Lightcone expansions of conformal blocks in closed form

We present new closed-form expressions for 4-point scalar conformal blocks in the s- and t-channel lightcone expansions. Our formulae apply to intermediate operators of arbitrary spin in general dimensions. For physical spin $\ell$, they are composed of at most $(\ell+1)$ Gaussian hypergeometric functions at each order.


Introduction
Conformal blocks are fundamental to the conformal bootstrap program [1,2], which aims to classify and solve conformal field theories (CFTs) by general consistency requirements, such as associativity of operator product expansion (OPE). In two dimensions, the conformal symmetry is infinite-dimensional, and the conformal bootstrap program can be carried out rather successfully [3]. The well-known examples include the minimal models, which describe the critical phenomena of 2d statistical models, such as the Lee-Yang, Ising, and Potts models [4]. On the other hand, the d > 2 conformal bootstrap is considerably more challenging as the conformal symmetry is usually finite-dimensional.
When performing operator product expansions in conformal field theories, the contributions of a primary and its descendants are related by conformal symmetry. A conformal block is defined by the contributions of a full conformal multiplet, which includes a primary and infinitely many descendants. The study of conformal blocks has a long history [5][6][7][8][9], which dates back to the 1970's when the conformal bootstrap proposal was just formulated. The understanding of conformal blocks was significantly advanced by the works of Dolan and Osborn [10][11][12], in which they found explicit analytic expressions in d = 2, 4, 6 dimensions, recursion relations and Casimir differential equations in general dimensions. These results had paved the road for the later revival of the d > 2 conformal bootstrap [13], where a new numerical conformal bootstrap method was proposed. Let us mention here the precise determinations of 3d Ising critical exponents [14][15][16][17], but refer to [18] for a comprehensive review on many other impressive results. 1 For the numerical conformal bootstrap, it may be adequate to be able to evaluate conformal blocks efficiently using the Zamolodchikov-like recursion relations [23][24][25] in the radial coordinates [26,27]. For the analytic conformal bootstrap, it is more desirable to have closed-formed expressions of conformal blocks to enable general analytic computations, with the dream in mind that the 3d Ising CFT will be eventually solved analytically. Beside those mentioned earlier, 4-point conformal blocks in position space have been studied in various approaches . In this paper, we will focus on conformal blocks with external scalars. In principle, spinning conformal blocks can be obtained from the scalar case using differential operators.
In the analytic conformal bootstrap, considerable recent efforts have been devoted to solving the crossing equations near the lightcone, where the correlator is dominated by the contributions of low twist operators. 2 The crossing equations follow from OPE associativity when applied to 4-point functions of scalar primaries. By considering the double lightcone limit, one can show that the leading contribution from the identity operator indicates the existence of infinitely many higher spin operators whose twists are asymptotic to the sum of two external scaling dimensions [64,65]. 3 The double-twist phenomena were noticed earlier in a more concrete context [67]. The same argument can be extended to subleading twists, but one need to be more careful about the potential mixing of different twist families. In some sense, these large spin sectors behave like mean fields or generalized free fields. Then the next step is to study the systematic corrections to the "free" theories [68][69][70][71][72][73][74][75][76][77][78][79][80]. Naturally, one can formulate the problem as a perturbation theory in large spin. As in many perturbation problems, the results are asymptotic series in the expansion parameter, i.e. 1/ [73]. However, it turned out that the results for the leading twist family of the 3d Ising CFT match with the numerical values down to spin two [72,77]. Therefore, the lightcone bootstrap should admit a convergent formulation.
proposed in [81], which upgrades the asymptotic expansion to a convergent integral transform and establishes the analyticity in spin assumed earlier [82]. 4 An alternative derivation was presented in [84] and a generalization to the spinning case was given in [85]. In two and four dimensions, general closed-form results of the Lorentzian inversion of a conformal block can be found in [86]. In general dimensions, the nonperturbative Lorentzian inversion in the lightcone limit has also been studied in [87,88] [49], which encodes the OPE data of the leading twist spectrum. It was also shown that the analyticity in spin extends to spin-0 in some cases [89][90][91][92], despite the presence of a singularity between = 0 and = 2 related to a poor Regge limit. More recently, a closely related dispersion relation for conformal field theory was proposed in [93], which expresses a correlator as an integral over the double discontinuity. 5 In this paper, we will focus on the 4-point functions of scalar primaries: where x i indicates the positions of the external scalars ϕ i , and x ij = |x i − x j | is the distance between two operators. In general, the external operators can have different scaling dimension, and their differences are denoted by The functional form of the 4-point function is determined by conformal symmetry up to a function of two conformally invariant cross-ratios: In the Lorentzian signature, the first cross-ratio u vanishes when the pair of operators in the s-channel OPE, i.e. ϕ 1 × ϕ 2 , are light-like separated, so we will call u → 0 the schannel lightcone limit. Analogously, v → 0 will be called the t-channel lightcone limit, which is related to ϕ 2 × ϕ 3 . Usually, correlators develop singularities in the lightcone limits, which are associated with ambiguities in time ordering when operators are timelike separated. It is sometimes convenient to switch to z,z, which are real, independent coordinates in the Lorentzian signature. 6 They are related to u, v by It is straightforward to obtain the lightcone expansions in terms of z,z from those in terms of u, v via (1.4). For an intermediate scalar, i.e. = 0, the double lightcone expansion of a 4point scalar conformal block is known in closed form in general dimensions 7 , given by hypergeometric functions of two variables [49]: where the series coefficients take the form The two-variable hypergeometric functions belong to Appell's function of type F 4 . One can carry out the k 1 or k 2 summation, which corresponds to a hypergeometric function of type 2 F 1 . Therefore, at each order of the s-or t-channel lightcone expansion, the dependence on the other cross-ratio is encoded in one Gaussian hypergeometric function.
The main goal of this paper is to extend (1.5) to the case of generic . For physical spin , we find that the s-and t-channel lightcone expansions of a generic 4-point scalar conformal block read: where for simplicity some normalization factors and the parameters in the 2 F 1 hypergeometric functions have been omitted. As generalizations of the = 0 case, the dependence on the other cross-ratio is now encoded in at most ( + 1) Gaussian hypergeometric functions. The precise formulae and the closed-form coefficients can be found in (3.11-3.15, 3.18) and (3.2, 3.3, 3.5, 3.9). The same formulae also apply to the case of generic , but the n-summation should be from 0 to 2k for (1.7) and from 0 to ∞ for (1.8). The double lightcone expansion can be obtained from either (1.7) or (1.8): where the series coefficients C k 1 ,k 2 are finite sums of A k,n in (3.16) or B k,n in (3.8). The double power series (1.9) furnishes a general, explicit solution to the quadratic and quartic Casimir equations, which are partial differential equations of two variables. For comparison, we will briefly discuss below other complete analytic formulae of 4-point scalar conformal blocks in general dimensions: • In Mellin space, 4-point scalar conformal blocks in general dimensions are known in closed form for non-negative integer . The essential part is known as the Mack polynomials [95] [12], which are the counterparts of A k,n , B k,n in position space 8 . Due to our choices of basis functions, A k,n , B k,n are much simpler than the results from the Mack polynomials at low orders of the lightcone expansions. In addition, A k,n , B k,n directly apply to generic , as opposed to only non-negative integer in the case of the Mack polynomials. 9 • As shown in [38], the quadratic Casimir equation can be mapped to the Schrödinger equations for integrable Calogero-Sutherland models. Then 4-point scalar conformal blocks are related to the Harish-Chandra functions, which can be written as double infinite sums of Gaussian hypergeometric functions in z,z with power law insertions [46]. To the best of our knowledge, while it is straightforward to derive the s-channel lightcone expansion, i.e. z 1, from the explicit results in [46], it is more difficult to obtain compact formulae for the t-channel lightcone expansion, i.e. 1 −z 1. Our s-channel lightcone expansion also seems to be simpler, at least at low orders. As solutions of the quadratic Casimir equation, the formulae in this paper provide alternative expressions for the Harish-Chandra functions in the small zz or (1 − z)(1 −z) expansions.

Lightcone limits of conformal blocks
In this section, we will give a brief overview of 4-point scalar conformal blocks in the lightcone limits. After performing the s-channel operator product expansion, the conformal invariant part decomposes into s-channel conformal blocks where P i is the product of two OPE coefficients associated with the intermediate primary operator O i , and τ i , i are the twist and spin of O i . Note that the primary operator is labeled by twist rather than the scaling dimension ∆, because twist appears naturally in the lightcone limit. The s-channel conformal block G encodes the contributions of O i and its descendants in the s-channel OPE, and satisfies the quadratic Casimir differential equation (2.5) As the differential operator D is symmetric in a, b, two independent solutions can be related by interchanging a, b. The boundary condition is given in the short distance limit: For physical operators, the spin is a non-negative integer, i.e. = 0, 1, 2, · · · . We will consider conformal blocks of generic using the Casimir equation (2.3), which can be considered as analytic continuation of the physical conformal blocks. Operators of generic spin are nonlocal. They play a significant role in the Lorentzian inversion formula [81] and can be understood as a combination of the spin-shadow and light transforms of local operators [85]. Analogous to the shadow transform, they can be viewed as Weyl reflections. From the perspective of conformal Casimirs, the eigenvalues are invariant under the Weyl reflections which correspond to the shadow, spin-shadow and light transforms. Note the spinshadow transform also appears in the closed-form expressions of conformal blocks in 2d and 4d.
In the s-channel lightcone limit u → 0, the coefficients of D u , D v in (2.4) are regular, so we can substitute D u with the leading exponent τ /2 and set u to 0. Then we obtain a second order differential equation in v for the leading term, whose explicit solution reads: The conformal blocks are normalized such that the double lightcone limit with where H n is the Harmonic number.
In the t-channel lightcone limit v → 0, the quadratic Casimir equation does not reduce to an equation for the leading term, but one can make use of both the quadratic and quartic Casimir equations to derive a closed equation [81]. As the resulting equation is of fourth order, it is more challenging to obtain closed-form solutions of the differential equation, in contrast to the standard second order equation from the small u limit. Nevertheless, a general closed-form expression for the small v limit was found in [49]: It is straightforward to verify that (2.10) solves the quartic differential equation. Here we have introduced a two-variable hypergeometric function, the Kampé de Fériet function. In our notation, the function is defined as x m y n m! n! . (2.12) Note that our definition is not standard. Usually the terms with (m + n) are on the left of those with m or n. In our notation, it is more clear that the n-summation terminates for physical spin due to the Pochhammer symbol (− ) n . Then for each n the m-summation corresponds to a 2 F 1 hypergeometric function. More explicitly, we can write (2.10) as (2.13) where the factorized coefficients are 14) and the basis functions are The subscript 0 indicates the lowest order in the small v expansion.

Lightcone expansions of conformal blocks
The lightcone limits of 4-point scalar conformal blocks are the leading terms of the lightcone expansions. In this section, we will consider the subleading terms. We will first generalize the formula for the v → 0 limit to all orders in the small v expansion. Then we will discuss the small u expansion. The double lightcone expansion can be readily derived from either of them.

The t-channel lightcone expansion
In the v → 0 limit, (1.5) should be equivalent to (2.10) with = 0. To show their equivalence, let us perform a linear transformation of the 2 F 1 function associated with the m-summation which is precisely g 0,n (u) with = n = 0 when k = 0. The coefficient of 2 F 1 also matches with B 0,n in (2.14) with = n = 0. In (3.1), we have intentionally preserved the k-dependence, which suggests the general basis functions should be where γ = τ − d + 2. For = 0, 1, 2, . . . , the small-v expansion of a generic conformal block then reads It is not surprising that conformal blocks can be expanded in terms of g k,n (u), since g k,n (u) ∼ u τ /2+n + · · · where the leading exponent increases with n. The nice feature is that the n-summation always terminates at n = when = 0, 1, 2, 3, . . . , generalizing a property of the small v limit to all the subleading orders. Note that for a generic the n-summation does not terminate for both the leading and subleading terms.
The next step is to find the general expression of B k,n . One can compute B k,n using the quadratic Casimir equation (2.3) order by order. For example, we find where B 0,n is defined in (2.14). In general, they are rational functions of ( , τ, a, b, d).
One can decompose them into factorized rational functions according to their poles, i.e. the zeros of the denominators. It turns out that B k,n can be expressed as where n i indicates (n 1 , n 2 , n 3 ). To arrive at (3.5), the poles from 1/(1 + a − b) m are particularly helpful. The last line of (3.5) also shares some nice features of the leading term in (2.14), where several terms coincide with the parameters in the associated 2 F 1 functions. Since (p!) −1 vanishes if p is a negative integer, there are additional conditions for non-vanishing summands n k ≥ n 2 , n 1 ≥ n 2 ≥ n 3 , n 2 + n 3 ≥ n 1 . (3.6) The dependence on n k is relatively simple, so it is straightforward to carry out the n k -summation. The (n 1 , n 2 , n 3 ) are more entangled with each other, but n 1 can be eliminated easily as well. So there remain at most two summations. Although (3.5) is guessed from low order expressions, we have tested it at much higher orders using the Casimir equation (2.3), which can be performed efficiently by setting ( , τ, a, b, d) to rational numbers. Then we can derive the double lightcone expansion from (3.3): where the coefficients C k 1 ,k 2 are sums of B k,n in (3.5): Before moving to the small u expansion, let us discuss some properties of the building block g k,n (u). After a linear transformation, g k,n (u) becomes The alternative expression (3.9) may look simpler than (3.2), but the parameters of the 2 F 1 function do not match with those in B k,n . One may wonder whether the terminating n-summation for non-negative integer is associated with the exponent of (1 − u). We will see that this is indeed the case. The boundary condition of conformal blocks is given in the limit u → 0, v → 1. The s-channel OPE in the u → 0 limit is dominated by the contributions of operators of low twist. Now we can also consider the dual limit u → 1 of conformal blocks, which reduces to expanding g k,n (u) around u = 1: The leading exponents are universal, which means that we are not able organize the spectrum according to the asymptotic behaviour in the u → 1 limit. In the first part, the leading exponent of (1 − u) is independent of n, which is in accordance with the facts that g k,n are natural basis functions for the lightcone expansion and g 0,n was first found by stripping off this part in the small u expansion. From the second part, we can see the n-summation terminates naturally for physical spin when the exponent of (1 − u) becomes zero.

The s-channel lightcone expansion
Above, we consider the small v expansion. Now we discuss the small u expansion. In general, the v-dependence can be encoded in (2k + 1) hypergeometric functions of type 2 F 1 at order u τ /2+k . As a generalization of the small u limit in (2.8), we can expand a generic conformal block as 10 where the building blocks 11 are At order k, the leading exponent of (1 − v) is ( − 2k), which decreases as −2k due to the second order nature of the Casimir differential equation (2.3). To derive the double lightcone expansion, we first perform a linear transformation. Then the small u expansion reads: The superscript (0) of f Then we can compute the coefficients A k,n by matching C k 1 ,k 2 in (3.16) with that in (3.8) order by order. At order k, we need to solve a system of (2k + 1) linear equations. The explicit solutions at low orders are A 0,0 = 1 , A 1,0 = (τ + − 1) 2 α=±a,±b (τ /2 + + α) (τ + 2 − 1) 2 (τ + 2 ) 2 (τ + − d/2 + 1) , which take simple factorized forms. From the concrete examples, we notice several interesting properties of A n,k : • They are symmetric in a 2 and b 2 .
• They are proportional to ab when n is an odd integer, so the odd-n cases vanish when two external operators have the same scaling dimension, i.e. ∆ 1 = ∆ 2 or ∆ 3 = ∆ 4 .
• If the spin is a non-negative integer, they always vanish when − n < 0 and the n-summation terminates at n = at arbitrary high orders 12 , which is similar to the small v expansion (3.3).
To find a general expression, we decompose the low order coefficients into factorized rational functions as in the case of B k,n . The dependences on a, b are again particularly useful, which suggests the factorized building blocks should be symmetric in ±a, ±b.
Then we are able to write these low order coefficients as double summations. In the end, we obtain a general formula for A k,n : where we have introduced n 1 = n/2 , n 2 = (n + 1)/2 (3.19) to encode the minor differences between even and odd n expressions, and x is the floor function. Due to (−n 1 ) m 1 +m 2 and 1/(k − n + m 1 )!, there are additional constraints for nonzero summands n 1 ≥ m 1 + m 2 , m 1 ≥ n − k , (3.20) which implies the general expression of A k,n becomes simpler when n is close to 0 or 2k. 13 Note that a, b only appear in the last line, and m 1 , m 2 are the arguments of the associated Pochhammer symbols. One may notice that A k,n have poles at τ + 2 = m with m = 1, 2, . . . , but they are due to the basis functions. In Appendix A, we discuss an alternative expansion without these spurious poles. As the concrete decompositions of A k,n are simpler than those of B k,n , we need to study higher order terms to see the general pattern of the a, b independent parts. For large k, it is much more efficient to compute A k,n by matching the double lightcone expansion coefficients (3.16) with (3.7) 12 At low orders, the n-summation terminate at n = 2k if 2k < . 13 The simplicity of A k,2k and A k,2k−1 was noticed earlier in [97].
than solving the Casimir differential equation (2.3). Since our results here are based on the formulae in Sec. 3.1, we discuss the expansion in small v before that in small u. We have also tested the formula (3.18) to much higher orders using the Casimir equation (2.3) with rational parameterizations.
Using the complete expression of the small u expansion, we can also expand the conformal blocks around the fully crossing symmetric point [98] where the cross-ratios are invariant under all the crossing transformations 1 ↔ 2, 1 ↔ 3, 1 ↔ 4. As we are in the Euclidean regime, let us assume is a non-negative integer, so the v → 1 limit is regular. The leading terms then read The fully-crossing symmetric point is interesting in that all the crossing constraints can be systematically solved by expanding the correlator, i.e. G(u, v), around this point order by order. Then we only need to expand the manifestly crossing symmetric correlator into physical conformal blocks.

Conclusion
In summary, we have presented new analytic expressions of 4-point scalar conformal blocks in the lightcone expansions in (3.3) and (3.11), with closed-form coefficients in (3.5) and (3.18). They provide explicit solutions of the Casimir differential equations in general dimensions for intermediate operators of arbitrary spin. They can be directly applied to the lightcone bootstrap to study the low twist spectra in general dimensions, and are particularly useful in d = 2, 4, 6 dimensions. Using the t-channel lightcone expansion (3.3) 14 , one can compute the Lorentzian inversion in the lightcone expansion by changing the integration variables to u, v, or make the substitutions (1.4). As infinite sums over the spectrum can lead to divergences related to enhanced singularities, one needs to regularize the results properly. When the precise forms of the enhanced singularities are known, we can add and subtract the corresponding infinite sums to obtain convergent results [77,81]. Otherwise, one should sum over the t-channel blocks before performing the lightcone expansion. For example, in the recent work [92], a nontrivial double infinite summation of the t-channel blocks was carried out to extract the exact lightcone limit. In [92], the small v expansion of the t-channel conformal blocks was computed order by order using the Casimir equation. Our formula for the s-channel lightcone expansion in (3.11) provides general closed-form expressions to arbitrary high order 15 , so should be helpful for similar summations in the analytic conformal bootstrap. Furthermore, (3.11) is clearly useful for studying crossing equations at subleading orders of the lightcone expansion, which encode additional constraints and the information of higher twist operators.
Since spinning conformal blocks can be generated from the scalar blocks using differential operators, it may also be interesting to revisit the spinning crossing constraints in the lightcone expansion, especially those associated with conserved currents and stress tensors [99,100].
One can computeĀ k,n from (3.7). It is easier to solveĀ k,n than A k,n using the small v expansion, as the exponents of the leading terms grow with n. We can solve the linear equations one by one, instead of (2k + 1) equations at the same time. The low order coefficients areĀ In general,Ā k,n is proportional to (− ) n , so the n-summation also terminates for physical spin . One can notice some similarities to the pole decomposition of B k,n in (3.5). After a linear transformation, the basis functions in (A.1) is the sum of v (a−b)+n (1 − v) −n 2 F 1 τ /2 + + a + k, τ /2 + − b + k 1 + a − b + n ; v , (A.5) which is dual to (3.9), and a part with a, b properly interchanged. The k dependence for small n or 2k − n is not hard to guess, but we do not find a relatively simple expression forĀ k,n . In the B k,n case, the poles from 1/(1 + a − b) m are particularly helpful, but they are absent in the small u expansion. Nevertheless,Ā k,n can be expressed as sums of A k,n or B k,n .