Finite Energy QCD Sum Rules Research Articles

QCD bounds on leading-order hadronic vacuum polarization contributions to the muon anomalous magnetic moment

QCD bounds on the leading-order (LO) hadronic vacuum polarization (HVP) contribution to the anomalous magnetic moment of the muon [aμHVP,LO, aμ=(g−2)μ/2] are determined by imposing Hölder inequalities and related inequality constraints on systems of finite-energy QCD sum rules. This novel methodology is complementary to lattice QCD and data-driven approaches to determining aμHVP,LO. For the light-quark (u, d, s) contributions up to five-loop order in perturbation theory in the chiral limit, LO in light-quark mass corrections, next-to-leading order in dimension-four QCD condensates, and to LO in dimension-six QCD condensates, we find that (657.0±34.8)×10−10≤aμHVP,LO≤(788.4±41.8)×10−10, bridging the range between lattice QCD and data-driven values. Published by the American Physical Society 2024

Nucleon Resonance Masses from QCD Sum Rules

We calculate the masses of the nucleon resonances [N+(1440), N-(1535), N- (1650), N+(1710), N+(1880), N-(1895), N+(2100)] using an adaptation of the method of finite energy QCD sum rules (FESR) introduced recently in the determination of the masses of the vector meson resonances. This variation of finite energy sum rules involves as usual integration over a closed contour consisting of a circle of a large radius R and a cut in the complex energy (squared) plane. A specific integration kernel is used, however, which minimizes the integral over the cut. We obtain remarkably stable results over a wide range R, where R is the radius of the integration contour. The sum rule predictions agree well with the experimental values. We make use of the nucleonic correlator discussed in our previous FESR calculation of the nucleon mass.

Nucleon axial coupling constant in a magnetar environment

Using appropriate QCD finite energy sum rules, we discuss the influence of an external magnetic field and baryonic density on the axial-vector coupling constant gA. This scenario corresponds to a magnetar environment. We found that gA decreases both as function of the magnetic field strength and the baryonic density. It turns out that at the nuclear density ρ0 the axial-vector coupling takes the value gA⁎≈0.92. Although gA decreases in general with the magnetic field intensity, gA⁎ does not change in a relevant way with the magnetic field.

Isovector meson masses from QCD sum rules

We present a calculation of the masses of the isovector mesons (vector, scalar and pseudoscalar including the established recurrences) using a new method of finite energy QCD sum rules. The method is based on the idea of choosing a suitable integration kernel which minimizes the occurring integral over the cut in the complex energy (squared) plane. We obtain remarkably stable results in a wide range [Formula: see text], where [Formula: see text] is the radius of the integration contour. The sum rule predictions agree with the experimental values within the expected accuracy showing that QCD describes single resonances.

Magnetic field dependence of nucleon parameters from QCD sum rules

Finite energy QCD sum rules involving nucleon current correlators are used to determine several QCD and hadronic parameters in the presence of an external, uniform, large magnetic field. The continuum hadronic threshold $s_0$, nucleon mass $m_N$, current-nucleon coupling $\lambda_N$, transverse velocity $v_\perp$, the spin polarization condensate $\langle\bar q\sigma_{12} q\rangle$, and the magnetic susceptibility of the quark condensate $\chi_q$, are obtained for the case of protons and neutrons. Due to the magnetic field, and charge asymmetry of light quarks up and down, all the obtained quantities evolve differently with the magnetic field, for each nucleon or quark flavor. With this approach it is possible to obtain the evolution of the above parameters up to a magnetic field strength $eB < 1.4$ GeV$^2$.

Precision Determination of the Light Quark Masses from QCD Finite Energy Sum Rules

The light quark masses are determined from a QCD finite energy sum rule, using the pseudoscalar correlator to six-loop order in perturbative QCD, with the leading vacuum condensates and higher order quark mass corrections included. Both the fixed order perturbation theory (FOPT) method and contour improved perturbation theory (CIPT) method are explored. The results in the latter framework exhibit good convergence and stability in the window s0 = 3.0 5.0 GeV2 for the strange quark and s0 = 1.5 4.0 GeV2 for the up and down quarks; where s0 is the radius of the integration contour in the complex s-plane. The results are: , and the sum is . These proceedings critically explore how the current results - computed precisely in a modern computer language, Mathematica compare to past determinations in literature.

Up- and down-quark masses from QCD sum rules

The QCD up- and down-quark masses are determined from an optimized QCD Finite Energy Sum Rule (FESR) involving the correlator of axial-vector current divergences. In the QCD sector this correlator is known to five loop order in perturbative QCD (PQCD), together with non-perturbative corrections from the quark and gluon condensates. This FESR is designed to reduce considerably the systematic uncertainties arising from the hadronic spectral function. The determination is done in the framework of both fixed order and contour improved perturbation theory. Results from the latter, involving far less systematic uncertainties, are: {overline{m}}_uleft(2 mathrm{GeV}right)=left(2.6pm 0.4right) MeV, {overline{m}}_dleft(2 mathrm{GeV}right)=left(5.3pm 0.4right) MeV, and the sum {overline{m}}_{ud}equiv left({overline{m}}_u+{overline{m}}_dright)/2 , is {overline{m}}_{ud}left(2 mathrm{GeV}right)=left(3.9pm 0.3right) MeV.

QCD determination of the magnetic field dependence of QCD and hadronic parameters

A set of four Finite Energy QCD sum rules are used to determine the magnetic field dependence of the sum of the up- and down-quark masses of QCD, $(m_u + m_d)$, the pion decay constant $f_\pi$, the pion mass $m_\pi$, the gluon condensate, $\langle 0| \alpha_s\, G^2|0\rangle$, and the squared energy threshold for the onset of perturbative QCD, $s_0$, related to the Polyakov loop of lattice QCD.

Determination of the temperature dependence of the up- and down-quark masses in QCD

The temperature dependence of the sum of the QCD up- and down-quark masses, (mu[Formula: see text]+[Formula: see text]md) and the pion decay constant, f[Formula: see text], are determined from two thermal finite energy QCD sum rules for the pseudoscalar-current correlator. This quark mass remains mostly constant for temperatures well below the critical temperature for deconfinement/chiral-symmetry restoration. As this critical temperature is approached, the quark mass increases sharply with increasing temperature. This increase is far more pronounced if the temperature dependence of the pion mass (determined independently from other methods) is taken into account. The behavior of f[Formula: see text](T) is consistent with the expectation from chiral symmetry, i.e. that it should follow the thermal dependence of the quark condensate, independently of the quark mass.

Tests of quark–hadron duality in τ-decays

An exhaustive number of QCD finite energy sum rules for [Formula: see text]-decay together with the latest updated ALEPH data is used to test the assumption of global duality. Typical checks are the absence of the dimension d = 2 condensate, the equality of the gluon condensate extracted from vector or axial vector spectral functions, the Weinberg sum rules, the chiral condensates of dimensions d = 6 and d = 8, as well as the extraction of some low-energy parameters of chiral perturbation theory. Suitable pinched linear integration kernels are introduced in the sum rules in order to suppress potential quark–hadron duality violations and experimental errors. We find no compelling indications of duality violations in hadronic [Formula: see text]-decay in the kinematic region above s [Formula: see text] 2.2 GeV2 for these kernels.

Phi meson spectral moments and QCD condensates in nuclear matter

A detailed analysis of the lowest two moments of the ϕ meson spectral function in vacuum and nuclear matter is performed. The consistency is examined between the constraints derived from finite energy QCD sum rules and the spectra computed within an improved vector dominance model, incorporating the coupling of kaonic degrees of freedom with the bare ϕ meson. In the vacuum, recent accurate measurements of the e+e−→K+K− cross section allow us to determine the spectral function with high precision. In nuclear matter, the modification of the spectral function can be described by the interactions of the kaons from ϕ→KK‾ with the surrounding nuclear medium. This leads primarily to a strong broadening and an asymmetric deformation of the ϕ meson peak structure. We confirm that, both in vacuum and nuclear matter, the zeroth and first moments of the corresponding spectral functions satisfy the requirements of the finite energy sum rules to a remarkable degree of accuracy. Limits on the strangeness sigma term of the nucleon are examined in this context. Applying our results to the second moment of the spectrum, we furthermore discuss constraints on four-quark condensates and the validity of the commonly used ground state saturation approximation.

Moments of ϕ meson spectral functions in vacuum and nuclear matter

Moments of the ϕ meson spectral function in vacuum and in nuclear matter are analyzed, combining a model based on chiral SU(3) effective field theory (with kaonic degrees of freedom) and finite-energy QCD sum rules. For the vacuum we show that the spectral density is strongly constrained by a recent accurate measurement of the e+e−→K+K− cross section. In nuclear matter the ϕ spectrum is modified by interactions of the decay kaons with the surrounding nuclear medium, leading to a significant broadening and an asymmetric deformation of the ϕ meson peak. We demonstrate that both in vacuum and nuclear matter, the first two moments of the spectral function are compatible with finite-energy QCD sum rules. A brief discussion of the next-higher spectral moment involving strange four-quark condensates is also presented.

Quark deconfinement and gluon condensate in a weak magnetic field from QCD sum rules

We study QCD finite energy sum rules (FESR) for the axial-vector current correlator in the presence of a magnetic field, in the weak field limit and at zero temperature. We find that the perturbative QCD and the hadronic contribution to the sum rules get explicit magnetic-field-dependent corrections and that these in turn induce a magnetic field dependence on the deconfinement phenomenological parameter ${s}_{0}$ and on the gluon condensate. The leading corrections turn out to be quadratic in the field strength. We find from the dimension $d=2$ first FESR that the magnetic field dependence of ${s}_{0}$ is proportional to the absolute value of the light-quark condensate. Hence, it increases with increasing field strength. This implies that the parameters describing chiral symmetry restoration and deconfinement behave similarly as functions of the magnetic filed. Thus, at zero temperature the magnetic field is a catalyzing agent of both chiral symmetry breaking and confinement. From the dimension $d=4$ second FESR we obtain the behavior of the gluon condensate in the presence of the external magnetic field. This condensate also increases with increasing field strength.

Determination of the gluon condensate from data in the charm-quark region

The gluon condensate, $$ \left\langle \frac{\alpha_s}{\pi }{G}^2\right\rangle $$ , i.e. the leading order power correction in the operator product expansion of current correlators in QCD at short distances, is determined from e+e− annihilation data in the charm-quark region. This determination is based on finite energy QCD sum rules, weighted by a suitable integration kernel to (i) account for potential quark-hadron duality violations, (ii) enhance the contribution of the well known first two narrow resonances, the J/ψ and the ψ(2S), while quenching substantially the data region beyond, and (iii) reinforce the role of the gluon condensate in the sum rules. By using a kernel exhibiting a singularity at the origin, the gluon condensate enters the Cauchy residue at the pole through the low energy QCD expansion of the vector current correlator. These features allow for a reasonably precise determination of the condensate, i.e. $$ \left\langle \frac{\alpha_s}{\pi }{G}^2\right\rangle =0.037\pm 0.015 $$ GeV4.

Chiral sum rules and vacuum condensates from tau-lepton decay data

QCD finite energy sum rules, together with the latest updated ALEPH data on hadronic decays of the tau-lepton are used in order to determine the vacuum condensates of dimension $d=2$ and $d=4$. These data are also used to check the validity of the Weinberg sum rules, and to determine the chiral condensates of dimension $d=6$ and $d=8$, as well as the chiral correlator at zero momentum, proportional to the counter term of the ${\cal{O}}(p^4)$ Lagrangian of chiral perturbation theory, $\bar{L}_{10}$. Suitable (pinched) integration kernels are introduced in the sum rules in order to suppress potential quark-hadron duality violations. We find no compelling indications of duality violations in the kinematic region above $s \simeq 2.2$ GeV$^2$ after using pinched integration kernels.

Analytical determination of the QCD quark masses

The current status of determinations of the QCD running quark masses is reviewed. Emphasis is on recent progress on analytical precision determinations based on finite energy QCD sum rules. A critical discussion of the merits of this approach over other alternative QCD sum rules is provided. Systematic uncertainties from both the hadronic and the QCD sector have been recently identified and dealt with successfully, thus leading to values of the quark masses with unprecedented accuracy. Results currently rival in precision with lattice QCD determinations.

Quark mass determinations in QCD

Recent progress on QCD sum rule determinations of the light and heavy quark masses is reported. In the light quark sector a major breakthrough has been made recently in connection with the historical systematic uncertainties due to a lack of experimental information on the pseudoscalar resonance spectral functions. It is now possible to suppress this contribution to the 1% level by using suitable integration kernels in Finite Energy QCD sum rules. This allows to determine the up-, down-, and strange-quark masses with an unprecedented precision of some 8–10%. In the heavy quark sector, the availability of experimental data in the vector channel, and the use of suitable multipurpose integration kernels allows to increase the accuracy of the charm- and bottom-quarks masses to the 1% level.

B meson decay constants f B c $$ {f}_{B_c} $$ , f B s $$ {f}_{B_s} $$ and f B from QCD sum rules

Finite energy QCD sum rules with Legendre polynomial integration kernels are used to determine the heavy meson decay constant $f_{B_c}$, and revisit $f_B$ and $f_{B_s}$. Results exhibit excellent stability in a wide range of values of the integration radius in the complex squared energy plane, and of the order of the Legendre polynomial. Results are $f_{B_c} = 528 \pm 19$ MeV, $f_B = 186 \pm 14$ MeV, and $f_{B_s} = 222 \pm 12$ MeV.

Dimuon production from in-medium rho decays from QCD sum rules

We compute the dimuon-excess invariant mass distribution at the rho-meson peak in the context of relativistic heavy-ion collisions at SPS energies. The temperature, T, and chemical potential, mu, dependent parameters describing the rho-meson --its width, mass and leptonic decay constant-- are determined from finite energy QCD sum rules. Results show that the rho-meson width increases whereas its mass and leptonic decay constant decrease near the (chemical potential-dependent) critical temperature T_c(mu) for chiral symmetry restoration/quark-gluon deconfinement. As a consequence, starting from T_c(mu), for a short lived cooling the main effect is a broadening of the dimuon distribution. However, when the evolution brings the system to a lower freeze-out temperature, with the rho-meson parameters approaching their vacuum values, the dimuon distribution shows a broad peak centered at the vacuum rho-meson mass. For even lower freeze-out temperatures the peak becomes less prominent since the thermal phase space factor suppresses the distribution for larger values of the invariant dimuon mass, given that the average temperature is smaller. The dimuon distribution exhibits a non-trivial behavior with mu. For small mu values the distribution broadens with increasing mu becoming a bit steeper at low invariant masses for larger values of mu. Our results are in very good agreement with data from the NA60 Collaboration.

Hadronic contribution to the muong−2factor

The lowest order hadronic contribution to the $g-2$ factor of the muon is analyzed in the framework of the operator product expansion at short distances, and a QCD finite energy sum rule designed to quench the role of the $e^+ e^-$ data. This procedure reduces the discrepancy between experiment and theory, $\Delta a_\mu \equiv a^{EXP}_\mu - a^{SM}_\mu$, from $\Delta a_\mu = 28.7 (8.0) \times 10^{-10}$ to $\Delta a_\mu = 19.2 (8.0) \times 10^{-10}$, i.e. without changing the uncertainty.