Conformal n-point functions in momentum space
Adam Bzowski∗ Department of Physics and Astronomy, Uppsala University, 751 08 Uppsala, Sweden.
Paul McFadden† School of Mathematics, Statistics & Physics, Newcastle University, Newcastle NE1 7RU, U.K.
Kostas Skenderis‡ STAG Research Center & Mathematical Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, U.K.
We present a Feynman integral representation for the general momentum-space scalar n-point function in any conformal field theory. This representation solves the conformal Ward identities and features an arbitrary function of n(n − 3)/2 variables which play the role of momentum-space conformal cross-ratios. It involves (n − 1)(n − 2)/2 integrations over momenta, with the momenta running over the edges of an (n − 1)-simplex. We provide the details in the simplest non-trivial case (4-point functions), and for this case we identify values of the operator and spacetime dimensions for which singularities arise leading to anomalies and beta functions, and discuss several illustrative examples from perturbative quantum field theory and holography.
I. MOTIVATION recent efforts, all the necessary prerequisites are now in place. Firstly, the momentum-space 3-point functions The structure of correlation functions in a conformal of general scalar and tensorial operators are known, in- field theory (CFT) is highly constrained by conformal cluding the cases where anomalies and beta functions symmetry. It has been known since the work of Polyakov arise as a result of renormalization [34–46]. Secondly, [1,2] that the most general 4-point function of scalar momentum-space studies of the 4-point function have yielded special classes of solutions to the conformal Ward primary operators ∆j , each of dimension ∆j, takes the form O identities [15, 32, 47–51]. Here, our aim is now to provide the general solution for the momentum-space n-point
∆1 (x1) ∆2 (x2) ∆3 (x3) ∆4 (x4) function. We start by providing a complete discussion of hO O O O i Y 2δ the 4-point function and an exploration of its properties, = f(u, v) x ij , (1) ij and we then present the result for the n-point function. 1≤i where xij = xi xj are the coordinate separations and | − | II. MOMENTUM-SPACE REPRESENTATION 4 ∆t X 2δij = ∆i ∆j, ∆t = ∆i. (2) 3 − − For scalar 4-point functions, our main result is the gen- i=1 eral momentum-space representation: The 4-point function is thus determined up to an arbi- ∆ (p1) ∆ (p2) ∆ (p3) ∆ (p4) trary (theory-specific) function f of the two conformal hhO 1 O 2 O 3 O 4 ii cross-ratios, Z ddq ddq ddq fˆ(ˆu, vˆ) = 1 2 3 . (4) 2 2 2 2 (2π)d (2π)d (2π)d Den (q , p ) arXiv:1910.10162v3 [hep-th] 3 Apr 2020 3 j k x13x24 x12x34 u = 2 2 , v = 2 2 . (3) x14x23 x13x24 d P Here, ... = ... (2π) δ( pi), d is the spacetime di- h i hh ii i This result straightforwardly generalizes to n-point func- mension, and the denominator is tions, which now involve an arbitrary function of n(n 2δ12+d 2δ13+d 2δ23+d 2δ14+d − Den3(qj, pk) = q q q p1 + q2 q3 3)/2 cross ratios. 3 2 1 | − | 2δ24+d 2δ34+d These results are easy to derive in position space where p2 + q3 q1 p3 + q1 q2 (5) the conformal group acts naturally. Yet for many mod- × | − | | − | ern applications, including cosmology [3–19], condensed where the δij are given in (2). We work in Euclidean matter [20–25], anomalies [26–28] and the bootstrap pro- signature throughout. As expected from (1), this 4-point gramme [29–33], it would be highly desirable to know the function depends on an arbitrary function fˆ(ˆu, vˆ) of two analogue of this result – and, indeed, the analogue of the variables: conformal cross-ratios themselves – in momentum space. 2 2 2 2 Despite the lapse of nearly five decades, such an under- q1 p1 + q2 q3 q2 p2 + q3 q1 uˆ = 2| − |2 , vˆ = 2| − |2 . (6) q p2 + q3 q1 q p3 + q1 q2 standing has yet to be achieved. Nevertheless, through 2| − | 3| − | 2 The role ofu ˆ andv ˆ is analogous to that of the position- p4 p3 space cross-ratios u and v defined in (3). These vari- p + q q 3 1 − 2 ables are thus the desired momentum-space cross-ratios, though notice they depend on the momenta q that are p2 + q3 q1 j − subject to integration in (4). p + q q q1 1 2 − 3 A. Proof of conformal invariance q2 The conformal invariance of (4) can be verified by direct substitution into the conformal Ward identities q3 (CWIs). Its Poincar´einvariance is manifest, and its scal- p1 p2 ing dimension is given by the sum of powers in (5) mi- nus 3d from the three integrals. This gives 2δt 3d = FIG. 1. The 3-loop tetrahedral integral (15), where each in- − − ∆t 3d, the correct result in momentum space. ternal line corresponds to a generalized propagator in (16). The− remaining CWIs associated with special conformal transformations are implemented by the second-order dif- ferential operator κ = P3 κ, where [36] K j=1 Kj Rather, for f(u, v) = uαvβ, the Fourier transform is given by (4) with κ κ ∂ ∂ α ∂ ∂ ∂ = p 2p + 2(∆j d) . (7) Kj j ∂pα ∂pα − j ∂pα ∂pκ − ∂pκ j j j j j ˆ δ12,δ34 δ13,δ24 δ14,δ23 α β f(ˆu, vˆ) = Cβ Cα−β C−α uˆ vˆ , (12) By acting with κ on the integrand of (4), one can show K where ˆ ! 3 κ f(ˆu, vˆ) X ∂ (qn)α d d 0 = µ δ,δ0 δ+δ0+2σ+d d Γ 2 + δ + σ Γ 2 + δ + σ K Den (q , p ) ∂qn Den (q , p ) Cσ = 4 π . 3 j k n=1 3 j k Γ( δ σ)Γ( δ0 σ) !# − − − − ∂fˆ ∂fˆ (13) αµκuˆ + αµκvˆ + αµκfˆ . (8) (n) ∂uˆ (n) ∂vˆ (n) × A B C This follows since the Fourier transform of a product is a convolution of Fourier transforms, and so we can write In order to write these coefficients explicitly, we define h 2(β+δ ) 2(β+δ ) 2(α−β+δ ) 2(α−β+δ ) 2kβ x 12 x 34 x 13 x 24 Aαµκ = n δκαδµ δµαδκ δµκδα , (9) F 12 34 × 13 24 (n) k2 β − β − β i n x2(−α+δ14)x2(−α+δ23) × 14 23 where the kn are the vectors featuring in (5), i.e., k1 = = [x2(β+δ12)x2(β+δ34)] [x2(α−β+δ13)x2(α−β+δ24)] p1 + q2 q3 along with cyclic permutations. The coeffi- 12 34 13 24 − F ∗ F cients in equation (8) are then [x2(−α+δ14)x2(−α+δ23)], (14) ∗ F 14 23 αµκ d αµκ d αµκ = ( + δ24)A + ( + δ34)A , (10) C(1) 2 (2) 2 (3) where denotes the convolution in all variables, namely ∗ 4 ddq αµκ αµκ R Q j ˆ (f g)(pk) = j=1 (2π)d f(qj)g(pj qj). With the f with C(2) and C(3) following by cyclic permutation of ∗ − the indices 1, 2, 3, while in (12), the momentum-space integral in (4) becomes αµκ αµκ αµκ αµκ αµκ Z d d d d q1 d q2 d q3 1 (1) = A(2) , (1) = A(3) A(2) , A B − Wα,β = d d d (αβ) , (15) αµκ αµκ αµκ αµκ (2π) (2π) (2π) = A , = A , Den3 (qj, pk) A(2) − (1) B(2) (3) αµκ = Aαµκ Aαµκ, αµκ = Aαµκ. (11) where A(3) (2) − (1) B(3) − (2) κ (αβ) 2γ12 2γ13 2γ23 2γ14 As the action of on the integrand of (4) is a to- Den (qj, pk) = q q q p1 + q2 q3 K 3 3 2 1 | − | tal derivative, the integral itself is then invariant. This 2γ24 2γ34 p2 + q3 q1 p3 + q1 q2 (16) proves the conformal invariance of the representation (4). × | − | | − | and B. The tetrahedron γ12 = δ12 + β + d/2, γ13 = δ13 + α β + d/2, − γ23 = δ23 α + d/2, γ14 = δ14 α + d/2, The momentum-space expression (4) is not the direct − − γ24 = δ24 + α β + d/2, γ34 = δ34 + β + d/2. (17) Fourier transform of the position-space expression (1). − 3 This is a 3-loop Feynman integral with the topology of a tetrahedron as presented in Fig.1. The four momenta (a) (b) (c) entering the vertices are those of the external operators, while the six internal lines describe generalized propa- gators in which the momenta are raised to the specific powers given in (17). FIG. 2. Simplifications of the kernel Wα,β : (a) where a C. Spectral representation propagator in (16) appears with γij = d/2 + n the loop order is reduced by one; (b) with two such propagators we obtain a 1-loop box; (c) for γij = −n, we obtain a 3-point function. Where convergence permits, the function fˆ(ˆu, vˆ) can be expressed as a double inverse Mellin transform over the monomialu ˆαvˆβ. The 4-point function (4) then ad- D. Singularities and renormalization mits the spectral representation For special values of the spacetime and operator di- ∆ (p1) ∆ (p2) ∆ (p3) ∆ (p4) hhO 1 O 2 O 3 O 4 ii mensions, momentum-space CFT correlators exhibit di- Z b1+i∞ Z b2+i∞ 1 vergences requiring regularization and renormalization. = 2 dα dβ ρ(α, β) Wα,β (18) (2πi) b1−i∞ b2−i∞ All divergences are local can be removed through the ad- dition of covariant counterterms giving rise to conformal for an appropriate choice of integration contour specified anomalies and beta functions for composite operators. by b and b . Here, W is a universal kernel corre- 1 2 α,β The renomalization of 3-point functions was studied in sponding to the tetrahedron integral (15) and ρ(α, β) [38–40]. For 4-point functions, a similar analysis holds as is a theory-specific spectral function derived from the we now discuss. Mellin transform of fˆ(ˆu, vˆ). Where the position-space Firstly, renormalizability requires that all UV diver- Mellin representation of a 4-point function is known – as gences should be either ultralocal, with support only is often the case for holographic CFTs [52–54] – the cor- when all four position-space insertions are coincident, responding ρ(α, β) in momentum space can be read off or else semilocal, meaning they are supported only in immediately using equations (12) and (13). the cases where either (i) x1 = x2 = x3 = x4, (ii) To evaluate the spectral integral, we close the contour 6 x1 = x2 = x3 = x4, or (iii) x1 = x2 = x3 = x4, along and sum the residues. For certain α and β, these residues with all related6 cases obtained by permutation.6 6 In mo- are simple to evaluate due to reductions in the loop order mentum space, ultralocal divergences are thus analytic of W . Such reductions arise whenever a propagator in α,β in all the squared momenta, while semilocal divergences the denominator (16) appears with a power γ = d/2+n, ij are analytic in at least one squared momentum. (Cases for some non-negative integer n. This can be seen by (i) and (ii) have a momentum-dependence matching that noting that, in a distributional sense as q 0, → of a 2-point function, while that of case (iii) corresponds d/2 to a 3-point function.) π n (d) lim = δ (q). (19) →0 qd+2n−2 4nn!Γ(d/2 + n) These divergences constitute local solutions of the CWI. Their form, as well as the d and ∆j for which they We then obtain a pole in α, β whose residue is given appear, can be predicted from an analysis of local coun- by a 2-loop integral as shown in Fig.2a. Where the terterms. Such counterterms exist only in cases where external dimensions permit, such poles can also coincide. 4 In Fig.2b, we illustrate the case where α β = δ14 = δ23, X − d + σj(∆j d/2) = 2n (20) creating a pair of delta functions δ(q2)δ(p2 + q3 q1) for − − j=1 which the residue is a 1-loop box. − Simplifications of a different kind occur whenever a for some n non-negative integer, with signs σj whose val- propagator in (16) appears with a vanishing power, or ues are either all minus, or else three minus and one plus. more generally for γij = n. This results in a contraction Ultralocal divergences are removed by counterterms − of the corresponding leg of the tetrahedron, producing a that are quartic in the sources ϕj for the operators ∆j . 1-loop triangle for which two of the legs are bubbles as These feature 2n fully-contracted derivatives whoseO ac- shown in Fig.2c. Evaluating the bubbles, one obtains tion is distributed over the sources, and exist whenever a pure 1-loop triangle whose propagators are raised to (20) is satisfied with all minus signs. Since the scaling di- new powers. This integral is equivalent to a general CFT mension of ϕj is d ∆j, this ensures the counterterm has 3-point function [36]. The locality can be understood by overall dimension −d. The appearance of ultralocal diver- noting that a denominator q−2n corresponds to a factor gences when this condition is satisfied can be seen by ex- −(d+2n) xij in position space, which is equivalent to a delta amining the region of integration where all three loop mo- function via the position-space analogue of (19). menta in the kernel Wα,β become large simultaneously. 4 φ4 φ4 φ4 φ4 φ4 φ4 φ4 φ4 φ4 φ4 φ4 φ4 FIG. 3. Three distinct topologies of Feynman diagrams contributing to the connected part of h:φ4 ::φ4 ::φ4 ::φ4 :i. 2 Re-parametrizing qj = λqˆj whereq ˆ = 1, as λ the III. FREE FIELDS 1 → ∞ denominator in (15) scales as λ6d−∆t (1 + O(λ−2)), while R 3d−1 the numerator contributes a Jacobian factor dλ λ . Consider a free spin-0 massless field φ and connected The λ integral is then logarithmically divergent precisely 4-point functions of the operators of the form φn. In all when (20) is satisfied with all minus signs. (For nonzero cases, the function f in position space is a sum of mono- n, the divergence derives from expanding the denomi- mials of the form uαvβ. For example, for the connected −2 2 2 2 2 nator to subleading order in powers of λ .) After the 4-point function :φ ::φ ::φ ::φ : conn, one has divergence is subtracted and the regulator removed, the h i 1 ∆ 1 ∆ 1 ∆ renormalized correlator has the expected nonlocal mo- u 6 φ2 v 6 φ2 w 6 φ2 f(u, v) + + (21) mentum dependence and obeys anomalous CWIs due to ∼ v w u the RG scale introduced by the counterterm, see [38]. 2 where ∆ 2 = d 2 is the dimension of φ , and to write f φ − Semilocal divergences are removed by counterterms fea- and fˆ succinctly we introduce the additional conformal turing one operator and multiple sources. For quar- ratios w andw ˆ defined by uvw = 1 andu ˆvˆwˆ = 1. Equa- tic counterterms, we have 2n fully-contracted deriva- tion (12) now yields the momentum space fˆ. In this case, tives whose action is distributed over ϕ1ϕ2ϕ3 ∆ . Such O 4 however, the prefactor in (12) vanishes as two out of the counterterms exist whenever (20) is satisfied with signs six gamma functions in the denominator of (13) diverge. ( +) (or some permutation thereof), ensuring the − − − This means we have to consider the regulated expression counterterm has dimension d. The resulting 4-point con- with regulated fˆ, namely tribution then has the momentum-dependence of a 2- 1 1 point function and corresponds to case (i) above. This 6 ∆φ2 − 2 2 uˆ counterterm effectively reparametrizes the source for fˆ(ˆu, vˆ) = 16˜ + 2 cycl. perms., (22) ∆4 vˆ and we obtain a beta function in the renormalized theory.O where ˜ = (4π)d/2Γ(d/2) and 2 cycl. perms. denotes two The appearance of a semilocal divergence in Wα,β when the ( +) condition is satisfied can be seen by remaining terms with cyclic permutations of the ratios, − − − uˆ vˆ wˆ uˆ. After this is substituted into (4) re-parametrizing the loop momenta in (15) as q1 = λqˆ1 7→ 7→ 7→ 2 and the momentum space integrals carried out, the limit withq ˆ1 = 1, q2 = λqˆ1 + p3 + `2 and q3 = λqˆ1 p2 + `3. − 0 should be taken. The λ integral is then logarithmically divergent when → this condition is satisfied, and has a semilocal momen- The appearance of the double zero in (22) reflects the 2 fact that the only Feynman diagram contributing to this tum dependence that is non-analytic in p4 only. For the correlator has the topology of a box. If instead we con- permuted cases featuring ∆j with j = 1, 2, 3 in place O sider the 4-point function of :φ4 :, the contributing Feyn- of ∆4 , the corresponding reparametrization is simply q O= λqˆ leaving the other loop momenta fixed. This man diagram topologies are as presented in Fig.3. Up j j ˆ difference reflects our use of momentum conservation to to an overall symmetry factor, the regulated f reads eliminate p4 in (15). After renormalization, the corre- 1 1 1 12 ∆φ4 6 ∆φ4 − 2 lator is again fully nonlocal and obeys anomalous CWIs ˆ 2 vˆ 2 4 uˆ f(ˆu, vˆ) c2 + ˜ c2 + reflecting the presence of the beta function, see [38]. ∼ uˆ vˆ 1 ∆ − 1 ! Besides the quartic counterterms discussed above, vˆ4 12 φ4 4 + ˜2c2 + 2 cycl. perms., (23) which contribute solely to 4- and higher-point functions, 3 uˆ we may also have cubic and quadratic counterterms. Their form is already fixed from the renormalization of 2- where ∆φ4 = 2(d 2). The constants cn are defined and 3-point functions [38–40, 55], but they nevertheless recursively through− contribute to 4-point functions as well [56]. In particular, cubic counterterms with two sources and one operator re- Γ(∆φ)Γ(n∆φ)Γ(1 n∆φ) cn+1 = cn d/2 − (24) move semilocal divergences of types (ii) and (iii). (4π) Γ((n + 1)∆φ)Γ(1 (n 1)∆φ) − − 5 with c1 = 1 and ∆φ = d/2 1. These coefficients arise z34 from the evaluation of effective− propagators. Denoting 2 the standard massless propagator as D1(p) = 1/p , the z24 effective propagator Dn(p) with n lines in Fig.3 is Z4 Z3 z z = Z d 14 23 cn d q Dn−1(q) Z Z Dn(p) = = . (25) 1 2 p2−2(n−1)∆φ (2π)d p q 2 z | − | 13 Finally, the disconnected part of any correlator can also be represented by the function fˆ. As an example, z12 consider a generalized free field of dimension ∆O, for O which the position-space 4-point function has FIG. 4. Equivalent electrical circuits where the impedances 1 ∆ are related by zij = ZiZj /Zt. Setting zi4 = si for i = 1, 2, 3 v 3 O 2 2 2 f(u, v) + 2 cycl. perms. (26) and (z12, z23, z31) = (z /s3, z /s1, z /s2) gives a mapping of ∼ u Schwinger parameters converting the contact Witten diagram The momentum-space expression is then proportional to (29) into the form (4) with fˆ(ˆu, vˆ) given in (30). 2∆O −d 2∆O −d p1 p3 δ(p1 + p2)δ(p3 + p4) plus permutations, and can be represented by a regulated fˆ with quadruple zero, tetrahedral network where the impedance connecting the " 1 ∆ − # vertices (i, j) is zij = ZiZj/Zt (see Fig.4). Since all vˆ 3 O fˆ(ˆu, vˆ) = ˜4 + 2 cycl. perms. . (27) products of the impedances on opposite edges are equal, 2 uˆ z = z12z34 = z13z24 = z14z23, we can re-parametrise the tetrahedron in terms of z and the three variables si = zi4 for i = 1, 2, 3. With this change of variables, the contact IV. HOLOGRAPHIC CFTS diagram (28) can be mapped to the form (4), with Holographic 4-point functions are obtained by evalu- uˆ−∆t/12+d/2 ating Witten diagrams in anti-de Sitter space. These fˆ(ˆu, vˆ) = 16c0 (2π)3d/2 W vˆ yield compact scalar integral representations for the Z ∞ −∆t/2+3d−1 momentum-space 4-point functions. Such expressions dz z Kδ13−δ24 (z) must again be special cases of the general solution (4) × 0 √ √ for some appropriate fˆ. This function can be found in Kδ23−δ14 (z uˆ)Kδ12−δ34 (z/ vˆ). (30) × several ways as we now discuss. Since exchange Witten diagrams can be reduced to a sum of contact diagrams This can be directly verified by Schwinger parametrizing [57, 58], we focus here on the latter deferring a complete the three Bessel functions in terms of the si then per- discussion to [59]. In the simplest case of a quartic bulk forming the Gaussian integrations over the momenta qi interaction without derivatives, we find in (4). (For full details, see [59]). Remarkably, this fˆ features precisely the same integral (the ‘triple-K’) that Φ = ∆ (p1) ∆ (p2) ∆ (p3) ∆ (p4) hhO 1 O 2 O 3 O 4 ii describes the momentum-space 3-point function [36]. Z ∞ 4 d−1 Y ∆j −d/2 An alternative derivation of (30) starts from the = cW dz z pj K∆j −d/2(pjz), (28) 0 j=1 position-space Mellin representation for the contact Wit- ten diagram [53]. 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