Borel Window Research Articles

Next-to-leading order (NLO) perturbative effects in QCD sum-rule analyses of light tetraquark systems: A case study in the scalar-isoscalar channel

QCD sum-rule mass predictions for tetraquark states provide insights on the interpretations and internal structure of experimentally-observed exotic mesons. However, the overwhelming majority of tetraquark QCD sum-rule analyses have been performed at leading order (LO), which raises questions about the underlying theoretical uncertainties from higher-loop corrections. The impact of next-to-leading order (NLO) perturbative effects are systematically examined in scalar (JPC=0++) isoscalar light-quark tetraquark systems where comprehensive LO sum-rule analyses have been performed and NLO perturbative corrections to the correlators have previously been calculated. Using the scalar-isoscalar state as a detailed case study to illustrate the differences between LO and NLO analyses, it is shown that NLO effects in individual Laplace sum-rules are numerically significant and have an important role in improving the reliability of the sum-rule analyses by widening the Borel window. However, ratios of sum-rules are found to be less sensitive to NLO effects with the additional advantage of cancelling the anomalous dimension that emerges from the NLO corrections. NLO mass predictions based on these sum-rule ratios are thus remarkably robust despite the slow perturbative convergence of the underlying correlator. The mass predictions 0.52GeV<mσ<0.77GeV for the lightest scalar-isoscalar σ state are in good agreement with the four-quark interpretation of the f0(500), and the relative coupling strengths of the f0(980) and f0(500) to the tetraquark current agree with the pattern found in chiral Lagrangian analyses. Effects of the σ resonance width are studied for different models, including resonance shapes inspired by chiral Lagrangians.

Wilson coefficients and four-quark condensates in QCD sum rules for medium modifications ofDmesons

Wilson coefficients of light four-quark condensates in QCD sum rules are evaluated for pseudo-scalar $D$ mesons, thus, pushing the sum rules toward mass dimension six. Contrary to the situation for $\bar{q}q$ mesons the impact of the four-quark condensates for vacuum as well as in-medium situations is found to be rather small within the Borel window used in previous analyses. The complete four-quark condensate contributions enable to identify candidates for an order parameter of spontaneous chiral symmetry breaking/restoration as well as to evaluate stability criteria of operator product expansions.

SU(3)-symmetry breaking effects and mass splitting in scalar and pseudoscalarDmesons from QCD sum rules

Motivated by the similar mass splitting in light-light and heavy-light meson system in $J^{P}=0^{-}$ and $J^{P}=0^{+}$ channel, taking into account the operator mixing to obtain an infrared stable OPE as well as a complete both non-perturbative and perturbative $\mathcal {O}(m_{q})$ corrections to the correlation function, the SU(3)-symmetry breaking effects in mass splitting in $0^{-}$ and $0^{+}$ channel of $D$ meson are analyzed by QCD sum rule approach based on a $c\bar{q}$ underlying structure. Within the sum rule we get, taking same threshold and Borel window for same channel, we find a correct mass splitting tendency in both channel which agrees with observed spectrum. We find $m_{D_{s}}-m_{D_{d}}\sim35\rm{MeV}$ in $0^{-}$ channel and $m_{D_{d}}-m_{D_{s}}\sim12\rm{MeV}$ in $0^{+}$ channel at scale $\mu=1\rm{GeV}$. These splitting can be accommodated with the different role of combination of mass effects and the parity-dependent "force" induced from non-perturbative QCD vacuum. When we scale up to $\mu=1.3\rm{GeV}$, the splitting remains unchanged which demonstrates a correct scale invariance. Combined with HQET, generalization to other channel of charmed meson and $b$-system are discussed briefly also.

Formula omitted] pentaquarks in QCD sum rules

We study pentaquark states with strangeness S = + 1 and I J π = 0 1 2 ± , 1 1 2 ± , 0 3 2 ± , 1 3 2 ± within the QCD sum rule technique. In order to obtain reliable results with this method, it is indispensable to establish a valid Borel window, where the operator product expansion (OPE) converges and the presumed ground state pole dominates the sum rules. By constructing the sum rules from the difference of two carefully chosen independent correlators and calculating the OPE up to dimension 14, such a Borel window can be established. This then leads to our main conclusion that the state with qantum numbers 0 3 2 + appears to be the most probable candidate for the experimentally observed Θ + ( 1540 ) . Furthermore, states with 0 1 2 − , 1 1 2 − , 1 3 2 + are also obtained at slightly higher mass regions.

Possible quantum numbers of the pentaquarkΘ+(1540)in QCD sum rules

The QCD sum rule technique is employed to investigate pentaquark states with strangeness $S=+1$ and $I{J}^{\ensuremath{\pi}}=0{\frac{1}{2}}^{\ifmmode\pm\else\textpm\fi{}}$, $1{\frac{1}{2}}^{\ifmmode\pm\else\textpm\fi{}}$, $0{\frac{3}{2}}^{\ifmmode\pm\else\textpm\fi{}}$, $1{\frac{3}{2}}^{\ifmmode\pm\else\textpm\fi{}}$. Throughout the calculation, emphasis is laid on the establishment of a valid Borel window, which corresponds to a region of the Borel mass, where the operator product expansion converges and the presumed ground state pole dominates the sum rules. Such a Borel window is achieved by constructing the sum rules from the difference of two independent correlators and by calculating the operator product expansion up to dimension 14. Furthermore, we discuss the possibility of the contamination of the sum rules by possible $KN$ scattering states. As a result, we conclude that the $0{\frac{3}{2}}^{+}$ state seems to be the most probable candidate for the experimentally observed ${\ensuremath{\Theta}}^{+}(1540)$, while we also obtain states with $0{\frac{1}{2}}^{\ensuremath{-}}$, $1{\frac{1}{2}}^{\ensuremath{-}}$, $1{\frac{3}{2}}^{+}$ at somewhat higher mass regions.

Spin-3/2pentaquark in QCD sum rules

The QCD sum rule method is formulated for the strangeness $+1$ pentaquark baryon with isospin $I=0$ and spin-parity ${J}^{\ensuremath{\pi}}={\frac{3}{2}}^{\ifmmode\pm\else\textpm\fi{}}$. The spin-$\frac{3}{2}$ states are considered to be narrower than the spin-$\frac{1}{2}$ ones, and thus may provide a natural explanation for the experimentally observed narrow width of ${\ensuremath{\Theta}}^{+}$. In order to obtain reliable results in QCD sum rule calculations, we stress the importance of establishing a wide Borel window, where convergence of the operator product expansion and sufficient low-mass strength of the spectral function are guaranteed. To this end, we employ the difference of two independent correlators so that the high-energy continuum contribution is suppressed. The stability of the physical quantities against the Borel mass is confirmed within the Borel window. It is found that the sum rule gives positive evidence for the $(I,{J}^{\ensuremath{\pi}})=(0,{\frac{3}{2}}^{+})$ state with a mass of about $1.4\ifmmode\pm\else\textpm\fi{}0.2\text{ }\text{ }\mathrm{GeV}$, while we cannot extract any evidence for the $(0,{\frac{3}{2}}^{\ensuremath{-}})$ state.

The meson [formula omitted] as a tetraquark state

We test the validity of the QCD sum rules applied to the meson Z+(4430), by considering a diquark–antidiquark type of current with JP=0− and with JP=1−. We find that, with the studied currents, it is possible to find an acceptable Borel window. In such a Borel window we have simultaneously a good OPE convergence and a pole contribution which is bigger than the continuum contribution. We get mZ=(4.52±0.09) GeV and mZ=(4.84±0.14) GeV for the currents with JP=0− and JP=1−, respectively. We conclude that the QCD sum rules results favors JP=0− quantum numbers for the Z+(4430) meson.

Sigma meson in pole-dominated QCD sum rules

The properties of the $\ensuremath{\sigma}$ meson are studied using the QCD sum rules for tetraquark operators. In the SU(3) chiral limit, we separately investigate SU(3) singlet and octet tetraquark states as constituents of the $\ensuremath{\sigma}$ meson and discuss their roles for the classification of the light scalar nonets $\ensuremath{\sigma}$, ${f}_{0}$, ${a}_{0}$, and $\ensuremath{\kappa}$ as candidates of tetraquark states. All of our analyses are performed in the suitable Borel window which is indispensable to avoid the pseudopeak artifacts outside of the Borel window. We can set up acceptably wide Borel windows after preparing favorable linear combinations of operators and including the dimension 12 terms in the operator-product expansion. Taking into account the possible large widths, we evaluate masses for singlet and octet states as 700--850 and 600--750 MeV, respectively, although the octet operators have a smaller overlap with the tetraquark states than the singlet case, which requires careful interpretations. The splitting of the singlet and octet states emerges from the number of the $\overline{q}q$ annihilation diagrams, which include the color singlet annihilation processes $qq\overline{q}\overline{q}\ensuremath{\rightarrow}(q\overline{q}{)}_{1}$ and the color octet annihilation processes $qq\overline{q}\overline{q}\ensuremath{\rightarrow}G(q\overline{q}{)}_{8}$. The mass of the $\ensuremath{\sigma}$ meson is evaluated as 600--800 MeV, which is much closer to the experimental value $\ensuremath{\sim}500\text{ }\text{ }\mathrm{MeV}$ than the mass evaluated by 2-quark correlator analyses, $\ensuremath{\sim}1.0\text{ }\text{ }\mathrm{GeV}$. This indicates that the tetraquark state shares a larger fraction in the $\ensuremath{\sigma}$ meson than ordinary two quark meson states.

Width of exotics from QCD sum rules : Tetraquarks or molecules?

(4250) using QCD sum rules. Extending previous analyses regarding these states as diquark-antiquark states or molecules of D mesons, we introduce the Breit-Wigner function in the pole term.We find that introducing the width increases the mass at small Borel window region. Using theoperator-product expansion up to dimension 8, we find that the sum rules based on interpolatingcurrent with molecular components give a stable Borel curve from which both the masses andwidths of these resonances can be well obtained. Thus the QCD sum rule approach strongly favorsthe molecular description of these states.

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.

Positive and negative-parity flavor-octet baryons in coupled QCD sum rules

We apply the method of the QCD sum rule, in which positive- and negative-parity baryons couple with each other, to the flavor-octet hyperons and investigate the parity splittings. We also reexamine the nucleon in the method, which was studied in our previous paper, by carefully choosing the Borel weight. Both in the nucleon and hyperon channels the obtained sum rules turn out to have a very good Borel stability and also have a Borel window, an energy region in which the OPE converges and the pole contribution dominates over the continuum contribution. The predicted masses of the positive- and negative-parity baryons reproduce the experimental ones fairly well in the $\ensuremath{\Lambda}$ and $\ensuremath{\Sigma}$ channels, if we assign the $\ensuremath{\Lambda}(1670)$ and the $\ensuremath{\Sigma}(1620)$ to the parity partners of the $\ensuremath{\Lambda}$ and the $\ensuremath{\Sigma}$, respectively. This implies that the $\ensuremath{\Lambda}(1405)$ is not the parity partner of the $\ensuremath{\Lambda}$ and may be a flavor-singlet or exotic state. In the $\ensuremath{\Xi}$ channel, the sum rule predicts the mass of the negative-parity state to be about 1.8 GeV, which leads to two possibilities; one is that the observed state with the closest mass, $\ensuremath{\Xi}(1690)$, is the parity partner and the other is that the parity partner is not yet found but exists around 1.8 GeV.

Exotic Hadron in Pole-Dominated QCD Sum Rules

We study pentaquark Θ(I = 0, J = 1/2) in the QCD sum rules emphasizing that we can not extract any properties of the pentaquark outside of the Borel window. To find the appropriate Borel window, we prepare a favorable set up of the correlators and carry out the operator product expansion up to dimension 15 within factorization hypothesis. Our procedures reduce the unwanted high energy contaminations and enhance the low energy correlation. In the Borel window, independent analyses for the chiral-even and odd sum rules give the consistent values of the Θ mass, 1.68±0.22 GeV, and the residue. The parity is found to be positive.

Pentaquark state in pole-dominated QCD sum rules

We propose a new approach in QCD sum rules applied for exotic hadrons with a number of quarks, exemplifying the pentaquark Theta^{+} (I=0,J=1/2) in the Borel sum rule. Our approach enables reliable extraction of the pentaquark properties from the sum rule with good stability in a remarkably wide Borel window. The appearance of its valid window originates from a favorable setup of the correlation functions with the aid of it chirality of the interpolating fields on the analogy of the Weinberg sum rule for the vector currents. Our setup leads to large suppression of the continuum contributions which have spoiled the Borel stability in the previous analyses, and consequently enhances importance of the higher-dimensional contributions of the OPE, which are indispensable for investigating the pentaquark properties. Implementing the OPE analysis up to dimension 15, we find that the sum rules for the chiral-even and odd parts independently give the Theta^{+} mass of 1.68 pm 0.22 GeV with uncertainties of the condensate values. Our sum rule indeed gives rather flat Borel curves almost independent of the continuum thresholds both for the mass and pole residue. Finally, we also discuss possible isolation of the observed states from the KN scattering state on view of chiral symmetry.

ρ−ωmixing in asymmetric nuclear matter via a QCD sum rule approach

We evaluate the operator product expansion (OPE) for a mixed correlator of the isovector and isoscalar vector currents in the background of the nucleon density with intrinsic isospin asymmetry (i.e., excess of neutrons over protons) and match it with its imaginary part, given by resonances and continuum, via the dispersion relation. The leading density-dependent contribution to $\ensuremath{\rho}\ensuremath{-}\ensuremath{\omega}$ mixing is due to the scattering term, which turns out to be larger than any density dependent piece in the OPE. We estimate that the asymmetric density of ${n}_{n}\ensuremath{-}{n}_{p}\ensuremath{\sim}2.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}{\mathrm{fm}}^{\ensuremath{-}3}$ induces the amplitude of $\ensuremath{\rho}\ensuremath{-}\ensuremath{\omega}$ mixing, equal in magnitude to the mixing amplitude in vacuum, with the constructive interference for positive and destructive for negative values of ${n}_{n}\ensuremath{-}{n}_{p}.$ We revisit sum rules for vector meson masses at finite nucleon density to point out the numerical importance of the screening term in the isoscalar channel, which turns out to be one order of magnitude larger than any density-dependent condensates over the Borel window. This changes the conclusions about the density dependence of ${m}_{\ensuremath{\omega}},$ indicating $\ensuremath{\sim}40\mathrm{MeV}$ increase at nuclear saturation density.

Vector mesons in nuclear medium with small three momentum, a QCD sum rule approach

Using the QCD Operator Product Expansion, we derive the real part of the transverse and longitudinal vector vector correlation function with the ϱ, ω quantum numbers to leading order in density and in q 2 at −gw 2 → ∞. To dimension 6, only twist-2 and 4 operators contribute. These OPE, through the energy dispersion relation, provide model independent constraints for the q dependence of the vector meson spectral density in nuclear medium. We further make a QCD sum rule type of analysis to extract the momentum dependence of the vector meson dispersion relation in medium. The contributions from twist-2 operators are added up to infinite order to check the validity of the OPE at the relevant Borel window.