Belle II Constraints on the Non-Minimal Universal Extra Dimensional Model

This paper interprets recent Belle II measurements of the $B^+\to K^+ \nu \bar{\nu}$ decay within a Non-minimal Universal Extra Dimensional model, finding that the data elevates the lower limit on the inverse compactification radius to approximately 900 GeV, whereas a variant of the model with nullified boundary terms fails to yield a comparable constraint.

The Gist

Imagine the universe as a giant, multi-story building. For decades, physicists have been convinced that this building only has one floor: the "Standard Model" floor, where all the known particles (like electrons and quarks) live and interact. But recently, a group of scientists at the Belle II laboratory in Japan looked at a very specific, rare event: a heavy particle called a B-meson decaying (falling apart) into a lighter particle and a pair of invisible "ghosts" (neutrinos).

They found something strange. The B-meson was doing this more often than the "one-floor" building rules predicted. It was like seeing a car drive through a wall that was supposed to be solid. This suggests there might be a hidden second floor, or even a whole extra dimension, that we can't see directly but can feel through these rare events.

This paper is an investigation into that possibility, using a specific blueprint called the Non-Minimal Universal Extra Dimensional (NMUED) model. Here is how the authors break it down, using simple analogies:

1. The "Hidden Floor" and the "Ghost" Particles

In this model, our universe has a tiny, curled-up extra dimension (like a very thin hose). If you zoom in enough, you'd see that particles can vibrate along this hose.

• The Zero-Mode: This is the particle we know and love (like a standard electron). It's the "ground floor" vibration.
• The KK-States (Kaluza-Klein modes): These are the "upper floors." Every time a particle vibrates up a level in this extra dimension, it becomes a heavier, copycat version of itself. These are the KK-states.
• The Problem: In the simplest version of this theory (called Minimal UED), all these copies are almost exactly the same weight. It's like a staircase where every step is the same height. This makes it hard to distinguish them in experiments.

2. The "Renovation" (Boundary Terms)

The authors of this paper are looking at a "renovated" version of the building called NMUED.

• Imagine the ends of that extra-dimensional hose (the boundaries) have been reinforced with special heavy weights.
• These weights are called Boundary Localized Terms (BLTs).
• The Effect: These weights change how the particles vibrate. Some "upper floor" copies become much heavier, while others become lighter. It's like adding heavy furniture to specific steps of the staircase, making the climb feel very different depending on where you are.

3. The Investigation: The B-Meson Mystery

The Belle II experiment saw the B-meson decaying into neutrinos more often than expected. The authors asked: "Could the hidden 'upper floor' particles (KK-states) be helping the B-meson decay faster?"

To answer this, they had to do some heavy math (calculating "loop diagrams," which are like complex detours particles take). They calculated how the presence of these extra-dimensional copies, influenced by the "renovation weights" (BLTs), would change the decay rate.

4. The Findings: How Heavy is the Building?

The main goal was to figure out how "tight" the extra dimension is curled up. This is measured by a value called $R^{-1}$ (the inverse of the radius).

• Think of $R^{-1}$ as the "stiffness" of the extra dimension. A high number means the dimension is very small and stiff; a low number means it's larger and looser.
• The Result:
  • If the "renovation weights" (BLTs) are set to specific, non-zero values, the math shows that the extra dimension must be quite stiff. The authors found a "safety limit": the dimension cannot be looser than a certain point, or the B-meson would decay too fast, contradicting the data.
  • They calculated that the "stiffness" ($R^{-1}$) must be at least around 900 GeV (a unit of energy/mass) for certain settings. This pushes the limit higher than some previous guesses.
  • The Twist: However, if they turned off the "renovation weights" (setting the BLTs to zero, returning to the simple, unrenovated model), the math failed to give a limit. In that simple case, the B-meson data didn't rule out any size for the extra dimension. The "renovation" was actually necessary to make the theory testable against this specific data.

5. The Conclusion

The paper concludes that:

1. The recent Belle II data is a powerful tool for testing these extra-dimensional theories.
2. The "Non-Minimal" version (with the boundary weights) can explain the data, but it forces the extra dimension to be quite small and heavy (high $R^{-1}$).
3. The "Minimal" version (without the weights) cannot be ruled out or confirmed by this specific data alone; it leaves the door open for the extra dimension to be almost any size.

In short: The authors used a rare particle decay as a magnifying glass to look for a hidden dimension. They found that if that dimension exists and has "special weights" at its edges, it must be very small and heavy. If it doesn't have those weights, this specific experiment can't tell us how big it is.

Technical Summary

Technical Summary: Belle II Constraints on the Non-Minimal Universal Extra Dimensional Model

Problem Statement
Recent experimental data from the Belle II Collaboration has reported a branching ratio for the decay $B^+ \to K^+ \nu\bar{\nu}$ with a $3.5\sigma$ significance, deviating by $2.7\sigma$ from Standard Model (SM) expectations. This discrepancy suggests potential New Physics (NP) in $b \to s \nu\bar{\nu}$ transitions. While various Beyond the Standard Model (BSM) frameworks have been proposed to explain such anomalies, the specific constraints imposed by this measurement on the Non-Minimal Universal Extra Dimensional (NMUED) model had not been comprehensively analyzed. Unlike the Minimal Universal Extra Dimensional (MUED) model, the NMUED framework incorporates Boundary Localised Terms (BLTs) at the orbifold fixed points, which significantly alter the mass spectra and interaction profiles of Kaluza-Klein (KK) states.

Methodology
The authors perform a theoretical analysis of the $B^+ \to K^+ \nu\bar{\nu}$ decay within the NMUED framework. The methodology involves:

1. Model Formulation: The study utilizes a 5D NMUED model where all SM fields propagate in a flat extra dimension compactified on an $S^1/Z_2$ orbifold. The model includes boundary localised kinetic terms (BLKTs) for fermions ($r_f$) and gauge/Higgs sectors ($r_V$), as well as boundary localised Yukawa terms ($r_y$). To simplify the analysis and avoid complex mode mixing, the authors impose the condition $r_y = r_f$.
2. Effective Hamiltonian Calculation: The transition is governed by the effective Hamiltonian for $b \to s \nu\bar{\nu}$. The authors compute the relevant Wilson Coefficient (WC), denoted as $X(x_t, r_f, r_V, R^{-1})$, which encapsulates both SM contributions and NP effects from KK modes.
3. Loop Diagrams: The NP contributions are derived from one-loop Feynman diagrams, specifically:
   • $Z$-penguin diagrams (involving KK excitations of top quarks, $W$ bosons, and Goldstone/Higgs bosons).
   • Self-energy diagrams.
   • Box diagrams (involving KK excitations of fermions and gauge bosons).
     The calculations explicitly account for the overlap integrals ($I_1^n, I_2^n$) arising from the non-trivial wavefunction profiles induced by BLTs, which distinguish NMUED predictions from MUED.
4. Numerical Analysis: The authors calculate the branching ratio by integrating the differential rate over the kinematically allowed region. They employ the flavio package to evaluate flavour observables, inputting the analytically derived modified loop functions specific to NMUED.
5. Constraints: The parameter space is constrained by:
   • The experimental measurement of $Br(B^+ \to K^+ \nu\bar{\nu})$ from Belle II.
   • The upper limit on $Br(B \to K^* \nu\bar{\nu})$ from the Belle collaboration.
   • Constraints from other rare $B$-decays ($B_s \to \mu^+\mu^-$, $B \to X_s \gamma$, $B \to X_s \ell^+\ell^-$).
   • Electroweak Precision Tests (EWPT) regarding the $S, T, U$ parameters.
   • Vacuum stability considerations, limiting the summation of KK modes to the first 10 levels.

Key Contributions

• First Comprehensive NMUED Analysis: This work presents the first detailed analysis of the $B^+ \to K^+ \nu\bar{\nu}$ branching ratio specifically within the NMUED framework.
• Derivation of New Functions: The authors derive explicit expressions for the loop functions $F(x_t^{(n)})$ and $G(x_t^{(n)}, x_e^{(n)})$ arising from $Z$-penguin and box diagrams, respectively, incorporating the effects of BLTs. These expressions differ significantly from those in the MUED model due to the inclusion of overlap integrals.
• Parameter Space Exploration: The study systematically explores the dependence of the branching ratio on the inverse compactification radius ($R^{-1}$) and the dimensionless BLT parameters ($R_V = r_V/R$ and $R_f = r_f/R$).

Results

• Lower Bounds on $R^{-1}$: The analysis establishes lower limits on the inverse compactification radius ($R^{-1}$) for various combinations of non-zero BLT parameters.
  • For specific positive BLT configurations (e.g., $R_V = 8, R_f = 16$), the lower limit is found to be approximately $711$ GeV.
  • For higher BLT values (e.g., $R_V = 24, R_f = 24$), the lower limit increases to approximately $902$ GeV.
• MUED Limit Failure: Crucially, when the BLT parameters are set to zero ($R_V = R_f = 0$), recovering the MUED scenario, the theoretical prediction curve for $Br(B^+ \to K^+ \nu\bar{\nu})$ does not intersect the upper limits of the experimental data (at both 95% and 99% confidence levels). Consequently, the authors cannot establish a lower bound on $R^{-1}$ for the minimal scenario using this specific decay channel.
• BLT Impact: The presence of non-vanishing BLT parameters allows for a relaxation of the mass spectrum degeneracy, enabling the model to accommodate the experimental data with specific parameter sets that yield lower bounds on $R^{-1}$ in the range of hundreds of GeV to $\sim 1$ TeV.
• Comparison with Previous Work: The derived bounds are consistent with previous constraints from other rare $B$-decays and electroweak observables within the NMUED framework.

Significance and Claims
The paper claims that while the $B^+ \to K^+ \nu\bar{\nu}$ decay provides valuable complementary insights into the NMUED parameter space, it does not independently yield the most stringent lower bound on $R^{-1}$ compared to other observables analyzed in previous studies (such as $B_s \to \mu^+\mu^-$).

The primary significance lies in the demonstration that the inclusion of Boundary Localised Terms fundamentally alters the phenomenological predictions of the Universal Extra Dimensional model. Specifically, the study highlights that:

1. The minimal scenario (MUED) cannot be constrained by the current $B^+ \to K^+ \nu\bar{\nu}$ data to set a lower bound on the compactification scale.
2. The non-minimal scenario (NMUED) allows for viable parameter regions where the lower bound on $R^{-1}$ is elevated to approximately $900$ GeV, depending on the BLT coefficients.
3. The analysis confirms that the NMUED framework remains a viable candidate for explaining potential deviations in flavour physics, provided the boundary terms are non-zero and appropriately tuned.

The authors conclude that their findings are consistent with existing experimental constraints and previous theoretical investigations into rare $B$-meson decays within the NMUED context.