The effect of the two-loop SMEFT RGEs at future colliders

This paper presents a comprehensive analysis of the impact of newly computed two-loop SMEFT Renormalisation Group Equations on future collider phenomenology, demonstrating through both bottom-up global fits and top-down model matching that these higher-order corrections induce non-negligible effects, particularly for specific operators and model couplings, thereby altering sensitivity predictions for the HL-LHC and FCC-ee.

The Gist

Imagine you are trying to predict the weather for next year. You have a very good model (the Standard Model of physics) that works perfectly for today's weather. But you suspect there are invisible forces—like New Physics—that might change the climate in the future.

To find these invisible forces, scientists use a tool called SMEFT (Standard Model Effective Field Theory). Think of SMEFT as a giant, complex spreadsheet. This spreadsheet tracks hundreds of "knobs" (called Wilson Coefficients) that, if turned, would signal the presence of new particles or forces.

The problem is, these knobs don't stay still. As you zoom out from the tiny world of atoms to the massive scale of particle colliders (like the HL-LHC or the future FCC-ee), these knobs interact with each other. They "mix" and "run" (change value) as you change the energy scale. This process is governed by Renormalization Group Equations (RGEs).

For a long time, scientists only calculated this mixing using a "one-loop" approximation. Think of this as calculating the weather using a simple map that shows major highways but ignores the side streets, traffic lights, and detours. It's a good first guess, but as our weather stations (colliders) become incredibly precise, we need a better map.

This paper is about drawing that better map: the "Two-Loop" map.

Here is what the authors did, explained through simple analogies:

1. The "Two-Loop" Upgrade: From Highways to Side Streets

In the world of particle physics, a "loop" represents a level of complexity in the calculation.

• One-Loop: You know that if you turn Knob A, Knob B changes because they are directly connected.
• Two-Loop: You realize that Knob A also changes Knob C, which then changes Knob B. It's a chain reaction.

The authors took the brand-new, complete mathematical formulas for these two-loop interactions and plugged them into their simulation. They found that the "side streets" (the two-loop effects) are not just minor detours; in some cases, they completely change the traffic flow.

2. Breaking the "Zero" Rules

In the old "One-Loop" map, there were many places where the spreadsheet said, "Nothing happens here." These were the "Zeroes."

• The Discovery: When they added the two-loop calculations, they found that hundreds of these "Zeroes" were actually fake.
• The Analogy: Imagine a traffic light that used to be permanently red (zero traffic). The new map reveals that if you wait long enough, or look at the right time of day, a few cars do actually pass through.
• The Result: Operators (knobs) that were previously thought to be isolated are actually talking to each other. For example, a knob related to the Higgs boson is now found to be secretly influencing a knob related to four-quark interactions.

3. The "Bottom-Up" Test: Fitting the Puzzle Pieces

The authors tested this new map against future data from the HL-LHC (a super-charged version of the Large Hadron Collider) and the FCC-ee (a future electron-positron collider).

• The "Individual" Fit: When they looked at single knobs one by one, the new map made the predictions slightly sharper. It was like adjusting the focus on a camera lens; the picture got a bit clearer, especially for knobs related to the Top Quark (the heaviest known particle) and the Higgs boson.
• The "Global" Fit (The Big Surprise): When they looked at all the knobs together (like trying to solve a giant jigsaw puzzle), the two-loop map caused a major shift.
  • The "Dilution" Effect: They found that for some knobs (like the one controlling how the Higgs interacts with gluons, $c_{\phi G}$), the constraints got weaker (the error bars got bigger).
  • Why? Because the new "side streets" allowed a poorly known knob (four-quark interactions) to "sneak in" and mess with the well-known knob. It's like trying to weigh a gold bar, but you realize the scale is also secretly weighing a hidden bag of sand attached to it. You can no longer be as sure about the weight of the gold.
  • The "Sharpening" Effect: Conversely, for other knobs, the new connections helped break "degeneracies" (situations where two different settings looked the same). This made the measurements for those specific knobs tighter and more precise.

4. The "Top-Down" Test: Looking for the Source

Finally, they asked: "If we assume a specific new theory exists (like a heavy new particle), does this new map help us find it?"

• They looked at a "dictionary" of possible new theories (the Granada Dictionary).
• The Result: For most theories, the two-loop map didn't change much. But for specific models involving heavy scalars (like a second Higgs boson), the new map improved their ability to constrain the theory's parameters by about 2–5%.
• The "Loop" Surprise: They also found that they could now detect couplings that only appear at the "one-loop" level (very subtle effects). This is like being able to hear a whisper in a noisy room that you previously couldn't hear at all.

The Takeaway

This paper is a "reality check" for the future of particle physics.

• Old View: "One-loop calculations are good enough; the two-loop stuff is just tiny noise."
• New View: "The two-loop stuff is not just noise. It changes the structure of the theory, breaks old assumptions, and significantly alters how precise our future measurements will be."

In summary: The authors built a more detailed, high-definition map of how particle physics parameters evolve. They found that this new map reveals hidden connections that can either blur our view of certain mysteries or bring others into sharp focus. As we build bigger, more powerful colliders, we can no longer ignore these "two-loop" side streets; they are essential for navigating the road to New Physics.

Technical Summary

1. Problem Statement

The search for New Physics (NP) via precision measurements requires theoretical predictions that match the increasing experimental precision of facilities like the High-Luminosity LHC (HL-LHC) and the Future Circular Collider (FCC-ee). Within the Standard Model Effective Field Theory (SMEFT) framework, Renormalization Group Equations (RGEs) are essential for evolving Wilson Coefficients (WCs) from a high-energy scale (where NP is generated) down to the electroweak scale (where measurements occur).

While one-loop SMEFT RGEs have been known for over a decade and are standard in global fits, the complete two-loop SMEFT RGEs for dimension-six operators were only recently computed. The central question addressed by this work is: Are two-loop corrections to operator running and mixing non-negligible in global SMEFT fits for future colliders? Specifically, do they alter mixing patterns, break degeneracies present at one-loop, and significantly impact sensitivity to UV parameters?

2. Methodology

The authors employed a comprehensive numerical approach combining theoretical computation with phenomenological fitting:

• Implementation of Two-Loop RGEs:

  • They obtained the two-loop RGE matrix from recent literature (in the Mainz basis) and translated it to the Warsaw basis.
  • They implemented this matrix into a modified version of DsixTools and subsequently rotated it to the SMEFiT basis (used for global fits).
  • The system was linearized by decoupling SM parameter running (evolved at five-loop order) from the dimension-6 WCs.
  • Numerical integration was performed from a high scale of $\mu_0 = 10$ TeV down to observable scales ($m_Z$, HL-LHC, FCC-ee energies).
  • Assumptions: Flavor symmetry $U(2)_q \times U(2)_u \times U(3)_d \times (U(1)_\ell \times U(1)_e)^3$ (with specific relaxations for global fits), retaining top Yukawa, Higgs self-coupling, and gauge couplings, while setting other Yukawas to zero.

• Phenomenological Analysis (Bottom-Up):

  • Individual Fits: Performed linear ($O(\Lambda^{-2})$) and quadratic ($O(\Lambda^{-4})$) fits for 61 independent Wilson coefficients using projected data from HL-LHC and FCC-ee.
  • Global Fits: Conducted a global fit with all 61 operators active simultaneously to assess correlations and degeneracy breaking.
  • Datasets: Combined current LHC measurements with aggressive projections for HL-LHC (single/double Higgs, top, dibosons) and FCC-ee (Z-pole, W-pair, Higgs, top).

• Phenomenological Analysis (Top-Down):

  • Analyzed scalar and fermion extensions of the SM catalogued in the Granada dictionary.
  • Performed matching at the one-loop level (including couplings that enter only via one-loop matching, previously unconstrained).
  • Compared sensitivity to UV couplings under three scenarios: No RGE, One-loop RGE, and Two-loop RGE.

3. Key Contributions

• First Systematic Assessment: This is the first study to quantitatively assess the impact of the complete two-loop SMEFT RGEs on global fits and specific UV models.
• Structural Analysis of Mixing: Detailed mapping of how two-loop corrections break "zeroes" present in the one-loop mixing matrix, revealing new operator connections.
• Granada Dictionary Extension: For the first time, constraints were placed on UV couplings that enter SMEFT matching only at the one-loop level (e.g., specific quartic scalar couplings).
• Scheme Dependence Note: The authors highlight that certain two-loop mixing effects (specifically involving $\gamma_5$ schemes) are scheme-dependent, noting that some effects (like $c_{QQ} \to c_{\phi G}$) vanish in the BMHV scheme.

4. Key Results

A. Structural Impact of Two-Loop RGEs

• Breaking of Zeros: The two-loop matrix retains 924 exact zeros (approx. 1/3 of the one-loop zeros). 1,896 previously vanishing entries acquire non-zero contributions.
• New Mixings: Notable new mixings include:
  • $c_{\phi G}$ mixing into four-heavy-quark operators (and vice versa).
  • Dipole operators ($c_{tG}, c_{tW}, c_{tZ}$) mixing into a broad range of four-fermion operators.
  • $c_{\phi t}$ (top Yukawa modifier) mixing into $c_{\phi D}$ and $c_{\phi \square}$.
• Magnitude of Corrections: While many corrections are small ($<1\%$), 68% of non-vanishing one-loop mixings receive corrections $\ge 5\%$, with a median of 27%. Some relative changes are extreme (e.g., 75,000% for specific elements), though these occur in matrix elements that were already small.

B. Bottom-Up Fit Results

• Individual Fits:
  • Two-loop effects generally improve sensitivity (tighten bounds) for specific operators, particularly $c_{\phi t}$ (top Yukawa), $c_{tG}$, and four-quark operators like $c_{QQ}^8$ and $c_{Qt}^8$.
  • Improvements range from 7% to 25% depending on the operator and collider (FCC-ee often shows larger gains due to Z-pole precision).
  • Quadratic fits show similar trends, as most coefficients remain in the linear regime.
• Global Fits:
  • Degeneracy Breaking: The most striking result is the loosening of bounds on $c_{\phi G}$ by a factor of $\sim 4$ at HL-LHC and 50% at FCC-ee. This is caused by two-loop mixing of poorly constrained four-quark operators into $c_{\phi G}$, creating new correlations that dilute the constraint.
  • Improved Sensitivity: Conversely, bounds on $c_{\phi D}$ and $c_{\phi t}$ tighten significantly (by 43–50%) because two-loop running breaks degeneracies among operators contributing to Electroweak Precision Observables (EWPOs).
  • Four-Heavy Quark Sector: Sensitivity improves at HL-LHC but often degrades at FCC-ee due to complex correlation patterns absorbing constraining power.

C. Top-Down (UV Model) Results

• Granada Dictionary: Two-loop RGEs improve sensitivity to specific UV couplings by 2–5%.
  • Significant improvements are seen for couplings generating contributions to $c_{\phi t}$, four-heavy-quark operators, and $c_{Q\ell}^{(-)}$.
  • Loop-Only Couplings: Remarkably, the study finds sensitivity to quartic scalar couplings that enter SMEFT only at one-loop matching (e.g., $L^{(2)}_\phi$, $(L^{(6)}_{\Xi_1})_{33}$), often below the naive perturbativity limit ($4\pi$). This demonstrates the unique power of FCC-ee to probe loop-induced NP.
• Specific Models:
  • 2HDM (Heavy Scalar Doublet): Two-loop RGEs provide a modest but visible improvement in sensitivity to the top-Yukawa-like coupling $(y_u^\phi)_{33}$ and the quartic $\lambda_\phi$, particularly when combined with one-loop matching.
  • $U + Q_1$ Vector-Like Fermions: No significant advantage from two-loop RGEs was found for this model; the sensitivity gain from one-loop RGEs was already dominant, and the richer dataset saturated the improvement.

5. Significance and Conclusion

• Necessity of Two-Loop Precision: The study confirms that two-loop RGEs are not merely formal corrections; they are phenomenologically relevant. They can significantly alter the interpretation of data, either by tightening constraints (breaking degeneracies) or loosening them (introducing new correlations).
• Impact on Future Colliders: For the FCC-ee and HL-LHC, ignoring two-loop effects could lead to biased interpretations of Wilson coefficients, particularly for gluon-Higgs operators ($c_{\phi G}$) and top-Yukawa modifiers.
• Guidance for Future Work: The authors conclude that a fully consistent two-loop analysis requires two-loop predictions for observables and two-loop matching conditions for UV models. However, this work establishes that two-loop RGEs must be included in the "standard" toolkit for precision SMEFT analyses to avoid systematic errors in the extraction of New Physics parameters.
• Scheme Dependence: The results highlight the importance of consistent renormalization schemes (e.g., $\overline{MS}$ vs. BMHV) when interpreting two-loop mixing effects, as some induced mixings are scheme-dependent.