Renormalization of mixing angles and computation of the hadronic $W$ decay widths

This paper proposes a model- and process-independent On-Shell renormalization scheme that eliminates the need for mixing matrix counterterms by utilizing a basis-free approach formulated entirely through self-energies, demonstrating its consistency by computing 1-loop hadronic $W$-boson decay widths in the Standard Model.

The Gist

Imagine you are trying to describe a complex dance routine involving three different pairs of dancers. In the Standard Model of particle physics, these "dancers" are fundamental particles like quarks, and the "routine" is how they interact and transform into one another.

Sometimes, a dancer doesn't just stay in their own lane; they mix with others. This is called particle mixing. To describe this mathematically, physicists use a "mixing matrix" (like the famous CKM matrix), which acts like a choreography chart telling us the probability of one dancer turning into another.

The Problem: The "Ghost" Counter-Terms

In physics, when we try to calculate these interactions with extreme precision (going beyond the basic rules to include quantum "loops" or corrections), we run into a problem: the numbers blow up to infinity. To fix this, we use a technique called renormalization. Think of renormalization as a way to subtract the infinite "noise" to get a clean, finite signal.

For decades, physicists have been trying to figure out how to "renormalize" the mixing matrix itself. The traditional approach was to treat the mixing angles (the numbers in the choreography chart) as independent variables that need their own "correction terms" (counter-terms).

The Analogy:
Imagine you are trying to fix a blurry photo of a dance. The traditional method says, "The blur is in the camera lens, so let's add a special filter just for the lens."
However, the author of this paper, Simonas Draukšas, argues: "Wait a minute. The blur isn't in the lens; it's in the way the dancers are standing relative to each other. If you just shift the dancers' positions slightly, the blur disappears without needing a special lens filter."

In physics terms, the "mixing" is just an artifact of how we choose to label our particles (our "basis"). If we change our labels, the mixing matrix changes. But the physical reality (the dance) shouldn't change just because we changed the labels. Therefore, trying to add a specific "correction" to the mixing matrix is like trying to correct a shadow by painting the wall; it's the wrong approach.

The Solution: The "No-Mixing" Trick

The paper proposes a clever new way to handle this, which the author calls a variant of the On-Shell scheme.

1. The Core Idea: Instead of trying to fix the mixing matrix (the choreography chart), the author suggests we simply don't fix it at all. We set the correction for the mixing matrix to zero ($\delta V = 0$).
2. How it works: If we don't fix the mixing, the "messiness" (the infinities and gauge dependencies) has to go somewhere else. The author shows that this messiness naturally moves into the masses of the particles.
   • Analogy: Imagine the dancers are wearing heavy, slightly uneven shoes. Instead of trying to fix the choreography chart to compensate for the shoes, we just acknowledge that the shoes themselves are slightly heavier on one side. We adjust the weight of the shoes (the mass counter-term) to keep the dance perfect.
3. The Result: By allowing the "mass counter-terms" to be slightly off-diagonal (meaning a particle's mass correction can slightly affect its neighbor), we absorb all the necessary corrections. The mixing matrix remains "pure" and needs no correction.

This approach is gauge-independent (it works regardless of the mathematical "perspective" you choose) and process-independent (it works for any reaction, not just one specific dance move).

The Test: The W-Boson Dance Floor

To prove this idea works, the author calculated the decay rates of the W-boson (a heavy particle that decays into quarks) using this new method.

• The Experiment: They calculated the probability of the W-boson decaying into different pairs of quarks (like up/down, charm/strange, etc.) using their new "no-mixing" rule.
• The Comparison: They compared their results with the old, traditional methods and other modern proposals.
• The Outcome: The numbers came out almost identical to the other methods.
  • Why this matters: It proves that the new method is numerically sound. It doesn't break the math or give weird answers. It just offers a cleaner, more logical way to get there. It's like finding a new route to the grocery store that takes the same amount of time but avoids a confusing roundabout.

The Big Picture: Why Should You Care?

1. Simplicity: The paper removes a layer of unnecessary complexity. It says, "Don't fix the mixing; fix the masses." This makes the math cleaner and easier to understand.
2. Universality: The method works for any particle mixing, not just quarks. It could be applied to neutrinos or even hypothetical particles in "Beyond the Standard Model" physics.
3. Consistency: It resolves a long-standing debate about whether mixing angles should be treated as physical things that need correction. The author argues they are not physical (they depend on your choice of coordinates), so they shouldn't have their own correction terms.

Summary in One Sentence

The author suggests that instead of trying to "fix" the mathematical chart that describes how particles mix, we should simply ignore the chart's imperfections and fix the particles' masses instead, resulting in a cleaner, more consistent way to calculate how the universe works at the smallest scales.

Technical Summary

1. Problem Statement

The renormalization of particle mixing matrices (such as the CKM matrix in the Standard Model) has historically been problematic. Traditional On-Shell (OS) schemes, which define counterterms based on physical pole masses and residues, often lead to gauge-dependent results for physical decay amplitudes when mixing matrices are involved.

• The Core Issue: Mixing angles are artifacts of basis transformations (rotating from flavor eigenstates to mass eigenstates). Treating them as independent parameters requiring renormalization introduces a specific basis choice, violating basis invariance.
• Previous Attempts: Various schemes have been proposed to remove gauge dependence (e.g., using self-energies at zero momentum, pinch techniques, or imposing specific renormalization conditions on form factors). However, many of these either:
  • Fail to be strictly "On-Shell" (relying on off-shell conditions).
  • Introduce additional, arbitrary field counterterms.
  • Depend on the specific physical process being calculated.
  • Lack numerical stability or flavor symmetry.

The author argues that mixing matrix counterterms ($\delta V$) are fundamentally unphysical and should be zero ($\delta V = 0$). The challenge is to construct a consistent OS scheme where mixing is renormalized without introducing a counterterm for the mixing matrix itself.

2. Methodology

The paper proposes a variant of the On-Shell scheme based on the principle that basis rotations should not be renormalized. Instead of renormalizing the mixing matrix, the scheme renormalizes the mass matrices directly, allowing for non-diagonal mass counterterms.

Key Theoretical Steps:

1. Basis Invariance Argument:

   • The author demonstrates that mixing matrices arise solely from diagonalizing mass matrices. Since basis rotations are non-physical, the bare mixing matrix should equal the renormalized one ($\hat{V} = V \implies \delta V = 0$).
   • Consequently, the effects of mixing must be absorbed into field renormalization and mass counterterms.

2. Electroweak Sector (Gauge Bosons):

   • Instead of renormalizing the Weinberg angle ($\theta_W$) via a counterterm $\delta \sin \theta_W$, the author introduces an off-diagonal mass counterterm ($\delta m^2_{AZ}$) in the $Z$-$\gamma$ mass matrix.
   • This off-diagonal term replaces the need for a Weinberg angle counterterm, ensuring the scheme remains strictly On-Shell and gauge-independent.

3. Fermion Sector (Quarks) - The Core Innovation:

   • In the quark sector, the On-Shell condition for the propagator creates a degeneracy between field renormalization counterterms ($\delta Z$) and mass counterterms ($\delta m$).
   • To break this degeneracy without introducing a mixing matrix counterterm, the author employs a fine-grained decomposition of the fermion self-energies ($\Sigma$).
   • Decomposition: The self-energies are expanded in terms of external masses ($m_i, m_j$) to isolate specific mass structures ($m_i - m_j$).
   • UV Subtraction: The author identifies that certain scalar functions in this decomposition contain UV divergences that must be subtracted to ensure the field renormalization counterterm is well-defined. This subtraction is performed by setting loop masses to zero in specific terms, a procedure shown to be gauge-independent.

4. Practical Prescription:

   • The paper derives a universal formula for the anti-hermitian part of the field renormalization counterterm ($\delta Z^A_{ji}$) for $i \neq j$.
   • This formula is expressed entirely in terms of self-energies evaluated on-shell.
   • Crucially, this prescription ensures that the CKM counterterm $\delta V$ is identically zero, while all gauge dependencies in the physical amplitude are canceled by the specific structure of $\delta Z^A$.

3. Key Contributions

• Model-Independent Prescription: A practical, universal formula (Eq. 2.95) for the anti-hermitian part of field renormalization that eliminates the need for mixing matrix counterterms.
• Consistent On-Shell Scheme: The scheme satisfies all standard requirements:
  • Gauge Independence: Physical amplitudes are independent of the gauge parameter $\xi$.
  • Process Independence: The counterterms are defined solely by 2-point functions (self-energies), not by specific decay processes.
  • Flavor Democracy: The treatment is symmetric with respect to flavors.
  • UV Finiteness: The resulting counterterms are UV finite (in the SM).
• Application to Weinberg Angle: The same logic is applied to the electroweak sector, replacing the Weinberg angle counterterm with an off-diagonal mass counterterm.

4. Numerical Results

The author computed the 1-loop hadronic $W$-boson decay widths ($W^+ \to u\bar{d}, c\bar{s}$, etc.) in the Standard Model to validate the scheme.

• Comparison: The results were compared against:
  • Traditional OS schemes (Ref [6]).
  • Alternative gauge-independent schemes (Refs [7, 12, 13, 20]).
  • $\overline{MS}$ schemes.
  • A scheme with no mixing on external legs.
• Findings:
  • The numerical differences between the proposed scheme and other established schemes are negligible (differences appear only in the 7th decimal place or higher).
  • The scheme produces results consistent with the precision of current calculations.
  • The electroweak corrections ($\delta_{EW}$) and QCD corrections ($\delta_{QCD}$) were calculated, showing that the new scheme handles gauge-dependent terms correctly, unlike the traditional scheme of Ref [6] which retains gauge dependence in the mixing counterterm.

5. Significance

• Theoretical Consistency: This work resolves the long-standing issue of gauge dependence in mixing matrix renormalization by adhering strictly to the principle of basis invariance. It proves that mixing matrices do not require their own counterterms.
• Practical Utility: The paper provides a "ready-to-use" prescription for calculating higher-order corrections in models with particle mixing (e.g., extended Higgs sectors, neutrino mixing) without the ambiguity of choosing a specific mixing scheme.
• Foundation for BSM Physics: By establishing a robust, process-independent framework for mixing, the scheme is particularly valuable for Beyond Standard Model (BSM) physics where mixing patterns may be more complex or non-unitary.
• Validation: The numerical agreement with existing literature confirms that the theoretical elegance of the "no mixing counterterm" approach does not come at the cost of numerical accuracy.

In summary, Draukšas presents a rigorous, gauge-invariant, and numerically stable On-Shell renormalization scheme that treats mixing angles as non-renormalized parameters, shifting the necessary counterterms to the mass and field sectors via a novel decomposition of self-energies.