Standard Model EFTs via on-shell methods

Abstract

We present the Standard Model Effective Field Theories (SMEFT) from purely on-shell arguments. Starting from few basics assumptions such as Poincaré invariance and locality, we classify all the renormalisable and non-renormalisable interactions at lowest order in the couplings. From these building blocks, we review how locality and unitarity enforce Lie algebra structures to appear in the S-matrix elements together with relations among couplings (and hypercharges). Furthermore, we give a fully on-shell algorithm to compute any higher-point tree-level amplitude (or form factor) in generic EFTs, bypassing BCFW-like recursion relations which are known to be problematic when non-renormalisable interactions are involved. Finally, using known amplitudes techniques we compute the mixing matrix of SMEFT marginal interactions up to mass dimension 8, to linear order in the effective interactions.

Amplitude bases in generic EFTs

“…For instance, such methods have been applied to compute the anomalous dimension in generic EFTs [6][7][8][9][10][11][12][13][14], and to understand the patterns of zeros in the one-loop anomalous dimension matrix for dimension-six operators in the SMEFT [15][16][17] and in Wilson coefficients when integrating out heavy modes [18]. In the study of gravitational binary systems, scattering amplitude techniques allowed to compute the conservative two-body scattering dynamics up to O(G 4 ) in all orders in velocity [19][20][21][22][23][24][25][26][27][28][29], and to lower orders in G including spin effects [22,[30][31][32][33][34][35][36][37], tidal effects [38][39][40][41][42] and higher-derivative interactions [43,44].…”

“…This means that from the S-matrix perspective the classification and enumeration of independent operators [45][46][47][48][49] is equivalent to finding a basis of kinematically independent polynomial structures in the amplitudes [50,51]. From these "building blocks", higher-point tree-level amplitudes can be constructed from the knowledge of the factorisation channels, using recursion relations [12,[52][53][54][55][56][57], and loop-level amplitudes through generalised unitarity techniques [58][59][60][61][62][63][64].…”

“…In this paper, we present an alternative method to classify generic contact terms in four-dimension, for any mass, spin and multiplicity (n ≥ 4, the classification for n = 3 is known [69,72,82,83]), starting from the graph method presented in [12], strengthen by the unbolding/bolding procedure of [72,78] which we implement at the level of graphs. In this algorithm, structures involving powers of any mass are always regarded as kinematically independent from structures with fewer powers of mass.…”

Constructing Generic Effective Field Theory for All Masses and Spins

“…On-shell scattering amplitude is efficient in dealing with some problems of EFT, such as calculating the running of EFT operators [13][14][15][16][17][18][19], deriving EFT selecting rules [20][21][22], and constructing scalar EFT with nontrivial soft-limit [23][24][25][26]. Especially it is very efficient in constructing EFT bases of massless fields (called amplitude bases) [19,[27][28][29][30].…”

“…On-shell scattering amplitude is efficient in dealing with some problems of EFT, such as calculating the running of EFT operators [13][14][15][16][17][18][19], deriving EFT selecting rules [20][21][22], and constructing scalar EFT with nontrivial soft-limit [23][24][25][26]. Especially it is very efficient in constructing EFT bases of massless fields (called amplitude bases) [19,[27][28][29][30]. A complete set of the amplitude bases without IBP and EOM redundancy can be systematically constructed by the semi-standard Young tableaus (SSYTs) of the global symmetry of massless spinors [31] (more applications can be found in [32,33]).…”