Two-flavor Chiral Perturbation Theory Research Articles

QCD θ-vacuum in a uniform magnetic field

We study the θ-vacuum of QCD using two-flavor chiral perturbation theory (χPT) in the presence of a uniform, background magnetic field calculating the magnetic field-dependent free energy density, the topological density, the topological susceptibility and the fourth cumulant at one-loop order. We find that the topological susceptibility is enhanced by the magnetic field while the fourth topological cumulant is diminished at weak fields and enhanced at larger fields when θ=0. However, in the QCD vacuum with θ≠0, the topological susceptibility can be either monotonically enhanced or diminished relative to their θ-vacuum values. The fourth cumulant also exhibits monotonic enhancement or suppression except for regions of θ near 0 and 2π, where it is both diminished and enhanced. Finally, the topological density is enhanced for all magnetic fields with its relative shift being identical to the relative shift of the up and down quark condensates in the θ-vacuum.

Diamagnetic and paramagnetic phases in low-energy quantum chromodynamics

While it is known that the QCD vacuum in a magnetic background exhibits both diamagnetic and paramagnetic characteristics in the low-energy domain, a systematic investigation of the corresponding phases emerging in the pion-dominated regime is still lacking. Here, within two-flavor chiral perturbation theory, taking into account the pion-pion interaction, we analyze the subtle interplay between zero- and finite-temperature portions in the magnetization and magnetic susceptibility. The dependence of the magnetic susceptibility on temperature and magnetic field strength in the paramagnetic and diamagnetic phase is non-monotonic. Our low-energy analysis complements lattice QCD that is currently operating at higher temperatures and stronger magnetic fields.

Condensates and pressure of two-flavor chiral perturbation theory at nonzero isospin and temperature

We consider two-flavor chiral perturbation theory (chi PT) at finite isospin chemical potential mu _I and finite temperature T. We calculate the effective potential and the quark and pion condensates as functions of T and mu _I to next-to-leading order in the low-energy expansion in the presence of a pionic source. We map out the phase diagram in the mu _I–T plane. Numerically, we find that the transition to the pion-condensed phase is second order in the region of validity of chi PT, which is in agreement with model calculations and lattice simulations. Finally, we calculate the pressure to two-loop order in the symmetric phase for nonzero mu _I and find that chi PT seems to be converging very well.

QCD at finite isospin density: Chiral perturbation theory confronts lattice data

We consider the thermodynamics of three-flavor QCD in the pion-condensed phase at nonzero isospin chemical potential (μI) and vanishing temperature using chiral perturbation theory in the isospin limit. The transition from the vacuum phase to a superfluid phase with a Bose-Einstein condensate of charged pions is shown to be second order and takes place at μI=mπ. We calculate the pressure, isospin density, and energy density to next-to-leading order in the low-energy expansion. Our results are compared with recent high-precision lattice simulations as well as previously obtained results in two-flavor chiral perturbation theory. The agreement between the lattice results and the predictions from three-flavor chiral perturbation theory is very good for μI<200 MeV. For larger values of μI, the agreement between lattice data and the two-flavor predictions is surprisingly good and better than with the three-flavor predictions. Finally, in the limit ms≫mu=md, we show that the three-flavor observables reduce to the two-flavor observables with renormalized parameters. The disagreement between the results for two-flavor and three-flavor χPT can largely be explained by the differences in the measured low-energy constants.

Two-flavor chiral perturbation theory at nonzero isospin: pion condensation at zero temperature

In this paper, we calculate the equation of state of two-flavor finite isospin chiral perturbation theory at next-to-leading order in the pion-condensed phase at zero temperature. We show that the transition from the vacuum phase to a Bose-condensed phase is of second order. While the tree-level result has been known for some time, surprisingly quantum effects have not yet been incorporated into the equation of state. We find that the corrections to the quantities we compute, namely the isospin density, pressure, and equation of state, increase with increasing isospin chemical potential. We compare our results to recent lattice simulations of 2 + 1 flavor QCD with physical quark masses. The agreement with the lattice results is generally good and improves somewhat as we go from leading order to next-to-leading order in chi PT.

Finite density two color chiral perturbation theory revisited

We revisit two-color, two-flavor chiral perturbation theory at finite isospin and baryon density. We investigate the phase diagram obtained varying the isospin and the baryon chemical potentials, focusing on the phase transition occurring when the two chemical potentials are equal and exceed the pion mass (which is degenerate with the diquark mass). In this case, there is a change in the order parameter of the theory that does not lend itself to the standard picture of first order transitions. We explore this phase transition both within a Ginzburg-Landau framework valid in a limited parameter space and then by inspecting the full chiral Lagrangian in all the accessible parameter space. Across the phase transition between the two broken phases the order parameter becomes an SU(2) doublet, with the ground state fixing the expectation value of the sum of the magnitude squared of the pion and the diquark fields. Furthermore, we find that the Lagrangian at equal chemical potentials is invariant under global SU(2) transformations and construct the effective Lagrangian of the three Goldstone degrees of freedom by integrating out the radial fluctuations.

SU(2)low-energy constants from mixed-action lattice QCD

An analysis of the pion mass and pion decay constant is performed using mixed-action Lattice QCD calculations with domain-wall valence quarks on ensembles of rooted, staggered n_f = 2+1 MILC configurations. Calculations were performed at two lattice spacings of b~0.125 fm and b~0.09 fm, at two strange quark masses, multiple light quark masses, and a number of lattice volumes. The ratios of light quark to strange quark masses are in the range 0.1 <= m_l / m_s <= 0.6, while pion masses are in the range 235 < m_\pi < 680 MeV. A two-flavor chiral perturbation theory analysis of the Lattice QCD calculations constrains the Gasser-Leutwyler coefficients bar{l}_3 and bar{l}_4 to be bar{l}_3 = 4.04(40)(+73-55) and bar{l}_4 = 4.30(51)(+84-60). All systematic effects in the calculations are explored, including those from the finite lattice space-time volume, the finite lattice spacing, and the finite fifth dimension in the domain-wall quark action. A consistency is demonstrated between a chiral perturbation theory analysis at fixed lattice spacing combined with a leading order continuum extrapolation, and the mixed-action chiral perturbation theory analysis which explicitly includes the leading order discretization effects. Chiral corrections to the pion decay constant are found to give f_\pi / f = 1.062(26)(+42-40) where f is the decay constant in the chiral limit. The most recent scale setting by the MILC Collaboration yields a postdiction of f_\pi = 128.2(3.6)(+4.4-6.0)(+1.2-3.3) MeV at the physical pion mass.

I=2ππS-wave scattering phase shift from lattice QCD

The ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}$ $s$-wave scattering phase shift is determined below the inelastic threshold using lattice QCD. Calculations were performed at a pion mass of ${m}_{\ensuremath{\pi}}\ensuremath{\sim}390\text{ }\text{ }\mathrm{MeV}$ with an anisotropic ${n}_{f}=2+1$ clover fermion discretization in four lattice volumes, with spatial extent $L\ensuremath{\sim}2.0$, 2.5, 3.0 and 3.9 fm, and with a lattice spacing of ${b}_{s}\ensuremath{\sim}0.123\text{ }\text{ }\mathrm{fm}$ in the spatial direction and ${b}_{t}\ensuremath{\sim}{b}_{s}/3.5$ in the time direction. The phase shift is determined from the energy eigenvalues of ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}$ systems with both zero and nonzero total momentum in the lattice volume using L\"uscher's method. Our calculations are precise enough to allow for a determination of the threshold scattering parameters, the scattering length $a$, the effective range $r$, and the shape parameter $P$, in this channel and to examine the prediction of two-flavor chiral perturbation theory: ${m}_{\ensuremath{\pi}}^{2}ar=3+\mathcal{O}({m}_{\ensuremath{\pi}}^{2}/{\ensuremath{\Lambda}}_{\ensuremath{\chi}}^{2})$. Chiral perturbation theory is used, with the lattice QCD results as input, to predict the scattering phase shift (and threshold parameters) at the physical pion mass. Our results are consistent with determinations from the Roy equations and with the existing experimental phase shift data.

The time-reversal- and parity-violating nuclear potential in chiral effective theory

We derive the parity- and time-reversal-violating nuclear interactions stemming from the QCD theta term and quark/gluon operators of effective dimension 6: quark electric dipole moments, quark and gluon chromo-electric dipole moments, and two four-quark operators. We work in the framework of two-flavor chiral perturbation theory, where a systematic expansion is possible. The different chiral-transformation properties of the sources of time-reversal violation lead to different hadronic interactions. For all sources considered the leading-order potential involves known one-pion exchange, but its specific form and the relative importance of short-range interactions depend on the source. For the theta term, the leading potential is solely given by one-pion exchange, which does not contribute to the deuteron electric dipole moment. In subleading order, a new two-pion-exchange potential is obtained. Its short-range component is indistinguishable from one of two undetermined contact interactions that appear at the same order and represent effects of heavier mesons and other short-range QCD dynamics. One-pion-exchange corrections at this order are discussed as well.

PandTViolating Form Factors of the Deuteron

We calculate the electric-dipole and magnetic-quadrupole form factors of the deuteron that arise as a low-energy manifestation of parity and time-reversal violation in quark-gluon interactions. We consider the QCD vacuum angle and the dimension-six operators that originate from physics beyond the standard model: the quark electric and chromoelectric dipole moments and the gluon chromoelectric dipole moment. Within the framework of two-flavor chiral perturbation theory, we show that in combination with the nucleon electric dipole moment, the deuteron moments would allow an identification of the dominant source(s) of symmetry violation.

The nucleon electric dipole form factor from dimension-six time-reversal violation

We calculate the electric dipole form factor of the nucleon that arises as a low-energy manifestation of time-reversal violation in quark–gluon interactions of effective dimension 6: the quark electric and chromoelectric dipole moments, and the gluon chromoelectric dipole moment. We use the framework of two-flavor chiral perturbation theory to one loop.

Hyperon electromagnetic properties in two-flavor chiral perturbation theory

The pion mass dependence of hyperon electromagnetic properties is determined using two-flavor heavy baryon chiral perturbation theory. Specifically we compute chiral corrections to the charge radii, magnetic moments, and magnetic radii of the spin one-half hyperons, as well as the charge radii, magnetic moments, magnetic radii, electric quadrupole moments, and quadrupole radii of the spin three-half hyperons. Results for the nucleon and delta are also included. Efficacy of the two-flavor theory is investigated by analyzing the role played by virtual kaons. For the electromagnetic properties of spin one-half hyperons, kaon loop contributions are shown to be well described by terms analytic in the pion mass squared. Similarly kaon contributions to the magnetic moments of spin three-half hyperons are well described in the two-flavor theory. The remaining electromagnetic properties of spin three-half resonances can be described in two-flavor chiral perturbation theory, however, this description fails just beyond the physical pion mass. For the case of experimentally known hyperon magnetic moments and charge radii, we demonstrate that chiral corrections are under reasonable control, in contrast to the behavior of these observables in the three-flavor chiral expansion. The formulae we derive are ideal for performing the pion mass extrapolation of lattice QCD data obtained at the physical strange quark mass.

Aspects of meson-baryon scattering in three- and two-flavor chiral perturbation theory

We analyze meson-baryon scattering lengths in the framework of covariant baryon chiral perturbation theory at leading one-loop order. We compute the complete set of matching relations between the dimension-two low-energy constants in the two- and three-flavor formulations of the theory. We derive new two-flavor low-energy theorems for pion-hyperon scattering that can be tested in lattice simulations.

Hyperon axial charges in two-flavor chiral perturbation theory

We use two-flavor heavy baryon chiral perturbation theory to investigate the isovector axial charges of the spin one-half hyperons. Expressions for these hyperon axial charges are derived at next-to-leading order in the chiral expansion. We utilize phenomenological and lattice QCD inputs to assess the convergence of the two-flavor theory, which appears to be best for cascades.

Hyperons in two-flavor chiral perturbation theory

We use two-flavor chiral perturbation theory to describe hyperons. We focus on the strangeness conserving sector, and, as an example, calculate hyperon masses. Convergence of this two-flavor chiral expansion for observables is improved over the three-flavor theory. The cost, however, is a larger number of low-energy constants that must be ultimately determined from lattice QCD data. A formula for the mass of the omega baryon is derived to sixth order in this expansion, and will aid lattice practitioners in scale setting or tuning the strange quark mass.

A framework for exploring the interactions and decays of hyperons with lattice QCD

In this work we outline a program for lattice QCD that would greatly contribute to understanding the strong and weak interactions of strange baryons. The low-energy effective field theory describing hyperon–nucleon scattering in QCD and partially-quenched QCD is constructed. This framework will allow lattice simulations of the low-energy scattering parameters to be extrapolated to the physical values of the quark masses. We explore the long-standing discrepancies in the P-wave amplitudes for non-leptonic hyperon decays using two-flavor chiral perturbation theory, and emphasize that the problem is not merely an artifact of perturbing about the chiral limit of the three-flavour theory. As energy is not conserved in the lattice simulation of the Λ → N two-point function, we include the surface terms in the effective field theory required to determine the underlying weak amplitude.

The pion mass in finite volume

We determine the relative pion mass shift $M_\pi(L)/M_\pi-1$ due to the finite spatial extent $L$ of the box by means of two-flavor chiral perturbation theory and the one-particle L\"uscher formula. We use as input the expression for the infinite volume $\pi\pi$ forward scattering amplitude up to next-to-next-to-leading order and can therefore control the convergence of the chiral series. A comparison to the full leading order chiral expression for the pion mass in finite volume allows us to check the size of subleading terms in the large-$L$ expansion.

Chiral extrapolation of strange matrix elements in the nucleon

In current lattice simulations of nucleon properties, the up and down quark masses are significantly larger than their physical values, while the strange quark can be included in simulations with its physical mass. When the up and down quark masses are much smaller than the strange-quark mass the chiral extrapolation of strange-quark matrix elements in the nucleon from the lattice up and down quark masses to their physical values can be performed with two-flavor chiral perturbation theory, thereby avoiding the slow convergence problem of the three-flavor chiral expansion. We explore the chiral expansion of several matrix elements of strange operators in the nucleon in two-flavor chiral perturbation theory and two-flavor partial-quenched chiral perturbation theory.

Virtual photons in the pion form factors and the energy–momentum tensor

We evaluate the vector and scalar form factor of the pion in the presence of virtual photons at next-to-leading order in two-flavor chiral perturbation theory. We also consider the scalar and tensor pion form factors of the energy–momentum tensor. We find that the intrinsic electromagnetic corrections are very small for the vector as well as the charged pion scalar form factor. The scalar radius of the neutral pion is reduced by two percent. We perform infrared regularization by considering electron–positron annihilation into pions and the decay of a light Higgs boson into a pion pair. We discuss the detector resolution dependent contributions to the various form factors and pion radii.