Crossing Symmetry in Alpha Space
Matthijs Hogervorst Balt C. van Rees


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  Crossing Symmetry in Alpha Space






Matthijs Hogervorst^a, Balt C. van Rees^b 
 

^a C.N. Yang Institute for Theoretical Physics, Stony Brook University, USA

^b Centre for Particle Theory & Department of Mathematical Sciences,

Durham University, Durham, UK









  We initiate the study of the conformal bootstrap using Sturm-Liouville theory, specializing to four-point functions in one-dimensional CFTs. We do so by decomposing conformal correlators using a basis of eigenfunctions of the Casimir which are labeled by a complex number α.  This leads to a systematic method for computing conformal block decompositions. Analyzing bootstrap equations in alpha space turns crossing symmetry into an eigenvalue problem for an integral operator K. The operator K is closely related to the Wilson transform, and some of its eigenfunctions can be found in closed form.
  
















§ INTRODUCTION


Symmetries and consistency conditions play an important role in quantum field theory. This is especially true in the realm of Conformal Field Theories (CFTs), which can be analyzed by combining constraints from  conformal invariance, unitarity and crossing symmetry. This set of ideas is known as the conformal bootstrap <cit.>. It was revived in <cit.> and has led to a wealth of numerical and analytical results about CFTs, see for instance <cit.>.[See <cit.> for an introductory discussion of the conformal bootstrap.] 

Since the bootstrap uses constraints coming from correlation functions, it is natural to express crossing symmetry as a sum rule in position space. This is not strictly necessary: for instance, some properties of CFT correlators are more transparent in Mellin space <cit.>. In the present paper we introduce alpha space, an integral transform for CFT correlators based on the Sturm-Liouville theory of the conformal Casimir operator. As we will explain, alpha space can be used to rephrase crossing symmetry as an eigenvalue problem.

To illustrate this idea, consider the toy crossing equation

∑_n=0^5 c_n   p_n(z) = ∑_n=0^5 c_n   p_n(1-z)
involving the following polynomials:[Up to a choice of normalization, these are the Kravchuk polynomials with N=5 and p = 1/2 <cit.>.]
p_n(z) = (-1)^n ∑_j=0^n 2^j 5-zn-j zj ,
    n = 0,…,5 .
How can we determine the set of all c_n that satisfy eq:cross0?  Since the p_n(z) are polynomials, various brute-force methods can be used. More elegantly, we can realize that the p_n form a complete basis for the space of polynomials of degree ≤ 5, orthogonal with respect to the inner product

∫d μf(z) g(z) ,
    ∫d μ= ∑_k=0^5 (-1)^k/2^k 5k  ∫dz   δ(z-k) .
This implies that the p_n(1-z) appearing in the RHS of the crossing equation can be decomposed as follows:
p_n(1-z) = ∑_m=0^5 Qmn   p_m(z)
for some 6 × 6 matrix Q. The latter can be easily computed using eq:ipk. Since z ↦ 1-z is an involution, we must have Q^2 = _6 × 6, as can be checked easily.  Eq. eq:cross0 can now be recast as

c_n = (Q ·c)_n 
hence our problem reduces to finding all eigenvectors of Q with eigenvalue +1. There are three such eigenvectors:

f_1 = p_0   (≡1)
  f_2 = 2p_1 - p_3 - p_4 ,
  f_3 = p_2 +2p_3 + 2 p_4 ,
so the most general solution to eq:cross0 is
∑_n=0^5 c_n p_n(z) = ∑_i=1^3 t_i f_i(z),   t_i ∈R .


In this paper we consider one-dimensional (defect) CFTs which are governed by crossing equations similar to eq:cross0. For definiteness, let us consider a four-point function F(z) of identical operators of dimension h_ϕ, admitting a conformal block decomposition

F(z) = ∫_0^∞d h   ρ(h) k_h(z) ,
  ρ(h) = ∑_n c_n δ(h-h_n) ,
where the k_h(z) are 2 conformal blocks:

k_h(z) = z^h _2F_1(h,h;2h;z) .
The spectrum {h_n} and the OPE coefficients c_n ≥ 0 are typically unknown. Bootstrapping entails computing or constraining these CFT data using the crossing relation

F(z) = (z/1-z)^2h_ϕ F(1-z) .
There are various technical differences between this d=1 bootstrap problem and the previous toy example. For one, h takes its values in the continuum R_≥ 0, whereas the toy example had a finite and discrete spectrum. Nevertheless, it is tantalizing to apply the logic from the toy example to the bootstrap. For instance, one could hope to constrain the density ρ(h) from eq:Fdef through a relation of the form

ρ(h) ?= ∫_0^∞d h'   Q(h,h'|h_ϕ) ρ(h')
for some continuous kernel Q(h,h'|h_ϕ) which plays the role of Q. Sadly Eq. eq:wrong cannot quite be true. The reason is that the conformal blocks k_h(z) don't form an orthogonal basis of functions on (0,1). The principal aim of this paper is to demonstrate that it is nevertheless possible to write down a qualitatively very similar relation. In order to do so we use a new basis of functions to transform our four-point function to a space that we denote as alpha space. In this space we can properly define (<ref>) in terms of a crossing symmetry kernel K which we will explicitly compute. We will discuss its main features and explain how the ordinary conformal block decomposition is recovered from an analytic continuation in alpha.

We stress that the philosophy of studying CFTs using crossing kernels — à la eq:wrong — is  not new. An early avatar  of this idea can be found in Eq. (2.66) of Ref. <cit.>. Nonetheless, we are not aware of earlier work where the relevant SO(d,2) or 2 crossing kernels have been worked out in detail. An exception is the 2d Liouville CFT, for which the crossing kernels have been computed <cit.> as the 6-j symbol of a class of representations of U_q(𝔰𝔩(2,R)), leading to a formal proof of consistency of the theory.[See also <cit.>.] The case of rational 2d CFTs (i.e. Virasoro minimal models) is also of interest, since in such theories the crossing kernel is realized as a finite matrix <cit.>.  We will comment on the group-theoretic interpretation of our crossing symmetry kernel in Sec. <ref>.

The outline of this paper is as follows. In Sec. <ref> we review the one-dimensional bootstrap problem and solve the Sturm-Liouville problem for the 2 Casimir operator. This allows us to construct a complete, orthogonal basis of eigenfunctions on the interval (0,1). In Sec. <ref> we use these basis functions to derive a crossing equation similar to eq:wrong, and we study the properties of the relevant kernel K. Sec. <ref> describes several possible applications of crossing kernels to the conformal bootstrap.

Note added: while preparing this manuscript we learned about Ref. <cit.>, which  discusses a crossing kernel approach to both SU(2) and conformal crossing symmetry equations and is tangentially related to this paper.



§ ONE-DIMENSIONAL BOOTSTRAP AND ALPHA SPACE


This section is devoted to the Sturm-Liouville theory of the conformal Casimir of 2, the conformal group in one spacetime dimension. One-dimensional CFTs arise in the description of line defects in higher-dimensional theories <cit.>. Although 1d CFTs are in many ways more tractable than d-dimensional systems, we also note that many salient features of the d-dimensional bootstrap already appear at the level of d=1. In addition the 1d conformal blocks appear naturally in the light-cone limit of the higher-dimensional crossing symmetry equations, where it becomes possible to obtain non-trivial analytic results <cit.>.



 §.§ Sturm-Liouville theory of the 2 Casimir



We will start by analyzing the four-point function of a single primary (or lowest-weight) operator ϕ(x) in a 1d CFT. The general case will be addressed in Sec. <ref>.  The only quantum number of ϕ is its scaling dimension h_ϕ, and conformal symmetry dictates that ϕϕϕϕ has the following form:
ϕ(x_1) ϕ(x_2) ϕ(x_3) ϕ(x_4) = (z)/|x_1 - x_2|^2h_ϕ  |x_3 - x_4|^2h_ϕ
where the points x_i ∈R lie on a line and z is the following cross ratio:

z |x_12| |x_34|/|x_13| |x_24| ∈(0,1)
writing x_ij = x_i - x_j.[Although a priori the variable z is not restricted to the unit interval, we require z ∈ (0,1) to guarantee OPE convergence on both sides of the bootstrap equation.] The function (z) admits the following conformal block (CB) decomposition:

(z) = ∑_  _ϕϕ^2   k_h_(z)
where the functions k_h(z) are the 1d conformal blocks defined in Eq. eq:CB. The sum runs over all operators  in the ϕ×ϕ OPE of dimension h_, and _ϕϕ is the ∈ϕ×ϕ OPE coefficient.  
Finally, crossing symmetry (invariance under the exchange x_i  x_j) of the ϕϕϕϕ correlator leads to the bootstrap constraint

(z) = (z/1-z )^2h_ϕ (1-z)
which must hold for all 0 ≤ z ≤ 1. 

We will not assume unitarity (i.e. reflection positivity) in this paper. Just for completeness, we recall that if the CFT in question is unitary, the decomposition eq:CBdec is constrained as follows:

  
  * the _ϕϕ must be real-valued, hence _ϕϕ^2 > 0;
  
  * there must be a contribution of the unit operator  with h_ = 0 and _ϕϕ = 1;
  
  * all other operators (including ϕ) have h_ > 0.



As noted in the introduction, it is conventional in the CFT literature to investigate the bootstrap equation eq:1dBS in position space. Here we will take a different approach. We start by remarking that the conformal blocks k_h(z) are eigenfunctions of a second-order differential operator D, the quadratic Casimir operator of 2:

D ·k_h(z) = h(h-1) k_h(z) ,
    D  =  z^2(1-z) ^2 - z^2  . 

In what follows, we will develop the Sturm-Liouville theory of the operator D on the interval (0,1).[Note added: although we have not attempted to do so, it is in principle possible to change the boundary conditions at z=1 <cit.>. We thank Miguel Paulos for pointing out this reference.]  As a first step, we notice that D can be written in the following suggestive form:

D ·f(z) = z^2 z [(1-z) f'(z) ].
This implies that D is self-adjoint with respect to the inner product

 f,g   = ∫_0^1  d z/z^2   f(z) g(z)
where f,g are functions (0,1) →C that are well-behaved near z=0 and z=1. Indeed, we have
 f, D ·g  -  D·f,  g   = ∫_0^1 d z    z [ (1-z) (fg' - f'g) ]
which is a boundary term. Of course, not all functions have a finite norm with respect to the inner product eq:innerProd. Requiring that a function f is square integrable leads to the following constraints on its asymptotics near z=0 and z=1:

f(z)   z →0   z^1/2+ 
  and
  f(z)   z →1   (1-z)^-1/2+'
for constants , ' > 0. In particular, this implies that in a unitary CFT all four-point functions (z) have a divergent norm with respect to eq:innerProd.

Our next order of business is to construct an orthogonal basis of eigenfunctions of D. We start by solving the eigenvalue equation D · f =  f. After writing = ^2 - 1/4 for convenience, we find that the general solution (for ≠ 0) is given by

f(z) = A_1() k_+(z) + A_2()k_-(z)
for two constants A_1,2() that are to be determined. 
In order to fix them, let's analyze the z → 0,1 asymptotics of f(z). First, we notice that the blocks themselves are logarithmically divergent near z = 1. To be precise, we have
k_+(z)   z →1   - Γ(1+2)/Γ^2(+) ln(1-z)   +   regular
and likewise for k_-(z). 
Requiring that eq:ans1 has a finite limit as z → 1 therefore determines the relative coefficient A_1()/A_2(). Fixing the overall normalization by imposing f(1) = 1, we arrive at the following eigenfunctions:[Remarkably, these are not the usual `shadow-symmetric' blocks obtained by integrating one-dimensional three-point functions over the real axis <cit.>. Indeed, in one dimension this integral is easily performed using the techniques of <cit.> and diverges logarithmically as z → 1, in contrast with our Ψ_(z).]


Ψ_(z) = 1/2 [ Q() k_+(z)  + Q(-) k_- (z)  ],
  Q() = 2Γ(-2)/Γ^2(- ) .

In what follows, it will be useful to rewrite Ψ_(z) as
Ψ_(z) = _2F_1(+,-  1;z-1/z )
using a hypergeometric identity. In particular, this makes it manifest that Ψ_(1) = 1. However, we have not yet inspected the asymptotics near z=0. Assuming that  is real, we find that Ψ_(z) z → 0  z^1/2-||, which means that the functions Ψ_ have infinite norm. The only way to avoid this problem is to assume that  is imaginary. In that case, we find that Ψ_ has the following asymptotics:
Ψ_(z)   z →0   |Q()|   √(z) cos(  Im()  lnz + const.  )     [ ∈i R]
implying that Ψ_ is rapidly oscillating near z=0. Notice that even for imaginary , the function Ψ_(z) is real-valued, since it is symmetric under → -. A plot of two different functions Ψ_(z) is shown in Fig. <ref>.


Since Ψ_(z) oscillates near z=0 at a rate that depends on , it is at least plausible that Ψ_,Ψ_ = 0 for ≠±$̱, cf. the Fourier transform onR. This is confirmed by an explicit computation, performed in Appendix <ref>. There it is shown that the inner product Ψ_, Ψ_behaves as a delta function on the imaginary axis. To be precise: iff()is defined oniRand has compact support,  we have
1/2π i∫_-i∞^i ∞d Ψ_, Ψ_ f()̱ = N()   f() + f(-) /2
where

N() = Γ()Γ(-)/2πΓ(+) Γ(-) = |Q()|^2/2≥ 0 .

Informally, Eq. eq:testInt shows that the functionsΨ_(z)are plane-wave normalized, having normN(). The fact that the RHS of eq:testInt contains the sum[f() + f(-)]reflects thatΨ_is even in, which carries over to the inner product Ψ_, Ψ_.

Summarizing, we have constructed a set of orthogonal eigenfunctionsΨ_(z)with respect to the inner product eq:innerProd. Naively, we can appeal to familiar arguments of Sturm-Liouville theory to argue that these eigenfunctions form a complete set. In other words, we can decompose a given functionf   :   (0,1) →Ras follows:

f(z) = 1/2π i∫_-i∞^i ∞d /N() f() Ψ_(z)
  ⇔  f() = ∫_0^1d z/z^2  f(z)Ψ_(z) .

This formula describes howf(z)is encoded by its “spectral density”f(), and vice versa. A mathematically rigorous way to obtain this identity will be described in the next section.

A sufficient condition for Eq. eq:sl2decomp to make sense is thatfbe square integrable:
 f,f  = ∫_0^1d z/z^2  |f(z)|^2 < ∞ .

An equivalent condition (see the next section) is that
1/2π i∫_-i∞^i ∞d /N()  |f()|^2

is finite. In Secs. <ref> and <ref> we discuss how these constraints can be loosened. 

Eq. eq:testInt shows that theΨ_(z)form a complete basis inspace. For reference, we remark that theΨ_(z)also obey a completeness relation in position space, namely
1/2π i∫_-i∞^i∞d/N() Ψ_(z) Ψ_(w) = z^2  (z-w)

as can be deduced from eq:sl2decomp.



 §.§ Alpha space as a Jacobi transform


The alpha space transformf(z) ↦f()is closely related to a known integral transform, known as the Jacobi transform. We will briefly describe this transform in the rest of this section, pointing to Refs. <cit.> as a  point of entry in the mathematics literature. The Jacobi transform is an integral transform that makes use of the Jacobi functions:
ϑ_^(p,q)(x) _2F_1((1+p+q) + , (1+p+q) - p+1;- x ),    x ≥ 0 .

The parametersp,q ≥0are fixed, whereas the label∈i Ris allowed to vary continuously. Notice thatϑ^(p,q)_(x)is even in, and therefore real-valued. Consider now a complex functionf(x), defined forx ≥0, decaying sufficiently fast asx →∞. We assign to it its Jacobi transformJfas follows:

f(x)  ↦  (Jf)() ∫_0^∞d x ω_p,q(x)  f(x) ϑ^(p,q)_(x) ,
  ω_p,q(x) = x^p (1+x)^q .ω_p,q(x)plays the role of a weight function in position space. A standard result —  see Theorem 2.3 of Ref. <cit.>— is thatfcan be restored from its Jacobi transform:

f(x) = 1/2π i∫_-i∞^i∞d /N_p,q()  (Jf)() ϑ^(p,q)_(x) + …
where
N_p,q() = 2Γ^2(1+p)Γ(± 2)/Γ((1+p+q) ±)Γ((1+p-q) ±) ,
  Γ(x± y) Γ(x+y)Γ(x-y) .

The dots in eq:inverseJ indicate that depending on the values ofpandqa finite number of terms must be added; equivalently, the integration contour incan be deformed to pick up poles coming from1/N_p,q().[A sufficient condition for such terms to be absent is p + q + 1 > 0 and p - q + 1 > 0.]

Properly speaking,Jfurnishes a map from the Hilbert spaceL^2(R_+,  ω_p,q(x) dx)to the space of functions oniRwhich are normalizable  with respect to the measured/N_p,q(). This map is an isometry: given two complex functionsf,g, the following Parseval formula holds:
∫_0^∞d x ω_p,q(x)  f(x)g(x)= 1/2π i∫_-i∞^i∞d /N_p,q() (J^f)() (J^g)() .

Specializing to the casef=g, this shows in which sense the Jacobi transform is unitary.

It is now straightforward to see that the alpha space transform for the2Casimir is a special case of the Jacobi transform withp = q = 0, after the change of variablex →(1-z)/z. The precise dictionary is given by
Ψ_(z) = ϑ_^(0,0)(1-z/z),
    ∫_0^∞ dx ω_0,0(x) = ∫_0^1dz/z^2 ,
    
N() = N_0,0() .

A direct consequence is the identity
f,g = 1/2π i∫_-i∞^i ∞d /N() f()g() .

It would be interesting to see if further theorems concerning the Jacobi transform can be recycled to prove results about alpha space densities in CFTs.

Our discussion has been quite abstract so far and at this stage the reader may want to experiment with some explicit alpha space computations. To do so, it is useful to know that the Jacobi transform essentially maps rational functions to polynomials. A precise statement is the following. Let

P_n^(p,q)(x) = (p+1)_n/n! _2F_1(-n,n+p+q+1;p+1;(1-x) )

be a Jacobi polynomial of degreen. Then for anyr,s ≥0we have <cit.>
    ∫_0^∞d x ω_p,q(x) 1/(1+x)^(p+q+r+s)+1 P_n^(p,r)(1-x/1+x) ϑ_^(p,q)(x) 
    
      = (-1)^n/n!Γ(p+1)Γ((r+s+1) ±)/Γ((p+ q+r+s)+1+n )Γ((p- q+r+s)+1+n )
    ×_n(;(p+q+1),(p-q+1),(r+s+1),(r-s+1)) .

The object on the last line is a Wilson polynomial <cit.>:
_n(;a,b,c,d) = (a+b)_n(a+c)_n(a+d)_n  _4F_3(-n,a+,a-,n+a+b+c+d-1 a+b,a+c,a+d;1).

Evidently_n(;a,b,c,d)is a polynomial of degreenin^2, and it can be shown that_ndepends symmetrically on its parametersa,b,c,d. Specializing to alpha space (p=q=0) whilst settingr→0,s →2ρ-2, the identity eq:jnaarw becomes
∫_0^1d z/z^2  z^ρ P_n(2z-1) Ψ_(z) = (-1)^n/n!Γ(ρ-±)/Γ^2(ρ+n) _n(;,,ρ-,32-ρ)

whereP_nis a Legendre polynomial of degreen. This formula can be used to find the alpha space counterpart of rather general functions in position space. As a simple example, we can setn=0to find the alpha space version of the functionz ↦z^ρ:
∫_0^1d z/z^2  z^ρ Ψ_(z) = Γ(ρ-±)/Γ^2(ρ) .

An additional example will be discussed in Sec. <ref>.



 §.§ Convergence of the alpha space transform


Before we turn to the application of alpha space to CFTs, let us comment on the convergence of the alpha space transformf(z) ↦f(). We have in mind a functionf(z)that has power-law growth atz=0andz=1, i.e.

f(z) z → 0 z^p  f(z) z → 11/(1-z)^q .

Moreover, we assume thatf(z)admits an expansion in powers ofz^haroundz=0, meaning that it is possible to writef(z) = ∑_n=1^∞c_n z^h_n. All of these conditions are certainly satisfied whenf(z)describes a CFT correlation function. 

Let us first consider the case wherep > 1/2andq<1. In that case, the integral defining its alpha space density
f() ∫_0^1 dz/z^2  f(z) Ψ_(z)

converges whenever|()| < p - þ, meaning thatf()is holomorphic on a finite strip. Moreover, using the alpha space transform of a single power law eq:powerlaw, it is possible to show thatf()extends to a meromorphic function on the entire complex plane, with poles at= h_n - 12+ Nforn=1,2,…(plus mirror poles on the left half plane).

Next, consider the casep < 1/2,q < 1. In this case, it is convenient to decomposef(z)as 

f(z) = f_sing(z) + f_reg(z)

where

f_sing(z) = ∑_h_n < 1/2 c_n z^h_n
f_reg(z) = ∑_h_n > 1/2 c_n z^h_n .

By construction, the regular piecef_reg(z)has a well-defined alpha space transform that extends to a meromorphic function onC. We can define the densityf_sing(α)termwise, by analytically continuing Eq. eq:powerlaw to arbitrary values ofρ.[Such analytic continuations may require a deformation of the alpha space integration contour away from the imaginary axis. Below we explain how to deal with such cases.] Concretely, we take the alpha space transform off(z)to be
f() = ∑_h_n < 1/2 c_n Γ(h_n - þ±)/Γ^2(h_n) + f_reg() .

If the leading term off_sing(z)is a constant, the above argument breaks down, since1/Γ^2(h)vanishes whenh →0. This is an order-of-limits issue, which can be avoided by writing1as the limit ofz^as→0. 

Finally, we consider the caseq > 1. For simplicity we considerp > þ, but the case of generalpis straightforward to treat using the above discussion. Given thatq>1, the integral definingf()diverges for all values of. We thus regulate this integral by cutting it off atz = 1-, writing
f_() ∫_0^1-dz/z^2  f(z) Ψ_(z) .

Notice that this regulator does not affect the analytic structure off(): all poles originate from the region of integration nearz=0. Now, to isolate divergent pieces inwe notice thatΨ_(z)admits an expansion in powers of(1-z)of the following form:
Ψ_(z) = ∑_k=0^∞ s_k() (1-z)^k

wheres_k()is a polynomial of degreekin^2. This implies thatf_()has the following structure of divergences:[To derive this formula, we are assuming that f(z) admits an expansion of the form

    f(z) = 1/(1-z)^q[ const. + ∑_n ≥ 1 a_n(1-z)^n ]

around z=1. If f(z) rather behaves as a more general sum of power laws

    f(z) = c_1/(1-z)^q_1 +  c_2/(1-z)^q_2 + …

Eq. eq:ct1 is modified in a straightforward fashion.
]f_() = [finite as → 0]  +  ∑_j=0^q-1t_j()/^q-1-j
wheret_j()is a polynomial in. Consequently, we takef()to be the finite piece off_()obtained by subtracting the divergent terms in eq:ct1. 




 §.§ Conformal block decomposition


As a first application of the alpha space formalism of the previous sections, we will show that it can be used to compute conformal block decompositions for CFT correlators. As a starting point, we have in mind a meromorphic spectral densityf(), even in, written in the following form:
f() = ∑_n -R_n/ - _n  +   (→ -)   +  entire.

The minus sign in front ofR_nis a choice of convention. We will assume that all poles_nlie on the positive real axis; in particular, we see that every pole has a corresponding mirror pole-_non the negative real axis.

Our goal is to compute the position space counterpart off():

f(z) = ∫_C[d]/N() f() Ψ_(z)

whereCis a contour parallel to the imaginary axis. Here and in what follows we write contour integrals as
∫[d ] = ∫_-i ∞^i ∞d /2π i
to avoid notational clutter. Notice that bothf()and the measureN()are even in, which means that we can replaceΨ_(z)by any linear combination of the conformal blockQ()k_+(z)and its shadowQ(-)k_-(z). Without loss of generality, let us attempt to close the contourCto the right, picking up all poles_non the right half plane. This means that we have to drop the shadow part∼Q(-)/N() ×k_-(z), as it grows exponentially on the right half plane, whereas the conformal block part decreases as() →∞. Consequently, we find that the position space version off()is given by

f(z) = ∫_C[d ]  f() Q()/N()k_+(z) = ∫_C[d ]  f() 2/Q(-)k_+(z)

using the second equality in eq:Ndef. 
In that case, we can rewritef(z)as

f(z) = ∑_n 2R_n/Q(-_n)  k_+_n(z) .

To pass from Eq. eq:tpint to eq:CBdec2, we used that1/Q(-)is analytic on the right half plane. But the sum appearing in the RHS is precisely a CB decomposition — cf. Eq. eq:CBdec— where then-th term corresponds to an exchanged operator_nof dimension[_n] = 1/2 + _n, having OPE coefficient
  _ϕϕ_n^2 = 2R_n/Q(-_n) .

SinceQ(-) > 0for all>0, we conclude that _ϕϕ_n^2is positive iffR_nis positive.


Above, we assumed that all_nwere positive. This means that only operators of dimension[_n] > 1/2appear in the CB decomposition eq:CBdec2. This condition can be loosened: an operator of dimensionh<1/2would simply correspond to a pole_*lying on the left half plane. We must in this case deform the contour to circle_*in the positive direction. Moreover,_*will have a mirror pole-_*on the right half plane, which must be circled in the negative direction, such that it does not give an anomalous contribution tof(z)— see Fig. <ref>. We will revisit this point in Sec. <ref>.




The casesh=0(corresponding to the unit operator) andh=1/2require special attention. As forh=0, notice that1/Q(-)has a pole at= -1/2,namely
1/Q(-) → -1/2  -1/ + 1/2 .

Consequently, it suffices forf()to be finite at= 1/2in order to generate a unit operator term. To be precise, if
f()  → -1/2  c + O[(+)]

and the contour is such that it wraps aroundα= -1/2in the sense described above, thenf(z) = 2c + other conformal blocks. A similar issue arises ifh=1/2, because1/Q(-)vanishes as→0. More precisely
1/Q(-) → 0 π + O(^2)

hence in order to obtain a contributionf(z) ∼c   k_1/2(z) ∼c √(z) + …in position space, we must have
f() = -c /2π^2 + O(^-1) .


  §.§.§ Examples


To develop some familiarity with the alpha space representation of correlation functions, we will compute the alpha space transform of some simple functions inz-space, and we use these results to compute the resulting conformal block decompositions. 


  *  Let's compute the alpha space transform of a single conformal block k_h(z) with h > 1/2:
k_h() = ∫_0^1 dz/z^2  k_h(z) Ψ_(z) = C(h)/^2 - (h-þ)^2 ,
  C(h) = -Γ(2h)/Γ^2(h) .

In order to derive this result, it's convenient to use the Mellin-Barnes formula
_2F_1(a,b;c;z) = Γ(c)/Γ(a)Γ(b)∫_-i∞^i∞ ds  Γ(-s)Γ(a+s)Γ(b+s)/Γ(c+s)(-z)^s

in order to expand both k_h(z) and Ψ_(z). Alternatively, Eq. eq:cbt is easy to check numerically inside the strip |()| < h-þ. 

Let us make two comments about the formula eq:cbt. First, although the integral in eq:cbt converges only in a finite strip, the RHS defines an analytic continuation to any value of . Moreover, the same formula defines an analytic continuation to values of h < 1/2. Second, k_h() has precisely one pole on the right half plane, at = h-1/2, in accordance with our discussion from the previous section.



  *  Let f_p(z) = z^p with p>1/2. We have already encountered this function in Eq. eq:powerlaw, finding that in alpha space it becomes
  f_p() = Γ(p-±)/Γ^2(p) .
  
  Let's use this to obtain the CB decomposition of f_p(z). First, we note that f_p() has poles at
  _n = p -  + n
    and  _n =  - p - n
      with  
  n ∈N .
  
  Closing the contour to the right, we only pick up the _n poles. The residue of the n-th pole is
  
  R_n = - Res f()|_ = _n = (-1)^n/n!Γ(2p-1+n)/Γ^2(p)
  and this pole corresponds to an operator of dimension h = 1/2 + _n  = p + n.
  Using the argument from the previous section, we conclude that
  
  f_p(z) = ∑_n=0^∞2R_n/Q(-_n) k_p+n(z) = ∑_n=0^∞(-1)^n/n!(p)_n^2/(2p-1+n)_n  k_p+n(z) .
  
  This confirms a known result, see for instance Eq. (4.15) from Ref. <cit.>.
  

  * 
  Let f_p,q(z) = z^p(1-z)^-q. It will be instructive to spend some time on the computation of the alpha space density f_p,q(). As a first step, we rewrite Ψ_(z) using the Mellin-Barnes representation eq:mbform.



  This means that we can write
  
    f_p,q()    = 1/Γ(±)∫[d s]  Γ(-s)Γ(± + s)/Γ(1+s)∫_0^1 d z/z^2(1-z/z)^s z^p/(1-z)^q
       = 1/Γ(±)Γ(p-q)∫[d s]  Γ(-s)Γ(± + s)/Γ(1+s)Γ(1-q+s)Γ(p-1-s)

  where in the first line we have interchanged the z and s integrals. What remains is a standard Mellin-Barnes integral, which evaluates to
  
    f_p,q() = Γ(p-1)Γ(1-q)/Γ(p-q) _3F_2( + , -,  1-q  1,  2-p;1 )  
    
        +  Γ(1-p)Γ(p-±)/Γ(±)Γ(p) _3F_2(p-12 + ,  p-12 - ,  p-q  p,  p;1 ).

which provides an analytic continuation to all , provided that q > p-1.[Interestingly, the above expression can be analytically continued to other values of p and q using hypergeometric identities, in particular Thm. (2.4.4) and Corrollary (3.3.5) from <cit.>. We can for instance write
 
    f_p,q() = Γ(1-q)Γ(p-±)/Γ(p)Γ(p-q) _3F_2( + , -,  q  p,  1;1 ) = Γ(p - ±)/Γ^2(p) _3F_2(p- + ,  p--,  q  p,  p;1 ).

 The _3F_2(1) hypergeometrics in these expressions converge when p > q resp.q > 1. 
   ]
 Notice that the first term above is analytic in , hence it does not contain any poles in .  However, it does influence the behaviour of f_p,q() at large . The second term contributes two series of poles, at ± = p -  + N. Closing the -contour to the right and computing residues, we arrive at the following conformal block decomposition:
  
  f_p,q(z) = ∑_n=0^∞(p)_n^2/n! (2p-1+n)_n _3F_2(-n,  2p-1+n, p-q  p, p;1 )  k_p+n(z) .
  
  This is a new result which would have been rather difficult to guess. For p=q, this reduces to Eq. (4.14) from <cit.>.
  
  

 §.§ Convergence and asymptotics


In Sec. <ref> we discussed the convergence of the alpha space transform in a general setting. In the present section, we will specialize to CFT correlation functions, and more particularly, we will relate the largebehaviour off()to the growth off(z)asz →1. Recall that at the extreme pointsz = 0andz = 1a crossing-symmetric four-point function in a unitary CFT behaves as

z    → 0 : (z) → 1 + …

z    → 1 : (z) →(z/1-z)^-2h_ϕ (1 + … )

Clearly such a function is not square integrable with respect to the inner product (<ref>). As we will now proceed to explain, an alpha space transform can nevertheless be defined also for such functions. We will show that divergences near the two endpointsz = 0andz=1translate very differently into alpha space and bear resemblance to the usual IR and UV divergences in Fourier space.

Let us first focus onz →0, which is the OPE limit, and suppose we try to transform a functionf(z)behaving likez^p (1 + …)for smallzto alpha space. For our inner product square integrability is lost as we dialpto a value less than or equal to1/2. In alpha space this is reflected by a pair of poles crossing the real axis, as follows from the correspondence between conformal blocks of dimensionhand poles atα= ±(h - 1/2). This forces the integration contour in the inverse alpha transform off the imaginary axis, since the correct position-space expression is recovered only if it wraps around the poles as indicated in Fig. <ref>. This is however the only modification necessary, and we conclude thatz →0singularities of power-law form can be entirely dealt with by augmenting the inverse alpha space transform (<ref>) with a contour prescription around the poles. This prescription works without issues for any0 < p < 1/2; the special casesp=0andp=1/2were discussed above in Sec. <ref>.

Now let us consider the limitz →1. For simplicity we will restrict ourselves to the (physically relevant) case of functionsf(z)analytic in0 < z < 1. First of all, sinceΨ_α(1) = 1we find that

f(1) = ∫ [d ] f()/N() ,

and similarly it follows fromD·Ψ_α(z) = (^2 - 1/4) Ψ_(z)that

D^n · f(1) = ∫ [d] (^2 - 1/4)^n f()/N() ,

which holds as long as theD^n ·f(z)remains square integrable. Supposingf(z)behaves as a power law nearz = 1, we see from

D ·( (1-z)^ρ (1 + … ) ) = ρ^2 (1-z)^ρ - 1(1 + …) ,

that acting with the Casimir operatorDworsens the behavior nearz = 1. For generic positiveρthere exists annsuch thatD^n ·f(1)ceases to be well-defined, and therefore the integral in (<ref>) should somehow suffer the same fate. Since we only modify the integrand with a polynomial factor, this can only happen if the integral stops converging. We conclude that the large alpha behavior reflects the `short-distance' behavior off(z)asz →1.[We can also offer a physical explanation. For fixed alpha the Ψ_a(z) oscillate very slowly near z = 1 and to probe this region we need to consider very short `wavelengths', corresponding to very large values of the `momentum' α.]

The above discussion also offers a way to make sense of power-law divergent densities in alpha space: we just dividef()by sufficiently powers ofα^2 - 1/4, perform the now-convergent integral overα, and act just as many times withDon the resulting position-space expression. This is in fact entirely analogous to the usual trick in Fourier space, where we habitually make sense of UV-divergent expressions likep^2 αwithα> 0by replacing powers ofp^2with a Laplacian operator,
∫ dx  e^i p x p^2 α( 1 + …) → (-□)^n ( ∫ dx   e^i p x p^2 α - 2 n( 1  + …) ) ,

withnchosen such that the integral becomes convergent at largep. 

The relation between largeαandzclose to 1 can be made more quantitative. Firstly, if a functionf(z)is infinitely differentiable atz = 1, then the preceding logic demonstrates thatf()/N()must fall off faster than any power for large imaginary alpha. This is exemplified by the alpha space transform ofz^ρgiven above, which falls off exponentially fast. Secondly, for the generic power-law behavior we find that if

f(z) = (1-z)^-ρ( 1 + O(1-z) )      then     f() = (-^2)^ρ-1(1-ρ)/(ρ)( 1 + O(^-2) ) ,

which can be found by subtracting the leading power using the alpha space transform of a known function. For example, for small enoughρone can use
∫d^2 z/z^2[ z^ρ(1-z)^-ρ - z^ρ] Ψ_ (z)    = (ρ - 1/2±)/^2(ρ)[ (1-ρ)(ρ)/Γ(1/2±) - 1 ] 

   = (-^2)^ρ-1(1-ρ)/(ρ)( 1 + O(^-2) )

which can be computed as a limit from the above examples.



  §.§.§ Application: OPE convergence

We can use the preceding result to discuss the asymptotic behavior of OPE coefficients in one-dimensional CFTs, i.e., to provide a one-dimensional analogue of the results of <cit.>. Such a result has been discussed previously in the context of the light-cone limit for higher-dimensional CFTs <cit.>. Here we offer an explanation based on the assumption of suitably nice asymptotic behavior in alpha space.

Consider once more a unitary CFT correlation function(z)with a corresponding alpha space expressionF()which is meromorphic with simple poles. Our preceding discussion leads us to conclude thatF() ∼(-^2)^2h_ϕ- 1for large imaginaryα, since(z) ∼(1-z)^-2 h_ϕasz →1. We will assume that this asymptotic behavior holds for all non-realαand so the `subtracted' function

F^(s)()  (^2)^- 2 h_ϕ + 1 - ϵ  F() 

vanishes asymptotically away from the real axis for anyϵ> 0. This means we can write a dispersion relation for it: we write

F^(s)() = ∮ [d]̱F^(s)()̱/ - 
and push the contour away from the point. With the arcs of the contour at infinity vanishing, we find contributions only from the cuts created by the power-law prefactor and the real axis whereF()has poles. The contributions from the cuts can be made manifestly finite by aligning them along the imaginary axis and keeping the contour some distance away fromα= 0. It follows that the contribution from the poles, which after picking up the residues can be written as
∑_n (_n^2)^- 2 h_ϕ + 1 - ϵ R_n ( 1/ - _n + (α↔ - α))

is necessarily finite as well. In a distributional sense, then, we expect the residue series to behave as
∑_n δ(h - h_n) R_n ∼ c(h_ϕ) h^4 h_ϕ - 2 .

By working out the example given previously we also find the prefactor:

c(h_ϕ) = 1/Γ^2(2 h_ϕ) .

We observe that the prefactor vanishes when2 h_ϕis a negative integer which is precisely when thez = 1singularity in(z)also disappears.

Finally we can use equation (<ref>) and to relate this result to the asymptotic behavior of the squared primary OPE coefficients themselves as
_ϕϕ_h(h)^2 ∼4^1 - h√(π)/Γ^2(2 h_ϕ)  h^4 h_ϕ - 3/2 .

agreeing with the lightcone bootstrap result, see e.g. <cit.>.[Strictly speaking there is a factor 2 mismatch between eq:ourres and formula (3.8) in <cit.>, due to the fact that in the d-dimensional lightcone results only even spins are allowed to contribute.] It is interesting to see that the leading exponential falloff arises from the prefactorQ(1/2 - h), and the falloff speed is independent of the external dimension.



 §.§ Alpha space for different external dimensions


So far we considered the case of a four-point function of identical operators. However, the Sturm-Liouville theory for the2Casimir operator applies just as well to four-point functions of different operators. In this section, we will briefly discuss this generalization.

Concretely, we have in mind a four-point function of primariesϕ_iof dimensionh_i,i=1,…,4. Conformal symmetry restricts this correlator to have the following form:
ϕ_1(x_1)ϕ_2(x_2)ϕ_3(x_3)ϕ_4(x_4) = (|x_24|/|x_14|)^h_12(|x_14|/|x_13|)^h_34z^h_12(z)/|x_12|^h_1 + h_2 |x_34|^h_3 + h_4
for some function(z), using the shorthandh_ij ≡h_i - h_j. The stripped correlator admits a conformal block decomposition of the following form:
(z) = ∑__ϕ_1 ϕ_2 _ϕ_3 ϕ_4   k^s_h_(z)

involving the mixed2conformal blocks

k^s_h(z) = z^h+a _2F_1(h +a,  h + b;  2h; z) ,
    
a = - h_12 ,   b = h_34 .

The sum in Eq. eq:mixedCB now runs over all operators that appear in both theϕ_1 ×ϕ_2andϕ_3 ×ϕ_4OPEs; the label `s' refers to thiss-channel.

The blocksk_h^s(z)are eigenfunctions of a mixed Casimir differential operatorD_a,b:

  D_a,b· f(z)   = w_s(z)^-1 d/d z[ w_s(z)(1-z)z^2 f'(z) ] +a(a+1) f(z) ,    w_s(z) = (1-z)^a+b/z^2+2a ,
  
  which means thatD_a,bis self-adjoint with respect to the inner product
f,g_s = ∫_0^1  d z  w_s(z) f(z)g(z) .

Analyzing the relevant Sturm-Liouville problem leads to the following basis of eigenfunctions:[The PDE D_a,b  f(z) = (^2-1/4) f(z) has a second solution, namely
  
    z^2a/(1-z)^a+b _2F_1(-a+, -a-  1-a-b;z-1/z) .

  This second solution ceases to be regular at z=1 when a + b > 0.]Ψ^s_(z)  =   _2F_1( + a + ,  + a -   1+a+b; z-1/z) = ϑ_^(a+b,a-b)(1-z/z). 
 
 In the second equality, we have rewrittenΨ_^s(z)as a Jacobi function, to make contact with the integral transform introduced previously.

To connect the eigenfunctionsΨ^s_(z)to the conformal blocks, we compute
Ψ_^s(z) = [ Q_s() k^s_+(z) + (→ -) ],
  
Q_s() = 2Γ(-2)Γ(1+a+b)/Γ( + a-)Γ(+b-) .

As in the case of equal external dimensions, we can decompose any functionf(z)—  normalizable with respect to eq:mixedNorm— in terms of the functionsΨ_^s(z), to wit:

f(z) = ∫[d ]/N_s() f() Ψ^s_(z)
  ⇔  f() = ∫_0^1 d z   w_s(z)  f(z) Ψ^s_(z)

where

N_s() = 2Γ(± 2)Γ^2(1+a+b)/Γ( + a ±)Γ( + b ±) = |Q_s()|^2/2 .

Some care must be taken when considering thecontour in Eq. eq:sturmS: when eithera,b ≤-, the contour must be deformed in the Mellin-Barnes sense because of poles in the factor1/N_s(). 



  §.§.§ Cross channel


Applying crossing symmetry to mixed four-point functions leads to a relation between two different four-point functions. In the case of the correlatorϕ_1 ϕ_2 ϕ_3 ϕ_4, the bootstrap equation of interest is

F_ϕ_1 ϕ_2 ϕ_3 ϕ_4(z) = (z/1-z)^2h_2  F_ϕ_3 ϕ_2 ϕ_1 ϕ_4(1-z)

whereF_ϕ_3 ϕ_2 ϕ_1 ϕ_4(z)is defined as in eq:mix4pt but withϕ_1 ϕ_3andh_1 h_3exchanged. Such mixed crossing equations have been used intensively in computing scaling dimensions and OPE coefficients for the 3dIsing andO(N)models <cit.>.

Like before, the correlatorF_ϕ_3 ϕ_2 ϕ_1 ϕ_4(z)appearing in the RHS of eq:mixcross admits a decomposition in conformal blocks and in plane-wave normalizable eigenfunctions of the conformal Casimir. However, care must be taken to use conformal blocks with dimensionsh_1 h_3exchanged, and likewise for the eigenfunctionsΨ_^s(z). To be completely explicit, this new conformal block decomposition reads:

F_ϕ_3 ϕ_2 ϕ_1 ϕ_4(z) = ∑__ϕ_2 ϕ_3 _ϕ_1 ϕ_4   k^t_h_(z)

with

k^t_h(z) = z^h+a' _2F_1(h +a',  h + b';  2h; z) ,
    
a' = h_23 ,   b' = h_14 .

Here and in what follows we use the `t' label for blocks and eigenfunctions in theϕ_2 ×ϕ_3 →ϕ_1 ×ϕ_4channel.
The appropriate eigenfunctions in thet-channel are

    Ψ_^t(z) ≡Ψ_^s(z) |_h_1  h_3   =    _2F_1( + a' + ,  + a' -   1+a'+b';z-1/z)
       = ϑ_^(a'+b',  a'-b')(1-z/z)

which satisfy
Ψ_^t(z) = [ Q_t() k^t_+(z) + (→ -) ],
  
Q_t() = 2Γ(-2)Γ(1+a'+b')/Γ( + a'-)Γ(+b'-) .

Finally, the decomposition of a functionf(z)in terms of the functionsΨ_^treads

f(z) = ∫[d ]/N_t() f() Ψ^t_(z)
  ⇔  f() = ∫_0^1 d z   w_t(z)  f(z) Ψ^t_(z)

where

w_t(z) = (1-z)^a'+b'/z^2+2a'  and  
N_t() = |Q_t()|^2/2 .


§ CROSSING KERNEL


So far, we have used Sturm-Liouville theory as a tool to represent conformal correlators as integrals over a set of basis functionsΨ_. In this section, we will use these integral representations to analyze crossing symmetry. In particular, we will compute thed=1crossing kernel and exhibit its properties. 



 §.§ General case


Let us start by considering a mixed four-point functionϕ_1 ϕ_2 ϕ_3 ϕ_4. For such a correlator, we can write down two inequivalent integral representations:

    ϕ_1 ϕ_2 ϕ_3 ϕ_4    ∼ (z) = ∫[d ]/N_s()  F_s() Ψ_^s(z) ,
    ϕ_3 ϕ_2 ϕ_1 ϕ_4    ∼ (z) = ∫[d ]/N_t()  F_t() Ψ_^t(z) .

The∼above denotes that we have omitted various unimportant scaling factors. The spectral densityF_s()encodes information about the CB decomposition in thes-channelϕ_1 ×ϕ_2 →ϕ_3 ×ϕ_4, whereasF_t()describes thet-channelϕ_1 ×ϕ_4 →ϕ_2 ×ϕ_3.

The two alpha space densitiesF_s,t()are related — at least implicitly — via the crossing equation eq:mixcross. Plugging Eq. eq:genrep into that equation, we find that
∫[d ]/N_s()  F_s() Ψ_^s(z) = (z/1-z)^2h_2∫[d]̱/N_t()̱  F_t()̱Ψ_^̱t(1-z) .

In order to find make the constraints onF_s,t()manifest, we can manipulate this alpha space bootstrap equation in various ways. For instance, it is possible to expresst-channel eigenfunctions in terms of thes-channel ones:
(z/1-z)^2h_2Ψ_^̱t(1-z) = ∫[d]/N_s()  K(,|̱h_1,h_2,h_3,h_4) Ψ_^s(z) .

The distributionK(,|̱h_1,h_2,h_3,h_4)introduced here relates eigenfunctions in thes- andt-channels, and we will refer to it as a crossing kernel. A schematic interpretation of Eq. eq:kerndef is given in Fig. <ref>.


 Using eq:kerndef, we can recast the crossing equation eq:alphaxing as
∫[d]/N_s()[F_s() - (K· F_t)() ] Ψ_^s(z) = 0

where we have introduced an integral operatorKwhich depends on theh_i:

 (K· f)() ∫[d]̱/N_t()̱  K(,|̱h_1,h_2,h_3,h_4) f()̱ . 

Recalling that theΨ_^s(z)form a complete basis inz-space, Eq. eq:interm1 can only be satisfied if

F_s() = (K· F_t)() .

The point of this identity is that it directly relates the two densitiesF_s,t(); once we compute the kernelK(,|̱h_i), Eq. eq:xint will be completely explicit. 

In the previous computation, we made an arbitrary choice by expressingΨ_^̱t(1-z)in terms of thes-channel functionsΨ_^s(z). It will be useful to go in the opposite direction as well, by writing
(z/1-z)^2h_2Ψ_^̱s(1-z) = ∫[d]/N_s() K(,|̱h_1,h_2,h_3,h_4) Ψ_^t(z)

which involves a second crossing kernelK(,|̱h_1,…,h_4). Using the same logic as before, we arrive at an alternate alpha space crossing equation:

F_t() = (K· F_s)() ,

where

(K· f)() ∫[d]̱/N_s()̱ K(,|̱h_1,h_2,h_3,h_4) f()̱ .

Bringing everything together, we have recast crossing symmetry as a system of integral equations in alpha space:

  F_s() = (K· F_t)() ,
    
  F_t() = (K· F_s)() .


 §.§ Identical operators


Let us briefly consider the case of the four-point functionϕϕϕϕof four identical primaries. In that case, there is only one spectral densityF()of interest, namely
ϕϕϕϕ ∼ (z) = ∫[d ]/N()  F() Ψ_(z) .

Rather than a system of coupled integral equations, one now finds an eigenvalue equation for the densityF():

  F() = (K_0 · F)()

where the integral operatorK_0is defined as

  (K_0 · f)()∫[d]̱/N()̱  K_0(,|̱h_ϕ) f()̱ ,
    
K_0(,|̱h_ϕ)  K(,|̱h_ϕ,h_ϕ,h_ϕ,h_ϕ) .


 §.§ Functional properties of the crossing kernels


In what follows, we will compute the crossing kernelsK(,|̱h_i),K(,|̱h_i)andK_0(,|̱h_i). Since this computation is somewhat technical, we will first derive several properties of these kernels.

Evidently, all of the kernels are even in their argumentsand$̱. Less trivially, we see that the kernels K and K are identical after exchanging the external dimensions h_1 and h_3:

K(,|̱h_1,h_2,h_3,h_4) = K(,|̱h_3,h_2,h_1,h_4)
as follows from Eqs. eq:kerndef, eq:kerndef2.

Next, from the structure of Eq. eq:mixsystem, we can surmise that

K ·K = K ·K = 𝕀 .
We have derived this with input from the bootstrap, but later we will rederive Eq. eq:inverses formally. For the case of identical operators, Eq. eq:inverses becomes

K_0^2 = 𝕀 .
Notice that Eqs. eq:inverses and eq:inverses2 only hold when restricted to some space of even functions, as the images of the integral operators K, K and K_0 are even by construction.

Both identities eq:inverses and eq:inverses2 are statements about integral operators. By acting with these operators on test functions — say, having compact support — we can turn them into orthogonality/completeness relations for the crossing kernels themselves. To make this concrete, let's define the distributions

    D_s(,|̱h_1,h_2,h_3,h_4)      N_s()^-1∫[dy]/N_t(y)  K(,y|h_i) K(y,|̱h_i) , 
    D_t(,|̱h_1,h_2,h_3,h_4)      N_t()^-1∫[dy]/N_s(y) K(,y|h_i) K(y,|̱h_i) 
       = D_s(,|̱h_3,h_2,h_1,h_4) .

Our claim is that D_s,t(,|̱h_i) behave as delta functions on the imaginary axis. Indeed, Eq. eq:inverses implies that

∫[d]̱    D_s(,|̱h_i) 
 D_t(,|̱h_i)  f()̱ =  f()+f(-)/2
where f() is arbitrary. This can be thought of as the “local” version of eq:inverses. In the case of identical operators, we simply have

∫[d ]̱   D_0(,|̱h_ϕ) f()̱ =  f()+f(-)/2
where

D_0(,|̱h_ϕ) =  N()^-1 ∫[dy]/N(y)   K_0(,y|h_ϕ)K_0(y,|̱h_ϕ) .
Eqs. eq:id11 and eq:id12 can be obtained as a limiting case of eq:KasDirac. Interestingly, Eqs. eq:KasDirac and eq:id11 imply that the distributions D_s,t(,|̱h_i) and D_0(,|̱h_ϕ) are identical and independent of external dimensions h_i resp. h_ϕ. As with the Fourier transform, the above identities mean that well-behaved functions f() can be decomposed in terms of the “basis functions” K, K and K_0, with computable coefficients.



 §.§ Computation of the crossing kernel


Let us now turn to the computation of the crossing kernel K(,|̱h_i). To do so, we can use the alpha space technology from Sec. <ref> to write down a position-space integral representation for K, namely

K(,|̱h_1,h_2,h_3,h_4) = ∫_0^1z  w_s(z)  (z/1-z )^2h_2 Ψ_^s(z) Ψ_^̱t(1-z) .
It will be convenient to employ standard Mellin representations for the functions Ψ_^s,t(z):

    Ψ_^s(z)    = Γ(1+a+b)/Γ( + a ±)∫[ds]  Γ(-s)Γ(+a+s±)/Γ(1+a+b+s)(1-z/z)^s  ,
    Ψ_^̱t(1-z)    = Γ(1+a'+b')/Γ( + a'  ±)∫[dt]  Γ(-t)Γ(+a'+t±)/Γ(1+a'+b'+t)(z/1-z)^t  .

Plugging these into eq:kintrep, one obtains an integral representation of the form
K(,|̱h_i) = ∫_0^1dz ∫[ds] ∫[dt] … .
Exchanging the order of the integrals, the z-integral yields a beta function, whereas the resulting t-intergral can be performed using the second Barnes lemma. What remains is the following Mellin representation:[A different-looking representation can be found by doing the s-integral first.]

    K(,|̱h_1,h_2,h_3,h_4) = Γ(1+a+b)Γ(1+a'+b')/Γ(  + a ±)Γ( + b'±)
    × ∫[ds]  Γ(-s)Γ(+a+s±)/Γ(1+a+b+s)Γ(2h_1-1-s)Γ(32-h_1-h_4+s ±)̱/Γ(2-h_1+h_2-h_3-h_4 + s) .

This integral can be performed by closing the contour and picking up poles on the right half plane, at s = N and s = 2h_1 - 1 + N. The result is a sum of two hypergeometric _4F_3(1) functions, and it can be cast into a standard form by introducing the Wilson functions of Ref. <cit.>:[Our conventions differ from those of <cit.> as follows: W_(|̱a,b,c,d) = ϕ_i(i;̱a,b,c,1-d).
    ]  
    
    W_(;̱a,b,c,d)   =
    Γ(d-a)/Γ(a+b)Γ(a+c)Γ(d±)̱Γ(d̃±) _4F_3(a +,̱  a-,̱ ã + , ã-  a+b,  a+c,  1+a-d;1 )    + (a  d)

    writing ã = (a+b+c-d)  and d̃ = (-a+b+c+d). It is useful to know that W_(;̱a,b,c,d) is even in its arguments  and $̱, and it depends symmetrically on its parameters{a,b,c,d}. A closed-form expression for the crossing kernel is then given by
    
    K(,|̱h_1,h_2,h_3,h_4) =  Γ(1-h_12 + h_34)Γ(1+h_14 + h_23 )
    × Γ(h_1 + h_2 - ±) Γ(32 - h_1 -h_4 ±)̱W_(;̱P)

with parametersP = P(h_1,h_2,h_3,h_4)specified by
P = { + h_14,   + h_23,   h_2 + h_3 - ,  3/2 - h_1 - h_4} .

The kernelK(,|̱h_1,h_2,h_3,h_4)admits an expression similar to eq:Kformula, the only difference being thath_1 h_3are swapped. For completeness, we print the formula for the identical-operator kernelK_0(,|̱ h_ϕ)here as well:

    K_0(,|̱h_ϕ) = Γ(2h_ϕ - ±)Γ(32 - 2h_ϕ±)̱W_(;̱P_0) ,
    P_0 = {, ,  2h_ϕ - , 3/2 - 2h_ϕ} .


 §.§ K and K as intertwiners


Having computed the crossing kernelsKandK, let us now revisit the alpha space crossing equation eq:mixsystem. Informally, it encodes thatKmaps a “t-channel” alpha space density to an “s-channel” one, and vice versa forK. In this section we will formalize this idea, making precise in which senseKandKintertwine between two different Hilbert spaces. 

First, let's introduce a Hilbert spaceH_s = H_s(h_1,h_2,h_3,h_4)fors-channel functions, consisting of all functionsf()that are even inandL^2with respect to the following inner product:

    (f,g)_s ∫[d]/M_s(;h_1,h_2,h_3,h_4) f()g() ,
    M_s(;h_1,h_2,h_3,h_4) = 2Γ^2(1-h_12+h_34)Γ(± 2)Γ(h_1 + h_2 - ±)/Γ( - h_12±) Γ( + h_34±)Γ(32-h_3 - h_4  ±) .

We have introduced an-independent factor in the measureM_s(;h_i)to simplify some formulas later on. The integration contour in eq:sHilb is to be understood in the Mellin-Barnes sense, which means that it may be deformed depending on the values of theh_i. Likewise, we introduce at-channel Hilbert spaceH_t(h_1,h_2,h_3,h_4)of even functions that are square-integrable with respect to
(f,g)_t ∫[d]/M_t(;h_1,h_2,h_3,h_4) f()g() ,
  M_t(;h_1,h_2,h_3,h_4) =  M_s(;h_3,h_2,h_1,h_4) .

We now claim that the following holds:

 Theorem 1.1: K is a unitary map H_t →H_s, and K  :  H_s →H_t is its inverse.
 
Unitarity here means thatKandKpreserve the inner products defined in Eqs. eq:sHilb and eq:tHilb, namely
(f,g)_t = (K· f,K· g)_s
  and  (f,g)_s = (K· f,K· g)_t .

The proof of this result follows from the properties of the  Wilson transform, introduced in Ref. <cit.>. This integral transform uses the Wilson functionsW_(;̱a,b,c,d)as a basis. The above result can straightforwardly be deduced from Theorem 4.12 of Ref. <cit.>. Consequently, we will not provide many details. However, it will be instructive to provide a sketch of a (constructive) proof. First, one establishes thatH_sis spanned by the following functions:
ξ_n^s(|h_i) = Γ(1-h_12 + h_34)Γ(h_1 + h_2 - ±)  _n(;P̃) ,
  
n ∈N.

The Wilson polynomials_nwere defined in Eq. eq:wdef, and the set of parametersP̃is given by
P̃(h_1,h_2,h_3,h_4) = { - h_12,   + h_34,   h_1 + h_2 - ,  3/2 - h_3 - h_4} = P(h_3,h_2,h_1,h_4)  .

Likewise,H_tis spanned by the functions
ξ_n^t(|h_i) = Γ(1+h_14 +h_23) Γ(h_2 + h_3 - ±)  p_n(;P) .

By linearity, it suffices to establish thatKandKact appropriately on these basis functions. To establish this, one proves first that
(ξ_m^s,ξ_n^s)_s = (ξ_m^t,ξ_n^t)_t ∝_mn
as well as

(K·ξ_n^t)() = (-1)^n  ξ_n^s() ,
  
(K·ξ_n^s)() = (-1)^n  ξ_n^t() .

Eq. eq:dualIP is a property of the Wilson polynomialsp_n<cit.>, and Eq. eq:eigX is a consequence of Theorem 6.7 of <cit.>.


A similar result holds for the case of identical operators. There one defines a Hilbert spaceH_0 = H_0(h_ϕ)of even functions that are finite with respect to
(f,g)_0 ∫[d]/M_0(;h_ϕ) f()g() ,
  M_0(;h_ϕ) = M_s(;h_ϕ,h_ϕ,h_ϕ,h_ϕ) .

Then the counterpart of the above theorem reads:

  Theorem 1.2: K_0 is a unitary map H_0 →H_0 obeying K_0^2 = id.
  Here unitarity means that
(f,g)_0 = (K_0· f,K_0· g)_0 .

The proof goes along the same lines as the general case discussed before. A basis forH_0is now spanned by the functions
ξ^0_n(|h_ϕ) = Γ(2h_ϕ - ±)  _n(;P_0)
 
 whereP_0was defined in eq:podef.  The operatorK_0maps theξ_n^0to themselves, up to a sign(-1)^n:
 
 (K_0 ·ξ_n^0)() = (-1)^n  ξ_n^0() .
 
Of course, the only permissible eigenvalues that could have appeared were±1, given thatK_0^2 = 𝕀. 




 §.§ Analytic structure of the crossing kernel


Since we have rephrased bootstrap equations as integral equations in alpha space, it will be instructive to analyze the analytic structure of the crossing kernelK(,|̱h_1,h_2,h_3,h_4). Let's first fix$̱ and investigate the properties of K as a function of , using Eq. eq:Kformula. Since the Wilson functions W_(;̱a,b,c,d) are analytic in  and $̱, the only poles inare due to the factorΓ(h_1 + h_2 - ±). ConsequentlyK(,|̱h_i)is a meromorphic function, with its only poles on the right half plane at= h_1 + h_2 - + N. The relevant residues are polynomials of degreenin^̱2, namely

    R_n(;̱h_1,h_2,h_3,h_4)     - Res  K(,|̱h_1,h_2,h_3,h_4) |_ = h_1 + h_2 - 1/2 + n
       = Γ(1-h_12+h_34)/n!(1+h_14+h_23)_nΓ(2h_1 + 2h_2 - 1+n)/Γ(2h_2+n)Γ(h_1 + h_2 + h_34 +n)
       × _n(;̱+h_14, +h_23,  h_1 + h_4 - ,  h_2 + h_3 - ) .

Next, remark that for generic values of,K(,|̱h_i)is a rather complicated function of$̱. Upon closer inspection it appears that at certain values _* the kernel K(_*,|̱h_i) becomes polynomial in $̱, up to a number of gamma functions. The relevant values= _*are organized in three families:
_n^I = 3/2 - h_3 - h_4 + n,
  _n^II =  - h_12 + n,
  _n^III =  + h_34 + n,
  
n ∈N .

For the first family, we find for instance

    K(_n^I,)̱   = k_n^I Γ(32 - h_1 - h_4 ±)̱/Γ(h_2 + h_3 - ±)̱ _n(;̱ + h_14, +h_23, 32 - h_1 - h_4, 32-h_2 - h_3 )
    where k_n^I is a constant that does not depend on $̱. For the second and third families, we find
    K(_n^II,)̱   = k_n^II Γ(32 - h_1 - h_4 ±)̱/Γ( + h_14±)̱ _n(;̱ - h_14, +h_23,  h_2 + h_3 - , 32-h_1 - h_4 ),
    
      K(_n^III,)̱   = k_n^III Γ(32 - h_1 - h_4 ±)̱/Γ( + h_23±)̱ _n(;̱ + h_14, -h_23,  h_2 + h_3 - , 32-h_1 - h_4 ).
    
    
    We can also consider the analytic structure ofK(,|̱h_i)as a function of$̱ for fixed . This is a simple exercise, given the relation eq:KandKw. We therefore refrain from printing explicit formulas. 
    
    
    
    
    
     §.§ Symmetries of the crossing kernel
    
    
    The crossing kernel obeys various identities which we will exhibit here. Since none of these results are used in the rest of this paper, this section can be skipped on a first reading.
    
    It will be convenient to strip off the gamma functions in Eq. eq:Kformula and to relabel the external dimensions as h_i → + γ_i. What remains is a single Wilson function, namely
    K̂(,|̱γ_1,γ_2,γ_3,γ_4) = W_(|̱ + γ_1 - γ_4,  + γ_2 - γ_3,  -γ_1 - γ_4,  + γ_2 + γ_3) .
    
    First, we recall that W_(;̱a,b,c,d) depends symmetrically on its parameters {a,b,c,d}, which implies that K̂(,|̱γ_i) obeys
    
        K̂(,|̱γ_1,γ_2,γ_3,γ_4)    = K̂(,|̱-γ_1,γ_2,γ_3,γ_4) =  K̂(,|̱γ_1,γ_2,-γ_3,γ_4)
           = K̂(,|̱γ_3,-γ_4,γ_1,-γ_2) 
           = K̂(,|̱-γ_1^♮,-γ_2^♮,-γ_3^♮,-γ_4^♮) ,
            γ_i^♮ =  -γ_i + ∑_j=1^4 γ_j  .
    
    A second type of symmetry can be found using the identity (see Lemma 5.3 of <cit.>)
    W_(;̱A+ω,A-ω,B+ρ,B-ρ) = W_ω(ρ;A+,A-,B+,̱B-)̱
    which descends to
    K̂(,|̱γ_1,γ_2,γ_3,γ_4) = K̂(γ_1,γ_3|,γ_2,,̱γ_4)  .
    
    A final relation follows from the “duality” property of the Wilson functions:
    W_(;̱a,b,c,d) = W_(̱;ã,b̃,c̃,d̃) ,
          [  ã; b̃; c̃; d̃ ] = (a+b+c+d) - [ d; c; b; a ]
    which implies that
    K̂(,|̱γ_1,γ_2,γ_3,γ_4) = K̂(,̱|γ_3,γ_2,γ_1,γ_4)  .
    
    
    The reader may notice that the above symmetries are reminiscent of those corresponding to the SU(2)6-j symbol <cit.>. In the SU(2) context, the transformations γ_1,3↦ - γ_1,3 are known as mirror symmetries and γ_i ↦γ_i^♮ is a Regge transformation;  Eqs. eq:row1 eq:col1 and eq:row2 are related to transformations that exchange rows and columns of the 6-j symbol. 
    
    A subset of the above symmetries lifts to the full crossing kernel K(,|̱h_i):
    
        K(,|̱h_1,h_2,h_3,h_4)    =K(,|̱h_3^♮,h_4^♮,h_1^♮,h_2^♮) ,   h_i^♮ = -h _i + ∑_j=1^4 h_j , 
           =K(,̱|1-h_3^♮,1-h_2^♮,1-h_1^♮,1-h_4^♮) , 
           =K(,̱|1-h_1,1-h_4,1-h_3,1-h_2) .
    
    Any two of these identities imply the third one. In conclusion, it appears that the automorphism group of the K(,|̱h_i) is isomorphic to the Klein four-group. In passing, we note that Eq. eq:fullK can also be derived by inspecting the integral representation eq:kerndef.
    
    
    
      §.§.§ Limit cases
    
    
    For bootstrap applications, one is often interested in four-point functions where some of the operators are identical. In that case, the discussion of the symmetries of the crossing kernel simplifies drastically. For a mixed four-point function of the form σσ, there are two relevant crossing kernels:
    
    K_m,1(,|̱h_σ,h_)  K(,|̱h_,h_σ,h_σ,h_) ,
      
    K_m,2(,|̱h_σ,h_)  K(,|̱h_σ,h_σ,h_,h_) .
    
    In this case, the content of Eq. eq:fullK reduces to
    
    K_m,1(,|̱h_σ,h_) = K_m,2(,̱|1-h_,1-h_σ) .
    
    Finally, when all external dimensions are identical, the relevant kernel is K_0(,|̱h_ϕ), which obeys
    
    K_0(,|̱h_ϕ) = K_0(,̱|1-h_ϕ) .
    
    
    § APPLICATIONS TO THE CONFORMAL BOOTSTRAP
    
    
    In Section <ref>, we reformulated crossing symmetry in the form of integral equations in alpha space, making use of the crossing kernel K(,|̱h_i). For definiteness, let us consider the identical-operator alpha space equation Eq. eq:idx:
    
    F() = ∫[]̱/N()̱  K_0(,|̱h_ϕ) F()̱ .
    
    In the bootstrap context, we can ask whether Eq. eq:eq (combined with unitarity) can be used to find useful constraints on F(). In this section we will sketch some ideas in this direction, making use of the properties of the crossing kernel as discussed in Sec. <ref>.
    
    
    
     §.§ (Dis)proving a false theorem
    
    
    We will start by outlining an simple idea for analyzing the alpha space crossing equation eq:eq. One can think of the RHS of eq:eq as a function of α ↦ ∫[d ]̱/N()̱  K_0(,|̱h_ϕ) F()̱
    and require that eq:afunc has exactly the same analytic structure as F(), appearing on the LHS of eq:eq. Taken at face value, this should lead to constraints of the poles and residues of F(), which correspond to CFT data.
    
    The function eq:afunc only depends on  through the crossing kernel K_0(,|̱h_ϕ). Using the results of Sec. <ref>, we see that the identical-operator kernel K_0(,|̱h_ϕ) has poles at _n = 2h_ϕ -  + n, n ∈N, with residues
    R_n(|̱h_ϕ) R_n(|̱h_ϕ,h_ϕ,h_ϕ,h_ϕ) = Γ(4h_ϕ - 1 + n)/n!^2 Γ^2(2h_ϕ + n) _n(;̱,,2h_ϕ - ,2h_ϕ - ).
    
    Plugging this result into eq:eq, we naively conclude that F() can only have poles at = _n, with their residues constrained as follows:
    
    - Res  F() |_ = _n?=∫[d ]̱/N()̱ R_n(|̱h_ϕ) F()̱ .
    
    
    Obviously, this conclusion is wrong: it says that any solution to crossing consists of a single tower of exchanged operators with dimensions 2h_ϕ + N. Although solutions of this form exist (e.g. in mean field theory), any interacting CFT correlator furnishes a counterexample to eq:wrongthm. From a mathematical point of view, we have arrived at eq:wrongthm using a doubtful manipulation:
    Res[ ∫[d ]̱/N()̱  K_0(,|̱h_ϕ) F()̱]_ = _n?=∫[d ]̱/N()̱ [ Res  K_0(,|̱h_ϕ) ]_ = _n F()̱ .
    
    This fails to hold at general , as the function eq:afunc is defined for real  only by analytic continuation. It would be interesting to see if this wrong argument can be refined to give useful bootstrap constraints, likely by deforming the contour in Eq. eq:eq, as discussed in Sec. <ref>.
    
    
    
    
    
     §.§ Split kernel
    
    
    A second idea is to close the $̱ contour in Eq. eq:eq to the right, picking up poles inβ.  Since the integrand appearing in the RHS of eq:eq equals
    K_0(,|̱h_ϕ) F()̱/N()̱
    poles in$̱ can come from three different factors. As mentioned, the poles in F(β)— and their residues — are unknown, but of physical interest. Next, 1/N(β) has poles at β = 1/2 + N, and K_0(,|̱h_ϕ) has poles at β = 3/2 - 2h_ϕ + N.[Note that the poles of K_0(,|̱h_ϕ) in β are related to the poles in  through Eq. eq:KandKw. In particular, the $̱ residues are Wilson polynomials in. Closing the contour means that we have to keep track of all of these different poles.
    
    We propose to modify Eq. eq:eq in a straightforward way, bypassing this bookkeeping exercise. The key point is that bothN(β)andF(β)are even inβ; in the definition eq:kerndef of the crossing kernel, it is therefore possible to replaceΨ^t_(̱1-z)byQ_t()̱ k^t_+(1-z), whereQ_tandk_^t(z)were defined in Sec. <ref>. Concretely, we recast the crossing equation as
    
        F() = ∫[d ]̱ (,|̱h_ϕ,h_ϕ,h_ϕ,h_ϕ) F()̱ ,
        (,|̱h_1,h_2,h_3,h_4) Q_t()̱/N_t()̱∫_0^1 z  w_s(z) (z/1-z)^2h_2Ψ_^s(z) k_ +^t(1-z) .
    
    We will from now on consider this “split” kernelK_split(,|̱h_i)with arbitrary external dimensions, although only the caseh_1 = … = h_4 ≡ h_ϕis of interest in the analysis of Eq. eq:eq.
    
    We claim that the split kerneldoes not have any poles on the right half plane()̱ > 0. That is to say, by closing the contour of eq:spliteq to the right, we only pick up poles coming fromF(β), as desired.
    
    The proof of this claim follows from a direct computation. The computation is very similar to the one from Sec. <ref>. The only difference is that we use a Mellin-Barnes representation for the cross-channel blockk_h^t(1-z), namely
    
        k_ + ^t(1-z) = Γ(1+2)̱/Γ( + a' + )̱Γ( - b' + )̱
        × ∫[d t] Γ(-t)Γ( + a'++̱t)Γ( - b' ++̱t)/Γ(1+2+̱t)(z/1-z)^++̱a'+t .
    
    As an intermediate step, we rewriteas a Mellin-Barnes integral:
    
        (,|̱h_i) = Γ(1-h_12+h_34)/Γ(1+h_14+h_23)2/Γ( - h_12±)Γ( + h_23 + )̱/Γ( - h_23 + )̱∫[ds]  Γ(-s)Γ( - h_12 +s ±)/Γ(1-h_12+h_34+s)
        × Γ(2h_1 - 1 -s)Γ(h_12 + h_3 + h_4 -1-s)Γ(32-h_1 -h_4 ++̱s)/Γ(h_1 + h_4 -  + -̱s) .
    
    Closing the contour to the left[Closing the contour to the right would mean picking up poles at s = 2h_1 - 1 + N and s = h_12 + h_3 + h_4 -1 + N. In the case of equal external dimensions, these two series of poles collide to form a single series of double poles.
    ] and picking up poles ats = -N,s = ± - + h_12 - N, we obtain the following closed-form formula for:
    
    K_split(,|̱h_1,h_2,h_3,h_4) = I_1(,|̱h_i) + I_2(,|̱h_i) +  I_2(-,|̱h_i)
    
    where
    
        I_1(,|̱h_i)    =  Γ(1-h_12 + h_34)/Γ(1+h_14+h_23)2/S(h_2 + h_4 +  -)̱S(h_2 + h_4 --)̱
               × Γ( + h_14+)̱Γ( + h_23+)̱Γ(32 - h_1 - h_4 + )̱/Γ( - h_12±)Γ(h_2 + h_3 -  -)̱
               × _4F̃_3[ + h_14+,̱  - h_23 + ,̱ 32 -  h_1 - h_4 + ,̱ 32 - h_2 - h_3 +  1+2,̱  2-h_2 - h_4 +  + ,̱  2-h_2 -h_4 - + ;1 ], 
        
          I_2(,|̱h_i)    = -Γ(1-h_12 + h_34)/Γ(1+h_14+h_23)2/S(h_2 + h_4 + -)̱
               × Γ(h_1 + h_2 -  + )Γ(h_3 + h_4 -  + )/S(2)Γ( - h_12 -)Γ( + h_34 -)Γ( + h_23+)̱/Γ( - h_23 + )̱
               × _4F̃_3[ - h_12+,  - h_34+,  h_1 + h_2 -  + ,  h_3 + h_4 -  +  1+2,  h_2 + h_4 ++,̱  h_2 + h_4 +-;1 ].
    
    Here we used the notationC(x) = cos(π x)/π,S(x) = sin(π x)/π, and the_4F̃_3(1)are regularized hypergeometric functions. 
    
    Above we claimed that(,|̱h_i)was analytic in$̱ on the right half plane. This is not completely manifest from the expressions in Eq. eq:Ksplitexp; in fact, it appears that both I_1 and I_2 have singularities at =̱ h_2 + h_4 ± + N. However, it can be shown (using hypergeometric identities, see e.g. <cit.>) that the residues in I_1(,)̱ and I_2(±,)̱ at these points exactly cancel. Equivalently, analyticity follows from a contour pinching argument applied to the Mellin-Barnes integral in Eq. eq:splitMB.
    
    In passing, we claim that  has the following symmetry:
    
      K_split(,|̱h_1,h_2,h_3,h_4) = K_split(,|̱h_3^♮,h_4^♮,h_1^♮,h_2^♮)
    
    cf. Eq. eq:k1r for the normal kernel.[
      We also note the existence of a rather mysterious relation between I_1(,|̱h_i) and I_2(,|̱h_i), namely
    
        I_2(,|̱h_1,h_2,h_3,h_4) = N_s()/N_t()̱C(β - h_23)S(h_2 + h_4 +  + )̱/C( + h_3 + h_4) S(2)̱   I_1(,̱|1-h_1,1-h_4,1-h_3,1-h_2) .
    ] To establish eq:kss, one develops an alternate Mellin-Barnes representation for , by changing the order of integration:
    
        (,|̱h_i) = Γ(1-h_12+h_34)/Γ(1+h_14+h_23)2/Γ( + h_34±)Γ( + h_14 +)̱/Γ( - h_14 + )̱
        ∫[dt] Γ(-t)Γ( + a' + +̱ t)Γ( - b' + +̱ t)/Γ(1+2+̱t)Γ(h_1 + h_3 - 1 - -̱ t ±)Γ(32 - h_1 - h_4 + +̱ t)/Γ(h_2 + h_3 -  - -̱ t) .
    
    Closing the contour to the right, we find a representation of  of the schematic form eq:K3t, with I_1,2(,|̱h_i) replaced by functions J_1,2(,|̱h_i) obeying
    
    I_k(,|̱h_1,h_2,h_3,h_4) = J_k(,|̱h_3^♮,h_4^♮,h_1^♮,h_2^♮) ,
       k =1,2.
    
    This proves Eq. eq:kss.
    
    Let us finally return to Eq. eq:spliteq. The modified falloff of the split kernel allows one to close the contour in the right β plane and pick up the poles, which we have just demonstrated can only come from F(β). Therefore, up to simple numerical factor the split kernel considered as a function of α for a fixed β is precisely the s-channel alpha space transform of a single t-channel conformal block. It is therefore of interest to consider the analytic properties of (,|̱h_i) in α as well. For example, for identical external dimensions h_i a contour pinching argument applied to Eq. eq:splitMB shows that (,|̱h_ϕ) has double rather than single poles at the double-trace values α = ± (2 h_ϕ -  + N), reflecting the logarithmic behavior of the k^s_β + 1/2(z) as z → 1 in position space. This most clearly demonstrates the impossibility of expressing physical conformal blocks in one channel as proper sums of blocks in the crossed channel and consequently the necessity of using a different basis of functions like our Ψ_α(z) to arrive at a meaningful crossing symmetry kernel.
    
    
    
     §.§ Using the ξ_n as a basis
    
    
    It appears that a special role is played by the alpha space functions ξ_n^s(|h_i), ξ_n^t(|h_i) and ξ_n^0(|h_ϕ), defined in Eqs. eq:xisdef, eq:xitdef, eq:xi0def. In fact, these basis functions furnish infinitely many solutions to crossing symmetry. To make this concrete, consider the mixed-correlator bootstrap equation eq:mixsystem, which is automatically solved if F_s,t() are chosen as follows:
    
    F_s() = ∑_n  even c_n  ξ_n^s(|h_i) ,
      
    F_t() = ∑_n  even c_n  ξ_n^t(|h_i) .
    
    It is crucial that the same coefficients c_n appear both in F_s() and F_t(), and that only ξ_n with even n appear. The reason is that the ξ_n^s,t() with odd n are antisymmetric under crossing. To understand this more intuitively, it is instructive to analyze the ξ_n in position space. Using Eq. eq:jnaarw, we find that the z-space versions of ξ_n^s(|h_i) and ξ_n^t(|h_i) are given by
    ξ_n^s(z|h_i)  
    ξ_n^t(z|h_i)  =
    n! Γ(2h_2 + n) Γ(h_1  + h_3 +h_24 + n)   z^2h_2 P_n^(a+b,a'+b')(1-2z) 
     P_n^(a'+b',a+b)(1-2z) .
    
    Given Eq. eq:xipos, it follows directly that
    ξ_n^s(z|h_i)  = (-1)^n (z/1-z)^2h_2 ξ_n^t(1-z|h_i)
    
    where we use that P_n^(p,q)(-x) = (-1)^n P_n^(q,p)(x).  Comparing to the crossing equation eq:mixcross, one confirms that the ξ_n with even (resp. odd) n are symmetric (resp. antisymmetric) under crossing symmetry.
    
    Next, we will consider the CB decomposition of the functions ξ_n, at least schematically. Notice that ξ_n^s(|h_i) only has poles at = h_1 + h_2 -  + N, as well as mirror poles on the left half plane. Given our discussion in Sec. <ref>, this implies that ξ_n^s(|h_i) has a CB decomposition consisting of operators of dimensions h_1 + h_2 + N. Such a conformal block decomposition looks similar to a mean-field solution, where only double-twist primaries [ϕ_1 ϕ_2]_n ∼ϕ_1 ^nϕ_2 contribute. Similarly, ξ_n^t has a CB decomposition with a spectrum given by h_2 + h_3 + N.  
    
    For definiteness, we will compute the CB decomposition of ξ^0_n(|h_ϕ) explicitly. The position-space version of ξ^0_n(|h_ϕ) is a limiting case of eq:xipos, namely
    ξ^0_n(z|h_ϕ) = n! Γ^2(2h_ϕ + n)   z^2h_ϕ P_n(1-2z)
    
    where P_n denotes a Legendre polynomial. As above, these functions are crossing (anti)symmetric for even (odd) n, as follows from 
    ξ^0_n(z|h_ϕ) = (-1)^n (z/1-z)^2h_ϕξ^0_n(1-z|h_ϕ) .
    
    The CB decomposition of ξ_n^0(z|h_ϕ) can be found using alpha space technology; in particular, its residues in alpha space are equal to Wilson polynomials evaluated at certain values of . The precise result is
    ξ^0_n(z|h_ϕ) = ∑_m=0^∞ A_m^(n) k^()_2h_ϕ + m(z)
    
    where
    
    A_m^(n) = Γ^2(2h_ϕ+n)n!  (-1)^m/m!(2h_ϕ)^2_m  /(4h_ϕ-1+m)_m _4F_3(-n,-m,n+1,4h_ϕ - 1+m 2h_ϕ,2h_ϕ,1;1 ).
    
    Notice that the coefficients A_m^(n) are sign-alternating: sgn( A_m^(n)) = (-1)^m, provided that h_ϕ > 0. This implies that the ξ_n do not correspond to unitary solutions of crossing.
    
    At least formally, it is possible to derive selection rules for alpha space densities using the functions ξ_n. We will focus on the identical-operator case for simplicity. Recall that the ξ_n^0 form a basis of the Hilbert space H_0 introduced in Sec. <ref>. This implies that if a density F() ∈H_0 is crossing symmetric, it must obey
      ∫[d ]/M_0(;h_ϕ) ξ_n^0(|h_ϕ) F() = 0
        for   n = 1,3,5,…
    This selection rule manifestly holds if F() is of the following form:
    F() = ∑_n  even c_n  ξ_n^0(|h_ϕ) 
    
    cf. Eq. eq:mixy. Of course, requiring that F() is normalizable imposes constraints on the growth of the coefficients c_n as n →∞. 
    
    Unfortunately an alpha space density of the form eq:evendec cannot belong to an interacting CFT: it would have a CB decomposition with exchanged operators of dimensions 2h_ϕ + N and nothing else — in particular, requiring that F() ∈H_0 rules out an identity operator contribution. These unphysical constraints on the spectrum of F are very similar to the issue encountered in Sec. <ref>. We also stress that eq:evendec generically corresponds to a non-unitary CB decomposition, in line with our remarks below Eq. eq:xicb.  Imposing unitarity leads to additional constraints on the coefficients c_n, and in future work it would certainly be interesting to examine these in detail.
    
    To better understand the role played by the ξ_n, we will briefly consider how these ideas apply to a mean-field correlator:
    
    F_MFT(z) = t_1 F_1(z) + t_2 F_2(z)
    
    with
    
    F_1(z) = z^2h_ϕ  and  
    F_2(z) = 1 +  (z/1-z)^2h_ϕ .
    
    Both pieces F_1,2 are crossing symmetric by themselves, but only their combination with t_2 ± t_1 ≥ 0 is unitary. This follows from the CB decompositions eq:powerCB and eq:pqCBdec1.[
      Here we are interested in the case p=q of Eq. eq:pqCBdec1, which reads   
        (z/1-z)^p = ∑_n=0^∞(p)_n^2/n! (2p-1+n)_n  k_p+n(z) .
    ] Separately, F_1 and F_2 contain contributions from an infinite tower of operators of dimension 2h_ϕ + n, but the contributions for odd (resp. even) n cancel out when t_1 = t_2 (resp.t_1 = -t_2). The combinations with t_1 = ± t_2 correspond to generalized free fields with bosonic (resp. fermionic) statistics.
    
    Can we decompose F_1 and F_2à la Eq. eq:evendec? As for F_1, we see by inspection that
    
    F_1(z) = 1/Γ^2(2h_ϕ) ξ_0^0(z|h_ϕ)
    
    consistent with the fact that F_1 is crossing symmetric and non-unitary. In particular, this shows that F_1() ∈H. Notice that this is only possible because F_1(z) has no unit operator contribution. Since F_2(z) does have a unit operator contribution, it follows that F_2(z) cannot be decomposed as in Eq. eq:evendec.
    Nevertheless, we compute
    (z/1-z)^2h_ϕ = ∑_n=0^∞ f_n  ξ_n^0(z|h_ϕ) ,
      
    f_n =  1/Γ^2(2h_ϕ)1+2n/n!(1-2h_ϕ+n)1/(2h_ϕ - n)_2n .
    
    Strictly speaking this holds only for h_ϕ < 1/2; for generic h_ϕ, eq:decLeg makes sense only after analytic continuation. 
    Notice that eq:decLeg contains terms with both even and odd n. This is consistent with the fact that [z/(1-z)]^2h_ϕ by itself has no definite crossing behaviour. Another interesting feature is that the f_n are not sign-definite; in fact, sgn(f_n) = (-1)^n provided that h_ϕ < 1/2. However, we know from Eq. eq:pqCBdec1 that [z/(1-z)]^2h_ϕ has a CB decomposition with positive coefficients. We conclude that there is a conspiracy between the coefficients f_n from Eq. eq:decLeg and the A_m^(n) from eq:xicb that guarantees that the full CB decomposition is unitary.
    
    The above example shows how the idea to draw selection rules from the ξ_n runs into problems when naively applied to CFT correlators. Nonetheless, it may be true that a modified version of Eq. eq:selrule holds after carefully regulating the identity operator contribution. We leave this question for future work.
    
    
    
    
    
    § DISCUSSION
    
    
    This paper has outlined how Sturm-Liouville theory provides a framework to study CFTs. Inspired by classic results <cit.>, we discussed the decomposition of a CFT four-point correlator in terms of a new basis of functions Ψ_α(z) and explained how the familiar conformal block decomposition can be obtained by analytic continuation in α. The alpha space decomposition allowed us to formulate crossing symmetry in terms of an eigenfunction problem for some integral kernels: in particular equation (<ref>) is a mathematically precise version of the abstract idea expressed by equation (<ref>) in the introduction. It features an explicitly known crossing symmetry kernel K_0(α,β|h_ϕ) whose properties we analyzed in some detail.
    
    In this paper we did not touch on the profound connection between the alpha space construction and the representation theory of the conformal group. Roughly speaking the dictionary is well-known: three-point functions map to Clebsch-Gordan kernels, conformal blocks are their square — as used in three-fold tensor products — and the crossing symmetry kernel is equal to a 6-j symbol for the conformal group. Moreover, the alpha space decomposition ought to correspond to tensor product decomposition into a direct integral over the principal unitary series of representations. We can however only make all these relations precise if we have a detailed knowledge of both the groups, the representations under consideration, and the Hilbert space of functions on which they act.[In this context it is important to note that the representations are only unitary in Lorentzian signature. In that case the conformal group is actually the universal cover of SL(2,R)<cit.>, which has a richer class of inequivalent unitary representations <cit.> (see also <cit.> for a detailed discussion of the 4d case).] For the case at hand the question appears to be partially solved in <cit.>, which showed that the Wilson functions W_(;̱a,b,c,d) indeed appear as 6-j symbols for representations of the 𝔰𝔩(2,R) conformal algebra. Surprisingly this connection works provided three of the four external dimensions transform in the discrete unitary series, in contrast with the older discussion of <cit.> which is based entirely on the principal unitary series.[This is related to our basis functions being different form the usual shadow-symmetric blocks of <cit.> which are in fact the correct squared Clebsch-Gordan coefficients for three unitary principal series.] It would be interesting to build on the results of <cit.> to explicitly connect all the dots between alpha space, one-dimensional unitary CFTs and representation theory. We hope to return to this problem in the near future.
    
    It is of clear interest to generalize our analysis to d ≥ 2 dimensions. This requires solving the Sturm-Liouville problem for the d-dimensional Casimir <cit.> on the square (0,1) × (0,1), or alternatively one could relate this kernel to a suitable set of 6-j symbols of the universal cover of SO(d,2). The higher-d alpha space picture will necessarily be more complicated, because both external and exchanged operators in higher-d CFTs can carry a nontrivial Lorentz spin. An obvious generalization pertains to superconformal field theories in various d <cit.>. Sturm-Liouville theory should also apply beyond four-point correlators in CFTs on R^d; for instance, one can consider its application to CFTs in the presence of boundaries or defects.
     
    Most of these problems are rather formal and group-theoretical in nature. In the framework of the conformal bootstrap, it is more exciting to investigate whether alpha space crossing equations can be leveraged to constrain CFT data, or — more ambitiously —  to solve bootstrap equations analytically.[See <cit.> for a connection between the conformal Casimir and integrability, which may be helpful in this context.]  In Sec. <ref> we discussed some tentative ideas in this direction. Together with recent developments in the realm of Mellin space and the lightcone bootstrap, we are optimistic that alpha space can become part of the analytic bootstrap toolkit.
    
    
    
    
    
    § ACKNOWLEDGMENTS
    
    This work originated from discussions with Leonardo Rastelli and Pedro Liendo in Stony Brook in 2011, and we would like to thank them for their valuable contributions in these early stages. Moreover, we gratefully acknowledge discussions with the participants of the `Back to the Bootstrap II' meeting in 2012 where an initial version of this work was first presented. We would like to thank Christopher Beem, Jyoti Bhattacharya, Liam Fitzpatrick, Simon Caron-Huot, Abhijit Gadde, Leszek Hadasz, Christoph Keller, Zohar Komargodski, Hugh Osborn, Slava Rychkov, Volker Schomerus, David Simmons-Duffin and Sasha Zhiboedov for more recent discussions and/or comments. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. This work was additionally supported by a grant from the Simons Foundation (#488659).
    
    
    
    § COMPUTING THE INNER PRODUCT Ψ_,Ψ_
    
    
    In this section, we will prove Eq. eq:testInt by computing the inner product Ψ_,Ψ_, as defined in Eq. eq:innerProd. Concretely, we must perform the following integral: 
    Ψ_,Ψ_ = ∫_0^1 d z/z^2 Ψ_(z) Ψ_(̱z)
    
    where we used that Ψ_(z) = Ψ_(z) for imaginary . As a first step, we write Ψ_(z) and Ψ_(̱z) using a Mellin-Barnes representation:
    Ψ_(z) = 1/Γ(±)∫[d s] Γ(-s) Γ(+s±)/Γ(1+s)(1-z/z)^s .
    
    Naively, the z-integral eq:inttodo is logarithmically divergent, the divergence coming from the region near z = 0. To resolve this divergence, we regulate Ψ_(̱z) by writing it as follows:
    Ψ_(̱z)  → z^ _2F_1(+,̱ -  1+;-1-z/z) =  z^ Γ(1+)/Γ(±)̱∫[d t] Γ(-t) Γ(+t±)̱/Γ(1++t)(1-z/z)^t
    for > 0. This behaves as O(z^1/2+) at small z. Evidently, in the limit → 0, the above function reduces to Ψ_(̱z).
    
    At this point, the inner product Eq. eq:inttodo is given by triple integral, schematically
    Ψ_,Ψ_ = ∫_0^1 d z ∫[d s] ∫[d t]   ( … ) .
    
    Since we have regulated the integrand, this integral converges and we can exchange the order of the different integrals. We do the z-integral first, which is a simple beta function integral. The result is
    
        … = Γ(1+)/Γ(±)Γ(±)̱∫[d s]  Γ(-s) Γ(± s + )/Γ(1+s)
        × ∫[d t]  Γ(-t) Γ(+t±)̱/Γ(1++t)Γ(1+s+t)Γ(-1-s-t+)/Γ() .
    
    We now do the t-integral, using the second Barnes lemma. This yields
    Ψ_,Ψ_ = lim_→ 0 Γ(1+)/Γ(±)Γ( +±)̱∫[d s] Γ(-s) Γ(± s + )Γ(- -s + ±)̱/Γ(-s+) .
     
     At this stage we can take the limit → 0 everywhere, except in the two factors Γ(- -s + ±)̱:
     … = 1/Γ(±)Γ(±)̱∫[d s]  Γ( + s ±)Γ(- - s + ±)̱ .
     
     This integral can be computed using the first Barnes lemma, yielding
     
        Ψ_, Ψ_ = 1/Γ(±)Γ(±)̱ lim_→ 0  Z_(,)̱ , 
        
        Z_(,)̱ =  1/Γ(2)Γ(++̱)Γ(-+̱)Γ(-++̱)Γ(--+̱) .
    
     To conclude, we need to analyze the limit → 0 of Z_(,)̱, which we claim is the sum of two Dirac delta functions:
     lim_→ 0 ∫[d ]̱  Z_(,)̱ f()̱ = Γ(± 2) [ f()+f(-) ] ,
     
     where f() is a test function.
     Notice that Eq. eq:toProveZ is sufficient to establish Eq. eq:testInt, after remarking that
      2Γ(± 2)/Γ(±)^2 = N() .
     
    
     The proof of eq:toProveZ goes as follows. We start by noticing that lim_→ 0 Z_(,)̱ vanishes, unless =̱±± n for some integer n. If n ≠ 0, the limit → 0 is finite, hence such points do not contribute to the integral in Eq. eq:toProveZ. Hence it suffices to consider the cases β = and β = -. For concreteness, let's consider β =, in which case we can approximate Z_(,)̱ by
     
     Z_(,)̱ →̱ Γ(± 2)  ω_(-)̱ ,
       ω_() = Γ(±)/Γ(2) .
     
     It is straightforward to see that ω_() behaves as a delta function along the imaginary axis, i.e.lim_→ 0 ∫[d ]  ω_() f() = f(0) .
     
     This follows from the fact that ω_() is peaked around = 0 with width  (taking  to be imaginary) together with the fact that
     ∫[d ]  ω_()  = 1/4^ →  1 .
     
     The same argument holds for the region where =̱ -. This allows us to conclude.
    
     
    utphys]