Conformal n-point functions in momentum space Adam Bzowski∗ Department of Physics and Astronomy, Uppsala University, 751 08 Uppsala, Sweden. Paul McFaddeny School of Mathematics, Statistics & Physics, Newcastle University, Newcastle NE1 7RU, U.K. Kostas Skenderisz 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<j≤4 where xij = xi xj are the coordinate separations and j − j 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 f^(^u; v^) = 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 j − j 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- × j − j j − j 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 f^(^u; v^) 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 u^ = 2j − j2 ; v^ = 2j − j2 : (6) q p2 + q3 q1 q p3 + q1 q2 standing has yet to be achieved. Nevertheless, through 2j − j 3j − j 2 The role ofu ^ andv ^ 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 ^ δ12,δ34 δ13,δ24 δ14,δ23 α β f(^u; v^) = Cβ Cα−β C−α u^ v^ ; (12) By acting with κ on the integrand of (4), one can show K where ^ ! 3 κ f(^u; v^) 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) !# − − − − @f^ @f^ (13) αµκu^ + αµκv^ + αµκf^ : (8) (n) @u^ (n) @v^ (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 ^ (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 j − j 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). × j − j j − j 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 f^(^u; v^) can be expressed as a double inverse Mellin transform over the monomialu ^αv^β. 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+i1 Z b2+i1 1 vergences requiring regularization and renormalization. = 2 dα dβ ρ(α; β) Wα,β (18) (2πi) b1−i1 b2−i1 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.