Regge trajectories for the (2, 0) theories

We investigate the structure of conformal Regge trajectories for the maximally supersymmetric (2, 0) theories in six dimensions. The different conformal multiplets in a single superconformal multiplet must all have similarly-shaped Regge trajectories. We show that these super-descendant trajectories interact in interesting ways, leading to new constraints on their shape. For the four-point function of the stress tensor multiplet supersymmetry also softens the Regge behavior in some channels, and consequently we observe that ‘analyticity in spin’ holds for all spins greater than −3. All the physical operators in this correlator therefore lie on Regge trajectories and we describe an iterative scheme where the Lorentzian inversion formula can be used to bootstrap the four-point function. Some numerical experiments yield promising results, with OPE data approaching the numerical bootstrap results for all theories with rank greater than one.


Introduction
The advent of Regge theory in the 1960s led to a profound improvement in our understanding of relativistic scattering amplitudes, relating in particular their high-energy behavior to the spectrum of resonances and bound states. Holography led to expectations that a similar structure should exist in conformal field theories (CFTs) [1][2][3][4], but it was only recently that these ideas became formalized non-perturbatively in a seminal paper by Caron-Huot [5]. The results of [5] indicate that a CFT spectrum organizes itself in Regge trajectories with spectra and OPE coefficients that are smooth functions of the spin . This picture elucidates the remarkable smoothness of numerically obtained OPE data, for example that of the three-dimensional Ising model analyzed in [6,7], and goes some way towards explaining the success of large spin perturbation theory [8,9], see for example [10,11]. As has become customary in the literature, we will use the expression "analyticity in spin" (of the OPE data) to refer to this circle of ideas.
One important caveat concerns the behavior for the lowest possible spins. For spins below some critical value it becomes much harder to use analyticity in spin to extract concrete information about the spectrum of the theory, see for example [11,12] for some attempts for the three-dimensional Ising and O(2) models. According to the analysis in [5] the exact value of * is related to the behavior of the correlation function in the Regge limit. For a generic unitary CFT, it was deduced in [5] that * 1 because its (suitably normalized) correlation functions are bounded by a constant in the Regge limit. A priori the spin 1 and spin 0 OPE data therefore do not need to smoothly connect to the higher spin OPE data.
It is worthwhile to investigate how analyticity in spin for conformal theories combines with supersymmetry, since the latter allows for extension of analyticity in spin. This is simply because a superconformal primary of spin * can have conformal primary descendants of spin > * and OPE coefficients related by supersymmetry. Since the scaling dimensions of these descendants are simply integer-shifted compared to that of the primary, and the coefficients of each descendant conformal block are often equal to that of the primary times a simple rational function of ∆ and , analyticity in spin of such a non-vanishing descendant trajectory would imply analyticity in spin of the primary trajectory 1 also below * ! JHEP01(2022)022 In this paper we undertake a study of the non-perturbative implications of analyticity in spin for supersymmetric conformal field theories (SCFTs). We have chosen to focus on the six-dimensional (2, 0) theories, but at a qualitative level our results certainly extend to theories in lower dimensions and likely also to theories with less supersymmetry -we comment on this further in section 7. 2 The particular four-point function we analyze is that of the superconformal primary of the stress tensor multiplet. This is the same fourpoint function as was analyzed holographically in [17,18], and with numerical bootstrap methods in [19]. Perturbatively, in a large c expansion, this correlator was also studied in [13], taking as input the tree-level results of [20,21]; and in [22][23][24]. The form of the superconformal Ward identities and superconformal blocks can be extracted from the more general analysis of [25].

Summary of results
Sections 3 and 6 contain our key results. The results of section 3 are summarized in figure 1. Unfortunately its explanation requires a minimal understanding of the rather technical superconformal block decomposition. We review this in section 2 and provide the essentials in the next paragraph.
We follow the notation of [19] and denote superconformal multiplets as X [p, q] ∆, with (∆, ) and [p, q] respectively corresponding to the conformal representation and the so(5) R-symmetry Dynkin labels of the superconformal primary, and with X ∈ {L, A, B, C, D} denoting the type of shortening condition. Long multiplets are denoted L and maximally short (half-BPS) multiplets are denoted D. For short multiplets we do not write ∆ because it is fixed by the other quantum numbers, and similarly we omit for the D-type multiplets because it is always zero. Using this language the stress tensor multiplet is known to be a D [2,0] multiplet and in its self-OPE we find the six non-trivial multiplets shown in figure 1.
(The Ward identities also allow for B[0, 0] multiplets but these contain higher spin currents and appear only in the free theory, as well as for D [2,2] and D[0, 2] multiplets which are excluded by Bose symmetry.) Out of the five types of short multiplets there are three, shaded in green, whose OPE coefficients are completely fixed by virtue of the chiral algebra construction of [26,27]. For the other short multiplets, which are shaded in orange, the coefficients are generally not known. For the long multiplets we know neither the scaling dimensions nor the coefficients.

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Given this rather involved structure of the superconformal OPE, it is a natural question to ask how the superconformal blocks organize themselves into Regge trajectories. The answer to this question turns out to be surprisingly involved. First of all, it is important to realize that a single Regge trajectory for superconformal primaries will induce many Regge trajectories corresponding to the conformal primary superdescendants. And because some of those descendant trajectories have higher spin, it becomes necessary to extend the superconformal trajectories to negative spin to get a complete picture. In doing so some obvious connections appear: the results that we review in section 2 (in particular table 1) indicate that the D [4,0] and D[0, 4] multiplets find a natural place in the (straight) Regge trajectories of the B-type multiplets: the latter as the continuation of the B[0, 2] multiplet to spin −1, and the former as the continuation of the B[2, 0] multiplet to spin −2. What is less obvious, however, is that the D[2, 0] multiplet itself does not fit into these trajectories. Instead, one finds that it is a combination of (i) the continuation to spin −2 of a long, unprotected Regge trajectory, and (ii) the continuation to spin −4 of the straight B [2,0] trajectory. Finally, the continuation of the B[0, 2] trajectory to spin −3 induces unwanted (descendant) blocks which can be canceled by the continuation to spin −2 of another unprotected long trajectory.
Altogether, then, analyticity in spin intertwines the straight and curved Regge trajectories in intricate ways, and multiplets with protected OPE coefficients or dimensions can appear on unprotected trajectories. This results in a non-trivial interplay between supersymmetry and analyticity in spin. 3 As we also review in section 2, the unfixed OPE data for the four-point function of the D[2, 0] multiplet is captured in a single function a(z,z). This function shares many similarities with an ordinary CFT four-point function and in particular has a standard conformal block decomposition in the s-channel. It is therefore natural to try to apply the Lorentzian inversion formula of [5] directly to this function. In section 4 we will set up the inversion procedure. We analyze numerous subtleties, leading ultimately to a picture of the analytic structure of the corresponding spectral density as shown in figure 6 on page 26. In line with our previous discussion, the axis extends to negative spins: in fact, the rather soft Regge behavior of a(z,z) leads one to conclude that * −3! In the figure we also observe two straight Regge trajectories corresponding to the two types of protected operators of figure 1, and the intersection of the leading long trajectory with a short trajectory at spin −2 as dictated by the analysis of section 3.
The observation that * < 0 means that, unlike in non-supersymmetric theories, all the physical supermultiplets appearing in the D[2, 0] self-OPE are expected to be reachable via Regge trajectories. This leads one to the appealing prospect that this four-point function can (at least approximately) be bootstrapped: we iteratively apply the Lorentzian inversion JHEP01(2022)022 formula to some initial trial spectrum until we hit a fixed point. Of course, with an OPE as in figure 1 the natural trial spectrum consists of the operators fixed by the chiral algebra.
In section 6 we present the initial results of such an approach. The first 'inversion' of the protected data in the t-channel yields an approximate s-channel spectrum whose long multiplets are of double-twist type, with anomalous dimensions that we estimate. We then refine our estimate by inverting the leading Regge trajectory several times until we hit a fixed point. Although this procedure ignores the subleading Regge trajectories, for sufficiently large c the resulting scaling dimensions and OPE coefficients nicely track the numerical bootstrap bounds of [19] -see the numerous figures in section 6 starting on page 39. We therefore believe that a more complete iterative scheme (i.e. inverting also contributions from subleading long Regge trajectories) would converge to the same 'extremal' solutions as those found with numerical bootstrap methods.

The four-point function
We consider the four-point function of the dimension four scalar which forms the bottom component of the D [2,0] stress tensor multiplet in the six-dimensional (2, 0) theories. Knowledge of this correlator allows for the computation of any four-point function involving only operators in the stress tensor superconformal multiplet [25], thus making it a natural object to study. Our conventions are exactly those of [19] from which we have lifted some of the equations and to which we refer the reader for more details. The essential summary is as follows. Our scalar transforms in the 14-dimensional [2,0] representation of the so(5) R-symmetry algebra. Since the four-point function features six different R-symmetry channels A R (z,z) with R labeling the representations on the right-hand side. Our conventions for the four-point function and the R-symmetry projectors are given in appendix A. As usual, conformal symmetry and the operator product expansion dictate that each of these admits a decomposition into ordinary conformal blocks: but supersymmetry imposes far stricter constraints. First of all, for this correlator, the information in all six channels is completely encoded by two functions: As indicated, the second function is independent ofz. The relation is through simple second-order derivative operators. To fix ideas let us quote them here in full [25]: A [2,2] (z,z) = 1 2

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with the operator and with u = zz and v = (1 − z)(1 −z) as usual. Equation (2.4) automatically resolves all the constraints of the superconformal Ward identities and was first published in [25]. The deeper reason for the appearance of a meromorphic function h(z) is the existence of a chiral algebra for the six-dimensional (2, 0) theories [26,27].

Superconformal block decomposition
A related consequence of supersymmetry is the grouping of ordinary conformal blocks into superconformal blocks. In the OPE under consideration there can appear eight types of supermultiplets [17,18,29,30]: As indicated, some multiplets can have non-zero spin (which are necessarily odd for the B[0, 2] multiplets and even for the other multiplets by Bose symmetry). The last type of multiplets are the unprotected or 'long' multiplets and can also have arbitrary ∆ (provided it lies above the unitarity bound, ∆ > + 6). The B[0, 0] multiplets contain higher spin currents and will therefore no longer be considered in this work. The contribution of each multiplet to the four-point function is most easily captured by stating their contribution to a(z,z) and h(z); this is given in table 1 with the listed 'atomic' contributions given by [19]: where G ( ) is an ordinary (non-supersymmetric) six-dimensional conformal block, but for a four-point function of operators with unequal scaling dimension ∆ i=1,...4 . We also introduced ∆ ij := ∆ i − ∆ j . The explicit form of the block is given in (A.4). We also use the notation for a block with ∆ 12 = ∆ 34 = 0. With (2.7) in hand one can verify, as was done in [19], that for each line in table 1 the application of the operators in equation (2.4) yields a finite sum of ordinary conformal blocks with the expected quantum numbers in each of the six R-symmetry channels -see the figures in the next section for examples.

OPE coefficients from the chiral algebra
The chiral algebra [26] underlying the (2, 0) theories [27] completely fixes the function h (z) in terms of a single parameter which one may take to be the c central charge. 4 This fixes 9) where β is an unphysical integration constant that does not appear in the correlation function, as is clear from equation (2.4). We fix it as β = −1/6 + 8/c such that the atomic decomposition of h(z) reads One can now use the block decomposition of h(z) to also completely fix the coefficients of the D[2, 0], D [4,0] and B[2, 0] multiplets in terms of c. According to table 1 the latter two multiplets also give a contribution to a(z,z). It is then useful to split off this 'chiral' contribution and write a(z,z) = a χ (z,z) + a u (z,z) , with a χ (z,z) capturing the completely known contribution of the D[4, 0] and B[2, 0] multiplets, and with an 'unknown' part a u (z,z) of the form The blocks that saturate the inequality correspond to the D[0, 4] or B[0, 2] multiplets, and all other blocks correspond to L[0, 0] ∆, multiplets.

Crossing symmetry equations
The crossing symmetry equations which arise from permuting the external operators lead to a set of linear algebraic relations for the functions A R (z,z). Substituting (2.4) into these relations, one finds that all the derivatives can be eliminated and (assuming the above form of h(z)) one also finds simple algebraic crossing equations for the function a(z,z) [19]. These read: The first of the crossing equations relates the t-and u-channel and is solved by demanding that the s-channel block decomposition (equations (2.12) and (2.13)) only contains even spin operators. The second one is less trivial as it relates the block decompositions in different channels. It will be used extensively below.

Regge trajectories and supersymmetry
As we mentioned in the introduction, a given superconformal multiplet contains several conformal primary operators whose Regge trajectories are naturally related by supersymmetry. It might then appear natural to focus on the Regge trajectories of the superconformal multiplets as a whole, and to ignore the trajectories of the superconformal descendants. For the D[2, 0] correlator one can do so by analyzing the Regge trajectories for all spins directly for the function a(z,z) instead of those for the six functions A R (z,z). Perhaps surprisingly, doing so would only paint an incomplete picture: for several superconformal Regge trajectories the arrangement of the conformal descendants, which is dictated by supersymmetry, is not automatically correct. Instead, the combined demands of analyticity in spin together with supersymmetry lead to non-trivial constraints for superconformal Regge trajectories. The aim of this section is to exhibit these constraints and argue for a particular structure of the superconformal trajectories that resolves them. We will first consider the contributions of the short multiplets, which lie on straight trajectories, and then move on to the long multiplets whose twist is not fixed by supersymmetry. The next two subsections detail the issues and are unavoidably a bit technical. The hasty reader may want to skip to subsection 3.3 for a summary of the issues we uncovered and their potential resolution.
All of the issues we discuss below will happen for low spins where, as mentioned in the introduction, the operators do not always manifestly lie on Regge trajectories. The threshold value * is claimed to be determined by the Regge behavior of the correlation function. In section 4.3 we will show that the Regge behavior of the six functions A R (z,z) is actually quite a bit softer than in a non-supersymmetric theory, leading to extended manifest analyticity in spin. Let us here already quote the upshot, which is that: A [4,0] has analyticity in spin for > −3 , A [2,2] has analyticity in spin for > −2 , A [2,0] has analyticity in spin for > −1 , 4] has analyticity in spin for > −1 , 2] has analyticity in spin for > 0 , A [0,0] has analyticity in spin for > 1 . If these inequalities are violated then the shape of the Regge trajectories for the six functions under consideration is less tractable, and a priori we cannot even exclude isolated contributions. 6 However, as long as these inequalities are obeyed we can safely draw conclusions on the shape of the Regge trajectories for the six functions under consideration.

Short multiplets and straight trajectories
Short multiplets with unknown coefficients. We start our exploration with the B[0, 2] −1 multiplets (for even > 0), which are short multiplets whose coefficients are not JHEP01(2022)022 fixed by the chiral algebra. Since their dimensions are fixed, they lie on straight Regge trajectories. We begin by plotting the conformal primary descendants and their trajectories for a few low-lying spins for each of the R-symmetry channels. This yields the black dots and solid black lines in figure 2.
Note that we have added a small horizontal split to overlapping Regge trajectories for presentational purposes: the twist of all trajectories are really even integers. For space reasons we have only added a (partial) legend to the [4,0]  The most interesting feature of these trajectories now follows from the dashed lines, which indicate the continuation of this trajectory to unphysical spins of the superconformal primary. We are forced to draw this continuation because analyticity in spin dictates that Regge trajectories cannot just end in the middle of the (∆, ) plane. After going down two units in spin along this dashed line we find the hypothetical "B[0, 2] −1 " multiplet and going down four units yields the "B[0, 3] −3 " multiplet. Neither of these multiplets exist, so what are we to make of them?

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Let us first consider the "B[0, 2] −1 " multiplet. In the plots we show gray, orange and white dots at the putative locations of its conformal primary operators. The white dots are easiest to explain: they correspond to the absence of an operator due to a kinematical zero. For example, the twist ten trajectory in the [0, 4] channel gives a contribution of the form with λ 2 B[0,2] −1 the contribution of this superconformal multiplet to (2.13). We see that this contribution vanishes at = 0 and this explains the corresponding white dot. The orange and the gray dots are then actual conformal primary blocks that, barring a dynamical zero in the overall OPE coefficient, need to be accommodated by other supermultiplets.
As the reader might have expected from table 1, the multiplet that comes to the rescue is the D[0, 4] multiplet. The operator contents of the latter is indicated by the squares in the figure and we see a nice one-to-one match between these and the gray and orange dots of the "B[0, 2] −1 " multiplet. Although this is not obvious from the figure, almost all the coefficients work out as well. For example, in the [2, 0] channel there is a twist 8 trajectory of the form: and in the limit → 0 this precisely matches a contribution to the D[0, 4] multiplet of the form: 10 (z,z) .

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This mismatch is why we colored this combination orange rather than gray in the figure. 7 Let us call this issue 1. We note that this issue is only present if we have analyticity down to spin 0 in the [0, 0] R-symmetry channel, which does not rigorously follow from the Regge limit analysis summarized in (3.1).
Let us now go down two more units in spin and investigate the "B[0, 2] −3 " multiplet which is indicated by the red dots in the figure. (Perhaps surprisingly, there is no kinematical zero that prevents any of these operators from appearing.) According to (3.1) we expect to have analyticity of the CFT data at all the red points except the scalars in the [0, 0] channel, but these blocks do not correspond to the operator content of any other superconformal multiplet. They are therefore genuinely unwanted contributions, and the fact that they do not vanish automatically is what we call issue 2. Of course, this issue would be resolved immediately if the overall coefficient vanishes, so if lim →−2 λ 2 B[0,2] −1 = 0, but we emphasize that this would be a dynamical constraint. We will find another potential dynamical resolution below.
Short multiplets with known coefficients. Our next set of multiplets contains short multiplets with a contribution to the chiral algebra: the B[2, 0] −2 and the D[4, 0] multiplets. We can repeat the above analysis for these operators, but with the one change that here the OPE coefficients are completely fixed: and therefore the analogue of equation ( whereas the D[4, 0] multiplet does not have such a scalar. We call this issue 3, although we must note once again that analyticity is not guaranteed from (3.1) for spin zero. We have two more issues to discuss. First, as for the previous figure, the further continuation down to a "B[2, 0] −4 " multiplet induces another unwanted scalar operator of dimension 8 in the [2, 0] channel with coefficient +10 (z,z) in the [0, 0] channel which has negative spin for = 0, 2. As we send → 2 this becomes a spin −2 block, which one may check vanishes identically in six dimensions and therefore does not contribute. But for → 0 we find a spin −4 block, which happens to be equal to ( This isolated operator once more cannot fit in a superconformal multiplet and so the block must somehow cancel: this is issue 4. Notice that in this case the OPE coefficient is known, The putative resolution of issue 2 therefore cannot work in this case, since 2/c > 0 in all but the generalized free theory. The last issue for this set of blocks is the D[2, 0] multiplet itself. It contributes three conformal blocks: 6 (z,z) , which we recognize as the contributions from the stress tensor, the superconformal primary, and the R-symmetry current. This block has an OPE coefficient equal to b −2 = 8/c as determined by the chiral algebra and so is present in any bona fide theory. It is however clear that neither of these three operators fits into any of the Regge trajectories we have drawn so far, and this is issue 5.

Long multiplets
Now let us analyze the Regge trajectories for the long multiplets L[0, 0] ∆, . For these multiplets the scaling dimensions are unknown, and likewise there is no a priori proof that the Regge trajectories (∆) have a particularly simple form. We can nevertheless explore the consequences if we assume that there exists a trajectory that extends to low spins. Just as for the protected multiplets, we will find issues corresponding to non-analyticity in the induced trajectories for the A R (z,z) as soon as any Regge trajectory for a supermultiplet crosses the lines with = 0, − 2, or = −4. For efficiency of presentation we would like to capture all the issues in one plot, and to do so we simply invented an otherwise random trajectory 8 that crosses all these three lines as shown in figure 4. We however stress that the issues we are about to list are local, in the sense that they hold for any trajectory crossing these low spins, and do not depend on the global shape of the trajectory. For example, in our view it would have been perfectly possible for a trajectory that intersects the = −4 line to be disconnected from a trajectory that extends to positive spins. The legend in the [4,0] channel indicates the scaling dimension that we have chosen for the first four long multiplets on our hypothetical trajectories. (Recall that ∆ > + 6 for an L[0, 0] ∆, multiplet by unitarity which is easily obeyed here.) Unlike the previous plots there are no exactly overlapping induced trajectories for any of the A R (z,z) and therefore there was no need to add a horizontal split by hand.
At low spins we encounter our first issue at spin 0. We see some kinematical zeroes, indicated by the white dots, which are analytic in spin and nothing to worry about. However the red dot indicates a problem: for a long supermultiplet with quantum numbers (∆, ) and generic there exists a conformal primary R-symmetry singlet of the form: with ξ( , ∆) a rational function of ∆ and that is too ugly to include here. In the limit → 0 we find: which is not the correct expression for a spin zero long multiplet, which instead has a contribution of the form: 9 We can call this issue 6. We stress that it exists for any long multiplet trajectory that crosses the = 0 line, provided analyticity holds down to spin zero in the singlet channel.  Going down in spin, we reach the orange points at spin −2, which correspond to issue 7. Again, a "L[0, 0] ∆,−2 " multiplet does not exist and so generically the combination of orange points should not actually be present in a physical theory. Notice that we have shown the situation for generic ∆; much like the function ξ( , ∆) vanishes at ∆ = 2 there are some zeroes in the coefficients for specific values of ∆ and then some of the orange points may disappear. This will be important below.
Finally if a long supertrajectory hits spin −4 then we induce a single scalar of dimension ∆ + 4 in the [0, 0] channel. This is the green dot and issue 8, which once more is only an issue if analyticity holds down to spin zero in this channel.

Resolving the issues
Let us recap. The issues we have collected are: We see that issues 1, 3, 6 and 8 all pertain only to scalars in the R-symmetry singlet channel. It is not entirely clear that we need to take them seriously: according to equation (3.1) analyticity in spin for A [0,0] (z,z) is guaranteed only down to spin > 1.
Although this would be one way to resolve the issues (or at least provide us with a license to ignore them), there is another option: issue 8 has the potential to resolve the other 3 issues. More precisely we simply postulate the existence of otherwise unknown unprotected Regge trajectories that cross the = −4 line at ∆ = 6 (to resolve issue 1), at ∆ = 4 (to resolve issue 3) and at ∆ =∆ with∆ the dimension of any generic long multiplet that crosses the = 0 line (to resolve issue 6). Whether this is the correct resolution, or whether there is another mechanism at play, is not something we can hope to address with our current knowledge of the (2, 0) theories.
More interesting resolutions can be found for the other issues. The first observation is a perfect cancellation between a special case of issue 7 and issue 2: if a long multiplet trajectory crosses = −2 exactly at ∆ = 4 then it induces precisely the same set of blocks as the "B[0, 2] −3 " multiplet. We can write that where, as always, the evaluation at negative spin is understood to be defined through analytic continuation. This means that issue 2 can be resolved not only by demanding that the B[0, 2] −3 multiplet has zero coefficient (as we hypothesized above), but also by a long trajectory hitting spin −2 exactly at ∆ = 4 with the right coefficient. Just as for the spin −4 long multiplets, at present our understanding of the (2, 0) theories is insufficient to know which of these two potential resolutions is realized. We do note, however, that hitting spin −2 at ∆ = 4 would not be entirely unreasonable for the first subleading trajectory. This trajectory asymptotes to ∆ = + 10 but according to a large spin analysis is expected to slope towards lower ∆ at lower spins. Again with our current knowledge of (2, 0) theories we cannot tell if this or other mechanisms are in place to resolve these issues.

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This leaves us with issue 5 which is arguably the most interesting. Our suggested resolution comes about by another special case of issue 7: an "L[0, 0] ∆=2, =−2 " long supermultiplet, or more precisely the analytic continuation of a regular long multiplet to = −2 and ∆ = 2, gives the contribution: 6 (z,z) , Therefore, if we add this multiplet with a (negative) coefficient −2/c then we reproduce all the stress tensor multiplet blocks with the right coefficient. The one mismatch is an additional scalar block of dimension 8 in the [2, 0] channel, but its coefficient −9/(350c) is precisely such that it cancels the unwanted conformal block of issue 4, which is thereby also resolved! It is therefore entirely natural to conjecture the following observation: The observation that the stress tensor multiplet lies on an unprotected trajectory implies a remarkable interplay between the long and protected multiplets that we could not have observed without appealing to analyticity in spin of descendants. It leads to the improved picture for the leading long trajectory in the (2, 0) theories shown in figure 5.
This picture is again heuristic: to draw the trajectory we took some reasonable guesses for the scaling dimensions of the first few long multiplets based on the numerical bootstrap results of [19] at c = 25. We then included the point ∆ = 2 at spin −2 and drew an otherwise arbitrary curve through all these points. As before, the solid part of the curve corresponds to long multiplets with physical spins 0 and the dotted part is the continuation to negative spin. The gray and orange dots correspond to respectively the resolution of issue 5 and 4. As for any long multiplet there is still an instance of issue 8 which is indicated by the red dot.
From the picture we also observe that the very leading Regge trajectory in the [0, 0] channel is given by a specific conformal primary descendant of dimension ∆ + 4 and spin +4 if the superconformal primary has dimension ∆ and spin . (This trajectory asymptotes to the line ∆ = + 8 which means that it is indeed more leading than any of the short trajectories.) According to our observation this is the trajectory that the stress tensor lies on. In this sense the (2, 0) theories would then be similar to a non-supersymmetric theory like the Ising or O(N ) models where there is ample numerical evidence that the stress tensor lies on the leading (non-straight) Regge trajectory.

Resolution in generalized free field theory
We can now investigate how the issues are resolved in theories whose spectrum and OPE coefficients we know exactly. There are only two such options: the theory of N free tensor multiplets, and generalized free field theory. However, the former is qualitatively different from the cases considered in this work since it is free, and will thus have an extra family of short multiplets (B[0, 0] 0 ), which contain conserved currents of spin larger than two. These multiplets give rise to various new conformal primary trajectories and change the aforementioned issues. In particular, the stress tensor fits as the continuation to = −2 of this trajectory.
We will then consider only the case of generalized free field theory. The corresponding four-point function is obtained just by Wick contractions, and solves the crossing equations with c = ∞. The function a(z,z) for this theory reads: An added advantage of this function is its very soft behavior in the Regge limit and in the second-sheet light-cone limit, which per the analysis of section 4.3 leads to an improvement over the generic behavior of equation ( issues are resolved for the leading trajectories. We note that the superconformal block decomposition of this correlator is easily found, and besides the protected multiplets we find towers of unprotected multiplets at the double twist values ∆ = 8 + + 2n, n ∈ N 0 . The corresponding coefficients can be found by applying the supersymmetric inversion formula of section 4. Let us start with the leading trajectory of the R-symmetry singlet channel, which has twist τ = ∆ − = 8. Operators on this trajectory get contributions from the two short trajectories as well as the leading long trajectory with ∆ = 8 + , 0. Taking the OPE coefficients of these operators we can reconstruct the leading trajectory of A [0,0] (z,z) using the superconformal blocks. Altogether we find the following conformal primaries where c • •,• are the coefficients of the expansion of the superconformal block of the respective supermultiplet in conformal blocks. We chose not to spell them out here, but they are completely known rational functions of ∆ and . Analyticity in spin of the leading Regge trajectory for 0 requires the above to be identical to (3.20) such that the OPE coefficients are an analytic function of . Indeed these two expressions agree due to the following relations between OPE coefficients and the superconformal block coefficients: The last equation means issues 3 and 8 (for the leading long trajectory) cancel out, while issue 2 and 7 (for the leading long trajectory) are resolved by the OPE coefficients vanishing. The same exercise can be done for the twist 10 conformal primaries in the R-symmetry singlet channel, and this time a relation between (a) the spin −4 long multiplet of the sub-leading trajectory (∆ = 10 + ), (b) the D[0, 4] multiplet, and (c) the short trajectory B[2, 0] at spin −1 conspire to solve issues 1 and 8 (for the subleading long trajectory). Carrying out these checks for higher twists, one finds that the OPE coefficients continue to conspire to resolve all issues; for example issue 6 for the leading long trajectory (∆ = 8 + ) is canceled by issue 8 for the sub-sub-leading long trajectory (∆ = 12 + ).

Shadow symmetry in all channels
We have seen that a single Regge trajectory at the level of supermultiplets induces several Regge trajectories for the A R (z,z) which have integer shifts in and ∆ and OPE coefficients determined by the super-Regge trajectory. What is not at all obvious is then whether shadow symmetry is automatically realized. To see that it is, let us focus on the long multiplets. Consider a long trajectory where the spin of the superconformal primary is given by a function s.c.p. (∆). To deduce the shadow symmetry of this function consider first the [4,0] channel. There we see that the super-trajectory induces a single regular conformal trajectory that is shifted by four units, so [4,0] This is a regular bosonic trajectory and therefore obeys ordinary shadow symmetry: [4,0] (6− ∆) = [4,0] (∆). The shadow symmetry of the super-trajectory is therefore with λ 2 s.c.p. ( , ∆) denoting the coefficient of the block corresponding to the superconformal primary (in the [0, 0] channel). The ordinary shadow symmetry [5] reads where K 0,0 ∆, is a kinematical factor defined in equation (4.17) below, and implies a slightly modified shadow symmetry for the superconformal primary: (3.26) Next, let us consider another trajectory, for example the superconformal primary itself in the [0, 0] channel. How can the above equations be compatible with ordinary shadow symmetry in this channel, which would send ∆ → 6 − ∆ and not ∆ → −2 − ∆? The answer is that another superconformal descendant comes to the rescue, as follows. To restore shadow symmetry we need a 'shadow superconformal primary' trajectory s.s.c.p. (∆) given by the shadow of the shadow: but it is precisely the one necessary to recover shadow symmetry between these two multiplets in the [0, 0] channel: with a little computation one finds that We have checked that similar relations exist between other superconformal descendants and that therefore the entire set of Regge trajectories in each of the six R-symmetry channels is shadow symmetric provided the relation (3.26) holds. To the best of our knowledge, the corresponding identities involving the coefficients of the different conformal blocks inside the superconformal multiplet have not been observed before. We expect it to be a very general property, valid for any superconformal algebra, that can for example be used as an additional verification in the computations of superconformal blocks, and perhaps provide extra constraints on the superconformal blocks of non-BPS operators. 11 It would be interesting to see if this property can be deduced more directly from properties of the superconformal algebra.

Supersymmetric inversion
The demonstration of analyticity in spin of the OPE data for any CFT proceeds via the so-called Lorentzian inversion formula of [5]. For the four-point function under consideration one could in principle apply the formula to the six different functions A R (z,z) and combine the data from these operations in order to get a complete picture for the Regge trajectories of the superconformal multiplets. A more elegant approach is to work directly with the function a(z,z). As we will discuss below, this function essentially has all the right properties for the Euclidean and Lorentzian inversion formulas to apply, and its block decomposition will give us direct access to all the OPE coefficients of operators that contribute to it.

Inversion formula for a(z,z)
As can be seen from equations (2.7), (2.12) and (2.13), the function a(z,z), multiplied by (zz) 6 , admits a conformal block decomposition as follows: (zz) 6 a(z,z) = (zz) 6 a χ (z,z) + (zz) 6 a u (z,z) , 11 In fact, we naturally would expect that a similar identity holds if we decompose ordinary conformal blocks into lower-dimensional conformal blocks. This would ensure that, if we for example analyze a correlation function of a three-dimensional theory with two-dimensional conformal blocks, the shadow symmetry of each three-dimensional trajectory separately suffices to ensure shadow symmetry of the (generically infinitely many) two-dimensional trajectories it induces. It would be interesting to verify this.

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It is therefore most natural to apply the inversion formula of [5] to (zz) 6 a(z,z). The inversion formula results in a function c(∆, ) whose meromorphic structure captures the OPE data. Normally one evaluates c(∆, ) at integer and, for generic values of ∆, a pole in c(∆, ) at some value ∆ * signifies the presence of a conformal block in the OPE decomposition, whose coefficient is simply given by (minus) the residue of the pole. In our case there is a small offset to take into account: a pole at generic ∆ * signifies the presence of a supermultiplet contributing as a at ∆ * −4, (for a long multiplet this means the primary has dimension ∆ s.c.p. = ∆ * − 4 according to table 1), and whose OPE coefficient is related to the residue by the factor 4/(∆ * − − 6)/(∆ * + − 2).
The decomposition (4.1) shows that we should treat (zz) 6 a(z,z) as a four-point function of non-identical scalar operators with ∆ 12 = 0 and ∆ 34 = −2. Furthermore, all the spins in its block decomposition are even integers and therefore there is no distinction between the t-and the u-channel contributions. The singularities of (zz) 6 a(z,z) are also those of a four-point function of non-identical scalars, as can be seen from its conformal block decomposition (4.1) and its crossing equation (2.14). All in all, the associated spectral density should then be computable through the Lorentzian inversion formula [5] as 12 where the most important factor involves the double discontinuity operation, which for a generic four-point function g(z,z) reads: where the e 2πi factors indicate an analytic continuation onto a secondary sheet, in this case aroundz = 1. The other factors are given by: and we recall that G ( ) is an ordinary bosonic conformal block. For our case one should set ∆ 12 = 0 and ∆ 34 = −2 in all of the above formulas.

Single-valuedness
In the derivations of the Lorentzian inversion formula [5,34] it is usually assumed that the function to be 'inverted' is a proper CFT four-point function. Here this is not exactly the case: although a(z,z) has a nice decomposition into s-channel conformal blocks, it has slightly awkward t-channel behavior. To see this, recall the crossing symmetry equation (2.14) and take z,z → 1. The functions a(1 − z, 1 −z) and C h (1 − z, 1 −z) are nicely behaved in that limit, but C h (z,z) has a logarithmic term not seen in an ordinary four-point function. Loosely speaking we can write that: 13 (4.6) The problem with this term is that it spoils Euclidean single-valuedness of a(z,z): when settingz = z * and taking z around 1 in the complex plane the function does not return to itself. It would be nice to investigate the true importance of Euclidean single-valuedness around 1 (and around ∞) for the validity of the Lorentzian inversion formula more generally. We can however show that it is unimportant in our case in a simpler way. We consider the function This function was introduced in [18] as the four-point function obtained by simple Wick contractions in the theory of N free tensor multiplets, with c = N (and it is also a fairly natural object from the holographic perspective). The conformal block decomposition of this function is rather easy to find. Furthermore, if we ignore its non-single-valuedness and blindly substitute it into the Lorentzian inversion formula (4.2) we get the right answer. 14 So at least for a free (z,z) the non-singlevaluedness is not an issue. The argument for the validity of the Lorentzian inversion formula for generic a(z,z) is then as follows. Let us define:ã (z,z) = a(z,z) − a free (z,z) . (4.8) Then (zz) 6ã (z,z) is single-valued and has a good conformal block decomposition (albeit without positive coefficients), so the Lorentzian inversion formula should be applicable. Of course we get a different densityc(∆, ) which includes the spectrum of a free (z,z) in addition to the physical spectrum of a(z,z). But the Lorentzian inversion formula is fundamentally a linear operation that in itself does not require single-valuedness. So we can splitã(z,z)

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again and apply the Lorentzian inversion formula to each term separately. Since we verified that the inversion of a free (z,z) works and the corresponding c free (∆, ) yields the correct conformal block decomposition, it must be that the inversion of a(z,z) gives 9) and this function has the right analyticity properties to recover the conformal block decomposition of a(z,z) and nothing more.

Behavior on the second sheet
As we stated above, the results of [5] tie the Regge behavior of correlation functions to the minimal spin for which the OPE data is guaranteed to be analytic in spin. Non-analyticity happens when the contributions to the integrals from 'arcs at infinity' are non-zero [5,34,35]. These arcs correspond to the case wherez → 0 on the secondary sheets reached by going aroundz = 1. If one sends z → 0 at the same rate asz then this is the Regge limit. It is customary to parameterize this limit as: For a general function g(z,z) the arguments of [5] imply that with a Regge behavior of the form g ∼ w 1− * , (4.11) the 'arcs at infinity' contributions from the Regge limit vanish if the integral 0 dw w − * −1 converges, so if > * . In this way softer Regge behavior implies a larger domain of analyticity in spin. Let us investigate the behavior of our correlation function in this limit. We begin by noting that the functions A R (z,z) have a decomposition into ordinary bosonic blocks, including an identity operator for the singlet channel. The non-supersymmetric arguments of [5] then go through: the A R (z,z) must be bounded by a constant in the Regge limit and analyticity in spin holds at least for any spin > 1. Supersymmetry allows us to do better. We can see this by eliminating the derivatives in (2.4) to find an expression for a(z,z) in terms of (z andz dependent) linear combinations of the A R (z,z) and h(z). Since we know h(z) exactly and can also bound the A R (z,z), it is not hard to deduce that in the Regge limit. This is a softer behavior in the Regge limit with respect to a bosonic correlator that grows as w 0 , ensuring analyticity in spin down to spin > −1. However, we expect analyticity to hold all the way down to spin > −3. One partial argument for this are the conformal primary trajectories in the R-symmetry singlet channel: as shown in section 3 a superconformal primary of spin has a descendant of spin + 4 in this channel. The > −3 result for the superconformal primaries then immediately follows from the non-supersymmetric result that analyticity in spin holds down to spin 1 for any conformal family.

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This expectation can be proven by bounding the growth of a(z,z) in the Regge limit directly, as done in [5], using the fact that the t-and u-channel OPEs converge in that limit. In appendix B we use the t-and u-channel OPEs in an expansion in the z andz variables to bound the growth of a(z,z) as along almost any direction in the complex w-plane, confirming analyticity in spin of c(∆, ) all the way for all spins greater than −3! 15 We note that the non-single-valuedness of a(z,z) is not important: since a free (z,z) behaves like w 5 in this limit, equation (4.13) also gives the leading behavior for the singlevalued correlatorã(z,z).
Let us finally discuss the Regge behavior of the six functions A R (z,z). A naive estimate arises by plugging the behavior (4.13) into equation (2.4), but the latter contains derivatives and if a function vanishes at least as fast w 4 then it is not mathematically guaranteed that its derivative vanishes at least as fast as w 3 . In appendix B we therefore explain how to bound the derivatives of a(z,z) directly, leading to the Regge behavior: This directly leads to the results quoted in equation (3.1) at the beginning of section 3.
The lightcone limit. Another potential contribution from the 'arcs at infinity' comes from a lightcone limit on the second sheet that corresponds toz → 0 holding z fixed. In this limit the convergence of the Lorentzian inversion formula depends on ∆ and ; more precisely, if g(z,z) ∼z τ * /2 in that limit, then we need It is not immediately clear what the correct value of τ * could be. On the first sheet τ * is equal to the lowest twist in the s-channel spectrum. In that case τ * ≥ d/2 − 1 would be appropriate for a generic CFT, because the s-channel identity operator needs to be subtracted, while in our case τ * = 8. As discussed in [34], this value again leads to analyticity in spin for > 1 because we can set ∆ = d/2, and in our case it would not spoil > −3. On the secondary sheets this is not necessarily true because the s-channel block decomposition no longer converges -instead one uses the t-channel conformal block decomposition (and unitarity) to bound the full correlator by its behavior on the first sheet. The best possible bound that can be rigorously proven in this way corresponds to τ * ≥ 0. This value is however unlikely: it would imply non-convergence of the Lorentzian inversion formula for ∆ < and invalidate the analyses of the first Regge trajectory in the literature.
(In our case we can show that τ * ≥ 4 instead of zero from (B.3) and (B.8), but the effect 15 To be precise, the z andz expansion can be used to show the given behavior along any direction with arg(w) = π/2. Using the ρ variables it might be possible to show the bound is valid also exactly along the imaginary axis. See appendix B for details.

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is similar.) It would be interesting to have a better handle on the lightcone behavior on the secondary sheets, but we will not consider possible lightcone contributions further in the remainder of this paper. See [34,[36][37][38] for previous discussions on the second sheet lightcone limit.

Analyticity properties of c(∆, )
As we explained in the previous subsection, conformal blocks in the decomposition of a(z,z) correspond to poles in the function c(∆, ). These are however not the only poles: there are also kinematical singularities. It is therefore worthwhile to take a look at the analytic structure of c(∆, ), which we will do in this subsection. In contrast with most other studies we will also be interested in what happens at negative spins . The result of our analysis is summarized in figure 6.

Kinematical singularities
Recall that the shadow symmetry of the OPE density is embodied in the following equation where The shadow symmetry for c(∆, ) follows automatically from c(∆, ) as defined through eqs. and one can recover the kinematical singularities of c(∆, ) by removing the K factor. From (4.2) we see that these singularities are encoded in the kernel block times a prefactor, more precisely +5 (0, −2; z,z). Using (5.11) it can be easily checked that this combination is shadow symmetric.
The generic pole structure of conformal blocks was discussed in [39,40], but it does not apply to even spacetime dimensions where we can also encounter double poles. In our case d = 6 and thus we will analyze the pole structure using the closed form of the six-dimensional conformal blocks directly.
We are only interested in the pole locations and below in table 2 we tabulate both the simple and double poles in +5 (0, −2; z,z). We have written these as poles in spin because we will soon be interested in the setup with fixed ∆ and negative .

Dynamical poles and analyticity in spin
Finally we have to take into account the dynamical poles in c(∆, ). The conformal block decomposition of (zz) 6 a(z,z) was given in equation (4.1). The short multiplets induce two straight Regge trajectories: one with known OPE coefficients at ∆ = + 8 and one with unknown OPE coefficients at ∆ = + 10. The long multiplets have unknown scaling dimensions, but as we discussed in the previous subsection we expect the leading long JHEP01(2022)022 pole locations in simple poles double poles  trajectory to smoothly continue to the stress tensor. If we take the various offsets into account this means that it must cross through the point ∆ = 6 and = −2 in c(∆, ). The numerical results of [19] further suggest that this trajectory asymptotes to the leading double twist trajectory, which here implies that its asymptotic twist equals 12.
In section 4.3 we analyzed the Regge behavior of the function a(z,z) and found that it behaves rather smoothly; if z = σw andz = w/σ then (zz) 6 a(z,z) ∼ w 4 as w → 0 on the secondary sheets. The integral in (4.2) therefore converges in that region as long as > −3, and the contributions from the 'arcs at infinity' in the Regge limit must also vanish for this range of spins.
An impression of the interplay of kinematic and dynamical poles is shown in figure 6. We omitted the subleading unprotected Regge trajectories in the figure. Note that the claim 1 implies the short B[2, 0] trajectory and the leading long trajectory intersect at spin −2 and ∆ = 6 in figure 6. The OPE coefficients of the two trajectories are such that the residue of c(∆, −2) at ∆ = 4 vanishes.

Practical supersymmetric inversion
In this section we will discuss some practical aspects of working with the inversion formula (4.2) for a(z,z). We will first discuss the use of the crossing equation and the simplifications that occur after substitution of the t-channel conformal block decomposition. This provides sufficient background for a small (and crude) numerical test of convergence for negative spins which we perform in subsection 5.2. It will be convenient to consider the small z expansion of the integrand of equation (4.2), and in subsection 5.3 we explain how this works and use it to recover the OPE coefficients of the short multiplets that contribute to the chiral algebra. This all provides sufficient background for more serious numerical experiments in section 6.
One subtle but important point of the inversion formula (4.2) is relegated to appendix C. In that appendix we discuss how to regulate divergences that arise in the z → 1 limit of the integral. Roughly speaking these divergences arise because the scaling dimensions are integers, and in practice we find badly divergent integrals of the type 1 dz (1 − z) −n . Although such divergences are invisible in the small z expansion discussed in subsection 5.3, it is of course of fundamental importance that we are able to tame them since otherwise the entire inversion formula would stop making sense.

The t-channel decomposition
The double discontinuity (4.3) vanishes for each s-channel block separately and the integral in (4.2) does not commute with the decomposition into these blocks. Instead one can employ crossing symmetry and consider the t-channel block decomposition. In typical 'experiments' one approximates the four-point function on the right-hand side of (4.2) with a finite sum of t-channel blocks and investigates the resulting approximation to c(∆, ).
Let us recall that the crossing equation (2.14) for a(z,z) reads: on the right-hand side to highlight that one part of this function is known and fixed by the chiral algebra as discussed in section 2. The Lorentzian inversion formula instructs us to take the double discontinuity of the right-hand side of equation (5.1). It is well-known that, in general conformal field theories, the contribution to the double discontinuity vanishes for the t-channel operators with doubletwist quantum numbers. In our case even more cancellations occur than the "double-twists" of the inverted correlator (zz) 6 a(z,z), although they do correspond to double-twists in the different R-symmetry channels.

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First consider a u (1 − z, 1 −z). Its block decomposition was given in equation (2.13) which we repeat here: It is not hard to verify that the blocks with ∆ = + 6 that saturate the inequality, which we recall correspond to the D[0, 4] multiplet (for = 0) and the B[0, 2] multiplets (for > 0), have vanishing integrated double discontinuity. Therefore only the long multiplets can contribute, and: 4) so the inequality constraint for ∆ has become strict. The double discontinuity of a single t-channel conformal block is where we have used the fact that is an even integer. Next we consider a χ (1 − z, 1 −z). This is a known function whose block decomposition was given in equation (2.12). For these blocks the integrated double discontinuity is not vanishing, 16 but instead it cancels almost entirely against the contribution from C h (1 − z, 1 −z). In fact we may write: withh(z) defined as the 'truncation' of h(z) to just the identity and the stress tensor block: Notice that the identity and the stress tensor are the only multiplets that do not contribute to a(1 − z, 1 −z) (excluding higher spin currents). The previous two paragraphs imply that, out of the all t-channel data, only the identity, the stress tensor multiplet, and the long multiplets contribute to the double discontinuity. The contribution of all the other short multiplets simply drops out. Notice that there is furthermore a contribution to the double discontinuity from C h (z,z) which is non-vanishing 16 The twists of these blocks are even integers. To see that the integrated double discontinuity is nonvanishing one can expand the integrand aroundz = 1 and will then find negative powers of 1 −z. This diverges asz → 1 and cancels with the vanishing factor from double discontinuity of even-twist blocks. See, for example, section 3.1 of [14] for more detailed explanation.

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and not subject to the above cancellations. In equations: where in passing to the final expression we have only kept the terms with non-vanishing integrated double discontinuity.

Convergence along shadow-symmetric line
As a first experiment we can try to see if we can numerically observe convergence for negative spins as predicted by the Regge behavior discussed in section 4.3. To do so we can start with (5.8) and plug in increasingly many t-channel blocks. Since we do not know the exact spectrum (and are only interested in a quick numerical check of the convergence properties anyway) we took a maximally crude approximation: apart from the leading Regge trajectory, for which we used some input from the numerical bootstrap results of [19], we took a 'shifted' mean-field-like spectrum of scalar operators where all the dimensions were shifted from their mean field values by 1. This was done to maximize their contribution to the dDisc, because with the shift the prefactor sin 2 (π(∆ + )/2) = 1. For the OPE coefficients we took the values obtained from the supergravity computation of [18] extrapolated to finite c. We did not add spinning operators since these numerically would have contributed much less to the double discontinuity anyway. We then inverted this spectrum using (4.2) and plot the resulting function c(∆, J) at the shadow-symmetric point where ∆ = d/2 + 10 −5 , taking care to regulate the z → 1 divergences as discussed in appendix C. The results are shown in figure 7. Note that we have divided out unphysical poles analyzed in subsection 4.4. Although our estimate is extremely crude, we do see convergence for negative spins and a breakdown not until spin −3 for c = 25 and around −4 for c = 8748. Note that the pole at = −5 is expected: it is the intercept of the leading short trajectory which we also show in figure 6.

Small z expansion
We would like to use the inversion formula (5.8) to extract anomalous dimensions and OPE coefficients. A practical way to do so is to use the small z expansion [5], which works as JHEP01(2022)022 shorts +1 long +10 long +20 long +30 long
(5.10) Here we have restricted the range ofz integration to z <z < 1 which brings in a factor of 2. We are also considering only the even spin c(h,h) and thus replaced 1 + (−1) = 2. In the limit z → 0 the bracketed expression above can be expanded in powers of z and then the outer integral simply converts these powers into poles of h. The OPE coefficients are then given by the coefficients of this power series expansion -up to a Jacobian to transform from (h,h) back to (∆, ).
When away from shadow symmetric line we can simplify (5.10) by discarding "half" of the kernel block. This follows from the fact that one can split a conformal block into two parts: where each of the G pure can be expanded into pure power terms in the limit z z 1 [5]. For example, the contribution from the first G pure to thez integral kernel in (5.10) has the JHEP01(2022)022 following expansion where the constant prefactor is inserted to simplify notation, and Similar manipulations can be done for the second pure power term, with one notable difference that the second expansion scales as z ∆−d/2 (1 + O(z)) which we can ignore when we are to the right of the shadow symmetric line. The expansion coefficients in (5.12) are easy to fix 17 and the first few coefficients with d = 6, ∆ 12 = 0, ∆ 34 = −2 are , (5.14) After substituting the inverted block with G pure and using the expansion (5.12), we can define a generating function as k 0,−2 2(h+j) (z) dDisc t (zz) 6 a(z,z) , (5.15) such that (5.10) can be rewritten as In the existing literature the small z expansion of the inversion formula is generally used to extract CFT data of the leading trajectory from the generating function. For such a case we would only need the leading (n = j = 0) term. We may write

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and the OPE data for the leading trajectory can then be extracted as The anomalous dimension can be determined by solving the equation (5.19) and the squared OPE coefficient can be calculated through (5.20) where the Jacobian factor is needed because λ 2 ∆, is the residue of c(∆, ) with respect to ∆ at fixed .
In our case the setup is slightly different. A quick look at figure 6 shows that for all non-negative spins the leading two trajectories are expected to be straight: they correspond respectively to the short multiplets belonging to the chiral algebra (with known coefficients) and the short multiplets at the unitarity bound of the long multiplet (with unknown coefficients). After that we find the leading unprotected trajectory. In equations this means that, if we write the generic power expansion ansatz,  a(z,z): taking all the prefactors into account it is easy to see that the contributions from a at +4, begin at z 4 , those from a at +6, at z 5 , and those from the long multiplets at even higher powers; for example, in mean field theory the leading unprotected trajectory begins to contribute at the z 6 term.
Next, let us try to discern the origin of the various powers in (5.22) from the t-channel perspective. (In the footnote below we discuss the obvious issue of convergence.) To do so we have to remember the inhomogeneous crossing symmetry equation for dDisc t (zz) 6 a(z,z) , which concretely speaking leads to the extra terms displayed in equation (5.8). These (known) terms already result in z 4 and z 5 terms (without logarithms) at small z, and in the next subsection we will show that the z 4 term precisely reproduces the OPE coefficients of the short multiplets that contribute to the chiral algebra. This leaves us with the sum over JHEP01(2022)022 t-channel blocks corresponding to the unknown operators. At small z, a single t-channel block looks like: (1 −z) , (5.23) where the prefactor on the left-hand side matches that of the crossing equation (5.8) and the explicit form of the H functions will not be important for us. Remarkably, this is precisely the structure that we would expect from equation (5.22): no contribution to the z 4 term, which is therefore determined entirely by the inhomogeneous terms in the crossing symmetry equation, a non-zero z 5 term, which means a non-zero contribution to the short multiplets with undetermined coefficients, and a logarithm signifying an anomalous dimension starting only at order z 6 . 18 In the next subsection we will first invert the z 4 term in order to recover the known OPE coefficients of the chiral algebra. In section 6 we will numerically estimate some of the unknown OPE data corresponding to the z 5 term and beyond.

Recovering the chiral algebra shorts
In agreement with the previous discussion we will take the coefficient of z 4 from the inhomogeneous terms in the crossing equation (5.8) and ignore any contribution from the unprotected multiplets. Substituting this into equation (5.15) for the generating function, the integral to do is then: and with a bit of algebra one then finds (5.26) It is easy to check that the result matches 2 b for all even spins, but is a nicer function of . 18 An important caveat to the above analysis is that an infinite sum over t-channel blocks need not have the same behavior as a single block, and so using equation (5.23) in the crossing symmetry equation does not constitute a completely reliable estimate for the small z limit of a(z,z). In fact, a comparison of the (known) z 4 term on both sides of the crossing equations does show that the t-channel blocks must sum up to give non-zero z 4 and z 5 terms in (zz) 6 a(z,z). However, it is also easy to see that these terms have vanishing double discontinuity and are therefore unimportant in the Lorentzian inversion formula. Also, each t-channel block only contributes with log(z) and integer z powers, while finite anomalous dimensions (with respect to exact double-twist dimensions) require higher powers of the log(z).

JHEP01(2022)022 6 Numerical approximations
In this section we will use the small z expansion of subsection 5.3 for some numerical experiments. Our main question is whether the input of some limited t-channel OPE data can produce a reliable approximation of the s-channel OPE data. We will mostly focus on the low-spin operators in the leading Regge trajectory which will allow us to make comparisons to the numerical bootstrap results of [19].

Inversion for higher-twist trajectories
In this section we generalize (5.18), which states how to extract the leading-twist data from the inversion formula, to higher-twist trajectories. This amounts to extracting the coefficient and powers of the z 5 and higher terms in C(z, h,h) shown in equation (5.22). These terms get contributions from the long multiplets, which we do not know exactly, and so we use an approximation scheme where we input only finitely many long t-channel blocks. For the z 5 term the scheme is obvious: we just sum the coefficients of all the z 5 terms of whatever t-channel blocks we put into the inversion formula. The resulting coefficient in the generating function directly provides an estimate for the OPE coefficients of the D[0, 4] and B[0, 2] short multiplets. The integrals to be done are detailed in subsection 6.1.3 below.
For the unprotected multiplets the ansatz (5.22) in principle dictates that we should compute: where the 'sub.' superscript indicates that we should subtract the contribution of the more leading terms in the small z expansion before taking the limit. This prescription however runs into the familiar problem that each t-channel block contributes only a log(z)z 6 term as z → 0, and such terms need to exponentiate to recover the expected zh (h) behavior, signaling that the t-channel sum does not commute with taking z → 0 term by term. In order to remedy this we will evaluate the right-hand sides of (6.1) at finite but small z.

Finite z inversion
The essence of the "finite-z inversion" method [7,10,12] is to replace equation (6.1) with an estimate at small but finite z 0 Notice that at finite z 0 higher order terms in the expansion (5.22) also contribute and this introduces h-dependence into the approximation of h n (h) which is therefore now denoted as h z 0 ,n (h,h). Equation (5.19) is modified into

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Since the exact answer is independent of h, a weak h-dependence of h z 0 ,n should be a sign of its good approximation to h(h). Note also that now we have distinct equations to solve for each spin -a significant difference compared to the more common (practical) analyses of the leading Regge trajectory. The value of z 0 in (6.2) is crucial to the approximation results we get, but before discussing the determination of z 0 , let us first introduce another variable y which turns out to be useful for the inversion calculation. The y variable is defined such that the inversion of the "generalized free" part in (5.8), i.e., the protected part of a(z,z) in the limit c → ∞, gives exactly zero anomalous dimensions even when using the finite-z inversion (6.2). Denoting the inversion result as C sub.
and by construction this gives for any z 0 . This also gives reliable results for anomalous dimensions of operators with large spin which asymptote to generalized free operators. The switch from z to y introduces a Jacobian factor and (6.2) becomes where the Jacobian is calculated by taking a derivative on both sides of (6.4), and 19 Although y is conceptually a nicer variable, from (6.4) we see that it depends on z, h,h, which makes it practically more complicated. Therefore, we will write y 0 = y(z 0 , h,h) and all the finite-y inversions in this paper will be performed at fixed z 0 rather than y 0 . To find the suitable value of z 0 (or y 0 ), we will plot both h z 0 ,n (h,h) (or h y 0 ,n (h,h)) and P z 0 ,n (h,h) (or P y 0 ,n (h,h)) as functions of z and look for a plateau, similar to the approach in [12]. Because of the relatively small amount of unprotected OPE data available to us, we will first pick a value for z 0 based on inversion of only the protected part of a(z,z) (called short inversion). After extracting the unprotected CFT data (see next paragraph for details) we will check if the selected z 0 still sits on a plateau when inverting both protected and unprotected part of a(z,z) (called long inversion). This is illustrated in figure 8 where we consider the leading long scalar multiplet with c = 98. By repeating this procedure for other values of central charge and spin, one will find different suitable values for z 0 . 19 Notice that we extract the coefficient of z hy 0 ,n instead of y hy 0 ,n , because, as can be seen from (6.4), y is defined to include the "generalized free" OPE coefficient. One could also have defined y dividing the right-hand side of (6.4) by the OPE coefficient, and then we would extract the coefficient of y hy 0 ,n . The vertical lines on both plots pass through log 10 z = −1.55, and this is our choice for the finite y inversion. The orange curves indicate that the plateau regions remain largely unchanged after adding the long multiplet contribution. We also included plots using the z variable for comparison.

Iterative inversion procedure
To test whether we can bootstrap the CFT data out of the protected sector we will adopt an iterative procedure. First we invert only the short multiplets to obtain a set of estimated data, denoted as iter 0 , for the multiplets on the leading long trajectory up to spin max = 16. 20 Then we invert these long multiplets from iter 0 to obtain a new set of CFT data, iter 1 , again for the leading long trajectory again up to max . Then we iterate for several times until the data converges and iter n will be our final numerical estimate. In practice we found that 2 n 5 is sufficient, depending on the central charge. As mentioned before the value of z 0 for finite-y(z) inversion should be determined separately for each spin. However, for large spins the anomalous dimensions are highly suppressed and vary little as we change z 0 , thus in practice we can use the same z 0 for spins 2 or 4 depending on the central charge.

Extract the OPE coefficients of the shorts at bound
With an estimate for the long multiplets in hand we can invert for the OPE coefficients of the non-chiral algebra short multiplets. 21 This can be done similarly as in section 5.3.1. The main differences are that we need to extract the coefficient of z 5 term rather than the z 4 term, and invert both protected and unprotected operators.
In equations we have It is straightforward to extend to higher max, but we find that this does not noticeably change the results.
We do expect, however, that including higher-twist long trajectories can improve the results significantly (see subsection 6.2). 21 Recall that the t-channel non-chiral algebra short multiplets have vanishing double discontinuity and therefore they do not participate in the iteration procedure, as discussed in section 5.1.

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and 2] . Inverting the protected part in a similar manner as before yields the contribution from the short multiplets in the t-channel: As for the long multiplets, using the double discontinuity expression (5.8) and the expression generating function (5.15), we find that they contribute: where we have used primed notation for t-channel quantum numbers. Note that the t-channel block is evaluated at z = 0, which gives a finite linear combination of eight hypergeometric functions 2 F 1 (. . . , 1 −z). Therefore, for each 6d conformal block thez integral breaks down into eight atomic integrals and they can be performed analytically using the method in [10]. Equation (6.10) presents a kind of 'supersymmetric sum rule' for the OPE data. We see no reason why it could not be absolutely convergent for all spins. One interesting application could then be the vanishing OPE coefficient of the D[0, 4] multiplet for the A 1 theory with c = 25 which was discussed already in [19]. In that case the positive contribution from equation (6.9) needs to be offset by the uniformly negative contributions from the long multiplets in equation (6.10). More generally, taking only finitely many t-channel blocks into account would provide an upper bound on the OPE coefficients.
We will plot all of this data as a function of the central charge c. As a reminder, we note that for the A N −1 and D N theories

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In each of the three cases we will take as our initial input the inhomogeneous contribution to the crossing equation, i.e., the last two lines on the right-hand side of (5.8). At first order in the iteration this yields something that bears qualitative similarities to the supergravity answer, but with the important difference that we work at finite c and finite z -so the familiar derivatives of conformal blocks now get replaced with an approximation consisting of regular conformal blocks. We then feed the answers for the leading Regge trajectory into the right-hand side of (5.8) and iterate a few times as described in the previous subsection.
Besides this straightforward iteration scheme, we have also attempted to improve our results for the A 1 theory with c = 25 with a small variation where we input the following numerical bootstrap data [19]: These are respectively the dimension and coefficient of the first scalar and the dimension of the first spin 2 operator. These values are rough extrapolations of the bootstrap bounds of [19] which are believed to be saturated for the (2, 0) theory with c = 25. To incorporate these values we have simply replaced the corresponding OPE data in the output of an iteration with (6.12) before feeding it back into the next iteration.

Dimensions of leading long multiplets
In figure 9 we present the conformal dimensions of the multiplets in the leading long Regge trajectory for the first few spins as functions of c −1/3 . 23 We include both the results from short inversion (the starting point of the iteration) and long inversion (the fixed point of the iterated inversion for the leading long trajectory). The shaded regions in the plots correspond to the numerical upper bounds from [19] which are available for = 0, 2, 4. 24 We also plot some holographic results: firstly the two-derivative supergravity solution [17,18], which agrees with large-spin perturbation theory, secondly the c −5/3 correction for the A n and D n theories obtained in [13] which arises from higher-derivative terms in the bulk, and thirdly the c −2 and c −8/3 corrections which arise from bulk loops and which were determined from the tree-level data also in [13]. 25 Although these curves only correspond to the first few terms in the large c expansion, we have nevertheless plotted them down to finite values of c. 26 Our best results are those for intermediate values of c where we differ significantly from the supergravity results and more closely trace the numerical bounds. For example, for 23 The reason of using c −1/3 is that this helps separate the large c results further than using c −1 , and c −1/3 is proportional to N −1 in the large N limit. 24 Note that the upper bound for leading long scalar at c = 25 in figure 9 is around 7.1 instead of 6.4 as shown in (6.12), because these gray shaded bounds are obtained with D[0, 4] multiplet present while the upper bound ∆0 6.4 is obtained by removing D[0, 4] by hand. See [19] for more details. Also, the steps occurring in the spin 4 bound are simply a numerical artifact due to an early termination of the binary search. 25 The computation in [13] left some coefficients undetermined. We only show a comparison with this data for higher spins where these coefficients have no effect. 26 We   Figure 9. Dimensions of the leading long multiplets with spin 0, 2, 4, 6, against c −1/3 . The gray shaded regions are numerical upper bounds (available for spin 0, 2, 4) from [19]. The orange numbers are central charges (a strict subset of the chosen values correspond to physical theories). The orange curves correspond to short inversions and the red curves correspond to stabilized iterative long inversions up to spin 16 (see text for a detailed explanation). For c = 25, we also perform a corrected iterative long inversion, taking into account the numerical results of [19], and the result is indicated by a triangular point. We also show 1/c results from two-derivative supergravity and higher order c α corrections (denoted as O(c α ) in the plot legend) from [13,23] whenever available. 10 2 c 10 3 the numerics indicate the existence of an extremal solution to the crossing equations with a fairly large ∆ 0 . The holographic results significantly underestimate this gap, but our repeated inversion formula tracks it much more reliably. Our iteration scheme is less successful for the only non-trivial theory with c < 98, which is the A 1 theory with c = 25: at this point our estimates sometimes even exceed the numerical bounds. This likely happens because the anomalous dimensions are too large and recovering them correctly in the s-channel requires the input of (many) more conformal blocks in the t-channel. 27 For comparison: in the three-dimensional Ising model the largest anomalous dimension for a physical operator on the leading Regge trajectory is approximately 0.036 (the stress 27 We have also attempted to include one more Regge trajectory in the t-channel but this did not significantly change the results. At a superficial level this is because anomalous dimensions and OPE coefficients go to zero quickly for higher trajectories, yielding suppressions in the Lorentzian inversion formula. This leads us to suspect that one may need to include more than just the double-twist Regge trajectories (of the external operators) in the t-channel, but we have not investigated this further.  Figure 10. OPE coefficients of leading long multiplets with spin 0, 2, 4, 6, against c −1/3 . The gray shaded regions are unpublished numerical upper bounds (available for spin 0, 2, 4) from [19]. The orange numbers are central charges (a strict subset of the chosen values correspond to physical theories). The orange curves correspond to short inversions and the red curves correspond to stabilized iterative long inversions up to spin 16 (see text for detailed explanation). For c = 25, we also perform a corrected (triangular point) iterative long inversion using the numerical results of [19]. tensor), whereas for c = 25 the scalar (with dimension ∆ 0 ) has an anomalous dimension of about 1.6.
As shown in the plot, the c −5/3 corrections are different for the A n and D n series of theories. This raises the interesting question whether our iterative inversion scheme could potentially also recover this difference. There might, for example, be different fixed points of our procedure which one could try to find by starting with different initial conditions, for example by an additional input of the c −5/3 corrections. In this way we could hope to iterate towards solutions that the numerical bootstrap cannot find. It would be very interesting to try this in future work.

OPE coefficients of leading long multiplets
In figure 10 we present the OPE coefficients of the leading long multiplets for the first few lowest spins as functions of c −1/3 . Similar to before, we show results from short inversion, the corrected and uncorrected long inversions, together with supergravity results and JHEP01(2022)022 numerical estimates. 28 The picture is rather similar: for intermediate central charges we much more accurately trace the numerical bounds (especially in the scalar sector) than the supergravity result. This corroborates our viewpoint that our iterative procedure converges towards the extremal solution.

OPE coefficients of non-chiral algebra short multiplets
In figure 11 we present estimates for the OPE coefficients of the short multiplets not fixed by the chiral algebra, for the first few lowest spins and as functions of c −1/3 . 29 Here the short inversion produces the same results as the one from supergravity and therefore we only need to show the long inversion results. We also show the results from [13] whenever applicable. The relative normalizations between that work and ours are as follows: Here we find only marginal improvement over the supergravity answers. Note that for A 1 theory (c = 25), λ 2 D[0,4] should vanish [19], but we were not able to recover this result from any estimate that involves only a single trajectory. Indeed, if we use (i) the numerical upper bounds on ∆ 0,2,4 , λ 2 L[0,0] =0,2,4 from [19], (ii) the assumption that large spin perturbation theory is reliable for 6, and (iii) convexity of the leading long trajectory then one cannot recover λ 2 D[0,4] = 0 at c = 25 by just inverting the data on the leading Regge trajectory. The difficulty of recovering this vanishing OPE coefficient is also confirmed by an analysis we present in appendix D, which shows that we need unrealistically large OPE coefficients if we only use a few t-channel blocks. 30 Finally, let us mention that a similar negative result holds for the recovery of the stress tensor multiplet. Recall that we argued in section 3 that the leading unprotected Regge trajectory should cross the stress tensor point at (∆, ) = (2, −2), which would correspond to an anomalous dimension equal to 4. In practice we observed that the finite y estimates rapidly stopped being sensible already at smaller negative spins: for example, we find a zero in the function y(z, h,h) which leads to a singularity in h y 0 (h,h). It is then not sensible to try even lower spins without adding many more t-channel blocks. 28 The numerical results are unpublished results from [19]. They correspond to upper bounds for the squared OPE coefficients under the assumption that the corresponding conformal dimensions saturate their own bounds. They are therefore similar to those shown in figure 12 of [19]. Since the inversion procedure does not produce exactly the same conformal dimensions, these upper bounds on OPE coefficients are not entirely applicable, but we do expect them to provide decent estimates. The bounds appear jittery due to an imprecise determination of the long dimensions following from an early termination of the binary search -see footnote 24, and also due to insufficient numerical precision as noted in [19]. 29 The numerical bounds displayed appear jittery due to "failed searches" which occur because the numerical precision used is barely sufficient to obtain these bounds as noted in [19]. 30 On the other hand, as explained in subsection 6.1.3, the fact that λ 2 D[0,4] = 0 translates into a specific sum rule for the other OPE data and one can ask whether this datum can be used as an input in order to improve our other estimates for this theory. We leave this as an interesting open question for future work.   Figure 11. OPE coefficients of non-chiral algebra short multiplets with spin 0, 2, 4, 6, against c −1/3 . The gray shaded regions are numerical upper bounds (available for spin 0, 2, 4) from [19]. The red numbers are central charges (a strict subset of the chosen values correspond to physical theories). Short inversions produce the same results as the supergravity ones. The red curves correspond to stabilized iterative long inversions up to spin 16 (see text for detailed explanation). For c = 25, we also perform a corrected (triangular point) iterative long inversion using numerical data from [19]. We also show higher order c α corrections (denoted as O(c α ) in the plot legend) from [13,23] whenever available.

Outlook
We have explored the consequences of analyticity in spin for the six-dimensional (2, 0) theories and found an interesting interplay between supersymmetry and Regge trajectories. Some numerical experiments allowed us to approximately bootstrap the four-point function of the stress tensor multiplet. Let us mention a few possibilities for further explorations. First, our numerical experiments can be extended. We can certainly consider other correlators or other theories, in different dimensions and with varying amounts of supersymmetry. Other possibilities include the incorporation of subleading Regge trajectories for the correlator at hand, or perhaps a multi-correlator study to improve the estimates. A related direction is the incorporation of known results into the numerics. For example, for the leading trajectory we claim to know both its location and coefficient at spin −2 and it would be nice to somehow use this information.

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It would be especially nice if we could use the c −5/3 results because it would allow us to distinguish between the A-and D-type theories at large c. Perhaps this can be done by finding different fixed points for very large c and then tracing them as c gradually decreased.
Second, it would be interesting to have a better idea of the Regge trajectories in supersymmetric theories for negative or non-integer spins. There should also be supersymmetric versions of the light-ray operators of [41] and it would be interesting to study their properties. Also, is there a more direct argument for the improved Regge behavior (and larger analyticity in spin) in supersymmetric theories or do we have to analyze individual multiplets for each correlator separately? Furthermore, the softer Regge behavior of supersymmetric correlators means the dispersion relations of [42,43], which reconstruct a correlator from its double-discontinuity should apply directly without any subtractions. It would be interesting to consider the convergent sum-rules of [43] in the case of the N = (2, 0) theories, and to solidify our understanding of the Regge intercept by resolving the questions we discussed in section 4.

Different amounts of supersymmetry and different dimensions.
Much of what was discussed here in the context of six-dimensional N = (2, 0) SCFTs is not unique to these theories. Four-point functions of BPS operators often enjoy an extended analyticity in spin due to a softer Regge behavior of the correlator that is inverted. Roughly, in dimensions d > 2 one expects analyticity in spin for > , with = 1 − d−2 2 N for d = 5, and = −1 for d = 5, from the fact that supersymmetry relates conformal primaries of spin one to conformal primaries with spin . 31 As a result, Regge trajectories of the superprimaries exchanged in such OPEs are more constrained than their bosonic counterparts, with analyticity in spin imposing constraints on short and long trajectories. We leave a detailed study of the interplay between analyticity in spin and supersymmetry for future work, and simply conclude with a few general comments on how the stress tensor fits into superconformal primary Regge trajectories in interacting theories.
In the case of SCFTs with more than eight supercharges we expect the stress tensor to fit into the leading long trajectory at a negative spin, similarly to what was discussed here. For maximally supersymmetric theories it corresponds to the spin −2 continuation of the leading long superprimary trajectory, which should have dimension 0 (−1) in four (three) dimensions. This follows from examining the half-BPS superblocks for the stress tensor supermultiplet correlator given in, e.g., [25,[45][46][47][48], and we expect this structure to hold for other half-BPS correlators. This fits well with the results for planar N = 4 Super-Yang-Mills, where complete Regge trajectories can be obtained numerically, for any value of the 't Hooft coupling, from the quantum spectral curve approach [49]. The Regge trajectory obtained in [49] is that of the leading unprotected single-trace conformal primary operators, in the 20 R-symmetry representation, that appear in the stress tensor superprimary's self-OPE. 32 For spin greater than two, the operators in this trajectory are 31 This can be seen from the structure of the superconformal multiplets in [44]. However, a precise statement requires an analysis of the Regge behavior of the correlator whose conformal block decomposition is being inverted. As far as we know this has been only done in four dimensions for chiral operators in the non-chiral channel [16], and N = 4 half-BPS operators [15]. 32 Note, however, that the leading singe-trace trajectory is not necessarily the leading Regge trajectory of the full non-perturbative CFT. This trajectory is also analyzed in [4].

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genuinely unprotected long operators, but at spin zero one finds the superprimary of the stress tensor multiplet itself. As pointed out in [49], the fact that the spin is an even function of the dimension, for this Regge trajectory, follows from shadow symmetry of the Regge trajectories in the 105 R-symmetry channel. We expect shadow symmetry in this channel (and all others) will follow from the structure of the superconformal blocks, as we observed in section 3.4 for the (2, 0) theories. Just like in the N = (2, 0) case, the stress tensor supermultiplet OPE in N = 4 features various short multiplets whose OPE coefficients are completely fixed from the chiral algebra of [26], see [47] for details. As argued in [15], the superprimary Regge trajectories are analytic for all spins greater than −3, and we expect other connections between long and short trajectories to follow from imposing simultaneously analyticity in spin and supersymmetry.
Analyticity in spin demands SCFTs with eight supercharges, or less, have a structure of superconformal Regge trajectories that looks closer to the non-supersymmetric case -the stress tensor supermultiplet should fit at non-negative spin in the leading long trajectory. For SCFTs with eight supercharges, we expect that the stress tensor superprimary (a scalar and singlet [44]) should fit into the leading super-Regge trajectory, which will have the first unprotected long operator at spin two. The stress tensor itself would fall in the leading Regge conformal primary trajectory, which has the first unprotected operator at spin four. At the level of kinematics this expectation can be checked from the known superblocks of half-BPS flavor current multiplets which were written down explicitly, for arbitrary dimensions, in [50,51]. The chiral algebra of [26] provides a check at the level of dynamics for the four-point function of flavor current multiplets in four-dimensional N = 2 SCFTs. Requiring the stress tensor to fit into the leading unprotected Regge trajectory fixes the OPE coefficient and scaling dimension of this trajectory at zero spin -see the blocks given in [45,52]. These values turn out to be precisely what is needed to ensure analyticity down to spin zero of the function (zz) 2 G(z,z) defined in [52], which admits a block decomposition and has a softer Regge growth (ensuring analyticity for > −1) similarly to (zz) 6 a(z,z) here. Once again, more interconnections between short and long trajectories will likely follow from analyticity in spin.

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was also performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.

B Regge bound of a(z,z)
In this appendix we show that the function a(z,z) is bounded in the Regge limit as (4.13). This limit corresponds to going around the branch cut starting atz = 1 and sending w, defined in (4.10), to zero along any direction in the complex-w plane. 34 Following [5], we do so by bounding the function on the secondary sheets (obtained by going around the branch cut in either direction according to the phase of w) by its (positive) value on the first sheet with real values of the cross-ratios.
The function a(z,z) admits a decomposition in powers of z andz following from its s-channel OPE, given in equations (2.11), (2.12) and (2.13), as where ∆ 8 + | | follows from the decomposition of a(z,z) in the blocks given in eq. (2.7), with the prime in ∆ to remind us the sum runs over both conformal primaries and descendants, and where runs over positive and negative spins of any parity. To bound a(z,z) on the second sheet we now need to show the coefficients b ∆ , are non-negative. The function a(z,z) is only related to a physical correlator through the inverse of the differential operator ∆ 2 as given in equation (2.5), so we do not know of a direct way to use reflection positivity to show positivity of the coefficients b ∆ , . However, reflection positivity would also follow from positivity of the coefficients of the decomposition of a(z,z) in blocks as given in (2.12) and (2.13), so all that remains is to show the blocks a at ∆, (z,z) in (2.7) have a positive decomposition in powers of z andz. We have checked this to be the case for all relevant scaling dimensions and for various spins in a series expansion in small z andz, but do not have a proof, due to the convoluted form of six-dimensional conformal blocks (A.4). We will proceed using the z andz coordinates and not the ρ variable of [54] since the expansion of z andz in powers of ρ andρ is not positive. 35 34 Compared to [5] we have re-scaled w such that 4w there = w here . 35 While (zz) 6 a(z,z) does not have an expansion in positive powers of ρ andρ it is still possible that (zz) 6 a(z,z)/(1 − z)/(1 −z) does, which given the form of the crossing equations would be a sufficient condition. However, due to the convoluted form of the blocks, we could only check this to a rather low order in the ρ andρ expansion. If the expansion is indeed positive, then following the same reasoning as below would yield the bound (4.13) for all values of arg(w). Two follow-up checks are needed: first, c (0) (∆, ) should give the correct residues at physical poles; second, the subtracted parts which are (the principal series integral of) c (−2) (∆, ) and c (−1) (∆, ) should not have a conformal block decomposition to make sure that nothing physical is subtracted.
The first check is straightforward. We simply calculate c (0) (∆, ) for the short multiplets and find that the residues at twist 8 and 10 in the sense of (zz) 6 a(z,z) match (5.26) and (6.9) respectively. The second check is however technically more involved because conformal blocks in 6d are complicated functions. Therefore we will instead do a simpler check to illustrate the essential idea.
Example: mean field theory in 4d. Let us now consider the four-point function φφφφ of mean field theory in 4d. We first use the inversion formula to reproduce the CFT data, and then focus on the cases with integral external dimensions. In this section the normalization of the conformal blocks is the same as that in [5].
For MFT we only need to invert the t-channel identity operator, and we get .

(C.2)
It it straightforward to check that when is an even integer, the residue of −c(∆, ) at ∆ = 2∆ φ + + 2n indeed gives the OPE coefficient of MFT given in [55].
When ∆ φ ∈ Z 2 it can be seen from the second line of (C.2) that c(∆, ) is divergent, since there is a fourth-order pole and only a second-order zero. To regulate c(∆, ) for any integer ∆ φ we expand (C.5) Now we check the poles of ∆ in c (−2) (∆, ; ∆ φ ). There are four sets of poles and they all come from the Γ-functions in the numerator of (C.4). From left to right, the first set of poles originate from the κ-factor. In general this factor reads but as shown in [34] they will be canceled exactly by the poles of the block G ( ) ∆ (z,z), thus nothing physical is subtracted. The third and fourth set of poles are physical double-twist poles and their shadows. One can check that the residues of the third set of poles vanish when ∆ φ ∈ Z ≥2 and the fourth set of poles are never picked up. We summarize these results in table 3 and note in passing that the check related to physical poles is the most important one.

D Exploring the required t-channel contributions
In this appendix we explore the issues with the A 1 theory a bit further. In particular, we would like to know whether the vanishing of the D[0, 4] OPE coefficient at c = 25 can be recovered at all from a sum over t-channel blocks, and if so what properties such a block decomposition has. Therefore we take the rather crude ansatz where we pack the contribution of the entire unprotected part of a(z,z) into the first few t-channel long blocks with some (unrealistically) large anomalous dimensions and OPE coefficients. Demanding then that λ 2 D[0,4] = 0 might give us an idea of what this implies for the unprotected data. Concretely we experimented with approximating the entire unprotected contribution in the t-channel by the following three groups of data: where the first two groups consider only multiplets on the leading long trajectory while the last one also takes into account of the subleading long trajectory, which we distinguish by adding a prime in the spin. The inversion results are shown in table 4. For all of these sets we have imposed λ 2 D[0,4] = 0. From the table we can draw some qualitative conclusions: • The OPE coefficient of the non-chiral algebra short multiplets are relatively stable across all three sets of input data.
• The dimensions of the long multiplets in all columns are lower than the inversion results in figure 9.
• Distributing the contributions to double discontinuity into more blocks on the leading long trajectory lowers the dimensions and OPE coefficients of the leading long multiplets. However, notice that in the "2 blocks" column ∆ 0 is still higher while ∆ 2 is JHEP01(2022)022  Table 4. Crude estimates of unprotected CFT data of A 1 theory for the first few lowest spins. The results are obtained by imposing λ 2 D[0,4] = 0 and approximating the entire unprotected part of a(z,z) by: a single scalar block on the leading long trajectory ("1"), one block of spin 0 and one block of spin 2 on the leading long trajectory ("2"), and blocks of spin 0 and spin 2 both on the leading and subleading long trajectories ("2+2"). The input CFT data are listed in (D.1). already lower than the numerical bootstrap's prediction ∆ 0 6.4, ∆ 2 9.4. All the OPE coefficients decrease after the redistribution.
• Distributing the contribution to double discontinuity into both leading and subleading long trajectories further slightly lowers ∆ 0 and increases ∆ 2 , which moves the results closer to numerical predictions. From the OPE coefficients, λ 2 L[0,0] 0 receives the most significant change, a decrease that is also consistent with our expectation. Therefore, we again see evidence that we should include contributions from the subleading long trajectory to fully recover the unprotected CFT data.
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