Interpreting the results on exclusive $c\rightarrow s\mu\nu$ modes

This paper argues that while a complex New Physics coupling proposed to explain mild deviations in $D\to K\mu\nu$ data conflicts with LHC high-$p_T$ constraints, other viable scenarios exist that are too subtle for integrated observables but may be detectable via $q^2$-binned angular distributions in specific $D_s$ and $\Lambda_c$ decay modes.

The Gist

The Big Picture: A Mystery in the Particle Zoo

Imagine the Standard Model of physics as the ultimate rulebook for how the universe's tiny building blocks (particles) behave. For decades, this rulebook has been incredibly accurate. But recently, a team of scientists at the BESIII experiment (a giant particle detector in China) looked at a specific event: a heavy particle called a D-meson decaying (breaking apart) into a lighter particle (a Kaon) and a muon (a heavy cousin of an electron).

When they looked at the details of how this happened—specifically, how the energy was distributed across different "bins" (like sorting marbles by size)—they found a slight glitch. The data didn't quite match the rulebook. It was off by about 2 sigma (which is like rolling a die and getting a "6" when you expected a "1" a few times in a row; it's suspicious, but not a slam-dunk proof of a new law of physics).

The First Attempt: The "Complex" Fix

The BESIII team suggested a fix: maybe there is a hidden "New Physics" force acting on these particles. They proposed that this new force is described by a complex number (a mathematical concept involving imaginary numbers). Think of this like trying to fix a wobbly table leg by adding a shim that has both a real thickness and an "imaginary" twist.

The Authors' Reaction:
The authors of this paper (Bečirević, Martines, et al.) said, "Hold on a second." They ran the numbers and found a problem.

They compared this "complex number fix" against data from the Large Hadron Collider (LHC) in Europe. The LHC smashes protons together at incredibly high speeds. If this "complex twist" existed, it would leave a huge, obvious fingerprint in the high-energy debris from the LHC collisions.

The Verdict: The LHC data acts like a security camera. It shows that the "complex number" solution would have triggered an alarm. Since the alarm didn't go off, the "complex number" fix is likely wrong. It's like trying to explain a missing cookie by saying you ate it, but the security camera clearly shows you never entered the kitchen.

The Second Attempt: The "Real Number" Compromise

If the "complex" fix is out, what's left? The authors suggest we look at real numbers (normal numbers, no imaginary twists). They propose that instead of one magic number fixing everything, we need two different new forces working together to explain the glitch.

They tested many combinations of these two forces. They found that while some combinations could mathematically explain the BESIII data without breaking the LHC rules, there's a catch: The forces have to be incredibly weak.

The Analogy:
Imagine the Standard Model is a loud rock concert. The BESIII data suggests there's a faint whisper in the crowd that doesn't fit the music.

• The "Complex" fix was like shouting a new lyric to cover the whisper, but the security guard (LHC) would have heard the shout and stopped it.
• The "Real Number" fix is like whispering a counter-melody. It fits the rules, but the whisper is so quiet that unless you have the most sensitive microphone in the world, you'll never hear it.

The Future: Where to Look Next?

So, is the mystery solved? Not really. The authors conclude that while these "weak whisper" scenarios are possible, they are too subtle to be seen with current tools that measure total averages (like counting the total number of cookies eaten).

However, there is a glimmer of hope. The authors suggest looking at very specific, detailed patterns in other particle decays, specifically:

1. $D_s \to \phi \mu \nu$: A decay involving a phi meson.
2. $\Lambda_c \to \Lambda \mu \nu$: A decay involving a Lambda baryon.

In these specific cases, the "whisper" might be loud enough to be heard if we look at the angular distribution (the direction the particles fly) rather than just the total count. It's like realizing that while you can't hear the whisper in a noisy room, you can hear it if you look at the specific pattern of dust motes dancing in a sunbeam.

Summary for the General Audience

1. The Glitch: Scientists saw a small mismatch in how a particle decays, suggesting new physics might exist.
2. The Failed Fix: A proposed solution involving "imaginary" math was ruled out because it would have been spotted by the massive LHC collider.
3. The New Theory: The mismatch might be caused by two very weak, "real" new forces working together.
4. The Problem: These forces are so weak that current experiments can't see them directly.
5. The Hope: Future experiments (like the High-Luminosity LHC) and more detailed studies of specific particle angles might finally catch a glimpse of these hidden forces.

In a nutshell: The universe might be hiding a tiny secret, but it's hiding so well that we need better microscopes and a lot more patience to find it. The "easy" explanations have been ruled out, leaving us with a more subtle, harder-to-detect mystery.

Technical Summary

1. Problem Statement

The paper addresses a recent anomaly reported by the BESIII collaboration regarding the exclusive semileptonic decays $D \to K \mu \nu$. While the total branching fractions and integrated forward-backward asymmetries ($A_{FB}$) are consistent with Standard Model (SM) predictions, the differential decay rate binned in $q^2$ (the squared four-momentum transfer) shows a deviation of nearly 2$\sigma$ from the SM.

BESIII initially proposed that this discrepancy could be resolved by introducing a complex-valued New Physics (NP) scalar coupling ($g_S$). The authors of this paper investigate whether this proposed solution is viable when subjected to constraints from both low-energy precision data and high-energy Large Hadron Collider (LHC) data.

2. Methodology

The authors employ a multi-faceted approach combining Low-Energy Effective Field Theory (LEFT), Standard Model Effective Field Theory (SMEFT), and global data analysis:

• Effective Lagrangian: They utilize the LEFT Lagrangian for $c \to s \mu \nu$ transitions, including dimension-six operators: Vector ($V_L, V_R$), Scalar ($S_L, S_R$), and Tensor ($T$).
• Low-Energy Constraints:
  • They re-analyze the BESIII $q^2$-binned data for $D^0 \to K^- \mu^+ \nu_\mu$ and $D^+ \to \bar{K}^0 \mu^+ \nu_\mu$.
  • They incorporate constraints from the purely leptonic decay $D_s \to \mu \nu$ and the branching fractions of $D \to K \mu \nu$.
  • They utilize form factors from three major Lattice QCD (LQCD) collaborations (ETMC, FNAL/MILC, HPQCD) to ensure theoretical robustness.
• High-Energy Constraints (LHC):
  • They match the LEFT couplings to SMEFT operators at the electroweak scale ($\mu \approx m_W$) and run them down to $\mu = 2$ GeV using renormalization group equations (RGE).
  • They constrain the NP couplings using LHC data on mono-lepton Drell-Yan processes ($pp \to \mu \nu + \text{jets}$), which are sensitive to four-fermion operators ($\psi^4$).
  • They utilize $Wh$ production data to constrain Higgs-current operators ($\psi^2 DH^2$), which specifically affect right-handed vector currents ($g_{VR}$).
• Scenario Testing: They test the viability of the BESIII proposal (complex $g_S$) and explore scenarios where two real-valued NP couplings are turned on simultaneously to satisfy all constraints.

3. Key Contributions and Results

A. Refutation of the Complex Scalar Solution

The authors demonstrate that the BESIII proposal of a complex scalar coupling ($g_S = g_{SL} + g_{SR}$) to explain the $D \to K \mu \nu$ anomaly is incompatible with LHC constraints.

• While a complex $g_S$ can fit the low-energy $q^2$ distribution, the required magnitude of the imaginary part ($\text{Im}(g_S)$) violates the bounds derived from the high-$p_T$ tail of the Drell-Yan process ($pp \to \mu \nu$).
• The tension between the low-energy fit and the high-energy bound is approximately 2$\sigma$, ruling out this specific solution.

B. Compatibility of Real-Valued Coupling Scenarios

The authors show that if the NP couplings are restricted to be real numbers, there are several plausible scenarios where two couplings are non-zero simultaneously. These scenarios can accommodate the low-energy $q^2$ deviation while remaining consistent with:

1. The integrated branching fractions and $A_{FB}$.
2. The leptonic decay $D_s \to \mu \nu$.
3. The high-energy LHC constraints (Drell-Yan and $Wh$ production).

C. Constraints on Specific Operators

• Right-Handed Vector Current ($g_{VR}$): This coupling is uniquely constrained by $Wh$ production data at the LHC, as it induces energy-enhanced contributions in this channel.
• Tensor and Scalar Couplings: The combination of LHC Drell-Yan limits and low-energy leptonic constraints severely restricts the allowed parameter space.
• Magnitude of Effects: The viable regions for the NP couplings (within the 2$\sigma$ allowed bands) are extremely small. Consequently, these couplings are too small to generate significant deviations in integrated observables (total rates or average asymmetries).

D. Future Prospects and Angular Observables

Since integrated observables are insensitive to these tiny couplings, the authors propose that the only viable way to test these scenarios is through differential $q^2$-binned angular observables in decays involving vector mesons or baryons:

• $D_s \to \phi(\to K K) \mu \nu$: The angular coefficients in the $q^2$ distribution could exhibit deviations of order 10% in the high-$q^2$ region.
• $\Lambda_c \to \Lambda(\to p \pi) \mu \nu$: Similar angular analyses could provide necessary discrimination.
• HL-LHC: The High-Luminosity LHC (HL-LHC) will significantly tighten constraints on these couplings (projected reach shown in Fig. 5), particularly for Drell-Yan and $Wh$ processes, potentially ruling out or confirming these subtle NP scenarios.

4. Significance

• Resolution of the Anomaly: The paper clarifies that the BESIII anomaly cannot be explained by a simple complex scalar coupling, a popular BSM hypothesis, due to high-energy constraints.
• Interplay of Scales: It highlights the critical necessity of combining low-energy flavor physics with high-energy collider data (SMEFT approach) to fully constrain New Physics. A solution that works at low energies often fails at high energies due to the energy-enhanced behavior of certain operators.
• Guidance for Future Experiments: The work shifts the focus from integrated branching ratios to differential angular distributions in specific decay modes ($D_s \to \phi \mu \nu$ and $\Lambda_c \to \Lambda \mu \nu$) as the most promising avenue for discovering these subtle NP effects.
• Theoretical Rigor: By utilizing multiple LQCD form factor sets and rigorous RGE running, the authors ensure that the tension between theory and experiment is not an artifact of theoretical uncertainties.

In summary, the paper concludes that while the $D \to K \mu \nu$ anomaly is real, the simplest BSM explanations are ruled out by LHC data. Any viable New Physics must be subtle, involving real-valued couplings that are only detectable through high-precision, differential angular measurements in future experiments.