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RG evolution and effect of intermediate new-physics on $ΔB=1$ four-fermion operators

Mathew Thomas Arun, Shyam M, Ritik Pal

TL;DR

This paper addresses how intermediate-scale baryon-number-conserving (BNC) new physics affects proton-decay predictions by studying the RG evolution of baryon-number-violating (BNV) dimension-6 four-fermion operators within the SMEFT framework. It derives 1-loop beta functions for the BNV coefficients including mixing with 22 BNC operators, with top-quark loops providing the dominant contributions, and evolves these Wilson coefficients from the electroweak scale up to intermediate scales of $10^4$, $10^6$, and $10^9$ GeV. The main finding is that BNC-BNV mixing can significantly lower the effective proton-decay scale $\Lambda/\sqrt{c}$, potentially by several orders of magnitude, thus reducing the need for an ultra-dlarge desert between the electroweak and BNV scales; representative results indicate a downward shift toward $\sim 10^7$ GeV in some scenarios. To facilitate broader usage, the authors provide a Python package that performs the RG evolution for generic BNC and BNV SMEFT operators with configurable thresholds, enabling systematic exploration of intermediate-scale new physics in nucleon-decay phenomenology.

Abstract

Motivated by the stringent experimental bounds on proton lifetime and the need for precise low-energy predictions, there has been renewed interest in the renormalization group (RG) evolution of Wilson coefficients for baryon number violating (BNV) operators and their characteristic new-physics scales. In this work, we analyze the RG running of dimension-6 four-fermion operators in the $\overline{\text{MS}}$ scheme that mediate nucleon decay channels such as $p \to e^+ π^0$, while systematically accounting for the impact of baryon number conserving (BNC) new-physics that can enter the theory at an intermediate scale as higher-dimensional effective field theory operator. These BNC operators mix with BNV ones at 1-loop and alter the RG flow. The running is performed from the electroweak scale up to representative intermediate scales of $10^4~\text{GeV}$, $10^6~\text{GeV}$, and $10^9~\text{GeV}$, corresponding to possible thresholds for new BNC degrees of freedom. Comparing the RG evolved coefficients with current experimental bounds on nucleon decay lifetimes, we find that the inclusion of BNC-BNV mixing, dominated by top quark loops, can significantly lower the effective proton decay scale to $\sim 10^7$ GeV, thus mitigating the need of a large desert. A Python package is provided to facilitate the RG evolution of nucleon-decay Wilson coefficients, allowing for the inclusion of generic BNC effects.

RG evolution and effect of intermediate new-physics on $ΔB=1$ four-fermion operators

TL;DR

This paper addresses how intermediate-scale baryon-number-conserving (BNC) new physics affects proton-decay predictions by studying the RG evolution of baryon-number-violating (BNV) dimension-6 four-fermion operators within the SMEFT framework. It derives 1-loop beta functions for the BNV coefficients including mixing with 22 BNC operators, with top-quark loops providing the dominant contributions, and evolves these Wilson coefficients from the electroweak scale up to intermediate scales of , , and GeV. The main finding is that BNC-BNV mixing can significantly lower the effective proton-decay scale , potentially by several orders of magnitude, thus reducing the need for an ultra-dlarge desert between the electroweak and BNV scales; representative results indicate a downward shift toward GeV in some scenarios. To facilitate broader usage, the authors provide a Python package that performs the RG evolution for generic BNC and BNV SMEFT operators with configurable thresholds, enabling systematic exploration of intermediate-scale new physics in nucleon-decay phenomenology.

Abstract

Motivated by the stringent experimental bounds on proton lifetime and the need for precise low-energy predictions, there has been renewed interest in the renormalization group (RG) evolution of Wilson coefficients for baryon number violating (BNV) operators and their characteristic new-physics scales. In this work, we analyze the RG running of dimension-6 four-fermion operators in the scheme that mediate nucleon decay channels such as , while systematically accounting for the impact of baryon number conserving (BNC) new-physics that can enter the theory at an intermediate scale as higher-dimensional effective field theory operator. These BNC operators mix with BNV ones at 1-loop and alter the RG flow. The running is performed from the electroweak scale up to representative intermediate scales of , , and , corresponding to possible thresholds for new BNC degrees of freedom. Comparing the RG evolved coefficients with current experimental bounds on nucleon decay lifetimes, we find that the inclusion of BNC-BNV mixing, dominated by top quark loops, can significantly lower the effective proton decay scale to GeV, thus mitigating the need of a large desert. A Python package is provided to facilitate the RG evolution of nucleon-decay Wilson coefficients, allowing for the inclusion of generic BNC effects.

Paper Structure

This paper contains 16 sections, 14 equations, 12 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Standard Model Effective Field Theory
  4. Baryon Number Conserving Operators:
  5. Baryon Number Violating Operators:
  6. Loop Diagrams from BNC-BNV Mixing and Their Beta Functions
  7. BNC-BNV Mixing
  8. Beta Function Calculation
  9. Running of the Beta Function
  10. Case 1: BNC operator scale at $10^4$ GeV
  11. Case 2: BNC operator scale at $10^6$ GeV
  12. Case 3: BNC operator scale at $10^9$ GeV
  13. Introduction to Python Package
  14. Summary
  15. Representative 1-loop mixing calculation
  16. ...and 1 more sections

Figures (12)

  • Figure 1: Representative 1-loop topologies for nucleon decay: the $(a)$ involves only BNV running, the $(b)$ loop gives the dominant (top-top) contribution to proton decay, and the $(c)$ loop gives the dominant (top-bottom) contribution to neutron decay.
  • Figure 2: Diagrammatic representation of BNV running (a) consists of tree (b), the SM interaction (c), and the BNC-BNV mixing (d) of first-generation BNV operators. The dominant loop contribution BNC-BNV mixing arises from top-top contractions, which feed into the anomalous dimensions of the BNV coefficients.
  • Figure 3: Running of the proton decay Wilson coefficients $C^{duql}_{1111}$, $C^{qque}_{1111}$, $C^{qqql}_{1111}$, and $C^{duue}_{1111}$ from $10^3$ GeV to $10^{4}~\text{GeV}$ as a function of the energy scale $\mu$. Solid blue lines show evolution without BNC interactions, while dotted orange lines include full BNC-BNV mixing. This running corresponds to the case with an initial value of $10^{-10}~\text{GeV}^{-2}$ (Set 1) for the Wilson coefficients $C^{duql}_{1331}$, $C^{qque}_{1331}$, $C^{qqql}_{1331}$, and $C^{duue}_{1331}$.
  • Figure 4: Comparison of the UV scale $\Lambda / \sqrt{c}$ for the four proton-decay operators in Eqs. (\ref{['eq:op_duql']}-\ref{['eq:op_duue']}), contributing to the decay channel $p \to \pi^0 e^+$ for case 1. The dotted line represents without RG running and the two different colour bars correspond to: (i) RG running without BNC-BNV mixing interactions (in blue) and (ii) RG running including BNC-BNV operator mixing (in orange), for different initial scale values (Sets) of each BNV Wilson coefficient. Results are computed using the beta functions in Eqs. (\ref{['eq:duqRGE']}-\ref{['eq:duuRGE']}), contributing to the decay channel $p \to \pi^0 e^+$ with the running scale $\mu$ from $10^3$ GeV to $10^4$ GeV.
  • Figure 5: The plot shows the dependence of the proton decay operator scale on the BNC-BNV parameter space for case 1. The x-axis represents the BNC operator Wilson coefficient ($C^{BNC}_{1313}$), while, the y-axis represent the BNV Wilson coefficient ($C^{BNV}_{1331}$).
  • ...and 7 more figures