Primary Observables for Indirect Searches at Colliders

We consider the complete set of observables for collider searches for indirect eﬀects of new heavy physics. They consist of SU (3) C × U (1) EM invariant interaction terms in the eﬀective Lagrangian renormalized at the experimental energy scale that parameterize deviations from the Standard Model. Under the assumption that the new physics arises from heavy particles with mass of order M , we show that the leading deviations from the Standard Model can be parameterized by a ﬁnite number of ‘primary’ operators, with the remaining operators given by ‘Mandelstam descendants’ whose eﬀects are suppressed by powers of Mandelstam variables divided by M . This classi-ﬁcation does not rely on any eﬀective ﬁeld theory power counting rule (as in SMEFT or HEFT) or assumptions about the UV physics (for example decoupling). We explicitly determine all 3 and 4-point primary operators relevant for Higgs signals at colliders. We give a detailed discussion of the methods used to obtain this result, including a new analytical method for determining the independent operators. The results are checked using the Hilbert series that counts independent operators. We also give a sketch of the phenomenology, including unitarity bounds on the interaction strengths and rough estimates of their importance for Higgs decays at the HL-LHC. This motivates further exploration of Higgs decays to Z ¯ ff , W ¯ ff (cid:48) , γ ¯ ff , and Zγγ .