Scalar-Vector Bootstrap

We work out all of the details required for implementation of the conformal bootstrap program applied to the four-point function of two scalars and two vectors in an abstract conformal field theory in arbitrary dimension. This includes a review of which tensor structures make appearances, a construction of the projectors onto the required mixed symmetry representations, and a computation of the conformal blocks for all possible operators which can be exchanged. These blocks are presented as differential operators acting upon the previously known scalar conformal blocks. Finally, we set up the bootstrap equations which implement crossing symmetry. Special attention is given to the case of conserved vectors, where several simplifications occur.

we perform OPEs, and the equivalence of different procedures puts constraints on which sets of data can correspond to consistent CFTs. In particular, for a four-point function we can divide the four operators into pairs in three different ways, or channels. Equivalence between these channels is called crossing symmetry, and the general endeavor of exploring the constraints on CFT data which are imposed by crossing symmetry is known as the conformal bootstrap program.
However, the consistency conditions from scalar correlators are only a small part of the (infinitely) many conditions that the bootstrap program imposes. One expects more interesting and universal bounds to arise from bootstrapping 4-point functions of operators with spin, such as the stress-tensor or conserved currents. The main obstacle in tackling these problems is that the full set of conformal blocks for spinning correlators is not readily available yet. Partial progress has been made in this direction. In [44] it was observed that there is a class of conformal blocks of tensor 4-point functions that can be related (via differential operators) to the well known scalar blocks of [41][42][43]. However, the class of conformal blocks derived in this way is associated to the exchange of traceless symmetric operators O, whereas tensor correlator bootstrap requires, in addition, the exchange of mixed-symmetric operators A. Later, in [45,46] it was shown that conformal blocks associated to A can be calculated as a (finite) sum of scalar blocks evaluated at zero spin which, in principle, can be done by a computer. However, the numerical evaluation of these blocks is quite resource intensive due to the fact that the number of terms in the sum increases rapidly with the spin of A. In numerical computations one might get away with if the maximum spin of A is not too large, but this approach is hopeless in the analytic bootstrap, where one needs to have control over the conformal blocks at very high spin [14,15]. Therefore the objective of this paper is to start building explicit closed form expressions of spinning conformal blocks that can be used in the analytic bootstrap and for efficient numerical evaluation 4 .
To start with, in section 2 we classify all of the tensor structures which can appear in the three-and four-point functions which concern us in this paper (namely three-point functions with either two scalars or a scalar and a vector, along with a third operator, and four-point functions with four, three, or two scalars with zero, one, or two vectors). We pay special attention to the information obtained from exchanging two operators, especially when the operators are identical. In section 2.5 we work out the extra information available when the vector operator is conserved. In this section we will determine the tensor structures which can appear in the fourpoint function of scalars and up to two vectors, and in the three-point functions which act as intermediate stages in the evaluation. The techniques are well established [41][42][43], and especially in [48], but we give a self-contained presentation in order to establish our conventions and to put emphasis on the properties that will be most relevant for our purposes. In subsequent sections we will compute the functions which multiply these tensor structures in terms of the underlying data of the CFT.

Embedding space
When considering the consequences of conformal invariance, it is often useful to make use of embedding space. This is a (D + 2)-dimensional space, with coordinates P A and metric on which the conformal group SO(D + 1, 1) acts linearly (we will be working with Euclidean signature in physical space throughout this paper). The D-dimensional physical space is identified with a null projective surface. The map to physical coordinates is given by while we can do the inverse map by sending a point in physical space to a particular point on the projective null line, Now consider a tensor function of three coordinates (to serve as an example) on embedding space, which is homogeneous in each variable (so that it is well defined on projective hypersurfaces), 5) and is transverse in the sense that We can map this function to a tensor function on physical space by where we use the map (2.3). Because we are mapping from a null hypersurface, different embedding space tensors can map to the same physical tensor if they are related by for any choice Λ A 2 ···A k ,B 1 ···B ℓ ,C 1 ···Cm , and similarly for each of the other indices. We will sometimes refer to this redundancy (somewhat sloppily) as gauge freedom.
The resulting function f a 1 ···a k ,b 1 ···b ℓ ,c 1 ···cm (x 1 , x 2 , x 3 ) transforms as a conformal tensor of weights ∆ 1 , ∆ 2 , and ∆ 3 under conformal transformations of x 1 , x 2 , or x 3 respectively. It turns out that a converse is also true; any function which transforms as a tensor of weights ∆ i can be obtained from a homogenous (of weights ∆ i ) transverse tensor in embedding space, unique up to equivalences of the form (2.8).
Thus, in order to determine the possible form of correlation functions of given operators, we need only determine the homogenous transverse tensors in embedding space up to the equivalences. In embedding space there are not many different objects we can build. Any scalar must be built out of scalar products of distinct P i 's, and it will be useful to define In physical space, this simply projects down to To ensure that free indices are transverse, it will also be useful to define (for distinct i, j, and k) which is transverse with respect to P A i , and antisymmetric in j and k, and projects down to and for distinct i and j, both which is transverse in both indices with respect to P i (or P j ) and projects to δ ab , and which is transverse to P A i in the first index and P B j in the second index, and projects down to (2.14) Note that these building blocks, particularly (2.10) are defined to be scale invariant.
Finally, note that if F A 1 ···A k (we suppress other indices for now) transforms in a given way under permutations, then its projection f a 1 ···a k will inherit the same transformation and will thus transform as the corresponding representation of the rotation group SO(D).
For example, if F A 1 ···A k is invariant under permutations of its indices, then f a 1 ···a k will be a symmetric tensor. If F A 1 ···A k is also traceless, then so will be f a 1 ···a k .
Appendix A contains several useful formulae and identities for these structures in physical space.

Two-point functions
As we will see in the next subsection, the primary operators we will need in this paper fall into two classes of irreducible representations of the rotation group SO(D). We either have totally symmetric traceless tensors of spin ℓ, O a 1 ···a ℓ (x), which includes scalars and vectors as special cases, or we have mixed symmetry tensors A a 1 a 2 b 1 ···b k (x) which are completely traceless, are antisymmetric in a 1 and a 2 , are totally symmetric in the b i , and which vanish when antisymmetrized over any three indices. In terms of Young tableaux, the O a 1 ···a ℓ are represented by a horizontal row of ℓ boxes, while A a 1 a 2 b 1 ···b k are represented by one row of k + 1 boxes and a second row with only one box (equivalently one column with two boxes and k columns of one box each). For each of these cases we construct projectors onto the given representation in Appendix B. For O a 1 ···a ℓ and A a 1 a 2 b 1 ···b k we use projectors given in (B.210) and (B.227) respectively. We can also write the projectors in embedding space by simply taking the expressions in Appendix B and replacing each δ ab with N (ij) AB , with i labeling the operator being projected, and j being an arbitrarily chosen other variable (the choice is not physically relevant and can be changed by a gauge transformation (2.8).
It is well known that we can diagonalize the space of primary operators with respect to the two-point correlation functions, so we will only need to compute the two-point function of either a pair or O operators or a pair of A operators. Indeed, if we have 16) then this must descend from a tensor H A 1 ···A ℓ ,B 1 ···B ℓ in embedding space. In order to get the symmetric traceless representation, we must be able to put the indices on projectors Π (ℓ) .
By transversality, each A index must be carried by either P 1 A , N AA ′ , or M AB . The first possibility is pure gauge and can be discarded. The second possibility, which projects down to δ aa ′ will be eliminated when multiplied by the projector Π (ℓ) , and so can also be discarded (though it will appear in the projectors themselves). This leaves only the third possibilty.
In order to get the correct homogeneity property we must include the appropriate power of P 12 . Finally, then, we are left with the form which projects down to The same reasoning gives 5 . This will be vanishing for any irreducible representation except for the symmetric traceless representation. Then the three-point correlator lifts to an embedding space tensor which projects down to Making use of the symmetries of Π (k) , one can show that any other arrangement of indices on the m (12) 's, e.g. replacing m (12) e2g2 m (12) f1h1 by m (12) e2h1 m (12) f1g2 , is equivalent to the one given.
Here λ 12O is a constant real number (in a unitary CFT), which is otherwise arbitrary.
If the two scalars are identical, then the the result has the form 22) and this result should be invariant under the exchange of x 1 and x 2 , which in turn forces ℓ to be even (otherwise the result changes sign under this exchange, since we get one factor of −1 from each k (312) ).

SV O
Next we consider the three-point function with one scalar φ, one vector v a , and one other operator. This correlator will lift to an embedding tensor F AB 1 ···Bm (P 1 , P 2 , P 3 ) where A is transverse to P 2 and the B indices are transverse to P 3 . The A index can only be carried by either K AB , and then the remaining B indices (after being projected by the appropriate rotation group projector) must be carried by K In the case where the third operator is totally symmetric, then there are two structures which can arise, with embedding space form If ℓ = 0, then we only have the first term labeled by a constant α φvO . If ℓ > 0, then we have two distinct possible tensor structures labeled by two real constant numbers α φvO and β φvO .

SV A
Similar considerations for the case where the third operator has mixed symmetry show that the three-point correlation function will have the form with γ φvA as a real constant.

Four-point functions
The case of four-point functions proceeds similarly, with the main difference being that there are cross-ratios in physical space. Then each tensor structure is accompanied by a function of the crossratios rather than by just a constant.

SSSS
For the case of four scalars, we have where q(u, v) is an (a priori) arbitrary function of the cross-ratios u and v. The factor multiplying q(u, v), which does the work in ensuring that the correlator scales correctly, will appear often, and so it is convenient to abbreviate it. Thus, we define and sometimes we will simply write X ∆ i for short.
In the case that all four scalars are identical, we have and invariance under exchange of x 1 with x 2 implies that q(u, v) = q(u/v, 1/v), (2.31) while under exchange of x 1 and x 3 we have Other permutations of the x i give no new information about the function q(u, v).

SV SS or SSSV
Let us now consider the four-point function of three scalars and one vector (in the second position to start). In principle the free index could be carried, in embedding space, by any of the three possibilities K so only two of the combinations are independent, and we can write (after projecting to physical space) (2.34) If φ 3 and φ 4 are identical, then symmetry under exchange of x 3 and x 4 implies that while if φ 1 and φ 3 are identical, then we have If all three scalars are identical, then both sets of constraints hold.
The situation when the vector is in fourth position is completely analogous (we put primes on the q ′ i to distinguish them from the SVSS functions), (2.37)

SV SV
Finally, consider a four-point function of two scalars and two vectors, In embedding space, the indices of the corresponding tensor can be either carried by M If the two scalars are identical, then x 1 -x 3 exchange gives constraints If the two vectors are identical, then exchanging x 2 and x 4 while also exchanging the indices a and b, gives Finally if we have two identical scalars and two identical vectors, then we can combine the constraints and determine Thus in this case we have one unconstrained function q 12 (u, v), and two constrained functions q 0 (u, v) and q 11 (u, v), with q 21 (u, v) and q 22 (u, v) determined in terms of the others.

Conserved vectors
Many of the structures discussed above simplify somewhat if we are dealing with conserved vectors, which obey ∂ a v a (x) = 0 inside correlation functions. From the vector-vector twopoint function, we have Thus we conclude that ∆ v = D − 1 for a conserved vector in D-dimensions, i.e. it saturates the unitarity bound.
Turning next to three-point functions, we have (for ℓ > 0) where we have made use of the fact that ∆ v = D − 1. For this expression to vanish, we For ℓ = 0, we simply set β φvO = 0 in the above equation, and require either α φvO = 0 or ∆ φ = ∆ O . Actually, we can assign some more physical significance to this case by first recalling that we expect each conserved primary vector operator to be associated to a one parameter continuous global symmetry of our CFT. Now pick a particular conformal weight ∆ and consider all scalar operators φ i that have that weight. Form a matrix α ij by taking three point functions with the conserved vector v a , (2.54) Then α ij is antisymmetric in its indices. Since we are free to make orthonormal (with respect to the normalized two-point functions) rotations on the space of φ i , we can always take a basis in which α ij is block diagonal, Here n is just the number of charged scalars with weight ∆. In this basis, we say that for i > 2n, φ i is neutral under the global symmetry. We can combine the others into complex combinations ϕ i = φ 2i−1 + iφ 2i , and we can say that ϕ i has charge 6 Q i .
For the other three-point functions, φvA , a similar calculation shows that conservation is automatic once we impose that ∆ v = D − 1. Conservation gives no other constraints in this case.
For four-point functions with conserved vectors, the coefficient functions must obey linear differential equations. For example, in the SV SS amplitude, if the vector is conserved then the functions q 1 and q 2 must obey And in the case of SV SV , if the vector at x 4 is conserved (so in particular ∆ 4 = D−1), 6 We have chosen a normalization for the charge that is convenient from the point of view of an abstract CFT, since it is given simply by the three-point function of primary fields (which have themselves been normalized by their two-point functions). However, it may well differ from other well-motivated normalizations. For example, in the case of a free complex scalar in D > 2 dimensions, and the usual global U(1) symmetry, our definition gives the scalar a charge of D−2 2 .
then we have and (2.58)

Shadow formalism
As an intermediate step in the calculation of conformal blocks, we will need to define shadow operators. Given any local primary operator O a 1 ···an (x) of conformal weight ∆, we can define its shadow operator which is a non-local operator that transforms as a primary operator of weight D − ∆ under conformal transformations, and under SO(D) rotations transforms in the same way as O.
When we insert O(x 1 ) in a correlation function, the prescription is to insert O(x 0 ), evaluate the correlation function, and then perform the integral above.

Mixing matrices
We would like to compute the constants which appear in three-point functions involving shadow operators. Since O is linearly related to O, the constant or constants appearing in a three-point function of O with two other operators will be linear combinations of the constants in the three-point function of O with those same two operators. We would like to determine the matrices which encode these linear combinations.

SS O
Consider first the case where O a 1 ···a ℓ is symmetric traceless, and the other two operators are scalars φ 1 and φ 2 . The three-point function with O is fixed up to a single constant and we expect that the shadow operator will similarly have Inserting the definition of the shadow operator (3.59) and performing the integral leads to 7 Details on the computation of these integrals are given in appendix E

SV O
Next, we consider symmetric traceless O(x 3 ), but in a three-point function with a scalar φ(x 1 ) and a vector v a (x 2 ). In this case, both OPE contribute: where the constants M s r can be read off from (3.63) and (3.64).

SV A
Finally, we turn to the mixed symmetry operator A b 1 b 2 c 1 ···c k (x 3 ) and its shadow which can appear in a three-point function with a scalar φ(x 1 ) and a vector v a (x 2 ). Similar techniques to those employed above give the relation between the three-point coefficients,

Shadow projectors
Given a primary operator O, define a shadow projector This should be interpreted as an operator that gets inserted into a correlation function, separating it into two correlation functions with an integral. When inserted into a given channel in a correlation function, it is designed to pick out the contribution of O and its descendants. N O is a normalization constant that we fix by demanding and so we need to take (see appendix E) .

(3.70)
Note that N O is independent of ∆ 1 and ∆ 2 , as it should be.
Similarly, for the mixed symmetry case we can define a projector, and N A can be computed to be . (3.72)

Conformal blocks
We next turn to four-point functions. These can be evaluated by first performing operator product expansions (OPEs) of the first two operators and the last two operators, and then evaluating the remaining two-point functions. Consider first a general OPE. Let's use notation whereā represents a multi-index, transforming as some representation of SO(D).
Then the OPE of two arbitrary operators has the form where the sum is in principle over all local operators Uc(x) in the theory, and the coefficients fc 12Uāb are functions of x 12 . Actually, for fixed representations only a finite number of tensor structures are compatible with the symmetries, so we can write this as a sum over tensor structures labeled by r, where the three-point tensor structures s rc ab (x 12 ) are universal quantities which depend on the conformal representations (meaning both the SO(D) representations and the conformal weights) involved, but are otherwise independent of the theory or the particular operators.
That dependence is entirely contained in the constants λ 12U r . Finally, there is one further simplification, which is that when U is a descendent of a primary operator O (and thus corresponds to some differential operator acting on O), then its coefficients in the φ 1 × φ 2 OPE is determined linearly in terms of the coefficients of O in the OPE. Thus the OPE can in fact be written as a sum over primary operators O, Again the differential operators C rc ab (x 12 , ∂ 2 ) are universal in the same sense as above. Now inserting this form of the OPE into the four-point function, we can write The functions W rs abcd depend only on the SO(D) representations and conformal weights of the φ i and of O. These functions are often called conformal partial waves (though this nomenclature is not universal). Conformal invariance can actually be used to further restrict the form of the W 's, so that we can write Here the sum p runs over allowed tensor structures, and the four-point tensor structures t p abcd (x i ) depend only on the SO(D) representations of the external operators, not the conformal dimensions, while the functions g rs p (u, v) depend on the full conformal representations (i.e. both SO(D) representations and conformal weights), but are themselves scalar functions of the cross-ratios u and v. These g rs p (u, v) are called conformal blocks, and our task in the rest of this section is to compute them for the scalar and vector four-point functions of interest.

General discussion
Our primary purpose in this paper involves specific examples of four-point functions, but let us first have a very brief general discussion. Roughly, the idea is that by inserting the projector P O into the correlator (4.76), we should pick out only the contribution from the primary O and its descendants. This is not quite correct, as explained in [45] and elsewhere; rather we must insert the projector and then pick out only the terms of the result which The remaining terms will transform with a phase e −2πi(∆ O +∆ 1 +∆ 2 ) under this rotation and these terms should be thrown away. This procedure is called monodromy projection. In practice, we can write the result before the monodromy projection as a certain double integral over Feynman-Schwinger parameters, and then the monodromy projection can be implemented simply as a modification of the integration region, along with some insertions of signs in the integrand.
Once we have successfully picked out the contributions from O and its descendants, we can read off g rs p (u, v) from the terms proportional to λ 12O r λ 34O s t p abcd . If we write the general four-point function as then we have

Scalars and vectors
Let's understand what we should then be computing for our examples of interest. First we will review the case of four scalars. In this case the exchanged primary must be traceless symmetric, with its representation labeled by a spin ℓ and dimension ∆ O . There is a unique three-point tensor structure for each ℓ, and a unique four-point tensor structure t = 1. In other words, the correlator should take the form Here we have shown on which parameters the conformal partial waves W or conformal blocks g can depend; often we will not indicate this explicitly. In terms of the function whose coefficients we will label α 12O and β 12O . There are also two four-point tensor struc- , with coefficient functions q 1 (u, v) and q 2 (u, v) respectively, and these are related to the conformal blocks g αλ 1 , g αλ 2 , g βλ 1 , and g βλ 2 by Having the vector in the fourth position is essentially the same upon interchanging , and Finally, for the case of two scalars and two vectors we found in (2.39) five four-point tensor structures, with associated coefficient functions q 0 and q ij . In this case the exchanged operator can either be traceless symmetric O of spin ℓ, or it can be a mixed-symmetry operator A whose representation is labeled by k. In the former case, each of the three point function has two tensor structures s α and s β , while in the latter case there is a unique three-point tensor structure (4.87) Hence, for generic (not necessarily identical) scalars and vectors, we have (4.89) Altogether there are twenty-five conformal block functions. 2 × 2 × 5 = 20 of them are associated with symmetric traceless exchange and will depend on the spin ℓ as well as the conformal weights ∆ i and ∆ O , while the other five are associated to mixed symmetry exchange, and will depend on ∆ i , ∆ A and k, which labels the mixed symmetry representation.

Exchange symmetries
As in the classification of tensor structures, the structure of conformal blocks can simplify significantly when some of the operators are identical, so that we have extra symmetry from exchanging those operators. Note however, that since the conformal block decomposition picks out a particular exchange channel, not all exchanges will give us constraints on individual conformal block functions. An exchange that results in a different exchange channel is called a crossing symmetry, and will constrain only the full sum of conformal blocks, not the individual conformal blocks themselves. Crossing symmetry is the subject of the next section, when we will set up the bootstrap. In the current subsection, however, we will consider the exchanges which don't mix channels, and so can constrain the blocks themselves. These can involve exchange of operator 1 with operator 2, of operator 3 with operator 4, or exchanging the pair (12) with the pair (34).
For example, consider the case of four scalars, with its unique conformal block function where for this section we will show explicit dependence on parameters. The four-point function will be invariant if we simultaneously exchange x 1 with x 2 and ∆ 1 with ∆ 2 . This leads to a constraint on the conformal blocks, and for (12) ↔ (34) exchange, if all four scalars are identical with weight ∆, then we have In the case of three scalars and a vector, we have a couple of options. If the vector is in the second position, then we have the 3 ↔ 4 exchange of scalars, which tells us that where r is either α or β. Note that in deriving these relations, we needed to transform the t p a under this exchange and then reëxpress the result in terms of our basis t p a again. Since our chosen basis t 1 a = k (214) a , t 2 a = k (234) does not behave particularly nicely (rather we chose it to make later computations with two scalars and two vectors slightly nicer), the resulting expressions are slightly messier than they would be in a basis like t ′ 1 which simply gets exchanged under 3 ↔ 4. Performing a (12) ↔ (34) exchange relates the SVSS conformal blocks to the SSSV conformal blocks, and Finally, for the SVSV case, the only useful exchange is (12) ↔ (34), which tells us for r and s being α or β, and for p being 0, 12, or 21. Similarly In particular, if we have identical scalars and identical vectors, then the g 0 , g 12 , and g 21 are only constrained to be symmetric in their upper indices (i.e. g αβ p = g βα p ), while the g 22 functions are determined by the g 11 's,

Computing the blocks
At the risk of cluttering notation, we will add a hat to the conformal block functions to denote the result obtained from insertion of the shadow projector, The actual conformal blocks g rs p (u, v) themselves are then obtained from the g rs p (u, v) by a monodromy projection, which now picks out the terms in g rs p (u, v) which transform with a phase e 2πi∆ O as u → e 4πi u, and throws away the terms which transform as e −2πi∆ O .
We will call the process of implementing the monodromy projection, going from g rs p to g rs p (i.e. removing the hat), doffing.

SSSS
We'll start by reviewing the computation of the conformal blocks for four scalar operators.
Here, on insertion of the shadow projector we have (4.104) from which we can identify λ 12O λ 34O g(u, v) with the quantity in square brackets.
As shown in Appendix B.1, we can write where p D,ℓ (t) is a polynomial of degree ℓ whose properties are explained in the appendix, and t = k (012) · k (034) Let us now define integrals For ℓ = 0 this integral is evaluated in (D.269).
With this definition and the expressions (3.70) and (3.62), we have Note that the prefactor (x 2 12 ) ∆ O /2 already has the desired behavior under the monodromy projection, so we will want to pick out the terms from the integral which are invariant under the monodromy.
Note that if we expand the polynomial p D,ℓ (t) using the explicit formulae in Appendix B.1 then the integral is simply a sum of terms of a form computed in Appendix D.2. For example, in the case ℓ = 0, then p D,0 (t) = 1, and we have (restoring the explicit parameter dependence) where f is defined in (D.272). Since the u ∆ O /2 factor already behaves correctly under the monodromy projection, then to obtain the conformal block g(u, v) we must restrict to the monodromy invariant piece of f , and this is given simply by a function f defined in (D.273).
Then g(u, v; ∆ i ; 0, ∆ O ) is given by doffing the expression (4.109), replacing f by f . Note also that this formula shows explicity that g(u, v; ∆ i ; ℓ, ∆ O ) doesn't depend on all four of the ∆ i individually, but only on the differences ∆ 1 − ∆ 2 and ∆ 3 − ∆ 4 . Because of this we can adopt some condensed notation that will be useful below, defining functions that are related to the standard blocks by shifting these two differences by integer amounts P and Q, g ℓ;P,Q (u, v) = g(u, v; ∆ 1 + P, ∆ 2 , ∆ 3 + Q, ∆ 4 ; ℓ, ∆ O ). (4.110) In this notation (which can also be used for g) the dependence on the ∆ i and ∆ O is left implicit.
In even dimensions 9 we can evaluate the integrals in f explicitly, with the result where the variables x andx are related to u and v via What about ℓ > 0? As indicated, for any fixed small ℓ we can of course expand p D,ℓ (t) into monomials and proceed as above. But in fact we can be a bit more clever than that and exploit the recursion relations (B.217) to expand the numerator of the integrand in (4.107). In the recursion relation we also need to expand t according to (4.106), and reabsorb the powers of (x 2 0i ) as shifts of the external operator dimensions. Finally, passing to the monodromy-projected answer, the result is [41] g ℓ;0,0 (u, v) (4.113) 9 In arbitrary dimensions there exists a closed form for the ℓ = 0 conformal block in terms of Appel functions [41], but for even dimensions the result can be expressed using the much more familiar 2 F 1 hypergeometric functions.
This recursion holds in any dimension. In D = 2 the recursion can actually be solved explicitly to get a closed form expression for g(u, v; ∆ i ; ℓ, ∆ O ) in terms of elementary hypergeometric functions, and in higher even dimensions solutions can also be constructed (by using a relation between the blocks in D + 2 dimensions and those in D dimensions).
For example, in D = 4, An alternative approach is to expand the polynomials p D,ℓ (t) in the integrals I (ℓ) to obtain an expression for conformal blocks with ℓ > 0 as a sum of ℓ = 0 blocks. This result (with or without hats) is At any rate, in subsequent sections we will assume that these SSSS conformal blocks are some known functions, and we will endeavor to compute the new conformal blocks in terms of these.

SV SS
Let's now move to the case with one vector. The most efficient way to proceed is to first note that we can relate the three-point function of a scalar, a vector, and a symmetric traceless tensor to the three-point function of two scalars and a symmetric traceless tensor [44].
Explicitly, we can define Then we can write and as can be verified by explicit computation.
The conformal blocks will be computed by the expression On the other hand, we have By expressing S α and S β in terms of S λ , and pulling the differential operators outside of the integral, we can express g rλ i in terms of differential operators acting on g. For example, to compute g αλ i we get This leads to Similarly, Note that as with the scalar blocks, the expressions only depend on the difference ∆ 1 −∆ v and ∆ 3 −∆ 4 , not on the weights individually. The other crucial property of these expressions is that the operators which act on the g on the right hand side involve only integer powers of √ u, so in particular they are all invariant under the monodromy projection. This means that when we implement the monodromy projection, all we have to do is remove the hats from the scalar blocks on the right hand side and from the new blocks on the left hand side. After these expressions have been thus doffed, we have relations between the full g rλ i blocks and the scalar blocks g.

SSSV
The case when the vector is in the fourth position is very similar. We have Note that because α 34 O is not simply proportional to α 34O (we should expand it using (3.63)), and similarly for the β's, it will now be the case (unlike for SVSS) that each conformal block will get contributions from both terms on the right-hand side.
The results (after also doffing the expressions) are again only depending on the differences ∆ 1 − ∆ 2 and ∆ 3 − ∆ v .

SV SV
At last we turn to the case of primary interest; two scalars and two vectors. We will label the scalars 1 and 3, and the vectors 2 and 4. As we have seen, the exchange operator can be either traceless symmetric O or a mixed symmetry operator A.
We'll start with the symmetric exchange. There are twenty different conformal blocks which can arise, g rs p , where r and s run over α and β (for ℓ > 0, or only α for ℓ = 0) and p runs over the five tensor structures of the four-point function, which we have labeled 0, 11, 12, 21, and 22. By inserting the shadow projector, we can get all the symmetric exchange blocks as contractions of S α 's and S β 's, which we can in turn write as differential operators acting on S λ 's. Finally, we impose the monodromy projection by doffing all expressions. In fact, the resulting expressions are more compact if we write them in terms of either g αλ i or g λα i . For the αα blocks we'll use the former representation, and we will further introduce shorthand g αλ i;ℓ;P,Q = g αλ i (u, v; ∆ 1 + P, ∆ 2 , ∆ 3 + Q, ∆ 4 ; ℓ, ∆ O ). (4.134) The final expressions are For ℓ = 0, this is the entire answer. For ℓ > 0, we can proceed similarly with the other blocks, obtaining the αβ, βα, and ββ components given in appendix F. To save space, we have omitted the arguments of the conformal blocks appearing above. For the SVSV blocks (i.e. the expressions on the left-hand-sides above) the arguments are unshifted, , while for the others we use the previously adopted condensed notation, along with The combination which occurs in the four-point function is It turns that if we write this combination in terms of scalar conformal blocks it has the remarkably simple form where and D ±± p are fairly simple differential operators whose explicit forms are given in a table in appendix H.
We turn next to the evaluation of blocks for exchanged of a mixed symmetry tensor A a 1 a 2 b 1 ···b k whose representation is labeled by a non-negative integer k (k = 0 corresponds to an antisymmetric two-index tensor). The contraction which we need is In appendix C we give more details and motivation for how we arrive at this expression.
This is the main formula that will allow us to relate mixed-symmetric conformal blocks to the ones for symmetric-traceless exchange. Notice that derivatives of the polynomials p(t) will always produce terms that appear in the conformal blocks of traceless-symmetric in analogy to our case. We believe that it will be possible to do this in more general situations, but this has not been definitively established. However, to support this conjecture, we present the contraction for [k + 1, 1, 1] in appendix C, which has an analogous form.
From these arguments, we obtain and also (4.152) and similarly for (S α a P Q • ℓ S λ RS ), etc., we can compute (recall that (4.148) appears inside a conformal integral like (4.122)) .
5 Setting up the bootstrap

General discussion
The picture now is that we are given explicit expressions for the conformal blocks, which de- However, in deriving this expression we made a choice to first perform the OPEs of φ 1ā with φ 2b and φ 3c with φ 4d , then evaluating the resulting two-point function. Starting with the same correlation function and the same CFT data, we could have evaluated instead the OPE of φ 1ā with φ 4d and φ 2b with φ 3c , or equivalently, we could have performed a 2 ↔ 4 crossing symmetry exchange before performing our OPEs. This should be an equivalent path to the same four-point correlator, and by comparing the two results we obtain a non-trivial constraint on the defining data of our CFT. 11 We have checked that our expressions for k = 0, 1 are consistent with the latest version of [46]. 12 In sections discussing very general four-point functions, such as this one, O will stand for all possible primary exchange operators, of arbitrary SO(D) representations, while in sections discussing particular assignments of representations, such as the following two subsections, O will refer only to traceless symmetric exchanges. In that case we will also have exchange operators A of mixed symmetry.
Let's recall how this works for four scalars operators φ i . We have (5.165) while in the other channel we have Comparing the two we learn that This constraint is of limited usefulness when the scalars are all distinct. A somewhat better case is when φ 2 and φ 4 are identical scalars, in which case we get In practice, it is not easy to extract information from this form either. Rather, the comparison becomes most powerful (at least in the absence of other information) when the first and third scalars are also identical and the theory is unitary. In this case we get For three scalars and one vector, there is no configuration which is quite as powerful.
For SVSS, we can consider 1 ↔ 3 exchange, which leads to (5.171) By grouping the two tensor structures, we get two scalar equations. Let's write them out just for the case that φ 1 and φ 3 are identical. We get and a similar equation where we act on the subscript p of by exchanging 1 ↔ 2.
By considering 2 ↔ 4 exchange, we would obtain equations relating a sum over g rλ p blocks of the SVSS correllator with the g λr p blocks of SSSV, or we could obtain an equation by considering 1 ↔ 3 exchange in the SSSV case.
Next we turn to our primary interest in this paper -the case of two scalars and two vectors.

SVSV case with generic vectors
Here we have abbreviated all the blocks on the left-hand side as while on the right-hand side we expand as we also recall that we use tensor structures In other words, t ′ 0 ab = t 0 ab and t ′ ij ab = t ji ab .
Grouping like tensor structures together now gives us five equations on the underlying data of the CFT. Note that by using (4.142), we can rewrite the symmetric exchange summands in terms of scalar blocks.
As with the case of four-scalars, the equations are much more constraining for the case where we have two identical scalars and two identical vectors. In this case, we saw in section 4.3 that g rs 22 (u, v) = g sr 11 (u, v) and g rs p = g sr p for the other p. This results in only three independent bootstrap constraints, and where we have defined for ℓ = 0) and using the notation of (4.142).

SVSV with conserved vectors
The situation is even more tractable in the case that the identical vectors are in fact For ℓ = 0, as reviewed in section 2.5, we can assume that either φ is neutral under the symmetry, or that is the real part of a complex scalar operator of charge Q. We'll focus on the latter case, and the former case can be recovered by setting Q = 0. Thus, there will be a unique scalar operator O that can be exchanged, with ∆ O = ∆ φ and α φvO = −Q. In this case we can split this piece out of the bootstrap constraints, much in the same way that the contribution from exchange of the identity operator is typically split off for the case of identical scalars in the SSSS bootstrap, and move it to the right-hand-side of the constraint equations.
For ℓ > 0 we also have another important result -the relation between α φvO and β φvO given in (2.53). Since we could in principle have either α φvO or β φvO vanishing, it is more useful to define a non-vanishing constant c O related to them by which defines constants a O and b O that depend only on the dimensions ∆ φ and ∆ O , and Now we can rewrite the bootstrap constraints as follows, where we have defined

Conclusions
Our primary goal in this paper was to set up, in explicit detail, the bootstrap equations suitable differential operators to scalar blocks using ideas from [44]. Furthermore we showed that the conformal blocks of the mixed operator A can also be written as differential operators acting on scalar blocks, if we allow them to have shifted spins k,k + 1,k + 2. We also found that writing the SVSV blocks in terms of lower spin ones (SVSS,SSSV) makes the expressions simpler. This could potentially be important for making the computation where the full four-point function can be computed directly. It would also be interesting to understand our results in the context of holography, and particularly to match our results with the bulk geometric quantities studied in [49]. 13 Bootstrapping non-conserved vectors could in principle be done using semi-definite techniques [8,13,23].
Going forward, there are two natural extensions to this work. The first is to actually apply our formalism to seek, both numerically and analytically, bounds on the data of some general class of CFTs. It would be particularly interesting to derive results for conserved vectors, which could constrain theories with continuous global symmetries and their spectra of scalars charged under the symmetry.
The second direction heading forward is to use the techniques developed in this work to set up the bootstrap for even more complicated four-point functions. The next one to attempt is probably four vectors, either conserved or not, and this case should be tractable by hand. Another possibility would be two scalars and two spin-two tensors, especially for the case where the tensors are conserved (e.g. stress-energy tensors). Finally, the most ambitious goal would be to bootstrap the four-point function of conserved stress-tensors (see for example the discussion in [50]). This is probably not feasible using our current techniques "by hand", but might be possible if we can computerize the necessary steps.

A Building blocks and identities
In this paper, the physical space is flat R D with Euclidean signature. Indices a, b, etc., are raised and lowered with the Kronecker delta δ ab . For two vectors x a i and x a j , we define Two particular structures play a significant role in constructing correlators in a conformal field theory, where x i , x j , and x k are assumed to be distinct points, and where again x i and x j are distinct.
From the fact that x ij + x jk = x ik , we can show that Using the basic identity that we can prove identities As special cases of these fomulae, we have One more useful identity is

B Lorentz representation projectors
We will be grouping tensor operators by their representations under SO(D). There is a large body of work on irreducible representations of SO(D) (for instance see the nice discussion in [46] and references therein), but we really don't need the full power of this theory for the current work.
Consider a tensor with n indices. It must transform as a sub-representation of the tensor product D ⊗n of n copies of the D-dimensional vector representation. To distinguish the different irreducible representations I which appear in the decomposition of D ⊗n , we can use projectors, Π I a 1 ···an b 1 ···bn . Being projectors, these must satisfy The projectors are built exclusively with Kronecker deltas δ a i b j , δ a i a j , or δ b i b j . Below, we will need the projectors for the totally symmetric traceless representation of spin ℓ (i.e. with ℓ indices), and also for a mixed symmetry representation with k + 2 indices which we will describe below.

B.1 Totally symmetric
Consider first the projector onto totally symmetric traceless representations, Π (ℓ) a 1 ···a ℓ b 1 ···b ℓ . By the symmetries of the problem, it must have the form where the A i are constants. For ℓ ≥ 2, taking the trace with δ b ℓ−1 b ℓ we get Thus tracelessness requires Finally, we can fix A 0 by the condition that Π 2 = Π, i.e.
In fact we only need to check the leading terms, not the subleading traceless terms, because the latter can't contribute to the former when we square. Then since we require A 2 0 = A 0 , and hence we should take A 0 = 1, and we can write These projectors obey certain recursion relations. With the explicit expressions for coefficients above, one can show that . (B.211) Now we can define polynomials p D,ℓ (t) by Explicitly, using (B.210), we have These are related to the more familiar Gegenbauer polynomials by They obey a simple differential identity, and also We can also prove a recursion relation for fixed D from (B.211), while demanding that it vanishes when we trace with δ c 2 d gives Finally, demanding that Π 2 = Π requires The unique non-vanishing solution to these constraints is that For k > 1, the following structure is the most general consistent with antisymmetry of the a i and c i , symmetry of the b i and d i , and symmetry between upper and lower indices, Demanding this vanish when we antisymmetrize over [a 1 a 2 b 1 ], when we trace with δ d k−1 d k , and when we trace with δ c 2 d k fixes everything up to one constant A 0 which can then be fixed by the condition that Π 2 = Π. The result is that while the A i are given by , (B.232) and the recursion solved by .

(B.234)
For D ≤ 4, the story so far is not quite complete.
In D = 2, these mixed symmetry tensors labeled by k are equivalent to spin-k symmetric traceless tensors, with the map In D = 3 similarly, there is an isomorphism between mixed symmetry labeled by k and traceless symmetric of spin k + 1, via and Finally, in D = 4 we don't have to worry about any isomorphisms of this sort, but we instead need to recognize that our mixed symmetry representations are in fact reducible.
To split the two pieces apart, we can define and then define As in the symmetric case, we will need to consider the result of contracting these projectors with vectors X and Y , so we consider the expression The free indices a and b can only be carried by a Kronecker delta δ a b or by the vectors X a and Y a . Moreover, the expression must be symmetric under simultaneous interchange of X with Y and a with b, and it must be identically zero when we contract with X a or with result of a particular differential operator (say in X) acting on the symmetric contraction of λ 1 indices and λ 1 is the length of the top row of the Young pattern [λ] (in our case this is k + 1). In the context of conformal blocks, the extra indices e j are contracted with m (10) , m (20) , and the indices g j with m (30) , m (40) . Furthermore, Thus a generic contraction with T i m 10 · · · m 10 m 20 · · · m 20 · T i (k (012) , k (034) ) · m (30) · · · m (30) m (40) · · · m (40) , (C.249) can include combinations of 1-and 2-index elements The basic building block for our integrals is along with the Feynman-Schwinger trick which uses the identity will be a conformal scalar of weight α, β, and γ under conformal transformations of x 1 , Then where we have also made use of the duplication formula for the gamma function, which in this case tells us Similarly, we will need to evaluate which for α + β + γ = D will be a conformal scalar of weight α (β) under conformal transformations of x 1 (x 2 ), and a traceless symmetric tensor of conformal weight γ under transformations of x 3 . We compute by doing a binomial expansion of the k (302) 's, where we used the identity We will also need one more result along these lines, In this case we made use of (A.202).

E Mixing matrices and normalization factors
For the case of two scalars and a symmetric traceless tensor, inserting (3.59) into (3.60) leads to where we use the notation and results for integrals defined in Appendix D.1, and we evaluated the sums, first over m and then over k, using the identities Now for a scalar, a vector, and a traceless symmetric tensor we have

Related to these integration techniques is the determination of the normalization factor
N O that appears in the shadow projector P O . As discussed in the main text, this is fixed by requiring , where we read off (3.70).