Operator Product Expansion Convergence Research Articles

Revisiting [formula omitted] as a 0+ tetraquark state from QCD sum rules

Stimulated by the renewed observation of Ds0⁎(2317) signal and its updated mass value 2318.3±1.2±1.2MeV/c2 in the process e+e−→Ds⁎+Ds0⁎(2317)−+c.c. by BESIII Collaboration, we devote to reinvestigate Ds0⁎(2317) as a 0+ tetraquark state from QCD sum rules. Technically, four different possible currents are adopted and high condensates up to dimension 12 are included in the operator product expansion (OPE) to ensure the quality of QCD sum rule analysis. In the end, we obtain the mass value 2.37−0.36+0.50GeV with the factorization parameter ρ=1 (or 2.23−0.24+0.78GeV with ρ=3) for the scalar–scalar current, which agrees well with the experimental data of Ds0⁎(2317) and could support its explanation as a 0+ scalar–scalar tetraquark state. The final result for the axial–axial configuration is calculated to be 2.51−0.43+0.61GeV with ρ=1 (or 2.52+0.76−0.52GeV with ρ=3), which is still consistent with the mass of Ds0⁎(2317) considering the uncertainty, and then the possibility of Ds0⁎(2317) as a axial–axial tetraquark state can not be excluded. For the pseudoscalar–pseudoscalar and the vector–vector cases, their unsatisfactory OPE convergence makes that it is of difficulty to find rational work windows to further acquire hadronic masses.

Unitarity and universality in nonrelativistic conformal field theory

We relate the notion of unitarity of a $SL(2,\mathbf{R})$ invariant field theory with that of a Schrodinger field theory using the fact that $SL(2,\mathbf{R})$ is a subgroup of Schrodinger group. Exploiting $SL(2,\mathbf{R})$ unitarity, we derive the unitarity bounds and null conditions for a Schr\"odinger field theory (for the neutral as well as the charged sector). In non integer dimensions the theory is shown to be non-unitary. Furthermore, the use of $SL(2,\mathbf{R})$ subgroup opens up the possibility of borrowing results from 1D $SL(2,\mathbf{R})$ invariant field theory to explore Schrodinger field theory, in particular, the sector with zero charge. We explore the consequences of $SL(2,\mathbf{R})$ symmetry e.g. the convergence of operator product expansion in the kinematic limit, where all the operators (neutral and/or charged) are on same temporal slice ($x=constant$), the universal behavior of weighted spectral density function, existence of infinite number of $SL(2,\mathbf{R})$ primaries, the analytic behavior of three point function as a function of spatial separation. We discuss the implication of imposing parity invariance ($\tau\to-\tau$) in addition to Schrodinger invariance and emphasize its difference from time reversal ($\tau \to-\tau$ with charge conjugation).

0+ tetraquark states from improved QCD sum rules: delving into X(5568)**Supported by National Natural Science Foundation of China (11475258, 11105223, 11675263) and the Project in NUDT for Excellent Youth Talents

In order to investigate the possibility of the recently observed X(5568) being a 0+ tetraquark state, we make an improvement to the study of the related various configuration states in the framework of the QCD sum rules. Particularly, to ensure the quality of the analysis, condensates up to dimension 12 are included to inspect the convergence of operator product expansion (OPE) and improve the final results of the studied states. We note that some condensate contributions could play an important role on the OPE side. By releasing the rigid OPE convergence criterion, we arrive at the numerical value for the scalar-scalar diquark-antidiquark 0+ state, which agrees with the experimental data for the X(5568) and could support its interpretation in terms of a 0+ tetraquark state with the scalar-scalar configuration. The corresponding result for the axial-axial current is calculated to be , which is still consistent with the mass of X(5568) in view of the uncertainty. The feasibility of X(5568) being a tetraquark state with the axial-axial configuration therefore cannot be definitely excluded. For the pseudoscalar-pseudoscalar and the vector-vector cases, their unsatisfactory OPE convergence make it difficult to find reasonable work windows to extract the hadronic information.

OPE convergence in non-relativistic conformal field theories

Motivated by applications to the study of ultracold atomic gases near the unitarity limit, we investigate the structure of the operator product expansion (OPE) in non-relativistic conformal field theories (NRCFTs). The main tool used in our analysis is the representation theory of charged (i.e. non-zero particle number) operators in the NRCFT, in particular the mapping between operators and states in a non-relativistic "radial quantization" Hilbert space. Our results include: a determination of the OPE coefficients of descendant operators in terms of those of the underlying primary state, a demonstration of convergence of the (imaginary time) OPE in certain kinematic limits, and an estimate of the decay rate of the OPE tail inside matrix elements which, as in relativistic CFTs, depends exponentially on operator dimensions. To illustrate our results we consider several examples, including a strongly interacting field theory of bosons tuned to the unitarity limit, as well as a class of holographic models. Given the similarity with known statements about the OPE in SO(2,d) invariant field theories, our results suggest the existence of a bootstrap approach to constraining NRCFTs, with applications to bound state spectra and interactions. We briefly comment on a possible implementation of this non-relativistic conformal bootstrap program.

Improved QCD sum rule study ofZc(3900)as aD¯D*molecular state

In the framework of QCD sum rules, we present an improved study of our previous work [Phys. Rev. D {\bf80}, 056004 (2009)] particularly on the $\bar{D}D^{*}$ molecular state to investigate that the possibility of the newly observed $Z_{c}(3900)$ as a $S$-wave $\bar{D}D^{*}$ molecular state. To ensure the quality of QCD sum rule analysis, contributions of up to dimension nine are calculated to test the convergence of operator product expansion (OPE). We find that the two-quark condensate $<\bar{q}q>$ is very large and makes the standard OPE convergence (i.e. the perturbative at least larger than each condensate contribution) happen at very large values of Borel parameters. By releasing the rigid OPE convergence criterion, one could find that the OPE convergence is still under control. We arrive at the numerical result $3.86\pm0.27 {GeV}$ for $\bar{D}D^{*}$, which agrees with the mass of $Z_{c}(3900)$ and could support the explanation of $Z_{c}(3900)$ in terms of a $S$-wave $\bar{D}D^{*}$ molecular state.

Study ofXc(3250)as aD0*(2400)Nmolecular state

We present a QCD sum rule analysis for the newly observed resonance $X_{c}(3250)$ by assuming it as a $D_{0}^{*}(2400)N$ molecular state. Technically, contributions of operators up to dimension 12 are included in the operator product expansion (OPE). We find that it is difficult to find the conventional OPE convergence in this work. By trying releasing the rigid OPE convergence criterion, one could find that the OPE convergence is still under control in the present work and the numerical result for $D_{0}^{*}(2400)N$ state is $3.18\pm0.51 {GeV}$, which is in agreement with the experimental data of $X_{c}(3250)$. In view of that the conventional OPE convergence is not obtained here, thus only weak conclusions can be drawn regarding the explanation of $X_{c}(3250)$ in terms of a $D_{0}^{*}(2400)N$ molecular state. As a byproduct, the mass for the bottom counterpart $\bar{B}_{0}^{*}N$ state is predicted to be $6.50\pm0.49 {GeV}$.

Operator product expansion convergence in conformal field theory

We clarify questions related to the convergence of the operator product expansion and conformal block decomposition in unitary conformal field theories (for any number of spacetime dimensions). In particular, we explain why these expansions are convergent in a finite region. We also show that the convergence is exponentially fast, in the sense that the operators of dimension above $\ensuremath{\Delta}$ contribute to correlation functions at most $\mathrm{exp}(\ensuremath{-}a\ensuremath{\Delta})$. Here the constant $a&gt;0$ depends on the positions of operator insertions and we compute it explicitly.

Transition magnetic moment ofΛ→Σ0in QCD sum rules

The $\ensuremath{\Lambda}\ensuremath{\rightarrow}{\ensuremath{\Sigma}}^{0}$ transition magnetic moment is computed in the QCD sum rules approach. Three independent tensor structures are derived in the external-field method using generalized interpolating fields. They are analyzed together with the $\ensuremath{\Lambda}$ and ${\ensuremath{\Sigma}}^{0}$ mass sum rules using a Monte-Carlo-based analysis, with attention to operator product expansion convergence, ground-state dominance, and the role of the transitions in the intermediate states. Relations between sum rules for magnetic moments of $\ensuremath{\Lambda}$ and ${\ensuremath{\Sigma}}^{0}$ and sum rules for transition magnetic moment of $\ensuremath{\Lambda}\ensuremath{\rightarrow}{\ensuremath{\Sigma}}^{0}$ are also examined. Our best prediction for the transition magnetic moment is ${\ensuremath{\mu}}_{{\ensuremath{\Sigma}}^{0}\ensuremath{\Lambda}}=1.60\ifmmode\pm\else\textpm\fi{}0.07{\ensuremath{\mu}}_{N}$. A comparison is made with other calculations in the literature.

NtoΔtransition amplitudes from QCD sum rules

We present a calculation of the $N$ to $\ensuremath{\Delta}$ electromagnetic transition amplitudes using the method of QCD sum rules. A complete set of QCD sum rules is derived for the entire family of transitions from the baryon octet to decuplet. They are analyzed in conjunction with the corresponding mass sum rules using a Monte Carlo-based analysis procedure. The performance of each of the sum rules is examined using the criteria of operator product expansion convergence and ground-state dominance, along with the role of the transitions in intermediate states. Individual contributions from the $u$, $d$, and $s$ quarks are isolated and their implications in the underlying dynamics are explored. Valid sum rules are identified and their predictions are obtained. The results are compared with experiment and other calculations.

Y(2175)state in the QCD sum rule

We study the mass of the state $Y(2175)$ of ${J}^{PC}={1}^{--}$ in the QCD sum rule. We construct both the diquark-antidiquark currents $(ss)(\overline{s}\overline{s})$ and the meson-meson currents $(\overline{s}s)(\overline{s}s)$. We find that there are two independent currents for both cases and derive the relations between them. The operator product expansion convergence of these two currents is sufficiently fast, which enables us to perform good sum rule analysis. Both the SVZ (Shifman-Vainshtein-Zakharov) sum rule and the finite energy sum rule lead to a mass around $2.3\ifmmode\pm\else\textpm\fi{}0.4\text{ }\text{ }\mathrm{GeV}$, which is consistent with the observed mass within the uncertainties of the present QCD sum rule. The coupling of the four-quark currents to lower lying states such as $\ensuremath{\phi}(1020)$ turns out to be rather small. We also discuss possible decay properties of $Y(2175)$ if it is a tetraquark state.

Octet baryon magnetic moments from QCD sum rules

A comprehensive study is made for the magnetic moments of octet baryons in the method of QCD sum rules. A complete set of QCD sum rules is derived using the external-field method and generalized interpolating fields. For each member, three sum rules are constructed from three independent tensor structures. They are analyzed in conjunction with the corresponding mass sum rules. The performance of each of the sum rules is examined using the criteria of operator product expansion convergence and ground-state dominance, along with the role of the transitions in intermediate states. Individual contributions from the $u$, $d$, and $s$ quarks are isolated and their implications in the underlying dynamics are explored. Valid sum rules are identified and their predictions are obtained. The results are compared with experiment and previous calculations.

Do the QCD sum rules support four-quark states?

We test the validity of the QCD sum rules applied to the light scalar mesons, the charmed mesons ${D}_{s0}(2317)$ and ${D}_{s1}(2460)$, and the $X(3872)$ axial meson, considered as tetraquark states. We find that, with the studied currents, it is possible to find an acceptable Borel window only for the $X(3872)$ meson. In such a Borel window we have simultaneously a good operator product expansion convergence and a pole contribution which is bigger than the continuum contribution. These results may indicate that light many-quark states cannot be considered as resonances separated from the continuum.