BCJ relation of color scalar theory and KLT relation of gauge theory

Abstract

We present a field theoretical proof of the conjectured KLT relation which states that the full tree-level scattering amplitude of gluons can be written as a product of color-ordered amplitude of gluons and color-ordered amplitude of scalars with only cubic vertex. To give a proof we establish the KK relation and BCJ relation of color-ordered scalar amplitude using BCFW recursion relation with nonzero boundary contributions. As a byproduct, an off-shell version of fundamental BCJ relation is proved, which plays an important role in our work. since on one side of the equation, the full supermultiplet of N = 8 SUGRA theory labeled by eight Grassmann variables η i , i = 1, ..., 8 contains both graviton and gluon, while one the other side of the equation the full supermultiplet of N = 4 SYM theory labeld by four Grassmann variables (η i , i = 1, ..., 4 for A and η i , i = 5, ..., 8 forÃ) contains both gluon and scalar. Specifically, we obtain expressions of amplitudes of the desired particle types by reading off the corresponding Grassmann variable η i 1 ...η ir 1 Recently, KK and BCJ like relations for the remaining rational function of one-loop gluon amplitudes have also been investigated in [27].

Note on symmetric BCJ numerator

“…(1.2) This duality between the color and kinematic factors was later found to be present in a variety of Yang-Mills theories [2][3][4][5][6][7][8][9][10] and, perhaps most surprisingly, was shown to be valid at least for the first few loop levels [2,3,[5][6][7][8][9][11][12][13][14][15][16][17][18][19]. The apparently symmetrical structure also suggests mirror versions of the existing color decomposition formulations.…”

“…Thus it is imaginable that there is a KLT-like relation for full Yang-Mills amplitudes. In fact the KLT-like construction for full Yang-Mills amplitudes was proposed long time before in [30] and has later been systematically proved in [13] using BCFW-recursion relation. Explicitly the relation is given by…”

“…As a third step, we expand A n using Kleiss-Kuijf(KK)-basis by KK relation [27] (KK relation for color ordered scalar amplitude was proved in [13]), with, for example, 1, n fixed at the first and the last position…”

Color/kinematics duality for general abelian orbifolds of $ \mathcal{N} $ = 4 super Yang-Mills theory

“…Thus, we must either require g = 0 and find the scalar theory of [44] (upon using the fact that σ IJK obeys the Jacobi identity to set σ IJK = σf IJK ) or set σ IJK = 0 and find the dimensional reduction of YM theory in D s = D + n s dimensions.…”

Algebras for amplitudes

“…The fact that this representation is possible for gauge theory amplitudes (and for amplitudes of the closely related theories to be discussed next) implies linear relations among color-ordered amplitudes, the BCJ relations. The color-kinematics duality has been discussed extensively in recent work, see [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38].…”