An algebraic approach to the scattering equations

Abstract

Abstract

We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.

Published for SISSA by Springer

Received: September 30, 2015 Accepted: November 27, 2015 Published: December 10, 2015

An algebraic approach to the scattering equations

Rijun Huang,a;1 Junjie Rao,a Bo Fenga;b and Yang-Hui Hec;d;e

aZhejiang Institute of Modern Physics, Zhejiang University,

Hangzhou, 310027, P.R. China

bCenter of Mathematical Science, Zhejiang University,

Hangzhou, 310027, P.R. China

cSchool of Physics, NanKai University,

Tianjin, 300071, P.R. China

dDepartment of Mathematics, City University,

London, EC1V 0HB, U.K.

eMerton College, University of Oxford,

Oxford, OX14JD, U.K.

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the nal integrand become immediate in this formalism.

Keywords: Scattering Amplitudes, Di erential and Algebraic Geometry, Gauge Symmetry

ArXiv ePrint: 1509.04483

JHEP12(2015)056

1The unusual ordering of authors is just to let authors get proper recognition of contributions under outdated practice in China.

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP12(2015)056

Web End =10.1007/JHEP12(2015)056

Contents

1 Introduction 1

2 Review of tree-level scattering equations 3

3 The mathematical framework 63.1 Warmup 7

4 Illustrative examples 84.1 Four-point amplitudes 94.1.1 Scalar 3 theory 94.1.2 Yang-Mills theory 104.1.3 Gravity 114.2 Five-point amplitudes 124.2.1 Scalar 3 theory 134.2.2 Yang-Mills theory 174.2.3 Gravity and n-point KLT relations 194.3 Six-point amplitudes 204.3.1 Scalar 3 theory 214.3.2 Yang-Mills theory 234.4 Seven-point amplitudes 25

5 Conclusions and outlook 29

1 Introduction

In last couple of years, amazing progress has been made by Cachazo, He and Yuan [CHY] in a series of papers [1{5], where tree-level amplitudes of a host of quantum eld theories can be calculated using solutions of a set of algebraic equations. These are called the scattering equations and appear in the literature in a variety of contexts [6{14].

The mysterious relationship between the CHY approach and the standard QFT paradigm has been explained from di erent points of view. In [15], using the BCFW on-shell recursion relation [16, 17] the validity of the CHY construction for 3 theory and Yang-Mills theories has been proven. A broader understanding is achieved using ambitwistor string theory [18{28], where using di erent world-sheet elds, di erent integrands in the CHY approach for di erent theories | which we will call CHY-integrands, a function of the coordinates zi in a Riemann surface | have been derived alongside with the natural appearance of scattering equations. A nice point of ambitwistor approach is that it provides the natural framework for loop scattering equations as studied in [21, 24], which

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lead to a breakthrough in [28]. A third understanding is given in [29], where inspired by the eld theory limit of string theory, a dual model has been introduced, based on which a direct connection between the CHY approach and the standard Feynman diagram method has been established not only at the tree-level in [30, 31], but also at the one-loop level (at least for 3 theory) in [32] (see also [33]).1

Although conceptually the CHY approach is remarkable and very useful for many theoretical studies of properties of scattering amplitudes, when applying to real evaluation, one faces the problem of solving scattering equations, which has (n 3)! solutions in

general. Furthermore, when n 6, one encounters polynomials of degree exceeding ve,

rendering analytic solutions in radicals hopeless. Nevertheless, while the solutions can be very complicated, when putting them back into the CHY integrand and summing up, one obtains simple rational functions. These observations have led people to wonder if there is a better way to evaluate the CHY-integrand without explicitly solving the scattering equations. In [35], using classical formulas of Vieta, which relate the sums of roots of polynomials to the coe cients of these polynomials, analytic expression can be obtained without solving roots explicitly. More general algorithms are given by two works. In one approach [36], using known results for scalar 3 theory, one can iteratively decompose the 4-regular graph determined by the corresponding CHY-integrand to building blocks related to 3 theory, thus nishing the evaluation. In another approach [30, 31], by careful analysis of pole structures, the authors wrote down a mapping rule, so that from the related CHY-integrand, one can read out contributions of corresponding Feynman diagrams.

Both approaches are powerful and have avoided the need of solving the scattering equations explicitly. Furthermore, based on these perspectives, especially the mapping rule, one can use Feynman diagrams to construct the CHY-integrand. These results produce a very interesting phenomenon: two di erent CHY-integrands can produce the same result. For example, there are two very di erent CHY-integrands for scalar 4 theory: one is given in [5], while another one is given in [30, 31]. We are naturally led to wonder how to explain the equivalence of di erent CHY-integrands.

In fact, as a rational function of coordinates zi on a Riemann surface, the equivalence can occur on three di erent levels.

1. At the rst level, their equivalence is pure algebraic, i.e., through some algebraic manipulations, one rational expression can be transformed to another one. For example, for 4-point amplitudes of 3 theory, on the one hand we have the integrand I1 =

1 z12z23z34z41

1

z12z23z34z41 where we have de ned zij = zi zj which gives a contribution

1s41 . On the other hand, we have the integrands I21 =

1

z13z32z24z41 which gives 1s41 . It

is easy to check algebraically that I1 = I21 I22. Equivalences at this level is of

course rather trivial and in order to proceed to the other two levels of equivalences, we need to change our viewpoint to algebraic geometry, i.e., to transform the scattering equations to a set of polynomials of (n 3) variables, de ning an ideals I;

1Recently, inspired by the development of CHY-approach, a new method to construct all loop integrands for general massless quantum eld theories has been proposed in [34].

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of 1

s12 +

1 z12z23z34z41

1 z12z24z43z31

which gives 1s12 and the integrand I22 =

1 z12z23z34z41

2. The di erence of two CHY-integrands can be written as J(I1 I2) = PQ where both P; Q are polynomials and J is the Jacobian we will review shortly. If P belongs to the ideal I, then for each solution of the scattering equations P = 0, thus at the

second level we say that I1 is equivalent to I2;

3. However, in practice, most of the time something more complicated happens and we nd that though P does not belong to the ideal I and J(I1 I2) = 0 when and only

when we sum over all solutions. If this happens, we say that I1 is equivalent to I2 at the third level. It is clear that this is the most involved situation, and indeed, in practice this is the most frequently encountered.

Motivated by the above considerations and bearing in mind that indeed the most conducive perspective on studying the scattering equations is through the language of algebraic varieties and polynomial ideals [37, 38], we turn to this method of attack. The above problem thus translates to nding the sum over the rational function PQ evaluated at the roots of a zero-dimensional ideal I, and testing whether the sum is zero. Luckily, there is a theorem in commutative algebra, due to Stickelberger, which addresses the situation [39]. We will discuss the theorem and the associated algorithm in illustrative detail. It turns out that this method not only checks the equivalence at the third level, but also evaluates the integration without solving the scattering equations. In this sense, it is in the spirit of the methods in [36] and [30, 31]. Although it is sometimes less e cient compared to these two methods, it does provide a very di erent angle to approach the problem and could have very advantageous repercussions.

The structure of the paper is as follows. We begin with a brief review of the tree-level scattering equations in section 2, before laying down the foundations of the theory of zero-dimensional ideals in section 3, especially that of companion matrices. We then illustrate the technique with ample computational examples in section 4, before concluding with remarks in section 5.

2 Review of tree-level scattering equations

In this section, we o er a brief review of tree-level scattering equations and the reader is referred to [1{3, 5] for details. The scattering equations are given by

Ea

Ea = 0;

Xa

Xa

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Xb[negationslash]=asabza zb= 0; a = 1; 2; : : : ; n ; (2.1)

where sab = (ka + kb)2 = 2ka [notdef] kb, and ka with a = 1; 2; : : : ; n are n massless momenta for n-external particles and zi are complex variables living on CP1 with n punctures. Although there are n equations, only (n3) of them are linear independent after using the momentum

conservation and massless conditions which translate to the following three relations

Xa

Eaz2a = 0 ; (2.2)

which are, in fact, the consequence of the SL(2; C) symmetry on the CP1. Because of this, we can insert only (n 3) delta-function. To make sure the result does not depend on

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Eaza = 0;

which three equations have been removed, we make following combination and de ne the measure2

(E) zijzjkzkiYa[negationslash]=i;j;k (Ea) ; (2.3)

with zij = zi zj. With the above, the general tree-level amplitude is given by

An =

[integraldisplay] [producttext]n

i=1 dzivol(SL(2; C)) (E)F(z) = [integraldisplay] [producttext]n

i=1 dzid! (E)F(z) ; (2.4)

where d! = dzrdzsdztzrszstztr comes after we use the Mobius SL(2; C) symmetry to x the location of three of the variables zr; zs; zt by the Faddeev-Popov method. Di erent QFTs give di erent forms of the CHY-integrand F(z). Invariance under the Mobius transformation

requires F(z) to have proper transformation behaviors, i.e., under z[prime] = az+bcz+d, we have

F(z) !

n

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Yi=1(czi + d)4 (ad bc)2

!F(z) : (2.5)

To simplify expression (2.4) further, we integrate out the delta-functions to arrive at the key expression

An =

Xsolzijzjkzkizrszstztr()i+j+k+r+s+t[notdef] [notdef]rstijk F; (2.6)

where three arbitrary indices i; j; k correspond to three removed scattering equations while three arbitrary indices r; s; t correspond to the above mentioned three xed locations. The sum is over the solution set of the scattering equations, which is generically a discrete set of points. Furthermore, in the above, the Jacobian matrix is calculated as (a for rows and b for column)

ab = @Ea@zb = 8

<

:

sabz2ab a [negationslash]= b

Pc[negationslash]=asacz2ac a = b; (2.7)

and [notdef] [notdef]rstijk is the determinant of after removing the i-th, j-th and k-th rows and r-th,

s-th and t-th columns.

Speci c examples. Now we list some examples in the literatures [2, 3] (more can be found in [5]). According to the CHY formula, the integrand unifying scalars(b = 0), gluons(b = 1) and gravitons (b = 2) is given by

Fb;n =

2b

Pf [prime]

[parenleftBig] [summationdisplay]

b; (2.8)

2Sn=Zn

Tr(T (1) : : : T (n))

z (1) (2) : : : z (n) (1)

2A nice explanation of this fact can be found in [15, 37].

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where the sum is over permutations on n elements by the symmetric group Sn, up to cyclic

ordering of Zn, is a 2n [notdef] 2n antisymmetric matrix de ned by =

A Ct

C B

[parenrightBigg]

(where t

is the transpose of the matrix), with A; B; C being n [notdef] n matrices with components

Aab =[braceleftBigg]

ka[notdef]kb

zazb

0 ; Bab = [braceleftBigg]

a[notdef] b

zazb

0 ; Cab = [braceleftBigg]

a[notdef]kb

zazb

Pc[negationslash]=a a[notdef]kczazcfor a [negationslash]= ba = b ; (2.9)

and Pf [prime] is the reduced Pfa an (square-root of the determinant) of de ned by

Pf [prime] = 2(1)i+j zi zj

Pf ijij ; (2.10)

where 1 i; j n and ijij is the matrix removing rows i; j and columns i; j. We recall

that the Pfa an of a 2n [notdef] 2n antisymmetric matrix can be computed as

Pf = 1

2nn!

X2S2n sgn()

n

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Yi=1 2i12i ; (2.11)

where sgn() is the signature of 2 S2n. Importantly, Pf ijij is non-zero on the solutions

of scattering equations, while Pf [prime] is independent of the choice of i; j.

Speci cally, we have that

For color-ordered bi-adjoint scalar 3 theory,

F3 = 1z212z223 [notdef] [notdef] [notdef] z2n1: (2.12)

For color-ordered Yang-Mills theory with ordering [notdef]1; 2; : : : ; n[notdef],

FYM = 1z12z23 [notdef] [notdef] [notdef] zn1Pf [prime] : (2.13)

For gravity,

FG = (Pf [prime] )2 = 4(zi zj)2Det( ijij) : (2.14)

Having presented the above examples, let us go back to (2.6). As is clear from the expression, the right hand side is a rational function in the complex variables zi. To

employ methods developed in algebraic geometry, we need to associate solutions to a zero-dimensional algebraic variety de ned by some polynomials. In other words, we should rewrite Ea de ned in (2.1) to an equivalent polynomial system. This has been done in a

beautiful paper [37], where it has been shown that scattering equations are equivalent to following set of polynomials

0 = hm

XS 2A;[notdef]S[notdef]=m

k2SzS ; 2 m n 2 ; (2.15)

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where the sum is over all n!

(nm)!m! subsets S of A = [notdef]1; 2; : : : ; n[notdef] with exactly m elements

and kS =

Pb2S kb and zS = Qb2S zb. The algebraic geometry, notably the a ne Calabi-Yau properties of (2.15), has been investigated in [38].

A very useful observation made in [15, 37] is that If all k2S [negationslash]= 0, then values of za are

all distinct. The set (2.15) has not xed gauge. One of the choice of gauge will be to set, as is standard with points on CP1, the three points z1 = 1, z2 = 1 and zn = 0. Under this

choice, the set of polynomial is reduced to

eh1 m n3 lim z1!1

hm+1

z1 =

XS

2A=[notdef]1;n[notdef];[notdef]S[notdef]=m

(kS + k1)2zS[notdef]z2=1;zn=0 ; (2.16)

In summary, de nes a zero-dimensional ideal in the polynomial ring in n 3 variables.

Then, using the standard B ezouts theorem, the number of points in this ideal (solutions of the scattering equation) is

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Qn3m=1 deg(

ehm) = (n 3)!.

Instead of computing the amplitude with formula (2.6) by summing over all solutions of scattering equations, we will show in next section that, using the companion-matrix method, we can compute the amplitude An =

PsolPQ as the trace of certain matrix com-

posed of so-called companion matrices Tzi

An = Tr(P [prime][notdef]zi!Tzi Q[prime]1[notdef]zi!Tzi ) ; (2.17) without the explicit solutions of scattering equations.

3 The mathematical framework

As mentioned in the introduction, it is expedient to consider the problem within the framework of ideal theory. Our problem is thus the following.

Problem: Let I = [angbracketleft]fi[angbracketright] be a zero-dimensional ideal in R = C[x1; : : : ; xn]

generated by fi=1;2;:::;k(x1; : : : ; xn) 2 R and let r(x1; : : : ; xn) be an arbitrary

rational function in the fraction eld of R. Because dimC I = 0, I = [unionsq]Nj=1[notdef]zj[notdef]

is a discrete set of, say N, points. We wish to evaluate

N

Xj=1 p(zj)

where each summand is an evaluation of p at one of the discrete set of zeros zj. In particular we wish to test whether this sum is 0. This is the level 3 equivalence mentioned in the introduction.

Of course, the idea is to solve this without explicitly nding the roots zj. This can be done using the technique of companion matrices [40] (cf. also [41]). Suppose a Grobner basis for I has been found for some appropriate monomial ordering and B is an associated monomial basis for I, which can be seen as a vector space of dimension d. Then the multiplication map by the coordinate variable xi

R=I ! R=I

Ti : f ! xif (3.1)

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is an endomorphism of quotient rings. In the basis B of monomials, this is a d [notdef] d matrix

and is called a companion matrix. Clearly, [notdef]Ti[notdef] all mutually commute and thus can be

simultaneously diagonalized. We have the following [39]:

Theorem 3.0 (Stickelberger) The complex roots zi of I are the vectors of simultaneous eigenvalues of the companion matrices Ti=1;:::;n, i.e., the corresponding zero dimensional variety consists of the points:

V(I) = [notdef]( 1; : : : ; n) 2 Cn : 9v 2 Cn8i : Tiv = iv[notdef] :

We point out that the original statement of the theorem is in terms of annihilators in algebraic number theory and is perhaps a little abstruse. Fortunately, the computational algebraic-geometry community has rephrased this into the readily usable form of companion matrices [40, 42]. In particular, we have the following important consequence:

Corollary 1 Our desired quantity

N

Xj=1r(zj) = Tr[r (T1; : : : ; Tn)]

where the evaluation of the rational function r on the matrices Ti is without ambiguity since they mutually commute.

We remark that because r is rational, whenever the companion matrices appear in the denominator, they are to be understood as the inverse matrix.

3.1 Warmup

Before proceeding to examples in our context, we present two simple exercises to demonstrate our algorithm. Computations can be made in Macaulay2 [42] or Singular [43], or the latters interface with Mathematica [44]. Let

I := [angbracketleft]xy z; yz x; zx y[angbracketright] R = C[x; y; z] : (3.2) We know, of course, that there are 5 roots

V(I) = [notdef](0; 0; 0); (1; 1; 1); (1; 1; 1); (1; 1; 1); (1; 1; 1)[notdef] : (3.3) Now we consider two functions, where one is polynomial and another, rational:

p(x; y; z) = 3x3y + xyz; Q(x; y; z) = 3x3y + xyz

2xy2 + 4z2 + 1 : (3.4)

It is easy to nd, after summing over the solutions, that

XV(I)p = 4 ;

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XV(I)

Q = 2021 : (3.5)

We now show how the companion matrices work without nding the roots (3.3) explicitly.

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In the lex ordering of x y z, the Grobner basis and the monomial basis are,

respectively,

GB(I) =

z3 z; yz2 y; y2 z2; x yz[angbracketrightbig]; B = [notdef]1; y; yz; z; z2[notdef] : (3.6)

Therefore, we have that, in the quotient ring R=I,

x:B = [notdef]yz; z; z2; y; yz[notdef] ; y:B = [notdef]y; z2; z; yz; y[notdef] ; z:B = [notdef]z; yz; y; z2; z[notdef] ; (3.7) so that

Tx =0

@

0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0

1

A; Ty =

0

@

0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0

1

A; Tz =

0

@

0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0

1

: (3.8)

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Therefore, the sum over the roots of p is

Tr 3T 3xTy + TxTyTz

[parenrightbig]

= 4 (3.9)

and we have nice agreement with (3.4).

For the Q, the numerator is N = 3T 3xTy + TxTyTz =

0

B

B

@

0 0 0 3 1 0 1 3 0 0 0 3 1 0 0 0 0 0 1 3 0 0 0 3 1

1

C

C

A

, while the denomi-

nator is D = 2TxT 2y + 4T 2z + I =

0

B

B

@

1 0 2 0 4 0 5 0 2 0 0 0 5 0 2 0 2 0 5 0 0 0 2 0 5

1

C

C

A

. Thus we calculate Tr(ND1) = 2021, which

is the right answer on comparing with (3.4).

Before going to examples of scattering equations, let us give some remarks. First, the theorem in its original form is for polynomial test functions r, while functions we will meet in scattering equations are rational functions, i.e., the form P

Q with both P; Q are polynomials. Luckily, the theorem and corollary can be generalized trivially since we can diagonalize companion matrices simultaneously because the next remark.

Now, there is a second part of the theorem which states that the companion matrices can be simultaneously diagonalized if and only if the ideal I is a radical ideal. That is, there are no multiple roots. However, as shown in [15], if all k2S [negationslash]= 0, the solutions of zi will

all be di erent, so we indeed have a radical ideal and nd simultaneous eigenvalues readily.

Third, since there are (n 3)! solutions, the size of Ti will be in general d = (n 3)!

which will become very large with n. Although with this counting, the e ciency of the method may be arguable, it does make the following property manifest: after summing over all solutions, the nal result must be rational functions of k; .

4 Illustrative examples

In the following, we will use several examples to demonstrate the companion matrix method. The n = 4 case is simple. The companion matrix is 1-dimensional, equaling to the single solution of scattering equations. We compute the amplitudes in scalar 3,

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Yang-Mills and gravity theories to show the validity of the method. For n = 5, we rst study the amplitude of scalar 3 theory, and show that the amplitude-level identity can be understood by the fact that the trace of matrix is a linear mapping, and use it the explain a 7-point identity proposed in [31]. For the amplitudes of Yang-Mills and gravity theories, we will show that the companion matrix method indeed produce the correct amplitudes.

For n = 6, the scalar 3 theory will be shown to detect the pole structures so that the amplitude can be constructed by setting appropriate kinematics. The Next-MHV gluon amplitude is also presented as an example to show the validation of companion matrix method in a more di cult situation. Finally, for n = 7 amplitudes of scalar 3 theory, we demonstrate that, when companion matrices are computed in the diagonal form, the diagonal elements of the integrand matrix (which we recall to be an (n 3)! [notdef] (n 3)!

matrix for n-points) have one-to-one mapping to the integrand computed at the (n 3)!

solutions of the scattering equations, so they are not only equivalent at the amplitude level, but also at the level of each solution as indicated by Stickelbergers theorem.

4.1 Four-point amplitudes

The n = 4 case is trivial. There is only 4 3 = 1 variable left, so the companion matrix is

just a complex number. Let us remove three scattering equations E1, E2, E4 and gauge- x three points z1 = 1; z2 = 1 and z4 = 0. The remaining one scattering equation is

E3 =

Xb[negationslash]=3s3bz3 zb= s13z3 z1+ s23z3 z2+ s34z3 z4= (s23 + s34)z3 s34z3(z3 1): (4.1)

We can de ne the ideal I =

(s23 + s34)z3 s34[angbracketrightbig]in C[z3]. It is a linear function, so the Grobner basis and monomial basis are trivially

GB(I) =

(s23 + s34)z3 s34[angbracketrightbig]; B = [notdef]1[notdef] : (4.2)

The polynomial reduction of z3B = [notdef]z3[notdef] over Grobner basis of ideal I gives the remainder

[braceleftBig]

s34 s23+s34

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o. Thus in the quotient ring, the companion matrix is given by

Tz3B =

s34

s23 + s34

!

Tz3 = s34

s23 + s34 : (4.3)

We now proceed to the three cases of concern.

4.1.1 Scalar 3 theory

For the 4-point amplitude in scalar 3 theory, we wish to compute (recall that the three points z1; z2; z4 have been gauge xed)

A4 =

Xsol z212z224z241 | [notdef]124124

1 z212z223z234z241

=

Xz32sol

1s34(z3 1)2 + s23z23 [summationdisplay]z32solP (z3)

Q(z3) ; (4.4)

where we have used the simpli cation

| [notdef]124124 = 33 =

s23

s34

z23

(z3 1)2 !

1

| [notdef]124124

=

z23(z3 1)2z23(s12 + s23) 2z3s12 + s12

; (4.5)

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so that the factor 1=z223z234 cancels the numerator of 1=[notdef] [notdef]124124. We see that the nal expres

sion is summed over the (discrete) solution set of the scattering equation which is rather

trivial here. The summand is a rational function in the free variable z3 which we de ne as

P=Q; of course, P = 1 here and Q will be used later.Finally, using the simple expression for the companion matrix Tz3 from (4.3), we have

Tr

P (Tz3) Q(Tz3)

=s23 + s34s23s34 =1s14 1s12 ; (4.6)

after some identities between Mandelstam variables have been used. This is indeed the same answer as the standard known result as given in the introduction.

4.1.2 Yang-Mills theory

For 4-point amplitude in Yang-Mills theory, we want to compute (under gauge- xing z1 =

1; z2 = 1; z4 = 0),

AYM4 =

=Tr

1 Tz3Tz3(s12+s23) 2Tz3s12+s12

PYM(z3) QYM(z3) : (4.7)

To avoid the divergence when taking the limit z1 ! 1, one of the removed rows(columns)

in should be 1, otherwise some terms in Pf [prime] 8[notdef]8 would lead to in nity. Let us then

choose the reduced Pfa an as

Pf [prime] 8[notdef]8 =

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Xsol z212z224z241 | [notdef]124124

Pf [prime] 8[notdef]8 z12z23z34z41

Xz32sol

2 z1 z2

Pf 1212 : (4.8)

The large z1 dependence of Pf [prime] 8[notdef]8 is then 1=z21, and together with the factor from the

scalar part, we obtain a nite integrand when taking the z1 ! 1 limit. Explicitly, the new

matrix

e 1212 is a 6 [notdef] 6 matrix,

e =

0

B

B

B

B

B

B

B

B

@

0 k3k4

z3z4 1k3z1z3

2k3 z2z3

Pc[negationslash]=3 3kcz3zc 4k3z4z3

k3k4z4z3 0 1k4z1z4 2k4z2z4 3k4z3z4

Pc[negationslash]=4 4kc z4zc

1k3 z1z3

1k4z1z4 0 1 2z1z2 1 3z1z3 1 4 z1z4

2k3 z2z3

2k4 z2z4

1 2z2z1 0

2 3 z2z3

2 4 z2z4

1

C

C

C

C

C

C

C

C

A

; (4.9)

Pc[negationslash]=3 3kcz3zc 3k4z3z4 1 3z3z1 2 3z3z2 0

3 4 z3z4

4k3z4z3

Pc[negationslash]=4 4kcz4zc 1 4z4z1 2 4z4z2 3 4z4z3 0

whose Pfa an is given by

Pf 1212 =

e 16

e 25

e 34

e 15

e 26

e 34

e 16

e 24

e 35 +

e 14

e 26

e 35 +

e 15

e 24

e 36

e 14

e 25

e 36 +

e 16

e 23

e 45

e 13

e 26

e 45 +

e 12

e 36

e 45

e 15

e 23

e 46

e 56 : (4.10)

The reduced Pfa an Pf [prime] 8[notdef]8 in this case is a rational function with denominator z23(z31).

Together with the factor 1=z23z34 = 1=z3(z3 1), they cancel the numerator of 1=[notdef] [notdef]124124,

leaving a z3 in the denominator of integrand.

{ 10 {

+

e 13

e 25

e 46

e 12

e 35

e 46 +

e 14

e 23

e 56

e 13

e 24

e 56 +

e 12

e 34

Therefore, it is immediate that the numerator of the integrand comes entirely from the numerator of the reduced Pfa an:

PYM = z23 s12

e 1;3

e 2;4 2

e 3;4 1;3 2;4 + 2

e 2;4 1;4 3;2 2

e 1;4 2;4 3;2 + 2

e 2;4 1;3 3;4

+ 2

e 2;4 1;4 3;4 2

e 1;4 2;4 3;4 + 2

e 2;3 1;3 4;2 2

e 1;3 2;3 4;2 + 2

e 1;2 3;2 4;2

+ 2

e 1;2 3;4 4;2 + 2

e 1;3 2;4 4;3

[parenrightbig]

+ z3 s12

e 1;4

e 2;3 + s12

e 1;3

e 2;4 + s12

e 1;2

e 3;4

2

e 3;4 1;4 2;3 + 2

e 3;4 1;3 2;4 2

e 2;4 1;3 3;4 2

e 2;4 1;4 3;4 + 2

e 1;4 2;3 3;4

+ 2

e 1;4 2;4 3;4 2

e 1;2 3;4 4;2 + 2

e 2;3 1;3 4;3 + 2

e 2;3 1;4 4;3 2

e 1;3 2;3 4;3

2

e 1;3 2;4 4;3 + 2

e 1;2 3;2 4;3

[parenrightbig]

s12

e 1;2

e 3;4 ; (4.11)

where

e i;j i j, i;j ikj. The denominator of the integrand, on the other hand, is QYM = z33(s12 + s23) 2z23s12 + z3s12 = z3Q ; (4.12)

where Q is the denominator of integrand for the scalar 3 theory from (4.4).In summary, by computing Tr(P Q1[notdef]z3!Tz3 ), we arrive at

AYM4 =

JHEP12(2015)056

e 1;3

e 2;4

e 1;4

e 2;3

e 1;2

e 3;4

s12

s23

e 1;4

e 2;3

s23

s12

e 1;2

e 3;4

+ 1

[parenleftBig]

2

e 3;4 1;4 2;3 + 2

e 3;4 1;3 2;4 2

e 2;4 1;3 3;4 2

e 2;4 1;4 3;4 + 2

e 1;4 2;3 3;4

s12

+ 2

e 1;4 2;4 3;4 2

e 1;2 3;4 4;2 + 2

e 2;3 1;3 4;3 + 2

e 2;3 1;4 4;3 2

e 1;3 2;3 4;3

2

e 1;3 2;4 4;3 + 2

e 1;2 3;2 4;3

[parenrightBig]

+ 1

s23

[parenleftBig]

2

e 3;4 1;4 2;3 + 2

e 2;4 1;4 3;2 2

e 1;4 2;4 3;2 + 2

e 1;4 2;3 3;4 + 2

e 2;3 1;3 4;2

2

e 1;3 2;3 4;2 + 2

e 1;2 3;2 4;2 + 2

e 2;3 1;3 4;3 + 2

e 2;3 1;4 4;3 2

e 1;3 2;3 4;3

: (4.13)

The pole structures are similar to the scalar 3 theory, while the terms without poles come from the gluon four-vertex. Of course, by momentum conservation and the property iki = 0, we can further write the above result as a function of all independent kinematics, for example by using identities jk4 = jk3 jk2 jk1 and 4k4 = 0. This result agrees

with the one computed directly by Feynman diagrams.

4.1.3 Gravity

For the 4-point amplitude in gravity, we want to compute

AG4 =

+ 2

e 1;2 3;2 4;3

[parenrightBig]

PG(z3) QG(z3) ; (4.14)

under the gauge- xing z1 = 1; z2 = 1; z4 = 0. As in Yang-Mills theory, we choose the

reduced Pfa an as

Pf [prime] 8[notdef]8 =

Xsol z212z224z241 | [notdef]124124

Det[prime]( 8[notdef]8) =

Xsol z212z224z241 | [notdef]124124

(Pf [prime] 8[notdef]8)2

Xz32sol

2 z1 z2

Pf 1212 ; (4.15)

{ 11 {

and as above, we know that the squared reduced Pfa an (Pf [prime] 8[notdef]8)2 is a rational function

with denominator z43(z3 1)2. This cancels the numerator of 1=[notdef] [notdef]124124, leaving a z23 in

the denominator of integrand, so that the numerator of integrand equals to the square of numerator of reduced Pfa an:

PG = (P YM)2 ; (4.16)

while the denominator of integrand is

QG = z43(s12 + s23) 2z33s12 + z23s12 = z23Q = z3QYM : (4.17)

Combining all together, we have

PG

QG =

2= s12s23s12 + s23 (AYM)2 : (4.19)

By BCJ relation [45], we can rewrite this to the familiar one

AG4 = s12AYM4(1; 2; 3; 4)AYM4(1; 2; 4; 3) ; (4.20)

in agreement with the known result by KLT relation [46{50].

4.2 Five-point amplitudes

For n = 5 amplitudes, there are ve scattering equations, but only two of them are independent. Under the gauge- xing z1 = 1, z2 = 1, z5 = 0, the Dolan-Goddards for

mula [15] gives:

f1 = s12 + s13z3 + s14z4 ; f2 = s45z3 + s35z4 + s25z3z4 : (4.21)

We can solve these two equations to get two solutions:

sol1 : z3 = s12s25 s13s35 + s14s45 p

2s13s25 ; z4 =

and

sol2 : z3 = s12s25 s13s35 + s14s45 + p

2s13s25 ; z4 =

{ 12 {

(P YM)2

(QYM)2 ; (4.18)

where Q is the denominator of integrand for scalar 3 theory from (4.4) and the expressions for P YM and QYM are given in (4.11) and (4.12). In the present case of n = 4, there is only one solution for scattering equations, and the companion matrix is really 1-dimensional in (4.3), so although in general Tr(M1M2) [negationslash]= Tr(M1) Tr(M2), here we simply have

Tr

P G QG

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(P YM)2

z3QYM =

QYM

(P YM)2

(QYM)2 = Q

z3

= Tr(Q) Tr

P YM QYM

s12s25 + s13s35 s14s45 + p

2s14s25 ;

s12s25 + s13s35 s14s45 p

2s14s25 ;

where = (s12s25 + s13s35 s14s45)2 4s12s13s25s35. We can see that, in general the solutions are not rational functions, as is to be expected from high degree polynomials,

though of course the nal result of the sum over these points will be. One can see that the cancelations and simpli cations will be very involved.

Let us turn to our companion matrix method. De ne ideal I =

f1; f2[angbracketrightbig]in the polynomial ring C[z3; z4], the Grobner basis of ideal I in Lexicographic order z3 z4 is given by

GB(I) =

s12s45 + s12s25z4 s13s35z4 + s14s45z4 + s14s25z24; s12 + s13z3 + s14z4 ; s45z3 + s35z4 + s25z3z4

[angbracketrightbig]

: (4.22)

The monomial basis in this Grobner basis is B = [notdef]1; z4[notdef]. Polynomial reduction of z3B and

z4B over GB(I) gives the companion matrices Tz3B = z3B, Tz4B = z4B as

Tz3 = s12s13 s14s13

s12s45 s13s25

s14s45s13s35

s13s25

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[parenrightBigg]

; Tz4 = 0 1

s12s45s14s25

s13s35s14s45s12s25

s14s25

[parenrightBigg]

; (4.23)

which are 2[notdef]2 matrices, in accordance with the number of solutions of scattering equations.

We note that the companion matrices actually formally \live" in the ideal I itself by satisfying scattering equations, i.e.,

f1 ! s12I2[notdef]2 + s13Tz3 + s14Tz4

= s12 0 0 s12

!+

s12 s14

s12s45 s25

s14s45s13s35

s25

!+0 s14

s12s45s25

s13s35s14s45s12s25

s25

!= 02[notdef]2 ; (4.24)

and likewise, s45Tz3 + s35Tz4 + s25Tz3Tz4 = 02[notdef]2. This is, of course, a general property by construction since the companion matrices are constructed as multiplication (on a particular basis), so that substituting into the de ning polynomials would vanish in the quotient ring. The situation is very much analogous to the classical result of Cayley-Hamilton that a matrix satis es its own characteristic polynomial. It is worth to emphasize this discussion as

Corollary 2 The companion matrices satisfy the de ning polynomials of the given ideal.

The above corollary shows some kind of equivalence between solutions of scattering equations and companion matrices of monomial basis over the Grobner basis of scattering equations. With these companion matrices, we now proceed to compute the trace of the integrands to obtain the amplitude for di erent theories.

4.2.1 Scalar 3 theory

The 5-point amplitude of scalar 3 theory is given by

A5 =

Xsol z212z225z251 | [notdef]125125

1 z212z223z234z245z251

1

=

Xsol

| [notdef]125125(z3 1)2(z3 z4)2z24 [summationdisplay] z3;z42sol

P (z3; z4)

Q(z3; z4) ;

(4.25)

where we have used that

| [notdef]125125 =[parenleftbigg]

s23

(z31)2

s34

(z3 z4)2

s35

z23

[parenrightbigg][parenleftbigg]

s24

(z41)2

s34

(z3 z4)2

s45

z24

s234 (z3 z4)4

{ 13 {

and as above, de ned the appropriate P and Q, which are, explicitly,

P = z23(z4 1)2 ;

Q =

s35(z3 1)2 + s23z23[parenrightBig](z3 z4)2[parenleftBig]s45(z4 1)2 + s24z24[parenrightBig]

+s34

hs45(z3 1)2z23(z4 1)2

+z24

z23 s24(z3 1)2 + s23(z4 1)2 [parenrightbig]

: (4.26)

Now, we wish to compute the trace of the matrix P Q1 upon substituting z3 and z4 by their associated companion matrices, instead of summing over all the complicated solutions of the scattering equations. In other words, we should replace the variables z3; z4

as Tz3; Tz4 in the integrand, i.e., P [prime] = P [notdef]z3!Tz3;z4!Tz4 , Q[prime] = Q[notdef]z3!Tz3;z4!Tz4 (Hereafter

we will always use P [prime]; Q[prime] to denote the matrices after replacing zi to Tzi). The product of variables z3; z4 changes to the product of matrices Tz3; Tz4, and since the companion matrices are commutable, their order does not matter in here. Then we should compute the inverse of matrix Q[prime], and the nal result is given by Tr(P [prime]Q[prime]1).

Recalling that the physical poles appearing in the color-ordered amplitude are s12; s23; s34; s45; s15, we can de ne them as the independent Mandelstam variables, and

rewrite all the other Mandelstam variables in P; Q; Tz3; Tz4 by using following identities:

s35 = s12 s34 s45 ; s24 = s15 s23 s34 ; s25 = s34 s15 s12 ;s14 = s23 s45 s15 ; s13 = s45 s12 s23 : (4.27)

After some algebraic manipulation, readily performed by Mathematica, we obtain

Tr(P [prime](Tz3; Tz4)Q[prime]1(Tz3; Tz4)) = 1

s15s23 +

which agrees with the known result [30, 31].

Let us further consider an example, corresponding to the two-cycles3

{(1; 2; 3; 4; 5); (1; 3; 5; 2; 4)[notdef], in the language of [30, 31, 36]. Using the CHY-integrand

de ned by above two-cycles, we have

A[prime]5 =

Xsol z212z225z251 | [notdef]125125

{ 14 {

+ s35(z3 1)2(z4 1)2[parenrightBig][bracketrightBig]

1 s15s34 +

JHEP12(2015)056

1 s12s34 +

1 s12s45 +

1s23s45 ; (4.28)

P1(z3; z4)

Q1(z3; z4) ; (4.29)

which is represented by the so-called pentacle diagram (shown in gure 1)from the view of integration rules. Using the mapping rule given in [30, 31], the answer is known to be zero. By directly computing the trace, we indeed nd that Tr(P [prime]1Q[prime]11) = 0 and con rms this result.

In fact, for this example, although CHY-integrands of A5 and A[prime]5 are di erent, after

simpli cation, their di erence appears only in the numerator, i.e.,

P1 = z3z4(1 z3)(1 z4)(z3 z4) ; Q1 = Q : (4.30)

3Each cycle de nes an expression, e.g., Cyclea(1, 3, 5, 2, 4) = 1/(z13z35z52z24z41), and the two-cycles denotes the expression given by CycleaCycleb.

1 z12z23z34z45z51

1

z13z35z52z24z41

1

5

2

4

3

Figure 1. The pentacle diagram representing the CHY-integrand de ned by the two-cycles

{(1,2,3,4,5),(1,3,5,2,4)[notdef].

Since the trace of matrix is a linear mapping, in particular Tr(M1+M2) = Tr(M1)+Tr(M2), relations between results of di erent integrands should also have hints in the integrand level. For example, let us consider the following three CHY-integrands de ned by three two-cycles 2 [notdef](1; 2; 3; 4; 5); (1; 2; 3; 5; 4)[notdef], 3 [notdef](1; 2; 3; 4; 5); (1; 2; 4; 5; 3)[notdef] and 4

{(1; 2; 3; 4; 5); (1; 3; 2; 5; 4)[notdef]. With some calculations, we nd

A5( 2) =Xsol z212z225z251 | [notdef]125125

1 z12z23z34z45z51

1

z12z23z35z54z41

XsolP2 Q ;

JHEP12(2015)056

A5( 3) =

Xsol z212z225z251 | [notdef]125125

1 z12z23z34z45z51

1

z12z24z45z53z31

XsolP3 Q ;

A5( 4) =

Xsol z212z225z251 | [notdef]125125

1 z12z23z34z45z51

1

z13z32z25z54z41

XsolP4Q (4.31)

where they share the same denominator Q, but di erent numerators

P2 = z3(z3 z4)(z4 1)2 ; P3 = z3(z3 1)(z3 z4)(z4 1) ;

P4 = z23(z3 z4)(z4 1)2 : (4.32)

After putting back the companion matrices, we nd that

A5( 2) = Tr(P [prime]2Q[prime]1) = 1 s12s45 +

1 s23s45 ;

A5( 3) = Tr(P [prime]3Q[prime]1) = 1s12s45 ; A5( 4) = Tr(P [prime]4Q[prime]1) =

1 s23s45 :

Realizing that the polynomials have the simple relation

P3 + P4 P2 = P1 ; (4.33)

we obtain the identity amongst these amplitudes as

Tr(P [prime]2Q[prime]1) = Tr((P [prime]3 + P [prime]4 P [prime]1)Q[prime]1) = Tr(P [prime]3Q[prime]1) + Tr(P [prime]4Q[prime]1) + Tr(P [prime]1Q[prime]1)

! A5( 2) = A5( 3) + A5( 4) + 0 : (4.34)

{ 15 {

Above example demonstrates an idea how to nd relations among di erent amplitudes. Starting from di erent CHY-integrands, we can equalize their denominators by multiplying proper polynomial both at the denominator and the numerator. After that, the relations among di erent amplitudes can be understood from the relations among di erent numerators.

Let us demonstrate above idea by another example, i.e., the 7-point amplitude-level identity given by eq. (3.7) of [31], viz., amplitude obtained from the CHY-integrand

1 z12z23z34z45z56z67z71

1

z12z27z74z46z65z53z31 (4.35)

is identical to the sum of following two amplitudes obtained from two CHY-integrand

1 z12z23z34z45z56z67z71

1 z12z56z37z46

[parenleftbigg]

1 z14z27z35 +

1 z25z74z31

[parenrightbigg]

: (4.36)

Under gauge- xing z1 = 1, z2 = 1, z7 = 0 and excluding the 1-st, 2-nd and 7-th scattering

equations, the Jacobian is

1

| [notdef]127127

=

Q3 i<j 6(zi zj)2Q : (4.37)

Thus we can immediately get the numerator of integrand after inserting the above three terms. The rst term gives

P1 = z3z5(z4 1)(z5 1)(z6 1)(z3 z6)

6

Yi=3zi(zi 1)

i[negationslash]=5

Q6i=3 z2i(zi 1)2

JHEP12(2015)056

Y3 i<j 6(zi zj) ; (4.38)

while the other two terms give

P2 = z4z5(z4 1)(z5 1)(z6 1)(z3 z6)

6

Yi=3zi(zi 1)

i[negationslash]=5

Y3 i<j 6(zi zj) ; (4.39)

P3 = z5(z4 1)(z6 1)(z3 z5)(z3 z6)

6

Yi=3zi(zi 1)

i[negationslash]=5

Y3 i<j 6(zi zj) : (4.40)

Note that

P1 P2 P3 (4.41)

= z5(z4 1)(z6 1)(z3 z6)(z4 z5 + z3z5 z4z5)

6

Yi=3zi(zi 1)

i[negationslash]=5

Y3 i<j 6(zi zj) ;

while the trace Tr((P [prime]1 P [prime]2 P [prime]3)Q[prime]1) is zero. Note also the following decomposition

z4 z5 + z3z5 z4z5 = (z3 1)(z5 z4) + z4(z3 z5) ; (4.42)

{ 16 {

so that we can write P1 P2 P3 = P4 + P5, with

P4 = z5(z3 1)(z4 1)(z6 1)(z3 z6)(z4 z5)

6

Yi=3zi(zi 1)

i[negationslash]=5

Y3 i<j 6(zi zj) ; (4.43)

P5 = z4z5(z4 1)(z6 1)(z3 z5)(z3 z6)

6

Yi=3zi(zi 1)

i[negationslash]=5

Y3 i<j 6(zi zj) ; (4.44)

which correspond to two-cycles

{(1; 2; 7; 4; 6; 5; 3); (1; 2; 5; 6; 7; 3; 4)[notdef] ; [notdef](1; 2; 3; 4; 5; 6; 7); (1; 3; 7; 2; 5; 6; 4)[notdef] (4.45) respectively with Tr(P [prime]4Q[prime]1) = 0, Tr(P [prime]5Q[prime]1) = 0.

We thus conclude that strictly speaking, the amplitude-level identity between (4.35) and (4.36) is up to some CHY-integrands which have vanishing amplitude. More explicitly, the identity (4.35)=(4.36)+(4.45) holds exactly at the integrand-level, while (4.45) has vanishing nal result, so that (4.35)=(4.36) holds at the amplitude-level. This provides the amplitude-level identity an explanation from the basic linearity of the trace.

4.2.2 Yang-Mills theory

For 5-point amplitude in Yang-Mills theory, we want to compute

AYM5 =

JHEP12(2015)056

PYM(z3; z4) QYM(z3; z4) ; (4.46)

under gauge- xing z1 = 1; z2 = 1; z5 = 0. Let us choose the reduced Pfa an as

Pf [prime] 10[notdef]10 =

Xsol z212z225z251 | [notdef]125125

Pf [prime] 10[notdef]10 z12z23z34z45z51

Xz3;z42sol

2 z1 z2

Pf 1212 ; (4.47)

where 1212 is a 8 [notdef] 8 matrix. As in the 4-point case, the large z1 dependence of Pf [prime] 10[notdef]10

is 1=z21, while 1=(z12z23z34z45z51) is also 1=z21. Together with the factor z212z225z251 in numerator, we get a nite integrand under the z1 ! 1 limit.

We now follow the standard computation procedure:

1. Write down the expressions for [notdef] [notdef]125125 and Pf [prime] 10[notdef]10, and work out P YM(z3; z4), QYM(z3; z4);

2. Replace the variables zis by companion matrices Tzi, as P [prime] = P [notdef]zi!Tzi , Q[prime] =

Q[notdef]zi!Tzi ;

3. Compute the inverse of Q[prime] and the trace Tr(P [prime]Q[prime]1).

The result for un-speci ed helicities is quite lengthy. For illustration, let us consider the 5-point amplitude with helicity AYM5(g1; g2; g+3; g+4; g+5). The polarization vector is de ned as

(k) = [angbracketleft]k[notdef] [notdef]r]

p2[k r] ; + (k) = [angbracketleft]

r[notdef] [notdef]k] p2[angbracketleft]r k[angbracketright]

; (4.48)

{ 17 {

and we choose the reference momenta as r1 = r2 = k3, r3 = r4 = r5 = k2. Thus settled,

the only surviving products of polarization vectors are (k1) [notdef] +(k4) and (k1) [notdef] +(k5).

After imposing momentum conservation for [notdef](ki) [notdef] kj to reduce the ambiguity, we can

simplify the 8 [notdef] 8 matrix

e 1212 as

0

B

B

B

B

B

B

B

B

B

B

B

B

B

@

0 k3k4

z3z4

k3k5z3z5 0 0

e 16 4k3z4z3 5k3z5z3

k3k4z4z3 0k4k5z4z5 1k4z1z4 2k4z2z4 3k4z3z4

e 27 5k1+ 5k3 z5z4

k3k5z5z3k4k5z5z4 0 1k2+ 1k4z1z5 2k1+ 2k4z2z5 3k1+ 3k4z3z5 4k1+ 4k3 z4z5

e 38

0 1k4z1z4 1k2+ 1k4z1z5 0 0 0 1 4z1z4 1 5 z1z5

0 2k4

z2z4 2k1+ 2k4z2z5 0 0 0 0 0

e 61 3k4z3z4 3k1+ 3k4z3z5 0 0 0 0 0 4k3z4z3

e 72 4k1+ 4k3z4z5 1 4z4z1 0 0 0 0

5k3z5z3 5k1+ 5k3z5z4

; (4.49)

JHEP12(2015)056

e 83 1 5z5z1 0 0 0 0

1

C

C

C

C

C

C

C

C

C

C

C

C

C

A

where

e 16 =

e 61 = 3k1z3 z1+ 3k4z3 z4

3k1 + 3k4

z3 z5

;

e 27 =

e 72 = 4k1z4 z1+ 4k3z4 z3

4k1 + 4k3

z4 z5

;

e 38 =

e 83 = 5k1z5 z1+ 5k3z5 z3

5k1 + 5k3

z5 z4

: (4.50)

This greatly simpli es the result of reduced Pfa an, which reads, after our gauge- xing,

Pf [prime] 10[notdef]10

N

D = 2

z4 2;1 + z4 2;4 2;1 z3z4(z41)(z3 z4)

[parenleftBig][parenleftBigg]

z3z4

[parenrightbig][tildewide]

1;5 3;1 4;1 z3z4

[parenrightbig][tildewide]

1;4 3;1 5;1

(z3z4)

e 1;4 3;1 5;3 z4

e 1;5 3;4 4;1 + z3

e 1;5 3;1 4;3 + z4

e 1;4 3;4 5;1

; (4.51)

where we recall again that

e i;j = i j, i;j = ikj. The factor of scalar part 1=(z12z23z34z45z51) after gauge xing is 1z4(z31)(z3z4) , and the Jacobian [notdef] [notdef]125125 is the same

as in the scalar theory,

| [notdef]125125

1 = z23z24(z3 1)2(z4 1)2(z3 z4)2

QYM ; (4.52)

where QYM is a polynomial of z3; z4 and Mandelstam variables, and it is also the denominator of integrand. The numerator of 1=[notdef] [notdef]125125 cancels the denominator of Pf [prime] and that

of scalar part, leaving a factor z3(z3 1)(z4 1) in the numerator. Combined with the

numerator N of Pf [prime] 10[notdef]10, they contribute to P YM = z3(z3 1)(z4 1)N .

Then it is straightforward to apply the replacements P [prime]YM(Tz3; Tz4) =

PYM(z3; z4)[notdef]zi!Tzi , Q[prime]YM(Tz3; Tz4) = QYM(z3; z4)[notdef]zi!Tzi , and compute the trace Tr(P [prime]Y M(Q[prime]YM)1). To make the computation more e cient, we can rstly apply the polynomial reduction of P YM(z3; z4), QYM(z3; z4) over GB(I). The remainders R(P YM); R(QYM) are polynomials of z4 only, since the monomial basis is [notdef]1; z4[notdef]. Then we can proceed by

replacing z4 ! Tz4 for the remainders, and compute the corresponding trace. This gives

{ 18 {

the same result as with the original P YM; QYM, but the computation would be much faster. With Mathematica, we obtain

Tr(P [prime]YM(Q[prime]YM)1) = 2

e 1;5 2;1 3;1 4;3 +

e 1;4 2;1 3;4 5;1

e 1;5 2;1 3;4 4;1 s12s34

+2

e 1;5 2;1 3;1 4;1 +

e 1;5 2;1 3;1 4;3

e 1;4 2;1 3;1 5;1

e 1;4 2;1 3;1 5;3

s12s45

e 1;5 2;4 3;4 4;1

e 1;5 2;4 3;1 4;3

e 1;4 2;4 3;4 5;1s15s34 : (4.53)

The missing of the pole terms 1=(s15s23); 1=(s23s45) (terms involving pole s23) is due to the choice of polarization vectors. However, the s23 pole do exist, hiding in 2;i; 3;i. Directly rewriting the spinor brackets for

e i;j; i;j and sij, and using the Schouten identities we get the famous MHV-amplitude [51, 52]

Tr(P [prime]YM(Q[prime]YM)1) = [angbracketleft]1 2[angbracketright]4

h1 2[angbracketright][angbracketleft]2 3[angbracketright][angbracketleft]3 4[angbracketright][angbracketleft]4 5[angbracketright][angbracketleft]5 1[angbracketright]

: (4.54)

+2

JHEP12(2015)056

4.2.3 Gravity and n-point KLT relations

For 5-point amplitude in pure gravity theory, under gauge- xing z1 = 1, z2 = 1, z5 = 0, we wish to compute

AG5 =

Xsol z212z225z251 | [notdef]125125

(Pf [prime] 10[notdef]10)(Pf [prime]

(Pf [prime] 10[notdef]10)2 : (4.55)

Let us consider the gravity amplitude AG5(1; 2; 3++; 4++; 5++), so that we can use

the same reduced Pfa an Pf [prime] 10[notdef]10 as in the Yang-Mills case. Here, we do not have the

factor of scalar part, but the square of the factor of the reduced Pfa an. The numerator of 1=[notdef] [notdef]125125 cancels the squared denominator of reduced Pfa an z23z24(z41)2(z3z4)2, leaving

a factor of (z3 1)2 in the numerator. Hence, we have QG = QYM, and P G = (z3 1)2N2

with N given in (4.51).

Thus, all the ingredients have been computed in the Yang-Mills situation above, and we only need to work out the trace Tr(P [prime]G(Q[prime]G)1), which gives a lengthy result:

h1 2[angbracketright]4 [parenleftBig] [angbracketleft]1 2[angbracketright]7 [angbracketleft]1 5[angbracketright] [angbracketleft]3 4[angbracketright] [2 1]4 [3 1]3 [4 2] [4 3]2 [5 1]3 + 971 more terms[parenrightBig]

h1 3[angbracketright] [angbracketleft]1 4[angbracketright] [angbracketleft]1 5[angbracketright] [angbracketleft]2 3[angbracketright]2 [angbracketleft]2 4[angbracketright]2 [angbracketleft]2 5[angbracketright]3 [angbracketleft]3 4[angbracketright] [angbracketleft]3 5[angbracketright] [angbracketleft]4 5[angbracketright] [2 1] [3 1]2 [3 2]2 [4 1] [4 2] [5 1] [5 2] [5 3] [5 4]

where we can see that all poles si;j; i; j = 1; : : : ; 5 appearing therein, indicating the colorless

structure of gravity amplitude.

This complicated expression can be simpli ed by non-trivially imposing momentum conservation and Schouten identities. Applying the algorithm described in the appendix of [53], for instance, we can simplify AG5(1; 2; 3++; 4++; 5++) to

h1 2[angbracketright]6 [4 3] [5 3] h1 4[angbracketright] [angbracketleft]1 5[angbracketright] [angbracketleft]2 4[angbracketright] [angbracketleft]2 5[angbracketright] [angbracketleft]3 4[angbracketright] [angbracketleft]3 5[angbracketright]

e 10[notdef]10) =

Xsol z212z225z251 | [notdef]125125

+ [angbracketleft]1 2[angbracketright]6 [4 3] [5 4] h1 3[angbracketright] [angbracketleft]1 5[angbracketright] [angbracketleft]2 3[angbracketright] [angbracketleft]2 5[angbracketright] [angbracketleft]3 4[angbracketright] [angbracketleft]4 5[angbracketright]

+ [angbracketleft]1 2[angbracketright]6 [5 3] [5 4]

: (4.56)

which agrees perfectly with the result given by KLT relation [46{50].

{ 19 {

h1 3[angbracketright] [angbracketleft]1 4[angbracketright] [angbracketleft]2 3[angbracketright] [angbracketleft]2 4[angbracketright] [angbracketleft]3 5[angbracketright] [angbracketleft]4 5[angbracketright]

More generally, for n-point amplitude, under the usual gauge- xing z1 = 1, z2 = 1, zn = 0, we wish to compute

AGn =

Xsol z212z22nz2n1 | [notdef]12n12n

(Pf [prime] 2n[notdef]2n)(Pf [prime]

e 2n[notdef]2n) =

Xsol z212z22nz2n1 | [notdef]12n12n

(Pf [prime] 2n[notdef]2n)2 : (4.57)

In order to write down the reduced Pfa an, we need to compute the Pfa an of a (2n 2) (2n2) matrix, which is quite complicated. Direct computation using the above formula

is obviously very di cult, just like the direct computation of gravity amplitude by Feynman diagram. So we would like to follow the KLT formalism, and compute the gravity amplitude as square of Yang-Mills amplitudes.

An important property of the reduced Pfa an is that, it can be expanded [3] as

Pf [prime] =

X 2Sn3

P 2Sn3 S[ [notdef] ]AYMn(1; ; n; n 1)(z1 z 2)(z 2 z 3) [notdef] [notdef] [notdef] (z n2 zn1)(zn1 zn)(zn z1); (4.58)

where ; are permutations of labels 2; 3; : : : ; n 2, and S[ [notdef] ] is the S-kernel. The

appearance of AYMn is a consequence of certain integrand summing over all (n3)! solutions

of scattering equations in the original derivation, and in the companion matrix method, it corresponds to the trace of that integrand when changing variables to companion matrices. In any event, it is a constant, and can be dragged out of the trace.

Using this expression, we can expand one Pf [prime] in the gravity amplitude,

AGn =

Xsol

JHEP12(2015)056

X 2Sn3

P (z3; z4; : : : ; zn1)

Q(z3; z4; : : : ; zn1)

[parenrightbigg] [notdef] [summationdisplay] 2Sn3S[ [notdef] ]AYMn(1; ; n; n 1) ; (4.59)

P

Q

z212z22nz2n1

| [notdef]12n12n

Pf [prime] 2n[notdef]2n z1 2z 2 3 [notdef] [notdef] [notdef] z n2;n1zn1;nzn1

: (4.60)

The trace Tr(P [prime](Tzi)Q[prime]1(Tzi)) for the set gives AYMn(1; ; n 1; n), and the summation

over permutations of can be taken out of the trace, and we thereby arrive at the KLT relation.4

4.3 Six-point amplitudes

We proceed onto six-point amplitudes, i.e., n = 6. Using the standard gauge- xing z1 = 1, z2 = 1, z6 = 0, Dolan-Goddards polynomial form [37] of the scattering equations is given by

f1 = s12 + s13z3 + s14z4 + s15z5 ; (4.61)

f2 = s123z3 + s124z4 + s125z5 + s134z3z4 + s135z3z5 + s145z4z5 ; (4.62)

f3 = s56z3z4 + s46z3z5 + s36z4z5 + s26z3z4z5 : (4.63)

We can thus de ne the ideal I =

f1; f2; f3[angbracketrightbig]in the polynomial ring C[z3; z4; z5]. The degree of ideal I is 6, so according to B ezouts theorem, it has 6 solutions, though it is not possible to obtain analytic expressions for these solutions, as already seen in the 5-point cases. Let us then consider the companion matrix method.

4Note that the ordering of set (or ) here de ned in [3] is the reverse of that de ned in [48].

{ 20 {

We generate the Grobner basis for I in Lexicographic ordering z3 z4 z5. Analytically, the explicit expression of GB(I) is rather complicated, especially in the presence of so many parameters sij in the ring. By varying the exponents to some high power, the polynomial reduction of the monomials za33za44za55 (with ai from 0 to some nite number, say 20) over GB(I) gives the monomial basis

B = [notdef]1; z5; z25; z35; z45; z55[notdef] :

The polynomial reduction of z3B; z4B and z5B over GB(I) gives the companion matrices Tz3; Tz4; Tz5, which are 6 [notdef] 6 matrices. Again, we need to compute P [prime] =

P [notdef]z3!Tz3;z4!Tz4;z5!Tz5 Q[prime] = Q[notdef]z3!Tz3;z4!Tz4;z5!Tz5 , and the nal amplitude is given by

A6 = Tr(P [prime]Q[prime]1), without summing over all solutions of scattering equations.

Since the operations we need are multiplication of matrices, taking inverse or trace of matrices, so in principle it can be done analytically. However, the symbolic manipulation for n = 6 case is quite complicated, especially when taking the inverse of matrix Q[prime] and simplifying the tedious trace result in Mathematica, so we introduce random numeric kinematics | i.e., by Monte Carlo assignments of the parametres sij | to get the nal result.

One will see that, as is customary with coe cient elds in polynomial rings, trying a few large prime numbers would su ce very quickly.

4.3.1 Scalar 3 theory

We can write the amplitude as

A6 =

Xsol z212z226z261 | [notdef]126126

1 z212z223z234z245z256z261

=

Xsol

JHEP12(2015)056

| [notdef]126126(z3 1)2(z3 z4)2(z4 z5)2z25 [summationdisplay] z3;z4;z52sol

P (z3; z4; z5)

Q(z3; z4; z5) ; (4.64)

1

where

126126 =

0

B

@

33 34 35

43 44 45

53 54 55

1

C

A

; [notdef] [notdef]126126 = Det( 126126) : (4.65)

Prime kinematic strategy. The idea is the following. Since we know for scalar 3 theory, the nal result of A6 should take the form

A6 =

X ; ic s 1s 2s 3 ; (4.66)

where s i are the independent Mandelstam variables of physical poles s12, s23, s34, s45, s56, s16, s123, s234, s345, and the summation is over all possible products of three physical poles, e.g., 1

s12s23s56 ,

1s12s45s234 , etc. So in total we have

= 84 terms, which we denote as S , = 1; 2; : : : 84, and the amplitude is expanded as A6 =

P84 =1 c S , where c is either 0 or 1.To each physical pole we now randomly assign a prime number, i.e., we are working with the much simpler polynomial ring C[z] instead of C(s)[z]. In this case, the computation

{ 21 {

9 3

[parenrightbig]

of Tr(P [prime]Q[prime]1) is trivial within seconds, and the result as well as S s are all numbers. Next,

we shall nd the solutions

P84 =1 c S = Tr(P [prime]Q[prime]1) for c s. However, doing this by brute-force is impossible since there are 84 c s and each one can take 0 or 1, so one would go through all 284 possibilities, which is far beyond any computational ability.

We therefore adopt the following strategy: instead of setting all coe cients to numbers, we can assign all physical poles to prime numbers except one pole. For example, we would leave s345, to detect rst the coe cients of S s which contains the pole

1s345 . Keeping one symbolic variable s345 would extend the computation time of Tr(P [prime]Q[prime]1) up to minutes, but it is still very manageable, while keeping two or more symbolic variables would make the computation of Tr(P [prime]Q[prime]1) in Mathematica very hard for a laptop.

Let us see the above strategy in action. Setting the kinematics (coe cient variables) as, e.g.,

s12 = 7 ; s23 = 37 ; s34 = 79 ; s45 = 97 ;

s56 = 131 ; s16 = 179 ; s123 = 181 ; s234 = 223 ;

while leaving s345 free, we get

Tr(P [prime]Q[prime]1) =

64909247478 1878479042622679

327369601739s345 : (4.67)

Among the S s, there are

JHEP12(2015)056

8 2

[parenrightbig]

= 28 terms containing physical pole s345, and the number

marked by 1

s345 in Tr(P [prime]Q[prime]1) should be expanded into these 28 terms.5 This is thus a problem in Egyptian fractions. By going through all 228 possibilities of c , we nd the unique expansion

32736 9601739

1s345 = [parenleftbigg]

17 [notdef] 79

+ 1

79 [notdef] 179

+ 1

7 [notdef] 97

+ 1

97 [notdef] 179

[parenrightbigg]

1s345 ; (4.68)

So mapping to the physical poles, we nd that

1 s12s34s345

1 s16s34s345

1 s12s45s345

1s16s45s345 (4.69)

is a part of A6.

Now, we try to get more poles. Taking the kinematics as, e.g.,

s12 = 7 ; s23 = 37 ; s34 = 79 ; s45 = 97 ;

s56 = 131 ; s16 = 179 ; s123 = 181 ; s345 = 251 ;

while leaving s234 free, we get

Tr(P [prime]Q[prime]1) =

35960 68541427s234

13829207594302048840293589 : (4.70)

The part marked by the physical pole s234 can be uniquely expanded as

35960 68541427

1s234 = [parenleftbigg]

1s234 ; (4.71)

5In fact, using the compatibility among poles, we can greatly reduce the number of possible combinations of poles. We will consider this fact in latter examples.

137 [notdef] 179

+ 1

79 [notdef] 179

+ 1

37 [notdef] 131

+ 1

79 [notdef] 131

[parenrightbigg]

{ 22 {

thus

1s34s56s234 ; (4.72)

is also a part of A6. With the same procedure, we nd that for physical pole s123,

1 s12s45s123

1 s16s23s234

1 s16s34s234

1 s23s56s234

1s23s56s123 (4.73)

is also part of A6. Finally, we need to determine the coe cients c of S s without physical

poles s123; s234; s345. There are in total 63

[parenrightbig]

1 s23s45s123

1 s12s56s123

= 20 terms. Taking the kinematics as, e.g.,

s12 = 7 ; s23 = 37 ; s34 = 79 ; s45 = 97 ;

s56 = 131 ; s16 = 179 ; s123 = 181 ; s234 = 223 ; s345 = 251 ;

computing the Tr(P [prime]Q[prime]1) and extracting the contributions from results (4.69), (4.72), (4.73), the remaining result can be uniquely expanded as

714874 46539628933 =

JHEP12(2015)056

17 [notdef] 79 [notdef] 131

137 [notdef] 97 [notdef] 179

; (4.74)

so the last part for A6 is

1s16s23s45 : (4.75)

Putting all the above together, we therefore conclude that

A6 =

[parenleftbigg]

1 s12s34s56 +

1 s12s34s56

1 s16s23s45 +

1 s12s45s123 +

1 s23s45s123 +

1 s12s56s123

+ 1

s23s56s123 +

1 s12s34s345 +

1 s16s34s345 +

1 s12s45s345 +

1 s16s45s345

+ 1

s16s23s234 +

1 s16s34s234 +

1 s23s56s234 +

1 s34s56s234

[parenrightbigg]

: (4.76)

This prime-numeric method can be applied to all the cases of n = 6 amplitudes of scalar 3 theory.

4.3.2 Yang-Mills theory

For Yang-Mills theory, when n = 6, we meet the rst \not so simple" gluon amplitude,i.e., the next-MHV amplitude, so it is worthwhile to verify the companion matrix method with this non-trivial example. To illustrate, let us consider the split helicity amplitude

AYM6 (g1; g2; g3; g+4; g+5; g+6), and choose the reference momenta as r1 = r2 = r3 = k4,

r4 = r5 = r6 = k3, so that only

e 1;5,

e 1;6,

e 2;5,

e 2;6 are non-zero. The object we want to

compute is

AYM6 =

Xsol z212z226z261 | [notdef]126126

Pf [prime] 12[notdef]12 z12z23z34z45z56z61

Xz3;z4;z52sol

PYM(z3; z4; z5) QYM(z3; z4; z5) : (4.77)

{ 23 {

Here both the Jacobian [notdef] [notdef]126126 and reduced Pfa an Pf [prime] 12[notdef]12 are very complicated, so

it is almost impossible to compute it analytically. As in the scalar 3 example, we can follow the semi-analytic procedure, and set the physical poles as some prime numbers, while keeping 12[notdef]12(all

e i;j = i j; i;j = ikj and kikj in ) analytic. In this case, the ideal and Grobner basis are just algebraic systems of polynomials with integer coe cients, while the elements of companion matrices are rational numbers. So the computation is very fast.

The Jacobian under the chosen gauge- xing is

| [notdef]126126

1 = z23z24z25(z3 1)2(z4 1)2(z5 1)2(z3 z4)2(z3 z5)2(z4 z5)2

D (z3; z4; z5) ; (4.78)

where D is polynomial in z3; z4; z5. The reduced Pfa an together with the factor of scalar part give

N (z3; z4; z5;

JHEP12(2015)056

e i;j; i;j; kikj)z3z4z25(z3 1)2(z4 1)(z5 1)(z3 z4)2(z3 z5)(z4 z5)2

(4.79)

under the chosen gauge- xing for some polynomial numerator N . So we have

PYM(z3; z4; z5) = z3z4(z4 1)(z5 1)(z3 z5)N ; QYM(z3; z4; z5) = D : (4.80) Note that N originates from the Pfa an of a 10 [notdef] 10 antisymmetric matrix, where by

de nition, each term in the Pfa an is a product of ve elements in the matrix. So each term in N is a product of ve elements selected from

e i;j, i;j, kikj, combined with a

e 2;6 1;2 3;1 4;1 5;1.

Finally we can take the replacement P [prime]YM = P YM[notdef]zi!Tzi , Q[prime]YM = QYM[notdef]zi!Tzi and compute the trace Tr(P [prime]YM(Q[prime]YM)1). It is given as

Tr(P [prime]YM(Q[prime]YM)1) = 446141149

e 2;6 1;2 3;2 4;1 5;1

1 877307

monomial of z3; z4; z5, for example, 2z33z44

e 2;5 1;3 3;2 4;1 6;2+500 more terms : (4.81)

Using the techniques shown in scalar 3 theory, we can uniquely decompose the rational numbers as

44 6141149 =

17 [notdef] 181 [notdef] 131

+ 1

181 [notdef] 37 [notdef] 131

; 1

877307 =

1181 [notdef] 37 [notdef] 131

; (4.82)

so we can conclude that

Tr(P [prime]YM(Q[prime]YM)1) =

[parenleftbigg]

1 s12s123s56 +

1 s123s23s56

[parenrightbigg]

e 2;6 1;2 3;2 4;1 5;1

1 s123s23s56

e 2;5 1;3 3;2 4;1 6;2 + 500 more terms : (4.83)

Rewriting them as spinor products and applying the simpli cation algorithm for spinor expression, we get a one-page long result, which remarkably agrees with the known answers [54, 55].

{ 24 {

4.4 Seven-point amplitudes

The companion matrices Tzi are simultaneously diagonalizable, and according to Stickelbergers theorem, the complex roots zi of ideal I are the vectors of simultaneous eigenvalues of the companion matrices Tzi. Thus when they are evaluated in the diagonal form, the matrices P [prime] = P [notdef]zi!Tzi , Q[prime] = Q[notdef]zi!Tzi , P [prime]Q[prime]1 are also diagonal, and it builds the one

to-one mapping between diagonal elements of (n 3)! [notdef] (n 3)! matrix P [prime]Q[prime]1 and the integrand P=Q evaluated at the (n 3)! complex solutions of scattering equations. To

demonstrate this, let us go through a 7-point example of scalar 3 theory.As usual, let us gauge xing z1 = 1; z2 = 1; z7 = 0, and the amplitude is given by

A7 =

Xsol z212z227z271 | [notdef]127127

1

z212z223z234z245z256z267z271 [summationdisplay]

z3;z4;z5;z62sol

P (z3; z4; z5; z6)

Q(z3; z4; z5; z6) ; (4.84)

JHEP12(2015)056

where

127127 =

0

B

B

B

@

33 34 35 36

43 44 45 46

53 54 55 56

63 64 65 66

1

C

C

C

A

; [notdef] [notdef]127127 = Det( 127127) : (4.85)

The Dolan-Goddard polynomial form [37] of the scattering equations is given by

f1 = s12 + s13z3 + s14z4 + s15z5 + s16z6 ; (4.86)

f2 = s123z3 + s124z4 + s125z5 + s126z6

+ s134z3z4 + s135z3z5 + s136z3z6 + s145z4z5 + s146z4z6 + s156z5z6 ; (4.87)

f3 = s1234z3z4 + s1235z3z5 + s1236z3z6 + s1245z4z5 + s1246z4z6 + s1256z5z6

+ s1345z3z4z5 + s1346z3z4z6 + s1356z3z5z6 + s1456z4z5z6 ; (4.88)

f4 = s67z3z4z5 + s57z3z4z6 + s47z3z5z6 + s37z4z5z6 + s27z3z4z5z6 : (4.89)

We can de ne the ideal I =

f1; f2; f3; f4[angbracketrightbig]in polynomial ring C[z3; z4; z5; z6], and generate the Grobner basis of I in Lexicographic order z3 z4 z5 z6. The degree of ideal I

is 24, so the variety of ideal I is given by 24 point solutions for which there are no closed form solutions.

Let us set the kinematics (all physical poles) as some prime numbers,

s12 = 5; s23 = 37; s34 = 43; s45 = 61; s56 = 97; s67 = 101; s17 = 139;

s123 = 151; s234 = 163; s345 = 191; s456 = 211; s567 = 223; s671 = 251; s712 = 263

(4.90)

in the following computation. The solutions of scattering equations fi = 0; i = 1; 2; 3; 4

requires computing the roots of equations of degree 24, which has no closed form in radicals. Doing it numerically, we get 24 solutions

{ 25 {

sol1 : z3 =20.9071, z4 =1.66835, z5 =7.08198, z6 =64.2332, sol2 : z3 =1.4223 0.318993

i, z4 =12.204 5.48743

i, z5 =0.342956 0.477119

sol3 : z3 =1.4223 + 0.318993

sol7 : z3 =4.92534 1.82303

sol11 : z3 =1.18325 1.93745

sol15 : z3 =1.78095 0.41639

sol16 : z3 =1.86192, z4 =0.877999, z5 =0.795994, z6 =0.601979,

sol17 : z3 =1.65547, z4 =1.9848, z5 =0.493798, z6 =2.31186,

sol18 : z3 =0.327576, z4 =0.0855936, z5 =0.212916, z6 =0.0559545,

sol19 : z3 =0.307828, z4 =0.645287, z5 =0.0420044, z6 =0.46483

sol20 : z3 =0.174313, z4 =0.120642, z5 =0.0855984, z6 =0.0445606,

sol21 : z3 =0.031819, z4 =0.15022, z5 =0.00455376, z6 =0.0545382,

sol22 : z3 =0.0191033, z4 =0.0765079, z5 =0.0145344, z6 =0.00921803,

sol23 : z3 =0.100486, z4 =0.0950558, z5 =0.00857861, z6 =0.0892275, sol24 : z3 =0.0162083, z4 =0.0167369, z5 =0.00970032, z6 = 0.0265855,

and the integrand summing over all solutions is given by

24

Xsoli; i=1

P (z3; z4; z5; z6)

Q(z3; z4; z5; z6) = 1:99605 [notdef] 106 : (4.91)

Let us now turn to the companion matrix method. The monomial basis over GB(I) is given by 24 elements

B =

n1; z6; z26; z36; z46; z56; z66; z76; z86; z96; z106; z116; z126 ;

z136; z146; z156; z166; z176; z186; z196; z206; z216; z226; z236

[bracerightBig]

: (4.92)

Accordingly, by polynomial reduction of ziB; i = 3; 4; 5; 6 over GB(I), we can get the companion matrices Tzi; i = 3; 4; 5; 6, which are 24 [notdef] 24 matrix and satisfying TziB = ziB.

In order to compute A7, we can proceed as usual by computing Tr(P [notdef]zi!Tzi Q[notdef]1zi!Tzi ), and

the result is

Tr(P [prime]Q[prime]1) = 19260317055974762778118

9649229470008137021319652355 1:99605 [notdef] 106 ; (4.93)

{ 26 {

i, z4 =12.204 + 5.48743i, z5 =0.342956 + 0.477119i, z6 =51.9097 + 32.886i,

sol4 : z3 =27.2316, z4 =1.76178, z5 =13.0497, z6 =12.5157, sol5 : z3 =1.34598, z4 =3.76733, z5 =1.28282, z6 =56.7763, sol6 : z3 =4.92534 + 1.82303

i, z4 =2.04236 + 0.47052i, z5 =0.12331 + 0.73366i, z6 =36.88 1.74857

i,

i,

i, z6 =51.9097 32.886

i,

sol8 : z3 =11.2804, z4 =3.5116, z5 =6.80042, z6 =1.26394, sol9 : z3 =1.19261, z4 =8.20104, z5 =3.07784, z6 =6.22689, sol10 : z3 =1.18325 + 1.93745

i, z4 =0.29585 0.48639

i, z5 =0.56997+1.11008i, z6 =0.154050.39359

i, z6 =0.15405+0.39359i,

i, z4 =2.04236 0.47052

i, z5 =0.12331 0.73366

i, z6 =36.88 + 1.74857

i, z4 =0.29585 + 0.48639i, z5 =0.569971.11008

sol12 : z3 =4.76521, z4 =3.05026, z5 =1.6908, z6 =0.488528,

sol13 : z3 =0.576445, z4 =3.05135, z5 =1.14498, z6 =0.712806, sol14 : z3 =1.78095 + 0.41639

i, z4 =2.1103 0.60663

i, z5 =0.52752 + 0.29927i, z6 =2.3283 1.39061

i, z6 =2.3283 + 1.39061i,

i,

i, z4 =2.1103 + 0.60663i, z5 =0.52752 0.29927

JHEP12(2015)056

i,

which agrees with the numeric result given by summing over all solutions of scattering equations. Again we see that, since the computation only involves basic manipulations on matrix, we are able to get the closed form result, and show that the nal result is rational functions of Mandelstam variables.

The companion matrices are simultaneously diagonalizable. We can choose Tz6 and compute its eigenvectors, since Tz6 is the simplest companion matrix by de nition. Such computation involves nding the roots of equations of degree 24, which prohibits analytic solution. Now, Tz6 has 24 column eigenvectors ui; i = 1; : : : ; 24, and from them we can de ne the transformation matrix U = (u1; : : : ; u24)24[notdef]24. Then T dzi = U1TziU; i = 3; 4; 5; 6

are all diagonal matrices, explicitly given as

diag(T dz3) = [notdef]20.9071 , 1.42230 + 0.31899

i , 1.42230 0.31899

i , 1.345982 , 4.92534 1.82303

i ,

1.98480 , 3.51160 , 3.05135 , 0.877999 , 3.05026 , 0.645287 , 0.295854 + 0.486386

i , 0.295854 0.486386

i , 0.0950558 , 0.0855936 , 0.150220 , 0.1206420 , 0.0167369 , 0.0765079[notdef] ,

diag(T dz5) = [notdef]7.08198 , 0.342956 + 0.477119

i , 0.342956 0.477119

i , 1.282815 , 0.123310 0.733658

i ,

i , 2.11030 0.60663

i ,

i ,

2.04236 + 0.47052

i , 1.76178 , 8.20104 , 2.11030 + 0.60663

JHEP12(2015)056

4.92534 + 1.82303

i , 27.2316 , 1.192609 , 1.78095 0.41639

i , 1.78095 + 0.41639i ,

1.65547 , 11.28035 , 0.576445 , 1.86192 , 4.76521 , 0.307828 , 1.18325 1.93745

i , 1.18325 + 1.93745i , 0.1004864 , 0.327576 , 0.0318190 , 0.174313 , 0.0162082 , 0.0191033[notdef] ,

diag(T dz4) = [notdef]1.66835 , 12.20402 + 5.48743

i , 12.20402 5.48743

i , 3.76733 , 2.04236 0.47052

0.569967 + 1.110081

i , 0.00857861 , 0.212916 , 0.00455376 , 0.0855984 , 0.00970033 , 0.0145344[notdef] ,

and

diag(T dz6) = [notdef]64.2332 , 51.9097 + 32.8860

i ,

i , 51.9097 32.8860

i , 56.7763 , 36.8800 + 1.7486

i , 2.32830 1.39061

i ,

i ,

0.123310 + 0.733658

i , 13.04970 , 3.07784 , 0.527517 0.299273

i , 0.527517 + 0.299273i ,

0.493798 , 6.80042 , 1.144984 , 0.795994 , 1.69080 , 0.0420044 , 0.569967 1.110081

0.154052 0.393588

i , 0.0892275 , 0.0559545 , 0.0545382 , 0.0445606 , 0.0265855 , 0.00921802[notdef] .

It can be checked directly that, each set of diagonal elements [notdef](Tz3)i;i; (Tz4)i;i; (Tz5)i;i; (Tz6)i;i[notdef] corresponds to a set of solution [notdef]zsolj3; zsolj4; zsolj5; zsolj6[notdef] of scattering equa

tions. Thus each diagonal element of matrix P [prime](T dzi)Q[prime]1(T dzi) is identical to the integrand P=Q evaluated at one solution of scattering equations, and the equivalence between results of these two methods is obvious.

With the arithmetic result, it is possible to determine the terms appearing in amplitude by setting appropriate kinematics. In fact, in this example, we know that the result should

{ 27 {

36.8800 1.7486

i , 12.51570 , 6.22689 , 2.32830 + 1.39061

2.31186 , 1.263937 , 0.712806 , 0.601979 , 0.488528 , 0.464830 , 0.154052 + 0.393588

i ,

be the sum

where naively the summation is over all possible products of 4 physical poles s i, i.e.,

4

[parenrightbig]

14 = 1001 terms with c ; = 1; : : : 1001; being either zero or one. By choosing 1001 di erent group of kinematics for physical poles, we get 1001 linear equations of (4.94), and solving them gives the c .

Indeed, the number of terms grows very fast with n in (4.94). The number of independent poles is n[prime] = (n1)(n2)

1 for massless theory, while the number of possible terms

in the expansion is n[prime]

n3

2 . For n = 8, the number is 15504, and for n = 9 the number is 296010. So it is not very doable when n is large. However, for 3 theory, the number of color-ordered diagram is much smaller, and the counting is 2n2(2n5)!!

(n1)! . So for n = 7, the

possible terms appearing in (4.94) is 42 (an auspicious number). For n = 8, the number is 132, and for n = 9, the number is 429, etc. If we restrict to the 42 possible terms in (4.94), then it is possible to compute the coe cients c by choosing one set of kinematics.

One can let each physical pole be assigned a random prime number, and compute Tr(P [prime]Q[prime]1) and then let Mathematica go through all 242 possibilities of c s to nd the summation

P42 =1c s 1s 2s 3s 4 = Tr(P [prime]Q[prime]1). If the prime numbers in kinematic variables are distributed randomly in a very large scale, e.g., primes between 2 to 10000, then usually we can nd one unique solution for c in the spirit of Egyptian fractions. This enables us to do one computation and x all coe cients.

For example, let us compute

A[prime]7 =

Xsol z212z227z271 | [notdef]127127

15 [notdef] 61 [notdef] 101 [notdef] 151

1

s12s67s123s567 ; (4.97)

and agrees with the result given by CHY mapping rules [30, 31].

There is a way to directly determine whether a certain term 1

s 1s 2s 3s 4 is present in

the result or not, by setting the kinematics s 1 = a, s 2 = a2, s 3 = a4, s 4 = a8, and others random primes not equaling to a. If this term exists, then the denominator has a factor a15. Again in the A[prime]7 example, if we instead set s12 = 5, s45 = 52, s67 = 54, s123 = 58, then

the result is 248

515[notdef]223 , thus

1s 1s 2s 3s 4 is a term in A[prime]7. However, if we set s12 = 5, s56 = 52, s67 = 54, s123 = 58, then the result is 284

513[notdef]61[notdef]223 . This indicates that

1

s12s56s67s123 is not

1s12s67s123 must

exist, which provides further information for detecting other existing terms. By this way, we can check all possible terms by setting kinematics for each one.

{ 28 {

a term in A[prime]7, while the 513 factor indicates that possible terms involving

A7 =

X ; ic s 1s 2s 3s 4 ; (4.94)

JHEP12(2015)056

1

z12z23z34z45z56z67z71z12z24z45z57z76z63z31 : (4.95)

With the kinematics shown in (4.90), we nd the unique decomposition

Tr(P [prime]Q[prime]1) = 284

1037296765 =

+ 1

5 [notdef] 101 [notdef] 151 [notdef] 233

; (4.96)

which indicates that

A[prime]7 = 1

s12s45s67s123 +

The number of solutions for scattering equations grows as (n3)!, while the companion

matrix grows as (n 3)! [notdef] (n 3)!. When n = 8, we need to invert the matrix Q[prime]120[notdef]120, and at n = 9, the matrix Q[prime]720[notdef]720, etc. This sets the limitation on the computation of

higher n.

5 Conclusions and outlook

In this paper, motivated by the explanation of equivalence of di erent integrands in the CHY setup, we propose a new method using companion matrices, borrowed from the study of zero-dimensional ideals in computational algebraic geometry, to evaluate the integrand. One advantage of the method is that the rationality of nal integral is obvious. Thus although our method may not be as e cient as the one proposed in [30, 31, 36], it does give a new angle to study the important problem of scattering amplitudes.

As shown in the plethora of examples, when the number of external legs grows, the analytic expression of companion matrix becomes harder. In fact, when n 6, the best

way to do it is by assigning the kinematic variables to random prime numbers in order to reconstruct the analytic result. The salient feature of our method is that it is purely linear-algebraic, involving nothing more than nding the inverse and trace of matrices. The linearity of the trace, for example, was demonstrated to immediately lead to non-trivial identities in the amplitudes.

Now, since the physical problem is very symmetric as can be seen by the polynomials given in (2.15) and (2.16), one is confronted with an immediate mathematical challenge. If we could analytically nd, say by induction, the Grobner basis and subsequent monomial basis for the polynomial form of the scattering equations in some appropriate lexicographic ordering, then one would nd a recursive way to construct the companion matrix explicitly, much like the recursive construction of tree-level amplitude by using BCFW deformation [16, 17]. Working out this construction is hard but worthwhile, as it would give explicit analytic results for the amplitudes and provide a deeper understanding of the CHY formalism.

Acknowledgments

We would like to thank Yang Zhang for discussion and sharing the similar idea of using computational algebraic geometry. We would also like to thank C. Baadsgaard, N. Bjerrum-Bohr, P. H. Damgaard for the discussion. RH acknowledges discussions with Qingjun Jin. BF is grateful to the Qiu-Shi Fund and the Chinese NSF under contracts No.11031005, No.11135006 and No.11125523; he would like to thank the hospitality of the Niels Bohr International Academy and City University, London. YHH would like to thank the Science and Technology Facilities Council, U.K., for grant ST/J00037X/1, the Chinese Ministry of Education, for a Chang-Jiang Chair Professorship at NanKai University as well as the City of Tian-Jin for a Qian-Ren Scholarship, as well as City University, London and Merton College, Oxford, for their enduring support. RH would also like to thank the supporting from Chinese Postdoctoral Administrative Committee.

{ 29 {

JHEP12(2015)056

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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{ 32 {

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