[Paper Review] Extending low energy effective field theory with a complete set of dimension-7 operators

Yi Liao, Xiao-Dong Ma+1 more|arXiv (Cornell University)|May 16, 2020

Particle physics theoretical and experimental studies49 references64 citations

TL;DR

This paper presents a complete and independent basis of dimension-7 operators in the low-energy effective field theory (LEFT), classifying them by baryon and lepton number violation. It performs tree-level matching between SMEFT and LEFT up to dimension-7, and applies the framework to study enhanced lepton-number-violating neutrino-photon scattering, showing cross sections significantly larger than in the SM even for new physics scales up to 100 TeV due to inverse electron mass enhancement at low energies.

ABSTRACT

We present a complete and independent set of dimension-7 operators in the low energy effective field theory (LEFT) where the dynamical degrees of freedom are the standard model five quarks and all of the neutral and charged leptons. All operators are non-Hermitian and are classified according to their baryon ($\Delta B$) and lepton ($\Delta L$) numbers violated. Including Hermitian-conjugated operators, there are in total $3168$, $750$, $588$, $712$ operators with $(\Delta B,\Delta L)=(0,0)$, $(0,\pm 2)$, $(\pm 1,\mp 1)$, $(\pm 1,\pm 1)$ respectively. We perform the tree-level matching with the standard model effective field theory (SMEFT) up to dimension-7 (dim-7) operators in both LEFT and SMEFT. As a phenomenological application we study the effective neutrino-photon interactions due to dim-7 lepton number violating operators that are induced and much enhanced at one loop from dim-6 operators that in turn are matched from dim-7 SMEFT operators. We compare the cross sections of various neutrino-photon scattering with their counterparts in the standard model and highlight the new features. Finally we illustrate how these effective interactions could arise from ultraviolet completion.

Motivation & Objective

• To construct a complete and independent basis of dimension-7 operators in the low-energy effective field theory (LEFT), where the dynamical degrees of freedom are SM quarks and leptons, excluding heavy particles like the Higgs and top quark.
• To perform tree-level matching between the SMEFT and LEFT frameworks, incorporating contributions from dimension-7 operators in both theories.
• To study the phenomenological implications of lepton-number-violating (LNV) dimension-7 operators in LEFT, particularly their role in generating effective neutrino-photon interactions.
• To illustrate how such dimension-7 operators can arise from UV completions via explicit model-building examples with heavy scalar and fermion states.

Proposed method

• Constructs a complete basis of dimension-7 operators in LEFT by classifying them according to baryon (∆B) and lepton (∆L) number violation, resulting in 3168, 750, 588, and 712 operators for (∆B, ∆L) = (0,0), (0,±2), (±1,∓1), and (±1,±1), respectively.
• Applies Lorentz symmetry and equations of motion (EoMs) with integration by parts (IBP) to reduce higher-derivative and higher-contraction terms to lower-dimensional operators, ensuring independence and completeness.
• Performs tree-level matching between SMEFT and LEFT by relating Wilson coefficients of dim-7 operators in both frameworks, including contributions from dim-7 terms in both theories.
• Analyzes the effective neutrino-photon interaction induced by LNV dim-7 operators in LEFT, tracing its origin to one-loop corrections from dim-6 operators that are matched from dim-7 SMEFT operators.
• Uses Fierz identities and algebraic manipulations to reduce redundant operator structures and define a minimal set of independent operators.
• Illustrates UV completion via explicit models—such as those with heavy scalar and vector-like fermion states—showing how dim-7 SMEFT operators can generate the required LNV interactions at low energy.

Experimental results

Research questions

• RQ1 What is the complete and independent basis of dimension-7 operators in the LEFT framework, classified by baryon and lepton number violation?
• RQ2 How do dim-7 operators in SMEFT match to those in LEFT at tree level, and what are the resulting Wilson coefficient relations?
• RQ3 What are the phenomenological consequences of lepton-number-violating dim-7 operators in LEFT for neutrino-photon scattering, and how do they differ from the SM?
• RQ4 How is the effective neutrino-photon interaction enhanced at low energies, and what is the role of the inverse electron mass in this enhancement?
• RQ5 Can explicit UV-complete models generate the required dim-7 operators that lead to LNV neutrino-photon couplings?

Key findings

• The complete basis of dimension-7 operators in LEFT comprises 3168, 750, 588, and 712 independent operators for (∆B, ∆L) = (0,0), (0,±2), (±1,∓1), and (±1,±1), respectively, with all operators being non-Hermitian.
• The tree-level matching between SMEFT and LEFT at dimension-7 includes contributions from dim-7 operators in both theories, enabling a consistent mapping of Wilson coefficients.
• Lepton-number-violating neutrino-photon interactions arise from one-loop corrections involving dim-6 operators in LEFT, which are induced by matching dim-7 SMEFT operators.
• The effective neutrino-photon scattering cross sections are significantly enhanced at low energies due to an inverse electron mass factor, leading to cross sections larger than in the SM even for new physics scales as high as 100 TeV.
• UV completion models—such as those with heavy scalar and vector-like fermion states—can generate the required dim-7 SMEFT operators, with explicit Wilson coefficient expressions derived via one-loop diagrams.
• In specific UV models, such as model (d2) with vector-like fermions and a scalar doublet, the Wilson coefficient for same-flavor neutrino-photon coupling does not vanish, unlike in other models, indicating a viable path to LNV at low energy.