Analytic expressions of amplitudes by the crossratio identity method
Eur. Phys. J. C
Analytic expressions of amplitudes by the crossratio identity method
Kang Zhou 0
0 Zhejiang Institute of Modern Physics, Zhejiang University , Hangzhou 310027 , China
In order to obtain the analytic expression of an amplitude from a generic CHYintegrand, a new algorithm based on the socalled crossratio identities has been proposed recently. In this paper, we apply this new approach to a variety of theories including the nonlinear sigma model, special Galileon theory, pure YangMills theory, pure gravity, BornInfeld theory, DiracBornInfeld theory and its extension, YangMillsscalar theory, and EinsteinMaxwell and EinsteinYangMills theory. CHYintegrands of these theories which contain higherorder poles can be calculated conveniently by using the crossratio identity method, and all results above have been verified numerically. In the past a few years, an elegant new formulation of the treelevel Smatrix in arbitrary dimensions for a wide range of field theories has been presented by Cachazo, He and Yuan (CHY) [15]. This formulation describes the scattering amplitude for n massless particles as a multidimensional contour integral over the moduli space of punctured Riemann spheres M0,n. It can be unified into a concise expression, n An = vol Si=L1(2d,ziC) δ(Ei ) In(k, , z)

i∈{1,2,...,n}\{r,s,t}
i∈{1,2,...,n}\{a,b,c}
⎞
dzi ⎠
where zi is the puncture location in CP1 for the i th
particle, and zi j is defined as zi j ≡ zi − z j . The second line in
(1) is obtained by fixing the gauge redundancy of the Möbius
S L(2, C) group. The δfunctions impose the scattering
equations
Ei ≡
j∈{1,2,...,n}\{i} zi j
= 0,
where si j ≡ (ki +k j )2 = 2ki ·k j are the ordinary Mandelstam
variables (in general, we use si j···k ≡ (ki + k j + · · ·kk )2
conventionally). These equations fully localize the integration
to a sum over (n − 3)! solutions, and no actual integration is
required for calculating the npoint amplitude. The Möbius
invariant integrand In(k, , z) is a rational function of the
complex variables zi , external momenta ki and polarization
vectors i . In(k, , z) depends on the theory under
consideration and carries all the information as regards the wave
functions of external particles.
Although conceptually simple and elegant, when applied
to practical evaluations, the essential step of finding the full
set of analytic solutions becomes a major obstacle, due to
the Abel–Ruffini theorem that there is no algebraic solution
to the general polynomial equations of degree five or higher
with arbitrary coefficients. Moreover, after summing over all
those solutions, one often ends up with a remarkably
simple result. It is natural to wonder if there were better
techniques to produce the analytic expression obtained by
summing over all solutions. To overcome this difficulty, many
methods have been proposed from various directions [6–18].
Among these approaches, one of the most efficient ways is
the integration rules proposed by Baadsgaard, BjerrumBohr,
Bourjaily and Damgaard [17,18]. Inspired by the
computation of amplitudes in the field theory limit of string theory,
they derived a simple set of combinatorial rules which
immediately give the result after integration for any Möbius
invariant integrand involving simple poles only. One can get the
desired final result after integration by applying these rules
directly rather than solving scattering equations. However,
the requirement that the CHYintegrand contains only
simple poles could not be satisfied in general. Logically, there
are two alternative issues to bypass this disadvantage. One
is to search the integration rules for higherorder poles [19],
the other is to decompose an integrand of higherorder poles
into that of simple poles [17, 20].
Recently, an algorithm of solving this problem has been
proposed by Cardona, Feng, Gomez and Huang, based on the
socalled crossratio identities [21]. These identities reflect
the relations between rational functions in terms of zi j with
different structures of poles. By applying the crossratio
identities iteratively, a systematic algorithm for reducing
CHYintegrands with higherorder poles has been established.
After decomposing the integrand into terms with simple poles
only, one can compute the desired analytic result via the
integration rules. This is the first systematic way to get the
analytic expression of an amplitude from a generic
CHYintegrand.
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 CHYintegrand?
In [22], it has been proved that any weighttwo rational
function of zi j can be decomposed as a sum of PTfactors with
kinematic coefficients via the crossratio identities within
finite steps. Since any term from a known CHYintegrand
can be expressed as a product of two weighttwo rational
functions, one can conclude that all known CHYintegrands
can be decomposed into terms of only simple poles by
applying the crossratio identity method.
To verify its validity, it is worth applying 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–Millsscalar theory (YMS),
Einstein–Maxwell theory (EM), Einstein–Yang–Mills theory
(EYM). All known CHYintegrands involving higherorder
poles are contained in the cases above. In the meanwhile,
theories corresponding to CHYintegrands with simple poles
only, such as the scalar theory with φ3 or φ4 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 . The second class includes
DBI, EM and a special case of YMS of which integrands
depend on antisymmetric matrices [ ]a,b:a , [X ]b as well as
. 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 . Computation shows that all amplitudes considered
in this paper can be calculated efficiently within finite steps.
This paper is organized as follows: In Sect. 2 we give
a brief review of the crossratio identity method. Based on
this approach, calculations of amplitudes of theories in the
three classes above are given in Sects. 3, 4 and 5 respectively.
Finally, we give a brief summary in Sect. 6.
2 Brief review of the crossratio identity method
For the reader’s convenience, we will give a brief introduction
to the crossratio identity method in this section [21], then
we will discuss its validity for general CHYintegrands.
2.1 The systematic decomposition algorithm
Firstly, we need to define the order of poles of an integrand. A
generic npoint CHYintegrand consists of terms as rational
functions of zi j in the form
I =
with the integer power βi j under the constraint of the
Möbius invariance: j βi j + j β ji = 4 for arbitrary
i ∈ {1, 2, . . . , n}. For a subset = {i1, i2, . . . , i } ⊂
{1, 2, . . . , n}, the pole index χ is defined as
= ⎝
i , j ∈
− 2(  − 1),
where   denotes the length of the set . If χ ≥ 0, a pole
sχ1+1 will arise in the result. It is straightforward to verify
χ = χ ¯ , which reflects the momentum conservation
constraint s = s ¯ , where the subset ¯ = {1, 2, . . . , n}\ is
the complement of . Thus, it is necessary to choose
independent . If a CHYintegrand has m independent subsets
1, 2, . . . , m with χ i ≥ 0, the order of the poles of the
integrand is defined as
ϒ [I] =
i=1
Then an integrand which results 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
crossratio identities to the integrand iteratively. The crossratio
identity for the set is given by
1 = −
i∈ \{ j } b∈ ¯ \{ p}
≡ In [ , j, p],
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 (  − 1)(n −   − 1) terms via the operation
I = In[ , j, p]I. This operation will not break the manifest
Möbius invariance, since the crossratio identity is of weight
zero under Möbius transformations for any node i with
i ∈ {1, 2, . . . , n}. Obviously, such an operation decreases
χ by 1; thus the order of the pole sχ1+1 has been reduced if
χ > 0.
The systematic reduction algorithm is presented in the
following:
1. Count the order of poles ϒ [I] of the integrand I. If
ϒ [I] > 0, find the full set of independent subsets with
χ i > 0 (say, there are m subsets):
2. Step 1: start from the first set 1, collect all  1(n− 1)
crossratio identities of 1 with different choices of j and
b as
The entire algorithm can be implemented in
Mathematica. For a given CHYintegrand, if this algorithm can be
terminated within finite steps, one can obtain an expression with
terms that have simple poles only and finally get the analytic
result by applying the integration rules.
2.2 The feasibility of this method
Given the decomposition algorithm, it is natural to ask
whether all Möbius invariant integrands can be computed
in this way. It is obvious that the algorithm can be performed
for any integrand. The question is, can it be terminated within
finite steps? An ideal situation is, we can always find
crossratio identities such that all terms satisfy ϒ [I ] < ϒ [I] at
each step, then the decomposition procedure can be
terminated after ϒ [I] steps at most. However, this happens for
some particular integrands rather than for all. Thus, we need
to consider if it is possible that at every step there are terms
carrying the structure of poles such that ϒ [I ] = ϒ [I], or
even ϒ [I ] > ϒ [I], for all choices of 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 crossratio 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 the χ for lengtht subsets of
{1, 2, . . . , n} is fully determined by the condition
βi j = 4:
j∈{1,2,...,n}
χ i = −2(t − 1)Cn2 + 2n C n−t .
n−2
 i =t
Thus, the sum of all χ , χtotal ≡ χ = t χt , is
invariant under any action which maintains the Möbius invariance.
If χtotal 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 χt > 0 if and only if n < t + 1, thus χtotal can
never be positive.
On the other hand, it has been proved that a weighttwo
rational function of zi j can always be transformed to the sum
of PTfactors zi1i2 zi21i3 ···zini1 with kinematic coefficients via
the crossratio identities within finite steps [22]. Any term
of a known CHYintegrand in the literature can be expressed
as a product of two weighttwo rational functions. Hence,
although it is not clear whether the CHYintegrand for any
physical theory can be decomposed as products of
weighttwo functions, one can use the crossratio identities to
decompose any known CHYintegrand into terms which contain
simple poles only.
In this paper, we will choose the original algorithm rather
than the one which decomposes a weighttwo function into
PTfactors, since the former is more convenient to be
impleIn[ 1, j, p] where j ∈
p ∈ {1, 2, . . . , n}\ 1.
I = In[ 1, j, p]I =
3. Step 2: decompose the CHYintegrand I by applying the
first crossratio identity
where the c are rational functions of the Mandelstam
variables.
4. Count all ϒ [I ].
• If all ϒ [I ] < ϒ [I], return c I .
• If there exists any ϒ [I ] ≥ ϒ [I], test the second
crossratio identity in step 2 and so on, until we find
a crossratio identity satisfying all ϒ [I ] < ϒ [I].
• If after running over all crossratio identities of the
set 1, there is still no such an identity satisfying
all ϒ [I ] < ϒ [I], then take the first identity in step
2 again but now we stop till an identity satisfies all
ϒ [I ] ≤ ϒ [I], and return c I .
5. Perform the procedure above for each I , 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 2, etc.
mented in Mathematica. Indeed, the feasibility of this
algorithm has not been proved, since the procedure of
decomposing a weighttwo function into PTfactors cannot ensure
ϒ [I ] ≤ ϒ [I] at each step. However, as can be seen in the
subsequent sections, all known CHYintegrands can be
computed by the original algorithm efficiently, i.e., the condition
ϒ [I ] ≤ ϒ [I] can always be satisfied, at least for all known
CHYintegrands.
3 Amplitudes of theories in the first class
For theories in this class, the most important object in the
construction of the integrand In is the 2n × 2n antisymmetric
matrix
where A, B and C are the n × n matrices given by
i = j,
i = j,
i = j,
i = j,
i = j,
i = j.
One also needs to introduce the reduced Pfaffian Pf =
(−)i+ j Pf iijj where iijj denotes the minor obtained by
deletzi j
ing rows and columns labeled by i and j , with i, j ∈
{1, 2, . . . , n}. On the support of scattering equations, the
reduced Pfaffian Pf is invariant with respect to the
permutation of particle labels. In addition, a useful factor is defined
as
C{i1,i2,...,is } =
σ ∈Ss /Zs zσ (i1)σ (i2), zσ (i2)σ (i3) · · · zσ (is )σ (i1)
where T I 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
Cii =
l=i
l=i,a
where momentum conservation and the gauge invariant
condition i · ki = 0 have been used. The new formula of Cii
gives the weight two for node i and weight zero for other
nodes, then the termwise Möbius invariance is guaranteed.
Throughout this paper, we choose
i = 1,
With these ingredients, we can now investigate theories in
the first class one by one.
3.1 Nonlinear sigma model We begin with the simplest case, the NLSM, whose standard Lagrangian in Cayley parametrization is L
NLSM = 81λ2 Tr(∂μU†∂μU),
U = (I + λ ) (I − λ )−1,
Here I is the identity matrix and the T I are generators of
U (N ). The CHYintegrand of NLSM is given by [5]
NLSM = λn−2 Cn (Pf A(k, z))2.
For this case, it is sufficient to calculate the colorordered
partial amplitude ANLSM(1, 2, . . . , n) defined via
In other words, we focus on the colorordered partial
integrand
(Pf A(k, z))2
INLSM(1, 2, . . . , n) = z12z23 · · · zn1 .
Here the coupling constant has been omitted.
We start from the 6point amplitude A6NLSM. By
definition, the corresponding colorordered partial integrand is
(k1 · k2)2(k3 · k4)2
INLSM(1, 2, . . . , 6) = z132z23z334z45z536z61
2(k1 · k2)(k2 · k4)(k2 · k3)(k3 · k4)
z122z14z223z324z45z536z61
2(k1 · k2)(k1 · k3)(k2 · k4)(k3 · k4)
z122z13z23z24z324z45z536z61
2(k1 · k3)(k1 · k4)(k2 · k3)(k2 · k4) .
z12z13z14z223z24z34z45z536z61
The pole structure of (22) is listed as follows:
It can be seen from the table that every term contains
higherorder poles which need to be decomposed. Via the
crossratio identity method, One can accomplish the
decomposition within three steps. Below is the table with #[ALL], the
number of resulting terms and #[H], the number of terms of
higherorder poles in each Round of decomposition:
Integrations of these terms can be bypassed by applying the
integration rules. Summing all terms from the final result, we
obtain
(s23 + s34)(s56 + s61)
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.
Then we turn to the 8point amplitude A8NLSM. The
integrand has 120 terms and all terms contain higherorder poles.
Performing the crossratio identity method, this integrand
can be decomposed into 4340 terms with simple poles only
within 6 steps. The table of #[ALL] and #[H] in each Round
of decomposition is given as follows:
Round 1 Round 2 Round 3 Round 4 Round 5 Round 6
From these terms with simple poles, we get the desired
analytic expression of the amplitude via the integration rules.
The final result can be simplified into the form
ANLSM(1, 2, . . . , 8) = Part1 − Part2 + Part3,
Part1 =
(s12 + s23)(s45 + s56)(s78 + s8123)
s123s456
(s23 + s34)(s56 + s67)(s81 + s1234)
s234s567
(s34 + s45)(s67 + s78)(s12 + s2345)
s345s678
(s45 + s56)(s78 + s81)(s23 + s3456)
s456s781
(s56 + s67)(s81 + s12)(s34 + s4567)
s567s812
(s67 + s78)(s12 + s23)(s45 + s5678)
s678s123
(s78 + s81)(s23 + s34)(s56 + s6789)
s781s234
(s81 + s12)(s34 + s45)(s67 + s7812)
s812s345
(s12 + s23)(s56 + s67)(s1234 + s4567)
s123s567
(s23 + s34)(s67 + s78)(s2345 + s5678)
s234s678
(s34 + s45)(s78 + s81)(s3456 + s6781)
s345s781
(s45 + s56)(s81 + s12)(s4567 + s7812) ,
Part2 =
(s12 + s23)(s45 + s56 + s67 + s78 + s8123 + s1234)
s123
(s23 + s34)(s56 + s67 + s78 + s81 + s1234 + s2345)
+ s234
(s34 + s45)(s67 + s78 + s81 + s12 + s2345 + s3456)
+ s345
(s45 + s56)(s78 + s81 + s12 + s23 + s3456 + s4567)
+ s456
(s56 + s67)(s81 + s12 + s23 + s34 + s4567 + s5678)
+ s567
(s67 + s78)(s12 + s23 + s34 + s45 + s5678 + s6781)
+ s678
(s78 + s81)(s23 + s34 + s45 + s56 + s6781 + s7812)
+ s781
+ s1234 + s2345 + s3456 + s4567.
Part3 = 2(s12 + s23 + s34 + s45 + s56 + s67 + s78 + s81)
(27)
This result has been verified numerically.
3.2 Special Galileon theory The next theory under consideration is SG. The general pure Galileon Lagrangian is L
Lm = φ det {∂μi ∂ν j φ}im, −j=11.
m=3
We restrict ourselves on the special situation in which there
exist constraints on the coupling constants gm such that all
amplitudes with an odd number of external particles vanish.
Then the CHYintegrand In of this theory is given by [5]
SG = (Pf A(k, z))4,
where the coupling constants have been omitted.
With this setup, we choose the 6point amplitude A6SG
as an example. The integrand has 15 terms, and all terms
contain higherorder poles. One can divide it into 3169 terms
with simple poles only within 10 steps. The table of #[ALL]
and #[H] in each Round of decomposition is given by
YM = Cn Pf
Although the final result is too lengthy to be presented, it has
been confirmed numerically.
3.3 Yang–Mills theory Then we turn to the pure YM case. The CHYintegrand of YM is [5]
Similar to the case of NLSM, it is sufficient to consider the
colorordered partial integrand
Pf (k, , z)
IYM(1, 2, . . . , n) = z12z23 · · · zn1 .
Let us take the 6point colorordered amplitude AYM
(1, 2, . . . , 6) as an example. The partial integrand IYM
(1, 2, . . . , 6) has 3420 terms and 1120 of them contain
higherorder poles. The decomposition procedure can be
terminated within 5 steps via the crossratio identity method,
and the analytic expression of AYM(1, 2, . . . , 6) is verified
numerically. The table of #[ALL] and #[H] in each Round
of decomposition is given by
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 crossratio identities. However, those of
polarization vectors can be chosen arbitrarily since they are
irrelevant to the crossratio identities and the integration. We
have verified the result with polarization vectors i · ki = 0
as well as i · ki = 0, and we find that the analytic expression
reproduces the value obtained from solving the scattering
equations numerically.
3.4 Born–Infeld theory Now we consider BI whose Lagrangian is given by L
−2
2 Fμν ) − 1 .
The CHYintegrand of BI is [5]
= n−2 Pf
For simplicity, we calculate the 6point amplitude A6BI.
The integrand contains 20400 terms and 18744 of them
involve higherorder poles. Using the crossratio identities,
one can reduce it to terms with simple poles within 10 steps.
The table of #[ALL] and #[H] in each Round of
decomposition is given by
Round 1 Round 2
#[ALL] 61200
#[H] 28616
Round 4 Round 5
This is the most complicated example in this paper; it takes
more than a day in Mathematica. The analytic expression
of the amplitude is confirmed by numerical verification.
3.5 Gravity The final theory under consideration in this section is GR. The CHYintegrand of this theory is the product of two inde
([ ]a,b:a )i, j =
C(i−n) j
(−C T )i( j−n) i ∈ {1, 2, . . . , n}, j ∈ {n + 1, n + 2, . . . , n + na },
B(i−n)( j−n) i, j ∈ {n + 1, n + 2, . . . , n + na }.
i, j ∈ {1, 2, . . . , n},
i ∈ {n + 1, n + 2, . . . , n + na }, j ∈ {1, 2, . . . , n},
respectively, and n = na +nb is the total number of particles.
[X ]b is an nb × nb matrix defined as
i = j,
i = j,
where I i ∈ {1, . . . , M } denotes the i th U (1) charge of the
U (1)M group. [ ]a,b:a is an (n + na ) × (n + na ) matrix
obtained from by deleting rows and columns labeled by
n + i for all i ∈ [na + 1, n]. More explicitly, its elements are
given by
pendent copies of the one for YM, each of which has its own
gauge choice for polarization vectors [5]
GR = Pf
(k, , z) Pf
The polarization tensor of a graviton is given by ζμν = μ ˜ν .
This integrand leads to amplitudes of gravitons coupled to
dilatons and Bfields.
We take the 4point amplitude A4GR as an example. The
integrand contains 484 terms, with 228 terms involving
higherorder poles. It can be decomposed into terms with
simple poles within 2 steps, as shown in the following
table:
Physically, polarization tensors of gravitons are traceless, i.e.,
they satisfy μ ˜μ = 0. However, as discussed before, their
values can be chosen without imposing any physical
constraint when performing the numerical verification.
4 Amplitudes of theories in the second class
In this section we move on to theories in the second class.
CHYintegrands of these theories require two new matrices
[X ]b and [ ]a,b:a as basic ingredients. a and b are two sets of
external particles, whose numbers are denoted by na and nb,
Among n external particles, only particles in subset a
contribute their polarization vectors to the matrix [ ]a,b:a . This
is the reason why the integrand including Pf [ ]a,b:a can
describe interactions between bosons with different spins.
It is worth emphasizing that all terms in Pf[X ]b are
manifestly invariant under the Möbius transformations. From the
formula of diagonal terms of the matrix C in (16), terms in
the expansion of Pf [ ]a,b:a also have the manifest Möbius
invariance, which ensures the feasibility of the integration
rules.
4.1 Special Yang–Millsscalar theory
The first theory under consideration in this section is the
special YMS case which describes the low energy effective
action of N coincident Dbranes. The Lagrangian of this
theory is
YMS = −Tr ⎛⎝ 41 Fμν Fμν + 21 Dμφ I Dμφ I − g42
I =J
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 Dbrane. The corresponding CHYintegrand
is [5]
IYMS(g, s) = Cn Pf[X ]s (z) Pf [ ]g,s:g(k, ˜, z), (39)
where g and s denote the sets of gluons and scalars,
respectively. Gluons have polarization vectors μ, while scalars do
not; thus their kinematical information can be combined into
the matrix [ ]g,s:g.YMASg(a1in,,2,w.e. .c,onn)s.ider the colorordered
partial amplitude A
The first example is the 6point partial amplitude AYMS
(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
higherorder poles. The decomposition procedure is shown
in the following table:
The second example is the 6point amplitude AYMS
(1g, 2g, 3sI1 , 4sI1 , 5sI2 , 6sI2 ), where 1g and 2g are gluons,
3sI1 , 4sI1 are scalars of one flavor and 5sI2 , 6sI2 are scalars
of another. The partial integrand contains 15 terms and 7 of
them contain higherorder poles. The decomposition can be
done within 3 steps as shown in the following table:
Analytic expressions of these two examples are verified
numerically.
4.2 Dirac–Born–Infeld theory We proceed to consider DBI whose Lagrangian is L
−2
(Pf A(k, z))2,
where I again labels the flavor of scalars. The CHYintegrand
of DBI is [5]
n−2 Pf[X ]s (z) Pf [ ]γ,s:γ (k, ˜, z)
where γ denotes the set of photons and s the set of scalars,
respectively.
DBI
Let us calculate the 6point amplitude A2γ 2sI1 2sI2 , which
contains two photons and four scalars carrying two flavor
indices. The integrand has 82 terms and all of them contain
higherorder poles. One can accomplish the decomposition
procedure via the crossratio identities within 10 steps, as
shown in the following table:
Again, this result is verified numerically.
4.3 Einstein–Maxwell theory The final theory in this section is EM which describes gravitons coupled to photons. The CHYintegrand of this theory is given by [5]
EM = Pf[X ]γ (z) Pf [ ]h,γ :h (k, , z) Pf
Here the set of gravitons is denoted by h, and that of photons
is denoted by γ . Equation (42) allows the photons to carry
more than one flavor in general. The polarization tensor of
a graviton is ζ μν = μ ˜ν , and the polarization vector of a
photon is ˜ν . The matrix (k, ˜, z) contains ˜ν for both
gravitons and photons, and the matrix [ ]h,γ :h (k, , z) contains
the remaining μ for gravitons.
EM
Our example is the 5point amplitude A3h2γ whose
external particles are three gravitons and two photons carrying the
same flavor index. The integrand has 5013 terms and 1171
of them contain higherorder poles. The decomposition
procedure can be done within 4 steps, as shown in the following
table:
Again, we have verified this result numerically.
5 Amplitudes of theories in the third class
Theories in the final class correspond to multitrace mixed
amplitudes. More precisely, an amplitude in this section
contains external bosons which belong to the set a ∪ bTr1 ∪
bTr2 · · · ∪ bTrm with bosons in set a of spin Sa and that in
the set bTri of spin Sb = Sa − 1. This structure leads to
the mixed color factor Tr1 ∪ Tr2 · · · ∪ Trm in the amplitude.
The kinematic part of the integrand for these theories can be
constructed in two equivalent ways; one is to introduce the
polynomial {i, j} P{i, j}, the other is to define the matrix .
Let us assume the amplitude contains m mixed traces, then
{i, j} P{i, j} is a sum over the perfect matching {i, j },
P{i, j} =
im−1< jm−1∈Trm−1
i1< j1∈Tr1
×Pf[ ]a,i1, j1,...,im−1, jm−1:h ,
sgn({i, j }) zi1 j1 · · · zim−1 jm−1
where ia and ja are labels of two external particles which
belong to bTra . This sum can be recognized as the reduced
Pfaffian of the matrix . The matrix can be constructed
from by performing the socalled 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 break the manifest Möbius invariance thus
are forbidden for the integration rules. Hence, we will use
{i, j} P{i, j} to express the integrands throughout this
section.
5.1 General Yang–Millsscalar theory Let us consider the general YMS with the Lagrangian L
gen.YMS = −Tr 41 F μν Fμν + 21 Dμφ I Dμφ I
I =J
which involves the general flavor group and a cubic scalar
selfinteraction. The trace is for the gauge group, and f I¯J¯K¯
and f I J K are the structure constants of gauge and flavor
groups respectively. The amplitudes of this theory can only
contain a single trace of the gauge group, and multitraces
for the flavor group, as can be seen from the general
CHYintegrand [5]
P{i, j}(sTr1 ∪ · · · ∪ sTrm , g),
where g denotes the set of gluons and sTri denotes the set of
scalars with the trace Tri .
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 with
Tr(T Ii1 T Ii2 ) replaced by δ Ii1 ,Ii2 . In other words, the
kinematic part of these amplitudes is included in the situations
we have calculated in the previous section. Thus, we choose
to compute a nontrivial case, the 7point colorordered
partial amplitude Agen.YMS(1g, (2s, 3s, 4s)Tr1 , (5s, 6s, 7s)Tr2 ),
which contains one gluon and six scalars with three scalars
carrying Tr1 and the rest three carrying Tr2. The integrand
has 21 terms and all of them contain higherorder poles.
The decomposition procedure can be done within 2 steps,
as shown in the following table:
Again, this analytic result is confirmed by the numerical
verification.
5.2 Extended Dirac–Born–Infeld theory
The second theory under consideration is the extended DBI,
which is described by the Lagrangian
2 ⎞
−det ημν − 4λ2 Tr(∂μU† ∂ν U) − 2 Wμν − Fμν − 1⎠ ,
where the matrix U( ) is defined in (18), and
∞ m−1 2(m − k) λ2m+1
m=1 k=0
2(m−k)−1).
The corresponding CHYintegrand is given by [5]
Iext.DBI(sTr1 ∪ · · · ∪ sTrm , γ ) = CTr1 · · · CTrm
P{i, j}(sTr1 ∪ · · · ∪ sTrm , γ ) (Pf A)2,
where γ denotes the set of photons and sTri denotes the set
of scalars with the trace Tri .
Our example is the 6point partial amplitude Aext.DBI
(1γ , (2s, 3s)Tr1 (4s, 5s, 6s)Tr2 ), which involves one photon
and five scalars, where two scalars carry Tr1 and three carry
Tr2. The integrand has 36 terms and 8 of them contain
higherorder poles. The decomposition procedure can be done within
3 steps as shown in the table
This result has been verified numerically.
5.3 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 CHYintegrand involving the
mixed traces is [5]
P{i, j}(gTr1 ∪ · · · ∪ gTrm , h) Pf ,
where the set of gravitons is denoted by h and the set of
gluons with the trace Tri is denoted by gTri .
We consider the 5point partial amplitude AEYM
((1g, 2g)Tr1 (3g, 4g, 5g)Tr2 ), of which all five external
particles are gluons with two of them carrying Tr1 and three
of them carrying Tr2. The original integrand has 239 terms
and 189 of them contain higherorder poles. The
decomposition procedure can be done within 3 steps, as shown in the
following table:
As all other examples, this analytic result is confirmed
numerically.
6 Conclusion
In this paper, we have applied the crossratio identity method
to CHYintegrands of various theories including: the
nonlinear sigma model, special Galileon theory, pure Yang–
Mills theory, pure gravity, Born–Infeld theory, Dirac–Born–
Infeld theory and its extension, Yang–Millsscalar theory,
and Einstein–Maxwell and Einstein–Yang–Mills theory. All
the integrands under consideration are computed
conveniently in this way, the decomposition procedures expend
10 steps at most. All the analytic results are verified
numerically, thus this method is confirmed for all examples of this
paper. Consequently, the crossratio identity method is valid
and effective for a wide range of CHYintegrands. An
interesting observation is that the condition ϒ [I ] ≤ ϒ [I] can
always be satisfied at each step, although its rigorous proof
is still absent.
In this paper, the most complicated example takes more
than a day in Mathematica. The reason of this low
efficiency is, the choices of i , j and p are tested by brute
force in the algorithm. Appropriate choices of the crossratio
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.
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
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Commons license, and indicate if changes were made.
Funded by SCOAP3.
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