CHY representations for gauge theory and gravity amplitudes with up to three massive particles

We show that a wide class of tree-level scattering amplitudes involving scalars, gauge bosons, and gravitons, up to three of which may be massive, can be expressed in terms of a Cachazo-He-Yuan representation as a sum over solutions of the scattering equations. These amplitudes, when expressed in terms of the appropriate kinematic invariants, are independent of the masses and therefore identical to the corresponding massless amplitudes.


Introduction
The new representations for tree-level gauge and gravitational amplitudes in arbitrary spacetime dimensions discovered by Cachazo, He, and Yuan [1,2], while as yet lacking a quantumfield-theoretic foundation, have illuminated and unified many previous results. The tree-level n-gluon amplitude for pure Yang-Mills theory is given by where {σ} is a set of n points on CP 1 , C({σ}) depends on group theory structure constants, and E({ǫ, k, σ}) depends on the momenta and polarizations of the gluons. The sum is over the (n − 3)! solutions of the scattering equations [3] b =a k a · k b σ a − σ b = 0, σ a ∈ CP 1 , a = 1, · · · , n . (1. 2) The structure of eq. (1.1) as a sum over products of color-and kinematic-dependent factors is closely related to BCJ color-kinematic duality [4]. For n = 4, the scattering equations have a single solution, so that eq. (1.1) implies that the four-gluon amplitude factors into a color-dependent term and a polarization-dependent term. The factorization of a wide class of four-point gauge theory amplitudes was first recognized over thirty years ago [5,6] following the discovery that radiative gauge theory amplitudes vanish in certain kinematic regions [7][8][9][10][11][12].
The tree-level gravity amplitude is and its structure clearly echoes the double-copy procedure of refs. [4,13,14]. The Kawai-Lewellen-Tye (KLT) relations of tree-level string theory [15] in the field theory limit [16,17] also follow from eq. (1.3). In particular, eq. (1.3) implies that the four-graviton amplitude factors into a pair of gauge-theory partial amplitudes. This has long been known from the field-theory limit of the four-point closed string amplitude [18] as well as from an explicit field theory calculation [19]. Many other four-point amplitudes involving gravitons have also been shown to factor into a pair of gauge theory amplitudes in an arbitrary number of spacetime dimensions [20][21][22][23].
In refs. [24,25], Cachazo, He, and Yuan extended their results to amplitudes for massless particles in a variety of theories using dimensional reduction and other techniques. Other work inspired by these developments includes refs. .
Most of the attention so far has been focused on amplitudes for massless particles. The factorization of four-point gauge theory amplitudes found in ref. [5,6], however, required that only one of the gauge bosons involved be massless, and the four-point gravitational amplitudes considered in refs. [20][21][22][23] involved massive particles as well. This suggests that CHY representations may be possible for various amplitudes containing massive particles. Dolan and Goddard proposed a modification of the scattering equations for massive particles with equal mass and derived a CHY representation for φ 3 theory [32,39]. In ref. [50], the scattering equations were generalized to arbitrary masses, and CHY representations were proposed for gauge theory amplitudes involving a pair of massive scalar particles in the fundamental representation and an arbitrary number of gluons, as well as for gravitational amplitudes involving a pair of massive scalar particles and an arbitrary number of gravitons.
In this paper, we derive CHY representations for a large class of tree-level gauge theory and gravitational amplitudes, with up to three massive particles. We require only that the remaining (massless) particles be flavor-preserving (as is generally the case). Furthermore, we show that these amplitudes, when written in terms of the appropriate kinematic invariants, are independent of the masses of the particles, and thus identical to purely massless amplitudes. Our approach is first to show that all the diagrams contributing to an amplitude with up to three massive particles (with the remaining particles being flavor-preserving) can be kinematically regarded as the dimensional reduction of diagrams of purely massless amplitudes, with non-zero momenta in the extra dimensions. Then, by different choices of the polarization vectors, we can then use eqs. (1.1) and (1.3) in higher dimensions to derive CHY representations for mixed amplitudes in lower dimensions, as was done in ref. [25]. In this way, we derive a variety of gauge theory amplitudes involving massive gauge bosons and scalar particles, and gravitational amplitudes involving gravitons, massive gauge particles, and massive scalars. We recover the results of ref. [50] as a special case.
This paper is structured as follows. In sec. 2, we show that the propagator matrix for an amplitude with up to three massive particles has the same rank as that for a purely massless amplitude, motivating the search for a CHY-type representation for such amplitudes. In sec. 3, we derive the scattering equations for amplitudes with up to three massive particles from massless scattering equations in higher dimensions. In sec. 4, we derive a CHY representation for the massive propagator matrix. In sec. 5, we dimensionally reduce massless gauge boson amplitudes to obtain CHY representations for amplitudes with massive and massless gauge bosons, and mixed amplitudes containing gauge bosons and adjoint or fundamental scalars. In sec. 6, we dimensionally reduce pure graviton amplitudes to obtain CHY representations for mixed amplitudes containing gravitons, massive or massless gauge bosons, and massive scalars. Sec. 7 contains our conclusions.

Propagator matrix
In this section, we briefly review the propagator matrix that appears in the amplitude for massless gauge bosons [54]. We then show that the propagator matrix for an n-point amplitude with up to three massive particles (assuming that the remaining n − 3 particles are flavor-preserving) is identical to that for an amplitude with all massless particles. Like the latter, therefore, it has rank (n − 3)!. This in turn suggests the possibility of finding a CHY representation in terms of a sum over the (n − 3)! solutions of the scattering equations for these amplitudes.
We begin with the tree-level n-gluon amplitude written as a sum over cubic diagrams [4] where c i , n i , and d i are the color factors, kinematic numerators, and denominators (arising from the propagators) associated with the diagram. The color factors c i of each of the cubic diagrams appearing in eq. (2.1) can be rewritten, using Jacobi identities, in terms of a basis of (n − 2)! color factors [55,56] where γ denotes a permutation of {2, · · · , n − 1} and The color factors c 1γn are associated with half-ladder diagrams, i.e., diagrams in which there is a line connecting particles 1 and n to which each of the other particles is directly connected.
The assumption of color-kinematic duality [4] is that the kinematic numerators n i obey the same Jacobi identities as c i , and therefore can be written in terms of (n − 2)! half-ladder numerators where we define the entries of the (n − 2)! × (n − 2)! propagator matrix [54] as the sum (weighted by the denominators 1/d i of the diagrams) over those cubic diagrams that contribute to both c 1γn and c 1δn .
More generally, we can define the propagator matrix for a gauge theory amplitude with particles other than gluons by using group theory relations to rewrite the color factors in with no explicit dependence on the masses m 1 , m 2 , and m n . (Of course, momentum conservation can be used to eliminate k n · k a , leaving n(n − 3)/2 independent invariants.) Because the inverse propagators d i , when written in terms of the invariants (2.9), do not depend on the masses, neither does the propagator matrix (2.6). And since the (n−2)!×(n−2)! propagator matrix for amplitudes with massless particles has rank (n − 3)! as a consequence of momentum conservation [54], the same is true for the propagator matrix for amplitudes with up to three massive particles. This suggests the possibility of expressing it as a sum over the (n − 3)! solutions of the scattering equations, which we will do in the next two sections.
On the other hand, if there are four or more massive particles in the amplitude, the rank of the propagator matrix will generally be larger than (n − 3)!. For example, the amplitude for four massive particles, one of which is flavor-preserving, has a 2 × 2 propagator matrix with non-vanishing determinant, and thus rank two.

Scattering equations
In order to obtain CHY representations for amplitudes with massive particles, we first need the appropriate scattering equations. In this section we show that the scattering equations for an n-point amplitude with up to three massive particles are identical to, and therefore have the same set of solutions as, the scattering equations for massless particles. We also show that the massive scattering equations for amplitudes with up to three massive particles can be derived from the massless scattering equations in higher dimensions.
The scattering equations for massless particles are given in eq. (1.2). In refs. [32,39], Dolan and Goddard proposed a modification of these equations for massive particles with equal mass. The scattering equations were subsequently generalized [50] for particles with distinct masses m a : provided n a=1 k a = 0 and that ∆ ab satisfy the constraints Furthermore, when the constraints (3.3) are satisfied, the three sums vanish identically (i.e., before imposing f a = 0), which implies that only n − 3 of the massive scattering equations (3.1) are independent.
Now consider an amplitude with three massive particles, m 1 , m 2 , and m n . The constraints (3.3) can be satisfied by choosing with the remaining ∆ ab vanishing. Since only n − 3 of the n scattering equations (3.1) are independent, we can eliminate those for a = 1, 2, and n. Since ∆ ab vanishes in the remaining equations (for a = 3, · · · , n − 1), they are identical to the scattering equations for massless particles, and hence have the same set of (n − 3)! solutions, when expressed in terms of the kinematic invariants (2.9). This will be crucial in the remainder of this paper.
Finally, we show that the massive scattering equations (3.1) are precisely the equations that arise if one regards the massive particles as massless particles in a higher-dimensional space.
In particular, the momentum K a for a massless particle in (d+M) dimensions can be written where k a and κ a are the components of momentum in d-and M-dimensional subspaces respectively. 2 We will refer to κ a as the internal momentum of a particle; the mass of the particle in d dimensions is given by m 2 imply that the scattering equations (3.1) hold in d dimensions with and the constraints (3.3) hold automatically by virtue of momentum conservation in the internal dimensions.
If M = 1, then momentum conservation in the internal space requires n a=1 (±m a ) = 0 (for some choice of signs), which is too restrictive for our purposes. If M ≥ 2, however, there is no constraint on the masses of the particles.
It must be stressed, however, that by no means can every massive amplitude be obtained from the dimensional reduction of a massless amplitude because momentum conservation (including conservation of internal momentum) must hold at each vertex of a diagram, placing severe constraints on the masses of the intermediate particles in any given diagram. A simple example will suffice to explain the difficulty. Consider the four-point amplitude involvingū, d, W + , and Z 0 . Since the Z 0 is flavor-and therefore mass-preserving, conservation of momentum at any vertex involving the Z 0 implies that (κ i + κ Z ) 2 = κ 2 i , where κ i is the internal momentum forū, d, or W + . Together with overall internal momentum conservation (κ Z = − i κ i ), this implies that κ 2 Z = 0, which implies that the Z 0 must be massless. For the amplitudes considered in this paper, however, with (up to) three massive particles, and the remaining (massless) particles flavor-preserving, internal momentum conservation at each vertex involving a massless (κ = 0) flavor-conserving particle holds automatically, and internal momentum conservation at the vertex involving the three massive particles, κ 1 + κ 2 + κ n = 0, is easily satisfied for appropriate choices of κ i .

Double-partial amplitudes
In this section, we show that the propagator matrix (2.6) for an amplitude with up to three massive particles can be written as a sum over the solutions of the scattering equations (3.1).
First we recall from sec. 3 that kinematically we can regard the scattering of n particles (up to three of them massive and the rest flavor-preserving) in d dimensions as the scattering of n massless particles in d + M dimensions, with the same propagator matrix.
Next, recall that Cachazo, He, and Yuan [2] demonstrated that the entries of the propagator matrix for massless amplitudes can be interpreted as double-partial amplitudes of a theory of massless scalar particles transforming in the adjoint of the color group U(N) × U(Ñ ) and presented a compact formula for computing them. First they defined an n × n matrix Φ({K, σ}) with entries where K a are the (d + M)-dimensional momenta of massless particles. When the scattering equations (3.7) are satisfied, this matrix satisfies n a=1 Φ ab = n a=1 σ a Φ ab = n a=1 σ 2 a Φ ab = 0, and therefore has rank (n − 3). Define the nonsingular matrix Φ ijk pqr by removing rows i, j, and k, and columns p, q, and r, and let |Φ| ijk pqr be its signed determinant. The combination where σ a,b ≡ σ a − σ b , was then shown to be independent of the choices of removed rows and columns. CHY then demonstrated that the entries of the propagator matrix can be computed using 1 (σ 1,γ(2) · · · σ γ(n−1),n σ n,1 )(σ 1,δ(2) · · · σ δ(n−1),n σ n,1 ) Now we need to rewrite eq. (4.3) in terms of d-dimensional quantities. Since det ′ Φ is independent of the choices of removed rows and columns, we can choose to remove the rows and columns associated with the massive particles (1, 2, and n). Since the remaining rows and columns correspond to massless particles (with κ a = 0), the numerators in Φ 12n 12n are given by K a · K c = k a · k c with a ∈ {3, · · · , n − 3} and c ∈ {1, · · · , n}, which are precisely the set of kinematic invariants (2.9). This implies that 3 provided that the determinant is expressed in terms of the invariants (2.9). Furthermore, as we saw in sec. 3, the massless scattering equations in d + M dimensions (3.7) are precisely equivalent to the massive scattering equations in d dimensions (3.1), and therefore have the same solutions. Thus, the propagator matrix for the massive amplitude is given by 1 (σ 1,γ(2) · · · σ γ(n−1),n σ n,1 )(σ 1,δ(2) · · · σ δ(n−1),n σ n,1 ) (4.5) where the sum is over the solutions of the scattering equations (3.1).
Finally, the expression (4.5) has no explicit dependence on the masses, nor does it have any implicit dependence on the masses since the solutions {σ} of the scattering equations, when expressed in terms of the kinematic invariants (2.9), are independent of the masses. Hence, eq. (4.5) for an amplitude with up to three massive particles is the same as that for massless amplitudes, as we already saw in sec. 2.

Gauge theory amplitudes
In this section, we demonstrate that various gauge theory amplitudes involving up to three massive particles can be expressed, like those for massless particles, as a sum over the (n−3)! solutions of the scattering equations. Moreover, the expressions for these amplitudes, when written in terms of the kinematic invariants (2.9), are independent of the masses of the particles.
We begin in d + M dimensions with the CHY representation for the tree-level scattering amplitude of n massless gauge bosons [1,2] A gauge = (−1) n−1 Here E a and K a are the (d+M)-dimensional polarizations and momenta of the gauge bosons, and the sum is over the solutions of the scattering equations (3.7). The first factor in the numerator is where c 1γn was defined in eq. (2.3), and det ′ Φ was defined in eq. (4.2). It is apparent that color-kinematic dual numerators n 1γn can be used to construct E({E, K, σ}) = δ∈S n−2 n 1δn σ 1,δ(2) · · · σ δ(n−1),n σ n,1 (5.3) so that eq. (4.3) implies that eqs. (5.1) and (2.5) are equivalent. The novelty of the CHY approach, however, is in providing an alternative definition for E({E, K, σ}) in terms of the Pfaffian of a matrix, so that eq. (5.1) yields the gauge theory amplitudes directly without resort to Feynman diagrams.
One first defines a 2n × 2n matrix where A, B, and C are the n × n submatrices (5.5) The matrix Ψ has two null vectors, so to obtain a nonvanishing Pfaffian, it is necessary to remove two of the first n rows and columns. It was shown in ref. [1] that the resulting "reduced Pfaffian" is independent (when the scattering equations are satisfied) of the choice of rows and columns deleted, but for our purposes (in order to make manifest the mass independence of the result) it will prove advantageous to choose ones corresponding to two of the massive particles. We therefore define Ψ 1,n as the matrix obtained by deleting the 1st and nth rows and columns from Ψ, in which case the CHY prescription is It was shown in refs. [1,32] that, with this definition, eq. (5.1) computes the amplitudes for massless gauge bosons in (d + M) dimensions.

Amplitudes for massive gauge bosons
We now use dimensional reduction to obtain the amplitude for massless and (up to three) massive gauge bosons in d dimensions. As in sec. 3, we set K a = (k a |κ a ), where the massless bosons (a = 3, · · · , n−1) have κ a = 0. We restrict all of the (d+M)-dimensional polarization vectors E a to the d-dimensional subspace: E a = (ǫ a |0, 0, · · · ). All of the (d + M)-dimensional dot products in the matrices B and C reduce to d-dimensional dot products, Every matrix element A ab that appears in Ψ 1,n (from which the 1st and nth rows and columns of Ψ have been eliminated) has either κ a = 0 or κ b = 0, so K a · K b = k a · k b , which belongs to the set of kinematic invariants (2.9). Thus, eq. (5.6) becomes so the d-dimensional amplitude for n gauge bosons, up to three of which may be massive, is where {σ} are solutions of the scattering equations (3.1). We showed in sec. 3 that these solutions, when expressed in terms of the invariants (2.9), are independent of the masses, and in sec. 4 that det ′ Φ is also independent of the masses. Hence, we conclude that the scattering amplitude (5.8) for n gauge bosons, up to three of which may be massive, is identical to that for n massless gauge bosons, provided it is expressed in terms of the invariants (2.9).
Taking σ 4 large but finite, we obtain where c 1234 = f a 1 a 2 b f ba 3 a 4 and c 1324 = f a 1 a 3 b f ba 2 a 4 , with f abc the three-gauge-boson vertex, and K is the totally permutation-symmetric expression that appears in the scattering amplitude for four open strings (cf. eqn. (7.4.42) of ref. [57]) Assembling the pieces, and taking the limit σ 4 → ∞, we obtain This expression exhibits the expected factorization into color-dependent and polarizationdependent factors [5,6].
Consider two amplitudes involving massive gauge bosons: W γ → W γ and W γ → W Z 0 . In each case we have c 1234 = c 1324 , so that eq. (5.13) reduces to both of which can be verified by a Feynman diagram calculation. 4 A Feynman diagram calculation also shows that eq. (5.13) is not valid for W Z 0 → W Z 0 , which is not surprising since this amplitude violates the condition m 3 = 0.
Amplitudes for two adjoint scalars and n − 2 gauge bosons Next we turn to the amplitude for two adjoint scalars (with masses m 1 and m n ) and n − 2 gauge bosons, one of which (m 2 ) may be massive. This amplitude may be obtained by dimensional reduction of eq. (5.1) with E 1 = E n = (0|1, 0, · · · ) and E a = (ǫ a |0, 0, · · · ) for a = 2, · · · , n − 1, as was done in ref. [25]. In that case, where all the particles were massless, Ψ became block diagonal, but the situation here is slightly more complicated because K a = (k a |κ a ) with κ a = 0 for a = 1, 2, and n.
Using eq. (5.17) in eq. (5.1), we obtain the d-dimensional amplitude for two massive adjoint scalars and n − 2 gauge bosons, one of which may be massive: For the same reasons as before, this amplitude is independent of the masses m 1 , m 2 , and m n when written in terms of the invariants (2.9).

Amplitudes for two fundamental scalars and n − 2 gauge bosons
The scalars in the amplitude (5.18) transform in the adjoint representation. The amplitude for scalars in the fundamental representation is essentially the same except that we replace c 1γn in eq. (5.2) with [50] where T a denote generators in the fundamental representation. In this way, we recover the results of ref. [50] for amplitudes with two massive scalars in the fundamental representation and n − 2 massless gauge bosons. 5 Again, we illustrate our result for the four-point amplitude with two fundamental scalars and two gauge bosons. On the single solution of the n = 4 scattering equations, we obtain Putting this together with eqs. (5.9) and (5.10), we obtain the amplitude Again we see the expected factorization into a product of a color factor and a polarizationdependent factor [5,6], valid provided one of the gauge bosons is massless.
Consider an amplitude involving a photon, W , and a pair of scalar quarks ψ d and ψū. In this case, each color factor t 1···4 is proportional to the ψ d W ψū coupling times the charge of the quark interacting with the photon, giving agreeing with a Feynman diagram evaluation [10]. Note that this amplitude vanishes when independent of the polarizations. This implies that the amplitude for ψ d ψū → W − γ, vanishes when cos θ = −1/3 (for massless quarks), where θ is the angle between ψ d and W − in the center-of-mass frame. Such "radiation zeros" were first found in calculations of dū → W − γ processes [7][8][9], leading to the discovery of four-point factorization [5,6], which was the forerunner of the BCJ relations [4]. Radiation zeros were also found to be present in higherpoint amplitudes under certain conditions [11,12].

Gravity amplitudes
In this section, we demonstrate that various gravitational amplitudes involving up to three massive particles can be expressed as a sum over the (n−3)! solutions of the scattering equations, and that the expressions for these amplitudes, when written in terms of the kinematic invariants (2.9), are independent of the masses of the particles.
We begin in d + M dimensions with the CHY representation for the tree-level gravitational scattering amplitude [1,2] A grav = (−1) n−1 for particles with momenta K a and whose polarization state is described by a tensor E µν a = E µ aẼ ν a , which includes the graviton, B-field, and dilaton. The factors det ′ Φ({K, σ}) and E({E, K, σ}) were defined in eqs. (4.2) and (5.6), and the sum is over the solutions of the scattering equations (3.7).
In ref. [25], dimensional reduction from d + M to d dimensions was used to obtain mixed amplitudes in Einstein-Maxwell theory with gauge group U(1) M . We will use the same approach to obtain various mixed amplitudes involving massive particles.
Amplitudes for two gauge bosons and n − 2 gravitons To obtain amplitudes for two (possibly massive) gauge bosons and n − 2 gravitons, we set K a = (k a |κ a ), with κ a = 0 for the gravitons (a = 2, · · · , n − 1). We choose the (d + M)dimensional polarization vectors to be E 1 = E n = (0|1, 0, · · · ) and E a = (ǫ a |0, 0, · · · ) for a = 2, · · · , n − 1, so that E({E, K, σ}) is given by eq. (5.17), while we setẼ a = (ǫ a |0, 0, · · · ) for all a so that E({Ẽ, K, σ}) is given by eq. (5.7). This yields the amplitude in d dimensions Although dimensional reduction gives the amplitude for two photons and n − 2 gravitons, it remains valid for any gauge theory since no non-abelian vertices are involved. For the same reasons described previously, this amplitude has no dependence (when expressed in terms of the invariants (2.9)) on the mass of the gauge bosons.
We illustrate eq. (6.2) for the case of the four-point amplitude gh → gh, describing the scattering of gravitons (h) from gauge bosons (g). Assembling eqs. (5.9), (5.11), and (5.20), we obtain where L was defined in eq. (5.21) andK indicates that ǫ is replaced byǫ in eq. (5.12). Equation (6.3) agrees with the result of a Feynman diagram calculation [20] in which the factorization into L andK was noted. As emphasized in ref. [22], factorization relates graviton scattering from W 's to Compton scattering from W 's, but only when the W magnetic moment is given by its standard model value g = 2.
Amplitudes for two scalars and n − 2 gravitons To obtain amplitudes for two massive scalars and n − 2 gravitons, we choose K a and E a to be the same as above, but now setẼ 1 =Ẽ n = (0|1, 0, · · · ) andẼ a = (ǫ a |0, 0, · · · ) for a = 2, · · · , n − 1. thus recovering the expression obtained in ref. [50]. As we have seen before, the amplitude is independent of the scalar mass, when expressed in terms of the invariants (2.9).
We illustrate eq. (6.4) for the case of the scattering ψh → ψh of gravitons from massive scalars [58]. Assembling eqs. (5.9) and (5.20), we obtain whereL indicates that ǫ is replaced byǫ in eq. (5.21). The factorization of this amplitude was first noted by the authors of ref. [20].
To illustrate eq. (6.8) for the four-point function, we evaluate its components on the single solution of the n = 4 scattering equations to obtain together with eqs. (5.9) and (5.11). Assembling the pieces and taking σ 4 → ∞, we obtain A gghg = (k 1 · k 3 ǫ 3 · k 2 − k 2 · k 3 ǫ 3 · k 1 ) 2k 1 · k 3 k 2 · k 3 k 3 · k 4 Tr(T a 1 T a 2 T a 4 )K (6.10) where all three gauge bosons can be massive. This result agrees with Feynman diagram computations of graviton photoproduction W γ → W h from massive vector bosons [20,23]. Equation (6.10) implies that W Z 0 → W h should also factorize.

Conclusions
In this paper we have derived CHY representations for a wide class of tree-level gauge and gravitational amplitudes containing up to three massive particles. These mixed amplitudes involve gravitons, massive and massless gauge bosons, and adjoint or fundamental massive scalar particles. All of these amplitudes can be obtained through the dimensional reduction of massless gauge and gravitational amplitudes in a higher dimension. Further, we demonstrated that amplitudes containing up to three massive particles, when written in terms of the kinematic invariants (2.9), are in fact independent of the masses, and so have the same form as purely massless amplitudes.
It is not so straightforward to obtain CHY representations for gauge and gravitational amplitudes with four or more massive particles. As explained in sec. 3, such amplitudes do not generically result from the dimensional reduction of massless amplitudes because of the difficulty of satisfying momentum conservation in the internal dimensions. Moreover, the propagator matrix in this case generally has rank greater than (n − 3)! and so cannot be represented as a sum over (n − 3)! solutions.
On the other hand, there is good reason to suspect the existence of CHY representations involving fermions, since the factorization of four-point gauge amplitudes also holds for amplitudes containing spin-one-half particles [5,6] and gravitational scattering from spinone-half particles [20][21][22][23] has also been shown to factorize. For progress in this area, see refs. [33,40,53].