One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations

Journal of High Energy Physics, Oct 2017

Abstract In this paper we reconsider the Cachazo-He-Yuan construction (CHY) of the so called scattering amplitudes at one-loop, in order to obtain quadratic propagators. In theories with colour ordering the key ingredient is the redefinition of the Parke-Taylor factors. After classifying all the possible one-loop CHY-integrands we conjecture a new one-loop amplitude for the massless Bi-adjoint Φ3 theory. The prescription directly reproduces the quadratic propagators of the traditional Feynman approach.

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One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations

HJE One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations Humberto Gomez 0 1 2 3 4 Cristhiam Lopez-Arcos 0 1 2 3 Pedro Talavera 0 1 2 0 Marti i Franques 1 , Barcelona 08028 , Spain 1 Campus Pampalinda , Calle 5 No. 62-00, Codigo postal 76001, Santiago de Cali , Colombia 2 Blegdamsvej 17 , DK-2100 Copenhagen , Denmark 3 Universidad Santiago de Cali, Facultad de Ciencias Basicas 4 Niels Bohr International Academy and Discovery Center, University of Copenhagen In this paper we reconsider the Cachazo-He-Yuan construction (CHY) of the so called scattering amplitudes at one-loop, in order to obtain quadratic propagators. In theories with colour ordering the key ingredient is the rede nition of the Parke-Taylor factors. After classifying all the possible one-loop CHY-integrands we conjecture a new oneloop amplitude for the massless Bi-adjoint 3 theory. The prescription directly reproduces the quadratic propagators of the traditional Feynman approach. Scattering Amplitudes; Di erential and Algebraic Geometry; Superstrings - 1 Introduction 2 3 4 5 6 7 8 4.1 4.2 5.1 5.2 5.3 of theories [4{7], even beyond eld theory [8, 9]. The main ingredient for this approach are the tree-level scattering equations [2] Ea := b6=a X ka kb = 0; ab ab := a b ; a = 1; 2; : : : ; n; where the a's denote punctures on the sphere. The tree-level S-matrix can be written in terms of contour integrals localized over solutions of these equations on the moduli space of n-punctured Riemann spheres An = Z d tnree ItCreHeY( ); where the integration measure, d tnree, is given by and the contour is de ned by the n 3 independent scattering equations d tnree = Qn Vol (PSL(2; C)) ( ij jk ki) Qn b6=i;j;k Eb Eb = 0; b 6= i; j; k : The integrand, ItCreHeY, depends on the described theory. There are other approaches that use the same moduli space [1, 10{12], but restricted to four dimensions. There have been developed several methods to evaluate the integrals, from di erent perspectives. Some approaches study the solutions to the scattering equations for particular kinematics and/or dimensions [4, 13{19], others work with a polynomial form [20{29], or formulating sets of integration rules [30{34]. A di erent approach was proposed in [35], taking the double covered version of the sphere, the so called -algorithm, which we will employ in this work. A generalization for loop level of the CHY formalism has been made. The ambitwistor and pure spinor ambitwistor worldsheet [36, 37] provided a prescription for a generalization to higher genus Riemann surfaces [38{40]. A di erent 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 propagators linear in loop momenta. In order to nd the equivalence with the usual Feynman propagators, (` + K) 2, two additional steps must be taken: the rst one is the use of partial fractions, and the second one is the shifting of loop momenta [39, 48]. Recently, one of the authors [49] proposed a di erent approach to obtain the quadratic Feynman propagators directly from the CHY-integrands for the scalar 3 theory.1 The motivation came by analysing a Riemann surface of genus two after an unitary cut, which look exactly like a tree level diagram before the forward limit, but instead of the two massive 1There are some overlapping ideas with the recent paper published by Farrow and Lipstein [50]. { 2 { particles associated to the loop momenta there are four massless particles. This new approach allows to work again with the scattering equations for massless particles, but at the expenses of increasing the number to n + 4. In addition there is also the need to introduce a new measure 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 for the one-loop Parke-Taylor factors. Splitting the massive loop momenta (`+; ` ) into the four massless ones ((a1; b1); (b2; a2)), we will have one-loop Parke-Taylor factors that will enter into CHY-integrands to lead directly to the usual Feynman propagators. The CHY-integrands in question are the ones for the Bi-adjoint manifestly tadpole-free.2 Outline. This paper is organized as follows. In section 2 we present our reformulation of the one-loop Parke-Taylor factors (PT). The expression is written in terms of the generalized holomorphic one-form on the Torus, !ia::jb. By exploiting algebraic identities we formulate the Theorem 1: each term in the PT factors can be decomposed into terms containing at least two !ia::jb factors, i.e. the PT factors are rearranged in an expansion 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 working with n + 4 massless particles, the contour integrals can be calculated using any of the existing methods of integration. As we have mentioned already we employ the so-called -algorithm, with the choice of a new gauge xing, to solve them. This allows to analytically evaluate arbitrary CHY-integrals using simple graphical rules. The classi cation is made tracing the structures de ned in section 2. As will become clear each element inside the partial amplitude have an unambigous correspondence with the elements of the CHYgraphs, starting with the n-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 3 theory at one-loop with quadratic propagators: rst we give a simple review at tree level and one-loop with linear propagators, then we propose our formula using our de nition for the PT factors. In order to support our proposition in section 5 we perform explicitly the calculation for the partial amplitudes of the three and four-point functions. We make an extensive use of the results of previous sections. In particular we emphasize the direct interpretation of the CHY-integrals in terms of Feynman diagrams. This mapping is codi ed in the following equality at the integrand level 2N1+1 R d sa1b1 R d tNre+e4 n- 2 tree-1 1 N 2As it will be explained, the number of !ia::jb is related with the polygon of the loop, for example, two !ia::jb in the left integrand can only generate a bubble or a triangle. { 3 { HJEP10(27)5 that constitutes one of the most important results of this work and it will be explained in detail during the course of this paper. In section 6 we comment on the issue of the external-leg bubble contributions. Diagrams involved are singular and need to be regularized. Next section, 7, is for illustrative purposes and is devoted to the i prescription and how to directly obtain it by dimension reduction. Finally, in section 8 we conclude by summarizing our ndings. For not disrupting the line of the paper more technical discussions have been gathered in some appendices: in appendix A we explicitly show the su cient form of the measure (3.5) to tackle the one-loop CHY-integral prescription. In particular how the momenta combination it contains 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 and the linear propagator prescription. We conclude by probing an statement of [47]. Before beginning section 2, we de ne the notation that is going to be used in the paper. Notation. For convenience, in this paper we use the following notation HJEP10(27)5 ij := i j ; !ia::jb := ab : ia jb integrands at two-loop. In addition, we de ne the ab's and !ia::jb's chains as Note that !ia::jb are the generalization of the (1; 0)-forms used in [51] to write the CHY(1.5) (1.6) (i1; i2; : : : ; ip) := i1i2 (i1; i2; : : : ; ip)a!:b := !ia1::bi2 = !ia1::bi1 ip 1ip ipi1 ; !a:b ip 1:ip !iap::bi1 !a:b ip 1:ip 1 !iap::bip ; (i1 ; i2 ; : : : ; i!m; im+1; : : : ; in!; : : : ip )a:b := i1i2 i2i3 !iam:bim+1 im+1im+2 !ian:bin+1 ip 1ip ipi1 : In order to have a graphical description for the CHY-integrands on a Riemann sphere (CHY-graphs), it is useful to represent each a puncture as a vertex, the factor 1 as a line and the factor ab as a dashed line that we call the anti-line. Additionally, since we often use the -algorithm3 [35] we introduce the color code given in gure 1 and 2 for a ab mnemonic understanding. Finally, we introduce the momenta notation kfa1;:::;amg = [a1; : : : ; am] := 2 sa1:::am := kfa1;:::;amg; ka1:::am := kai kaj : m X kai ; i=1 m X ai<aj 3It is useful to recall that the -algorithm xes four punctures, three of them by the PSL(2; C) symmetry and the last one by the scale invariance. { 4 { unfixed massless puncture massless puncture fixed by PSL(2,C) massive puncture fixed by PSL(2,C) massless puncture fixed by scale symmetry similar to ones given in [39, 42, 43]. After that, we carry out some manipulation in order to write algebraic identities that will be very convenient to perform computations using the -algorithm. Before de ning the PT factor at one-loop, it is useful to remind that, at tree level in the CHY approach, it is given by the expression PTtree[ ] = 1 ( 1; 2; : : : ; n) ; where is a generic ordering and n is the total number of particles. Following the ideas presented in [42, 43], we formulate: De nition. We de ne the Parke-Taylor factor at one-loop with ordering as PT1a-1lo:ao2p[ ] := X at least two !ia:1j:a2 's. As it will be discussed later, the task of performing CHY-integrals using the PT de ned in (2.2) is not simple. The di culty of these computations resides in the number of !ia:1j:a2 's, more !ia:1j:a2 's imply that the singular solutions of the scattering equations do not contribute. In fact, the minimum number of !ia:1j:a2 's so the CHY-integrals become simpler is two, as we are going to explain in section 3. Nevertheless, it is always possible to manipulate algebraically PT1a-1lo:ao2p[ ] and to decompose it as a linear combination of terms that contain in !ia:1j:a2 with two as its lowest power. Theorem 1. The PT1a-1lo:ao2p[ ] factor, which was de ned in (2.2), admits a power expansion We shall only sketch some examples in order to illustrate this theorem, leaving the complete, technical proof, together with the precise construction using the Schouten-like identity for the ij 's to the appendix B. { 5 { (2.1) (2.2) The previous theorem allows to clarify, that the cancellations of the tadpoles in the bi 3 theory can follow directly from an algebraic property of the one-loop PT factors and not necessarily from the anti-symmetry of the structure constant in the cubic vertex. Before proceeding we shall introduce the following de nitions: Dtay1p:ae2 0[1; : : : ; n]p ! := (1; : : : ; p) !1a:11:a2 !pa:1p:a2 + (2; : : : ; p + 1) !2a:12:a2 !pa+1:1a:2p+1 + + (n; 1; : : : ; p with 1 < p n; Notice, that we have de ned the factors, Dtay1p:ae2 0, Dtay1p:ae2 I and Dtay1p:ae2 II, with the particular ordering f1; 2; : : : ; ng, nevertheless, their de nitions for another ordering are straightforward. These terms also carry the cyclic permutation invariance from the PT factor. 2.1 Examples computation. Two-point In this section we give some non-trivial examples in order to illustrate the above proposition. The following identities are purely algebraic and they can be proven after a, somehow, direct PT1a-1lo:ao2p[1; 2] = 1 12 !2a:11:a2 + 1 21 !1a:12:a2 = PTtree[1; 2] Dtay1p:ae2 0[1; 2]2!; where let us remind, Dtay1p:ae2 0[1; 2]2! = (1; 2) !1a:11:a2 !2a:12:a2 . Three-point PT1a-1lo:ao2p[1; 2; 3] = 1 12 23 !3a:11:a2 + 1 23 31 !1a:12:a2 + 1 31 12 !2a:13:a2 = PTtree[1; 2; 3] h 2Dtay1p:ae2 0[1; 2; 3]3! + Dtay1p:ae2 I[1; 2; 3]2!i = PTtree[1; 2; 3] h Dtay1p:ae2 0[3; 2; 1]3! + Dtay1p:ae2 I[3; 2; 1]2!i : { 6 { (2.3) (2.4) (2.5) (2.6) (2.7) PT1a-1lo:ao2p[1; 2; 3; 4] Note that the (1=2) factor in Dtay1p:ae2 II[4; 3; 2; 1]2! comes from the fact there is a double counting, i.e. Dtay1p:ae2 II[4; 3; 2; 1]2! = (4 ; 3!; 2 ; 1!)a1:a2 + (3 ; 2!; 1 ; 4!)a1:a2 loop. Notice that the one-loop CHY-integral prescription given in [49] has the particular structure4 In = In := Z dD` (2 )D In ; 1 2n+1 Z d sa1b1 Z where n is the number of massless external particles and the d tnr+ee4 is the tree level measure de ned in [3] d tnr+ee4 := EA := QnA+=41 d A Vol (PSL(2; C)) xing PSL(2;C) B=1 B6=A n+4 kA kB = 0; X AB with the identi cation 4For more details see appendix A. fkn+1; kn+2; kn+3; kn+4g := fka1 ; ka2 ; kb1 ; kb2 g; f n+1; n+2; n+3; n+4g := f a1 ; a2 ; b1 ; b2 g; fEn+1; En+2; En+3; En+4g := fEa1 ; Ea2 ; Eb1 ; Eb2 g: { 7 { (3.1) (3.2) (3.3) (3.4) n - p = 1 2 (n- p) anti - lines 2 n - p 2 Additionally, let us remind the measure, d , it is given by the expression5 Notice that in (3.2), without loss of generality, we have xed f 1; b1 ; b2 g and fE1; Eb1 ; Eb2 g. This measure is introduced to take the forward limit of the four on-shell loop momenta. The motivation comes because momenta will appear combined in a particular way after the integration of the A's. In appendix A we give a more detailed explanation of how this particular combination appears. It is useful to recall that the integration over the A's variables is a contour integral, therefore we do not worry to write the measure (2dD)`D . which is localized over the solution of the scattering equations, i.e. EA = 0. In addition, in this paper we are just interested to focus at the integrand level, in other words in In, The classi cation is going to be made by taking into account the CHY-integrands appearing in the Dtay:pbe de nitions given in (2.3), (2.4) and (2.5). i.e. ILa1:a2 ( ) = PTtree[1; 2; : : : ; n] IRb1:b2 ( ) = PTtree[1; 2; : : : ; n] Dtay1p:ae2 [n; n 12 !1b1:2:b2 : 1; : : : ; 1]; In order to not saturate the notation on the CHY-graphs, we introduce the de nition given in gure 3. 3.1 One loop integrands classi cation As it is going to be checked in section 5, the terms Dtay:pbe there is a relation between them and Feynman diagrams. Basically, the number of !ia:1j:a2 's in 's have a physical meaning, as IL a1:a2 = PTtree[1; 2; : : : ; n] Dtay1p:ae2 [n; n corresponds to the number of legs attached to the loop in the Feynman diagram, and the term in IR b1:b2 = PTtree[1; 2; : : : ; n] ij !ib:1j:b2 with a single !ib:1j:b2 will be in charge of its ordering. In fact, the CHY-integrand, I(aLa11;:ab12;b2;a2)2 , IRb1:b2 can be represented as a linear combination of the three CHY-graphs given in gure 4. In particular, in gure 5 we consider the simplest cases. Therefore, the Dirac delta functions in (3.5) are well de ned. 5In this paper we are considering that the D-dimensional momentum space is real, i.e. ki 2 R D 1;1. { 8 { n-C1HY TREnE the scale symmetry [35]. Clearly, we are using a di erent gauge than the one introduced in [49]. This new gauge will allows us to work a la Feynman, i.e. each cut on the CHY-graph becomes a Feynman propagator, quadratic in momenta. The CHY-graphs in gure 5 come from the following set of integrands PTtree[1; : : : ; n] Dtay1p:ae2 0[n; : : : ; 1]n! PTtree[1; : : : ; n] 12 !1b1:2:b2 PTtree[1; : : : ; n] Dtay1p:ae2 I[n; : : : ; 1](n 1)! PTtree[1; : : : ; n] 12 !1b1:2:b2 := I(I) CHY[1; : : : ; n]1:2 + ; PTtree[n; : : : ; 1] Dtay1p:ae2 I[1; : : : ; n](n 1)! PTtree[1; : : : ; n] 12 !1b1:2:b2 := I(CIHI)Y[1; : : : ; n]1:2 + ; := I(CIHIIY)[1; : : : ; n]1:2 + ; where ellipsis stand for additional terms with the same structure6 as those given in the graphs of gure 4. Bearing in mind that the -algorithm is able to solve the CHY-graphs up to an overall sign the rst point to tackle is how to x this ambiguity.7 Let us consider the integrands, I(CIH)Y[1; : : : ; n]1:2; I(CIHI)Y[1; : : : ; n]1:2 and I(CIHIIY)[1; : : : ; n]1:2 above. Pulling out the common factors they become I(CIH)Y[1; : : : ; n]1:2 = ( 1) ( 1)n+1 I(CIHI)Y[1; : : : ; n]1:2 = ( 1) ( 1)n+1 I(CIHIIY)[1; : : : ; n]1:2 = ( 1) ( 1)n+1 " " " 2 2 2 an1a22 2 2 2 We claim that the terms in the square brackets have a direct representation in terms of the CHY-graphs in gure 5. In the following we make use of the -algorithm to perform the set of integrals side the brackets in (3.7), ( 1)n+1, is checked numerically afterwards.8 I(CIH)Y[1; : : : ; n]1:2, I(CIHI)Y[1; : : : ; n]1:2 and I(CIHIIY)[1; : : : ; n]1:2. The conjectured overall sign inProposition 1. There is an equality among the CHY-integral of (I), (3.7), and the one-loop n-point Feynman integrand I(CIH)Y[1; : : : ; n]1:2 := d sa1b1 d tnr+ee4 I(I) CHY[1; : : : ; n]1:2 = 3 Proof. The result follows from a direct calculation, I(CIH)Y[1; : : : ; n]1:2 = sa1b1 d tnr+ee4 I(I) CHY[1; : : : ; n]1:2 = = a1b1 a2b2 b1b2 1 Qin=1 ia1 ia2 1b1 2b2 23 n1 6Sometimes, it is necessary to manipulate the integrand to obtain the CHY-graphs in gure 4. 7Notice that the overall sign can be xed by the technique developed in [33], where the anti-lines have been considered. ( 3; E3)g, namely, the Faddeev-Popov determinant becomes ( 12 23 31)2. 8For the numerical checking we have xed the punctures and scattering equations, f( 1; E1), ( 2; E2), where in the last step we have mapped each element in the expression inside the bracket to the CHY-graph: the rst factor is the box, namely the loop momenta sector, the second one is in charge of connecting the loop momenta to all the external points, and the third factor is the one that gives the ordering. With these identi cations all the integrands of -algotithm [35], we introduce the \nearest neighbour" gauge xing. One can show that it is necessary to perform a total of (n + 1)-consecutive cuts, each one introducing a single propagator, like in the Feynman diagrams. For instance, using the -rules, there is only one non zero cut on the CHY-graph in (3.9), as it is shown on left graph in (3.10). this type will look alike but with the external points permuted. In order to compute the R d tnr+ee4 integral within the 1 This rst cut gives the factor ka1b1 dubbed cut-2. This cut is simple to compute and its result is given by as it is shown on right graph in (3.10), see also gure 3. 0 B B B B B B B B B B B 2) times this procedure we are led with = 1 ka1b1 1 2 n 2 [ 1 n -2 1 cut-2 0 B B B B B B B B B B B cut-3 (3.10) (3.11) where clearly the resulting graph keeps the same form but with one less puncture. Iterating and the resulting CHY-graph contains now a massive puncture with momentum, ka1 + kb1 , which is connected with a double line to the puncture 1. By scale symmetry we x the nearest neighbour to 1 (next to its right), i.e. n, such The resulting CHY-graph in (3.10) contains also only one non zero cut, which we have 1 ka1b1 cut-(n+1) 1 C C 2CC C C A (3.12) This nal resulting graph has just one non-zero cut, as it is shown in (3.12), which we have called \cut-(n+1)" and its result by the -rules is ka2b2 Summarizing, so far we have proven the equality where the momentum conservation condition, Pin=1ki+ka1+ka2+kb1+kb2 = 0, has been used. Carrying out the last integration, R d sa1b1 , we are left with the following expression HJEP10(27)5 1 Z 2n+1 d sa1b1 2n+1 nally identi ed the algebraic expression with the pictorical one-loop npoint Feynman integrand. This equality, which corresponds to our initial claim, has also been checked numerically. As a nal remark for this rst proposition, we can generalize the previous calculation to the case of attaching CHY tree-level graphs instead of points as in (3.9). Schematically it is 1 2N+1 d sa1b1 Z where N is the total number of external particles. The gluing process to join the CHY tree-level graphs at the one-loop skeleton can be found in [46, 51]. These kind of graphs can be solved in the same fashion as we did for the n-point case. Proposition 2. The CHY-integral of (II), (3.7), identically vanishes. 1 2n+1 Z d I(CIHI)Y[1; : : : ; n]1:2 := sa1b1 d tnr+ee4 I(CIHI)Y[1; : : : ; n]1:2 = 0 : (3.14) 1-C1HY TREE 2-CHY T2REE Z n-CHY TREEn 2n+1 " n- 3 cut-1 1 2 Z Z where N is the total number of external particles. These type of graphs, with tree-level CHY-graphs attached, can be solved in a similar way as we did for the case in (3.17). Applying the -rules on the graph in (3.15) to compute the integral, R d tnr+ee4, one obtains that there is just one non-zero cut Furthermore all possible cuts on the resulting CHY-graph in (3.16) are zero, therefore one can conclude that the integral, I(CIHI)Y[1; : : : ; n]1:2, vanishes, i.e. I(CIHI)Y[1; : : : ; n]1:2 = 2n+1 Z This calculation can be generalized to the case of attaching CHY tree-level graphs instead of points. Schematically it is = = 2n+1 Z Proof. As in the previous case, the proof is straightforward, but a little tedious. 1 BB n -3 1 1 2 1 2 1-1C1HY TREE To compute this integral we use the previous results together with the cross-ratio identity It is straightforward to check that by multiplying the CHY-integrand (or graph) in (3.20) times the above identity and using Proposition 2, the integral, I(CIHIIY)[1; : : : ; n]1:2, becomes I(CIHIIY)[1; : : : ; n]1:2 = sa1b1 d tnr+ee4 I(CIHIIY)[1; : : : ; n]1:2 1 1 2n+1 Z d = = I(CIH)Y[1; : : : ; n]1:2 + I(CIHI)Y[1; : : : ; n]1:2 I(CIH)Y[1; : : : ; n]1:2 ; 2 4 l Proposition 3. There is a correspondence between the CHY-integral of (III), (3.7), and the one-loop n-point Feynman integrand 1 2n+1 Z I(CIHIIY)[1; : : : ; n]1:2 := d sa1b1 d tnr+ee4 I(CIHIIY)[1; : : : ; n]1:2 = ( 1) 3 = = 2n+1 2n+1 Z 1 Z 2n+1 " d d sa1b1 d tnr+ee4 ( 1)n+1 Z Proof. The result follows from a direct calculation, I(CIHIIY)[1; : : : ; n]1:2 = sa1b1 d tnr+ee4 I(CIHIIY)[1; : : : ; n]1:2 1 2 and consequently 1 2n+1 d sa1b1 Z n-3 This result can also be generalized for external CHY tree graphs. Schematically it is 1 2N+1 where N is the total number of external particles and again we can solve these type of CHY-graphs like we have done in the previous cases. Now that we have a classi cation for the type of integrands that will appear in our calculations we can employ the results in the computation of the integrands for the Biadjoint The scattering amplitudes at tree level for the massless bi-adjoint 3 scalar theory are given by the elements [4] mtnree[ j ] := 1 where and the measure, d tnree, is given in (3.2). In [4], it was shown that the integral, mtnree[ j ], is composed by the sum over all the trivalent Feynman diagrams containing two planar embeddings, consistent with the ( ) ordering respectively.9 Speci cally mtnree[ j ] reduces to the sum over all elements contained in the intersection among these two planar ordering. Schematically it can be written as [4] mtnree[ j ] = ( 1)n 3+n j mtnree[ j ] \ mtnree[ j ] ; example, let us consider the element, mt4ree[1234j1243], it reads where n j is determined by the number of ips between two permutations ; [4]. For mt4ree[1234j1243] = mt4ree[1234j1234]\mt4ree[1243j1243] = 9There are many techniques to compute mtnree[ j ], such as ones given in [4, 18, 20, 24, 25, 29, 31{ 33, 51, 52]. = 1 2 (4.1) (4.2) (4.3) 4 3 (4.4) which is the right answer. In the previous example one can appreciate the advantage of (4.3), since the calculation for both orderings is exactly the same but with external legs permuted to the given ordering. Therefore, we only have to do the full calculation in the canonical ordering (i.e. (1,2,. . . ,n)). This will prove extremely useful for the one-loop calculations. The CHY prescription to obtain the scattering integrands at one-loop, respecting the planar orderings ( , ), is given by the elements where with and the measure d 1n-+lo2op m1n-loop[ j ] = 1 where, without loss of generality, we have xed f `+ ; ` ; 1g and fE`1+-loop; E`1-loop; E11-loopg. As it has been shown in [39, 42, 43], the CHY-integral in (4.5) reproduces the linear propagators in the internal loop momentum ` , i.e. in the Q-cut representation [53{55]. Therefore, as it is very well known, these results match with the traditional Feynman propagators just after using the partial fraction identity and performing a shift in the loop momentum. 4.2 A new proposal In this section we propose a new CHY prescription for the S-matrix at one-loop of the bi-adjoint 3 scalar theory, that takes into account the planar orderings. In this proposal we will be able reproduce directly the quadratic propagators, in the same way as the traditional Feynman approach. Borrowing the line of reasoning in [49] and the prescription (4.5) we arrive at: matrix elements at one-loop for the 3 bi-adjoint massless theory. We are going to present HJEP10(27)5 several examples in order to support this statement. Notice that despite the similarity among the prescriptions (4.5) and (4.9), there are signi cant di erences between them: i) The total number of punctures do not match, namely in (4.5) there are n + 2 punctures, out of which two are massive, while in (4.9) there are n + 4 massless punctures. ii) The scattering equations are neither the same, although in [41] it was shown that the massive scattering equations in (4.8) can be obtained from (3.3) after dimensional reduction. Additionally, iii) the nal outcomes are di erent, as it is going to be shown later, (4.5) produces linear propagators in ` , while (4.9) is able to reproduce the quadratic propagators as the traditional Feynman approach. After some manipulations I1C-HloYop[ j ] in (4.10) becomes (4.9) (4.10) (4.11) I1C-HloYop[ j ] := PT1a-1lo:ao2p[ ] = X X 2cyc( ) 2cyc( ) 1 PT1b-1l:obo2p[ ] PTtree( 1; : : : ; n; a1; b1; b2; a2) PTtree( 1; : : : ; n; b1; a1; a2; b2): The integral Z As a consequence the integral, R d tnr+ee4 I1C-HloYop[ j ], is just a sum over trivalent tree level planar Feynman diagrams. However, this is not a very e cient and useful way to proceed because there is a large number of singular Feynman diagrams that do not contribute to the partial Amplitude, i.e. these diagrams must cancel out among them. For example, consider the CHY-integrand I1C-HloYop[1; 2; 3j1; 2; 3] = PTtree(1; 2; 3; a1; b1; b2; a2) PTtree(1; 2; 3; b1; a1; a2; b2)+ (4.12) where ellipsis stand for terms obtained under cyclic permutations. The CHY-graph for the rst term of the expansion in (4.12) is represented on the left hand side of gure 6. d t3r+ee4 PTtree(1; 2; 3; a1; b1; b2; a2) PTtree(1; 2; 3; b1; a1; a2; b2) ; contains the sum over all possible trivalent planar Feynman diagrams, [4, 29, 31, 35], that have been depicted with a grey circle and blue lines in gure 6(b). In particular, in gure 7(a) we give one example. Clearly, by performing the R d sa1b1 integral of this diagram, one obtains a tadpole, such as it is shown in gure 7(b). This singular diagram must not contribute to the amplitude and therefore it cancels out with another one which comes from the next contributions. This kind of analysis is tedious since the number of tree level diagrams in (4.11) is large. making the forward limit, i.e. by integrating R d (ka1 + kb1 )2. In order to handle this group of cancelling diagrams without going through the detailed analysis described above, we shall rely on the ndings of sections 2 and 3 to classify the Feynman diagrams at one-loop from the CHY approach. In the next section we are going to show several examples where this new technology is applied and the conjecture over M1n-loop[ j ] will be checked. Finally, note that the method used in [42, 43] could be applied here with a small variation: the o -shell momenta, ` and ` , are split into two on-shell momenta, (ka1 ; kb1 ) and (ka2 ; kb2 ), as is shown in gure 6. 5 Examples In the following section we will use all the previous results to calculate the particular cases n = 3 and n = 4. Since n = 3 is simpler, it contains only two possible orderings, we will show all the contributions by direct calculation of the integrands in (2.7). Before giving the examples, it is useful to introduce the following notation := 3T-RFEEEY + 3T-RFEEEY + + 3T-RFEEEY l 3-FEY TREE 2-FEY TREE 4-FEY TREE 1-FEY TREE 5-FEY TREE n-FEY TREE 6-FEY TREE 2T-RFEEEY l 4-FEY TREE 1-FEY TREE 5-FEY TREE n-FEY TREE 6-FEY TREE 2-FEY TREE 4-FEY TREE 1-FEY TREE 5-FEY TREE l nT-RFEEEY 6-FEY TREE 1 1 (b) 1 l 2 3 2 3 2-FEY TREE 4-FEY TREE 1-FEY TREE 5-FEY TREE n-FEY TREE 6-FEY TREE : (5.1) ated to the CHY-graph (The grey circle means the sum over all possible trivalent vertices.). 2 3 2 2  Ω sa1b1 2 2 1 1 (a) 1 2 3 2 1 2 1 5.1 In this case there are just two partial amplitudes, one coming from expressions for PT1a-1lo:ao2p[1; 2; 3], one written in terms of the Dtay:pbe and another from M31-loop[1; 2; 3j3; 2; 1]. In addition, as inferred from (2.7), we have two 's from the ordering (1,2,3), and the other in terms of the ordering (3,2,1). In the following we disentangle each Let us start with the rst CHY-integrand I1C-HloYop[1; 2; 3j1; 2; 3] = 2 1 2 3 3 + 0 + cyc(1; 2; 3) : + cyc(1; 2; 3)] where we have introduced the shorthand notation I(CIH)Y[i; (j; k)]i:j := ( 1) ( 1) I(CIH)Y[(k; i); j]i:j := ( 1) ( 1) " " 2 2 In terms of the CHY-graphs one has d sa1b1 That can be simpli ed further using (3.22) Z d t3r+ee4< > d 1 2 1 2 1 2 1 2 sa1b1 3 + 0 1 2 3 8 > > > > > > > 2 3 1 3 1 2 = = 1 s23 1 s13 9 > > > > > > > > ; > > 1 2 1 (ka2 + kb2 )2 (k1 + ka1 + kb1 )2 (5.6) 1 (ka2 + kb2 )2 (k1 + k3 + ka1 + kb1 )2 : (5.7) 1 24 1 24 sa1b1 sa1b1 Z Z d t3r+ee4 0 d t3r+ee4 0 over R d ka1 2 ) (D)(kb2 + kb1 external vectors11 ( Notice that by performing the integral R d , i.e. at the forward limit, the momentum conservation condition becomes, k1 + k2 + k3 = 0 (s12 = s13 = s23 = 0), and the expressions (5.6), (5.7) are ill de ned. This fact indicates that these type of terms need some regularization.10 One way to obtain a well de ned result by integrating (5.6) and (5.7) is to regularize the forward limit condition, i.e. instead to consider the measure, dDka2 dDkb2 (D)(ka2 +ka1 ) (D)(kb2 +kb1 ), we allow for the measure dDka2 dDkb2 (D)(ka2 + 2 ), where is an in nitesimal vector ( 2 0) orthogonal to the ki = 0). Using this new measure the momentum conservation condition becomes, k1 + k2 + k3 = (s12 = s13 = s23 = 2), and now we are able to integrate (5.6) and (5.7). Considering the leading order term one has From the general result in section 3.1 the rst integral reduces to 1 Z 24 while the rest of terms can be cast in the general form depicted in gure 4(a), and their computations is totally similar to the one presented in section 3.1 HJEP10(27)5 s13 (ka2 + kb2 )2 (k1 + k3 + ka1 + kb1 )2 = = 1 s23 1 s13 2 l 1 `2 (` + k1)2 = 1 `2 (` + k1 + k3)2 = 10Notice that in the linear propagator approach these kind of diagrams are absent [42, 43], see section 6. 11Let us recall this condition depends of the dimension of the momentum space. 3 ; 1 3 3 2 ; (5.8) l 1 : Finally, adding all the partial results one gets which is the expected answer, and has also been checked numerically. In section 6 we will brie y discuss about the external-leg bubbles contributions in M31-loop[ j ]. To compute the next partial amplitude we use the second expression for PT1a-1lo:ao2p[1; 2; 3] in (2.7). The CHY-integrand for this case is = I(CIH)Y[3; 2; 1]3:2 +I(CIHIIY)[3; 2; 1]3:2 +I(CIH)Y[3; (2; 1)]3:2 +I(CIH)Y[(1; 3); 2]3:2 = I(CIH)Y[3; (2; 1)]3:2 +I(CIH)Y[(1; 3); 2]3:2 +cyc(3; 2; 1) ; where in the rst line we used the inversion property of the PT factors, and in the third one the result, I(CIHIIY)[3; 2; 1]3:2 = I(CIH)Y[3; 2; 1]3:2 + (zero after integration), given in (3.22). Translating to CHY-graphs the amplitude becomes, M31-loop[1; 2; 3j3; 2; 1] = d sa1b1 M31-loop[1; 2; 3j3; 2; 1] = > 0 > > > > 8 > > > > > : > > 8 < > > > > > > :3 > > > > > > > > ; ; The integrals entering in (5.11) were already computed in the example above, (5.6){(5.8). Collecting then we can write the total result as which has been checked numerically. Hitherto we have found the Feynman diagrams expansion for the canonical ordering, M31-loop[1; 2; 3j1; 2; 3] and its opposite ordering, M31-loop[1; 2; 3j3; 2; 1], in (5.9), (5.12) respectively. With those it is now straightforward to verify the relation ( 1) M31-loop[1; 2; 3j1; 2; 3] \ M31-loop[3; 2; 1j3; 2; 1] = M31-loop[1; 2; 3j3; 2; 1] ; (5.13) (5.10) (5.11) 9 > > > > > > > > > > ; (5.12) this paper as which is the one-loop equivalent to (4.3) [4]. In [42] a general relation at one-loop was conjectured, but in the Q-cut representation, i.e. using the prescription presented in section 4.1. Although we have not got a general proof for (5.13), we have a strong numerical evidence that the conjecture formulated in [42] can be extended to the new proposal formulated in M1n-loop[ j ] = ( 1)n j M1n-loop[ j ] \ M1n-loop[ j ] : In this case we will have to deal with three di erent independent orderings, We will calculate explicitly the rst one and rely heavily in the use of (5.14) in order to infer the rest of them. The CHY-integrand for the partial amplitude M41-loop[1; 2; 3; 4j1; 2; 3; 4] reads After expanding the terms we obtain several cancellations due to the identity, I(CIHIIY)[ ] = I(CIH)Y[ ], which was proven in (3.22) in the simplest case. After a rather cumbersome calculation the above amplitude can be cast in terms of CHY-graphs as M41-loop[1; 2; 3; 4j1; 2; 3; 4] = d sa1b1 Z d t4r+ee4 < 2 8 > > > > > > > > > > : + 0 1 2 1 2 4 3 4 3 All the integrals in (5.16) can be computed using the Propositions 1, 2, 3 and the technology described in their proofs. The result is, following the same order: one box, four triangles, two bubbles and the external-leg bubbles The latter must be regularized. In terms of I1C-HloYop[1; 2; 3; 4j1; 2; 3; 4] = PTtree[1; 2; 3; 4]PTtree[1; 2; 3; 4] 3Dtay1p:ae2 0[1; 2; 3; 4]4! + 2Dtay1p:ae2 I[1; 2; 3; 4]3! 12 !1b1:2:b2 + 23 !2b1:3:b2 + 34 !3b1:4:b2 + 41 !4b1:1:b2 : 1 2 1 2 4 4 3 1 2 1 2 1 2 3 4 Feynman diagrams one has 2 1 2 4 3 4 2 4 3 0 B1 B B B 4 2 1 C C A 1 3 where the grey circle in the third graph inside of the bracket means the sum over all possible trivalent planar diagrams. This result was checked numerically. In order to calculate analytically the next contributions we are going to use the conjecture in (5.14) M41-loop[1; 2; 3; 4j4; 3; 2; 1] = ( 1) M41-loop[1; 2; 3; 4j1; 2; 3; 4] \ M41-loop[4; 3; 2; 1j4; 3; 2; 1] M41-loop[1; 2; 3; 4j1; 2; 4; 3] = ( 1) M41-loop[1; 2; 3; 4j1; 2; 3; 4] \ M41-loop[1; 2; 4; 3j1; 2; 4; 3] + cyc(1; 2; 3; 4)CC ; 1 3 2 4 3 1 = = 1 2 1 2 4 3 4 3 1 2 1 2 4 3 4 3 0 B1 B B B 4 4 2 3 1 3 2 These results were checked numerically concluding that, in fact, the conjecture in (5.14) works perfectly. With this procedure we can easily go to higher point cases, since we can always construct the CHY-graphs for M1n-loop[1j1], where 1 means canonical ordering, and know how to calculate all the CHY-integrals. General structure of M1N-loop[1N j1N ] From the CHY-graphs representation found for M31-loop[13j13] and M41-loop[14j14] in (5.4) and (5.16) respectively, where 1N means the canonical ordering (1; 2; : : : ; N ), it is direct to obtain the general expression for M1N-loop[1N j1N ]. To be precise, up to global sign, one has M1N-loop[1N j1N ] = 1 2N+1 Z d sa1b1 Z X Gchy1a;::2:;N [[i]] + cyc(1; : : : ; N ) ; + cyc(1; 2; 3; 4)C 3 + cyc(1; 2; 3; 4)CC ; where the set, Gchy1a;::2:;N , is de ned as 8 > being Gchy1n;:::;N [[i]] an element in Gchy1n;:::2;N . For instance 2 Gchy11;2;3;4 = < 1 Gchy1n;:::2;N := < All possible CHY-graphs with the form n- 2 ; (5.21) Therefore, it now is clear that (5.20) is in agreement with (5.4) and (5.16). Using the same techniques developed in the proof of Proposition 1, we can compute the CHY-integral for a generic element in Gchy1n;:::2;N . Thus, we assert the following general result at the integrand level 1 2 1 1 N 2 2 3 n 9 > > > > > > = > > > > > > ; 2 1 9 > > > = : 4 > 3 >; > = N Pn (5.23) 8 > > > > > > : 1 2 4 tree-2 2 3 (5.22) HJEP10(27)5 1 2N+1 Z d Z sa1b1 d tNre+e4 n- 2 where the grey circles mean the sum over all possible trivalent planar diagrams and the symbol, \Pn", means the loop circle is a regular polygon of n edges, where we de ne P2 as integrands that contribute to M1N-loop[ j ]. Finally, using the general results given in (5.20) and the \map" in (5.23), the structure of M1N-loop[1N j1N ] becomes simple. In [56], we also analyse the general case for the 6 External-leg bubbles As it is known, in the linear propagators formalism the external-leg bubbles vanish. To illustrate that, let us consider the following Feynman integrand l 1 = where we have applied the partial fraction identity, A1B = A (B A) + B (A B) . Performing 1 1 a shifting over the loop momentum variable in the second term in (6.1), i.e. ` ! `~ = ` + ki, one obtains12 k2) : i Since we are interested on the external leg bubbles,13 we assume that ki is an external massless on-shell particle and therefore (6.2) vanishes. This is the reason, in principle, why in the linear propagator formalism the external-leg bubbles do not appear [43]. In [42] it was also argued that the external-leg bubbles contribution must be regularized. In the proposal we present in this work, the external-leg bubbles appear in a natural way and been also singular their contribution need to be regularize. Notice that from the map (5.23), it is straightforward to identify the external-leg bubbles contributions in M1N-loop[1N j1N ]. This contribution is given by the expression sa1b1 Z where Gchy01;::E:;NLB is de ned as Gchy01;::E:;NLB := < 0 8 > > > > > > : 2 1 2 2 j=1 N ; 0 Gchy10;:::;N , and this is the generator of the external-leg bubbles (ELB). Finally, we believe perhaps it would be interesting to nd a regularization method in the CHY coordinates. 7 Feynman i prescription (6.2) 3 (6.3) (7.1) d tNre+e4 4 X Gchy01;::E:;NLB[[j]] + cyc(1; : : : ; N )5 ; 1 N 2 9 3 >>> = > > > ; : In order to obtain the full form of the traditional Feynman propagators, we show that our proposal is able to reproduce the Feynman i . This term can be included at each propagator in a pragmatic and simple way, by dimensional reduction in the momenta of the auxiliary punctures, i.e. (ka1 ; ka2 ; kb1 ; kb2 ). than the rest of the kinematic data, namely, Let us consider that the momenta of the auxiliary punctures has one more dimension KaM1;2 := (ka1;2 ; kaD1+;21); KbM1;2 := (kb1;2 ; kbD1;+21); kM := (k ; 0); `M := (` ; 0); = 1; : : : ; D ; = 1; : : : n ; where D is the number of physical dimensions and n is the number of external particles. The forward limit measure, d , is modi ed in the following way d := dD(ka1 +kb1 ) (D)(ka1 +kb1 `)dD+1Ka2 dD+1Kb2 (D+1)(Ka2 +Ka1 ) (D+1)(Kb2 +Kb1 ): 12We are assuming that the integration measure over ` is invariant under this transformation. 13On this example we are considering that the propagator, K12 , is regularized. After performing all integrals, the propagators become 1 (Ka1 + Kb1 + P )2 = 1 1 where P M = (p ; 0) is a momentum vector given by the sum of external momenta, i.e P M = kM1 + kMi = (k 1 + Finally, with the identi cation, 2 kaD1+1 k b1 + k i ; 0) := (p ; 0), therefore (Ka1 + Kb1 ) P = (ka1 + kb1 ) p. D+1 = i , we obtain the Feynman propagators in full form. 8 Along the line of reasoning introduced in [49], we have proposed a reformulation for the one-loop Parke-Taylor factors given in [39, 42, 43]. Exploiting the algebraic (Schouten-like) identity between the ij 's in the PT factors, we were able to show that they can be expanded in such a way that no tadpoles integrands will appear later, so for bi-adjoint 3 theory this cancellation is not necessarily related to the anti-symmetry of the structure constant in the cubic vertex. The construction also allowed us to build and classify systematically all the contributing CHY-integrals to the one-loop n-point case. These new PT factors were used to calculate the partial amplitudes for the bi-adjoint 3 theory at one-loop. It can be seen that the prescription presents advantages over the previous ones: In this approach the CHY-integrals are supported over n + 4 massless scattering equations only, where n is the total number of external particles. Thus, all the known techniques to compute these type of integrals can be used [4, 18, 20, 24, 25, 29, 31{ 33, 51, 52]. It gives directly the quadratic Feynman propagators, unlike the already known prescriptions, which must apply the partial fractions identity, namely, their results are written in the Q-cut language [53]. The corresponding CHY-graphs are well suited to be easily solved using the -algorithm allowing to calculate the integrands for higher points cases in a similar fashion to Feynman diagrams. In addition, with the technology developed in this work, we are able to compute the CHY-integrals directly in the forward limit, up to external-leg bubble con gurations. The reason is because the singular solutions of the scattering equations do not contribute. In addition, it is straightforward to note that if one takes all the permutations instead of the cyclic ones, i.e, IL a1:a2 = X 2Sn 1 1 where it with, IR 1 := 1 and Sn 1 is the set of all permutations of f2; 3; : : : ; ng, and by integrating b1:b2 = PT1b-1l:obo2p[1; 2; : : : ; n], one obtains the n-gon with the canonical ordering, which is a consequence of Proposition 1. Therefore, we can say that the calculations in [49] are a particular case of our prescription. At this time, most of computations performed at loop level using the CHY prescription have been obtained in the Q-cut language [38{43, 46{48, 51, 57{59], i.e. linear propagators in the loop momenta. Additionally, it is very well known that the PT factors is one of the most important ingredient to de ne several types of theories in the CHY approach, most notably Yang-Mills and Einstein gravity (from the Kawai-Lewellen-Tye (KLT) relations point of view). We are con dent that by using the new formulation of the PT factors at one-loop, as it has been proposed in this work, we will be able to extend our ideas beyond the bi-adjoint 3 case, in particular for the Yang-Mills theory (see appendix A and eq. (A.13)), Recently, many works about the Bern-Carrasco-Johansson (BCJ) duality and KLT kernel in the CHY context have been published [60{67]. In particular, at one-loop all found results have been written in the Q-cuts representation [62, 63]. Thus, following the lines of the new proposal developed here, it would be very interesting to obtain result in terms of the conventional propagators, (` + K) 2, and to compare with the technologies presented in [62, 63, 68{70]. Moreover, extensions to higher loops are being developed [56]. In addition, it would be fascinating to found the origin of this new prescription or its relationship with the Ambitwistor string theory [36, 71{73]. Acknowledgments H.G. would like to thank to E. Bjerrum-Bohr, J. Bourjaily, and P. Damgaard for discussions. H.G. is very grateful to the Niels Bohr Institute | University of Copenhagen for hospitality and partial nancial support during this work. We thank to S. Mizera and P. Damgaard for useful comments. The work of H.G. is supported by USC grant DGI-COCEIN-No 935-621115-N22. P.T. is partially supported by MINECO grant FPA2016-76005-C2-1-P. A CHY-integrands at one-loop In this appendix we will give a simple way to construct CHY-integrands that have a particular dependence on the loop momenta after integration, i.e. they come as couples, (ka1 +kb1 ) As it was mentioned previously, the proposal given in this paper follows the idea presented in [49]. The main idea that motivated the one-loop calculation in [49] is that the and (ka2 + kb2 ). CHY-integral 1 2n+1 Z dDka2 dDkb2 (D)(ka2 + ka1 ) (D)(kb2 + kb1 ) Z 1 2 1 1 5 2 2 4 (A.1) 3 + per(1; : : : ; n) by the integral, R dD(ka1 + kb1 ) (D)(ka1 + kb1 is just the unitary cut of the two-loop diagram [51], at the integrand level, as it has been represented above. Therefore, in order to obtain a one-loop integrand, we multiply by the factor, (ka1 + kb1 )2, and we make the identi cation, ka1 + kb1 = `. This process is performed `). This would be a simple explanation why the measure, d sa1b1 , is introduced in that particular way. Note that the (A.1) CHY-integrand is a generalization for the one found in [39, 47] Z 6 which is the one that reproduces only linear propagators. Generalizing the (A.1) idea, our proposal is In := 1 2n+1 Z So, a natural question is: what must be the form of the integrands, fIL to obtain a function of the couples, (ka1 + kb1 ) and (ka2 + kb2 ) ? Before giving an answer of this question, it is useful to remind our one-loop Parke-Taylor factors construction. a1:a2 ; IR b1:b2 g, in order In [42, 43], the planar one-loop Parke-Taylor factors for linear propagators were presented, and they can be written like PT1-loop[ ] := 1 (`+; ` ) X 1 Following the previous proposal and the generalization of (A.2) given in (A.1), in this paper we proposed the planar Parke-Taylor factors at one-loop for quadratic propagators as PT1a-1l:oao2p[ ] X X 1 1 n 1 n n 1 n It is not obvious that by using these integrands we will obtain a functional dependence of the loop momenta like (ka1 + kb1 ) and (ka2 + kb2 ), nevertheless, it is not di cult to show that this turn out to be the case. First of all, as in (4.11), it is straightforward to check that the integrands in (A.4) can be written as Each term in the (A.5) sums is called a partial planar one-loop Parke-Taylor factor, and we denote them as PT1L-loop[ ] := PTtree[ 1; : : : ; n; a1; b1; b2; a2] = PT1R-loop[ ] := PTtree[ 1; : : : ; n; b1; a1; a2; b2] = (a1; b1; b2; a2) ( 1; : : : ; n; a1; a2) (a1; b1; b2; a2) ( 1; : : : ; n; b1; b2) (a1; a2) (b1; b2) PT1R-loop, for two generic orderings14 and , it can be represented as Next, by taking the CHY-integral of the product of these two factors, PT1L-loop and 1 1 α1 (A.5) : ; (A.6) (A.7) β 2 βn ; (A.8) (A.9) (A.10) Z d tnr+ee4 PT1L-loop[ 1; : : : ; n] PT1R-loop[ 1; : : : ; n] α 1 β 1 α 2 αn \ 2 2 1 1 1 1 β 1 β 2 β 2 2 α 2 αn \ 2 2 1 1 1 1 where we have used the intersection property [4] from section 4, and the grey circles mean the sum over all possible trivalent planar diagrams. Clearly, in (A.7) we have shown that the CHY-integral, R d tnr+ee4 PT1-loop[ ] L R PT1-loop[ ], is in fact a function of the two o -shell momenta which come from the combinations of four on-shell momenta: (ka1 + kb1) and (ka2 +kb2). This implies that the whole construction developed in this paper is well de ned, i.e. the three types of CHY-integrands: ILa1:a2( ) ILa1:a2( ) ILa1:a2( ) IRb1:b2( ) IRb1:b2( ) IRb1:b2( ) = PT1L-loop[ ] PT1R-loop[ ] ; PT1a-1l:oao2p[ ] ; 14Note that this is a general case of the example shown in (4.12). give a functional dependence of the loop momenta like (ka1 + kb1 ) and (ka2 + kb2 ). In addition, from the identities X 2Sn 1 X 2Sn 1 PT1L-loop[ 1; : : : ; n 1; n] = PT1R-loop[ 1; : : : ; n 1; n] = where Sn 1 is the group of (n 1)-permutations, the CHY-integrand in (A.1), ; gives also a functional dependence of (ka1 + kb1 ) and (ka2 + kb2 ). Therefore, we have found an answer for the question formulated previously, to obtain of the factors, PT1L-loop and PT1R-loop, respectively. a CHY-integral that is able to give a functional dependence of the momenta, (ka1 + kb1 ) and (ka2 + kb2 ), the integrands, (a1;b1;b2;a2) and (a1b;b11:b;b22(;a)2) , must be a linear combination ILa1:a2 ( ) IR Finally, from the ideas given in this appendix, in order to reproduce the planar contribution at one-loop (with quadratic propagators) for Yang-Mills theory, we propose the following prescription [56] A1Y-Mloop(1; 2; : : : ; n) = Z Z d sa1b1 2n+1 na2;b2j 1 njb1;a1 Z d tnr+ee4 PT1a-1l:oao2p[1; 2; : : : ; n] (or PT1R-loop), as it is claimed in this appendix. where na2;b2j 1 njb1;a1 are the BCJ numerators, which must be found (see [62, 63] for linear representation). Additionally, some progress towards the construction of the partial non-planar one-loop Parke-Taylor factors is in development. In [56], we have shown that those non-planar Parke-Taylor factors can be written as a linear combination of PT1L-loop B Proof of the one-loop Parke-Taylor factor expansion The path we found to write (2.2) as a sum of terms with a minimum number of two !ia:1j:a2 's was not a straightforward one, but it allowed us to see that the space for the one-loop CHY diagrams with a xed number of external points is bigger than the one of Feynman diagrams. We have seen that the diagrams we have encounter so far can be written in terms of the diagrams we computed in section 3, even diagrams that cannot be solved using the -algorithm, which happens to be the case for the original Parke-Taylor factor. Our proof will be supported only in the use of the Schouten-like identity (A.11) (A.12) 1 = ac bd ad bc ; ab cd (B.1) which give us a cross-ratio to relate the diagrams algebraically, without the use of the scattering equations. Our starting point is the expression (2.2) for the one-loop Parke-Taylor factor with ordering , it can be rewritten as follows PT1a-1lo:ao2p[ ] := PTtree[ ] n 1 !a1n::a21 : X Since the proof works the same way for any ordering, we can take = (12 : : : n), then HJEP10(27)5 (B.2) (B.4) One thing to attempt would be to solve the diagram corresponding to the rst term and then take cyclic permutations of the result, but it leads to singular cuts, so we cannot apply the -algorithm. Since the PTtree is a global factor, we can perform our analysis ! and then apply the cyclic permutations to get the whole PT1-loop. 12!1a:12:a2 = 12 a1a2 : 1a1 2a2 We want to nd terms with the higher order of !'s, since we are missing 2n 2 factors in the denominator, we use the cross-ratios (B.1) to obtain them. Now we have (B.4) times \1" 12 a1a2 1a1 2a2 23 a1a2 + 2a2 3a1 2a1 3a2 34 a1a2 + 3a2 4a1 3a1 4a2 n1 a1a2 + na2 1a1 : (B.5) na1 1a2 Expanding all the products and performing the sum over cyclic permutations will give us more than just the term with n !'s, it will give all the correct terms15 down to !2, but we will still have terms linear in !. Actually, those linear terms belong to the inverse ordering. Schematically, the expansion now looks like this PT1a-1lo:ao2p[1; 2; : : : ; n] = PTtree[1; 2; : : : ; n] n : : : )(n)(!! : : : !)(n) ( 1n + (n + 2 1) : : : !)(n)(!! : : : !)(n 2) + : : : + : : : ( !)(2)( !)(2) PT1a-1lo:ao2p[n; n (B.6) write PT1a-1lo:ao2p[n; n diagram. where the round brackets mean closed cycles, and the super indices on them mean the number of factors inside. All the terms inside the square brackets belong to the (12. . . ,n) ordering, so we put a super index 1 on them. A somehow unexpected result, is that we can 1; : : : ; 1] as a sum of all the terms on the square brackets (i.e. from its inverse order), but with all the coe cients equal to 1. The expression we have, again 15By correct we mean that each one of these terms will give the integrand for a contributing Feynman 1; : : : ; 1] = PTtree[1; 2; : : : ; n] ( : : : )(n)(!! : : : !)(n) : : : !)(n)(!! : : : !)(n 2) + : : : + : : : + ( !)(2)( !)(2) 1n : To prove the previous relation we apply the inverse procedure with the Schouten like cancel out the denominators will appear and we will arrive to the PT1a-1lo:ao2p[n; n identity, we dismantle the numerator by mixing the ij 's with the a1a2 's, factors that Replacing (B.7) in (B.6) we will have an expression with no linear terms in !. Its coe cients are the only ones modi ed PT1a-1lo:ao2p[1; 2; : : : ; n] = PTtree[1; 2; : : : ; n] (n 1) ( : : : )(n)(!! : : : !)(n) + (n + 1 2) : : : !)(n)(!! : : : !)(n 2) + : : : + : : : ( !)(2)( !)(2) 1n : Now this one-loop Parke-Taylor factor will enter into CHY integrands that can be easily solved using the -algorithm, these give also the correct contributions for the bi-adjoint 3 scalar theory. C From quadratic to linear propagators in the CHY-graphs The computational techniques developed in this work can be applied to the linear propagators approach as well. CHY-graphs, meaning Schematically, the CHY-graphs that lead to linear propagators can be obtained from the ones related to quadratic ones just by replacing the box loop by a \line loop" in the n-2 1 2 l n-3 l+ l(n- p- 1) anti- lines where, `+ = ` := `, and ` is the o -shell loop momentum, `2 6= 0. 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Humberto Gomez, Cristhiam Lopez-Arcos, Pedro Talavera. One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations, Journal of High Energy Physics, 2017, 175, DOI: 10.1007/JHEP10(2017)175