[Paper Review] $ \bar D^*D$ and $\bar B^*B~(1^{++})$ molecules at N2LO from QSSR

F. Fanomezana, Stéphan Narison+1 more|arXiv (Cornell University)|Sep 30, 2014

Quantum Chromodynamics and Particle Interactions3 references3 citations

TL;DR

This paper presents next-to-next-to-leading order (N2LO) QCD spectral sum rule calculations for the $\bar{D}^*D$ and $\bar{B}^*B$ $1^{++}$ molecule states, including perturbative QCD corrections up to $\alpha_s^2$ and non-perturbative condensates up to dimension eight. It finds masses of $3738(152)$ MeV and $10687(232)$ MeV, respectively, in good agreement with the $Z_c(3900)$ and $Z_b(10610)$ experimental states, supporting their molecular interpretation.

ABSTRACT

We use QCD spectral sum rules (QSSR) and the factorization properties of molecule currents to estimate the masses and couplings of the $\bar D^*D$ and $\bar B^*B~(1^{++})$ molecules at N2LO of PT QCD. We include in the OPE the contributions of non-perturbative condensates up to dimension-eight. With the Laplace sum rules approach (LSR) and in the $\overline{MS}$-scheme, we obtain $M_{D^*D}=3738(152)$ MeV, which agrees within the errors with the newly discovered $Z_c$(3900). For the bottom channel, we find $M_{B^*B}=10687(232)$ MeV in good agreement with the observed $Z_b$(10610). Couplings of these states to the currents are also extracted. Our results are improvements of the LO ones in the existing literature.

Motivation & Objective

• To improve the theoretical prediction of the masses and couplings of $\bar{D}^*D$ and $\bar{B}^*B$ $1^{++}$ molecule states beyond leading order (LO) in perturbative QCD.
• To incorporate higher-order QCD corrections (N2LO) and non-perturbative condensates up to dimension eight in the operator product expansion (OPE).
• To test whether the observed $Z_c(3900)$ and $Z_b(10610)$ states can be interpreted as molecular states via comparison with QCD spectral sum rules.
• To extract the coupling constants of these states to their interpolating currents using the Laplace sum rule approach in the $\overline{MS}$ scheme.

Proposed method

• Employing QCD spectral sum rules (QSSR) with the Laplace sum rule (LSR) method to relate the correlation function of the molecule current to physical observables.
• Using the factorization approximation for the two-point correlation function of the molecule current, expressed as a convolution of spectral functions of vector and pseudoscalar quark currents.
• Including perturbative QCD corrections up to $\alpha_s^2$ (N2LO) and non-perturbative condensates up to dimension eight, such as $\langle \bar{q}q \rangle$, $\langle g^2 G^2 \rangle$, $\langle g\bar{q}Gq \rangle$, $\rho\langle \bar{q}q \rangle^2$, $\langle g^3 G^3 \rangle$, and $\rho\langle \bar{q}q \rangle \langle g^2 G^2 \rangle$.
• Transforming the on-shell quark masses to the $\overline{MS}$ scheme using the relation involving $\alpha_s$ and logarithmic terms in the renormalization scale $\nu$.
• Applying the Laplace sum rule to extract the mass $M_H$ and coupling $f_H$ via the ratio of moments and the exponential weight $e^{-t\tau}$, with stability in $\tau$ and $t_c$.

Experimental results

Research questions

• RQ1 What is the mass of the $\bar{D}^*D$ $1^{++}$ molecule state when calculated at N2LO in perturbative QCD with full non-perturbative condensate contributions?
• RQ2 How do the N2LO corrections affect the coupling of the $\bar{D}^*D$ molecule to its interpolating current compared to LO results?
• RQ3 Does the predicted mass of the $\bar{B}^*B$ $1^{++}$ molecule state at N2LO agree with the observed $Z_b(10610)$ resonance?
• RQ4 How do the radiative corrections in the PT series influence the convergence and stability of the sum rule predictions for both charm and bottom channels?
• RQ5 Can the observed $Z_c(3900)$ and $Z_b(10610)$ states be consistently interpreted as molecular states based on QSSR predictions with N2LO accuracy?

Key findings

• The predicted mass of the $\bar{D}^*D$ $1^{++}$ molecule state is $3738(152)$ MeV, which is in good agreement with the experimental $Z_c(3900)$ resonance.
• The predicted mass of the $\bar{B}^*B$ $1^{++}$ molecule state is $10687(232)$ MeV, showing good agreement with the observed $Z_b(10610)$ state.
• The coupling of the $\bar{D}^*D$ molecule to its current is $f_{D^*D} = (11.57 \pm 2.17) \times 10^{-2}$ MeV, comparable to the LO result for the $X_c(3872)$ state.
• The coupling of the $\bar{B}^*B$ molecule is $f_{B^*B} = (1.22 \pm 0.51) \times 10^{-2}$ MeV, consistent with the LO prediction for the $X_b(10144)$ state.
• The N2LO corrections increase the mass by approximately 0.46% from LO to NLO and 0.35% from NLO to N2LO in the bottom channel, indicating slow convergence.
• The radiative corrections to the coupling are larger than in the mass ratio, with a 10.1% increase from LO to NLO and 9.4% from NLO to N2LO in the bottom channel.