Heavy-light Bootstrap from Lorentzian Inversion Formula

We study heavy-light four-point function by employing Lorentzian inversion formula, where the conformal dimension of heavy operator is as large as central charge $C_T\rightarrow\infty$. Implementing Lorentzian inversion formula back and forth reveals the universality of lowest-twist multi-stress-tensor $T^k$ as well as large spin double-twist operators $[\mathcal{O}_H\mathcal{O}_L]_{n',J'}$. In this way, an algorithm is proposed to bootstrap heavy-light four-point function with extracting relevant OPE coefficients and anomalous dimensions. Following the algorithm, examples of $d=4$ are exhibited up to triple-stress-tensor, and moreover, general dimensional heavy-light bootstrap up to double-stress-tensor is discussed with ending up presenting an infinite series representation of lowest-twist double-stress-tensor OPE coefficient. Exact expressions of lowest-twist double-stress-tensor OPE coefficients in $d=6,8,10$ are also obtained as further examples.


Introduction
AdS/CFT correspondence (holography) serves as a bridge connecting gravity theories in anti-de Sitter (AdS) spacetime and strong-coupled CFT living in the AdS boundary [1][2][3], enabling us to exploit conformal field theories (CFT) with sparse spectrum [4] at strong coupling without counting on any specific CFT theories. From AdS side, investigations of Witten diagrams could enlighten us the organization and universality associated with CFT correlation functions. On the other hand, although directly studying strongly-coupled CFT is a hard task, recent development of conformal bootstrap makes it achievable. Without referring to any Lagrangian, conformal bootstrap focuses on conformal symmetry itself [5] combined with crossing symmetry [6][7][8][9][10], unitarity [11,12] and other physical consistency conditions, from which properties of conformal dimensions and operator product expansion (OPE) coefficients can be efficiently explored. In turn, the progress of strongly-coupled CFT can be expected to shed a light on some essential aspects of quantum gravity.
In parallel to numerical bootstrap which aims to precisely constrain and even determine those CFT data for numerous specific models like Ising model (see [13] for a recent review), analytic bootstrap has been developed to probe universality of CFT data provided with some parametric limit. By analyzing the singularities from crossing symmetry near light-cone limit, the universal spectrum and OPE coefficients of large spin operators were understood [8,9]. This progress boosted the large spin perturbation theory [10,[14][15][16], those universal spectrum can be expanded as inverse of spin 1/J, which surprisingly remains intact down to finite spin [17,18]. This incredible validity can be explained by analyticity in spin which was made manifest by Caron-Huot Lorentzian inversion formula [19][20][21]. The Lorentzian inversion formula encapsulates the large spin systematics and allows us to compute OPE coefficients and anomalous dimensions more efficiently even with finite spin [22,23].
Naturally, Lorentzian inversion formula was applied to investigate quantum gravity and AdS/CFT, for example, it allows us to study correlators up to loop level in supergravity [24,25] and to understand the growth of extra dimension in AdS/CFT [26]. However, these explorations only involve pure AdS and do not include any heavy states. Undoubtedly e.g. information loss and black hole collapse [27][28][29][30], entanglement entropy [31][32][33][34] and chaos [35], which are well-understood in AdS 3 /CFT 2 thanks to Virasoro symmetry in CFT 2 .
Roughly speaking, at large central charge limit C T → ∞, the heavy-light four-point func-tion (conformal dimension of heavy operator is heavy as ∆ H ∼ C T while ∆ L ≪ C T ) is all exchanged multi-stress-tensors T n are packaged universally [28,31,36,37]. However, the Virasoro symmetry is not available in d ≥ 3 CFT. Studying heavy-light four-point functions in d ≥ 3 CFT is thus necessary. Holographically, the underlying exchanged operators in this channel were studied considerably recently by either computing the bulk phase shift [38][39][40][41] or adopting Hamiltonian perturbation theory [16,28] in [42,43]. In parallel, to search for universality of multi-stresstensor T n OPE coefficients in high dimensions similar to CFT 2 , [44] initiated holographic studies as hints, and it is evidently that the lowest-twist multi-stress-tensor OPE coefficients exhibit universality by only depending on ∆ H , ∆ L and C T . However, a CFT origin of this universality is not clear. By studying stress-tensor commutation relation without calling holography, [45] shows that Virasoso-like structure indeed exists at light-cone limit. Before long, lowest-twist double-stress-tensor OPE coefficients were conjectured in [46] by equating two channels provided with borrowing some of holographic anomalous dimensions in [42,43]. It turns out that lowest-twist double-stress-tensor OPE proposed in [46] is exactly same as one found from holography [47]. A recent progress was made in [48] where in d = 4 lowest-twist double-stress-tensor OPE coefficients, triple- at same order can all be extracted by using crossing symmetry back and forth, without any holography coming in. Even some results in d = 6 were also achieved up to T 2 [48].
Remarkably, it can be verified that the data exchanged in consistent with predictions from holography [42,43].
These results are exciting, but can still be improved. At the first glimpse, the groundwork of [48] is exponentiated heavy-light four-point function ansatz near light-cone limit be only valid at eikonal limit (Regge limit [49]), however, it is equal to those extracted near light-cone limit [48]. Such a connection between universality in eikonal region and lowesttwist region was also discussed in [47]. In the end, the framework of [48] could not exhibit 1 Recently, the exponentiated Virasoro block is proved in [37] lowest-twist multi-stress-tensor universality in general. In this paper, we apply Lorentzian inversion formula to heavy-light four-point functions back and forth, and we surprisingly find these questions are answered by Lorentzian inversion formula partly, with agreement of those results existed in the literature.

Generalities
In this section, we assemble some preliminaries that will be used throughout this paper, includes conformal blocks, Lorentzian inversion formula and some background knowledge of heavy-light four-point function.

Conformal blocks
block is the solution of the quadratic Casimir equation where In d = 4, the closed form of conformal block for scalar four-point function where k a,b β (x) is SL(2,R) block and is given by The conformal block (2.4) is symmetric under (z →z,z → z). However, in general dimensions, the exact solutions are hard to come by.
Fortunately, we can probe some useful information hidden in conformal blocks by series expansion without knowing exact conformal block. The colinear expansion around z → 0 is very useful for our purpose in this paper. The leading term is Specializing (2.6) in d = 4 and comparing with exact block in d = 4 (2.4), it is obvious that this expansion has lost control of another part of z power, therefore sometimes (2.6) is called power law [19] in the sense that it only captures the essential power z (∆−J)/2 . Group theoretically, the full colinear expansion is expected to take the form as where for simplicity we write ∆ − J = τ and ∆ + J = β. The coefficients B a,b n,m can be solved by quadratic Casimir equation, see, e.g. [19] and Appendix A.1.

Lorentzian inversion formula
Lorentzian inversion formula is a powerful formula to extract the OPE data associated with [19][20][21]. The formula is given as where µ a,b (z,z) is given by . (2.10) Moreover, dDisc represents the double-discontinuity, which is defined by expectation value of multiplication of 14 and 23 commutators and can be computed by where G and G are two different analytic continuations forz around 1. Notice that in Lorentzian inversion formula (2.8), there is a conformal block with spin and conformal dimension interchanged G a,b J+d−1,∆−d+1 which is called funny block and is related to lighttransform [21]. Notably, the formula is analytic in spin except for (−1) J , which could be set to 1 in this paper since exchanged operators can only have even spin. Practically, we should expand G(z,z) by cross-channel conformal blocks, for a certain block with (∆, J) it should be Then we could integrate z andz to obtain c(∆, J).
The OPE coefficients are encoded in c(∆, J) by [19] c ∆,J = −Res ∆=∆ ′ c(∆ ′ , J) . (2.13) This implies that c(∆ ′ , J) has poles around physical operators (2.14) In fact, z integral in Lorentzian inversion formula is responsible for creating these poles, whilez integral provides other factors. To end this subsection, we would like to mention that an integral formula againstz from [19] would be useful throughout our calculation . (2.15)

Heavy-light four-point function
where z,z are cross ratios, and expand it in terms of conformal blocks. Typically, in most cases, we still prefer similar (z,z) rather than (1 −z, 1 − z) in t-channel. Thus for clarity, we should clarify the notation used throughout this paper where c ∆,J is the OPE coefficient.

HHLL t-channel: the cross-channel of HHLL s-channel
where the exchanged operators in cross-channel are denoted with prime, the crosschannel OPE coefficients are denoted with tilde and a = b = (∆ L − ∆ H )/2.
which is actually equivalent to HHLL t-channel but with different conformal frame for latter convenience. 20) which is actually equivalent to HHLL s-channel but with different conformal frame for latter convenience.

HLLH t-channel: cross-channel of HLLH
3. We always use s-channel, i.e. HHLL s-channel and HLLH s-channel to indicate what is the underlying OPE expansion. t-channel terminology, as the cross-channel of schannel, will only be used when we are implementing Lorentzian inversion formula.
Usually, in large C T CFT, the OPE coefficients should be expanded in powers of 1/C T . If we are only interested in O(1) OPE, then target theory is called generalized free field theory.
In generalized free field theory, operators that can be exchanged in HLLH s-channel (let us assume ∆ H ∼ ∆ L ≪ C T for the moment) are double-twist operators [O H O L ] n ′ ,J ′ [5,8,9,50] [ where n ′ is integer. Thus it would be more convenient to denote the OPE coefficients with twistsc n ′ ,J ′ . There are an infinite number of double-twist operators, and they are contributing to identity exchange in HHLL s-channel. The exact free OPE coefficients can be computed by Euclidean inversion formula elegantly and in fact they are well-known [50] c free n ′ , It behaves like J ′∆ L −1 at heavy-limit and large J ′ limit [42,43,46,48]. Typically, as one goes to next order and even higher of large C T expansion, not only OPE coefficients will be corrected by 1/C n T with n ≥ 1, but also double-twist operators will acquire anomalous dimensions suppressed by 1/C n T with n ≥ 1. From holographic viewpoint, these corrections and anomalous dimensions come from tree-level exchange (n = 1) and loop effects of Witten digrams (n > 1). When an additional parametrically large conformal dimension ∆ H ∼ C T is available in the spectrum, higher order 1/C T suppressions have their chance to be compensated by ∆ H , consequently, OPE corrections and anomalous dimensions may have O(1) and can not be neglected. Instead, OPE coefficients and anomalous dimensions for double-twist operators exchanged in HLLH s-channel could be expanded by ∆ H /C T . Follow the convention from [42,43,46,48] and for latter convenience, we introduce a parameter µ Naturally, we can organize the double-twist OPE coefficients and anomalous dimensions bỹ It is worth commenting that the expansion (2.24) is an natural organization: presumably, we can start from full 1/C T expansion and collect those terms having enough power of ∆ H to reorganize an expansion by arranging µ order. For O(µ),c n ′ ,J ′ andγ (1) n ′ ,J ′ are contributed by single-stress-tensor exchange in HHLL s-channel which is shaped by Ward identity and is proportional to µ . (2.25) Thenc (1) n ′ ,J ′ andγ (1) n ′ ,J ′ could be extracted [42,43] by using the impact parameter representation at Regge limit [38][39][40][41]. By dimensional analysis, O(µ k ) corrections of HLLH s-channel OPE coefficients and anomalous dimensions are contributed by multi-stress-tensor exchange T k in HHLL s-channel, however, we almost know nothing about their OPE coefficients beyond single-stress-tensor. Hence beyond O(µ), the expansion (2.24) can only be calculated by holographic studies either by bulk phase shift [42,43] or Hamiltonian perturbation theory [42]. Those holographic investigations are limited to Regge limit where OPE coefficients and anomalous dimensions are restricted to large spin limit ∆ H ≫ J ′ ≫ 1, in which the holographic investigations also suggest [42,43] (2.26) These data calculated from bulk is universal in the sense that any higher-derivative gravity corrections will be suppressed by 1/J ′ . Typically, in this paper, we will show the large spin behavior (2.24) for HLLH s-channel data is indeed valid from CFT point of view by using Lorentzian inversion formula.
On the other hand, in HHLL s-channel, we expect the dominate exchanged operators are multi-stress-tensors T k , for example Similar to organization of HLLH s-channel data, the OPE coefficient for T k could be organized by factorizing µ out However, as we mentioned previously, the multi-stress-tensor OPE coefficients are beyond our knowledge, impeding our efforts on understanding O(µ k ) correction of double-twist operators from pure CFT point of view. The efforts were made recently toward understanding multi-stress-tensor OPE coefficients holographically in [44]. By treating heavy operator as a black hole, the heavy-light four-point function could be understood as a two-point function under this black hole, a technique was then developed in [44] to read off multi-stress-tensor OPE coefficients. The main conclusion of [44] is that they found, by considering arbitrary higher-derivative gravity models, the lowest-twist multi-stress-tensor OPE coefficients are universal. Although by applying crossing symmetry at light-cone limit with help of holography [46] or exponentiated HHLL block ansatz [48], [46] and [48] successfully extracted lowest-twist double-stress-tensor OPE coefficients as well as some low-lying double-twist for which a precise agreement with holographic results [42,43,47] was observed, an insightful CFT understanding of this universality is still lacking. In this paper, we would employ Lorentzian inversion formula to fill this gap up to some extents. Considering it can be observed in [46,47,47,48,51] that multi-stress-tensor OPE coefficients have integer ∆ L poles in even dimension, we will assume ∆ L is neither an integer nor half-integer (see section 3.2) throughout this paper except for section 3.2. The origin of such poles could be easily observed in our framework and we will leave the comments in section 3.2. For clear and as a guide for readers , we list the main conclusion of this paper below provided with two assumptions Assumption: a. O L belongs to an non-even-integer multiplet: additional light operators with conformal dimension∆ L = ∆ L +2q (where q is an integer) are not available in the spectrum.
b. ∆ L is not integer and half-integer.
Main conclusion: 1. We can bootstrap heavy-light four-point function by implementing Lorentzian inversion formula back and forth.
2. The large spin limit of double-twist OPE coefficients exchanged in HLLH s-channel are universal.
3. The lowest-twist multi-stress-tensor OPE coefficients exchanged in HHLL s-channel are universal.

Bootstrapping heavy-light: the algorithm
In this subsection, we present the generic algorithm for bootstrapping heavy-light fourpoint functions. By bootstrapping heavy-light, we mean, ambitiously, we would like to have a machine that both details of HHLL s-channel and HLLH s-channel can come out by following the algorithm. The machine should be Lorentzian inversion formula. The idea is that we could implement Lorenztian inversion formula back and forth to extract all universal CFT data, i.e. · · · HHLL → HLLH → HHLL · · · Typically, Lorentzian inversion formula is powerful to probe the universality of double-twist operators at large spin limit [17,18], elegantly and systematically capturing the large spin perturbation systematics [10,[14][15][16], in which finite spin makes sense at the end of the day [17,18,22,23]. More surprisingly, in this section, we will show that for heavy-light four-point function where ∆ H is comparable to C T charge, the Lorentzian inversion formula tells us the multi-stress-tensor exchanged in HHLL s-channel is universal and allows us to have an algorithm computing multi-stresstensor OPE coefficients alongwith computing HLLH s-channel double-twist data at large spin limit.

HLLH large spin behavior
To exhibit that Lorentzian inversion formula can encode the multi-stress-tensor data, we Nevertheless, it is not necessary to know everything there, for example, the recursion coefficients B a,b in (2.7) actually plays no essential role for our purpose, since the recursion coefficients turn out to contribute O(1). Generally, in Lorentzian inversion, we should consider following terms where m can be integers from −n to n, andB a,b n,m is some linear combination of B a,b . It turns out the contribution ofB is of order 1, i.e. O(1) at heavy and large spin limit, hence it is of no importance for large J ′ power behavior and can be slipped off here for simplicity.
On the other hand, the HHLL t-channel twist τ = ∆ − J conformal block is given by To extract large J ′ limit data, we can take the light-cone limitz → 1 of HLLH s-channel, in which z andz dependence is factorized, then using (2.15) to integrate againstz yields following function to be integrated against z The z dependence in (3.3) will not introduce additional J ′ and ∆ H dependent factors, and it does nothing but tells us the underlying exchanged operators are double- Hence, the large J ′ behavior is encoded in the remaining factor I (a,a) τ −∆ H −∆ L (β ′ + 2m) lying in the double-twist operator trajectories. For our purpose, we are supposed to take both the heavy and large J ′ limit. Taking the limit is a little bit subtle here. Precisely we should consider ∆ H ≫ J ′ ≫ 1. To achieve such a limit, we parameterize ∆ H ∼ J ′ /ξ and take .

(3.4)
Recall that the free OPE coefficients go like J ′∆ L −1 [42,43,46,48], we immediately havẽ for any twist n ′ , where the superscript τ denotes that it is contributed by twist τ conformal block in the cross-channel. However, there is a gap in this rough proof, which is the large J ′ behavior ofB a,b n,m . By solving quadratic Casimir as in Appendix A.2, we find that for double-twist operators the heavy and large J ′ limit ofB a,b n,m is Thus it does nothing to do with final large J ′ behavior of HLLH OPE and anomalous dimension.

Finding lowest-twist multi-stress-tensor
Next, we would like to show that knowingc t-channel data allows us to find lowest-twist multi-stress-tensor T k+1 exchanged in HHLL s-channel in Lorentzian inversion formula. The ingredient is the HLLH t-channel heavy block. The HLLH s-channel heavy block with twist n ′ can be deduced from (2.7), i.e.
HLLH t-channel heavy block. Note we are restricted in large J ′ limit where the summation of J ′ can be replaced by integration, we thus have It is worth noting that (3.8) only makes sense for z → 0, since HLLH s-channel four-point function evaluated at large J ′ limit by integrating against J ′ is only consistent withz → 1 limit, namely z → 0 after crossing. In other words, the large J ′ data of HLLH s-channel evaluated before forces that we can only probe the lowest-twist data in HHLL s-channel.
Then as soon as we knowc 2) ) order of G HLLH by expanding with respect to anomalous dimension in (3.7). Practically, the expansion up to O(µ (k+1)(d−2) ) is permitted, since dDisc only keeps power m ≥ 2 of log m , while the unknown informationc is attached to linear log which will always be killed by dDisc. This is analogous to one-loop investigation of supergravity correlator, in which the one-loop effect can be computed by squaring the tree-level data due to the same reason here [24,25].
Note as forB a,b in (3.1), B a,b is also of order O(1) at heavy and large J ′ limit and hence does not contribute any J ′ dependence. Precisely, B a,b is is given by for which the detail is presented in Appendix A.1. Then after integration against J ′ , the only relevant factor is the power of z All other factors likez dependence, summing n ′ and other ∆ L dependent coefficients are not relevant for our purpose, since the pole that signals the exchanged operators is encoded in z dependence. We keep a Gamma function for later comments in section 3.

Then
Lorentzian inversion formula provided with (3.11) now is where F is some unknown but regular factors (up to ∆ L poles) independent of z. It is obvious from (3.12) that it encodes the OPE coefficients for lowest-twist multi-stress-tensor τ = (k + 1)(d − 2) and we are allowed to compute them by using Lorentzian inversion formula as soon as we know allc It is worth noting that one has to be cautious of the procedure discussed in this subsec- the conformal dimension ofÕ L is∆ L = ∆ L + 2: they share same conformal dimension, twist and spin. In this way, the OPE coefficientsc is apparent that, e.g. γ n ′ ,J ′ . Hence the simple multiplications (3.9) are not trustable any more 2 . Similar mixing problem appears in the efforts toward understanding loop level of supergravity correlators, e.g. [24,25,52,53]. Therefore, an assumption should be made throughout this paper: there are no other light operators having conformal dimension∆ L = ∆ L + 2q where q is an integer. This is assumption a listed in section 2.3, we shall call this assumption non-even-integer multiplet assumption.

The universality
Now as assumption a in section 2.3 is made, we are ready to show the main conclusions of this paper listed in section 2.3. The assumption b restricting ∆ L to non-integer and nonhalf-integer could actually be quickly observed in factor of (3.11), nevertheless, we leave this to section 3.2.
The input is OPE coefficients of single stress-tensor that is completely fixed by Ward 2 We would like to thank Simon Caron-Huot for pointing this out to us.
identity (2.25). For convenience, we present it here again .  thus the universality of heavy-light four-point function we present here is valid atz → 1 limit of HLLH s-channel. Such a limit does not have any constraints on z and thus is way beyond light-cone limit for which we also require z → 0. It is worth noting that z → 1 limit is not forbidden by our construction, in this way, we could say this universality holds at both light-cone limit and Regge limit. This explains why the results of double-twist data obtained by bulk phase shift in eikonal limit is consistent with light-cone limit treatment [47,48].

Comments on ∆ L poles
Before we finally propose the algorithm for bootstrapping heavy-light four-point function, we would like to leave a subsection commenting on the ∆ L poles and explaining why the assumption b in section 2.3 is necessary. The holographic calculations in even dimensions [44,51] implies that the multi-stress-tensor would be suffering from poles of 1/(∆ L − n) where n is integer. This phenomenon can also be observed from recent CFT investigations [46,48].
Now the pattern of such poles is clear: 1. In even dimensions, all multi-stress-tensor OPE coefficients suffer from integer ∆ L poles.
As discussed in [44], these poles exist because HHLL s-channel double-twist operators  [51,54]. It thus deserves future investigations [55]. On the other hand, it turns out that when ∆ L approaches a certain pole, the relevant operators acquire anomalous dimension for which the ratio between this anomalous dimension and OPE coefficient could be determined by Residue around that pole of relevant multi-stress-tensor OPE coefficient [51]. We can also understand, from viewpoint of Lorentzian inversion formula, that this anomalous dimension should emerge. Note the relevant term in dDisc is where p is the upper bound of involved poles, by expanding around a certain pole where · · · denotes other irrelevant terms and the divergence term should be expected to be canceled by another set of operators. log z implies that the corresponding multi-stresstensor or HHLL s-channel double-twist (now they mix with each other) acquire anomalous dimension. We hole our framework could also inspire the understanding of this anomalous dimension and verify the Residue relation proposed in [51] in the future.

The algorithm
In this subsection, with assumptions listed in section 2.3 in hands, we would explicitly propose the algorithm to bootstrap heavy-light four-point function below. In this section, we follow the algorithm introduced in the previous section to solve the heavy-light four-point function in four dimension up to T 3 as an explicit example.

O(µ) double-twist
In d = 4, the closed form of conformal block is known as (2.4) which simplifies things a lot.
Since the conformal block (2.4) is explicitly invariant under interchanging z andz, making it possible to just use a half of it, thus we only need to evaluate where G(z,z) is single-stress-tensor conformal block, which in d = 4 is specifically given by (still evaluate a half of (2.4)) where we have served µ as expansion parameter and thus slipped it off here as the organization in section 2.3. (4.2) should be automatically separated into two parts, one is free of log and one contains log z. The former would be evaluated to contribute the O(µ) correction of HLLH s-channel double-twist OPE coefficients, and the latter reflects that the HLLH s-channel double-twist operators acquires anomalous dimension at order O(µ). Evaluating the part without log and taking both the heavy limit ξ → 0 and large spin limit in which we set τ ′ = ∆ H + ∆ L + 2n ′ . Note the free OPE coefficients (2.22) with heavy and large spin limit specializing in d = 4 arẽ Then taking the Residue of interested twists integer n ′ and dividing by free OPE coefficients (4.4) leads toc This result exactly agrees with examples of low-lying n ′ obtained in [42,48].
The computation for log part is similar but more involved. Notably, in previous work on computing anomalous dimension via Lorentzian inversion formula, there is no z integral needs to be done. In most cases, one could just evaluates thez integral and the remaining z-dependence will be same as z-dependence of integral associated with OPE data up to an overall log z. Therefore, by definition, the anomalous dimension can be easily worked out by dividing the z-dependence and pushing everything onto double-twist trajectories.
However, in our case, discrepancy emerges for z-dependent of log part and OPE part, which is manifest in (4.2). The trick here is simply ignoring the overall log z and integrating the remaining factor against z. This integration does a job to make the double-twist trajectories visible. Subsequently, we should take the Residue to specify the value on the doubletwist trajectories and then divided it by free OPE coefficients to end up with anomalous dimension. The limits ξ → 0, J ′ → ∞ should be taken, we thus find Thus we end up with the anomalous dimension as It is matching with those examples obtained in [48].

Lowest-twist double-stress-tensor
Now we are ready to bootstrap the lowest-twist double-stress-tensor with (4.5) and (4.7) in hands. From (2.4), the full HLLH s-channel block in d = 4 with bare double-twist operators at the heavy-limit is given by As a warm-up exercise, we would present the HLLH s-channel four-point function at O(µ) order. We present contribution of the twist n ′ , then we should sum over n ′ . For a certain twist n ′ and J ′ we have where the superscript denotes that it is O(µ) order of HLLH s-channel. Substituting (4.4), (4.5) and (4.7), integrating J ′ from 0 to ∞ and summing over all twists n ′ yields (We also need to takez → 1 limit in the end for the consistency with large J ′ limit) which is obviously consistent with the HHLL t-channel single-stress-tensor block (4.2). This is the double-check of this approach.
Then we move to the HLLH s-channel four-point function at the order of O(µ 2 ), specifically, what we are looking at is where we have already shut down the contribution fromc (2) n ′ ,J ′ andγ (2) n ′ ,J ′ since they will be killed by dDisc. In fact, evenc (1) n ′ ,J ′ is useless here in the sense that it gives us linear log. Integrating J ′ , summing n ′ and turning to cross-channel, we thus have (for simplicity we only keep log 2 (1 −z) that survives under dDisc) (4.12) The pole ∆ L − 2 in T 2 OPE observed in [44] already appears here. Then we just need to work out the Lorentzian inversion formula (2.8) with leading z → 0 term Nevertheless, it is worth noting that we should not apply (2.15) anymore, since now noz → 1 limit is assumed. In other words, what we are interested in is finite J result. Following formula would be useful (4.14) The trick to do the integral is to expand the hypergeometric function as a series which makes the integral doable, and then sum the infinite series back to a finite result. Meanwhile, the integral of z is not necessary, since we know it will give rise to the pole ∆ − J − 4, we only need to slip off z and assign the value ∆ = J + 4 to the rest. After some algebra, we have  One can straightforwardly verify that (4.15) is exactly same as the holographic result in [47] and also as conjectured in [46].

O(µ 2 ) double-twist and lowest-twist T 3
To go further and work on O(µ 2 ), a practical problem arises. Typically, there are infinite number of lowest-twist double-stress-tensors with different spin J, and one has to sum all of them for the purpose of using Lorentzian inversion formula. This would be a hard-core task, and [46,48] have done this by taking advantage of a complicated hypergeometric identity. In fact, the summed block exhibits a nice pattern at limitz → 1 with respect to HHLL t-channel. Based on this nice pattern, [48] proposed an ansatz to write down all multi-stress-tensor blocks, from which the computation was carried out to obtain HLLH s-channel data and HHLL s-channel T 3 OPE coefficients that are partly overlapped with this section [48]. The summed lowest-twist double-stress-tensor four-point function is given by (after crossing) [46,48] Then exactly as in (4.1) and previous subsections, we work out the integral and take heavy and large spin limit followed by taking the corresponding Residue to have the correction of double-twist OPE coefficients (4.17) and the correction of double-twist anomalous dimensions which agrees with results obtained from Hamiltonian perturbation theory [42], and the low-lying examples n ′ = 0, 1, 2, 3 exactly match those obtained in [48].
Then we would like to have attempt at solving T 3 OPE coefficients. Expanding the HLLH s-channel heavy block associated with twist n ′ and spin J ′ up to O(µ 3 ) leads to (ignoring linear log term) By substituting the known data (4.5), (4.7), (4.17) and (4.18), we are allowed to integrate J ′ and sum n ′ to obtain G HLLH . Although the expression of G HLLH is too cumbersome and complicated to be presented here, it is for sure that log 3 (1 −z) is involved. After doing the double-discontinuity, we are still left with log(1 −z). In this way, at the order T 3 , we have to face with following integral  2 2β−1 Γ(β + 1 2 ) 2 Γ(k + β) 2 (γ + ψ(α + k + 2)) √ π(α + k + 1)Γ(k + 1)Γ(β)Γ(2β + k) . (4.21) Thus we are hindered to have lowest-twist T 3 OPE coefficients with symbolic J dependence.
Nevertheless, for specific J, the integral is easy to evaluate and we could steadily have many . Typically, G T n takes the form of that ansatz, where the undetermined coefficients could be fixed by drawing references from some low-lying J OPE coefficients of T n . Thus the HHLL ansatz proposed in [48] is undoubtedly important for improving our algorithm, which could largely promote the efficiency. When it comes to summing twists n ′ , no difficulty appears in examples d = 4. However, we will see that this issue is inevitable in the next section. Some other issues exist and for the moment we are not aware of the resolution. For examples, we will see in next section that in general dimension even O(µ) order double-twist OPE coefficients can not be solved!

O(µ 2 ) bootstrap in general dimension
In this section, we would employ our algorithm to push on O(µ 2 ) heavy-light bootstrap in general dimensions. The main results are as follows: 1. We find a series representation of O(µ) correction to HLLH s-channel double-twist OPE coefficients in general dimension. Nicely, O(µ) order of HLLH s-channel doubletwist anomalous dimension is found with a closed form as 3 F 2 function.

For lowest-twist double-stress-tensor OPE coefficient in general dimension, an infinite
series representation is given.

A warm-up: free double-twist OPE
As a warm-up, we would like to reproduce the double-twist free OPE coefficients in this subsection. The key ingredient is HLLH s-channel funny block in general dimension, which is an infinite series and each term is shown in (3.1). For each term, we could take advantage of the nice formula (2.15) to integrate it and take the interested limit ξ → 0 followed by J ′ → ∞, in which we would like to recap the fact that onlyB n,n survives at heavy-limit as in (3.6), and we find where we assume ∆ ′ − J ′ = ∆ H + ∆ L + 2n ′ andB n,n can be found in (3.6). We are happy that the summation over n is not hard, we find By taking the Residue around integer n ′ , it is straightforward to find which can be verified to be consistent with heavy and large J ′ limit of (2.22) and comes back to (4.4) as soon as d = 4 is specified.

O(µ) double-twist
Now we turn to compute O(µ) correction of HLLH s-channel data. The essential ingredient is the form of G T . Since we are only interested in large J ′ limit, we could adopt the colinear block (2.6) in the cross-channel, we thus have The next step is to address k 0,0 d+2 (1 − z). The strategy is to expand it as an infinite series around z → 0, and in the end sum back. Notice that the involved hypergeometric function is of the type 2 F 1 (β, β, 2β, 1 − z), specifically, β = (d + 2)/2, we should take following series As expected, we have log free part and log part responsible for OPE and anomalous dimension respectively. Then we would like to obtain anomalous dimension at first by following the strategy demonstrated in section 4. For each k and n in the heavy and large spin limit we find .

(5.6)
Fortunately, it is not difficult to sum n and k in (5.6) After taking the Residue and dividing by free OPE (5.3), it gives rise tõ which is precisely what [42] obtained by using holographic technique of Hamiltonian perturbation theory.
For log free part, follow similar analysis, we find The difficulty thus arises. To our knowledge, we can only do summation over n in (5.9).
When it comes to k, polygamma functions are involved and the summation is hard to carry out. Nevertheless, we could take limit and Residue for each k, in which a truncation in k summation becomes manifest k max = n ′ , and we end up with .

(5.10)
The simplest case would be the leading-twist n ′ = 0, in general dimension we havẽ When d is even, it is not hard to implement the summation. Particularly, specializing (5. (5.12)

An infinite series of lowest-twist T 2
In this section, we would like to see whether we can have access to something on T 2 OPE in general dimension. Although we do not even have a closed form for O(µ) double-twist OPE coefficients, they are not necessary to come in as we discussed in section 4: they are suppressed by double-discontinuity. Now to implement Lorentzian inversion formula we need the full heavy-block (3.7) with summing n.
Thanks to the heavy-limit where we have (3.10), we thus find the HLLH s-channel four-point function with bare double-twist operators is which gives us the relevant term in (4.8) when specializing in d = 4. 4 Subsequently we will have exactly (4.11) withc (1) n ′ ,J ′ being slipped off (since it is irrelevant). However, we immediately encounter the problem. Following the algorithm, we are required to sum over twists n ′ . Unfortunately, considering the anomalous dimension in general dimension (5.8) is a generalized hypergeometric function without any simple identity to simplify, we are not likely to accomplish the summation. Nevertheless, as before, we could keep n ′ and apply Lorentzian inversion formula to each term with n ′ . Although the process is very complicated and it is not appropriate to write all of them down, we mange to have a final answer for lowest-twist T 2 OPE contribute from each twist n ′ by following the standard steps as shown before. Hence, we end up with an infinite series representation for lowest-twist double-stress OPE coefficients where H(∆ L , J) is given by .

(5.15)
However, it is rather difficult to start with the infinite series (5.15) trying to work out examples with specific dimensions due to the existence of generalized hypergeometric function.
Instead, one should start with anomalous dimension (5.8). We find, for even dimension, (5.8) could be reduced to be a nice finite series for which summing n ′ to obtain G (2) HLLH is manageable. Thereafter, lowest-twist T 2 OPE coefficients with symbolic J can be steadily extracted by following the standard integration technique. We present some low-lying examples d = 6, 8, 10 in Appendix B. It should be commented that it seems even dimension is special, while odd dimension is harder to handle. This is consistent with holographic treatment of multi-stress-tensor OPE in [44,51] where only even dimension case could be truncated to finite series such that the framework is applicable.

Conclusion and future directions
In this paper, we studied heavy-light four-point functions by implementing Lorentzian inversion formula back and forth. Focusing on non-degenerate scalar fields and assuming ∆ L is not integer and half-integer, we generally show (but not a serious proof) that Lorentzian inversion formula can probe the universality of lowest-twist multi-stress-tensor exchanged in HHLL s-channel and large spin OPE coefficients and anomalous dimensions of double-twist operators exchanged in HLLH s-channel. This universality holds at the regionz → 1 with respect to HLLH s-channel. Moreover, an algorithm for computing these data was proposed. In this way, we could state that we can bootstrap heavy-light four-point functions. Although now we can claim that universality of lowest-twist multi-stress-tensor in heavylight four-point function is understood by Lorentzian inversion formula up to some extent, many related valuable questions are still far from clear. We would like to point out some important future directions • The efficiency of our algorithm is somehow limited. [48] suggests that first few twists n ′ of double-twist HLLH s-channel data and some low-lying spin J of lowest-twist multi-stress-tensor OPE are enough to maintain the cycle of crossing back and forth and extract more data. It is thus important to investigate the necessary minimum number of twists n ′ and spin J examples in order to maintain the algorithm, which could, enhance the efficiency and allow us to go to higher orders.
• It is clear from Lorentzian inversion formula that lowest-twist multi-stress-tensor OPE coefficients are suffering from some ∆ L poles. These poles are expected to be can-  [44,51].
• In order to touch specific CFTs or supergravities, it is also necessary to get rid of noneven-integer multiplet assumption. It is thus very important and interesting to include other light operators, forming a class of light operator where double-twist operators are mixed. In this situation, there should be extra index such that the double-twist OPE coefficients and anomalous dimensions in HLLH s-channel are matrixes and an appropriate diagonal basis is required.
• Our results achieve a precise agreement with [48], verifying the exponential ansatz in some sense. We wish, similar to virasoro block in d = 2 [37], we could somehow directly solve the universal heavy-light conformal block of HHLL s-channel which is supposed to be exponentiated. This might be possible by using 6j symbol [56].

Acknowledgement
We are grateful to Simon Caron-Huot for useful discussions. This work is supported in part At First, we would like to keep track of full B a,b n,m without any limits taken. The logic is simple, we just throw (2.7) into quadratic Casimir equation (2.2) with (2.3) and organize the resulting equation as recursion equation. We will frequently use two derivative identities for k a,b β (z). The first one is which connects second derivative to first derivative without shifting β. The second identity relates firs derivative of k a,b β to k a,b β with β shifted by −2, 0, 2, namely Then after expanding z series and taking advantage of (A.1) and (A.2) to remove all derivatives, the Casimir equation becomes where all A, B, C are given by In addition, another important identity is necessary [19] k a,b By using this identity (A.5) to remove all extra 1/z, the equation (A.3) boils down to a recursion relation that could be solved for B a,b n,m with boundary condition B a,b 0,m = δ 0m . Take examples for n = 1, we find By solving B a,b n,m order by order and taking the relevant limits, the formula (3.10) would come out. However, this approach is not convincing enough in the sense that we could not find a well-organized closed formula as a solution of the full recursion (A.3) .
In fact, we can restrict onto bare double-twist trajectories and take the heavy-limit at the very beginning and surprisingly the infinite recursion equation would be self-consistently truncated to be finite and simple one. Taking the heavy-limit reduces (2.7) to (3.7) with vanishing γ(µ), i.e.
We should emphasize that in above recursion (A.8) we have already specify b = a = 1/2(∆ L − ∆ H ) as before, and in particular τ = ∆ H + ∆ L + 2n ′ where n ′ is arbitrary twist. Then we can continue, take heavy and large spin limit in above A in recursion equation (A.8). We find for n > m > −n However, this shall not be the end of story. The reduced block that needs to be solved (A.7) suffers from ambiguity of m. To be precise, for example, relevantz m−1 in (A.7) could either be k a,b β+2m /z or k a,b β+2(m−1) . Fortunately, this ambiguity is of no significance here, because we could always use (A.5) to state k a,b β+2m /z and k a,b β+2(m−1) is equivalent provided with the coefficients in (A.5) is vanishing in the heavy-limit. Till now, the proof of (3.10) is completed.
A.2B a,b n,m Now we turn to draw (3.6) forB a,b n,m . We have to remind that this subsection is not a serious proof, but should be served as a strong evidence that (3.6) is correct. In fact, as soon as we solve B a,b n,m in (2.7) from (A.3), we could multiply (2.7) by the overall factor κ a,b (β ′ )/κ a,b (β ′ + 2m)(1 − z) a+b (1 − z/z) d−2 , then we re-expand z, organize resulting expansion as (3.1) by using (A.5) and turn (∆ → J + d − 1, J → ∆ − d + 1), the coefficientsB a,b n,m could thus be read off [19]. Take the heavy and large spin limit, we can observe (3.6). As in previous subsection on B a,b n,m , this approach is not satisfactory since we are not allowed to solve (3.6) in an apparent way.
A better way is to take the heavy-limit at the first place. One should note we have a factor κ a,b (β ′ )/κ a,b (β ′ + 2m) attached to each m which is a a little bit annoying and unnatural. For now, we simply do not consider this factor and aim to solve an auxiliary where the coefficients are given bỹ Then we take the heavy and large spin limit for these coefficients within double-twist trajectories. For n > m > −n, first three terms in the right hand side of (A.14) tend to zero.
Furthermore, for m = n, only the first term in the right hand side of (A.14) makes sense, although it is not zero and actually diverges as J, it expressesB a,b n,n in terms ofB a,b Then we would like to recover the factor κ a,b (β ′ )/κ a,b (β ′ + 2m) and translateB toB. One may naively multiply κ a,b (β ′ )/κ a,b (β ′ − 2n), which, however, identically vanishes at heavy and large spin limits. This subtlety arises because of the ambiguity ofz m exactly as previous subsection. Now we are not lucky enough to make k a,b β+2m /z and k a,b β+2(m−1) equivalent, since the factor κ a,b (β ′ )/κ a,b (β ′ + 2m) is different for each of them. The only possibility such that we have nontrivial result is 5 We then should adopt (A.5) n times to remove all additional 1/z factor, and multiplying each term with corresponding κ a,b (β ′ )/κ a,b (β ′ +2m) factor. Note the factor κ a,b (β ′ )/κ a,b (β ′ +2m) goes like ξ −2m , while coefficients for second and third term in the right hand side of (A.5) behave as ξ and ξ 2 respectively, we finally find the only surviving term isB a,b n,−n k a,b β+2n , thus B a,b n,n =B a,b n,−n ,B a,b n,m<n = 0 , (A. 19) which is precisely (3.6).

B More examples for double-stress-tensor
In this subsection, we present some low ip . We then just list other a    The case d = 6 was obtained recently in [48], which is exactly same as ours.