One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations
OOne-loop Parke-Taylor factors for quadratic propagatorsfrom massless scattering equations
Humberto Gomez, a,b
Cristhiam Lopez-Arcos b and Pedro Talavera c a Niels Bohr International Academy and Discovery Center, University of CopenhagenBlegdamsvej 17, DK-2100 Copenhagen , Denmark, b Universidad Santiago de Cali, Facultad de Ciencias Basicas,Campus Pampalinda, Calle 5 No. 62-00, C´odigo postal 76001, Santiago de Cali, Colombia c Institut de Ciencies del Cosmos, Universitat de Barcelona (IEEC-UB),Marti i Franques 1, Barcelona 08028, Spain
E-mail: [email protected], [email protected], [email protected]
Abstract:
In this paper we reconsider the Cachazo-He-Yuan construction (CHY) of the socalled scattering amplitudes at one-loop, in order to obtain quadratic propagators. In theorieswith colour ordering the key ingredient is the redefinition of the Parke-Taylor factors. Afterclassifying all the possible one-loop CHY-integrands we conjecture a new one-loop amplitudefor the massless Bi-adjoint Φ theory. The prescription directly reproduces the quadraticpropagators of the traditional Feynman approach. a r X i v : . [ h e p - t h ] N ov ontents Φ scalar theory 17 at One-Loop 184.2 A New Proposal 19 M − loop3 [1 , , | , ,
3] 215.1.2 Obtaining M − loop3 [1 , , | , ,
1] 245.2 Four-point 255.3 General Structure of M − loop N [ N | N ] 27 i(cid:15) prescription 298 Discussions 30A CHY-integrands at One-loop 31B Proof of the one-loop Parke-Taylor factor expansion 35C From quadratic to Linear propagators in the CHY-graphs 37 Our most accurate knowledge on quantum field theory in Minkowski (3+1)-d is almost entirelybased on perturbation methods. At leading order, quantities such as scattering amplitudesare computed by adding tree-like diagrams. Even this elementary manipulation becomes aformidable task at rather low multiplicity kinematics and become only technically feasible byusing the Weyl-van der Waerden spinor calculus.– 1 –he use of on-shell methods for the calculation of scattering amplitudes has come intoattention since the last decade, following Witten’s seminal work [1]. A remarkable develop-ment following these ideas is the approach by Cachazo-He-Yuan (CHY) [2, 3], it has the greatadvantages of being applicable to several dimensions and also to a large array of theories[4–7], even beyond field theory [8, 9]. The main ingredient for this approach are the tree-levelscattering equations [2] E a := (cid:88) b (cid:54) = a k a · k b σ ab = 0 , σ ab := σ a − σ b , a = 1 , , . . . , n, (1.1)where the σ a ’s denote punctures on the sphere. The tree-level S-matrix can be written interms of contour integrals localized over solutions of these equations on the moduli space ofn-punctured Riemann spheres A n = (cid:90) Γ dµ tree n I CHYtree ( σ ) , (1.2)where the integration measure, dµ tree n , is given by dµ tree n = (cid:81) na =1 dσ a Vol (PSL(2 , C )) × ( σ ij σ jk σ ki ) (cid:81) nb (cid:54) = i,j,k E b (1.3)and the contour Γ is defined by the n − E b = 0 , b (cid:54) = i, j, k . (1.4)The integrand, I CHYtree , depends on the described theory. There are other approaches that usethe same moduli space [1, 10–12], but restricted to four dimensions.There have been developed several methods to evaluate the integrals, from differentperspectives. Some approaches study the solutions to the scattering equations for particularkinematics and/or dimensions [4, 13–19], others work with a polynomial form [20–29], orformulating sets of integration rules [30–34]. A different approach was proposed in [35],taking the double covered version of the sphere, the so called Λ-algorithm, which we willemploy in this work.A generalization for loop level of the CHY formalism has been made. The ambitwistorand pure spinor ambitwistor worldsheet [36, 37] provided a prescription for a generalizationto higher genus Riemann surfaces [38–40]. A different approach was also developed in [41–43],where the forward limit with two more massive particles, playing the role of the loop momenta,were introduced. The scattering equations for massive particles were already studied in[44, 45]. Another alternative approach using an elliptic curve was developed in [46, 47].The previous prescriptions give a new representation of the Feynman integrals with prop-agators linear in loop momenta. In order to find the equivalence with the usual Feynmanpropagators, ( (cid:96) + K ) − , two additional steps must be taken: the first one is the use of partialfractions, and the second one is the shifting of loop momenta [39, 48].– 2 –ecently, one of the authors [49] proposed a different approach to obtain the quadraticFeynman propagators directly from the CHY-integrands for the scalar Φ theory . Themotivation came by analysing a Riemann surface of genus two after an unitary cut, whichlook exactly like a tree level diagram before the forward limit, but instead of the two massiveparticles associated to the loop momenta there are four massless particles. This new approachallows to work again with the scattering equations for massless particles, but at the expensesof increasing the number to n + 4. In addition there is also the need to introduce a newmeasure of integrations that guarantees the cut and then take the forward limit.In the present work, we follow the line of thought of [49] and propose a reformulation forthe one-loop Parke-Taylor factors. Splitting the massive loop momenta ( (cid:96) + , (cid:96) − ) into the fourmassless ones (( a , b ) , ( b , a )), we will have one-loop Parke-Taylor factors that will enter intoCHY-integrands to lead directly to the usual Feynman propagators. The CHY-integrands inquestion are the ones for the Bi-adjoint Φ scalar theory. Outline
This paper is organized as follows. In section 2 we present our reformulation of the one-loopParke-Taylor factors (PT). The expression is written in terms of the generalized holomorphicone-form on the Torus, ω a : bi : j . By exploiting algebraic identities we formulate the Theorem1: each term in the PT factors can be decomposed into terms containing at least two ω a : bi : j factors, i.e. the PT factors are rearranged in an expansion manifestly tadpole-free .In section 3 we write, classify and match with their Feynman integrands counterparts,some general type of CHY-integrands that can appear at one-loop level. Since we are workingwith n + 4 massless particles, the contour integrals can be calculated using any of the existingmethods of integration. As we have mentioned already we employ the so-called Λ-algorithm,with the choice of a new gauge fixing, to solve them. This allows to analytically evaluatearbitrary CHY-integrals using simple graphical rules. The classification is made tracing thestructures defined in section 2. As will become clear each element inside the partial amplitudehave an unambigous correspondence with the elements of the CHY-graphs, starting with then-gon, then following with the ones with tree level structures attached to their corners.Section 4 shows our proposal for the partial amplitude of the Bi-adjoint Φ theory atone-loop with quadratic propagators: first we give a simple review at tree level and one-loopwith linear propagators, then we propose our formula using our definition for the PT factors.In order to support our proposition in section 5 we perform explicitly the calculationfor the partial amplitudes of the three and four-point functions. We make an extensive useof the results of previous sections. In particular we emphasize the direct interpretation ofthe CHY-integrals in terms of Feynman diagrams. This mapping is codified in the followingequality at the integrand level There are some overlapping ideas with the recent paper published by Farrow and Lipstein [50]. As it will be explained, the number of ω a : bi : j is related with the polygon of the loop, for example, two ω a : bi : j in the left integrand can only generate a bubble or a triangle. – 3 – N +1 (cid:82) d Ω × s a b × (cid:82) dµ tree N +4 � � � � � � � �
12 3 n - N tree - - - - n = � � � � � � � � � l N P n tree - - ntree - that constitutes one of the most important results of this work and it will be explained indetail during the course of this paper.In section 6 we comment on the issue of the external-leg bubble contributions. Diagramsinvolved are singular and need to be regularized. Next section, 7, is for illustrative purposesand is devoted to the i(cid:15) prescription and how to directly obtain it by dimension reduction.Finally, in section 8 we conclude by summarizing our findings.For not disrupting the line of the paper more technical discussions have been gathered insome appendices: In Appendix A we explicitly show the sufficient form of the measure (3.5) totackle the one-loop CHY-integral prescription. In particular how the momenta combination itcontains arrises. Proof of Theorem 1 is casted in Appendix B where it is discussed at length.Appendix C collects the relation between some techniques developed across the paper andthe linear propagator prescription. We conclude by probing an statement of [47].Before beginning section 2, we define the notation that is going to be used in the paper. Notation