Evaluation of conformal integrals
EEvaluation of conformal integrals
Adam Bzowski, a Paul McFadden b and Kostas Skenderis. c a Institute for Theoretical Physics, K.U. Leuven, Belgium. b Theoretical Physics Group, Blackett Laboratory, Imperial College, London, UK. c STAG research centre and Mathematical Sciences, University of Southampton, UK.
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We present a comprehensive method for the evaluation of a vast class ofintegrals representing 3-point functions of conformal field theories in momentum space.The method leads to analytic, closed-form expressions for all scalar and tensorial 3-pointfunctions of operators with integer dimensions in any spacetime dimension. In particular,this encompasses all 3-point functions of the stress tensor, conserved currents and marginalscalar operators. a r X i v : . [ h e p - t h ] J un ontents K integrals and their divergences 3 n <
0) 173.3.2 Linearly divergent integrals ( n ≥ n <
0) 183.3.3 Integrals with n = 0 183.3.4 Integrals with n > d = 4 20 I ν +1 { ννν } K and momentum-space integrals 29C Half-integral betas 29D Derivation of scheme-changing formula 30 – 1 – Introduction
Conformal invariance imposes strong constraints on the form of correlation functions in anyfield theory. In particular, 2- and 3-point functions of the stress tensor, conserved currentsand scalar operators are uniquely fixed up to a few constants. The standard approach,developed over several decades [1–4], proceeds in position space and leads to position-space expressions. Recently, however, a pressing need for closed-form momentum-space expressions for correlators has arisen in various applications including cosmology [5–17],the analysis of general properties of CFTs [18–22], and condensed matter physics [23, 24].In the papers [25, 26] we initiated a comprehensive study of momentum-space 3-pointfunctions in any conformal field theory. In [25], we expressed all 3-point functions involvingthe stress tensor, conserved currents and scalar operators of arbitrary dimension in terms ofa class of scalar integrals we call triple- K integrals . For special combinations of the operatorand spacetime dimensions, these triple- K integrals contain divergences necessitating theirregularisation and renormalisation. (For this same reason one cannot generally obtain themomentum-space correlators via a straightforward Fourier transform.) In [26], we presenteda complete classification of the divergences and their renormalisation for purely scalar 3-point functions, and in [27, 28] we discuss the corresponding renormalisation procedure fortensorial 3-point functions.Having expressed all 3-point functions in terms of triple- K integrals, the next step is tocalculate them. In this paper we present a complete method for systematically computingall the triple- K integrals that arise in the evaluation of scalar and tensorial 3-point functionsof operators with integer dimensions . This broad class includes many operators of physicalinterest such as the stress tensor, conserved currents, and marginal scalar operators.In cases where the spacetime dimension is odd, all triple- K integrals can be evaluatedtrivially in terms of elementary functions. We therefore focus our attention on cases wherethe spacetime dimension is even. For purely scalar correlators, the only non-trivial newcases to be analysed are those for which both the following conditions hold:(i) The spacetime dimension d ≥ j ∈ Z are integers satisfying ∆ j ≥ d/ + ∆ > ∆ , ∆ + ∆ > ∆ , ∆ + ∆ > ∆ . (1.1)The first condition arises through a special symmetry property of the triple- K integralwhich allows us to relate cases with ∆ j < d/ j > d/
2. As for the secondcondition, if any of the triangle inequalities are violated then we have no need to evaluatethe triple- K integral: in such cases the triple- K integral diverges and the renormalised3-point function is simply proportional to the leading divergence, as described in [26]. Crucially this leading divergence, and hence the 3-point function, can be extracted through With integer ∆ j , the violation of condition (ii) implies the presence of (+ + − )-type singularities in theterminology of [26]. – 2 – simple series expansion of the integrand without evaluating the full triple- K integral.Explicit expressions for the renormalised 3-point functions in all such cases may be foundin appendix A of [26].As we will discuss in section 2, conditions (i) and (ii) can be re-expressed in terms of theparameters appearing in the corresponding triple- K integrals. This equivalent form of theconditions (corresponding to (a)–(c) on page 5) is very useful, since it also parametrisesthe non-trivial triple- K integrals arising for correlators with tensorial structure. Thus,to complete the analysis of all 3-point functions – both scalar and tensorial – featuringoperators of integer dimensions, we must solve all triple- K integrals for which this latterset of conditions hold. As we will show, all such triple- K integrals can be reduced in afinite number of steps to a single master integral . This integral can be evaluated in closedform and contains only a single special function, the dilogarithm Li .The organisation of this paper is as follows. First, in section 2, we review the triple- K integral and elaborate in greater detail on role of conditions (i) and (ii). We alsorecall the singularities that can arise in triple- K integrals, their regularisation and howto pass between different schemes. (This material is largely complementary to that in[26] although here we take a somewhat more mathematical focus.) Next, our reductionscheme expressing the relevant triple- K integrals in terms of the master integral is presentedin section 3. The scheme is recursive in nature and is based on simple identities betweenBessel functions. Since both scalar and tensorial 1-loop 3-point massless Feynman integralscan be re-expressed as triple- K integrals, our procedure generalises and simplifies othermomentum-space recursion schemes such as those presented in [29–31]. For purposes ofillustration, we apply our reduction scheme to evaluate all the triple- K integrals arisingin 3-point functions of the stress tensor and conserved currents in four-dimensional CFTs.The evaluation of the master integral is then undertaken in section 4. Its solution relies onexpressing the master integral in terms of hypergeometric and Appell functions. (Similarmethods for representing momentum integrals have appeared elsewhere in the literature,see for example [32–39].)Overall, our focus will be on computing the triple- K integrals rather than the 3-pointfunctions themselves. This approach makes sense as triple- K integrals are the simple build-ing blocks from which the generally more-complicated 3-point functions are constructed.After the relevant triple- K integrals have been evaluated via our recursive procedure, thefull scalar and/or tensorial 3-point functions can be reconstructed as described in [25, 26]. K integrals and their divergences A triple- K integral is a function of three momentum magnitudes p , p , p , defined as I α { β β β } ( p , p , p ) = p β p β p β (cid:90) ∞ d x x α K β ( p x ) K β ( p x ) K β ( p x ) . (2.1)Here K ν ( z ) denotes a modified Bessel function of the second kind, or Bessel K functionfor short. The constants α and β j are fixed numbers relating to the physical input. For– 3 –xample, in the case of scalar operators O , O , O of respective dimensions ∆ , ∆ , ∆ in d -dimensional conformal field theory, the unique 3-point function in momentum space is (cid:104)(cid:104)O ( p ) O ( p ) O ( p ) (cid:105)(cid:105) = c I α { β β β } ( p , p , p ) , (2.2)with α = d − , β j = ∆ j − d , j = 1 , , , (2.3)where c denotes an unspecified theory-dependent constant. Triple- K integrals are thusindeed conformal integrals as per our title; they satisfy appropriate dilatation and specialconformal Ward identities as discussed in [25, 26]. For tensorial 3-point functions, triple- K integrals with various different α and β j can arise for a given set of operator and spacetimedimensions ∆ j and d ; see [25] for a full description. In physical situations the momentummagnitudes obey the triangle inequalities p i + p j ≥ p k for all i, j, k = 1 , , (cid:80) j p j = 0. For the purposes of this paper, however, it will besufficient to assume the p j are simply real positive numbers.As noted above, the triple- K integral is related to massless 1-loop 3-point Feynmanintegrals in momentum space. The exact relation, derived in appendix A.3 of [25], isreproduced in appendix B for convenience. Triple- K integrals are also naturally relatedto holographic 3-point functions, since the AdS bulk-to-boundary propagator in Poincar´ecoordinates is proportional to the Bessel K function. The representation (2.2) thus arisesin holographic calculations of 3-point functions [40] (see also appendix D of [26]).As it stands, the triple- K integral (2.1) converges in the range α + 1 > | β | + | β | + | β | , (2.4)with fixed p , p , p >
0. Outside this range, the triple- K integral can be defined throughits unique analytic continuation. In fact, we will always regard the triple- K integral asa maximally extended analytic function which, on its domain of convergence, agrees with(2.1).The triple- K integral defined in this manner still exhibits singularities at special valuesof α and β j , as shown in [26]. These special values correspond to solutions of the condition α + 1 ± β ± β ± β = − n, n = 0 , , , . . . (2.5)The triple- K integral becomes singular if there exists any choice of independent signs andnon-negative integer n such that the above condition is satisfied. It is often useful to referto these singularities by their associated set of signs; thus, for example, if (2.5) is satisfiedwith a + + − choice of signs then we will call the resulting singularity a (+ + − ) singularity.Note that there may exist more than one choice of signs for any specific value of n .For physical applications, if the triple- K integral diverges we have to regulate andrenormalise. If the condition (2.5) holds, we introduce the regulated parameters α (cid:55)→ ˜ α = α + u(cid:15), β j (cid:55)→ ˜ β j = β j + v j (cid:15), (2.6)– 4 –here u and v j , j = 1 , , K integral I α { β β β } ( p , p , p ) (cid:55)−→ I ˜ α { ˜ β ˜ β ˜ β } ( p , p , p ) (2.7)as a function of the regulator (cid:15) with all momenta fixed. The divergence of the triple- K integral manifests itself as a pole at (cid:15) = 0. Most of the integrals considered in this paperare divergent, meaning that such a regularisation is usually necessary.A full analysis of the singularity structure of triple- K integrals and the associatedrenormalisation procedure for scalar 3-point functions was carried out in [26]. In particular,the singular part of a triple- K integral can always be extracted through a simple seriesexpansion of its integrand, meaning the singularities can be determined without a fullevaluation of the integral. In certain special cases, knowing these singularities alone issufficient to determine the renormalised correlator. More generally, however, to determinethe renormalised 3-point function we also need to know the finite part of the regulatedtriple- K integral as (cid:15) →
0. While the general method for obtaining this finite part hasbeen sketched in [25], our aim here is to present a more thorough analysis.Let us return now to the conditions (i) and (ii) specified in the introduction. Caseswhere all operator dimensions are integral but the spacetime dimension is odd are trivialsince all the β j are half-integer. (Recall (2.3) for scalar correlators; for tensorial correlatorsthe β j also turn out to be half-integer, see [25].) When all the β j are half-integer (as also oc-curs for operators of half-integral dimension in an even-dimensional spacetime), the Bessel K functions in the integrand of the triple- K integral (2.1) reduce to elementary functions.The entire triple- K integral can then be evaluated in terms of elementary functions; thegeneral result is listed in appendix C.For operators of integer dimension in a even -dimensional spacetime, the β j are insteadall integers and the Bessel K functions are no longer elementary. We can however restrictour attention to cases where all the β j ≥
0. Since Bessel K functions are even in theirindex, i.e., K β ( x ) = K − β ( x ), it follows immediately that I α {− β ,β ,β } = p − β I α { β β β } , (2.8)with similar identities holding for β and β . We can therefore relate any triple- K integralin which some of the β j are negative to an equivalent triple- K integral in which all β j ≥ j ≥ d/ K integrals in the rest of the paper will be phrased in termsof the parameters α and β j , it will be useful to re-state conditions (i) and (ii) directly interms of these parameters. This leads us to an equivalent set of conditions:(a) All the β j are non-negative integers (i.e., β j ∈ , , . . . );(b) The combination α + 1 − β − β − β = − n , (2.9)is an even integer, i.e. , n is an integer of any sign or zero;– 5 –c) The following inequalities hold α + 1 + β + β > β , α + 1 + β + β > β , α + 1 + β + β > β . (2.10)Our reasons for writing the conditions in this precise form is partly for later convenience,as will become apparent. Nevertheless, for scalar correlators with α and β j as in (2.3),note that given condition (a), condition (b) implies that the spacetime dimension is aneven integer. Condition (a) then yields that all ∆ j are integers satisfying ∆ j ≥ d/
2, whichis equivalent to condition (i). Condition (c) is directly equivalent to condition (ii). Moregenerally, for tensor correlators where a range of α and β j parameters can appear for agiven set of operator and spacetime dimensions, it will be more convenient simply to usethe conditions (a)–(c) in place of (i) and (ii).The relation between the conditions (a)–(c) and the singularity condition (2.5) for thetriple- K integral should also be noted. Condition (c) forbids the appearance of (+ + − ),( − + +) and (+ − +) solutions of (2.5). Solutions of type (+ + +) are also forbidden inview of condition (a). Conversely, given conditions (a) and (b), violations of condition(c) implies that (+ + − ) singularities (and permutations) are necessarily present, andpotentially also (+ + +) singularities. As mentioned in the introduction, if either (+ + +)or (+ + − ) solutions (in any permutation) are present, the renormalised 3-point functionsare simply given by the leading divergence of the triple- K integral as (cid:15) →
0. As shown in[26], this leading divergence is nonlocal in the momenta and satisfies the complete set ofhomogeneous conformal Ward identities. Thus, when condition (c) is violated, we have noneed to evaluate the corresponding triple- K integral: the leading singularity, and hence therenormalised 3-point function, can be extracted through a series expansion of the integrand.For scalar 3-point functions a complete listing of the renormalised correlators is given inappendix A of [26], while the equivalent analysis for tensorial correlators is given in [27, 28].To re-iterate, our goal in this paper will be to evaluate all triple- K integrals assumingthe conditions (a)–(c) hold. Our reduction scheme will enable all such integrals to bereduced to a single master integral of the form I ˜0 { ˜1˜1˜1 } = I u(cid:15) { v (cid:15), v (cid:15), v (cid:15) } , (2.11)which can be evaluated in terms of elementary functions plus the dilogarithm. Beforepresenting this reduction scheme, however, let us now review in greater detail the singu-larity structure of the regulated triple- K integrals and their dependence on the choice ofregularisation scheme. As discussed above, we wish to define the triple- K integral through its maximal analyticcontinuation agreeing with (2.1) on its domain of convergence. We must therefore showthat the integral is analytic with respect to the four parameters α and β j for j = 1 , , α, β , β , β ) ∈ C , we wish to show thatthe triple- K integral, regarded as a function( α, β , β , β ) (cid:55)−→ I α { β β β } ( p , p , p ) (2.12)– 6 –ith fixed positive p , p , p >
0, is analytic in the region of convergenceRe( α − β t + 1) > , Re β j > , j = 1 , , , (2.13)where β t = β + β + β .By Hartogs’ theorem a complex function of many variables is analytic if it is analyticin each variable separately. Analyticity in a single variable can be proven by means ofMorera’s theorem, i.e. , by showing that an indefinite integral of the function (2.12) exists.For this one shows that the integral over any closed curve in the parameter space vanishes.For concreteness consider a closed curve C in the complex plane of β . Since for x > | x α | ≤ x | α | and | K β ( x ) | ≤ K | β | ( x ), using Fubini’s theorem we can write (cid:90) γ d β (cid:90) ∞ d x x α p β p β p β K β ( p x ) K β ( p x ) K β ( p x )= (cid:90) ∞ d x x α p β p β K β ( p x ) K β ( p x ) (cid:90) γ d β p β K β ( p x )= 0 (2.14)In the last line we used the analyticity of the function β (cid:55)→ p β K β ( p x ) and Cauchy’stheorem. The triple- K integral is therefore indeed analytic with respect to β and anidentical argument applies to the remaining parameters.Since the triple- K integral is analytic with respect to all its parameters in its non-empty domain of convergence (2.13), the method of analytic continuation can be applied.Outside the domain of convergence (2.13), the unique analytically continued function mayexhibit singularities; a possibility we would now like to explore. As discussed above, singularities in the triple- K integral arise if the condition (2.5), α + 1 ± β ± β ± β = − n (2.15)is satisfied for at least one independent choice of signs and a non-negative integer n [26].The singular behaviour of the triple- K integral arises from the divergence at its lower limit x = 0. Indeed, as K ν ( x ) ∼ x − / e − x for large x , the integral always converges at x = ∞ .To determine the pole structure of the singularity, it suffices to expand the integrandabout x = 0 using (A.4) and (A.6). Since we regard expressions such as (2.1) through theirmaximal analytic extensions, the integral diverges if there exists a term of order 1 /x in theexpansion of the integrand. Indeed, each power term x a integrates to (cid:90) µ − x a d x = µ − ( a +1) a + 1 , (2.16)where the upper limit of the integral is arbitrary. In particular, the divergent part ofthe integral cannot depend on µ . While the convergence of the integral requires a > −
1, the right-hand side of the above expression is an analytic function of a ∈ C with a– 7 –ingle pole at a = −
1. It therefore defines an analytic extension of the integral for any a ∈ C \{− } . Thus, while the triple- K integral naively diverges if the expansion of itsintegrand contains terms of the form x a with real a < −
1, its value in such cases is infact uniquely defined through the analytic continuation (2.16). The condition (2.5) simplyenumerates all possible instances where terms of the form 1 /x appear in the expansion ofthe integrand. The coefficients of the various poles then follow from the series expansions(A.4) and (A.6) of the Bessel functions.If all the β j are non-integer, the series expansion of the Bessel functions just consistsof powers x a for various exponents a . If, however, some β j are integral, logarithms canappear in the series expansion of the integrand according to (A.6). The integral (cid:90) µ − x a log n x d x = ( − n n !( a + 1) n +1 µ − ( a +1) n (cid:88) j =0 ( a + 1) j log j µj ! , (2.17)then generates a pole of order n + 1 at a = −
1. As the function is analytic away fromthe singularity, the order of the pole does not depend on the ‘direction’ of approach. Inparticular, the series expansion around a = − (cid:90) µ − x a log n x d x = ( − n n !( a + 1) n +1 + ( − n log n +1 ( µ − ) n + 1 + O ( a + 1) . (2.18)As we can see, the only divergent term is the leading pole of order n + 1, and in agreementwith our expectations, the scale µ is absent in this term. For physical applications it is convenient to analyse divergent integrals by introducing aregulator. We start with a divergent triple- K integral with fixed parameters α and β j andsatisfying at least one of the conditions (2.15). We then regulate the integral by shifting itsparameters by small amounts proportional to a regulator (cid:15) according to the formula (2.7), I α { β β β } (cid:55)−→ I α + u(cid:15) { β + v (cid:15),β + v (cid:15),β + v (cid:15) } . (2.19)The fixed, but otherwise arbitrary numbers u , v , v , v specify the direction of the shift.In general the regulated integral exists, but exhibits singularities when (cid:15) is taken to zero.As we will see, not all choices of directions parametrised by the constants u and v j actuallyregulate the integral, but ‘good’ choices exist and there are many of them.In the triple- K representation (2.2) for the 3-point function of scalar operators in aCFT, the relation between the parameters in the integral and the physical dimensions isgiven by (2.3). In this case, the shift in spacetime and conformal dimensions on going tothe regularised theory are d (cid:55)→ d + 2 u(cid:15), ∆ j (cid:55)→ ∆ j + ( u + v j ) (cid:15), j = 1 , , . (2.20)Certain regularisation schemes may be more useful than others (see the related discussionin [26]). Some particularly useful choices are:– 8 –. u = v = v = v . In terms of the physical dimensions, this scheme corresponds toshifting d (cid:55)→ d + 2 u(cid:15), ∆ j (cid:55)→ ∆ j + 2 u(cid:15), j = 1 , , . (2.21)Its most important feature is the fact that dimensions of sources corresponding toCFT operators (namely d − ∆ j ) do not change. While useful from a physical per-spective, this scheme suffers from the drawback that it does not regulate the triple- K integral in all cases.2. v j = 0 for all j = 1 , ,
3. This scheme preserves the β j and hence the indices of theBessel functions in the triple- K integral. It is therefore particularly useful in caseswhere the indices of Bessel functions are half-integral, as discussed in appendix C.3. − u = v = v = v . As we will see later, the first step in our evaluation of the masterintegral will take place in this scheme. Many other triple- K integrals with integerindices for the Bessel functions are also naturally evaluated in this scheme. In this section we want to answer two important questions regarding the regulated triple- K integrals. Firstly, we want to find a simple method for extracting the terms divergentin (cid:15) for any triple- K integral without actually evaluating the entire integral. Secondly,since a regulated triple- K integral depends on parameters u and v j , we want to extractthe dependence of a finite order (cid:15) part of the integral on these parameters. Throughoutthe section we will therefore assume that the unregulated parameters α and β j satisfy thesingularity condition (2.15) for at least one choice of signs and non-negative integer n .These two problems are closely related as can be anticipated from physical reasoning.Indeed, in any local quantum field theory one would expect that the divergent termsin regulated correlation functions should be computable without knowledge of the entirecorrelator. Furthermore, such divergences should be removable by local counterterms andhence should be of a certain special form. Finally, one would expect scheme-dependentterms to be related to divergent terms, since both are related to the scheme dependenceintroduced by counterterms.As an example, let us consider the master integral I ˜0 { ˜1˜1˜1 } . From the analysis of section2.3, the master integral exhibits a double pole in the regulator so we can write I ˜0 { ˜1˜1˜1 } = I ( − { ˜1˜1˜1 } (cid:15) + I ( − { ˜1˜1˜1 } (cid:15) + I (0)˜0 { ˜1˜1˜1 } + O ( (cid:15) ) . (2.22)The first problem requires finding a simple procedure to evaluate I ( − { ˜1˜1˜1 } and I ( − { ˜1˜1˜1 } . Inprinciple all terms here, including the finite one, depend moreover on u and v j . For thesecond problem we want to extract these u and v j -dependent terms from the finite part I (0)˜0 { ˜1˜1˜1 } . We will refer to such contributions as scheme-dependent terms.This notion of scheme-dependent terms is not however very well defined, as one canalways include in such terms any scheme-independent expression. Hence, what we are reallylooking for is a general procedure for changing the regularisation scheme. Let us assume a– 9 –riple- K integral is evaluated in two different schemes with the corresponding values of the u and v j parameters being u, v j and ¯ u, ¯ v j . We then look for a simple procedure to evaluatethe difference I α + u(cid:15) { β j + v j (cid:15) } − I α +¯ u(cid:15) { β j +¯ v j (cid:15) } = I (¯ u, ¯ v j ) (cid:55)→ ( u,v j ) α { β j } (2.23)up to and including finite terms of order (cid:15) . As we will see in section 4.4.1, the masterintegral can be evaluated explicitly in a specific regularisation scheme with − u = v = v = v . Since ultimately we are interested in the master integral evaluated in an arbitraryscheme, however, we need a scheme-changing procedure expressed by means of (2.23).In [26], we presented a procedure for changing scheme based on the use of differentialoperators reducing the degree of divergence of a triple- K integral. In this manner anydivergent triple- K integral can be reduced to finite, scheme-independent integrals and theresulting expressions can be integrated back (with respect to the momenta) to find the mostgeneral form of the scheme-dependent terms. The undetermined constants of integrationthat arise in this method can be fixed through the explicit computation of triple- K integralsin the simplifying limit where some of the momenta become small.In this paper we present an alternative procedure for changing the regularisationscheme. The method is rather simpler and faster as it does not require solving any differen-tial equations. Moreover, only the integration of power functions is required, in contrast tosmall-momentum limits of triple- K integrals as in [26]. In the following two subsections wefirst review the method for extracting the divergences of triple- K integrals, then proceedto the problem of changing the regularisation scheme. As discussed in section 2.3, all divergences of the triple- K integral follow from the x = 0region of integration. To extract the divergences, then, we can expand all three Bessel K functions appearing in the integrand of the regulated triple- K integral and integrate theresulting expression from zero to some arbitrary cut-off µ − . Using analytic continuationto define the integral, only terms for which the power of x is close to − I (div)˜ α { ˜ β j } = (cid:88) w ∈ R n ∈{ , ,... } (cid:90) µ − d x x − w(cid:15) log n x c − w(cid:15),n ( p , p , p ) , = (cid:88) w ∈ R n ∈{ , ,... } ( − n n !( w(cid:15) ) n +1 µ − w(cid:15) n (cid:88) j =0 ( w(cid:15) log µ ) j j ! c − w(cid:15),n ( p , p , p ) , (2.24)where c − w(cid:15),n represents the coefficient of x − w(cid:15) log n x in the expansion of the integrandof the regulated triple- K integral. There are only finitely many nonvanishing terms of thisform.The coefficients c − w(cid:15),n can be read off from the standard series expansion of theBessel K functions, (A.4, A.6). In general they may be singular at (cid:15) = 0. However, inthe scheme where all v j = 0, the β j parameters are not regularised meaning ˜ β j = β j is– 10 –ndependent of (cid:15) . In this scheme then all c − w(cid:15),n are finite as (cid:15) →
0, and all poles in thetriple- K integral follow from integrating the various logarithmic terms in (2.24). From thisperspective the v j = 0 scheme is rather convenient for practical calculations, as discussedfurther in appendix A of [26].If all v j (cid:54) = 0, on the other hand, we can express the c − w(cid:15),n in terms of the Besselexpansion coefficients (A.5), a σj ( β ) = ( − j Γ( − σβ − j )2 σβ +2 j +1 j ! . (2.25)In this case no logarithmic terms appear in the integrand of (2.24) and the expression canbe simplified as follows. Consider all choices of signs σ , σ , σ ∈ {± } and all non-negativeintegers n , n , n such that the condition (2.15) is satisfied with n = (cid:80) j n j , i.e. , α + 1 + (cid:88) j =1 ( σ j β j + 2 n j ) = 0 . (2.26)As no logarithms are present, we use (2.24) with n = 0 leading to I (div)˜ α { ˜ β j } = (cid:88) cond µ − w(cid:15) w(cid:15) (cid:89) j =1 p (1+ σ ) ˜ β j +2 n j j a σ j n j ( ˜ β j ) , (2.27)where the sum is taken over all σ j and n j , j = 1 , , w = u + (cid:88) j =1 σ j v j . (2.28) By construction, (2.24) encodes all divergent contributions to the regulated triple- K inte-gral, i.e. , I ˜ α { ˜ β j } = I (div)˜ α { ˜ β j } + O ( (cid:15) ) . (2.29)In fact, I (div) also contains all scheme-dependent contributions to the triple- K integral: ∂∂u (cid:16) I ˜ α { ˜ β j } − I (div)˜ α { ˜ β j } (cid:17) = ∂∂v (cid:16) I ˜ α { ˜ β j } − I (div)˜ α { ˜ β j } (cid:17) = O ( (cid:15) ) . (2.30)The scheme-changing expression in (2.23) is then given by I (¯ u, ¯ v j ) (cid:55)→ ( u,v j ) α { β j } = I (div) α + u(cid:15) { β j + v j (cid:15) } − I (div) α +¯ u(cid:15) { β j +¯ v j (cid:15) } + O ( (cid:15) ) . (2.31)This formula, derived in appendix D, provides an effective way to change the regularisationscheme. If a triple- K integral is known in one regularisation scheme, then by adding theabove expression one can find the value of the integral in any other scheme. Crucially(2.31) contains only a finite number of terms from the series expansion of the integrand,and hence is easily computed for any given triple- K integral.– 11 –et us illustrate this method of changing the regularisation scheme with a workedexample. Our goal will be evaluate the triple- K integral I { } regularised in a genericscheme. This example can also be found in [26], although our present method is rathersimpler as only elementary integrals of powers need to be evaluated.The divergent part of the integral, regulated in a general scheme, reads I (div)˜2 { ˜1˜1˜1 } = 2 v t (cid:15) Γ(1 + v (cid:15) )Γ(1 + v (cid:15) )Γ(1 + v (cid:15) ) (cid:90) µ − d xx − u − v t ) (cid:15) = 1( u − v t ) (cid:15) + (cid:20) v t v t − u ( γ E − log 2) − log µ (cid:21) + O ( (cid:15) ) , (2.32)where v t = (cid:80) j v j . As we expect, the scale µ only shows up in the finite part of thisexpression. Notice also that the coefficient of log µ is independent of u and v j , and hence thescheme-changing term (2.31) does not depend on µ . Using (2.32), we can now immediatelywrite down the required expression for the triple- K integral in some scheme ( u, v j ) givenits value in another scheme (¯ u, ¯ v j ), namely I (div)2+ u(cid:15) { v (cid:15), v (cid:15), v (cid:15) } = I (div)2+¯ u(cid:15) { v (cid:15), v (cid:15), v (cid:15) } + 1 (cid:15) (cid:20) u − v t ) − u − ¯ v t ) (cid:21) + ( γ E − log 2) (cid:20) uv t − u − ¯ u ¯ v t − ¯ u (cid:21) + O ( (cid:15) ) . (2.33) In this section we present the complete reduction scheme allowing for the evaluation of anyintegral satisfying conditions (a) - (c) from section 2.1. The reduction scheme relies on ourknowledge of a single master integral I u(cid:15) { v (cid:15), v (cid:15), v (cid:15) } , (3.1)which we will return to evaluate in section 4.All the integrals accessible through our reduction scheme exhibit either a single or adouble pole in the regulator, or else possess a finite (cid:15) → I { } and the finite integral I { } . As a first step, integrals with single poles can then berelated to I { } while finite integrals can be related to I { } . Both the latter integralscan then be related to the original master integral I { } in a second step. (Integrals withdouble poles meanwhile reduce to I { } in one step.)In the remainder of this section, then, our goal will be to reduce every triple- K integralsatisfying conditions (a) - (c) in section 2.1 to one of the three integrals above according toits degree of divergence. For ease of reference, we have gathered the key equations relat-ing the different master integrals in the table below, along with their explicit expressionsalthough we will not need these in the following.– 12 –ntegral Order of divergence Reduction to I { } Explicit expression I { } I { } I { } From the definition of the triple- K integral (2.1) it follows that for any permutation σ ofthe set { , , } , I α { β σ (1) β σ (2) β σ (3) } ( p , p , p ) = I α { β β β } ( p σ − (1) , p σ − (2) , p σ − (3) ) . (3.2)Also, as noted earlier, since Bessel functions are even in their index, i.e., K − ν ( x ) = K ν ( x ),we have I α {− β β ,β } = p − β I α { β β β } . (3.3)By combining these two relations we can always order the β j parameters and assume β ≥ β ≥ β ≥ . (3.4)We will assume such an ordering from now on.Consider a triple- K integral satisfying conditions (a) - (c) from section 2.1. As we willexplain shortly, the degree of divergence of the regulated integral can be discerned fromthe values of three constants n , n and n defined as follows,2 n = β + β + β − α − , (3.5)2 n = β + β − β − α − , (3.6)2 n = β − β − β − α − . (3.7)For integer α, n , n , n with n <
0, we will find the triple- K integral possesses a simplerepresentation in terms of elementary functions and dilogarithms and can be reduced tothe master integral I { } .Let us first show however that this last condition is equivalent to conditions (a) -(c) presented in section 2.1. Indeed, conditions (a) and (b) together imply that α , n , n = n − β and n = n − β are all integers. Condition (c) is equivalent to n < n < β and β are integer if n , n and n are. Integer α thenimplies integer β as well.In the following subsections we will present a (non-unique) reduction scheme leading toanalytic expressions for all integrals under considerations. We divide all triple- K integralsinto three classes, depending on the order of singularity. More precisely, we have thefollowing cases: • If n <
0, the triple- K integral I α { β j } is finite and expressible in terms of I { } .– 13 – If n ≥ n < − − − ) form of the condition (2.5) is satisfied;the triple- K integral I ˜ α { ˜ β j } has a single pole in the regulator (cid:15) and is completelyexpressible in terms of I { } . • If n ≥ n ≥ − − +) and ( − − − ) conditions in (2.5) are satisfied;the triple- K integral I ˜ α { ˜ β j } has a double pole in the regulator (cid:15) and is completelyexpressible in terms of the master integral I { } .Before we discuss these cases further, let us make a few remarks. The values of n and n can be either positive or negative, but the corresponding ( − − − ) and/or ( − − +)conditions are only satisfied when n and/or n are non-negative. We will always assumethat n <
0, which ensures that the ( − + +) condition is never satisfied.If the ( − + +) condition were to be satisfied, the procedure we present is not sufficientfor the evaluation of the finite part of the triple- K integral. Nevertheless, as explained inthe introduction, in this case the renormalised 3-point function is simply proportional tothe leading divergence of the triple- K integral. This leading divergence can be extractedas described in section 2.5.1; see also section 4.3.4 and appendix A of [26] for exact results.Here, we will concentrate instead on cases when the finite part of a triple- K integral isindeed necessary for the evaluation of the corresponding 3-point function.Notice also that due to the ordering (3.4), we always have n ≥ n ≥ n . (3.8)The first equality only occurs if β = 0 and the second equality is only possible if β = β =0. From (3.8), it follows that if the (+ − − ) condition is satisfied, the ( − + − ) condition istoo. Similarly, if the ( − + − ) condition is satisfied, so is the ( − − +) condition. It is crucialhere that α and all the β j are integers. In general other conditions besides these may besatisfied, but in such cases our reduction scheme will not be applicable. Let us now list the important identities between triple- K integrals we will use for thedevelopment of a reduction scheme. Two relations we have already mentioned are I α { β σ (1) β σ (2) β σ (3) } ( p , p , p ) = I α { β β β } ( p σ − (1) , p σ − (2) , p σ − (3) ) , (3.9) I α {− β β ,β } = p − β I α { β β β } . (3.10)Three further important relations involving derivatives are I α { β β β } = − p ∂∂p I α − { β +1 ,β β } , (3.11) I α +1 { β +1 ,β ,β } = (cid:18) β − p ∂∂p (cid:19) I α { β β β } , (3.12) I α +2 { β β β } = K j I α { β β β } , (3.13)– 14 –here in the last equation the index j takes values j = 1 , , j denotes the conformalWard identity operator introduced in [25]K j = K j,β j = ∂ ∂p j − β j − p j ∂∂p j . (3.14)This operator depends on a single parameter β j which in this paper will always be equalto the corresponding β j of the triple- K integral upon which K j acts.Finally, we have the useful identities I α − { β β β } = 1 α − β t (cid:2) p I α { β − ,β ,β } + p I α { β ,β − ,β } + p I α { β ,β ,β − } (cid:3) , (3.15) I α − { β β β } = 1 α − β t B β β β I α − { β − ,β − ,β − } , (3.16)where β t = β + β + β and for later convenience we have definedB β β β = p (cid:18) β − − p ∂∂p (cid:19) (cid:18) β − − p ∂∂p (cid:19) + cyclic permutations. (3.17)Notice that among these identities only (3.15) decreases the value of α . In the remain-ing identities the operators appearing on the right-hand sides act to increase the value of α while either increasing or decreasing β j by integer amounts. The action of these operatorscan be summarised as follows:change in α change in β j equation- β j (cid:55)→ − β j (3.10) α (cid:55)→ α + 1 β j (cid:55)→ β j − α (cid:55)→ α + 1 β j (cid:55)→ β j + 1 (3.12) α (cid:55)→ α + 2 - (3.13) α (cid:55)→ α − β j (cid:55)→ β j + 1 (3.15) α (cid:55)→ α + 1 all three β j (cid:55)→ β j + 1 (3.16)In the following two subsections we will prove the identities listed above. Relation (3.11) follows directly from a simple property of the Bessel function, ∂∂p [ p ν K ν ( px )] = − xp ν K ν − ( px ) . (3.18)If, however, we first use equation (3.10), followed by (3.11) then (3.10) again, we derivethe second identity (3.12), I α +1 { β +1 ,β ,β } = − p β +11 ∂∂p (cid:104) p − β I α { β β β } (cid:105) = (cid:18) β − p ∂∂p (cid:19) I α { β β β } , (3.19)– 15 –urthermore, we can apply this operation and its permutations repeatedly to obtain I α + k t { β j + k j } = ( − k t (cid:89) j =1 p β j + k j ) j (cid:18) p j ∂∂p j (cid:19) k j (cid:104) p − β p − β p − β I α { β j } (cid:105) , (3.20)where k t = k + k + k and k j are non-negative integers.Both the reduction relations (3.11) and (3.12) obtained by differentiation happen toincrease α by one, while changing the value of one of the β j by ±
1. When they are combinedtogether, they increase the value of α by two and the resulting expression is equal to (3.13). To derive relation (3.15) consider the following integral (cid:90) ∞ d x ∂∂x (cid:16) x α (cid:89) j =1 p β j j K β j ( p j x ) (cid:17) = ( α − β t ) I α − { β β β } − (cid:2) p I α { β − ,β ,β } + p I α { β ,β − ,β } + p I α { β ,β ,β − } (cid:3) . (3.21)In the domain of convergence (2.4) the boundary term vanishes by simple power counting.Hence, by analytic continuation, relation (3.15) holds whenever its two sides remain finite.The second equation (3.16) follows from a combination of (3.15) with (3.12) appliedtwice to each integral on the right-hand side.Relation (3.15) is the only relation reducing the value of α . It is closely related toDavydychev’s recursion relation (3.4) introduced in [29] as well as equation (36) of [41].Indeed, using equation (A.3.17) of [25] one can rewrite Davydychev’s J integral defined in(2.1) of [29] as J ( δ , δ , δ ) = 4 π Γ( δ )Γ( δ )Γ( δ )Γ(4 − δ t ) I { − δ − δ , − δ − δ , − δ − δ } . (3.22)Note that the rather complicated form of equation (3.4) in [29] is a consequence of thespecific index structure in the triple- K integral above. Conversely, triple- K integrals con-veniently resolve the complicated structure of linear dependencies in [29] leading to a morenatural representation of the 3-point function. The relation between triple- K integrals and1-loop integrals in momentum space is summarised in appendix B.Let us also comment on the validity of equation (3.15). The left-hand side divergesfor α − β t = − n , where n is a non-negative integer. Nevertheless, it can still be used forregulated integrals with ˜ α − ˜ β t = − n + ( u − v t ) (cid:15) so long as (cid:15) (cid:54) = 0. If n = 0, however,(3.15) evaluates only the divergent part of the left-hand side, assuming the integrals onthe right-hand side are only known up to the finite part of order (cid:15) . For example, (3.15)relates the divergent part of I ˜2 { ˜1˜1˜1 } to the finite integral I { } . It is therefore impossibleto use (3.15) to retrieve the finite part of I ˜2 { ˜1˜1˜1 } from a knowledge of I { } . In this section we now present the complete reduction scheme. We will analyse cases inorder of increasing complexity. The integrals that can be evaluated through the schemeare presented graphically in figure 1. – 16 – igure 1 : A summary of all triple- K integrals that can be obtained from the masterintegral I { } using the reduction scheme. Depending on the values of n and n the inte-grals are either finite (represented by squares), linearly divergent (circles), or quadraticallydivergent (diamonds). Each point represents an infinite series of integrals having the samevalues of n and n . In particular the master integral I { } belongs to the series denotedby the red diamond at ( n , n ) = (1 , I { } belongs tothe series denoted by the blue circle at (0 , − I { } to the series denoted bythe green square at ( − , − n ≥ n , no integral can appear in the excludedregion. n < Let us start with finite integrals satisfying n <
0. We can simply use (3.12) and (3.13) incombination to write the reduction formula, I α { β β β } = ( − β t K | n |− j,β j (cid:34) p β p β p β (cid:18) p ∂∂p (cid:19) β (cid:18) p ∂∂p (cid:19) β (cid:18) p ∂∂p (cid:19) β I { } (cid:35) (3.23)– 17 –here j can take any of the values j = 1 , ,
3. It is crucial here that n is strictly less thanzero. This formula expresses any finite integral under consideration in terms of the finite I { } integral. n ≥ n < Every linearly divergent integral under consideration satisfies n ≥ n <
0, and canbe reduced to the I { } integral. Notice that these restrictions imply β >
0. If β = 0,we obtain n = n ≥
0, a contradiction.The reduction procedure consists of two steps. First we reduce the triple- K integral I ˜ α { ˜ β ˜ β ˜ β } to an integral of the form I β +1 { β β β } with all beta parameters equal by meansof (3.20) and (3.13), I ˜ α { ˜ β ˜ β ˜ β } = ( − k + k K | n |− j, ˜ β j (cid:34) p β p β (cid:18) p ∂∂p (cid:19) k (cid:18) p ∂∂p (cid:19) k × (cid:16) p − β − v (cid:15) p − β − v (cid:15) I β +1+ u(cid:15) { β + v (cid:15),β + v (cid:15),β + v (cid:15) } (cid:17)(cid:105) , (3.24)where j takes values j = 1 , , k = β − β , k = β − β . (3.25)If β = 1 then the integral on the right-hand side is the familiar I { } integralregularised in a generic scheme. If β >
1, then the integral on the right-hand side has theform I n +1 { nnn } with n ≥
2, and the equation (3.16) can be used recursively. One has I n +1+ u(cid:15) { n + v (cid:15),n + v (cid:15),n + v (cid:15) } = B n + v (cid:15),n + v (cid:15),n + v (cid:15) I n + u(cid:15) { n − v (cid:15),n − v (cid:15),n − v (cid:15) } − n + 2 + (cid:15) ( u − v t ) , (3.26)where the operator B is defined in (3.17). The recursion is well defined since n ≥ I u(cid:15) { v (cid:15), v (cid:15), v (cid:15) } . Notice that atall stages of the calculation one needs to keep track of subleading terms in the regulator (cid:15) .Indeed, since the integrals are divergent, subleading terms in (cid:15) may combine with divergentpieces producing additional contributions to the finite part. n = 0 This interesting class of integrals arises for 3-point functions of marginal scalar operators ineven-dimensional CFTs. The triple- K integral exhibits a double pole and can be reducedto the master integral I { } . The method is almost identical to that presented in theprevious subsection.First, however, consider the special case n = 0, which implies β = 0. Here, we havethe relation I ˜ α { ˜ β ˜ β } = ( − β + β +1 p β p β (cid:18) p ∂∂p (cid:19) β − (cid:18) p ∂∂p (cid:19) β − p ∂∂p × (cid:104) p − − v (cid:15) p − − v (cid:15) I u(cid:15) { v (cid:15), v (cid:15), v (cid:15) } (cid:105) , (3.27)– 18 –here it is important that β ≥ β ≥
1. Indeed, if in addition to β = 0 we also had β = 0,(3.7) would then imply n = 0 contradicting our assumption n < n = 0 and n >
0, implying β >
0. Inthis case we can use a modification of (3.24) to write I ˜ α { ˜ β ˜ β ˜ β } = ( − k + k p β p β (cid:18) p ∂∂p (cid:19) k (cid:18) p ∂∂p (cid:19) k × (cid:104) p − β − v (cid:15) p − β − v (cid:15) I β − u(cid:15) { β + v (cid:15),β + v (cid:15),β + v (cid:15) } (cid:105) , (3.28)where k and k are defined in (3.25). For β = 1 the integral on the right-hand side isthe master integral I { } regularised in a generic scheme. If β >
1, the integral on theright-hand side has the form I n − { nnn } with n ≥
1, and as previously (3.16) can be usedrecursively. We find I n − u(cid:15) { n + v (cid:15),n + v (cid:15),n + v (cid:15) } = B n + v (cid:15),n + v (cid:15),n + v (cid:15) I n − u(cid:15) { n − v (cid:15),n − v (cid:15),n − v (cid:15) } − n + (cid:15) ( u − v t ) , (3.29)where the operator B was defined in (3.17). The recursion is well defined since n ≥ n > The reduction procedure for integrals with n > I { } .We will consider two cases. The first consists of all integrals satisfying n > n > i.e. , integrals lying in the interior of the upper-right wedge in figure 1. The second caseconsists of integrals satisfying n = n > i.e. , integrals lying on the diagonal line infigure 1.We will show that in the first case the application of (3.15) leads to a sum of integralswith the values of their corresponding constants n decreased by one. Similarly, in thesecond case, we will show that the original integral can be re-expressed as a combinationof triple- K integrals with values of n that have been decreased by one. In both cases therecursion is such that values of n remain negative throughout.In each step of the reduction procedure the values of n and n are thus decreaseduntil we reach the point where n = 0, allowing the results of the previous subsection tobe utilised. Case n > n . If n > n we use equation (3.15), which can be written as I ˜ α { ˜ β ˜ β ˜ β } = 1 − n + ( u − v t ) (cid:15) (cid:104) p I ˜ α +1 { ˜ β − , ˜ β , ˜ β } + p I ˜ α +1 { ˜ β , ˜ β − , ˜ β } + p I ˜ α +1 { ˜ β , ˜ β , ˜ β − } (cid:105) . (3.30)Due to the ordering (3.4) we have n ≥ n >
0, and hence the use of (3.30) is alwaysjustified. – 19 –ssume that a given integral on the left-hand side of (3.30) possesses the associatedconstants n , n , n . We want to calculate the values of the corresponding constants – say n (cid:48) , n (cid:48) , n (cid:48) – for the integrals on the right-hand side, and to confirm that they decrease.Indeed, n > n is equivalent to β >
0, and for the integrals on the right-hand sidewe then have Integral n (cid:48) n (cid:48) n (cid:48) I ˜ α +1 { ˜ β − , ˜ β , ˜ β } n − n − n − I ˜ α +1 { ˜ β , ˜ β − , ˜ β } n − n − n I ˜ α +1 { ˜ β , ˜ β , ˜ β − } n − n n As we can see n (cid:48) ≤ n < n decreases by one. The repeateduse of the equation (3.30) will thus lead to a combination of integrals satisfying either n = 0 or n = n with n < Case n = n . The remaining case is the analysis of n = n >
0. This, however, impliesthat β = 0 and the last integral in (3.30) therefore has a negative value of its β coefficient.In this case we can flip the sign using (3.3) leading to a modification of (3.30) for β = 0, I ˜ α { ˜ β ˜ β , v (cid:15) } = 1 − n + ( u − v t ) (cid:15) (cid:104) p I ˜ α +1 { ˜ β − , ˜ β ,v (cid:15) } + p I ˜ α +1 { ˜ β , ˜ β − ,v (cid:15) } + p v (cid:15) I ˜ α +1 { ˜ β , ˜ β , − v (cid:15) } (cid:105) . (3.31)The last integral on the right-hand side satisfies n (cid:48) = n , n (cid:48) = n − n (cid:48) = n − n decreased by oneand satisfy n (cid:48) ≥ n (cid:48) ≥
0. This concludes the reduction procedure. d = 4 I { } , whose value is given in (4.2). Where a ‘cell’ of the diagram contains two entriesthe corresponding arrows carry two operators: of these two operators, the upper one leadsto/is applied to the upper entry and lower operator to the lower entry.All integrals are implicitly assumed to be regulated in a single but arbitrary schemewith fixed values of u and v j . Calculating the parameters n and n , equations (3.5) and(3.6) then identify the degree of divergence of all integrals in the table. Red entries thenindicate integrals exhibiting a double pole in the regulator; blue entries those that arelinearly divergent; and green, finite integrals.The operators L j and M j are the differential operators featuring in (3.11) and (3.12),defined as L j = − p j ∂∂p j , M j = 2 ˜ β j − p j ∂∂p j . (3.32)– 20 – { } M (cid:47) (cid:47) I { } L (cid:47) (cid:47) M (cid:15) (cid:15) I { } M (cid:15) (cid:15) I { } L (cid:47) (cid:47) M (cid:15) (cid:15) I { } (cid:106) (cid:106) L (cid:47) (cid:47) M M (cid:15) (cid:15) I { } L (cid:47) (cid:47) M M (cid:15) (cid:15) I { } M (cid:15) (cid:15) I { } L L (cid:47) (cid:47) M (cid:15) (cid:15) I { } I { } L L (cid:47) (cid:47) M M (cid:15) (cid:15) I { } I { } L L (cid:47) (cid:47) M M (cid:15) (cid:15) I { } M (cid:15) (cid:15) I { } L (cid:47) (cid:47) M M (cid:15) (cid:15) I { } L L (cid:47) (cid:47) M M (cid:15) (cid:15) I { } I { } L L (cid:47) (cid:47) M M (cid:15) (cid:15) I { } M (cid:15) (cid:15) I { } I { } L L (cid:47) (cid:47) I { } I { } L L (cid:47) (cid:47) I { } L (cid:47) (cid:47) I { } Table 1 : Reduction scheme for the integrals required to calculate all 3-point functions ofconserved currents and the stress tensor in d = 4. The operators L j and M j are defined by L j = − p − j ∂/∂p j and M j = 2 ˜ β j − p j ∂/∂p j . Further details may be found in section 3.4.The corresponding regulated value ˜ β j is equal to that of the integral on which M j is acting.The dotted line in table 1 indicates the use of (3.15). In this section we derive the expression for the master integral I { } , or rather its regu-larised version (2.19), I u(cid:15) { v (cid:15), v (cid:15), v (cid:15) } , (4.1)where u and v j , j = 1 , , K integral satisfying conditions (a) - (c) from section 2.1 can– 21 –e expressed in terms of the master integral I { } . For convenience we will also presentexpressions for the two auxiliary master integrals featured in figure 1, I { } and I { } .Our evaluation of the master integral is based upon the relation between triple- K integrals and hypergeometric functions. In the context of 1-loop 3-point integrals in mo-mentum space similar relations have been analysed in a number of papers, for example[32–39]. The conformal case, however, corresponds to massless integrals and hence theresulting expressions can usually be simplified to more elementary functions. The earliestexamples containing dilogarithms can be traced back to [42, 43]. The master integral takes the following form I u(cid:15), { v (cid:15), v (cid:15), v (cid:15) } = I ( − (cid:15) + I ( − (cid:15) + I (scheme) + I (non-local) + I (scale-violating) + O ( (cid:15) ) , (4.2)where I ( − = 12( v t − u ) (cid:88) j =1 p j u − v t + 2 v j , (4.3) I ( − = 14 (cid:88) j =1 p j log p j u − v t + 2 v j + u (1 − γ E + 2 log 2) − v t v t − u ) (cid:88) j =1 p j u − v t + 2 v j , (4.4) I (scheme) = 18 (cid:88) j =1 v j u − v t + 2 v j p j log p j + 18 [ u (1 − γ E + 2 log 2) − v t ] (cid:88) j =1 p j log p j u − v t + 2 v j + [ v t − u (1 − γ E + log 2)] + u ( γ E − log 2) + π v tt v t − u ) (cid:88) j =1 p j u − v t + 2 v j , (4.5) I (non-local) = − (cid:112) − J (cid:20) π − p p log p p + log X log Y − Li X − Li Y (cid:21) , (4.6) I (scale-violating) = 116 (cid:2) ( p − p − p ) log p log p + ( p − p − p ) log p log p + ( p − p − p ) log p log p (cid:3) . (4.7)The meaning of various parts is as follows: • I ( − and I ( − are the coefficients of the divergent parts of the integrals. These canbe extracted straightforwardly using the method of section 2.5.1. • I (scheme) denotes the finite part of the integral depending on the regularisation pa-rameters u and v j . (Note that the coefficients I ( − and I ( − depend on u and v j aswell.) The remaining pieces I (non-local) and I (scale-violating) do not depend on either u or v j . • I (non-local) contains the essential non-local part of the integral, as well as the onlyspecial function, the dilogarithm Li . This part is scale-invariant, i.e. , all logarithmsdepend on the ratios p /p and p /p only.– 22 – I (scale-violating) contains the non-local yet scale-violating part of the integral. Afterrenormalisation, these terms are related to beta functions and conformal anomalies.In writing the master integral, we used the definitions J = ( p + p − p )( p − p + p )( − p + p + p )( p + p + p ) , (4.8) X = − p + p + p − √− J p , Y = − p + p + p − √− J p , (4.9) v t = v + v + v , v tt = v + v + v . (4.10)The decomposition we have made is by no means unique. It simply organises the finalresult neatly. We discuss further properties of the master integral in the following section. The physical implications of the double logarithms of momenta in the scale-violating partof the master integral were discussed in [26]. In this section we will concentrate on the‘non-local’ part I (non-local) containing the dilogarithms.The quantity J defined in (4.8) has a geometric interpretation. Physically, the p j = | p j | are the magnitudes of three d -dimensional vectors p , p , p satisfying p + p + p = 0due to momentum conservation. Scalar products between these momentum vectors can beexpressed in terms of the p j according to p · p = 12 ( p − p − p ) (4.11)and similarly for other products. Equation (4.8) can then be rewritten as J = ( p + p − p )( p − p + p )( − p + p + p )( p + p + p ) , = − p − p − p + 2 p p + 2 p p + 2 p p = 4 (cid:2) p p − ( p · p ) (cid:3) = 4 · Gram( p , p ) , (4.12)where Gram is the Gram determinant. The area of a triangle with side lengths p , p and p is therefore (1 / √ J . For physical momentum configurations obeying the triangleinequalities one has J ≥
0, with J = 0 holding if and only if the momenta p j are collinear.The definitions of X and Y in (4.9) agree with those in [29]. In the literature, e.g. ,[32, 44], an alternative choice of variables in place of X and Y is commonly encountered;usually these are denoted by z and its complex conjugate ¯ z , defined as any pair of solutionsto the quadratic equations z ¯ z = p p , (1 − z )(1 − ¯ z ) = p p . (4.13)The relation between z, ¯ z and X, Y is then simply z = X and ¯ z = 1 − Y . In such arepresentation the ‘non-local’ part of the master integral thus reads I (non-local) = 18 (¯ z − z ) p (cid:20)
12 log( z ¯ z ) (log(1 − z ) − log(1 − ¯ z )) + Li z − Li ¯ z (cid:21) . (4.14)– 23 –nfortunately it is not obvious in either representation that I (non-local) is real andsymmetric under any permutation of momenta p j , j = 1 , ,
3. Nevertheless, the function isindeed completely symmetric by virtue of certain identities between dilogarithms, see forexample [34, 45].Every integral which can be obtained from the master integral by means of the reduc-tion scheme discussed in section 3 contains a piece proportional to I (non-local) . It is thenrather convenient to define the quantity L = π − p p log p p + log X log Y − Li X − Li Y = −
12 log( z ¯ z ) (log(1 − z ) − log(1 − ¯ z )) − Li z + Li ¯ z. (4.15) L is purely imaginary and completely symmetric under any permutation of p j , j = 1 , , L turn out to be extremely simple, ∂ L ∂p = 2 p √− J (cid:20) p log p + 12 ( p − p − p ) log p + 12 ( p − p − p ) log p (cid:21) = 2 p √− J (cid:2) p log p + p · p log p + p · p log p (cid:3) , (4.16)where in the last line we used the physical interpretation of the parameters p j as magnitudesof vectors p j satisfying p + p + p = 0. To evaluate derivatives of L with respect to other p j one simply needs to permute the momenta accordingly. In the course of the reduction scheme, we expressed all integrals satisfying conditions (a)- (c) from section 2.1 in terms of three integrals: I { } , I { } and I { } . The lattertwo auxiliary integrals are related to the master integral I { } by the identities (3.16) and(3.13) respectively, I { } = 12 p p p (cid:20) p ∂ ∂p ∂p + p ∂ ∂p ∂p + p ∂ ∂p ∂p (cid:21) I u(cid:15) { v(cid:15), v(cid:15), v(cid:15) } + O ( (cid:15) ) , (4.17) I u(cid:15) { v (cid:15), v (cid:15), v (cid:15) } = K j, v j (cid:15) I u(cid:15) { v (cid:15), v (cid:15), v (cid:15) } , (4.18)where the operator K j, v j (cid:15) = K j is the conformal Ward identity operator given in (3.14)and j takes an arbitrary value j = 1 , ,
3. The first integral is finite and hence does notrequire a regulator. The second integral exhibits a single pole. Using the definition (4.15)for L , their exact expressions can be derived and read I { } = L √− J , (4.19) I u(cid:15) { v (cid:15), v (cid:15), v (cid:15) } = 1( u − v t ) (cid:15) + 2 p p p ( − J ) / L + uu − v t (log 2 − γ E ) − J (cid:2) p ( p + p − p ) log p + p ( p + p − p ) log p + p ( p + p − p ) log p (cid:3) + O ( (cid:15) ) , (4.20)– 24 –he first integral I { } is known in the literature (see e.g. , [29, 35, 43]) and represents,for example, the 3-point function of φ in the 4-dimensional theory of free massless scalars. To evaluate the master integral we used a sequence of mathematical identities. This se-quence, to be explained in the following subsections, is:1. We first evaluate integrals of the form I ν +1 { ννν } for any ν ∈ R . Integrals of this formcan be expressed in terms of hypergeometric functions (Legendre functions).2. Substituting ν = − (cid:15) and expanding in (cid:15) , we evaluate the triple- K integral I (cid:15) {− (cid:15), − (cid:15), − (cid:15) } . The result contains a single special function, the dilogarithm.3. Using (3.3), we obtain I (cid:15) { − (cid:15), − (cid:15), − (cid:15) } = ( p p p ) − (cid:15) I (cid:15) {− (cid:15), − (cid:15), − (cid:15) } (4.21)which is the master integral in a regularisation scheme with − u = v = v = v .4. Finally, we change to an arbitrary regularisation scheme specified by general u and v j parameters according to the method described in section 2.5.2. I ν +1 { ννν } To evaluate I ν +1 { ννν } we start with the representation of the triple- K integral in terms ofthe generalised hypergeometric function Appell F [45, 46], I α { β β β } = 2 α − p α [ A ( λ, µ ) + A ( λ, − µ ) + A ( − λ, µ ) + A ( − λ, − µ )] , (4.22)where A ( λ, µ ) = (cid:18) p p (cid:19) λ (cid:18) p p (cid:19) µ Γ (cid:18) α + λ + µ − ν (cid:19) Γ (cid:18) α + λ + µ + ν (cid:19) Γ( − λ )Γ( − µ ) × F (cid:18) α + λ + µ − ν , α + λ + µ + ν λ + 1 , µ + 1; p p , p p (cid:19) . (4.23)This representation is not very useful for numerical evaluation, but provides a good start-ing point for formal manipulations. For triple- K integrals of the form I ν +1 { ννν } , the Ap-pell function simplifies to regular hypergeometric functions. Using the reduction formulae(A.11) to (A.14), we find I ν +1 { ννν } = 2 ν − Γ( ν ) π sin( πν ) (cid:20) p ν p p XY F ν (cid:18) p p p X Y (cid:19) − p ν p Y F ν (cid:18) p p Y (cid:19) − p ν p X F ν (cid:18) p p X (cid:19)(cid:21) + 2 ν − π Γ (cid:0) ν + (cid:1) sin ( πν ) ( p p p ) ν ( (cid:112) − J ) − (2 ν +1) , (4.24)– 25 –here F ν ( x ) = F (1 , ν + 1; 1 − ν ; x ) (4.25)and the variables X, Y are as defined in (4.9) while J is given in (4.8). Note that this par-ticular combination of parameters in the hypergeometric function also appears in Legendrefunctions.In terms of the parameters n , n , n defined in (3.5) to (3.7), any integral of the form I ν +1 { ννν } with integer ν satisfies n = ν − n = − n = − ν −
1. Such integralstherefore lie on the horizontal line n = − n <
0. Note that themaster integral is not in this class. Nevertheless, by using the inversion trick (3.3) we canwrite I − ν +1 { ννν } = ( p p p ) ν I − ν +1 {− ν − ν − ν } . (4.26)The integral on the right-hand side can be expressed through (4.24), while the integral onthe left-hand side satisfies n = 2 ν − n = ν − n = − <
0. By this method wecan then directly generate all integrals lying on the line n = ( n − / For generic values of ν the expression (4.24) is finite. However, we are interested in ν = n + (cid:15) close to an integer value of n where the expression becomes singular. In such cases (4.24)can be series expanded around ν = n up to terms vanishing as (cid:15) →
0. The expansion canbe obtained by representing the hypergeometric function in terms of its usual power series, F ν ( x ) = Γ(1 − ν )Γ(1 + ν ) ∞ (cid:88) k =0 Γ(1 + ν + k )Γ(1 − ν + k ) x k . (4.27)In the case of interest, ν = − (cid:15) , the relevant expansion is F − (cid:15) ( x ) = 1 + F (1) − ( x ) (cid:15) + F (2) − ( x ) (cid:15) + O ( (cid:15) ) , (4.28)where the expansion coefficients can be resummed as F (1) − ( x ) = 1 − (cid:18) − x (cid:19) log(1 − x ) , (4.29) F (2) − ( x ) = 2 + (cid:18) − x (cid:19) (cid:2) − log(1 − x ) + log (1 − x ) + Li x (cid:3) . (4.30)Combining everything we obtain an analytic expression for I (cid:15) {− (cid:15), − (cid:15), − (cid:15) } up toterms vanishing as (cid:15) →
0. Using (3.3), we then arrive at I (cid:15) { − (cid:15), − (cid:15), − (cid:15) } = ( p p p ) − (cid:15) ) I (cid:15) {− (cid:15), − (cid:15), − (cid:15) } . (4.31)Following the method described in section (2.5.2), we can now change the regularisationscheme from − u = v = v = v to a general scheme with arbitrary u and v j , j = 1 , , Discussion
In this paper we showed how to evaluate the integrals needed for the computation ofmomentum-space 3-point functions of operators of integer dimension in any CFT. Togetherwith the results in [25, 26], one can now obtain explicit expressions for all such scalar 3-point functions and for all tensorial correlators that do not require renormalisation. Theresults here are also sufficient for the computation of tensorial correlators that do requirerenormalisation, once this renormalisation has been carried out as discussed in [27, 28].After reducing to triple- K integrals, the most nontrivial cases are those with dimensionssatisfying the triangle inequalities in (1.1) and we developed a comprehensive procedurefor the evaluation of all such integrals. We showed that all such integrals can be reduced tothe master integral I { } and we computed this integral, with the answer given in (4.2).In all remaining cases the computation is more straightforward. If the spacetimedimension is odd, then all Bessel K functions appearing in the triple- K integrals becomeelementary and can be computed straightforwardly (the result is given in appendix C). Ifthe spacetime dimension is even and the conditions above are not satisfied, the correlatorcan be extracted from the divergent part of the triple- K integrals, as discussed in [26]. Acknowledgements
AB and KS would like to thank the Galileo Galilei Institute in Florence for support andhospitality during the workshop “Holographic Methods for Strongly Coupled Systems” andPM thanks the Centre du Recherches Mathematiques, Montreal. KS gratefully acknowl-edges support from the Simons Center for Geometry and Physics, Stony Brook Universityand the “Simons Summer Workshop 2015: New advances in Conformal Field Theories”during which some of the research for this paper was performed. AB is supported by the In-teruniversity Attraction Poles Programme initiated by the Belgian Science Policy (P7/37)and the European Research Council grant no. ERC-2013-CoG 616732 HoloQosmos. ABwould like to thank COST for partial support via the STMS grant COST-STSM-MP1210-29014. PM is supported by the STFC Consolidated Grant ST/L00044X/1. AB and PMwould like to thank the University of Southampton for hospitality during parts of this work.
A Useful formulae
The Bessel function I , also known as the modified Bessel function of the first kind, isdefined by the series I ν ( x ) = ∞ (cid:88) j =0 j !Γ( ν + j + 1) (cid:16) x (cid:17) ν +2 j , ν (cid:54) = − , − , − , . . . (A.1)The Bessel function K , or modified Bessel function of the second kind, is defined by K ν ( x ) = π νπ ) [ I − ν ( x ) − I ν ( x )] , ν / ∈ Z , (A.2) K n ( x ) = lim (cid:15) → K n + (cid:15) ( x ) , n ∈ Z . (A.3)– 27 –or x >
0, the finite point-wise limit exists for any integer n .The series expansion of the Bessel function K ν for ν / ∈ Z is given directly in terms ofthe expansion (A.1) via the definition (A.2). In particular K ν ( x ) = ∞ (cid:88) j =0 (cid:104) a − j ( ν ) x − ν +2 j + a + j ( ν ) x ν +2 j (cid:105) , ν / ∈ Z , (A.4)where the expansion coefficients read a σj ( ν ) = ( − j Γ( − σν − j )2 σν +2 j +1 j ! , σ ∈ {± } . (A.5)For non-negative integer index n , the expansion reads instead K n ( x ) = 12 (cid:16) x (cid:17) − n n − (cid:88) j =0 ( n − j − j ! ( − j (cid:16) x (cid:17) j + ( − n +1 log (cid:16) x (cid:17) I n ( x )+ ( − n (cid:16) x (cid:17) n ∞ (cid:88) j =0 ψ ( j + 1) + ψ ( n + j + 1) j !( n + j )! (cid:16) x (cid:17) j , (A.6)where ψ is the digamma function. At large x , the Bessel functions have the asymptoticexpansions I ν ( x ) = 1 √ π e x √ x + . . . , K ν ( x ) = (cid:114) π e − x √ x + . . . , ν ∈ R . (A.7)Appell’s F function can be defined by the double series [45, 46] F ( α, β ; γ, γ (cid:48) ; ξ, η ) = ∞ (cid:88) i,j =0 ( α ) i + j ( β ) i + j ( γ ) i ( γ (cid:48) ) j i ! j ! ξ i η j , (cid:112) | ξ | + (cid:112) | η | < , (A.8)where ( α ) i is a Pochhammer symbol. Notice that F ( α, β ; γ, γ (cid:48) ; ξ, η ) = F ( β, α ; γ, γ (cid:48) ; ξ, η ) = F ( α, β ; γ (cid:48) , γ ; η, ξ ) . (A.9)The series representation, however, is not very useful as in our case ξ = p p , η = p p (A.10)and the series only converges when p > p + p , which is opposite to the triangle inequalityobeyed by physical momentum configurations.– 28 –he following reduction formulae can be found in [47] or [45] F (cid:18) α, β ; α, β ; − x (1 − x )(1 − y ) , − y (1 − x )(1 − y ) (cid:19) = (1 − x ) β (1 − y ) α − xy , (A.11) F (cid:18) α, β ; β, β ; − x (1 − x )(1 − y ) , − y (1 − x )(1 − y ) (cid:19) = (1 − x ) α (1 − y ) α F ( α, α − β ; β ; xy ) , (A.12) F (cid:18) α, β ; 1 + α − β, β ; − x (1 − x )(1 − y ) , − y (1 − x )(1 − y ) (cid:19) = (1 − y ) α F (cid:18) α, β ; 1 + α − β ; − x (1 − y )1 − x (cid:19) , (A.13) F (2 ν − , ν ; ν ; x ) = (1 − x ) − ν . (A.14) B Triple- K and momentum-space integrals Let K d { δ δ δ } denote a massless scalar 1-loop 3-point momentum-space integral, K d { δ δ δ } = (cid:90) d d k (2 π ) d k δ | p − k | δ | p + k | δ . (B.1)Any such integral can be expressed in terms of triple- K integrals and vice versa. For scalarintegrals, the relation reads K d { δ δ δ } = 2 − d π d × I d − { d + δ − δ t , d + δ − δ t , d + δ − δ t } Γ( d − δ t )Γ( δ )Γ( δ )Γ( δ ) , (B.2)where δ t = δ + δ + δ . Its inverse reads I α { β β β } = 2 α − π α +1 Γ (cid:18) α + 1 + β t (cid:19) (cid:89) j =1 Γ (cid:18) α + 1 + 2 β j − β t (cid:19) × K α, { ( α +1+2 β − β t ) , ( α +1+2 β − β t ) , ( α +1+2 β − β t ) } , (B.3)where β t = β + β + β .All tensorial massless 1-loop 3-point momentum-space integrals can be also expressedin terms of a number of triple- K integrals when their tensorial structure is resolved bystandard methods, e.g. , [42, 48]. For exact expressions in this case see appendix A.3 of[25]. C Half-integral betas
Bessel K functions with half-integral indices reduce to elementary functions meaning thecorresponding triple- K integrals can be evaluated with ease. We present below a completeexpression valid for any triple- K integral in which all β j are positive half-integral numbers.– 29 – Bessel function K with a half-integral index is equal to K β ( x ) = e − x √ x | β |− (cid:88) j =0 c j ( β ) x j , β ∈ Z + 12 , (C.1)where the coefficients are c j ( β ) = (cid:114) π (cid:0) | β | − + j (cid:1) !2 j j ! (cid:0) | β | − − j (cid:1) ! . (C.2)Any triple- K integral for which all β j are half-integer then evaluates to I α { β β β } = | β |− (cid:88) k =0 | β |− (cid:88) k =0 | β |− (cid:88) k =0 Γ (cid:18) α − − k t (cid:19) p + k t − αt × p β − k − p β − k − p β − k − c k ( β ) c k ( β ) c k ( β ) , (C.3)where k t = k + k + k and p t = p + p + p , and the value of α is arbitrary.For 3-point functions featuring operators of integral dimensions in odd -dimensionalspacetimes α is half-integer. In such cases the gamma function in the expression abovemay become singular, provided the condition (2.15) is satisfied. Assuming α is half-integer,one can regulate (C.3) in a scheme with all v j = 0 by shifting α (cid:55)→ ˜ α = α + u(cid:15) . The usualexpansion of the gamma function can then be applied. To change the regularisation schemeto one with non-vanishing v j we then follow the procedure of section 2.5.2. D Derivation of scheme-changing formula
Our goal in this section is to derive the formula (2.31) for changing the regularisationscheme. Following the discussion in section 2.5.2, our first move is to split the regulatedtriple- K integral into three parts, I ˜ α { ˜ β j } = I (div)˜ α { ˜ β j } + I (lower)˜ α { ˜ β j } + I (upper)˜ α { ˜ β j } , (D.1)where I (div) is given by (2.24) and I (upper)˜ α { ˜ β j } = (cid:90) ∞ µ − d x x ˜ α (cid:89) j =1 p ˜ β j j K ˜ β j ( p j x ) , (D.2) I (lower)˜ α { ˜ β j } = (cid:90) µ − d x x ˜ α (cid:89) j =1 p ˜ β j j K ˜ β j ( p j x ) − I (div)˜ α { ˜ β j } . (D.3)Both I (upper) and I (lower) as defined here are clearly finite in the limit (cid:15) → I (upper) converges for any ˜ α and ˜ β provided µ − > I (lower) is finite since all divergent termshave been explicitly subtracted in its definition.We now want to show that the finite pieces ( i.e., terms of order (cid:15) ) in I (upper) and I (lower) are independent of u and v j . The finite part of the difference between a regulated– 30 –riple- K integral and its divergent part I (div) is then scheme independent, reproducing(2.30) and (2.31).To show the finite part of I (upper) is scheme independent, note thatlim (cid:15) → I (upper)˜ α { ˜ β j } = (cid:90) ∞ µ − d x lim (cid:15) → x ˜ α (cid:89) j =1 p ˜ β j j K ˜ β j ( p j x ) = (cid:90) ∞ µ − d x x α (cid:89) j =1 p β j j K β j ( p j x ) . (D.4)For any µ − >
0, the exchange of the integral and the limit here is justified by the dominatedconvergence theorem. The finite part of I (upper) is therefore scheme independent since theright-most expression is independent of u and v j .Similar arguments can be used to show that the finite part of I (lower) is scheme in-dependent. First, we series expand all Bessel functions in the integrand of the triple- K integral. In the following, it will be useful to denote the coefficient of x A + B(cid:15) log N x inthis expansion as c A + B(cid:15),N . Note that the value of A is independent of u and v j , whereas B = B ( u, v j ) is scheme dependent. We can then write I (lower)˜ α { ˜ β j } = (cid:88) A, B ∈ R A (cid:54) = − N ∈{ , ,... } (cid:90) µ − d x x A + B ( u,v j ) (cid:15) log N x c A + B ( u,v j ) (cid:15),N , (D.5)where the sum is taken over all terms appearing in the series expansion of the triple- K integrand except for those of the form x − O ( (cid:15) ) ( i.e., for which A = − I (div) but were subtracted in the definition of I (lower) (see (D.3)). Our exchangein the order of summation and integration going from (D.3) to (D.5) is justified by Fubini’stheorem, noting that the sum converges absolutely. The natural analytic continuation of(2.17) can then be used to extend the result to any values of α and β j .For purposes of illustration let us now concentrate on two special cases; the generalcase can then be handled by similar arguments. As a first case, consider the regularisationscheme where all v j = 0. From the series expansion (A.6), all c A + B(cid:15),N in (D.5) are thenfinite and scheme-independent. The remaining integral is given by (2.18), and since A (cid:54) = −
1, the finite part of I (lower) is indeed scheme-independent as we wished to show.As our second case, consider the opposite situation where all v j (cid:54) = 0. The c A + B(cid:15),N then vanish for all
N > c A + B(cid:15), can be expanded in terms of the Bessel expansioncoefficients a σj ( β ) in (2.25). The result takes a form similar to (2.27), namely I (lower)˜ α { ˜ β j } = (cid:88) cond (cid:48) µ − W W (cid:89) j =1 p (1+ σ ) ˜ β j +2 n j j a σ j n j ( ˜ β j ) , (D.6)where W = ˜ α + 1 + (cid:88) j =1 ( σ j ˜ β j + 2 n j ) . (D.7)This time, however, the summation runs over all terms in the complement of those presentin (2.27), which we have indicated with a prime on the summation sign. In other words,– 31 –he sum runs over all σ , σ , σ ∈ {± } and non-negative integers n , n , n such that thecondition (2.26) is not satisfied.Clearly 1 /W has a finite and scheme-independent limit as (cid:15) → β j is non-integer, a σ j n j ( ˜ β j ) has a finite and scheme-independentlimit as (cid:15) → β j ∈ Z , on the other hand, a σ j n j ( ˜ β j )may diverge as (cid:15) →
0. More precisely, a − n j ( ˜ β j ) has a finite, scheme-independent limit if n j < β j but otherwise diverges, while a + n j ( ˜ β j ) always diverges for integer β j . Nevertheless,such divergences do not lead to a divergence in I (lower) since the divergent contributionsfrom a + n j ( ˜ β j ) and a − n j + β j ( ˜ β j ) cancel. To see this, consider for concreteness the case β ∈ Z .Denoting the values of (D.7) with given n and σ = ± W ± n , the corresponding contributions to (D.6) take the form µ − W + n W + n p β +2 n a + n ( ˜ β ) + µ − W − n β W − n + β p β + n )1 a − n + β ( ˜ β ) . (D.8)By inspection, however, this expression has a finite and scheme-independent limit as (cid:15) → a + n ( ˜ β ) can be matched with its corresponding a − n + β ( ˜ β ), the expression (D.6)therefore has a finite and scheme-independent limit. The finite part of I (lower) is thus againscheme-independent as we wished to show. References [1] A. M. Polyakov,
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