Amplitude bases in generic EFTs

Abstract

We present an algorithm to find a basis of kinematically independent structures built of (massless and massive) spinor helicity variables in four dimensions. This method provides a classification of independent contact terms for the scattering amplitudes with generic masses, spins and multiplicity, in any effective field theory (EFT). These contact terms are in one-to-one correspondence with a complete set of marginal operators in the EFT. As basic applications of our method, we classify the D 2n F 4 contact terms in SU(N ) Yang-Mills theory for n ≤ 8, dimension-six operators involving five W ± , Z and γ vector bosons, and spin-tidal effective interactions for spin-1 massive particles in gravitational theories.

Constructing Generic Effective Field Theory for All Masses and Spins

“…While our paper was being finalized, Ref. [41] appeared, which presents a similar topic. This work uses a graphic method to construct the massive amplitude bases, which is equivalent to the Young Tableaux method used here.…”

“…Nevertheless, some of the assumptions in Ref. [41] are only numerically checked without rigorous proof. On the contrary, our work has a solid mathematical foundation.…”

“…1324 [42][31] Zµ Z ν W + µ W − ν 13 24 [43][21] Z µ ZµW +ν W − ν 34 [42][31][21] Źµν Zµν W +ρ W − ρ − i µνρσ ZµγZ γ ν W + ρ W − + µν Z ρ W − ρ − 2W − µ W + ρν Z ρ − 2ZµW + ρν W −ρ ) 23 24 [41][31] Źµν (Zµν W +ρ W − ρ − 2W − µ Zρν W +ρ − 2W + µ Zρν W −ρ ) 14 24 [32][31] Ẃ +µν (W − µν Z ρ Zρ − 4ZµW − ρν Z ρ ) 13 23 [42][41] Ẃ −µν (W + µν Z ρ Zρ − 4ZµW + ρν Z ρ ) 12 24 [43][31] Ẃ +µν (Zµν W −ρ Zρ − 2ZµZρν W −ρ − 2W − µ Zρν Z ρ ) 12 23 [43][41] Ẃ −µν (Zµν W +ρ Zρ − 2ZµZρν W +ρ − 2W + µ Zρν Z ρ ) − µν W +ρ Zρ + 2ZµW − ρν W +ρ − 2W + µ W − ρν Z ρ ) 13 23 24 [41] Ẁ +µν (Zµν W −ρ Zρ + 2ZµZρν W −ρ − 2W + µ Zρν Z ρ ) 12 23 34 [41] Zµν (W + µν W −ρ Zρ + 2ZµW + ρν W −ρ − 2W − µ W + ρν Z ρ ) 12 13 24 [43] Zµν (Zµν W −ρ W + ρ + 2W + µ Zρν W −ρ − 2W − µ Zρν W +ρ ) 323] 24 [41][21] i Źµν W − µ (D ρ Zν )W + ρ 424] 23 [31][21] i Źµν W + µ (D ρ Zν )W − ρ 134] 24 [32][31] iDρ( Ẃ +µρ g νσ + Ẃ +ρσ g µν + Ẃ +ρν g µσ + Ẃ +σµ g νρ + Ẃ +νµ g ρσ )W − µ Zν Zσ 234] 13 [42][31] i Ẃ −µν Zν (DµW + ρ )Z ρ 423] 13 24 [21] i( Ẁ −µρ g νσ + Ẁ −ρσ g µν + Ẁ −ρν g µσ + Ẁ −σµ g νρ + Ẁ −νµ g ρσ + Ẁ −σν g µρ )(DσZν )ZρW + µ 424] 13 23 [21] i Ẁ +µν (DρZµ)Zν W −ρ 234] 13 24 [31] i Zµν W −And for vertex D 2n W + W + W − W − , we have 2 physical dim-4 operators, and 18 dim-6 operators,Amplitude Operator D 2n W + W + W − W − 13 24 [42][31] W +µ W +ν W − µ W − − µν W +ρ W − ρ − 2W − µ W − ρν W +ρ − 2W + µ W − ρν W −ρ ) 23 24 [41][31] Ẃ +µν (W + µν W −ρ W − ρ − 4W − µ W + ρν W −ρ ) 14 24 [32][31] Ẃ −µν (W − µν W +ρ W + ρ − 4W + µ W − ρν W +ρ ) 12 24 [43][31] Ẃ −µν (W + µν W −ρ W + ρ − 2W + µ W + ρν W −ρ − 2W − µ W + ρν W +ρ ) 13 24 34 [21] Ẁ −µν W − µν W +ρ W + ρ 13 24 2 [31] Ẁ −µν W + µν W −ρ W + ρ 12 24 34 [31] Ẁ +µν (W − µν W −ρ W + ρ + 2W + µ W − ρν W −ρ − 2W + µ W − ρν W +ρ ) 12 13 24 [43] Ẁ +µν W + µν W −ρ W − −µρ g νσ + Ẃ −ρσ g µν + Ẃ −ρν g µσ + Ẃ −σµ g νρ + Ẃ −νµ g ρσ )W − µ W + ν W + σ 423] 13 24 [21] i( Ẁ −µρ g νσ + Ẁ −ρσ g µν + Ẁ −ρν g µσ + Ẁ −σµ g νρ + Ẁ −νµ g ρσ + Ẁ −σν g µρ )(DσW + ν )W + ρ W − µ 234] 13 24 [31] i Ẁ +µν W − µ (Dν W − ρ )W +ρ 232] 423] 13[41] (i µνρσ + g µσ g νρ − g µρ g νσ + g µν g ρσ )(D ξ W − µ )W − ρ (Dν W +ξ )W +σ 424] 242] 13 [31] W +ρ W − ρ (DµW −ν )(Dν W +µ )Finally, for vertex D 2n ZZZZ, we only have 1 physical dim-4 operator, and 4 dim-6 operators,…”

“…µ (Dν W + ρ )Z ρ 434] 12 23 [31] i Zµν Zµ(DρW + ν )W −ρ 232] 423] 13 [41] (i µνρσ + g µσ g νρ − g µρ g νσ + g µν g ρσ )(D ξ W + µ )W − ρ (Dν Z ξ )Z σ 424] 242] 13 [31] Z ρ W + ρ (DµW −ν )(Dν Z µ ) ρ 323] 24 [41][21] i Ẃ +µν W − µ (D ρ W + ν )W − ρ 134] 24 [32][31] iDρ( Ẃ ρ Zρ − 4ZµZρν Z ρ )1324 34 [21] Zµν Zµν Z ρ Zρ 232] 423] 13[41] (DµZ ν )(Dν Z µ )ZρZ ρ…”

Theoretical developments in the SMEFT at dimension-8 and beyond

“…Moreover, a systematic construction of three and four-point non-factorizable amplitudes was presented in [76]. Systematic algorithms for the construction of independent massive amplitudes were further in [77][78][79], while their derivation from the Higgsing of massless amplitudes was studied in [80]. This approach also yields all-order results in v/Λ, whose powers are all absorbed in constant amplitude coefficients.…”