Pion Decay Constant Research Articles

Scalar radius of the pion in the Kroll-Lee-Zumino renormalizable theory

The Kroll-Lee-Zumino renormalizable Abelian quantum field theory of pions and a massive $\ensuremath{\rho}$-meson is used to calculate the scalar radius of the pion at next-to-leading (one-loop) order in perturbation theory. Because of renormalizability, this determination involves no free parameters. The result is $⟨{r}_{\ensuremath{\pi}}^{2}{⟩}_{s}=0.40\text{ }\text{ }{\mathrm{fm}}^{2}$. This value gives for ${\overline{\ensuremath{\ell}}}_{4}$, the low energy constant of chiral perturbation theory, ${\overline{\ensuremath{\ell}}}_{4}=3.4$, and ${F}_{\ensuremath{\pi}}/F=1.05$, where $F$ is the pion decay constant in the chiral limit. Given the level of accuracy in the masses and the $\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\pi}$ coupling, the only sizable uncertainty in this result is due to the (uncalculated) next-to-next-to-leading order contribution.

Form-factors and current correlators: chiral couplingsL10r(μ) andC87r(μ) at NLO in 1/NC

Using the resonance chiral theory Lagrangian, we perform a calculation of the vector and axial-vector two-point functions at the next-to-leading order (NLO) in the 1/NC expansion. We have analyzed these correlators within the single-resonance approximation and have also investigated the corrections induced by a second multiplet of vector and axial-vector resonance states. Imposing the correct QCD short-distance constraints, one determines the difference of the two correlators Π(t) ≡ ΠVV(t)−ΠAA(t) in terms of the pion decay constant and resonance masses. Its low momentum expansion fixes then the low-energy chiral couplings L10 and C87 at NLO, keeping full control of their renormalization scale dependence. At μ0 = 0.77 GeV, we obtain L10r(μ0) = (−4.4±0.9) · 10−3 and C87r(μ0) = (3.1±1.1) · 10−5.

Isotropic and anisotropic lattice spacing corrections forI=2ππscattering from effective field theory

The calculation of the finite lattice spacing corrections for $\mathrm{I}=2$ $\ensuremath{\pi}\ensuremath{\pi}$ scattering is carried out for isotropic and anisotropic Wilson lattice actions. Pion masses and decay constants are also determined in this context. These results correct the phase shift calculated from the lattice, which is connected to the scattering length and effective range in this low energy scattering process. When in terms of the lattice-physical parameters for either Wilson action, these lattice spacing effects first appear at the next-to-leading order counterterms.

πandσmesons at finite temperature and density in the Nambu–Jona-Lasinio model with dimensional regularization

Dynamical symmetry breaking and meson masses are studied in the Nambu--Jona-Lasinio model at finite temperature and chemical potential using dimensional regularization. Since the model is not renormalizable in four space-time dimensions, physical results and parameters depend on the regularization method. Following the imaginary-time formalism, we introduce the temperature $T$ and the chemical potential $\ensuremath{\mu}$. The parameters of the model are fixed by calculating the pion mass and decay constant in dimensional regularization at $T=\ensuremath{\mu}=0$.

Extension of the Nambu–Jona-Lasinio model predictions at high densities and temperatures using an implicit regularization scheme

Traditional cutoff regularization schemes of the Nambu--Jona-Lasinio model limit the applicability of the model to energy-momentum scales much below the value of the regularizing cutoff. In particular, the model cannot be used to study quark matter with Fermi momenta larger than the cutoff. In the present work, an extension of the model to high temperatures and densities recently proposed by Casalbuoni, Gatto, Nardulli, and Ruggieri is used in connection with an implicit regularization scheme. This is done by making use of scaling relations of the divergent one-loop integrals that relate these integrals at different energy-momentum scales. Fixing the pion decay constant at the chiral symmetry breaking scale in the vacuum, the scaling relations predict a running coupling constant that decreases as the regularization scale increases, implementing in a schematic way the property of asymptotic freedom of quantum chromodynamics. If the regularization scale is allowed to increase with density and temperature, the coupling will decrease with density and temperature, extending in this way the applicability of the model to high densities and temperatures. These results are obtained without specifying an explicit regularization. As an illustration of the formalism, numerical results are obtained for the finite density and finite temperature quark condensate and applied to the problem of color superconductivity at high quark densities and finite temperature.

Consequences of the partial restoration of chiral symmetry in an AdS/QCD model

Chiral symmetry is an essential concept in understanding QCD at low energy. We treat the chiral condensate, which measures the spontaneous breaking of chiral symmetry, as a free parameter to investigate the effect of partially restored chiral symmetry on the physical quantities in the framework of an AdS/QCD model. We observe an interesting scaling behavior among the nucleon mass, pion decay constant, and chiral condensate. We propose a phenomenological way to introduce the temperature dependence of a physical quantity in the AdS/QCD model with the thermal AdS metric.

Role of theσresonance in determining the convergence of chiral perturbation theory

The dimensionless parameter $\ensuremath{\xi}={M}_{\ensuremath{\pi}}^{2}/(16{\ensuremath{\pi}}^{2}{F}_{\ensuremath{\pi}}^{2})$, where ${F}_{\ensuremath{\pi}}$ is the pion decay constant and ${M}_{\ensuremath{\pi}}$ is the pion mass, is expected to control the convergence of chiral perturbation theory applicable to QCD. Here we demonstrate that a strongly coupled lattice gauge theory model with the same symmetries as two-flavor QCD but with a much lighter $\ensuremath{\sigma}$-resonance is different. We first confirm that the leading low-energy constants appearing in the chiral Lagrangian are the same when calculated from the $p$-regime and the $ϵ$-regime as expected. However, $\ensuremath{\xi}\ensuremath{\lesssim}0.002$ is necessary before 1-loop chiral perturbation theory predicts the data within 1%. For $\ensuremath{\xi}>0.0035$ the data begin to deviate dramatically from 1-loop chiral perturbation theory predictions. We argue that this qualitative change is due to the presence of a light $\ensuremath{\sigma}$-resonance in our model. Our findings may be useful for lattice QCD studies.

Precision tests with Kl3 and Kl2 decays

In 2000, the analysis indicated that the unitarity relation, |Vud|2 + |Vus|2 + |Vub|2 = 1, might be broken at the 2.3σ level. At that time, however, the |Vus| was inferred from the old experimental data. Since then, a great experimental and theoretical effort has been invested to understand the source of that deviation. Thanks to the new and improved measurements by BNL-E865, KLOE, KTeV, ISTRA+ and NA48 the old Kl3 decay rate got shifted so that the new |Vus| is now consistent with unitarity. On the theory side, much progress in the lattice QCD community has been made in order to tame the systematic uncertainties related to the computation of the Kl3 form factors.This joint progress allowed to assess the validity of the CKM unitarity relation at the level less than 1%. Undergoing lattice studies will corroborate and improve this scenario. The key challenge will be to simulate lighter pions in the region in which ChPT predictions apply. Also interesting is the recent progress in accurately computing the kaon and pion decay constants on the lattice, which then give us access to |Vus| and |Vud| from the corresponding leptonic decays.In addition, we argue that the Kl3 and Kl2 decays offer the possibility to test various scenarios of physics beyond Standard Model.

Optimised Dirac operators on the lattice: construction, properties and applications

We review a number of topics related to block variable renormalisation group transformations of quantum fields on the lattice, and to the emerging perfect lattice actions. We first illustrate this procedure by considering scalar fields. Then we proceed to lattice fermions, where we discuss perfect actions for free fields, for the Gross-Neveu model and for a supersymmetric spin model. We also consider the extension to perfect lattice perturbation theory, in particular regarding the axial anomaly and the quark gluon vertex function. Next we deal with properties and applications of truncated perfect fermions, and their chiral correction by means of the overlap formula. This yields a formulation of lattice fermions, which combines exact chiral symmetry with an optimisation of further essential properties. We summarise simulation results for these so-called overlap-hypercube fermions in the two-flavour Schwinger model and in quenched QCD. In the latter framework we establish a link to Chiral Perturbation Theory, both, in the p-regime and in the ϵ-regime. In particular we present an evaluation of the leading Low Energy Constants of the chiral Lagrangian – the chiral condensate and the pion decay constant – from QCD simulations with extremely light quarks.

Quark mass and the masses of Goldstone bosons

Based on the Dyson-Schwinger Equations (DSEs) of QCD in the "rainbow" approximation, the fully dressed quark propagator S(f)(p) is investigated, and then an algebraic parametrization form of the propagator is obtained as a solution of the equations. The dressed quark amplitudes A(f) and B(f) which built up the fully dressed quark propagator, and the dynamical running masses M(f), which is defined by A(f) and B(f) for light quarks u, d and s, are calculated, respectively. Using the predicted current masses m(f), quark local vacuum condensates, and our predicted value of pion decay constant, the masses of Goldstone bosons K, pi and eta and their in-medium values are also evaluated. Our predictions fit to data and to many other different calculations quite well. The numerical results show that the mass of quark is dependent of its momentum p(2). The fully dressed quark amplitudes A(f) and B(f) have correct behaviors and can be used for many purposes in our future researches on non-perturbative QCD.

Finite-volume correction to the pion decay constant in the ϵ-regime

In the chiral limit of QCD, the pion decay constant F can be extracted from lattice gauge theory by means of a coupling to isospin chemical potential. Here we compute the leading correction due to finite volume in the -expansion of chiral perturbation theory. A comparison is made to recent Monte Carlo data.

Pion form factor in the chiral limit of a hard-wall AdS/QCD model

We describe a formalism to calculate form factor and charge density distribution of the pion in the chiral limit using the holographic dual model of QCD with hard-wall cutoff. We introduce two conjugate pion wave functions and present analytic expressions for these functions and for the pion form factor. They allow us to relate such observables as the pion decay constant and the pion charge electric radius to the values of chiral condensate and hard-wall cutoff scale. The evolution of the pion form factor to large values of the momentum transfer is discussed, and results are compared to existing experimental data.

Pion scattering poles and chiral symmetry restoration

Using unitarized chiral perturbation theory methods, we perform a detailed analysis of the $\ensuremath{\pi}\ensuremath{\pi}$ scattering poles ${f}_{0}(600)$ and $\ensuremath{\rho}(770)$ behavior when medium effects such as temperature or density drive the system towards chiral symmetry restoration. In the analysis of real poles below threshold, we show that it is crucial to extend properly the unitarized amplitudes so that they match the perturbative Adler zeros. Our results do not show threshold enhancement effects at finite temperature in the ${f}_{0}(600)$ channel, which remains as a pole of broad nature. We also implement $T=0$ finite-density effects related to chiral symmetry restoration, by varying the pole position with the pion decay constant. Although this approach takes into account only a limited class of contributions, we reproduce the expected finite-density restoration behavior, which drives the poles towards the real axis, producing threshold enhancement and $\ensuremath{\pi}\ensuremath{\pi}$ bound states. We compare our results with several model approaches and discuss the experimental consequences, both in relativistic heavy ion collisions and in $\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ and $\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ reactions in nuclei.

Soliton in the global color model with a sophisticated effective gluon propagator

With a sophisticated effective gluon propagator, Maris-Tandy model, we solve the Dyson-Schwinger equation to get the quark propagator and then study the soliton solution in the global color model (GCM). Along the constraints on the parameters fitted to the pion decay constant, we take several sets of parameters and find that some of the properties of soliton can be produced in the GCM soliton model with a special choice of parameters. We also discuss the influences of the parameters and the ultraviolet perturbative term on the property of the soliton. We find that the interaction among quarks is the one with self-adjusting characteristic and only the fine-tuned interaction can generate an appropriate solition, but not that much stronger attraction produces more stable soliton.

Scalar meson properties from D-meson decays

Decay amplitudes of D ( D s ) → f 0 ( 980 ) X , X = π , K , are compared to experimental branching ratios with the aim of singling out the poorly known D → f 0 ( 980 ) transition form factor in these amplitudes. Since the other elements of the amplitudes are either calculable in an effective QCD theory using operator product expansion or are known from experiment (e.g. the pion and kaon decay constants), we can take advantage of these reactions to constrain the transition form factors obtained in relativistic quark models [O.M.A. Leitner, B. El-Bennich, B. Loiseau and J.-P. Dedonder, hep-ph/0609062 ; B. El-Bennich, O.M.A. Leitner, B. Loiseau and J.-P. Dedonder, hep-ph/0609154 ]. In these models, the f 0 ( 980 ) wavefunction requires an unknown size parameter for both its non-strange u ¯ u ( d ¯ d ) and strange s ¯ s components, which we fit to the D ( D s ) decay data.

Hadron spectrum, quark masses, and decay constants from light overlap fermions on large lattices

We present results from a simulation of quenched overlap fermions with Luescher-Weisz gauge field action on lattices up to 24{sup 3}48 and for pion masses down to {approx_equal}250 MeV. Among the quantities we study are the pion, rho, and nucleon masses; the light and strange quark masses; and the pion decay constant. The renormalization of the scalar and axial vector currents is done nonperturbatively in the RI-MOM scheme. The simulations are performed at two different lattice spacings, a{approx_equal}0.1 fm and {approx_equal}0.15 fm, and on two different physical volumes, to test the scaling properties of our action and to study finite volume effects. We compare our results with the predictions of chiral perturbation theory and compute several of its low-energy constants. The pion mass is computed in sectors of fixed topology as well.

Quark gap equation within the analytic approach to QCD

The compatibility between the QCD analytic invariant charge and chiral symmetry breaking is examined in detail. The coupling in question incorporates asymptotic freedom and infrared enhancement into a single expression, and contains only one adjustable parameter with dimension of mass. When inserted into the standard form of the quark gap-equation it gives rise to solutions displaying singular confining behavior at the origin. By relating these solutions to the pion decay constant, a rough estimate of about 880 MeV is obtained for the aforementioned mass-scale.

A new chiral two-matrix theory for Dirac spectra with imaginary chemical potential

We solve a new chiral random two-matrix theory by means of biorthogonal polynomials for any matrix size N. By deriving the relevant kernels we find explicit formulas for all ( n , k ) -point spectral (mixed or unmixed) correlation functions. In the microscopic limit we find the corresponding scaling functions, and thus derive all spectral correlators in this limit as well. We extend these results to the ordinary (non-chiral) ensembles, and also there provide explicit solutions for any finite size N, and in the microscopic scaling limit. Our results give the general analytical expressions for the microscopic correlation functions of the Dirac operator eigenvalues in theories with imaginary baryon and isospin chemical potential, and can be used to extract the tree-level pion decay constant from lattice gauge theory configurations. We find exact agreement with previous computations based on the low-energy effective field theory in the two special cases where comparisons are possible.

Towards a determination of the chiral couplings at NLO in 1/NC: L8r(μ) and C38r(μ)

We present a dispersive method which allows to investigate the low-energy couplings of chiral perturbation theory at the next-to-leading order (NLO) in the 1/N(C) expansion, keeping full control of their renormalization scale dependence. Using the resonance chiral theory Lagrangian, we perform a NLO calculation of the scalar and pseudoscalar two-point functions, within the single-resonance approximation. Imposing the correct QCD short-distance constraints, one determines their difference Pi(t)=Pi_S(t)-Pi_P(t) in terms of the pion decay constant and resonance masses. Its low momentum expansion fixes then the low-energy chiral couplings L_8 and C_38. At mu_0=0.77 GeV, we obtain L_8(mu_0)^{SU(3)} = (0.6+-0.4)10^{-3} and C_{38}(mu_0)^{SU(3)}=(2+-6)10^{-6}.

Chiral Restoration in a Nuclear Medium

Using 500 MeV (d,He π−) pion transfer reactions in recoil free kinematics, pionic 1sstates were populated in the Sn isotopes and their binding energies and widths determined by precision missing mass spectroscopy. Using these data and corresponding ones from iso-scalar light nuclei nuclei, O, Ne and Si, we determined the pion nucleus s-wave strength parameters, b0, b1, ReB0, and ImB0. By comparison of the iso-vector pion nucleon strength, determined from pionic hydrogen X-ray spectroscopy b 1 , with the b1 in a nuclear medium scaled to the density ρ(0), we deduced a scaling factor, the square of the pion decay constant in the vacuum and in nuclear medium, as R = b 1 /b1 = f 2 π(ρ0)/f 2 π = 0.64. Thus from the observed increase of the pion s-wave iso-vector strength in a nuclear medium a reduction of f π, the order parameter of chiral symmetry breaking, is indicated in accordance with theoretical expectations. This finding is supported by recent π and π− scattering experiments. A short outlook is given on a future program at RIBF in RIKEN for precision studies of deeply bound 1s-states in heavy nuclei.