$b \to c$ semileptonic sum rule: exploring a sterile neutrino loophole

This paper investigates whether a massive sterile neutrino could explain the discrepancy between $B$-meson and $\Lambda_b$-baryon decay anomalies by revisiting the $b \to c$ semileptonic sum rule, concluding that the induced effects are too small to resolve the tension and thus the sum rule remains a robust consistency check for experimental data.

The Gist

Imagine the universe is a giant, complex puzzle, and physicists have been trying to fit the pieces together for decades. The "Standard Model" is the picture on the puzzle box—it's our best guess at how everything works. But recently, some pieces (specifically, how certain heavy particles called B-mesons decay) haven't seemed to fit quite right. They are wobbling a bit, suggesting there might be a hidden piece we haven't found yet.

This paper is like a group of detectives (the authors) checking if a specific "hidden piece"—a Sterile Neutrino—could be the culprit causing the puzzle to look broken.

Here is the story of their investigation, broken down into simple concepts:

1. The Mystery: The "Wobbly" Puzzle Pieces

In the world of particle physics, there are particles called B-mesons (the heavy ones) that decay into lighter particles. Sometimes they turn into a D-meson or a Lambda-b baryon (a cousin of the proton).

• The Problem: When these B-mesons decay, they usually shoot out a "tau" particle and a "neutrino" (a ghostly particle that barely interacts with anything).
• The Clue: Experiments show that B-mesons turning into D-mesons happen more often with tau particles than our standard rules predict. It's like a coin that is supposed to be fair landing on "Heads" 60% of the time.
• The Twist: However, when B-mesons turn into the "cousin" particle (Lambda-b), the coin seems fair. It lands exactly where the rules say it should.

This creates a tension: Why is one type of decay "wobbly" while the other is "steady"?

2. The Suspect: The "Sterile Neutrino"

Physicists suspect a new type of particle might be hiding in the shadows. They call it a Sterile Neutrino.

• The Analogy: Imagine a standard neutrino is a ninja who is invisible but leaves footprints (interacts with other particles). A Sterile neutrino is like a ghost that doesn't just leave footprints; it's so "sterile" it doesn't interact with anything at all, except through gravity or very specific new forces.
• The Theory: The authors asked: "What if the B-meson isn't just shooting out a normal neutrino, but sometimes shooting out this heavy, invisible ghost? Could that explain why the numbers look weird?"

3. The Test: The "Sum Rule" (The Accounting Ledger)

To check if this ghost is real, the authors used a mathematical tool called a Sum Rule.

• The Analogy: Think of the B-meson decay like a family budget.
  • You have a total amount of money (the total decay rate).
  • You spend it on three things: D-mesons, D*-mesons, and Lambda-b baryons.
  • The "Sum Rule" is a strict accounting law: If you spend X on D-mesons and Y on D-mesons, you must spend exactly Z on Lambda-b baryons.*
  • In the Standard Model, this math works perfectly. If the "wobbly" D-meson data is real, the Lambda-b data should also be wobbly to keep the books balanced.

But the experiments show the D-meson is wobbly, and the Lambda-b is steady. The books don't balance!

4. The Investigation: Does the Ghost Fix the Ledger?

The authors ran the numbers to see if adding a Sterile Neutrino to the mix could fix this broken accounting.

• The Scenario: They imagined the B-meson decaying into a tau and a heavy sterile neutrino. This changes the energy and speed of the particles, potentially altering the "spending" on the ledger.
• The Result: They crunched the numbers with super-computers and complex formulas. They found that even if this ghostly neutrino exists, its effect on the "Sum Rule" is tiny.
  • The Metaphor: It's like trying to balance a massive ship's ledger by adding a single grain of sand to the cargo. The sand (the sterile neutrino) is there, but it's too light to fix the weight difference caused by the wobbly D-meson data.

5. The Conclusion: The Ledger is Still Broken (But Robust)

The paper concludes that the "Sterile Neutrino" loophole is not the answer to the mystery.

• The Takeaway: The fact that the Sum Rule holds up so well (even with this new ghost particle theory) actually makes the experimental data more reliable. It tells us that the "wobble" in the D-meson data is a real, stubborn problem that cannot be easily explained away by just adding a heavy neutrino.
• The Future: While this specific ghost didn't fix the puzzle, the authors note that if these ghosts do exist, they would leave a unique "fingerprint" on the energy distribution of the decay. Future experiments (like the ones at the Large Hadron Collider) need to look for these specific fingerprints, rather than just counting the total number of decays.

In a Nutshell

The universe is acting a bit weird with how heavy particles decay. Physicists wondered if a new, invisible "ghost" particle (the sterile neutrino) was messing up the math. They did the math and found that the ghost is too weak to explain the weirdness. This means the mystery is still unsolved, but we now know for sure that the "ghost" theory isn't the easy way out. The puzzle pieces are still wobbly, and we need to keep looking for the real solution!

Technical Summary

1. Problem Statement

The paper addresses a current tension in flavor physics regarding Lepton Flavor Universality (LFU).

• The Anomaly: Measurements of the ratios $R_{D^{(*)}} = \mathcal{B}(B \to D^{(*)}\tau \bar{\nu}_\tau) / \mathcal{B}(B \to D^{(*)}\ell \bar{\nu}_\ell)$ show a $\sim 4\sigma$ deviation from Standard Model (SM) predictions.
• The Counterpart: In contrast, the baryonic ratio $R_{\Lambda_c} = \mathcal{B}(\Lambda_b \to \Lambda_c\tau \bar{\nu}_\tau) / \mathcal{B}(\Lambda_b \to \Lambda_c\ell \bar{\nu}_\ell)$ is consistent with the SM.
• The Sum Rule: These observables are linked by a $b \to c$ semileptonic sum rule. In the heavy quark limit (small velocity), the relation is:
  $$\frac{R_{\Lambda_c}}{R_{\Lambda_c}^{SM}} - \frac{1}{4}\frac{R_D}{R_D^{SM}} - \frac{3}{4}\frac{R_{D^*}}{R_{D^*}^{SM}} = \delta$$
  where $\delta$ is expected to be negligible for SM physics. However, substituting current experimental values yields $\delta \simeq -0.41 \pm 0.24$, indicating a mild tension.
• The Hypothesis: The authors investigate whether this tension could be explained by the emission of a massive sterile neutrino ($N_R$) instead of (or in addition to) the standard massless left-handed neutrino. Since neutrinos are not directly detected in these decays, their chirality and mass are unconstrained by current data. If $N_R$ is emitted, it modifies the total decay rates and could potentially violate the sum rule in a way that reconciles the $B$-meson and $\Lambda_b$-baryon data.

2. Methodology

The authors employ an Effective Field Theory (EFT) approach to model New Physics (NP) contributions involving a sterile neutrino.

• Effective Hamiltonian: They introduce dimension-six operators describing the transition $b \to c \tau \bar{N}_R$. The Hamiltonian includes:
  • Vector operators ($O_{V_R}'$)
  • Scalar operators ($O_{S_L}', O_{S_R}'$)
  • Tensor operators ($O_{T}'$)
    The SM contribution corresponds to the limit where all Wilson coefficients (WCs) for these new operators vanish.
• Differential Decay Rates: They derive the full differential decay rates ($d\Gamma/dq^2$) for $B \to D^{(*)}\tau \bar{N}_R$ and $\Lambda_b \to \Lambda_c \tau \bar{N}_R$. These expressions explicitly depend on the sterile neutrino mass ($m_{N_R}$) and the WCs.
• Sum Rule Derivation:
  • They demonstrate that in the heavy quark limit (HQET), the sum rule holds exactly for the differential rates even with a sterile neutrino, provided the Isgur-Wise functions are universal.
  • They extend this to total decay rates by integrating over the phase space.
  • They define a deviation parameter $\delta_{NR}$ for the presence of sterile neutrinos:
    $$\delta_{NR} = \frac{R'_{\Lambda_c}}{R_{\Lambda_c}^{SM}} - \frac{1}{4}\frac{R'_D}{R_D^{SM}} - \frac{3}{4}\frac{R'_{D^*}}{R_{D^*}^{SM}}$$
    where $R'$ includes contributions from both SM neutrinos and sterile neutrinos.
• Constraints: The analysis incorporates constraints from LHC high-$p_T$ mono-$\tau$ searches and $b \to s$ flavor-changing neutral currents (FCNC) arising from $SU(2)_L$ invariance. They consider two scenarios for the NP mass scale: a "relaxed" bound (2 TeV Leptoquark mediator) and a "stringent" bound (EFT limit > 10 TeV).

3. Key Contributions

1. Formalism for Sterile Neutrinos: The paper provides the first comprehensive derivation of the $b \to c$ semileptonic sum rule in the presence of a massive right-handed (sterile) neutrino, including the full dependence on $m_{N_R}$ and the interference between SM and NP amplitudes.
2. Kinematic Threshold Analysis: They explicitly analyze the kinematic limits where the sterile neutrino mass becomes large enough to close specific decay channels (e.g., $B \to D^* \tau \bar{N}_R$ becomes forbidden for $m_{N_R} \gtrsim 1.49$ GeV). They show that while this technically violates the sum rule construction, the phase space suppression renders the effect negligible.
3. Quantitative Evaluation of Violation: The authors calculate the magnitude of $\delta_{NR}$ across the allowed parameter space of Wilson coefficients and sterile neutrino masses.

4. Results

• Magnitude of Violation: The induced violation of the sum rule, $\delta_{NR}$, is found to be negligible compared to current experimental uncertainties.
  • Even with maximal allowed Wilson coefficients (under relaxed LHC constraints), the total deviation $\delta_{NR}$ remains at the level of $\sim 0.1$ or less.
  • In most scenarios, $\delta_{NR}$ is positive. While specific interference terms (e.g., $V_R' T'$) can produce negative contributions, the positive contributions from pure tensor ($T'T'$) and vector ($V_R'V_R'$) terms dominate, keeping the net result positive or very small.
• Mass Dependence:
  • The deviation peaks around $m_{N_R} \approx 0.8 - 1.1$ GeV but drops significantly as $m_{N_R}$ increases due to phase space suppression.
  • For $m_{N_R} > 1.4$ GeV, the deviation becomes extremely small ($\ll 0.01$) because the decay channels are kinematically suppressed.
• Comparison with Experimental Tension: The current experimental tension suggests $\delta \simeq -0.41$. The sterile neutrino mechanism cannot generate a negative $\delta_{NR}$ large enough to resolve this tension. The maximum negative contribution achievable is far smaller than the required $-0.41$.
• Differential Distributions: While the total rates (and thus the sum rule) are robust, the paper notes that the differential decay rates ($d\Gamma/dq^2$) would show distinct kinematic distortions (a shift in the threshold $q^2_{min} = (m_\tau + m_{N_R})^2$) compared to the SM.

5. Significance and Conclusion

• Robustness of the Sum Rule: The primary conclusion is that the $b \to c$ semileptonic sum rule is a robust consistency check. The presence of a massive sterile neutrino does not provide a "loophole" to explain the discrepancy between $R_{D^{(*)}}$ and $R_{\Lambda_c}$. The violation induced by such a particle is too small to account for the observed experimental tension.
• Implications for New Physics: The results suggest that if the $R_{D^{(*)}}$ anomaly is real, it likely stems from New Physics that affects the $B$-meson and $\Lambda_b$-baryon decays differently (e.g., via form factor dependencies or specific operator structures not captured by a simple sterile neutrino emission) or that the experimental measurements require re-evaluation.
• Future Directions: The authors emphasize that while the sum rule remains valid, the differential distributions offer a promising avenue for discovery. A massive sterile neutrino would alter the $q^2$ spectrum of the visible final states. Detecting this would require dedicated experimental analyses of the differential decay rates, accounting for the different kinematics of massive neutrino emission.

In summary, the paper effectively rules out the massive sterile neutrino as a simple explanation for the $R_{D^{(*)}}$ vs. $R_{\Lambda_c}$ tension via the sum rule mechanism, reinforcing the need for more complex New Physics models or further experimental scrutiny.