Mellin bootstrap for scalars in generic dimension

Journal of High Energy Physics, Jun 2018

Abstract We use the recently developed framework of the Mellin bootstrap to study perturbatively free scalar CFTs in arbitrary dimensions. This approach uses the crossing-symmetric Mellin space formulation of correlation functions to generate algebraic bootstrap equations by demanding that only physical operators contribute to the OPE. We find that there are no perturbatively interacting CFTs with only fundamental scalars in d > 6 dimensions (to at least second order in the perturbation). Our results can be seen as a modest step towards understanding the space of interacting CFTs in d > 6 and are consistent with the intuition that no such CFTs exist.

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Mellin bootstrap for scalars in generic dimension

HJE Mellin bootstrap for scalars in generic dimension John Golden 0 1 Daniel R. Mayerson 0 1 0 University of Michigan , 450 Church Street, Ann Arbor, MI 48109-1020 , U.S.A 1 Department of Physics and Leinweber Center for Theoretical Physics We use the recently developed framework of the Mellin bootstrap to study perturbatively free scalar CFTs in arbitrary dimensions. This approach uses the crossingsymmetric Mellin space formulation of correlation functions to generate algebraic bootstrap equations by demanding that only physical operators contribute to the OPE. We nd that there are no perturbatively interacting CFTs with only fundamental scalars in d > 6 dimensions (to at least second order in the perturbation). Our results can be seen as a modest step towards understanding the space of interacting CFTs in d > 6 and are consistent with the intuition that no such CFTs exist. Conformal Field Theory; Field Theories in Higher Dimensions - 1 Introduction and summary 2 3 2.1 2.2 3.1 3.2 3.3 3.4 Bootstrap A De nitions B Simpli cations Setup, assumptions, and simpli cations The Mellin bootstrap Simpli cations t-channel contributions from t-Channel Contributions from Bootstrap equations to O( 2) Summary and extensions 2 B.1 The coe cients c O;l B.2 S1: s-channel contributions for heavier operators B.3 S2: t-channel contributions for l0 > 0 B.4 S3: t-channel contributions for heavier scalars (l0 = 0) B.5 Contributions from and 2 The program of the conformal bootstrap was introduced in the 1970s [1, 2] and has recently regained attention [3{6]. This program is largely based on the fact that the consequences of conformal symmetry are su ciently stringent to signi cantly constrain the space of possible CFTs. Two particularly important consequences are the operator product expansion (OPE) and crossing symmetry. Considerable insights can be gained by studying the constraints of crossing symmetry on theories which, by construction, satisfy the OPE (e.g. [7]). This is done by constructing an ansatz in the form of an OPE of a four-point correlation with generic coe cients, and then imposing crossing symmetry and exploring the solution space. The recent Mellin bootstrap, pionereed by Gopakumar, Kaviraj, Sen, Sinha in [8, 9],1 approaches the problem from the complementary perspective of assuming crossing symmetry and then imposing the correct OPE structure. In this case, one constructs an ansatz that explicitly satis es crossing symmetry but then the correct operator spectrum has to be imposed as a consistency condition. Demanding that the \exchanged" 1Some earlier works important for the development of the Mellin bootstrap are [10{12]. Wilson-Fisher xed points studied in [8, 9, 13],2 it was shown that, to a certain order in , this in nite set of algebraic equations in the Mellin bootstrap truncates to an analytically solvable nite set. It is interesting to consider whether this simpli cation of the Mellin bootstrap is a feature of perturbatively free CFTs in general. In this note, we begin by investigating theories with a single fundamental that are perturbatively close to a free point in a general dimension d (which can be a function of the perturbation parameter as in the Wilson-Fisher xed point but does not necessarily have to be). We show that the Mellin bootstrap again reduces to a nite set of solvable analytic equations in this more general case. Furthermore, the solutions to these equations constrains the dimensions of and the lowest excited scalar operator \ 2", as well as the OPE coe cients C T , C and C 2 . In particular, we nd that the scalar theory is forced to be the free scalar theory in arbitrary dimension d > 6, to the perturbative order we are able to calculate. This provides evidence that there are no interacting CFTs with a single fundamental scalar in d > 6 dimensions. Our results can also easily be generalized to the case of an arbitrary number of fundamental scalars, again providing evidence that there are no interacting CFTs with any number of fundamental scalars in d > 6 dimensions. Our results are perhaps not surprising: for perturbatively free scalar theories, Lagrangian methods give an easy argument that no non-trivial marginal operator can be constructed in d > 6 for scalar theories - therefore no interacting scalar theory would be expected to exist. Furthermore, recall that the superconformal algebra does not close in d > 6, so there are no superconformal eld theories in this regime [16]. However, the Mellin bootstrap approach allows us to approach the question of existence of CFTs from a fundamentally di erent viewpoint not involving Lagrangians at all. This approach also begins to clarify the criteria under which the Mellin bootstrap reduces to a nite set of equations. It would be interesting to expand our results to include fermions, other (fundamental) elds, and to connect these techniques with the analysis of generalized Wilson-Fisher xed points in [17, 18]. This paper is organized as follows: section 2 brie y introduces the Mellin bootstrap and our CFT setup, including our assumptions and the resulting simpli cations that occur that reduce the bootstrap equations to a nite number of algebraic equations. Section 3 discusses solving these bootstrap equations to show that the theory we obtain is forced to be free. The necessary de nitions of functions etc. can be found in appendix A while the simpli cations are discussed in more detail in appendix B. 2 Setup, assumptions, and simpli cations The objects of study for this note are unitary, perturbative CFTs containing one fundamental scalar as the lowest-dimension operators. (We will discuss in section 3.4 how to extend our results to an arbitrary number of fundamental scalars.) 2A similar -expansion analysis for the O(N ) model was made using the Mellin bootstrap in [14]; the saturates the unitary bound and is a free scalar for any [19]. We have a similar expansion for the dimension of the lowest excited operator, which we denote as \ 2" following standard convention: is to emphasize the fact that d can be an analytic function of , as is the case in the Wilson-Fisher CFT). The = 0 limit corresponds to the free scalar CFT in d0 dimensions. The dimension of is parameterized as = d 2 2 2 + l + l( 1 ) + O( 2 ); Cl = C(0) + C( 1 ) + O( 2 ): l l which is given by the schematic form3 We will also consider the lowest dimension (primary) operator Jl of spin l (for l 6= 0; 2), and has dimension Jl and OPE coef cient C Jl given by: We make the following assumptions about the theory: A1: there is a conserved stress tensor with = d and spin l = 2 (i.e. the operator Jl=2 has l(=i)2 = 0 for all i). A2: there is only one fundamental scalar, , of dimension given in (2.1).) Note that we do not assume any other symmetry of the theory, such as an overall Z2, as is often done. We will instead keep our analysis generic and then note where a Z2 symmetry (i.e. C = 0) allows us to make even stronger claims. We do note that we have assumed that all relevant quantities in (2.2){(2.7) allow for an analytic expansion in ; this is an assumption we make on the nature of the perturbative expansion.4 We do not need to assume that is positive; in our analysis we will see that the only possible solution for d0 > 6 is the free theory which saturates the unitarity bound in d dimensions, regardless of the sign of . (By contrast, note that > 0 is necessary in the d = 6 analysis 3We have in mind the symmetric, traceless current operator of spin l that is a conserved current in the free theory. The exact form of these operators can be found in e.g. [20]. 4We thank M. Paulos for stressing this point to us. { 3 { of [9]; there, when already at O( 1 ).) review now. can be seen to violate the unitarity bound Our goal is to determine the values of the expansion coe cients in eqs. (2.1){(2.5) given the assumptions A1 and A2. We will do so using the Mellin bootstrap, which we Our analysis is an extension of the Mellin bootstrap techniques introduced in [8] and described in detail in [9]. We are interested in the four-point function (with identical i, which in position space is determined completely by the function A(u; v) where u and v are the standard conformal cross ratios: h (x1) (x2) (x3) (x4)i = x122 x342 A(u; v); u = As described in the Introduction, the Mellin bootstrap involves constructing h crossing symmetry explicitly satis ed, but without explicit agreement with the OPE. In other words, it is not guaranteed that the terms that appear in the expansion are primaries and their descendants. Polyakov introduced this crossing symmetric construction in [1] and noted that in this expansion terms proportional to u and u log u necessarily appear. These terms correspond to an operator of dimension 2 appearing in the spectrum. Since there is not generically an operator of this dimension in an interacting CFT, these terms must cancel when summed over all channels. Requiring that this is the case imposes non-trivial constraints on the CFT. However, actually calculating the coe cients of the spurious u and u log u terms is very di cult in position space, so this approach did not recieve much attention until Gopakumar, Kaviraj, Sen, and Sinha realized [8] that the spurious terms in position space become spurious poles, with calculable coe cients, in Mellin space. We now brie y describe this calculational scheme. Appendix A contains explicit de nitions for all of the functions that follow. The (reduced) Mellin amplitude M(s; t) of the four-point function A(u; v) is de ned by A(u; v) = Z i1 ds dt i1 2 i 2 i usvt 2 ( t) 2(s + t) 2( s)M(s; t); where the form of the measure has been chosen to highlight the pole at s = particular, the expansion of M(s; t) around s = has a constant and linear term in (s ), so the overal integral has a single and double pole at this point. The residues at these poles give rise to precisely the u and u log u terms that indicate the unphysical operator described in the previous paragraph. Therefore, requiring that the coe cients of the u and u log u terms be zero then corresponds to ensuring that the constant and linear terms vanish: (2.8) (2.9) i with (2.10) . In M(s = ; t) = 0 and = 0: (2.11) { 4 { This is the most general representation of `the Mellin bootstrap constraints'. While we will not make use of it here, it is interesting to note that there are in fact an in nite class of these constraints, as 2 ( s) has spurious poles at s = + n; 8 n 2 Z+. Now we must cast M(s; t) in a form where we can impose (2.11) for all values of t. First o , we can decompose M(s; t) into what can be thought of as exchange (Witten) diagrams M (s;=lt=u)(s; t) where an operator O ;l is exchanged between pairs of 's; these exchanges can happen in each of the s; t; u-channels. This gives us the explicitly crossing symmetric representation M(s; t) = X c ;l M (s;)l(s; t) + M (t;)l(s; t) + M (u;)l(s; t) : (2.12) and O ;l (from the vertices of the exchange diagram). Fortunately there is a convenient basis of orthogonal functions, the continuous Hahn polynomials fQl (t)g, which allow us to partially separate the s M (s=t=u)(s; t). In this decomposition, only a single term contributes in the s-channel: and t dependence in M (s;)l(s; t) = q(s;)l(s)Ql2 +l(t); where q(s;)l(s) = q(2;;ls) + (s )q(1;;ls) + : : : (as s ! ): q(2;;ls) and q(1;;ls) are labeled this way as they correspond to the double and single poles in (2.10), respectively. The contributions from the t and u channels turn out to be identical, so we need only include the t-channel twice. The in nite number of contributions in the t channel which must cancel against the s-channel contribution are: (2.13) (2.14) (2.15) (2.16) (2.17) where also: q(t);ljl0 (s) = q(2;;ltj)l0 + (s )q(1;;ltj)l0 + : : : (as s ! ): Note that there is a polynomial ambiguity that we have omitted from (2.13) and (2.15), which stems from an ambiguity in the Mellin space Witten diagrams related to contact terms [9, 12, 13, 15]. This ambiguity was shown to lead to di erent results for certain quantities in the Mellin bootstrap approach compared to the regular bootstrap approach at high orders in [15]. While the presence of this ambiguity is thus certainly important, it is also expected that it will only start contributing substantially to the relevant Mellin bootstrap results at higher orders in than we will be considering (speci cally, O( 4 )) [15]. Summing up the contributions from the di erent channels then gives us a more concrete formulation of the bootstrap equations X c ;lq(a;;ls) + 2 X c ;l0 q (a;t) ;ljl0 ! = 0; the constraint M(s = ; t) = 0. = 0 and a = 2 corresponds to M (t;)l(s; t) = l0 X q(t);ljl0 (s)Ql20 +l0 (t); l0 { 5 { The bootstrap (2.17) represents an in nite number of equations (one for each l), and each equation involves an in nite number of terms (from the sums over ; l0). A number of non-trivial simpli cations occur for every l, resulting in only a nite number of terms contributing up to O( 2 ). These terms involve the unknown parameters in eqs. (2.1){(2.5) and form a solvable nite system of equations. These simpli cations are: S0: the identity operator only contributes to the simple pole constraint, so we can separate out that contribution and write the bootstrap equations as HJEP06(218) 2q(1=;t)0;lj0 + X 6=0 X 6=0 c ;lq(2;;ls) + 2 X c ;l0 q c ;lq(1;;ls) + 2 X c ;l0 q l0 l0 (2;t) ;ljl0 (1;t) ;ljl0 ! ! = 0; = 0: S1: in the s-channel, only the lowest dimension operator of spin l contributes to P c ;lq(2;;ls) up to O( 2 ) (for l = 0, we will consider the two lowest operators explicitly). For l = 2 the in nite sum reduces to the single term X c ;l=2q ;l=2 = c =d ;l=2q(a;s) (a;s) =d ;l=2 + O( 2 ); 6=0 as the lowest dimension operator of spin l = 2 is the stress tensor with = d . For l = 0 the in nite sum reduces to X c ;l=0q ;l=0 = c = ;l=0q(a;s) (a;s) = ;l=0 + c = 0;l=0q(a;s) = 0;l=0 + O( 2 ) 6=0 as the two lowest dimension scalars that can contribute are itself and 2. For other l, the lowest dimension operator is the operator Jl, so the in nite sum reduces to: 6=0 X c ;lq(a;;ls) = c ;lq(a;s) = Jl ;l + O( 2 ): S2: in the t-channel, for l0 > 0, we have c ;l0 q scalars contribute to the sum Pl0 c ;l0 q (a;;ltj)l0 = O( 2 ); i.e. to this order in , only (a;;ltj)l0 . This in nite sum over l0 thus reduces to: X c ;l0 q l0 (a;;ltj)l0 = c ;l0=0q ;ljl0=0 + O( 2 ): (a;t) (2.18) (2.19) (2.20) (2.21) (2.22) (2.23) S3: furthermore, in the t-channel, for \heavier" scalars of c ;0q(a;;ltj)0 = O( 2 ). This reduces the in nite sum over in the t-channel to: > 0 , we have X c ;l0=0q ;ljl0=0 = c = ;l0=0q(a;t) (a;t) = ;ljl0=0 + c = 0;l0=0q(a;t) = 0;ljl0=0 + O( 2 ): (2.24) { 6 { Derivations of these simpli cations are in appendix B. With these simpli cations, the bootstrap equations for generic l become c Jl ;lq and in the special cases l = 0 and l = 2 we have 2q0(1;0;tj0) + c ;0 q(1;s;)0 + 2q(1;t) ;0j0 cd ;2qd(2;;s2) + 2c ;0q(2;t) cd ;2qd(1;;s2) + 2q0(1;2;tj0) + 2c ;0q(1;t) c ;0 q(2;s;)0 + 2q(2;t) ;0j0 left-hand side of these equations vanish up to the given order in will give us non-trivial constraints on the coe cients appearing in (2.1){(2.7). 3 Bootstrap In this section we describe the bootstrap in considerable detail in order to highlight the di erences between the generic d case and the special d = 4 studied in [9]. Before we do so, let us describe our result in physical terms, as the preceeding notation is a bit cumbersome. Consider, as an example, the schematic form of (2.28): T(s-channel) + 1 t-channel + (t-channel) + (2t-channel) = O( 2 ) where by \O(channel)" we refer to an overall contribution of the form c O;lqchOan;lnel. The other bootstrap equations take a similar form, with the s-channel contribution coming from either in the l = 0 case, T for l = 2, and generic Jl for l > 2, and the t-channel identity operator only arises in the simple-pole constraints. For d0 6, all of these terms can make nite, non-zero contributions through O( 1 ) (though note that they do not all contribute in all cases | for example (t-channel) = 0 in the Z2-symmetric case as C = 0). The essential story for the d0 6 regime is that the complicated interplay between non-trivial s- and t-channel contributions produces a set of linear equations to get cancellations up to O( 2 ). This set of linear equations can then be used to determine operator dimensions, e.g. for the Wilson-Fisher xed point in d = 4 and the 3 theory in d = 6 . We will show that for d0 > 6, the and 2 t-channel contributions necessarily begin O( 2 ), so they drop out of the bootstrap equations and all we are left with is the tchannel identity to cancel the s-channel operator. Without these non-trivial t-channel contributions, the only possible solution to the resulting set of equations is the free theory. 5d0 = 2 is also obviously a special case, but for the sake of brevity we will not comment on those subtleties here. { 7 { (2.25) (2.26) (2.27) (2.28) (2.29) (2.30) cases5 (3.1) While we have emphasized d0 > 6 so far, the 2 t-channel contribution in fact drops out already at d0 > 4. We can see from a simple series expansion that the normalization coe cient c 0;0 already exhibits a special value at d0 = 4: The fact that these poles behave di erently in d0 > 4 is not particularly surprising. We know from [9] that at least in d0 = 6, the 2 operator indeed does only contribute at higher order (otherwise it would compete with the C exchange in the non-Z2 invariant theory); however, what we nd here is that the 2 operator cannot contribute to the relevant orders for any d0 > 4 (in particular d0 = 4n for n > 1 is not special in any way). 3.2 t-Channel Contributions from Now, we turn to the contributions from in the bootstrap equations. Once again, we rst note that a simple series expansion shows that the normalization coe cient c ;0 exhibits a special value at d0 = 6: c ;0 / > < 8 However, for d0 > 6, q(a0;t;l)j0 has an in nite sum of contributions from poles of the integrand of that go as O( 0). In particular, this in nite sum is not convergent. This means that in d0 > 6, we have (schematically): where we have made explicit the factor of C(0) that is included in the normalization c ;0. There are no other in nities that this could cancel with until the in nite tower of heavier operators enters at O( 2 ) in the bootstrap expressions; thus, we are forced to set C = O( 3 ) in order for the bootstrap equations to make any sense to O( 2 ). Note that this suppresses the contribution of from all the bootstrap equations (2.25){(2.30). { 8 { It is interesting to compare this with the 2 case, as the t-channel contributions take an apparently similar form: However, let us emphasize that the reason must drop out of the bootstrap for d0 > 6 2 is di erent than : we saw that the contribution of the suppressed to O( 2 ) for d0 > 4; here, the contribution of 2 operator is automatically in d0 > 6 is actually in nite, forcing us to set the OPE coe cient C = O( 3 ). This could not have happened for the 2 operator, as the 2 OPE is already non-zero in the free theory. 3.3 Bootstrap equations to O( 2 ) Above, we have discussed the t-channel contributions from and 2 and found that they all vanish. Now, we can use this to solve the bootstrap equations, which at this point consist solely of operators in the s-channel either cancelling o t-channel identity operator contributions (in the simple-pole constraints) or being set directly to O( 2 ) (in the doublepole case). Let us consider the l = 2 bootstrap equations (2.27){(2.28) rst. The lowest order term of (2.27) is at O( 1 ), while (2.28) has terms at O( 0) and O( 1 ). Demanding that (2.27) and (2.28) hold up to O( 1 ) thus gives us three equations to solve. The solution is ( 1 ) = 0; C(0) = C2;free(0); 2 C( 1 ) = C20;free(0); 2 where C2;free( ) is the square of the (appropriately normalized) OPE coe cient in the free C2;free( ) = d (d These results match those found in [9] for d = 4 the l = 2 case. The results of the bootstrap give: The l 6= 0; 2 bootstrap equations (2.25){(2.26) for Jl proceed entirely analogously to theory where:6 (3.7) (3.8) (3.9) (3.10) (3.11) Next, we can consider the l = 0 bootstrap equations (2.29){(2.30). Equation (2.30) has terms at O( 0) and O( 1 ), which can be set to zero to obtain: 0 C(0) = 2; C( 1 ) = 2 0( 1 ) 0 2Hd0=2 3 Hd0 4 + d 2 0 3d0 8 7d0 + 12 ; (3.12) 6Of course, (3.11) agrees with (3.9) for l = 2. { 9 { where Hn is the n-th harmonic number. This again matches [9] when d = 4 Finally, we turn to equation (2.29), where the crucial distinction between d0 = 4 and 2 contributions plays a critical role. Using (3.4), we nd that the O( 1 ) The d0 = 4 value again matches that of [9]. Because c 0;0q extra order of in d0 > 4, we see that 0( 1 ) is forced to vanish in this case, giving the free (2;t) 0;0j0 is suppressed by an To summarize, we have found that the quantities in question satisfy (for d0 > 6): (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) = ;free + O( 2 ); C = 0 + O( 3 ); f0;2;lg = f0;2;lg;free + O( 2 ); Cf0;2;lg = Cf0;2;lg;free + O( 2 ): Therefore we conclude that any CFT in d0 > 6 with a conserved stress tensor and a single fundamental scalar must be perturbatively free up to O( 2 ). Furthermore, for Z2symmetric theories the range of validity for these results extends to d0 > 4, since the 2 operator drops out already at that dimension. We stress that these results only depend on the value of d0, and in particular not on the sign of . These results can also be generalized to higher orders in , as well as to include multiple scalars. Higher orders in . While our results are certainly sugggestive, they are at relatively low orders in . To what extent can we push this to deeper orders? We have found two regimes in which we can go further: d = 4n + 2 + O( 1 ); or d = 2n + 2 + O( 1 ) and the CFT is Z2-symmetric; for n > 1; n 2 Z+. As described in appendix B (see especially B.1), when either of these is true, the simpli cations S1-S3 actually hold to O( 4 ) (note this also requires l( 1 ) = 0, which is guaranteed by (3.14)). This means that the right hand side of (2.25){(2.30) is O( 4 ) and we can then perform the bootstrap analysis above at orders O( 2 ) and O( 3 ) as well. The analysis of the equations at these orders proceeds entirely analogously as above, and results in determining the quantities in question to higher order in : 7 = ;free + O( 4 ); C = 0 + O( 5 ); f0;2;lg = f0;2;lg;free + O( 4 ); Cf0;2;lg = Cf0;2;lg;free + O( 4 ): 7Note that we expect the polynomial ambiguities in the Mellin space Witten diagrams in (2.13) and (2.15) to start contributing at O( 4 ) [15], which would complicate the analysis beyond this order in even further. Bootstrap for multiple scalars. Our analysis was done assuming only one fundamental that saturates the unitary bound when the theory is free at = 0. However, it is fairly straightforward to see that we can relax this assumption to include an arbitrary number N of fundamental scalars (i), with dimensions: (i) = d 2 2 We can perform N versions of the four identical scalar bootstrap that we have performed above in the case of one scalar. The simpli cations S1-S3 still hold with arbitrary index structures on the heavy operators Ok;l;m. The rest of the bootstrap proceeds analogously - for instance, the l = 2 bootstrap equations will give e.g. (1(i)) = 0 for every i. The only possible extra non-triviality that one might worry about is the existence of a conserved spin-1 current (such as in the O(N ) model). However, since our results force the scalar theory into a free theory, the relevant spin 1 current is automatically conserved, anyway, without needing any further input.8 Our analysis is thus insensitive to any demand of global symmetries the theory could have, as the resulting theory is always free. We thus conclude that our analysis can be extended to the case of N fundamental scalars (i) with arbitrary indices placed on the relevant operators. It is surprising that with very little input or assumptions, we are able to see clear evidence that no perturbative interacting CFTs with any number of fundamental scalars exist in d0 > 6, without needing any notion at all of the Lagrangian. Perhaps by considering more of the in nite class of spurious poles in the Mellin bootstrap one could further relax our assumptions to fully explore the world of CFTs in d0 > 6. Acknowledgments We thank Marco Baggio, Anthony Charles, Ben Safdi, and Miguel Paulos for useful discussions, and especially Henriette Elvang for many helpful suggestions and guidance during this project. This work was supported by the U.S. Department of Energy under grant DEFG02-95ER40899 and JG was supported in part by a Van Loo Postdoctoral Fellowship. A De nitions In this appendix, we collect a number of necessary and useful de nitions and expressions we need; for more details and the derivations of these expressions, we refer the reader to [9] where they originally appeared. We will often use the notation: and the de nition of the Pochhammer symbol: 8Note that the spin-2 conserved current, the stress-energy current, is not automatically conserved. We need to demand its conservation in order to nd the condition (1()i) = 0. h = d 2 ; (a)b = (a + b) (a) : (A.1) (A.2) The coe cient c ;l is given by: c ;l = C ;lN ;l; where C ;l is the square of the OPE coe cient C normalization coe cient: O ;l , and N ;l is the The function q(s;)l(s) is given by: from which the coe cients q(2;;ls) and q(1;;ls) can be extracted using: q(2;;ls) = q(s;)l(s = ); q(1;;ls) = These are given by: q(2;;ls) = q(1;;ls) = (l (l + l h) + 2 The function q(t);ljl0 (s) is given by: q(t);ljl0 (s) = l(s) 1 X X g Here, we used the sub-expressions: 2 h) + 2 2 ) + 2) + 2 q s ; 2 2 + (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) # ; 1 : (A.9) (A.10) (A.11) ( s)m ( s)l0 2k m 2l(s)l2 ( 1 )ql!(2s + l 1)q Qi4=1 (li) (2s + l 1)l (k + q + s + + 2 (s)q2q!(l q)! ) (k + q + s + (q + 2s + 2k + + + 2 + 2 + 2) (k + s + (k + s + " 3F2 q; 1 k q ;l0 ( ) = l(s) = 2 i(( (2s + l ( 1 )l4ll! h)2 2) ( ) ( )(h + 1)l0 (h 1)l2 (2s + 2l ; 1 s 2 ( 2k q 2 + 2s ; 1 k 2) 2( 4(l + s) 1) (2s + l 2) 1) ; and the de nition of the complicated summation: X g and nally the shorthand notations: l0! l0 The expression (A.9) simpli es considerably for l0 = 0 (with The coe cients q(2;;ltj)l0 and q(1;;ltj)l0 can be extracted from q(t);ljl0 (s) in a similar way as in (A.6). For completeness, we also give the expression for the continuous Hahn polynomials Ql (t), which we give in a convenient form: Q 2 l +l(t) = (2 2l( ) 2 l + l 1)l 3F2 " l; 2 + l ; 1; + t # ; 1 : These satisfy the orthonormality condition: 1 Z +i1 2 i i1 B Simpli cations dt 2 ( + t) 2( t)Ql2 +l(t)Ql20 +l0 (t) = l ( ) l;l0 : Here we give more information about the simpli cations S1{S3 (the simpli cation S0 relies only on the fact that we are working with identical scalars, therefore the derivation given in [9] follows through). These simpli cations deal with the contributions of generic operators Ok;2m;l that contains k 's, l uncontracted derivatives, and 2m contracted derivatives. Such an operator will have spin l and dimension: Ok;2m;l = k d 2 2 + l + 2m + k(1;2)m;l + k(2;2)m;l 2 + O( 3 ): (B.1) the coe cients k(i;)2m;l. B.1 The coe cients c O;l For a given k; m; l, one possible such operator is (schematically) k 1@a1 there could possibly be other operators (by distributing the derivatives di erently among the 's).9 However, any such operators Ok;2m;l and Ok;2m;l will only di er by the value of 0 First of all, let us consider the coe cients c O;l for the heavy operators Ok;2m;l with dimension (B.1). These coe cients are given by (A.3); we discuss the factor C Ok;2m;l and N ;l separately. e.g. the three point function k vanishes for k > 2); this implies that: In the free theory, the three point function of Ok;2m;l for k > 2 vanishes (because HJEP06(218) Next, we note that the normalization factor N ;l of every heavy operator with k = 2 (or a higher power) as the C coe cient is the square of the Ok;2m;l OPE coe cient (which is reasonable to assume goes as O( 1 )). For k = 2, we can consider only (primary) operators where the derivatives hit only one of the 's, i.e. of the schematic form For m > 0, these operators identically vanish in the free theory so that we again have (B.2). Only the operators with k = 2; m = 0 (i.e. the Jl operators) may have: C Ok;2m;l = O( 2 ); C O2;0;l = O( 0): goes as:10 while when k > 2, we have: N ;l = ( 1 ) k=2;2m;l 2 g(h0)O( 2 ) + O( 4 ); 1 2 N ;l = (k 2)(h0 1) m 2 g~(h0)O( 0); N ;l = O( 2 ): c O;l = O( 2 ): (B.2) (B.3) (B.4) (B.5) (B.6) (B.7) where g(h0) and g~(h0) are unimportant additional factors involving h0. Note that the inverse gamma function has a zero when its argument is a negative integer; the only way to achieve this here is if k = 2n1; h0 = n2 with n1; n2 integers, or if h0 = 2n + 1 with n integer (since k is always an integer). Remembering that d = 2h, this is precisely the condition presented in (3.16). So, when (3.16) holds, we have: 9Note that, in [9], only operators with k = 2 were possible; this is because they explicitly consider the 4 theory where 3 is a descendant operator. Note that we could possibly be \overcounting" the number of operators when considering generic Ok;2m;l (i.e. also considering some descendants), but this is irrelevant since we prove we can ignore their contribution to the bootstrap equations anyway. 10We already set ( 1 ) = 0 here for simplicity; this does not change the leading power of in N ;l. Putting everything together, we can conclude that for any heavy operator Ok;2m;l, Since (B.7) holds for these operators, heavier operators in the s-channel begin contributing For the simpli cation S2, we want to prove that operators Ok;2m;l0 for l0 > 0 (with k 2 We rst note that the integrand in the integral expression for q(a;t) O;ljl0 (for a = 1; 2) has When (3.16) holds and when 2(1;0);l = l( 1 ) = 0, (B.6) and (B.4) imply that we have: B.2 S1: s-channel contributions for heavier operators = 0 for heavier operators Ok;2m;l with k > 2 and/or m > 0, Expanding (A.5) around gives at O( 2 ), i.e. we have: and m 0) satisfy: for l = 0 and l = 2.11 the following poles: I. II. III. IV. = = 2 O h + l0 = h + l0 + 2n3 = h 1 + n4; h + 2n2 (n4 < l0) In poles II and III, n2; n3; can be any non-negative integer; these are in nite families of poles; for the poles IV, n4 is a non-negative integer in the range 0; 1; ; l0 1. The integral expression for q(a;t) but the residues can be seen to combine to give vanishing contributions up to O( 0) (for O;ljl0 will then have contributions from the residues at each of these poles, both q(1;t) and q(2;t) separately). The cancellations that occur for any Ok;2m;l0 are: ResII(n2=n3+1) + ResIII(n3) = O( 0); ResIV (n4) = O( 0); n3 6= m n4 = 0; 1; ; l0 1; 2: (B.12) (B.13) 11In checking these simpli cations for l 6= 2, it is computationally favorable to already set ( 1 ) = 0 (as brie y mentioned in [9]). The only place where we need to keep ( 1 ) explicitly non-zero is in checking the simpli cations for q(2;t) when l = 2. The rest of the poles cancel in slightly di erent ways for m = 0 and m > 0: Thus, we can conclude that q(a;t) holds. to O( 4 ). When (3.16) holds and l O;ljl0 = O( 0). Together with (B.7), this proves that (B.11) ( 1 ) = 0, (B.7) implies that (B.11) becomes suppressed HJEP06(218) B.4 S3: t-channel contributions for heavier scalars (l0 = 0) For the simpli cation S3, we want to show that for operators Ok;2m;l0 with l0 = 0 (i.e. scalar operators) that are heavy, i.e. k > 2 and/or m > 0, we have: c O;l0=0q(a;t) O;ljl0=0 = O( 2 ): The analysis proceeds in a similar way to S2. The poles I, II, III (with l0 = 0) are present in the integrand for q(a;t), while the poles IV are now absent. The pole cancellations (B.12) still holds. For m > 0, the cancellations (B.15) still hold. The nal poles present satisfy: (B.17) (B.18) (B.19) ResI + ResII(n2=0) = O( 0) ResII(n2=0) = O( 0) (m = 0); (m > 0): We again have (B.7), so we can conclude that (B.17) indeed holds. When (3.16) holds and l ( 1 ) = 0, (B.7) implies that (B.17) becomes suppressed to O( 4 ). B.5 Contributions from and 2 in (B.18), so that all of the poles of q(a;t) For the operators and 2 in the t-channels, the cancellations (B.12) again holds for q(a;t) f ; 2g;lj0. The only other possible contributions can come from ResI and ResII(n2=0). For d0 larger than the special dimension 4 (for 2) or 6 (for ), these are also suppressed as f ; 2g;lj0 go as O( 0). However, there is a discontinuity for q(2;t) f ; 2g;0j0 in this cancellation for d0 = 4 (for 2), as discussed in section 3.1. For there are in fact discontinuities at both d0 = 4 (for ResII(n2=0)) and d0 = 6 (for ResI ). However these discontinuities end up playing no signi cant role as the operator does not contribute for the Z2 case at d0 = 4 and then, as discussed in section 3.2, the normalization coe cient c ;0 = O( 1 ) in d0 = 6. Open Access. 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John Golden, Daniel R. Mayerson. Mellin bootstrap for scalars in generic dimension, Journal of High Energy Physics, 2018, 66, DOI: 10.1007/JHEP06(2018)066