Analytic expressions of amplitudes by the cross-ratio identity method

In order to obtain the analytic expression of an amplitude from a generic CHY-integrand, a new algorithm based on the so-called cross-ratio identities has been proposed recently. In this paper, we apply this new approach to a variety of theories including: non-linear sigma model, special Galileon theory, pure Yang-Mills theory, pure gravity, Born-Infeld theory, Dirac-Born-Infeld theory and its extension, Yang-Mills-scalar theory, Einstein-Maxwell theory as well as Einstein-Yang-Mills theory. CHY-integrands of these theories which contain higher-order poles can be calculated conveniently by using the cross-ratio identity method, and all results above have been verified numerically.


I. INTRODUCTION
In the past a few years, an elegant new formulation of the tree-level S-matrix in arbitrary dimensions for a wide range of field theories has been presented by Cachazo, He and Yuan (CHY) [1][2][3][4][5]. This formulation describes the scattering amplitude for n massless particles as a multidimensional contour integral over the moduli space of punctured Riemann spheres M 0,n . It can be unified into a concise expression z rs z st z tr i∈{1,2,...,n}\{r,s,t} dz i z ab z bc z ca i∈{1,2,...,n}\{a,b,c} where z i is the puncture location in CP 1 for the i-th particle, and z ij is defined as z ij ≡ z i −z j .
The second line in (1) is obtained by fixing the gauge redundancy of the Möbius SL(2, C) group. The δ-functions impose the scattering equations E i ≡ j∈{1,2,...,n}\{i} Although this algorithm can be applied to any Möbius invariant integrand in principle, an important question is, can it be terminated within finite steps for any CHY-integrand?
In [22], it has been proved that any weight-two rational function of z ij can be decomposed as a sum of PT-factors with kinematic coefficients via the cross-ratio identities within finite steps. Since any term from a known CHY-integrand can be expressed as a product of two weight-two rational functions, one can conclude that all known CHY-integrands can be decomposed into terms of only simple poles by applying the cross-ratio identity method.
To verify its validity, it is worth appling this new method to integrands of various theories and checking the result numerically. In this paper we consider the following theories: nonlinear sigma model (NLSM), special Galileon theory (SG), pure Yang-Mills theory (YM), pure gravity (GR), Born-Infeld theory (BI), Dirac-Born-Infeld theory (DBI) and its extension, Yang-Mills-scalar theory (YMS), Einstein-Maxwell theory (EM), Einstein-Yang-Mills theory (EYM). All known CHY-integrands involving higher-order poles are contained in the cases above. In the meanwhile, theories corresponding to CHY-integrands with simple poles only, such as the scalar theory with φ 3 or φ 4 interaction, will not be considered in this paper.
We divide them into three classes according to different building blocks of integrands. The first class includes NLSM, SG, YM, BI as well as GR. Integrands of these theories can be constructed from a 2n × 2n antisymmetric matrix Ψ. The second class includes DBI, EM and a special case of YMS of which integrands depend on antisymmetric matrices [Ψ] a,b:a , [X ] b as well as Ψ. The third class includes the general YMS, the extended DBI and EYM, which contains the mixed traces of the generators of Lie groups. Integrands of these theories are related to a polynomial {i,j} ′ P {i,j} , or equivalently, an antisymmetric matrix Π.
Computation shows that all amplitudes considered in this paper can be calculated efficiently within finite steps.
This paper is organized as follows: In section (II) we give a brief review of the cross-ratio identity method. Based on this approach, calculations of amplitudes of theories in the three classes above are given in sections (III), (IV) and (V) respectively. Finally, we give a brief summary in section (VI).

II. BRIEF REVIEW OF THE CROSS-RATIO IDENTITY METHOD
For reader's convenience, we will give a brief introduction to the cross-ratio identity method in this section [21], then we will discuss its validity for general CHY-integrands.
A. The systematic decomposition algorithm Firstly, we need to define the order of poles of an integrand. A generic n-point CHYintegrand consists of terms as rational functions of z ij in the form where |Λ| denotes the length of the set Λ. If χ Λ ≥ 0, a pole 1 s χ+1 Λ will arise in the result. It is straightforward to verify χ Λ = χΛ which reflects the momentum conservation constraint s Λ = sΛ, where the subsetΛ = {1, 2, . . . , n} \ Λ is the complement of Λ. Thus, it is necessary to choose independent Λ's. If a CHY-integrand has m independent subsets Λ 1 , Λ 2 , . . . , Λ m with χ Λ i ≥ 0, the order of poles of the integrand is defined as Then an integrand which result in simple poles only must satisfy Υ[I] = 0.
In order to apply the integration rules, it is necessary to decompose an integrand with Υ[I] > 0 into terms with Υ[I ′ ] = 0. This can be achieved by multiplying the cross-ratio identities to the integrand iteratively. The cross-ratio identity for the set Λ is given by where j and p are selected manually. This identity holds on the support of the scattering equations and the momentum conservation constraint. One can expand the original I into 2.
Step 2: decompose the CHY-integrand I by applying the first cross-ratio identity where c ℓ 's are rational functions of Mandelstam variables. 5. Perform the procedure above for each I ′ ℓ , and repeat the same operation iteratively, then end with the expression such that the order of poles of each term is zero.
6. If after some steps, there always exist terms with the order of poles no less than Υ[I], restart the algorithm by starting from Λ 2 , etc. , for all choices of Λ i , j and p. If this happens, the corresponding integrand cannot be calculated by the method introduced in the previous subsection. In order to fully understand the cross-ratio identity method, one needs to prove that the situation above can be excluded in general, or clarify when such a situation might arise.
Actually, the sum of χ Λ 's for length-t subsets of {1, 2, . . . , n} is fully determined by the condition j∈{1,2,...,n} β ij = 4 as Thus, the sum of all χ Λ 's χ total ≡ Λ χ Λ = t χ t is invariant under any action which maintains the Möbius invariance. If χ total is positive for some integer n, it is impossible to decompose the corresponding integrand into terms with simple poles only. Fortunately, a little algebra leads to the conclusion that χ t > 0 if and only if n < t + 1, thus χ total can never be positive.
On the other hand, it has been proved that a weight-two rational function of z ij can always be transformed to the sum of PT-factors 's with kinematic coefficients via the cross-ratio identities within finite steps [22]. Any term of a known CHY-integrand in the literature can be expressed as a product of two weight-two rational functions. Hence, al-though it is not clear whether the CHY-integrand for any physical theory can be decomposed as products of weight-two functions, one can use the cross-ratio identities to decompose any known CHY-integrand into terms which contain simple poles only.
In this paper, we will choose the original algorithm rather than the one which decomposes a weight-two function into PT-factors, since the former is more convenient to be implemented in Mathematica. Indeed, the feasibility of this algorithm has not been proved, since the procedure of decomposing a weight-two function into PT-factors cannot ensure Υ[I ′ ℓ ] ≤ Υ[I] at each step. However, as can be seen in the subsequent sections, all known CHY-integrands can be computed by the original algorithm efficiently, i.e., the condition Υ[I ′ ℓ ] ≤ Υ[I] can always be satisfied, at least for all known CHY-integrands.

III. AMPLITUDES OF THEORIES IN THE FIRST CLASS
For theories in this class, the most important object in the construction of the integrand where A, B and C are n × n matrices given by and One also needs to introduce the reduced Pfaffian Pf ′ Ψ = (−) i+j z ij PfΨ ij ij where Ψ ij ij denotes the minor obtained by deleting rows and columns labeled by i and j, with i, j ∈ {1, 2, . . . , n}.
On the support of scattering equations, the reduced Pfaffian Pf ′ Ψ is invariant with respect to the permutation of particle labels. In addition, a useful factor is defined as where T I 's are generators of the Lie group under consideration.
The diagonal terms of the matrix C will break the manifest Möbius invariance, since they are not of a uniform weight under Möbius transformations. To keep the validity of the integration rules, they need to be rewritten as where momentum conservation and the gauge invariant condition ǫ i · k i = 0 have been used.
The new formula of C ii gives the weight two for node i and weight zero for other nodes, then the term-wise Möbius invariance is guaranteed. Throughout this paper, we choose With these ingredients, we can now investigate theories in the first class one by one.

A. Non-linear sigma model
We begin with the simplest case, the NLSM, whose standard Lagrangian in Cayley parametrization is where Here I is the identity matrix and T I 's are generators of U(N). The CHY-integrand of NLSM is given by [5] For this case, it is sufficient to calculate the color-ordered partial amplitude In other words, we focus on the color-ordered partial integrand Here the coupling constant have been omitted.
We start from the 6-point amplitude A NLSM

6
. By definition, the corresponding colorordered partial integrand is The pole structure of (22) For this simple example, the full computation takes less than a minute in Mathematica.
One can see the manifest cyclic symmetry in (23), which is the characteristic of the colorordered partial amplitude. This analytic result is confirmed numerically against the one obtained from solving scattering equations numerically.
This result has been verified numerically.

B. Special Galileon theory
The next theory under consideration is SG. The general pure Galileon Lagrangian is with We restrict ourselves on the special situation in which there exist constraints on coupling constants g m 's such that all amplitudes with an odd number of external particles vanish.
Then the CHY-integrand I n of this theory is given by [5] where the coupling constants have been omitted.
With this setup, we choose the 6-point amplitude A SG 6 as an example. The integrand has 15 terms, and all terms contain higher-order poles. One can divide it into 3169 terms with simple poles only within 10 steps. Although the final result is too lengthy to be presented, it has been confirmed numerically.

C. Yang-Mills theory
Then we turn to the pure YM. The CHY-integrand of YM is [5] I YM = C n Pf ′ Ψ(k, ǫ, z) .
Let us take the 6-point color-ordered amplitude A YM (1, 2, . . . , 6) as an example. The partial integrand I YM (1, 2, . . . , 6) has 3420 terms and 1120 of them contain higher-order poles. The decomposition procedure can be terminated within 5 steps via the cross-ratio identity method, and the analytic expression of A YM (1, 2, . . . , 6) is verified numerically. The It is worth noticing that, when checking the result numerically, the values of external momenta must satisfy the momentum conservation constraint, which is necessary for the derivation of the cross-ratio identities. However, those of polarization vectors can be chosen arbitrarily since they are irrelevant to the cross-ratio identities and the integration. We have verified the result with polarization vectors ǫ i · k i = 0 as well as ǫ i · k i = 0, and find that the analytic expression reproduces the value obtained from solving scattering equations numerically.

D. Born-Infeld theory
Now we consider BI whose Lagrangian is given by The CHY-integrand of BI is [5] For simplicity, we calculate the 6-point amplitude A BI 6 . The integrand contains 20400 terms and 18744 of them involve higher-order poles. Using the cross-ratio identities, one can reduce it to terms with simple poles within 10 steps. This is the most complicated example in this paper, which takes more than a day in Mathematica. The analytic expression of the amplitude is confirmed by numerical verification.

E. Gravity
The final theory under consideration in this section is GR. The CHY-integrand of this theory is the product of two independent copies of the one for YM, each of which has its own gauge choice for polarization vectors [5] The polarization tensor of a graviton is given by ζ µν = ǫ µǫν . This integrand leads to amplitudes of gravitons coupled to dilatons and B-fields.
We take the 4-point amplitude A GR 4 as an example. The integrand contains 484 terms, with 228 terms involving higher-order poles. It can be decomposed into terms with simple poles within 2 steps, as shown in the following Physically, polarization tensors of gravitons are traceless, i.e., they satisfy ǫ µǫ µ = 0. However, as discussed before, their values can be chosen without imposing any physical constraint when performing the numerical verification.

IV. AMPLITUDES OF THEORIES IN THE SECOND CLASS
In this section we move on to theories in the second class. CHY-integrands of these theories require two new matrices [X ] b and [Ψ] a,b:a as basic ingredients. a and b are two sets of external particles, whose numbers are denoted by n a and n b respectively, and n = n a + n b is the total number of particles. [X ] b is an n b × n b matrix defined as where I i ∈ {1, ..., M} denotes the i-th U(1) charge of the U(1) M group.
where the gauge group is U(N), and the scalars carry a flavor index I with I ∈ {1, ..., M} from the M-dimensional space transverse to the D-brane. The corresponding CHY-integrand is [5] I YMS (g, s) = C n Pf[X ] s (z) Pf ′ [Ψ] g,s:g (k,ǫ, z) , where g and s denote the sets of gluons and scalars respectively. Gluons have polarization vectors ǫ µ 's while scalars do not, thus their kinematical information can be combined into the matrix [Ψ] g,s:g . Again, we consider the color-ordered partial amplitude A YMS (1, 2, . . . , n).
The first example is the 6-point partial amplitude A YMS (1g, 2g, 3g, 4g, 5s, 6s), where external particles 1g, 2g, 3g and 4g are gluons, while 5s, 6s are scalars of the same flavor. The partial integrand has 222 terms and 68 of them contain higher-order poles. The decomposition procedure is shown in the following The second example is the 6-point amplitude A YMS (1g, 2g, 3s I 1 , 4s I 1 , 5s I 2 , 6s I 2 ), where 1g and 2g are gluons, 3s I 1 , 4s I 1 are scalars of one flavor and 5s I 2 , 6s I 2 are scalars of another.
The partial integrand contains 15 terms and 7 of them contain higher-order poles. The decomposition can be done within 3 steps as shown in the following We proceed to consider DBI whose Lagrangian is where I again labels the flavor of scalars. The CHY-integrand of DBI is [5] where γ denotes the set of photons and s the set of scalars respectively. Again, this result is verified numerically.

C. Einstein-Maxwell theory
The final theory in this section is EM which describes gravitons coupled to photons. The CHY-integrand of this theory is given by [5] where the set of gravitons is denoted by h, and that of photons is denoted by γ. The expression (42) allows the photons to carry more than one flavor in general. The polarization tensor of a graviton is ζ µν = ǫ µǫν , and the polarization vector of a photon isǫ ν . The matrix Ψ(k,ǫ, z) containsǫ ν for both gravitons and photons, and the matrix [Ψ] h,γ:h (k, ǫ, z) contains the remaining ǫ µ 's for gravitons.
Our example is the 5-point amplitude A EM 3h2γ whose external particles are three gravitons and two photons carrying the same flavor index. The integrand has 5013 terms and 1171 of them contain higher-order poles. The decomposition procedure can be done within 4 steps, as shown in the following where i a and j a are labels of two external particles which belong to b Tra . This sum can be recognized as the reduced Pfaffian of the matrix Π. The matrix Π can be constructed from Ψ by performing the so-called squeezing operation iteratively. Terms in the expansion of {i,j} ′ P {i,j} respect the Möbius invariance automatically, while terms in the expansion of Pf ′ Π break the manifest Möbius invariance thus are forbidden for the integration rules.
Hence, we will use {i,j} ′ P {i,j} to express integrands throughout this section.

A. General Yang-Mills-Scalar theory
Let us consider the general YMS with the Lagrangian which involves the general flavor group and a cubic scalar self-interaction. The trace is for the gauge group, and fĪJK and f IJK are the structure constants of gauge and flavor groups respectively. Amplitudes of this theory can only contain a single trace of the gauge group, and multi-traces for the flavor group, as can be seen from the general CHY-integrand [5] I gen.YMS (s Tr 1 ∪ · · · ∪ s Trm , g) = C n C Tr 1 · · · C Trm {i,j} ′ P {i,j} (s Tr 1 ∪ · · · ∪ s Trm , g) , where g denotes the set of gluons and s Tr i denotes the set of scalars with the trace Tr i .
Obviously, the simplest example is that the scalars belong to two traces and each trace contains two scalars. However, these amplitudes correspond to the special case of YMS Again, this analytic result is confirmed by the numerical verification.

B. Extended Dirac-Born-Infeld theory
The second theory under consideration is the extended DBI, which is described by the where the matrix U(Φ) is defined in (18), and The corresponding CHY-integrand is given by [5] I ext.DBI (s Tr 1 ∪ · · · ∪ s Trm , γ) = C Tr 1 · · · C Trm where γ denotes the set of photons and s Tr i denotes the set of scalars with the trace Tr i .
Our example is the 6-point partial amplitude A ext.DBI (1γ, (2s, 3s) Tr 1 (4s, 5s, 6s) Tr 2 ), which involves one photon and five scalars, where two scalars carry Tr 1 and three carry Tr 2 . The integrand has 36 terms and 8 of them contain higher-order poles. The decomposition procedure can be done within 3 steps as shown in the This result has been verified numerically.

C. Einstein-Yang-Mills theory
The final theory in this section is the Einstein-Yang-Mills theory, which describes the interaction between gravitons and gauge bosons. The general CHY-integrand involving the mixed traces is [5] I EYM (g Tr 1 ∪ · · · ∪ g Trm , h) = C Tr 1 · · · C Trm {i,j} ′ P {i,j} (g Tr 1 ∪ · · · ∪ g Trm , h) Pf ′ Ψ , where the set of gravitons is denoted by h and the set of gluons with the trace Tr i is denoted by g Tr i .
We consider the 5-point partial amplitude A EYM ((1g, 2g) Tr 1 (3g, 4g, 5g) Tr 2 ), of which all five external particles are gluons with two of them carrying Tr 1 and three of them carrying In this paper, the most complicated example takes more than a day in Mathematica.
The reason of this low efficiency is, the choices of Λ i , j and p are tested by brute force in the algorithm. Appropriate choices of the cross-ratio identities at each step can minimize the number of steps of the decomposition, which is crucial for practical calculations. Thus, how to optimize these choices to improve the efficiency is an important future project.