New Predictions for the Lifetimes of Doubly Heavy Baryons and the $B_c$ Meson

This contribution presents updated theoretical predictions for the lifetimes of all weakly decaying doubly heavy baryons ($bb$, $cc$, $bc$) as well as the $B_c$ meson by incorporating higher-order QCD corrections, comparing various mass schemes, and establishing specific lifetime hierarchies for different ground-state spin configurations.

The Gist

Imagine the universe as a massive, chaotic construction site. On this site, there are tiny, heavy workers called quarks. Normally, these workers form teams of three to build particles called baryons (such as protons and neutrons).

Most of these teams consist of one heavy worker and two light ones. But sometimes nature builds a rare, doubly heavy team: two heavy workers and one light one. These are called doubly heavy baryons. There is also a special pair of heavy workers that join only with each other to form a meson (the $B_c$ meson).

These heavy teams are unstable. They do not last forever; they eventually decay into lighter particles. The big question physicists ask is: How long does each specific team last before it decays?

This article is like a very precise, high-tech stopwatch and a set of blueprints. The authors, Lovro Dulibić, Blaženka Melić, and Ivan Nišandžić, have updated the calculations to predict exactly how long these rare, doubly heavy teams survive.

Here is a breakdown of their work using simple analogies:

1. The "Heavy Quark Expansion" (The Rulebook)

To predict how long a team lasts, scientists use a method called Heavy Quark Expansion (HQE). Think of it as a rulebook for calculating decay.

• The Main Rule: The most important factor is simply how heavy the workers are. Heavier workers generally decay faster.
• The Fine Print: But it is not just about weight. The arrangement of the workers matters. If the two heavy workers hold hands tightly (a specific spin configuration), or if the light worker stands in a particular spot, this changes how the team decays.
• The "Spectator" Effect: Imagine the two heavy workers are the ones doing the heavy lifting (decaying), while the light worker just watches (the "spectator"). Sometimes the light worker accidentally bumps into the heavy ones, accelerating the process. Sometimes the light worker gets in the way and slows it down. The article calculates exactly how much this "bumping" changes the lifetime.

2. The New "High-Definition" Calculations

Previous versions of this rulebook were somewhat blurry. This article sharpens the image by adding NNLO and NLO corrections.

• Analogy: Imagine you are trying to predict the speed of a car.
  • Old Way: You only considered the engine size (the basic mass).
  • This Article: You added aerodynamics, tire friction, air resistance, and even the tiny vibrations of the engine. You did not just look at the main engine; you considered the "Darwin term" (a subtle quantum effect related to how the heavy workers jitter) and "Penguin terms" (strange, loop-like interactions happening in the background).
• Result: These new, high-resolution calculations make the predictions much more reliable, especially for teams containing Charm quarks, which are lighter and harder to predict than Bottom quark teams.

3. The Three Types of Teams They Studied

The authors calculated lifetimes for three different types of doubly heavy teams:

• The Double-Bottom Team ($bb$): Two very heavy Bottom workers and one light worker.

  • Prediction: The neutral version of this team ($\Xi^0_{bb}$) decays the fastest. The two charged versions ($\Xi^-_{bb}$ and $\Omega^-_{bb}$) last about equally long, slightly longer than the neutral version.
  • Why? The neutral team has a "weak exchange" interaction where the workers swap places in a way that accelerates decay.

• The Double-Charm Team ($cc$): Two Charm workers and one light worker.

  • Prediction: The positively charged team ($\Xi^{++}_{cc}$) is the most long-lived. The neutral one ($\Xi^+_{cc}$) is the least long-lived.
  • Reality Check: Scientists at the LHCb experiment have already measured the lifetime of the $\Xi^{++}_{cc}$. The authors' new, more precise calculation (which includes the new "Darwin term" corrections) brings their prediction much closer to the actual measurement than their previous attempts.

• The Mixed Team ($bc$): One Bottom worker and one Charm worker.

  • The Puzzle: This is the trickiest. The two heavy workers can hold hands in two different ways (Spin 0 or Spin 1). The article does not yet know which one is the "ground state" (the most stable version).
  • The Solution: They calculated the lifetimes for both possibilities.
  • The Twist: They found a way to tell the difference! The neutral version of the "Spin 0" team ($\Xi^0_{bc}$) should live significantly longer than the neutral version of the "Spin 1" team ($\Xi'^0_{bc}$). If future experiments measure these lifetimes, they can finally say which version of the team actually exists in nature.

4. The $B_c$ Meson (The Special Pair)

They also looked at the $B_c$ meson, which is just a Bottom and a Charm worker forming a pair.

• The Surprise: When they included the new "Darwin term" (the jitter effect mentioned above), their prediction for how long this pair survives became shorter than what is actually observed in experiments.
• The Implication: If you remove this specific "Darwin" correction from the math, the prediction matches the experiment perfectly. This suggests that although the mathematics is very advanced, there may be something about how this specific "jitter" works in a system of two heavy particles that we do not yet fully understand. It is a puzzle for future physicists to solve.

5. The "Mass Scheme" Problem

In physics, one must decide how to define the "weight" of a particle. It is like asking: "Is the weight of a suitcase measured when the handle points up or down?"

• The authors tested three different ways to define this weight (called MS-, Kinetic-, and $\Upsilon$-schemes).
• Good News: Although the numbers changed slightly depending on which "ruler" they used, the relative order of who lives longer than whom remained the same. This gives them confidence that their predictions are solid, regardless of the specific ruler used.

Summary

This article is an important update to the "rulebook" on how rare, doubly heavy particles die.

1. They added highly precise corrections (like adding air resistance to a car speed calculation).
2. They predicted lifetimes for all possible doubly heavy teams ($bb$, $cc$, and $bc$).
3. They found a specific difference in the lifetimes of neutral $bc$ teams that could help scientists understand the internal structure of these particles.
4. They highlighted a small discrepancy in the $B_c$ meson prediction, suggesting that a small puzzle piece is still missing regarding how these heavy particles jitter.

Essentially, they have created a more accurate map of the "heavy particle zoo" and told us exactly how long each rare creature is expected to live before it disappears.

Technical Summary

Technical Summary: New Predictions for the Lifetimes of Doubly Heavy Baryons and the $B_c$ Meson

Problem and Motivation
The inclusive weak decays of heavy hadrons provide a primary testing ground for the Heavy-Quark Expansion (HQE) and the Heavy-Quark Effective Theory (HQET). While the expansion parameter $1/m_b$ ensures good convergence for bottom hadrons, the charm sector ($1/m_c$) presents significant challenges due to the proximity of the charm mass to the non-perturbative QCD scale, leading to slower convergence and enhanced sensitivity to higher-dimensional operators. Doubly heavy hadrons ($bb$, $cc$, $bc$) offer a unique intermediate regime to test the HQE. While the lifetime of the $\Xi_{cc}^{++}$ has been measured, the lifetimes of other doubly heavy baryons ($\Xi_{bb}$, $\Omega_{bb}$, $\Xi_{cc}$, $\Omega_{cc}$, and the $bc$ sector) as well as the $B_c$ meson remain theoretical predictions. In the $bc$ sector, a specific ambiguity exists regarding the spin assignment of the ground state of the $bc$ diquark ($S_{bc}=0$ or $S_{bc}=1$), which influences the matrix elements of the operators and the predictions for the lifetimes.

Methodology
The authors perform a comprehensive update of lifetime predictions using the HQE framework, where the total decay width is expanded in inverse powers of the heavy quark mass ($1/m_Q$). The analysis integrates several key theoretical advances:

1. Perturbative Corrections:

   • Dimension 3 (Leading Order): Updated to Next-to-Next-to-Leading Order (NNLO) in $\alpha_s$.
   • Dimension 5 (Chromomagnetic): Updated to Next-to-Leading Order (NLO).
   • Dimension 6 (Spectator): Contains the complete set of known NLO corrections for heavy-light spectator contributions, including penguin terms. For $bc$ baryons, penguin contributions to heavy-heavy ($bc$) spectator operators are included, although complete NLO corrections for two massive external quarks are currently unavailable.
   • Dimension 7: Estimates for subleading spectator operators are included.
   • Darwin Term: The Wilson coefficient for the dimension-six Darwin term is included at Leading Order (LO).

2. Mass Schemes: To assess theoretical stability, predictions are calculated in three heavy-quark mass schemes:

   • $\overline{\text{MS}}$ scheme.
   • Kinetic mass scheme (using a low-scale cutoff $\mu_{kin}$).
   • $\Upsilon$ mass scheme (specifically for $bc$ baryons and the $B_c$ meson), relating quark masses to meson masses (mass differences of $\Upsilon(1S)$ and $B/D$ mesons).

3. Non-perturbative Matrix Elements:

   • Adoption of a non-relativistic diquark-quark picture, where the heavy pair forms a compact color-antitriplet diquark.
   • Heavy-Light Matrix Elements: Taken from hyperfine mass splittings using the De Rujula–Georgi–Glashow constituent quark model, which links baryon wave functions at the origin to meson decay constants and mass splittings.
   • Heavy-Heavy Matrix Elements: Derived from potential model analyses of heavy baryon spectroscopy.
   • $bc$ Spin Ambiguity: Predictions are provided for both scalar ($S_{bc}=0$, unprimed states) and vector ($S_{bc}=1$, primed states) diquark configurations.

Main Contributions and Results

• Doubly Bottom Baryons ($B_{bb}$):

  • Predicted lifetime hierarchy: $\tau(\Xi_{bb}^0) < \tau(\Xi_{bb}^-) \simeq \tau(\Omega_{bb}^-)$.
  • The splitting is driven by a substantial weak exchange contribution to the width of $\Xi_{bb}^0$, enhanced by the spin-1 $bb$ diquark configuration.
  • Results are presented in the $\overline{\text{MS}}$ and kinetic schemes, showing mild scheme dependence for ratios.

• Doubly Charm Baryons ($B_{cc}$):

  • Predicted hierarchy: $\tau(\Xi_{cc}^+) < \tau(\Omega_{cc}^+) < \tau(\Xi_{cc}^{++})$.
  • The inclusion of NNLO corrections to the leading dimension-3 term increases the total decay widths by approximately 30% for $\Xi_{cc}^{++}$, 5% for $\Xi_{cc}^+$, and 15% for $\Omega_{cc}^+$.
  • The updated prediction for $\tau(\Xi_{cc}^{++}) \approx 0.27$ ps (kinetic scheme) moves closer to the experimental value of $0.256 \pm 0.024 \pm 0.014$ ps.

• Bottom-Charm Baryons ($B_{bc}$):

  • Predictions are provided for both scenarios $S_{bc}=0$ and $S_{bc}=1$.
  • $S_{bc}=0$ (Unprimed): Hierarchy $\tau(\Xi_{bc}^0) \lesssim \tau(\Omega_{bc}^0) < \tau(\Xi_{bc}^+)$.
  • $S_{bc}=1$ (Primed): Hierarchy $\tau(\Xi_{bc}^{\prime 0}) < \tau(\Omega_{bc}^{\prime 0}) < \tau(\Xi_{bc}^{\prime +})$.
  • The authors identify the neutral baryons ($\Xi_{bc}^0$ vs. $\Xi_{bc}^{\prime 0}$) as the most sensitive test to distinguish the diquark spin. The predicted lifetime of $\Xi_{bc}^0$ is significantly larger than that of $\Xi_{bc}^{\prime 0}$ due to a large weak exchange contribution, which is enhanced in the case of the vector diquark.

• $B_c$ Meson:

  • The analysis includes NNLO corrections to the leading term and NLO to the chromomagnetic term.
  • A significant result is the large impact of the Darwin term ($\hat{\rho}_D^3$). The matrix element for the $B_c$ is approximately five times larger than for heavy-light $B_q$ mesons due to the compact nature of the $B_c$ state (large wave function at the origin).
  • Tension with Experiment: The full prediction including the Darwin term yields $\tau(B_c) \approx 0.29$ ps, which is in tension with the experimental value of $0.510 \pm 0.009$ ps. However, excluding the Darwin contribution leads to $\tau(B_c) \approx 0.46$ ps, which is compatible with the experiment.

Significance and Claims
The work claims to place lifetime predictions for all doubly heavy baryons on a "common foundation" by using a consistent non-perturbative framework (diquark model) and incorporating the latest perturbative corrections (NNLO for dimension 3, NLO for dimension 5 and spectator terms).

The authors claim that:

1. The HQE provides an "adequately controlled description" of the lifetimes of doubly heavy hadrons, despite remaining uncertainties dominated by hadronic input parameters and renormalization scale variation.
2. The inclusion of higher-order perturbative corrections reduces the spread between predictions in different mass schemes, although residual scheme dependence remains.
3. The differing lifetime predictions for the neutral $bc$ baryons ($\Xi_{bc}^0$ vs. $\Xi_{bc}^{\prime 0}$) offer a potential experimental method to resolve the ambiguity regarding the ground-state diquark spin.
4. The large discrepancy in the $B_c$ lifetime prediction when the Darwin term is included points to a possible issue with the current theoretical treatment of this specific operator in the $B_c$ system and suggests the need for further investigation of the non-perturbative matrix element or higher-order corrections.

The work does not propose new experimental setups but emphasizes that ongoing measurements at LHCb will provide critical tests of the HQE and the theoretical determination of hadronic matrix elements.