EFT anomalous dimensions from the S-matrix
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Springer
Received: June 5, 2020 Accepted: August 22, 2020 Published: September 25, 2020
Joan Elias Mir´ o,a James Ingoldbya and Marc Riembaub a
Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151, Trieste, Italy b Universit´e de Gen`eve, Ernest-Ansermet 24, 1211 Gen`eve, Switzerland
E-mail: [email protected], [email protected], [email protected] Abstract: We use the on-shell S-matrix and form factors to compute anomalous dimensions of higher dimension operators in the Standard Model Effective Field Theory. We find that in many instances, these computations are made simple by using the on-shell method. We first compute contributions to anomalous dimensions of operators at dimension-six that arise at one-loop. Then we calculate two-loop anomalous dimensions for which the corresponding one-loop contribution is absent, using this powerful method. Keywords: Beyond Standard Model, Effective Field Theories, Renormalization Group, Scattering Amplitudes ArXiv ePrint: 2005.06983
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)163
JHEP09(2020)163
EFT anomalous dimensions from the S-matrix
Contents 1 Introduction
1 4 5 5 7 9 9 11 12
3 Structure of the anomalous dimension matrix
14
4 The 4.1 4.2 4.3
16 17 19 21
‘easy’ two-loop anomalous dimensions h · i2 → h · i3 : 4-point to 3-point h · i → h · i2 : 5-point to 4-point 1 → h · i: 6-point to 5-point
5 Conclusions
22
A Phase space integrals A.1 The 2 → 2 case A.2 The 3 → 2 case
23 25 25
B Amplitudes
27
1
Introduction
The discovery of the Higgs particle [1, 2], Higgs coupling measurements, and a plethora of beyond the Standard Model (SM) searches suggest that there is an energy gap between the electroweak scale and any new physics scale. Physics within this energy gap is appropriately described by the SM, together with a tower of operators encoding the deformations generated by the new dynamics. Given such separation of scales, the Renormalisation Group (RG) mixing of higher-dimension SM Effective Field Theory (EFT) operators can lead to important physical effects on precision observables. The RG running of SM EFT operators down from a hypothetical new physics scale to the EW scale is governed by the operator anomalous dimensions. The anomalous dimension of an operator can be computed using a number of different methods. For instance, the S-matrix elements, the effective action, or form factors, are
–1–
JHEP09(2020)163
2 One-loop RGEs of dimension-six operators 2.1 Contributions at order λ 2.1.1 ∂ 2 H 4 operators 2.1.2 The rest of the order λ corrections 2.2 Contributions from gauge interactions 2.2.1 IR divergences 2.2.2 Gauge mixing 2.3 Non-minimal form factors
subject to a Callan-Symanzik equation, a.k.a. renormalisation group equation (RGE), that depends on the anomalous dimensions. In this work we will compute the anomalous dimensions of SM dimension-six operators through the RGE’s satisfied by the form factors. The form factors (FFs) are defined by FO (~n) ≡
n| O(0) |
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