Group constraint relations for five-point amplitudes in gauge theories with SO(N) and Sp(2N) groups

In this paper,linear constraint relations among loop-order five-point color-ordered amplitudes in $SO(N)$ and $Sp(2N)$ gauge theories are derived with the group-theoretic method. These constrains are derived up to four-loop order. It is found that in both theories, there are $n=6,22,34,44,50$ linear constraint relations at $L=0,1,2,3,4$ loop orders. Then the numbers of independent color-ordered five-point amplitudes are respectively $n_{ind.}=6,12,22,34,50$ at each loop order.


I. INTRODUCTION
There has been a great advance in our understanding about perturbative amplitudes in gauge and gravity theories during last decade. One aspect of the advance is the revealing of many different kinds of relations between color-ordered gauge amplitudes and double-copy relations between gauge and gravity amplitudes. For gauge amplitudes, one important discovery is the Bern-Carrasco-Johansson(BCJ) relation [1]. These relations have been generally proved for tree level gauge amplitudes [2][3][4]. Subsequently, tree level BCJ relations have been proved in N=4 SYM [5],noncommutative U(N) gauge theory [6] and string theories [7][8][9].
In the past several years, a group-theoretic method has been proposed to derive linear constraints among loop order color-ordered amplitudes. One virtue of these constraints is that they hold for any helicity configuration of external particles. Some nontrivial all loop relations have been derived for four-point color-ordered amplitudes in SU(N) gauge theories [26]. Subsequently, the all loop relations for five-and six-point color-ordered amplitudes have also been derived [27,28]. These loop-order constraint relations generalize known treelevel and one-loop SU(N) gauge amplitude relations. Importantly, some of the nontrivial loop-order constraints are precisely those derived by direct calculations of loop amplitudes in SU(N) gauge theories [24,29,31]. Recently, this group-theoretic method has been applied to study color-ordered amplitudes in SO(N) and Sp(2N) gauge theories [30]. Linear constraint relations for four-point amplitudes in both kinds of theories are derived up to four-loop order.
In this paper, based on our previous work [30], we continue to use the group-theoretic method to derive linear constraints on five-point color-ordered amplitudes in SO(N) and Sp(2N) gauge theories through four loop. All particles in the scattering amplitudes are adjoint particles. It is found that, at L = 0, 1, 2, 3, 4 loop orders, there are respectively n = 6, 22, 34, 44, 50 group-theoretic constraint relations among five-point color-ordered am- where {c λ } is a color basis and {t i } is a trace basis. a λ is a kinematic factor corresponding to c λ and A i is the color-ordered amplitude corresponding to t i .
In the fundamental representation, the generators of SO(N) algebra are N × N antisymmetric, traceless matrices, which are denoted by {T a } (a = 1, 2, ..., N(N − 1) 2) [30]. For five-point amplitudes, a trace basis consists of both single-trace and double-trace elements.
A typical element in a trace basis is Tr(T a 1 T a 2 T a 3 T a 4 T a 5 ), where a i is the color quantum number of the i-th particle. For simplicity, we use Tr(12345) instead of Tr(T a 1 T a 2 T a 3 T a 4 T a 5 ) Linear constraints among Sp(2N) color-ordered amplitudes {Ā (L,m) k } are studied in Section IV.

III. CONSTRAINTS ON SO(N ) FIVE-POINT AMPLITUDES
In this section, we derive the linear constraints on SO(N) five-point amplitudes with group-theoretic method. This method is based on the fact that the full gauge amplitudes can be decomposed by two ways: f -based and trace-based decompositions. As in eq.(1), the tree-level five-point full amplitude can be decomposed as c 0 λ is a tree-level color basis element, which is a product of several SO(N) structure constants. A SO(N) structure constant can always be expressed as a trace of its generators, And with following two useful identities for the trace of SO(N) generators, a color basis can always be expressed as linear combination of a trace basis.
At tree level, there are six elements in a color basis {c 0 λ }, which can be chosen as {i 3 f 12a f a3b f b45 , all other permutations of (234)}. The expansion matrix between this color basis and trace basis is M 0 , defined as Combining with eq.(6), we can obtain the following relation Then right null vectors of M 0 lead to linear constraints among color-ordered amplitudes The six tree-level null vectors of M 0 , r where we use (..., 0 3 , ...) to denote (..., 0, 0, 0, ...). They imply six constraints among twelve five-point color-ordered tree amplitudes. For example, the constraint equations implied by the first and last vectors are The first equation is obviously the same as tree-level dual Ward identity or KK-like relation in SU(N) gauge theories. Using the reflection relation we can see that the second equation is also a dual Ward identity or a KK-like relation. In fact, all six constraint relations are dual Ward identities or KK-like relation, which is the same as the SU(N) case [27]. There are six constraint relations among twelve tree-level five-point amplitudes, so the number of independent tree amplitudes is six.

A. constraints among SO(N ) one-loop five-point amplitudes
In this section, we first explain the procedure to derive constraint relations on L-loop amplitudes. In order to find these constraints, we should find a complete (or overcomplete) L-loop color basis, expand elements of the color basis by L-loop trace basis and then find the null vectors of the expansion matrix. Each null vector leads to a constraint relation among L-loop color-ordered amplitudes.
It is easy to construct a L-loop trace basis. But constructing a L-loop color basis is not easy. There is an assumption that all (L + 1)-loop color factors can be obtained from L-loop color factors by attaching a rung between two of its external legs [26]. This assumption can be checked explicitly at lower loop orders (L ⩽ 4) for SU(N) and is thought to be correct at L > 4 loop orders [26]. Here we adopt the same assumption and assume that all possible elements of (L + 1)-loop color basis can be obtained from L-loop elements by attaching We consider the effect of this attaching process on the trace basis. Let where the explicit forms of A, B, C, D, E, F are given in the Appendix.
Suppose {c L α } is a complete color basis for L-loop five-point amplitudes and they can be expressed by a L-loop trace basis According to eq.(17), the attaching procedure transforms L-loop trace basis where G is a (22L + 12) × (22L + 34) transformation matrix. After the attaching procedure, L-loop color basis and trace basis are transformed to (L+1)-loop color basis and trace basis, and then we have which means This is the relation between L-loop and (L+1)-loop null vectors, which can be used to derive higher loop null vectors from lower loop ones.
Now we derive the one-loop null vectors from tree-level null vectors. The transformation matrix G (0,1) between trace bases of tree amplitudes and one-loop amplitudes is Substituting G (0,1) and tree null vectors (13) into eq. (22), we can obtain 22 one-loop null vectors, which can be written as a matrix r (1) = R (1) . Each column in matrices The matrices m are given by The number of one-loop five-point color-ordered amplitudes is 34 (22L + 12). So the number of independent color-ordered amplitudes is 12. From eq.(4), the one-loop color decomposition of five-point amplitude is The null vectors allow one to choose leading single-trace amplitudes, A (1,1) 1 through A (1,1) 12 , as the one-loop independent amplitudes. And all other subleading single-trace or doubletrace amplitudes can be written as linear combinations of them.
In this section we consider the constraints on two-loop amplitudes. The transformation matrix between one-loop and two-loop trace bases is Substituting G (1,2) and one-loop null vectors into eq.(22), we can obtain 34 two-loop null vectors. These null vectors can be written as r (2) = R 1 , R . Each column in The explicit expressions of matrices m   In this section, we discuss the constraints among three-and four-loop five point amplitudes. The transformation matrix G (2,3) between two-loop null vectors and three-loop null vectors is By solving eq. (22), we can obtain 44 three-loop null vectors. There are 78(22L + 12) threeloop color-ordered amplitudes, so the number of independent three-loop amplitudes are 34.
We do not list the null vectors explicitly.
The transformation matrix G (3,4) between two-loop null vectors and three-loop null vectors is By solving eq.(22), we can obtain 50 four-loop null vectors. There are 100(22L + 12) fourloop color-ordered amplitudes, so the number of independent four-loop amplitudes are 50.
We do not list the null vectors explicitly.

IV. CONSTRAINTS ON Sp(2N ) FIVE-POINT AMPLITUDES
In this section, we use the same procedure to derive group-theoretic constraints among The six tree-level null vectors of M 0 is also the same as SO(N) case, The transformation matrixḠ (0,1) between trace bases of tree amplitudes and one-loop amplitudes isḠ The explicit forms ofĀ,B,C are given in the appendix. Solving recursive equation (22), we can obtain 22 one-loop null vectors, which can be written as a matrixr (1)

Each column in matricesR
(1) are defined as The transformation matrixḠ (1,2) between trace bases of Sp(2N) one-loop amplitudes and two-loop amplitudes isḠ The explicit forms of the block matrices,Ā, ⋯,F , are given in the appendix. Substitutinḡ G (1,2) and one-loop null vectors into eq.(22), we can obtain 34 two-loop null vectors. These null vectors can be written asr (2) . Each column inR 3 ,m In this appendix, we give the explicit forms of the block matrices in transformation matrix G (L,L+1) in both theories. Let quantity e 1i take one when we attach legs (1, i) and otherwise take zero. Define Λ ij = e 1i + e 1j , ∆ ij = e 1i − e 1j and Θ ij,kl = 3Λ ij − Λ kl . For SO(N) case, the block matrices A, ⋯, F are of the following forms For Sp(2N), the block matricesĀ, ⋯,F are related with those of SO(N) and they satisfy the following equationsĀ