A Mellin space approach to the conformal bootstrap

Journal of High Energy Physics, May 2017

We describe in more detail our approach to the conformal bootstrap which uses the Mellin representation of CFT d four point functions and expands them in terms of crossing symmetric combinations of AdS d+1 Witten exchange functions. We consider arbitrary external scalar operators and set up the conditions for consistency with the operator product expansion. Namely, we demand cancellation of spurious powers (of the cross ratios, in position space) which translate into spurious poles in Mellin space. We discuss two contexts in which we can immediately apply this method by imposing the simplest set of constraint equations. The first is the epsilon expansion. We mostly focus on the Wilson-Fisher fixed point as studied in an epsilon expansion about d = 4. We reproduce Feynman diagram results for operator dimensions to O(ϵ 3) rather straightforwardly. This approach also yields new analytic predictions for OPE coefficients to the same order which fit nicely with recent numerical estimates for the Ising model (at ϵ = 1). We will also mention some leading order results for scalar theories near three and six dimensions. The second context is a large spin expansion, in any dimension, where we are able to reproduce and go a bit beyond some of the results recently obtained using the (double) light cone expansion. We also have a preliminary discussion about numerical implementation of the above bootstrap scheme in the absence of a small parameter.

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A Mellin space approach to the conformal bootstrap

Received: March Mellin space approach to the conformal bootstrap Rajesh Gopakumar 0 1 2 4 Apratim Kaviraj 0 1 2 3 Kallol Sen 0 1 2 3 Aninda Sinha 0 1 2 3 Open Access 0 1 2 c The Authors. 0 1 2 0 The University of Tokyo Institutes for Advanced Study , Kashiwa, Chiba 277-8583 , Japan 1 C.V. Raman Avenue , Bangalore 560012 , India 2 Survey No. 151, Shivakote, Hesaraghatta Hobli, Bangalore North 560 089 , India 3 Centre for High Energy Physics, Indian Institute of Science 4 International Centre for Theoretical Sciences (ICTS-TIFR) We describe in more detail our approach to the conformal bootstrap which uses the Mellin representation of CF Td four point functions and expands them in terms of crossing symmetric combinations of AdSd+1 Witten exchange functions. We consider arbitrary external scalar operators and set up the conditions for consistency with the operator product expansion. Namely, we demand cancellation of spurious powers (of the cross ratios, in position space) which translate into spurious poles in Mellin space. We discuss two contexts in which we can immediately apply this method by imposing the simplest set of constraint equations. The rst is the epsilon expansion. We mostly focus on the Wilson-Fisher xed point as studied in an epsilon expansion about d = 4. We reproduce Feynman diagram results for operator dimensions to O( 3) rather straightforwardly. This approach also yields new analytic predictions for OPE coe cients to the same order which t nicely with recent numerical estimates for the Ising model (at mention some leading order results for scalar theories near three and six dimensions. The second context is a large spin expansion, in any dimension, where we are able to reproduce and go a bit beyond some of the results recently obtained using the (double) light cone expansion. We also have a preliminary discussion about numerical implementation of the above bootstrap scheme in the absence of a small parameter. Conformal Field Theory; AdS-CFT Correspondence 1 Introduction The philosophy outlined The s-channel The t-channel The u-channel The bootstrap constraints The case of identical scalars Double and single poles 4.2 Identity operator contribution 4.3 The bootstrap constraints 5 Scalar dimensions and OPE coe cients Higher spin anomalous dimensions and OPE coe cients Justi cation for truncating operator sums A summary and comparison of results Anomalous dimensions OPE coe cients Comparisons with numerics in the 3d Ising model dimensions | a non-unitary example -expansion in other dimensions Large spin asymptotics Strongly coupled theories Weakly coupled theories CFTs close to a free theory Theories in 4 Witten diagrams & conformal blocks in Mellin space The spectral function representation Adding in the t; u channels The bootstrap strategy implemented A The Mack polynomial B The continuous Hahn polynomials A key normalisation D t integrals in the crossed channels E q sums in the t channel F Simpli cations for the F.1 s-channel F.2 Crossed channels Spin-0 exchange Spin-`0 > 0 exchange G Large ` behavior of Q ;` H Comparision with numerical results Introduction Quantum Field Theory (QFT) is one of the most robust frameworks we have in theoretical physics. Its versatility is attested by the fact that it plays a central role in many contexts in high energy physics, condensed matter physics and statistical physics. Thanks to the work of Wilson and others [1{8], QFT was understood beyond a perturbative Feynman diagram expansion. The central role in this modern understanding is played by scale xed points of the Renormalisation Group (RG) When combined with d dimensional Poincare invariance, these xed points are believed to have an enhanced SO(d; 2) conformal invariance [9]. The resulting CFTs while being dynamically nontrivial are also strongly constrained by the conformal symmetry. The conformal bootstrap is the philosophy that these constraints are strong enough to largely determine the dynamical content of the CFT viz. the spectrum of operator dimensions of primaries and their three point functions. The presence of a convergent OPE then implies that all other correlators can be xed in terms of this data [10{12]. 2 [13], employs the associativity of the four point function, as we describe below. Recently, making use of the progress in nding e cient expressions for conformal blocks [14, 15], this approach was revived for d > 2 [16] where associativity constraints, often together with positivity on the squares of OPE coe cients, were implemented numerically through linear programming and semi-de nite programming, together with judicious truncation of the operator spectrum [17{31]. This has led to remarkably precise bounds on low-lying operator dimensions in a number of nontrivial CFTs. This includes, famously, the 3d Ising model [32{34] which is in the same universality class as the critical point of the liquid-vapour transition of water. There are also very strong indications of such theories living at special points (\kinks") in the numerically allowed regions of parameter space. This suggests that these theories are special in some way and perhaps amenable to analytic treatment. These numerical methods have also been extended to supersymmetric theories [35{37]. Furthermore, there are also certain analytic results available at large spin [38{48, 83, 84]. However, the existing approaches do not appear to be well suited for extracting analytic results in general. Also limited progress has been made in the case where external operators carry spin, see e.g. [49{55]. Recently, using the conformal invariance of the three point function, the leading order ) dimensions were calculated for the Wilson-Fisher xed point [56]. This approach was further generalized to extract leading order anomalous dimensions for other theories in [57{60]. Results have also been dimensions of almost conserved higher spin currents [61{63]. These results crucially rely on the use of the equations of motion or a higher spin symmetry, that is present when the coupling constant goes to zero. It is not immediately obvious how to systematize these approaches to subleading orders. In [64], a dispersion relation based method of Polyakov [12] (which had built in crossing symmetry) was re-visited and it was found that this approach could be extended to get the subleading order anomalous dimension for the 2 operator.1 In spite of this encouraging result (though it took more than 40 years to reach here!), it was again not clear how to extend this dispersion relation based approach to operators with spin or to make it a starting point for a systematic algorithm. A major stumbling block was the reliance on momentum space where the underlying conformal symmetry is not fully manifest. In this paper, we will describe in more detail a novel approach to the conformal bootstrap that was recently outlined in [65]. This approach is calculationally e ective and at the same time conceptually quite suggestive. It combines two important ingredients. The rst goes back to an alternative approach to the above dispersion based one, also attempted by Polyakov in his original bootstrap paper [12]. He outlined a general way in which demanding consistency of the operator product expansion with crossing symmetry gave rise to constraints on operator dimensions and OPE coe cients. This was then implemented in position space which made the symmetries more manifest compared to momentum space. The idea behind this approach was to expand the CF Td four point function not in terms of the conventional conformal blocks but rather in terms of a new set of building blocks with built-in crossing symmetry from the beginning. We will see, in our modern incarnation, that these new building blocks can be chosen to be essentially tree level Witten exchange diagrams in AdSd+1. This is very suggestive of a reorganisation of the CFT in terms of a dual AdS description though this will not be the main thrust of the present work. The second ingredient we introduce is to implement the above bootstrapping procedure in Mellin space rather than position space as used in [12]. The position space approach 1It was also shown how the leading order anomalous dimension at O( 2) for large spin operators could be extracted using large spin bootstrap arguments based on [83, 84]. made the equations in [12] quite cumbersome and not explicit, especially for exchanges involving spin. We are familiar with this from the complicated form that Witten diagrams take in position space. The technology of the Mellin representation has been developed quite a bit in recent years starting from the work of Mack [66{73]. As has been amply stressed in these works, Mellin space is very natural for a CFT and plays a role analogous to momentum space in usual QFTs. This enables one to exploit properties such as meromorphy and more generally, features of scattering amplitudes (to which Mellin space amplitudes naturally transition to, in an appropriate at space limit). This, we will see, brings us big calculational gains. We will be able to reproduce many of the analytic results available in the literature for the conformal bootstrap in a fairly straightforward manner. In addition, we will be able to derive new results which we subject to various cross checks. We also give some preliminary evidence that this approach might also be workable into a useful computational scheme, complementary to existing ones. In the rest of this section we give a broad sketch of the new philosophy that we adopt and state some of the new results obtained with this approach. We rst describe the ideas in position space and only later translate them into Mellin space. The philosophy outlined Consider a four point function (of four identical scalars, for de niteness - we will consider the general case in section2). In essence, we expand this amplitude in a new basis of building blocks as follows A(u; v) = hO(1)O(2)O(3)O(4)i = X c ;` W (s;)`(u; v) + W (t;)`(u; v) + W (u;`)(u; v) : (1.1) Here (u; v) are the usual conformally invariant cross ratios, whose dependence captures the nontrivial dynamical information of the four point function (we have suppressed a trivial additional dependence on positions which is predetermined). In the second line we sum over the entire physical spectrum of primary operators generically characterised by the operator dimensions ( ) together with the spin (`) quantum numbers. The building block W (s;)` can, for the moment, be viewed as the Witten exchange function in AdSd+1 | it will be de ned more precisely later. This is diagrammatically represented in gure 1. It involves the four identical scalars with an exchange in the s channel of a eld of spin ` and corresponding to a dimension . Similarly, for the t and u-channels. The to-be determined coe cients c ;` will turn out to be proportional to the (square) of the three point OPE The idea behind this expansion, which we will contrast below to the usual conformal block expansion, is that we are expanding in a basis which 1. Is conformally invariant, as Witten exchange diagrams are; 2. Is consistent with factorisation, in that the individual blocks factorise on the physical operators with the right residues corresponding to three point functions; 3. Is crossing symmetric by construction since we are summing over all three channels. to an operator of dimension and spin `. The Witten exchange diagrams satisfy the second criterion since they arise from a local eld theory in AdS. This will be much more explicitly seen in the Mellin representation. The last criterion ensures that we don't need to check channel duality since that is built in. But what is not obvious now is that expanding the resulting amplitude in any one channel, say the s-channel, is consistent with the operator product expansion. In other words, if we expand A(u; v) in powers of u, it is not guaranteed that all the powers that appear are those of the physical primary operators together with their descendants. In fact, generically, such an expansion will have spurious power law dependence. For instance, with identical external scalars (of dimension ), we will see that there are pieces which go like u dimension 2 ln (u). The u would indicate the presence of an operator with , which generically does not exist in the theory.2 These are often called \double-trace operator" (\O2") contributions in the AdS/CFT literature since these are there interpreted as contributions from two particle states whose energy is almost the same (in a large N limit) as the two external (single) particle states.3 We will then obtain constraints on operator dimensions as well as the coe cients c ;` (and thus the OPE coe cients) from requiring that such spurious powers vanish. Note that these are strong constraints implying an in nite number of relations since there is a full function (of v) multiplying these powers. Though we will not make use of them in this work, there are additional spurious powers (and logs) of the form u +n ln (u). These can viewed as contributions from descendants as well as other double-trace primaries (what would have been \O@2nO" in a weakly coupled theory). One would obtain additional constraints from requiring their vanishing but we will not explore the consequences of this in this paper (see [87]). We should stress that the Witten exchange diagrams are being employed as a convenient kinematical basis for this expansion, for an arbitrary CF Td. We are not assuming (and it does not have to be) that the theory has an AdSd+1 gravity dual. We could have 2There could be special operators in interacting superconformal theories | \chiral primaries" | for which there indeed are physical operators with dimension 2 . Such cases would have to be treated specially, perhaps using mixed correlators or by focussing on other spurious powers. 3The logarithmic dependence is a consequence of having identical scalars. If we had generic dimensions i for the external operators, the spurious powers would take the form u 1+2 2 and u 3+2 4 corresponding to the two sets of double trace operators associated with the external states in the s-channel. The logarithm arises in the coincident limit . We also emphasise that the logarithmic dependence has nothing to do with anomalous dimensions since we are not making any expansion in a small parameter (yet). alternatively used conformal blocks as a basis of expansion. But as will become clearer in Mellin space these are not very well behaved at in nity.4 In contrast, Witten exchange diagrams will be polynomially bounded and thus a better basis for expansion. We note that since each Witten exchange diagram contains the conformal block contribution of the exchanged operator and since we are summing over the full primary operator spectrum we are not undercounting in this basis. In particular, what would have been double trace operators are included separately in the sum | this is di erent from what we do in AdS/CFT where we only sum over single trace primaries. In this context note also that contact four point Witten diagrams make no appearance in our approach. We do not have to include them since it is known that they are decomposable into the double trace conformal blocks and thus, in our context, have purely spurious power law contributions. Another important point to note is that we are implicitly assuming that the sums over ( ; `) in the spurious pole cancellation conditions are convergent or can be analytically continued. In the examples we have considered in this paper, the spurious poles have gotten contributions from only a small set of operators and hence we did not have to worry about convergence. In the usual case with conformal blocks, convergence is demonstrated in [100]. A preliminary discussion on convergence in Witten diagram expansion can be found in [65]. It would be good to investigate the issue more generally. As for the physical contributions, once the spurious pole cancellation has been achieved, the remaining sum is just the usual sum over the physical conformal blocks which is believed to be convergent in a nite domain. Let us contrast this approach to the more \conventional" bootstrap approach to CFTs [10, 11, 13] where we expand the four point function A(u; v) = X C ;`G(t);`(u; v) = In this expansion, the function G ;`(u; v) represents the conformal block due to a primary operator of dimension and spin `. It satis es the usual quadratic Casimir di erential equation [85]. The corresponding OPE coe cient is given by C ;`. We choose to expand in terms of the conformal block in a particular channel, say, s-channel, as in the rst line. The conformal blocks are 1. Are conformally invariant by construction; 2. Are consistent with factorisation, in that the individual blocks give the factorised contribution on physical operators with the right residues; 3. Are consistent with the OPE by construction since we are summing over all physical operators in any given channel. 4Polyakov made a similar observation in terms of the behaviour of these blocks in the spectral parameter The last criterion now ensures compatibility with the operator product expansion in terms of the powers that appear, but it is now not guaranteed that the resulting amplitude is crossing symmetric. In other words, the equality of the rst line with the second line does not automatically follow. Demanding this associativity of the OPE is the nontrivial requirement which constrains operator dimensions and the OPE coe cients C ;`. Recent progress has come from e cient ways to translate the constraints of associativity and positivity (which follows from unitarity) into inequalities which can be numerically Coming back to our approach, to convert Polyakov's scheme into a calculationally e ective tool we mix in our second ingredient which is the Mellin representation of CFT amplitudes. The position space amplitude A(u; v) has the Mellin representation A(u; v) = s)M(s; t) The additional kinematic factors in the measure are de ned for convenience [66]. M(s; t) is the (reduced) Mellin amplitude for the original conformal amplitude A(u; v). Mellin amplitudes are ideally suited to our present purpose since they share many of the features of momentum space for standard S-matrix amplitudes. In particular, the contributions of di erent operators show up as poles with a factorisation of the residues into lower point amplitudes. Moreover, our building blocks, the Witten exchange diagrams, are complicated in position space but are analytically easier to deal with in the Mellin representation. In fact, they can be viewed as the meromorphic piece of the conformal blocks in Mellin space, which are also known explicitly since the work of Mack. They therefore have the same poles as the corresponding conformal blocks together with the same residues thus exhibiting the needed factorisation. In fact, as mentioned above, the Witten blocks are better behaved in Mellin space compared to conformal blocks: the latter have exponential dependence on the Mellin variables compared to the former which are polynomially bounded. We can now translate the presence of spurious power law (as well as log) dependence in position space into Mellin space. The u ln u) behaviour arise from spurious poles (and double poles) at s = where s is the Mellin variable conjugate to the cross ratio u. Therefore, we now demand that these residues vanish identically, i.e. as a function of the other Mellin variable, t. This gives an in nite set of constraints on operator dimensions and OPE coe cients.5 Here another advantage of the Mellin representation makes its appearance. In analogy with partial wave expansions for momentum space scattering amplitudes, there is a natural decomposition of the residues into a sum over a basis of orthogonal polynomials in the t-variable. These polynomials (known as the continuous Hahn polynomials in the mathematics literature) go over to the generalised Legendre (or Gegenbauer) polynomials in an appropriate at space limit. This decomposition makes the imposition of our in nite set of conditions more tractable analytically. One big simpli cation is that operators of spin-` contribute (in the s-channel) only to the orthogonal 5As mentioned above, there are additional spurious powers which lead to subsidiary spurious poles (double as well as single) at s = + m (with m = 1; 2; : : :). polynomial of degree `, as one might expect in analogy with at space scattering. In the tchannel an in nite number of spins do contribute but this happens in a relatively controlled way. This makes it natural to impose the vanishing residue conditions independently for each partial wave `.6 This feature of the Mellin space approach to the conformal bootstrap makes it very close in spirit to the at space S-matrix bootstrap (see [74, 75] for one recently proposed way of connecting the two). It turns out to be simplest to implement this schema when there is a small parameter to expand in. We will focus here on two such examples. The rst is the canonical expansion in d = (4 ) dimensions for a single real scalar at the Wilson-Fisher xed point. The second is the large spin limit (in any dimension) in scalar theories with a twist gap. In the former case, we will see that, in the Mellin partial wave decomposition, there are some signi cant simpli cations when we impose the vanishing constraints. By assuming contributions we nd that the lowest couple of orders in get contribution only from the 2 primary exchange, in addition to the identity operator. This enables us to recover known results for the anomalous scaling dimensions ( 0, respectively) of both The anomalous dimensions of these operators are known upto O( 5) [76{78]. = 1 0 = 2 We also nd the OPE coe cient C 2 = C0 with a new result for the O( 2) piece C0 = 2 In fact, if we take the input from Feynman diagram calculations of the O( 3) contribution 0 then we can make a new prediction (5.35) for the corresponding O( 3) contribution to C0 as well. By moving onto the partial wave ` we again nd that, in the s-channel, it is @` ) that contribute to the rst two non-vanishing orders in . Once again, to this same order in the t-channel it is only the 2 (and identity) contribution that is needed. This enables one to obtain, in a fairly easy way, the nontrivial results for the anomalous dimensions of these operators ` = d (3) being given in (5.33). The O( 3) term matches with the nontrivial Feynman diagram computations of [79]. Our approach gives the OPE coe cients too with equal ease unlike other methods. We thus obtain C J` = C` to O( 3) as given in (5.36), (5.40) 6In the discussion section we will mention an alternative procedure for imposing the constraints which setting each of the terms to zero. This set of conditions is linearly related to the set of conditions from the the central charge cT which is related to C2 (by (5.42)). Our result = 1 agrees with previous calculations to O( 2) [80{82] and gives a new prediction at O( 3). As we describe in section 5.4.3 these results, after setting = 1, compare very well with some of the numerical results obtained for the 3d Ising model. The second context is of the large spin asymptotics (in a general dimension d, for Wilson-Fisher like xed points) we consider the two regimes that have been analysed in the literature through the (double) lightcone expansion. Our techniques here allow us to reproduce results in both the large and small twist gap regimes. Thus we reproduce the results of [83, 84] for the anomalous dimensions of the operators J ` in (6.15) and the leading corrections to the OPE coe cients in (6.17). In an opposite \weakly coupled" regime [43] we reproduce again the anomalous dimensions of the operators J ` in (6.23) together with a new determination of the coe cients in (6.24). The plan of the paper is as follows. In section 2 we discuss both Witten diagrams and the usual conformal blocks in Mellin space. We also discuss the spectral function representation of Witten diagrams that we employ in the rest of the paper. In section 3 we explain how to implement the bootstrap in Mellin space. In section 4 we turn to the identical scalar case which sets up the explicit -expansion calculation in section 5. Section 6 deals with large spin asymptotics both for strongly coupled theories and weakly coupled theories. We conclude in section 7 with a preliminary discussion on numerics and future directions. There are several appendices containing useful identities and intermediate results. Witten diagrams & conformal blocks in Mellin space In this section we begin the process of migrating to Mellin space by carrying over the familiar conformal blocks and the associated Witten exchange diagrams from position space. We will consider the somewhat more general case of arbitrary scalar external operators and de ne our amplitudes, setting notation in the process. Let A(x1; x2; x3; x4) denote the four point function of four scalar operators in a CFT (the scalar Oi has dimension Here we have pulled out overall factors in the four point function appropriate for an schannel decomposition and de ned The cross ratios (u; v) are de ned in the standard way as = u = 2) ; bs = x212x2324 ; v = i = , we recover the previous expression eq. (1.3). Note that we are making a particular choice here so that even when we consider t; u-channel exchange diagrams, we will still be using the convention (2.4) with the overall factors as in (2.1). Though we will not be utilising them very much, let us discuss how the conformal blocks look in Mellin space [66, 86]. Under the transform of eq. (2.4), the conformal blocks G(s;)`(u; v) ! B(s;)`(s; t): The Mellin space conformal blocks B(s;)`(s; t) take the form [86] t) (s + t + as) (s + t + bs)M(s; t) The corresponding Mellin amplitude reads as7 A(u; v) = f ig(s; t)M(s; t): = ei (h = ei 2 (2s 2s = 2h h;`(s) is de ned, for later use, by the equality between the rst and second lines. The Gamma functions in the numerator of h;`(s) exhibit poles at both 2s = d the spacetime dimension). Since we would like to project out the contribution of the shadow poles the prefactor in brackets was introduced in [86] so that it has zeroes precisely at these unphysical values. This cancellation of poles is made manifest in the third line. The projection, however, leads to an exponential dependence on s at large values of this Mellin variable. The crucial piece of the conformal blocks in Mellin space are the P (s) h;`(s; t) | the so-called Mack Polynomials which are of degree ` in the Mellin variables (s; t). In addition to the dependence on , they also depend on the external scalars through as; bs, but we suppress this dependence, so as not to clutter notation.8 We merely signal this dependence through the superscript which indicates that we are considering parameters relevant to an s-channel. The explicit form of these polynomials is given in appendix A.9 12; t = 4). See also [85]. 8Both the Mack Polynomials as well as h) and this is re ected in their subscript. 9Our normalisation of the Mack Polynomials agrees with that of Mack and di ers from that of [70] by a factor of ( have a parametric dependence which is naturally in the ` + 2m giving residues which are kinematic polynomials in the variable t determined by the spin ` of the intermediate state and the level m of the conformal descendants [70]. B(s;)`(s; t) = The dots refer to the entire function piece of the block in eq. (2.6). That is the part which has an exponential behaviour at in nity. The Q`;m(t) polynomials are single variable specialisations of the Mack Polynomials. following i.e. h;` s = Q`;0(t) = h;` s = The Q`;0(t) turn out to be a family of orthogonal polynomials (continuous Hahn Polynomials) whose properties are given in appendix B. These can be viewed as the generalisations of the Legendre/Gegenbauer polynomials that accompany the partial wave decomposition for scattering amplitudes. Just as for the conformal blocks, we can consider the Mellin version of the contribution from Witten exchange diagrams under the transform (2.4) W (s;)`(u; v) ! M (s;)`(s; t): Witten exchange diagrams in Mellin space have been investigated in the literature [67{70]. It is known that they have the same poles and residues as the corresponding conformal blocks. However, they are polynomially bounded for large (s; t), in contrast to the exponential dependence of conformal blocks B(s;)`(s; t). They therefore take the form M (s;)`(s; t) = + R` 1(s; t) where R` 1(s; t) is a polynomial10 of degree at most (` 1) in (s; t). Note that the rst term is identical to that in (2.7). The second term is an additional polynomial ambiguity coming from freedom in the choice of three point vertices in the bulk AdS in de ning the exchange diagram. The meromorphic piece is however xed to be the same as that of the conformal blocks (2.7). Since our interest is to use an appropriate basis, we will choose the ambiguity to our convenience. A particularly simple choice of basis would, for instance, be to only use the meromorphic piece of the conformal block i.e. just the rst term in (2.11). 10Even though we will not need the form of R` explicitly, we can x it by writing the Mellin amplitude as a meromorphic piece plus a regular piece. The Mellin amplitude (2.23), which is in terms of a spectral function, hence xes R` implicitly. We can write this sum (for any `) in terms of a nite sum of hypergeometric functions. Our choice will actually involve the additional polynomial piece R` 1(s; t) as well. In the case of a scalar exchange, however, such terms don't enter and the answer for the corresponding sum in (2.11) is particularly simple [67{69] M (s;)`=0(s; t) = ; 1 : (2.12) In forthcoming work [87] we will employ this direct method to explicitly write down the Witten exchange function. In the current paper, we use an alternative approach to writing the exchange diagram in terms of a spectral function representation. While this introduces some additional terminology, it will have some advantages for implementing our bootstrap philosophy. The spectral function representation Our starting point will be the spectral representation of the Witten exchange function in position space (in, say, the s-channel). Following (2.1) we de ne W (s;)`(xi) = The spectral representation is then a decomposition in terms of conformal partial waves (see for e.g. [88], section 6). This follows from a \split" representation of the bulk-to-bulk propagator in terms of two bulk-to-boundary propagators with a spectral parameter is integrated over. The latter can be expressed in terms of conformal partial waves W (s;)`(u; v) = The conformal partial waves F (;s`)(u; v) are closely related [70, 85] to the conformal blocks being just linear combinations of a block of ctitious dimension = h + and its shadow with dimension d = h (s;)`( ) = F (;s`)(u; v) = n( ; `) = + `)=2, 1 = (h + `)=2 and x;y = (x + y) (x The spectral function (s;)`( ) itself is the dynamical piece that contains information about the exchanged operator with dimension . We can further break it as conformal group, see for e.g. appendix B of [89])) `( ) = (h + `) (h + and a piece which is dynamical (i.e. (s;)`( )) and thus knows about . The explicit ex(s;)`( ) = ) ( 3+ 4 h+`+ ) ( 3+ 4 h+` : (2.19) contribute.11 The interpretation as a spectral function comes from the fact that we can evaluate the integral along the imaginary axis by closing the contour (when the integral is well behaved at in nity) on, say, the right half plane and picking up the residues at the simple poles. These then correspond to the primary operators which are exchanged whose contribution, along with their conformal descendants, is captured by the conformal block in (2.15). The contour is chosen to enclose either an operator or its shadow, but not both. The superscript in (s) (and (s)) signi es the channel and is re ected in the dependence on the i in (2.19). Let us see which primaries contribute. The spectral function in (2.19) has simple h corresponding to the operator ( ; `) (and its shadow). But it also has simple poles at h 2 + ` + 2n and h 4 + ` + 2m where n and m are non-negative integers. In a generic theory there are no operators of this literature. This is because in a weakly coupled (\generalised free eld") theory, like in the large N limit, these would be the dimensions of double trace operators of the schematic form O1@`(@2)nO2 (and similarly with O3 and O4). It is known [90, 91] that precisely these double trace primary operators (of spin `) do contribute to the Witten diagram (in the s-channel) and the spectral function merely reproduces this fact. Note that when we contour on the right half plane only the poles with the plus sign will (typically) The full spectral function (with the Plancherel measure) thus takes the form (s;)`( ) = ) ( 3+ 4 h+`+ ) ( 3+ 4 h+` We have seen that this has the right behaviour to reproduce the known properties of the Witten exchange diagram. There is one subtlety though that we should mention in this form of the spectral function. There are a nite number of extra poles in the integral from the denominator factors of (h 1)` whose contributions need to be cancelled by adding 11Note that this would imply that there are no poles coming from 1;bs in the integrand. For general as, there may be a nite set of poles on the right but an in nite set of poles on the left. Our choice of contour is such that the entire in nite chain is on one side of the contour. lower order spin terms (`0 < `). These have been explicitly studied in [88]. However, these additional terms will not contribute to the terms of interest to us which are the residues at the double trace poles which come purely from the above piece. agrees with what was constructed by Polyakov for what he called the `unitary' amplitude. Polyakov, of course, did not come to this from Witten exchange diagrams. He rst constructed a spectral function for what he called the `algebraic amplitude' which is nothing other than the conformal block itself. This turns out to have the form Al;g`( ) = This algebraic spectral function Al;g`( ) is designed to reproduce the conformal block on the l.h.s. if we insert it in the r.h.s. of (2.14) instead of (s;)`( ). We see that now the integral over the right half plane only gets contributions from the single pole at after using (2.15), (2.16). Note that Al;g`( ) in (2.21) di ers from (s) ( ) given in (2.20) by the four numerator -functions in the latter which were the double trace contributions. Al;g`( ) su ers from the problem that it diverges as one goes to i1. As we will see later, this is related to the poor behaviour of the conformal block in Mellin space, at in nity along the imaginary axis. To cure this problem, Polyakov prescribes adding certain additional factors of -functions to the numerator. These turn out to be precisely (in his case, for identical scalars) the double trace ones which appear in the numerator of (2.20) so that we indeed get the spectral function appropriate to the Witten exchange diagram! We can now translate this spectral representation to Mellin space. We use the fact that the partial wave appearing in (2.14) has the Mellin representation [85] F (;s`)(u; v) = in terms of the Mack Polynomials as well as the Mellin measure f ig(s; t) in (2.4). channel Witten exchange diagram in Mellin space to be Then, combining (2.22) with (2.14) we have the spectral representation for the s(s;`)(s) de ned in (2.6) and the standard M (s;)`(s; t) = Adding in the t; u channels Our discussion was for the s-channel contribution W (s;)`(u; v) to (1.1) and the corresponding M (s;)`(s; t) in Mellin space. It is not di cult to extend the discussion to the other two channels. The main point to keep in mind is that our conventions are chosen, for de niteness, for an s-channel expansion. Thus we pull out the same external factor as in (2.1) when we are considering the reduced amplitude in the t and u-channels also. This is even though the natural de nition for the Witten diagram in the t-channel would involve an interchange of subscripts (2; 4) (and similarly (2; 3) for the u-channel) of the s-channel which gives, for instance, answer (2.13) W (t;)`(xi) = where at = 4) and bt = 12 2). If we recast this in the form (2.1) by pulling out the same external factor as in that equation, then this corresponds to multiplying W (t;)`(u; v) by an extra factor of u 2 ( 3+ 4)v 21 ( 2+ 3). In a similar way, an extra factor of 1 1 u 2 ( 1+ 4) multiplies W (u;`)(u; v). Here, both W (t;;`u)(u; v) are obtained from W (s;)`(u; v) by the interchange of labels (2; 4) (and (2; 3), respectively). We can translate this to Mellin space in a straightforward manner. Thus the t-channel partial wave (the analogue of (2.22) reads, with the above prefactor, as 1 1 = u 2 ( 3+ 4)v 2 ( 2+ 3) (at + s + t) (bt + s + t) Here the integrand on the r.h.s. is obtained from the corresponding one of the schannel (2.22), with the interchange (s $ t) i.e. of labels (2; 4). The superscript t on the Here `0 denotes the spin in the t-channel. But now observe that by shifting variables we can make the r.h.s. of (2.25) now in the same form as (2.22) i.e. Similarly, in the u-channel, we have (t;)`0 (t) = (u;`)0 (s + t) = `0) +s+t (bs + s + t) (as + s + t) 3) and bu = 12 ( 2 : (2.28) This was for the partial waves in the t; u-channels. We can now employ the corresponding versions of (2.14) to write the expressions for the corresponding Witten exchange diagrams in the spectral representation in Mellin space i.e. the counterparts of (2.23). Combining (2.25) with the analogue of (2.14), we nd M (t;)`0 (s; t) = M (u;)`0 (s; t) = And similarly in the u-channel with (2.29) Here the spectral weights, (t);`0 ( ); (u) subscripts (2 $ 4) and (2 $ 3) respectively. The bootstrap strategy implemented ;`0 ( ) are given by (2.20) with the exchange of With all this machinery in place, we are now ready to come to the crux of our strategy. As mentioned in the introduction, we write the four point function as a sum over a set of crossing symmetric Witten exchange diagrams as in (1.1). In position space this can be written, using the spectral representation (2.14), as A(u; v) = X Z i1 1 1 + c(t);` (t);`( )u 2 ( 3+ 4)v 2 ( 2+ 3)F (;t`)(u; v) 1 + c(u;)` (u;)`( )u 2 ( 1+ 4)F (;u`)(u; v) : Here the sum over ; ` is over the entire physical (primary) operator spectrum of the CFT. Note that we have, in general, to-be-determined coe cients c(s;;`t;u) which are mutually related by exchanges of the labels (e.g. (2 $ 4) or (2 $ 3)). This ensures that the full amplitude is crossing symmetric. Since we are not making an expansion of the amplitude in terms of conformal blocks in a xed channel, we are not guaranteed that this expansion will have the right power law dependences on the positions (or equivalently, cross-ratios) that is consistent with the OPE. For instance, in the case of identical scalars we see from (4.3) that the spectral func(s) ( ) has double poles (at h + is the dimension of the common external scalar). When we perform the integral, this double pole gives rise to u terms in the sum, as well as u terms. Both of these dependences would imply the presence of an operator with dimension 2 in the spectrum which is generically not the case. More generally, we will have spurious power laws of the form u 1+2 2 and u 3+2 4 when we expand (3.1) in the s-channel. There are generically no operators corresponding to dimensions ( 1 + 4). Thus we have to demand that these terms identically vanish after including the contributions from the other channels and on summation over ( ; `). As discussed, it will be easier to implement this in Mellin space. In other words, we look at the total Mellin space amplitude corresponding to (3.1) which we obtain by putting together (2.23), (2.31) and (2.32) M(s; t) = c(s;)`M (s;)`(s; t) + c(t);`M (t;)`(s; t) + c(u;)`M (u;)`(s; t) c(s;)` (s;)`( ) (s;`)(s)P (;s`)(s; t) + c(u;)` (u;)`( ) (u;`)(s + t)P (;u`) s and s = 12 The de nition of the Mellin transform in (2.4) imply that the spurious powers in 4). When the external scalars are identical, these two sets of spurious . It is important to note that these are statements about the full Mellin space amplitude and not just the reduced one, M(s; t). In other words, recalling the notation of (2.4) we need to examine the spurious f ig(s; t)M(s; t). In particular, for identical scalars, the . So we will need to look at the s) piece of constant as well as terms linear in (s ) of M(s; t) to isolate the poles of interest to us. In either case, the residues at these spurious poles will be a function of t and we will obtain an in nite number of constraints on our CFT by setting these identically to zero. Below, we will individually look at the Mellin amplitudes in each channel, for non-identical scalars, and isolate the residues. We can then add them all up and nd the conditions for consistency with the OPE. In the following section we will examine the special features that arise for identical scalars. The s-channel We start with the unitary block in the s-channel (i.e. the Mellin transform of the Witten exchange diagram) given in eq. (2.23). where, as in (2.20), we have the spectral function M (s;)`(s; t) = (s;)`( ) = (s;`)(s) = 2 = (h + `)=2; 2 = (h Here we have introduced, for compactness, the notation [85]: We are to carry out the integral by closing the contour on the right half plane. The \physical pole" in the spectral function is the one at = ( h).12 In this case the factors of cancel out with the corresponding factors in s) in the denominator of f ig(s; t) of eq. (2.4).13 The residue at this physical pole in has factors of -functions from the numerator of (3.5) which give rise to the physical pole in the s-variable (as well as for the shadow) i.e. 2s = 2s = (2h ` + 2n; (n = 0; 1; 2 : : :) : When we do the s integral (again closing the contour appropriately) of the Mellin amplitude, this gives rise to the physical contribution with a u `)=2+n dependence but does not pick up the shadow. spectral function (3.4) However, there are other poles in which give rise to the spurious poles in s that we described earlier. For instance, when we consider the poles from the numerator of the the residue will get a contribution from the numerator of (3.5) / The denominator of (3.5) then cancels with the Mellin measure but the above piece gives at s = 12 spurious poles at s = 21 2) + n. By a similar argument there are spurious poles also 4) + n. Instead, if we had taken the poles from the numerator of (3.5) i.e. = 2s then the residue contribution from the numerator of (3.4) would be again / s). Thus the residues of the Mellin amplitude evaluated on these second set of poles gives a factor of two to the previous contribution.15 We have already discussed (see around (2.20) and footnote 10) the absence of any role from all the other poles of the spectral function. 2. The two variable . We use the general relation (2.9) to obtain = 4 `( 1 + 12We are assuming 0. If not, we have to deform the contour so that we include this pole but not that of the shadow operator which would now lie on the right half plane [66]. 13For the scalar, this can be explicitly seen in the denominator factors in (2.11) which cancel against f14iAg(ss;wte)llaatsthaeshpahdyoswicaplipecoele i(nhs. ` 1 2 s) which will always be understood to be present but which we will ignore since we will choose to close the Mellin contour so as to exclude this set of poles. 15The poles at s = 12 2) + n with (n > 0) actually come with a multiplicity. But since we will be As a result the nett residue of the Mellin amplitude at the unphysical pole is M (s)(s; t) s= 12 ( 1+ 2) Here we have de ned s) s= 12 ( 1+ 2) q(s;)` = denotes contribution from the physical pole as well as the other spurious pole at s = 12 ( 3 + 4) which gets an identical contribution to above with (1; 2) replaced by (3; 4). The case of identical scalars will be discussed separately in the next section. In the next subsections we look at the t; u-channels and similar pole contributions 1 that lead to anomalous u 2 ( 1+ 2) behaviour in the amplitude. Demanding that these cancel against the above s-channel contribution will give us constraints. Note that the cancellation conditions involve a whole function of t. To facilitate the comparison, we have expanded the functional dependence on t in terms of the orthogonal polynomials Q`;01+ 2+`(t). We will expand the amplitudes in the other channels in terms of the same polynomials and use the orthogonality to set the nett coe cients of each Q`;01+ 2+`(t) to zero.16 We also note another important utility of this particular orthogonal decomposition | a spin ` Witten diagram contributes only to the orthogonal polynomial labelled by the same `. This property makes this decomposition the analogue of the usual partial wave decompositions for at space scattering amplitudes. The contributions from a speci c spin is, however, a special feature of the s-channel decomposition and will not hold in the t-channel (see (3.16)). The t-channel The unitary block in the t-channel is given by (2.31) M (t;)`0 (s; t) = polynomials Q`;03+ 4+`(t). (t);`( ) = dimension. This is mostly negative except for comes positive. 1:733) where this ips sign and bepn iS4 i42 -pn in2 pS function of the dimension of the exchanged operator dimensions ( ). All of these plots are mostly positive except for near the unitarity bound given by d 2 + `. The sign ip near the unitarity bound is given by the inset plots. This will demonstrate how one may hope to see a bound arising from these numerics. 0:518 which is the value for the 3d-Ising model. As is clear from the spin block plots, close to the unitarity bound ( the spin blocks are negative but positive elsewhere. The scalar block on the other hand is positive only for a small region of 1:733) (see In the rest of the range of 0, the scalar block is negative. If we assume that the non-zero spin operators, which contribute most to the constraints, are close to the unitarity bound (as happens for the 3d Ising model), then their contribution to the constraint equation will be negative. So the only way to satisfy the constraint equation would be if the scalar block were to give a positive contribution. This gives 1:733) which is indeed the case for the 2 operator. Note that we used just one constraint for illustrative purposes to demonstrate that investigating numerics along these directions ought to be a promising future endeavour. Of course, one needs to demonstrate, since there is an in nite sum over the spectrum the resulting numerics converge. Our preliminary investigations, of theories living at the border of the known allowed regions, using presently available numerical methods, does suggest that the approach above will lead to convergent numerics.27 A more thorough investigation of this issue should be carried out. Higher orders in : we have had striking success in using our approach to obtain results to O( 3) | therefore it is natural to ask how to go to the next order in the -expansion. Indeed one would like to know if there is a systematic approach that allows one to go to any arbitrary order in the expansion, if one so desired.28 Once we set up the formalism obtaining the O( 3) results was conceptually and mathematically straightforward (though, perhaps a bit tedious) and needed very few and rather mild assumptions. The two main inputs were the existence of a conserved stress tensor and the leading behaviour of OPE coe cients for higher order operators which we know from the perturbation expansion | for instance C 2 begins at O( 2) since it is the square of the OPE coe cient which is assumed to be O( ). We had pointed out in section 5, that at O( 4), the constraint equations at the spurious pole (s = ) involved an in nite number of operators. However, it is plausible that, by appropriately combining the enormous amount of information in the additional constraints at s = + n, one can give an algorithm to continue to higher orders in a controlled manner. Furthermore, we have only investigated the case of identical scalars. Thus we may need to combine the information from other spurious poles with correlators of other scalar operators. At present, our approach yielded information about operators which were bilinear in the elementary scalar . It is possible to extend our results to operators with higher powers of [87], for which some information is known in the -expansion [101{103]. At some stage we expect the non-unitary behaviour in 4 to show up for some large dimension operator [104]. Our approach, however, did not crucially rely on unitarity as the non-unitary example showed and should be able to capture this behaviour in 4 dimensions. It will also be interesting to use our approach to study the theories in higher dimensions considered in [105{107]. Other small expansion parameters: we have studied our equations with two small expansion parameters, namely and 1` , and found quite remarkable simpli cations. It will therefore be interesting to investigate our equations when we have other small expansion parameters. These could be, for example, large dimensions for external operators, large spacetime dimension d limit, strong or weak coupling limits and of course, large N . In these cases one might hope to have similar simpli cations which organise the bootstrap conditions so that there is a controlled way of incorporating the contributions from di erent families of operators. Recently in [108], a systematic procedure has been outlined to solve the conventional boostrap equations in the large spin limit using \twist blocks". These twist blocks resum the contribution of 27We thank Slava Rychkov for suggesting this check. 28Note that the -expansion, like any perturbative QFT expansion, is only an asymptotic one and needs to be Pade-resummed to obtain something useful. The question is whether there is an in-principle systematic method to obtain the n-th term in this expansion using our Mellin space approach. all operators of degenerate twist and di erent spins and appear to be a useful way to compute the anomalous dimension of large spin operators with arbitrary twists. One could attempt a similar procedure to approximate the Mack polynomials in this limit and set up the analogous equations in Mellin space. Technical hurdles: in order to use our constraint equations systematically, one bottleneck is the integration over the spectral parameter in the crossed channel and another is the lack of a compact expression for the Mack polynomials. In the way we have currently set-up the equations there were coincidences which led to remarkable (almost) cancellations between various -poles. This fact enabled us to go to higher than what one may have naively expected from [12, 64]. In fact, as we saw in our calculation, in the crossed channel only the 2 operator contributed to yield the nontrivial results at O( 3). In [87] we will show how to get rid of the integration over the spectral parameter leading to an enormous simpli cation in the form of the equations. This should enable a systematic investigation of a plethora of questions, some of which have been indicated above. It will also be desirable to have a better understanding of the Mack polynomials to see if more compact representations for them exist, compared to the present one. One could also perhaps try to see if there is a geometric way (in AdSd+1) of understanding our consistency conditions.29 This could lead to a new way of doing quantum eld theory for critical phenomena which uses a di erent set of diagrams rather than Feynman diagrams to systematize general perturbative expansions. Connection with AdS/CFT: our building blocks are Witten diagrams in Mellin space. This suggests the tantalizing possibility of AdS/CFT playing an important role to understand the Wilson-Fisher xed point. Of course, any such dual string theory is likely to be in the quantum regime. In a companion paper [93], the present method is extended to O(N ) both in the -expansion as well as in the large-N limit showing that it works analogously, yielding the rst few subleading orders. A systematic study of our constraints in the large-N limit will be an important question to investigate in the future in order to explicate the connection with a weakly coupled string theory/Vasiliev theory. We do not have further insights to o er at this stage, but clearly it will be fascinating to unearth a direct connection between string theory and the 3d Ising model. Acknowledgments Special thanks to J. Penedones for collaboration during the initial stages of this work and for discussions. We acknowledge useful discussions with S. Giombi, T. Hartman, J. Kaplan, I. Klebanov, G. Mandal, S. Minwalla, D. Poland, S. Pufu, Z. Komargodski, J. Maldacena, H. Osborn, E. Perlmutter, L. Rastelli, M. Serone, S. Wadia and especially S. Rychkov. We also thank all our other colleagues at IISc, ICTS and TIFR-Mumbai for numerous discussions and encouragement during various stages of this work. R.G. acknowledges the 29There is a geometric way of understanding the conventional conformal blocks in terms of geodesic Witten diagrams [109]. support of the J. C. Bose fellowship of the DST. A.S. acknowledges support from a DST Swarnajayanti Fellowship Award DST/SJF/PSA-01/2013-14. This work would not have been possible without the unstinting support for the basic sciences by the people of India. The Mack polynomial The Mack polynomials P (;s`)(s; t) are explicitly known [66, 70, 85], albeit in terms of a multiple sum Here we employ the notation in [85] 1 = 1 = 2 = 2 = 1;bs are de ned in (2.17) with as; bs de ned in (2.2) and the li-s are given by, l1 = 2 l3 = 2 + bs + k + m ; l2 = 2 + as + k + m l4 = 2 The continuous Hahn polynomials In (2.9) we specialised the two variable Mack polynomials de ned in appendix A to obtain polynomials of degree ` in one variable. Q`;+0`(t) = P (+s)` h;` s = Note that we have suppressed the dependence on the parameters as; bs These polynomials have a number of remarkable properties which enable us to simplify the bootstrap conditions. The rst is that the multiple sum that de nes the Mack polynomials in (A.1) collapses into a simple single sum which is, in fact, a familiar 3F2 hypergeometric function. Q`;+0`(t) = Here and in the rest of this appendix, for generality, we replace the s-channel parameters (as; bs) by arbitrary parameters (a; b). The second remarkable fact is that the Q`;0 polynomials are orthogonal and known in the mathematics literature as the continuous Hahn polynomials [110]. They obey the orthonormality condition, b)Q`;+0`(t)Q`0+;0`0(t) = `( =2) `;`0 ; `( =2) = `+ =2;a `+ =2;b We note another useful property of the Q`;0 polynomials (in the special case of identical scalars) which follows from properties of the hypergeometric function 3F2. The transformation n; k1; k2; 1 = factor (k3 obtain the relation If we take n = `, k1 = + ` 1, k2 = b + 2 + t, k3 = 2 a and k4 = 2 + b, we see that this Q`;+0`(t) = ( 1)`Q`;+0` Now let us use this to show that for identical scalars, and for an even spin ` exchange in the s-channel, we have the t-channel expansion coe cient in (3.18) equal to that in the u-channel (3.23). In the t-channel, q`(t)(s) for identical scalars is given by (4.7), q(t);`j`0( =2) = `( =2) 1 while in the u-channel q(u;)`j`0(s) is given by, q(u;)`j`0( =2) = `( =2) 1 2( =2 + t) ( 2 (`t0)( )P (;t`)0( =2 ( t) ( =2 + t + 2 ) ( =2 + t + 2 (`u0)( )P (;u`)0( =2 Now if we use the identity (B.6), the above two expressions become equal under the exchange t $ t. Hence one can conclude in general for identical scalars and even spin exchange in s-channel, that the t-channel is equal to the u-channel. Finally the more general polynomials Q`;+m`(t) , Q`;+m`(t) = P +` h;` s = appear at the descendant poles. However, analogues of the above nice properties of Q`;0(t) are not known for the polynomials with m > 0. We have been expanding the amplitude in terms of Witten diagram blocks as in (1.1) with to-be determined coe cients c ;`. These c ;` are proportional to the (square of the) OPE coe cients C ;` which appear in the conventional conformal block expansion of (1.2). We need to x the relative normalisation between the two if we are to be able to compute the OPE coe cients C ;`. We will do so in this appendix. This is simplest to do in position space. The conformal blocks G ;`(u; v) are normalised Thus we need to ensure that as (u ! 0; v ! 1) = C ;`u 2 (1 By explicitly evaluating the l.h.s. we will be able to obtain the relative normalisations. We will actually carry this out for non-identical scalars for generality. The Witten diagram in position space has the Mellin representation is obtained from the general expression (2.4) by plugging in M (s)(s; t) as the reduced mellin amplitude. The latter is given in the spectral representation by (2.23). In the integral over the spectral parameter, we now focus on the contribution from the physical pole at h. This gives c ;`W (s;)`(u; v) = c ;` bs) (s + t + as) (s + t + bs) ) ( 1 + 2h + ` ( 2h + ` + ( 2h + `+ Thus we will multiply the above expression by a factor of two. To proceed we expand vt in 1 v. This will give, contribution to the spectral integral from a = 2h ` s. Evaluating the residue 2 ` . Actually, there there is another ` which gives an identical contribution. ( 2h + ` + ( 2h+`+ Here we have used the relation + t + bs Q`;0(t; as; bs) : h;` s = = 2 2`( 1)`Q`;0(t) : The binomial expansion coe cient ( 1)m t = ( 1)m O(tm 1) is a polynomial in t of degree m. Therefore we can rewrite it as a sum over the orthogonal polynomials Q`0;0(t) for 0 Q`;0(t) = 2`t` + O(t` 1) we must have ( 1)m t m m. Since Q`;0 is normalized such that Qm;0(t) + . Using the orthonormality of the Q`;0 polynomials as given in appendix B, we can evaluate the contribution to (C.4) which goes as (1 expansion. Doing the t integral gives the contribution c ;`W (s;)`(u; v) = u 2 ` (1 ( 2h + ` + The omitted terms denote higher powers of u and 1 N 1;` = ( 2h + ` + t integrals in the crossed channels This section will deal with performing the t integrals in (4.7). We will carry out the integrals for generic `; `0. The technique can be generalised to the non-identical scalar cases of (3.18) and (3.23) but we will not do so here since we only need the identical scalar results. Let Q`;+0` = 2` 2( =2 + `) ( + ` 2( =2) ( + 2` 1; =2 + t Now the hypergeometric function can be written as a sum as below, 1; =2 + t ; 1 = q(t);`j`0( =2) = `( =2) 1 (t);`0( )P (;t`)( =2 2`(( =2)`)2 ( `)q( + ` (( =2)q)2 q! In the second equality we have used the explicit forms of the Mack Polynomials in (A.1) that, we will use the integral representation of the 2F1 function, which is given by, With this let us evaluate the t-channel q(t);` as given in (3.18), Shifting the variables t ! t s and using the mapping, a = 2 + +k ; b = 2 + +k ; c = 2k + + + + 2 + 2 ; (D.7) denotes that we have shifted the contour along the real axis so that all the poles of the same sign lie on the same side of the contour. We can further use, z) =2 t = X above, by picking up the power of zq. So we have, The idea is to use the above as a generating funtion for the part 2 + t q in the t-integral z) =2 t : 2F1(a; b; c; z) = 2 i (a) (b) (c ds (s) (c b + s) (a s) (b s)(1 = (1 z) 2k + + + 2 + 2 + 2 Collecting the powers of zq, we nally have, q + + 2k + + + 2 + 2 2 + k)(1 z) =2 t Using this in (D.3) we get, q(t);`j`0( =2) 2`(( =2)`)2 ( `)q( + ` 1)q (( =2)q)2 q! The above expression also gives q(u;)`j`0 in the u-channel. = 0. Also q + + 2k + + + 2 + 2 2 The hypergeometric 3F2 simply reduces to 1. Also note that q(t);`j`0=0( =2) = 2`(( =2)`)2 `( =2)( + ` (( =2)q)2 q! = (h + )=2 and 1 = 2 = 2`(( =2)`)2 ( `)q( +` 1)q (( =2)q)2 q! ` 2 X ( 1)q2 ` (2` + 1) (` + 1 + q) q=0 q! (` + 1 q) (` + 1)3 (2 + q) 1;0 1;0=Qi (li) is 1. So q sums in the t channel In this section, we will demonstrate, how the expression in (4.8) can lead to simple expressions, in an expansion. Here we will show this for the leading term in (4.9). Let us write down the full expression, given by, h)2)`!q! (` + ( ) ( ) ( + q) ( + q) = h+ = h 2 . The expression can be expanded in . The integral can be carried out by evaluating residues at only the poles = 2 discussed in section 5.3 and appendix F, the other poles are subleading. The leading term is given by, 10 + 16` + (1+2`) 45 E + 18H2` + 18Hq 18H`+q + 36 (` + 1) where, note that the problematic terms are Hq and H`+q. Using the identity, we can pull out the k sum and rst perform the q sum over these term. After the q sum, we can perform the sum over k to get, q=0 q! (` + 1 q) (` + 1)3 (2 + q) (1 + 2`)(18Hq 18H`+q) = 9 22+` (3=2 + `) The remaining terms can be handled with the usual sum over q to obtain, q=0 q! (` + 1 q) (` + 1)3 (2 + q) 10 + 16` + (1 + 2`) 45 + 18H2` + 36 (` + 1) 2 `(1 + 2`)(2`)! `(1 + `)(`!)3 Adding these two separate contributions, we nd that, q(2;;`tj)0 = 2 X ( 1)q2 ` (2` + 1) (` + 1 + q) q=0 q! (` + 1 q) (` + 1)3 (2 + q) 10+16`+(1+2`) 45 E + 18H2` +18Hq 18H`+q + 36 (` + 1) indicate subleading terms in . One must be careful while handling the above expression, since with the normalization inside c ;` that multiplies this, the whole thing starts from O( 2). Simpli cations for the dimensions rested on several simplications, which occur when we Taylor expand our equations in . In this appendix we will address all of them. Recall that the s-channel has P c ;`q(s;)`, which is a sum over all operators of spin `. The rst simpli cation here is that only the lowest dimension operator of spin ` contributes to the sum to the order we consider. This is the operator with dimension ` = 2+` For the 4 theory, we are considering, there are higher dimension operators with ` + 2 + 2m + m + O( 2). These operators have the generic form O2m;` (@2)m@` . Using 2m;` = the equation of motion, O2m;` b @c , where among the a + b + c (= 2m derivatives, 2m 2 derivatives are contracted and ` derivatives carry indices. We will show that these operators are suppressed in an expansion. We will demonstrate this only for the q(2;;`s) term. The q(1;;`s) follows a similar logic. Using from (4.6) for 4(1 + m + `) Here C2m;` is the OPE coe cient of this operator. Now these operators do not exist in the free theory. Since Om;` has four -s it is easy to guess that the 3-point function starts hence C2m;` ). The ope coe cient C2m;` goes as square of this quantity and O( 2). Accounting for this, we must have, 2m;`;` = O( 4) : There can be other \heavier" operators too, with spin `, also contributing to the sum (s) . However, such operators are composites with a higher number of -s, for . Now such operators will have the OPE coe cient C ;` which are even further suppressed in . So the corresponding q(s;)` will begin from O( 6) or beyond. Because of this, when we considered only up to the O( 3) term, keeping only the ` operator had su ced. 2m;0;0 = ( 1)2m+1 ( m + 1) 2 2(2 + 2m) 2 4(1 + m) Then, again since C2m;0 only go to O( 2) order, by keeping just one operator. O( 2), we have c2m;0q(s) Crossed channels In the crossed channels, we have P we had considered only a certain set of poles and neglected others under the assumption that the other set of poles would contribute at a higher order in the expansion. Here we will put these assumptions on solid grounds. We will rst go through the case when the exchanged operator is a spin 0 particle (scalar) and then move on to substantiate the same arguments for the case of spin `0 exchanged operators. In the subsection below, we will rst demonstrate the case of the scalar exchange (for identical external scalars, which ;`0 c ;`q(t);`j`0 , which is a sum over all operators in the O( 3). Thus, for ` = 0 we could is the case of interest). We begin by explicitly writing down the spin 0 contribution for the t channel (since the crossed channels give identical contributions for the identical scalar case, we can just consider either one of them) in (D.10), 2 `c ;0 ( ` + q) (2` + 2s) (2s + ` + q h)2)`!q! ( `) (` + s)2 (q + s)2 (2s + ` 1) ( ) ( ) + q + s) ( We will show that the leading dependence comes from only the 2 operator and also considering only two poles in the spectral integral is su cient. This happens because of cancellations. Let us introduce the notation, a nontrivial cancellation among residues of poles for every operator. We will demonstrate this with the log term, which is q(2;;`tj)0. The power law term which is q(1;;`tj)0 will have similar f (q; s) = 1)q (q + s `!q! (` + s)2 (q + s)2 (h + q + 2(s The poles of the above integral, lying on the positive real part of the - contour, c ;0q(t);`j0(s) = Note that for n2 q the poles III become double poles. = 2 1. Lowest dimension scalar: consider the operator 2 with dimension 0 = 2 + (1) + O( 2). This is the only operator that contributes in the crossed channels up to the O( 3) order. In our computation we used only the poles h. Let us see why the other poles do not contribute. Let us assume n1 < q and n2 < q + 1. Then residues at the poles II and II give (where we have used h = 2 =2 and the familiar dimension of , = 1 =2 + (2) 2), Res =2 h+2n1 = Res =h 2 +2s+2n2 = C2;0( 1) n1+1( 0(1))2(1 + 0(1))2f2n1(q; 1) (q C2;0( 1) n2( 0(1))2(1 + 0(1))2f2+2n2(q; 1) (q Here C2;0 is the OPE coe cient of 2 . The subleading terms in are cumbresome quite nicely this cancellation is till the O( 3) order. Hence we have, Res =2 h+2n + Res =h 2 +2s+2n 2 = O( 4) ; with n being a positive integer. This cancellation is true for any function f (q; s). When the poles III become a double pole for n2 q, the individual residues start from a di rent order, viz Res =2 Res =h 2 the cancellation (F.8) till O( 3) would still hold. O( 1). However 2. Heavier scalars: now let us look at heavier scalars with dimensions of the form 2m;0 = 2 + 2m + + O( ). Here m is a positive integer. It was argued in section F.1 that OPE coe cients of such operators begin from C2m;0 With this taken into account one makes the following observations: O( 2) at least. Res = h + Res =2 h+2m + Res =h 2 Res =2 h+n + Res =h 2 Res =2 h = O( 5) +2s+2m 2 = O( 4) +2s+2n 2 = O( 4) n 6= m : Hence we see, none of the higher dimensional scalars can contribute to the crossed Spin-`0 > 0 exchange law term c ;`0 q (1;;`sj)`0 we can simply put s = For spin `0 operators too, there are analogous cancellations just like what we have seen above. Let us take the general expression (D.10). To avoid tedious expressions, we will not give the explicit forms of the individual residues. We will just indicate the pairs of poles that cancel each other under -expansion. For both the log term c ;`0 q , which will give us the following poles, h) | from the denominator of the spectral weight (t);`0 ( ) h + 2n) | from the numerator factors of the spectral weight = (h + `0 + 2n) | from the other factors in numerator = (h 2 + `0) | from the denominator Pochhammer terms in the spectral weight . operators, with dimensions following cancellations, 1. Lowest dimension spin `0: the lowest dimension operators with spin `0 are the J`0 `0 = `0 + 2 + O( 2) . For these operators we nd the Res = h + Res =2 +`0 h + Res =h 2+`0 = O( 4) Res =2 h+`0+2n+2 + Res =h+`0+2n = O( 4) where n = 0; 1; 2; Res =h 1 Res =h 2 Res =h+`0 3 = O( 4) : 2. Higher dimensional spin `0 operators: there are heavier operators with spin `0 as we discussed in F.1. These operators, labelled O2m;`0 have the dimensions 2m;` = ` + 2 + 2m + m + O( 2) . We had argued that since these operators must have the composition coe cients go like C2m;`0 operators, we have the cancellations, c , or a higher number of -s, their OPE O( 2) or higher. Taking this into account for these Res =2 +`0 h + Res =h 2+`0 = O( 4) Res = h + Res =2 h+`0+2m + Res =h+`0+2m 2 = O( 4) Res =2 Res =h 1 Res =h 2 Res =h+`0 3 = O( 4) : Thus we conclude none of the operators in the crossed channels contribute except the 2 operator for which only two poles are su cient up to the O( 3) order. Large ` behavior of Q ;` In this appendix we will derive the large ` approximation for Q`;0. Let us start by deriving an approximation for the 3F2 hypergeometric function. It has the integral representation, n; k1; k2; 1 = Now as n ! 1, we have [111], 2F1( n; k2; k3; z) z) k1+k4 12F1( n; k2; k3; z)dz : (G.1) (k2)(nz) k2 + (k3 k2)( nz)k2 k3(1 z)k3 k2+n : For identical external scalars, the Q`;0 has a 3F2 which is equal to the above under the map So we get, n; k1; k2; 1 = (k1) ( k2 + k3) ( k2 + k4) Finally using this in (B.2) and putting the values of k1;2;3;4, we get the large ` approxima1, k2 = s + t, k3 = 2 and k4 = 2 . Since n second term in the parentheses above. Then (G.1) becomes, n; k1; k2; 1 = (k3) (k4) 1 Z 1 Carrying out the z integral it gives, (k3) (k4) k3 k2 we can neglect the (k3) (k4) (k1 Comparision with numerical results In this appendix we compare the results obtained in section 5.2 with numerical data found in [112]. To compare, we have taken the OPE coe cients (5.36) and obtained their square roots. Then we re-expand in to O( 3) to obtain f J` , and put = 1. As expected the match worsens as we go higher in spin. 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Rajesh Gopakumar, Apratim Kaviraj, Kallol Sen, Aninda Sinha. A Mellin space approach to the conformal bootstrap, Journal of High Energy Physics, 2017, 27, DOI: 10.1007/JHEP05(2017)027