefficients agree with the earlier holographic computation [23]. A potentially important step, which we have not considered in the present note, is to link L m to a general differential operator acting on O. In [34], using the T O OPE, the authors show how to recast the d = 4 averaged null energy (ANEC) operator as a differential operator, given as a series expansion and then resum. (See also [36][37][38] for recent related discussions.) Note that the ANEC operator can be related to L −1 , after integrating the transverse coordinates out. Considering a more general computation to obtain a differential form of L m in d = 4 CFTs with a large central charge can be useful.
A differential form of the operator L m should in principle allow one to compute the following more general object: with the k-stress tensors lightcone effective projector P given by A direct k > 1 computation is more complicated, but we expect that, similar to the d = 2 case, the computation can be simplified in the geodesic limit, ∆ L → ∞, leading to a possible exponentiation in the lightcone limit. Moreover, it might be possible to derive certain near-lightcone null-state equations for this universal class of d = 4 CFTs via an algebraic approach. We hope to discuss these possibilities somewhere else.
It would be also interesting to explore other possible constructions of the lightcone operator L m that leads to an effective algebra and compare to the form proposed here.
Let us end by mentioning another general question that has not been addressed: what precisely is the validity of the lightcone universality in d = 4 CFTs?