Doubly heavy spin-$\frac {3}{2}$ baryons spectrum in the ground and excited states

This study utilizes QCD sum rules with nonperturbative effects up to dimension ten to predict the masses and residues of ground and excited spin-3/2 doubly heavy baryons ($\Xi^*$ and $\Omega^*$ containing $cc$, $bc$, or $bb$ quarks), providing essential theoretical guidance for future experimental searches.

The Gist

Imagine the universe is built from a giant, cosmic LEGO set. The smallest, most fundamental bricks are called quarks. Usually, these bricks snap together in groups of three to form particles called baryons (like the protons and neutrons that make up your body).

Most of the time, these LEGO bricks are "light" and fast. But sometimes, nature tries to build something heavier and more exotic: a baryon made of two heavy bricks (called "charm" or "bottom" quarks) and one light brick.

For decades, physicists struggled to find these "double-heavy" LEGO creations. They were like rare, elusive toys hidden in a massive toy box. Recently, scientists finally found a few of them, but they are still hunting for the rest of the family, especially the "spinning" versions (spin-3/2) and the excited versions (where the bricks are vibrating or jumping around).

This paper is a theoretical treasure map created by two physicists, M. Shekari Tousi and K. Azizi. They didn't build these particles in a lab; they built them in their minds using a powerful mathematical tool called QCD Sum Rules.

Here is a simple breakdown of what they did and why it matters:

1. The "Recipe" for Heavy Particles

Think of a baryon like a cake. To know how heavy the cake is, you need to know the ingredients (quarks) and the oven temperature (energy).

• The Ingredients: The authors looked at baryons with two heavy quarks (like two chocolate chips) and one light quark (like a strawberry).
• The States: They didn't just look at the "plain cake" (the ground state). They also predicted the weight of:
  • The "Spinning Cake" (1P): Where the ingredients are swirling around faster.
  • The "Bouncing Cake" (2S): Where the ingredients are jumping up and down (radial excitation).

2. The Tool: QCD Sum Rules (The "Crystal Ball")

How do you weigh a particle you can't see yet? You use QCD Sum Rules.
Imagine you are trying to guess the weight of a sealed box. You can't open it, but you can shake it, listen to the sound it makes, and feel how it vibrates.

• The "Physical Side": This is what the box should sound like if it contains a specific particle (based on known physics).
• The "QCD Side": This is the math of how the tiny quarks inside interact.
• The Match: The authors tried to make the "sound" of the math match the "sound" of the particle. When they matched, they could calculate the exact weight (mass) and the "stickiness" (residue) of the particle.

3. The "Super-Resolution" Upgrade

Previous scientists tried to do this math, but they only looked at the first few layers of the LEGO structure (low-dimensional operators). It was like trying to describe a complex building by only looking at the foundation.

• What this paper did: They zoomed in much deeper. They included effects up to dimension ten.
• The Analogy: If previous studies were looking at a photo with 100 pixels, this study looked at a photo with 10,000 pixels. This allowed them to see the "fuzziness" of the quantum world much more clearly, leading to much more precise predictions.

4. The Results: A Menu for Future Experiments

The authors produced a detailed menu of predicted masses for six different types of these heavy baryons (named $\Xi^*$, $\Omega^*$, etc., with combinations of charm and bottom quarks).

• The "Residue": They also calculated something called the "residue." Think of this as the volume knob on a radio. If the residue is high, the particle is loud and easy to detect. If it's low, it's a whisper. This tells experimentalists at giant machines like the LHC (Large Hadron Collider) exactly how hard they need to listen to find these particles.

5. Why Does This Matter?

• The Hunt: Experimentalists are currently scanning the universe looking for these particles. They have a list of suspects, but they don't know exactly what they weigh. This paper gives them a Wanted Poster with accurate descriptions.
• The Gap: There is a gap between what we know (the math of quarks) and what we see (the particles in the detector). This paper helps bridge that gap.
• The Future: Once these particles are found, scientists can study how they decay (fall apart). This helps us understand the "glue" (strong force) that holds the universe together.

In a Nutshell

This paper is a high-precision forecast for a new class of cosmic LEGO structures. The authors used advanced math to predict exactly how heavy these "double-heavy" particles are and how "loud" they will be when they appear in experiments. Their work is a guidebook for the next generation of particle hunters, telling them exactly where to look and what to expect when they finally catch these elusive, heavy baryons.

Technical Summary

1. Problem Statement

The study addresses the theoretical characterization of doubly heavy baryons containing two heavy quarks ($c$ and/or $b$) and one light quark ($u, d, s$). While experimental progress has been made in identifying the ground state of the doubly charmed $\Xi_{cc}$ baryon by the LHCb collaboration, the spectroscopic properties of the spin-3/2 ($J^P = 3/2^+$) members of these families, as well as their excited states (orbital $1P$ and radial $2S$), remain largely unexplored experimentally.

Existing theoretical predictions for these states often suffer from:

• Limited precision due to truncation of nonperturbative contributions in the Operator Product Expansion (OPE) (typically only up to dimension 5 or 6).
• A lack of consistent calculations for residues (coupling constants), which are essential for predicting decay widths and branching ratios.
• Insufficient data on excited states ($1P$ and $2S$) for spin-3/2 configurations.

2. Methodology

The authors employ the QCD Sum Rules method, a nonperturbative approach based on the QCD Lagrangian, to calculate the masses and residues of the target baryons.

• Target Particles: The study focuses on the spin-3/2 doubly heavy baryons: $\Xi^*_{cc}, \Xi^*_{bc}, \Xi^*_{bb}, \Omega^*_{cc}, \Omega^*_{bc},$ and $\Omega^*_{bb}$.
• Interpolating Currents: A specific interpolating current $\eta_\mu$ is constructed to generate spin-3/2 states. The current is symmetric under the exchange of identical heavy quarks and carefully designed to isolate the spin-3/2 component while minimizing contamination from spin-1/2 states.
• Correlation Function: A two-point correlation function $\Pi_{\mu\nu}(q)$ is constructed using the time-ordered product of the interpolating currents.
  • Physical Side: Evaluated by inserting a complete set of hadronic states (ground state $1S$, first orbital excitation $1P$, and first radial excitation $2S$). The residues ($\lambda$) and masses ($m$) are extracted by matching Lorentz structures ($g_{\mu\nu}$ and $\not{q}g_{\mu\nu}$).
  • QCD Side: Evaluated in the deep Euclidean region using the OPE. The calculation includes quark and gluon propagators.
• Key Innovation (OPE Expansion): Unlike previous studies limited to dimension 5 or 6, this work extends the OPE to include nonperturbative operators up to dimension 10. This includes contributions from quark condensates ($\langle \bar{q}q \rangle$), gluon condensates ($\langle \frac{\alpha_s}{\pi} G^2 \rangle$), and higher-dimensional mixed condensates.
• Factorization Assumption: For dimension $\ge 6$, higher-dimensional condensates are estimated using the vacuum saturation hypothesis, modified by a phenomenological parameter $\kappa$ (varied from 1 to 5) to account for non-factorizable effects.
• Sum Rule Extraction: The Borel transformation is applied to suppress higher excited states and continuum contributions. The masses and residues are extracted by equating the physical and QCD sides of the sum rules for the selected Lorentz structures.

3. Key Contributions

1. Extended OPE Precision: The inclusion of nonperturbative operators up to dimension 10 significantly reduces theoretical uncertainties compared to previous sum rule analyses.
2. Comprehensive Excited State Analysis: The paper provides the first consistent QCD sum rule predictions for the masses and residues of spin-3/2 doubly heavy baryons in the $1P$ (orbital) and $2S$ (radial) excited states, in addition to the ground state ($1S$).
3. Residue Calculations: A major contribution is the calculation of the residues ($\lambda$) for all states. These values are critical inputs for future calculations of strong and weak decay widths, which are currently unavailable for these excited states.
4. Systematic Uncertainty Analysis: The authors rigorously analyze the dependence on auxiliary parameters (Borel mass $M^2$, continuum threshold $s_0$) and input parameters (quark masses, condensates), providing robust error estimates.

4. Key Results

The study presents numerical predictions for the masses (in GeV) and residues (in GeV$^3$) of the six baryon families across three states ($1S, 1P, 2S$).

Selected Mass Predictions (Ground State $1S$):

• $\Xi^*_{cc}$: $3.68^{+0.15}_{-0.14}$ GeV
• $\Xi^*_{bc}$: $6.95 \pm 0.20$ GeV
• $\Xi^*_{bb}$: $10.28 \pm 0.30$ GeV
• $\Omega^*_{cc}$: $3.74 \pm 0.15$ GeV
• $\Omega^*_{bc}$: $7.01^{+0.20}_{-0.19}$ GeV
• $\Omega^*_{bb}$: $10.34 \pm 0.30$ GeV

Excited State Trends:

• The mass splitting between the ground state ($1S$) and the first orbital excitation ($1P$) is approximately 0.17–0.26 GeV.
• The mass splitting between the ground state ($1S$) and the first radial excitation ($2S$) is approximately 0.31–0.48 GeV.
• The calculated residues generally increase with the excitation level, though with larger relative uncertainties than the masses.

Comparison with Other Models:

• The ground state mass predictions show excellent agreement with Lattice QCD results and other QCD sum rule studies (e.g., Refs. [13, 32]).
• The results for excited states ($1P, 2S$) are consistent with hypercentral constituent quark models and relativistic quark models within error margins, though some deviations exist with specific previous sum rule studies (e.g., Ref. [31]), attributed to the extended OPE and different current choices in this work.

5. Significance

• Experimental Guidance: The predictions serve as a crucial roadmap for experimental facilities like the LHCb and future colliders. By providing precise mass windows and identifying the expected residues, the study aids in the search for these elusive excited states.
• Decay Dynamics: The calculated residues are fundamental for determining decay widths and branching ratios. Without these values, theoretical models cannot accurately predict how these baryons decay, hindering their identification in experimental data.
• Theoretical Advancement: By pushing the OPE to dimension 10, the authors demonstrate that higher-order nonperturbative effects are necessary for high-precision spectroscopy of heavy baryons, setting a new standard for future QCD sum rule analyses.
• Bridging Perturbative and Nonperturbative QCD: The work reinforces the utility of QCD sum rules in bridging the gap between fundamental QCD parameters and observable hadronic properties, particularly in the heavy quark sector where experimental data is sparse.

In conclusion, this paper provides the most precise and comprehensive QCD sum rule analysis to date for the spin-3/2 doubly heavy baryon spectrum, covering ground and excited states, and offers essential parameters for the next generation of heavy flavor physics experiments.