Constraining new physics effective interactions via a global fit of electroweak, Drell-Yan, Higgs, top, and flavour observables

1. Problem Statement

The Standard Model (SM) of particle physics is highly successful but incomplete, failing to explain phenomena such as dark matter, baryon asymmetry, and neutrino masses. While direct searches for New Physics (NP) at the Large Hadron Collider (LHC) have not yet yielded definitive signals, indirect searches via precision measurements offer a powerful alternative.

The challenge lies in systematically interpreting a vast and diverse set of experimental data (Electroweak, Drell-Yan, Higgs, Top, and Flavour) to constrain potential NP effects. Previous studies often:

• Focused on specific sectors (e.g., only Higgs or only Flavour).
• Neglected the interplay between different observables.
• Ignored the Renormalization Group Evolution (RGE) of Wilson coefficients from the high NP scale ($\Lambda$) down to experimental scales.
• Did not simultaneously float Standard Model parameters and hadronic uncertainties, leading to underestimated theoretical errors.

This paper addresses these gaps by performing a comprehensive global fit of the Standard Model Effective Field Theory (SMEFT) that integrates all major experimental sectors while consistently handling RGE effects and parameter uncertainties.

2. Methodology

Framework: SMEFT and Flavour Symmetries

The authors utilize the SMEFT framework, extending the SM Lagrangian with dimension-6 operators ($d=6$) suppressed by a cutoff scale $\Lambda$. The Lagrangian is truncated at linear order in Wilson coefficients ($C_i/\Lambda^2$).
Two distinct flavour symmetry assumptions are tested at the UV scale $\Lambda$:

1. $U(3)^5$ Symmetry: Assumes NP is flavour-blind (universal across all three generations). This reduces the number of independent Wilson coefficients to 41 (or 40 excluding $C_\phi$).
2. $U(2)^5$ Symmetry: Assumes NP respects the hierarchy of the first two generations but treats the third generation differently. This is motivated by the SM flavour structure. The analysis considers two bases for defining the third generation:
   • UP Basis: Third generation aligned with up-type Yukawa couplings.
   • DOWN Basis: Third generation aligned with down-type Yukawa couplings.
     This scenario involves 124 independent Wilson coefficients.

Renormalization Group Evolution (RGE)

A key technical feature is the consistent inclusion of leading-order (LO) RGE.

• Wilson coefficients are defined at the UV scale $\Lambda$ and evolved down to the electroweak scale ($\mu_W \approx M_W$) using LO RGE equations.
• At $\mu_W$, the SMEFT is matched to the Low Energy Effective Theory (LEFT) by integrating out heavy SM particles ($W, Z, H, t$).
• LEFT coefficients are further evolved to the specific low-energy scales relevant for flavour observables (e.g., $m_b$).
• This approach accounts for operator mixing and resums large logarithmic corrections ($\ln(\Lambda/\mu)$).

Fitting Procedure

• Framework: The analysis is performed using the open-source HEPfit framework.
• Data Set: The global fit includes:
  • Electroweak (EW): Precision observables at the $Z$-pole (LEP/SLC), $W$ mass and width (LEP2, ATLAS, CMS, Tevatron), and lepton universality tests.
  • Drell-Yan (DY): High-$p_T$ dilepton and single-lepton processes at the LHC.
  • Higgs: Signal strengths, cross-sections, and decay branching ratios (ATLAS/CMS 8 TeV and 13 TeV data).
  • Top Quark: Differential and inclusive cross-sections, charge asymmetries, and associated production ($t\bar{t}Z, t\bar{t}\gamma$, etc.).
  • Flavour: $\Delta F = 2$ observables (meson mixing, CP violation) and $\Delta F = 1$ processes (rare decays like $B \to X_s \gamma$, $B_s \to \mu^+\mu^-$, $K \to \pi \nu \bar{\nu}$).
• Parameter Treatment: Unlike many previous fits, this study floats all relevant SM parameters (quark masses, CKM matrix elements) and hadronic parameters (lattice QCD bag parameters, decay constants) simultaneously with the SMEFT Wilson coefficients.
• Statistical Approach: A Bayesian analysis using Markov Chain Monte Carlo (MCMC) to derive posterior probability densities. Fits are performed for:
  • Individual fits: One Wilson coefficient active at a time.
  • Global fits: All coefficients active simultaneously.
• Benchmark Scales: Results are presented for NP scales $\Lambda = 10$ TeV and $3$ TeV (with RGE) and $\Lambda = 1$ TeV (without RGE).

3. Key Contributions

1. First Comprehensive Global Fit: This is the first study to simultaneously fit EW, Drell-Yan, Higgs, Top, and Flavour observables within a single SMEFT framework while floating SM and hadronic parameters.
2. Consistent RGE Implementation: The paper rigorously implements the evolution of Wilson coefficients from the UV scale to low-energy scales, demonstrating how RGE mixing significantly alters constraints (e.g., Higgs observables constraining gluonic operators via mixing).
3. Flavour Symmetry Comparison: It provides a detailed comparison between the $U(3)^5$ and $U(2)^5$ scenarios, highlighting how relaxing flavour universality introduces new operators (like third-generation dipoles) that are tightly constrained by flavour data.
4. Basis Dependence: The study explicitly compares the "UP" and "DOWN" bases for the $U(2)^5$ scenario, showing how the choice of flavour alignment drastically affects constraints on dipole operators.
5. Quantification of Correlations: The analysis maps the complex correlation structure between Wilson coefficients in global fits, identifying "flat directions" that are lifted by specific combinations of observables.

4. Key Results

$U(3)^5$ Scenario (Flavour Blind)

• Individual Fits: Strong constraints are obtained for many operators. For example, $C_{\phi G}$ (Higgs-gluon) is constrained to $\Lambda/\sqrt{|C|} \gtrsim 19$ TeV, and $C^{(3)}_{lq}$ to $\gtrsim 25$ TeV.
• Global Fits: Correlations between coefficients (e.g., between bosonic operators and four-fermion operators) relax individual bounds. However, meaningful constraints persist:
  • $\Lambda/\sqrt{|C_{\phi G}|} \gtrsim 14$ TeV.
  • $\Lambda/\sqrt{|C^{(3)}_{lq}|} \gtrsim 5$ TeV.
  • Other coefficients yield bounds in the 2–3 TeV range.
• RGE Impact: RGE significantly tightens constraints for operators like $C_G$ and $C_W$, which mix into Higgs and EW observables.

$U(2)^5$ Scenario (Third Generation Special)

• Flavour Constraints: The inclusion of flavour observables is decisive. Operators involving the third generation (e.g., $C^{[33]}_{dB}, C^{[33]}_{dW}, C^{[33]}_{dG}$) are constrained to extremely high scales (tens of TeV) by processes like $B \to X_s \gamma$.
• Basis Dependence:
  • In the UP basis, dipole operators are strongly constrained by flavour data.
  • In the DOWN basis, these constraints are relaxed by nearly two orders of magnitude because the flavour misalignment suppresses contributions to $b$-quark transitions.
• Global Fit Challenges: The global fit for $U(2)^5$ involves $\sim 200$ parameters. Strong correlations and the large parameter space lead to a "dilution" of sensitivity. Many coefficients cannot be constrained individually in the global fit, often hitting the perturbativity prior ($4\pi$).
• Protection Mechanism: For operators modifying neutral/charged currents of light quarks ($C^{(1),(3)}_{\phi q}$), the $U(2)^5$ symmetry provides protection, yielding bounds comparable to the $U(3)^5$ case. However, for third-generation specific interactions, flavour data provides the dominant constraint.

5. Significance and Conclusion

• Validation of SMEFT: The results confirm that for generic strongly-coupled NP scenarios, current data pushes the NP scale well above the TeV scale ($\gtrsim 10$ TeV), ensuring the internal consistency of the SMEFT expansion.
• Role of Flavour: The study underscores that flavour observables are not just complementary but often the dominant constraint for operators involving the third generation, especially in non-universal NP scenarios.
• Future Prospects: The paper highlights that while current data is powerful, the "dilution" effect in global fits with many parameters suggests a need for:
  • Higher precision in theoretical predictions (especially for hadronic parameters).
  • New observables to break degeneracies in the global fit.
  • Improved experimental precision in the flavour sector.
• Methodological Benchmark: By providing a state-of-the-art framework that includes full RGE, floating SM parameters, and a broad spectrum of observables, this work sets a new standard for future SMEFT global analyses.

In summary, this paper demonstrates that a global, consistent approach is essential to accurately probe New Physics. It reveals that while some operators are tightly constrained, the interplay of correlations and the specific flavour structure of NP can significantly alter the reachable sensitivity, necessitating a multi-pronged experimental and theoretical effort.