Isospin Limit Research Articles

Two-loop representations of low-energy pion form factors and ππ scattering phases in the presence of isospin breaking

Dispersive representations of the ππ scattering amplitudes and pion form factors, valid at two-loop accuracy in the low-energy expansion, are constructed in the presence of isospin-breaking effects induced by the difference between the charged and neutral pion masses. Analytical expressions for the corresponding phases of the scalar and vector pion form factors are computed. It is shown that each of these phases consists of the sum of a “universal” part and a form-factor dependent contribution. The first one is entirely determined in terms of the ππ scattering amplitudes alone, and reduces to the phase satisfying Watson’s theorem in the isospin limit. The second one can be sizeable, although it vanishes in the same limit. The dependence of these isospin corrections with respect to the parameters of the subthreshold expansion of the ππ amplitude is studied, and an equivalent representation in terms of the S-wave scattering lengths is also briefly presented and discussed. In addition, partially analytical expressions for the two-loop form factors and ππ scattering amplitudes in the presence of isospin breaking are provided.

A note on isospin violation in [formula omitted] decays

We discuss the size of isospin violating effects of both electromagnetic (e≠0) and strong (mu≠md) origin in Pℓ2(γ) decays. Our results are relevant for the extraction of |Vus/Vud| from the measured ratio of the Kμ2(γ) and πμ2(γ) decay widths combined with FK/Fπ obtained in lattice calculations in the isospin limit. We point out that strong isospin-breaking corrections, neglected in previous studies, enter at the same level as the electromagnetic effects. The updated value for the ratio of CKM elements reads |Vus/Vud|=0.2316(12).

Isospin breaking and chiral symmetry restoration

We analyze quark condensates and chiral (scalar) susceptibilities including isospin breaking effects at finite temperature $T$. These include $m_u\neq m_d$ contributions as well as electromagnetic ($e\neq 0$) corrections, both treated in a consistent chiral lagrangian framework to leading order in SU(2) and SU(3) Chiral Perturbation Theory, so that our predictions are model independent. The chiral restoration temperature extracted from $<\bar q q>= <\bar u u + \bar d d >$ is almost unaffected, while the isospin breaking order parameter $<\bar u u - \bar d d >$ grows with $T$ for the three-flavour case SU(3). We derive a sum rule relating the condensate ratio $<\bar q q>(e\neq 0)/<\bar q q>(e=0)$ with the scalar susceptibility difference $\chi(T)-\chi(0)$, directly measurable on the lattice. This sum rule is useful also for estimating condensate errors in staggered lattice analysis. Keeping $m_u\neq m_d$ allows to obtain the connected and disconnected contributions to the susceptibility, even in the isospin limit, whose temperature, mass and isospin breaking dependence we analyze in detail. The disconnected part grows linearly, diverging in the chiral (infrared) limit as $T/M_\pi$, while the connected part shows a quadratic behaviour, infrared regular as $T^2/M_\eta^2$ and coming from $\pi^0\eta$ mixing terms. This smooth connected behaviour suggests that isospin breaking correlations are weaker than critical chiral ones near the transition temperature. We explore some consequences in connection with lattice data and their scaling properties, for which our present analysis for physical masses, i.e. beyond the chiral limit, provides a useful model-independent description for low and moderate temperatures.

Cold Neutrons Trapped in External Fields

The properties of inhomogeneous neutron matter are crucial to the physics of neutron-rich nuclei and the crust of neutron stars. Advances in computational techniques now allow us to accurately determine the binding energies and densities of many neutrons interacting via realistic microscopic interactions and confined in external fields. We perform calculations for different external fields and across several shells to place important constraints on inhomogeneous neutron matter, and hence the large isospin limit of the nuclear energy density functionals that are used to predict properties of heavy nuclei and neutron star crusts. We find important differences between microscopic calculations and current density functionals; in particular, the isovector gradient terms are significantly more repulsive than in traditional models, and the spin-orbit and pairing forces are comparatively weaker.

Hadron Spectrum from Lattice QCD

In order to claim that we understand strong interactions quantitatively, we should be able to derive key properties of hadrons, such as their mass, “ab initio” from the interaction of quarks and gluons. In particular the phenomenon of “dimensional transmutation”, i.e. of having a massive proton even with exactly massless quarks and gluons, is a distinctive property of QCD. In this talk I summarize recent efforts, in the context of the Budapest-Marseille-Wuppertal collaboration, to compute the masses of the flavor non-singlet mesons and of the light baryon octet and decuplet in the isospin limit of QCD on the lattice, with fully controlled systematics.

Connected parts of decuplet electromagnetic properties

We determine the electromagnetic properties of decuplet resonances using chiral perturbation theory at next-to-leading order. Utilizing a partially quenched charge matrix, we isolate and remove the quark disconnected contractions. This allows us to compare the physical meson loop contributions to the connected contributions, of which the latter are currently calculated using lattice QCD. Finally we determine linear combinations of decuplet resonance and hyperon electromagnetic properties that are exactly independent of disconnected contractions in the isospin limit.

Dispersive approach to chiral perturbation theory

We generalize the reconstruction theorem of Stern, Sazdjian, and Fuchs based on the dispersion relations to the case of the ($2\ensuremath{\rightarrow}2$) scattering of all the pseudoscalar octet mesons $(\ensuremath{\pi},K,\ensuremath{\eta})$. We formulate it in a general way and include also a discussion of the assumptions of the theorem. It is used to obtain the amplitudes of all such processes in the isospin limit to the one-loop order (and can be straightforwardly extended to two loops) independently on the particular power-counting scheme of the chiral perturbation theory in question. The results in this general form are presented.

Flavor twisted boundary conditions, pion momentum, and the pion electromagnetic form factor

We investigate the utility of partially twisted boundary conditions in lattice calculations of meson observables. For dynamical simulations, we show that the pion dispersion relation is modified by volume effects. In the isospin limit, we demonstrate that the pion electromagnetic form factor can be computed on the lattice at continuous values of the momentum transfer. Furthermore, the finite volume effects are under theoretical control for extraction of the pion charge radius.

Ginsparg-Wilson pions scattering in a sea of staggered quarks

We calculate isospin 2 pion-pion scattering in chiral perturbation theory for a partially quenched, mixed action theory with Ginsparg-Wilson valence quarks and staggered sea quarks. We point out that for some scattering channels, the power-law volume dependence of two-pion states in nonunitary theories such as partially quenched or mixed action QCD is identical to that of QCD. Thus one can extract infinite-volume scattering parameters from mixed action simulations. We then determine the scattering length for both 2 and $2+1$ sea quarks in the isospin limit. The scattering length, when expressed in terms of the pion mass and the decay constant measured on the lattice, has no contributions from mixed valence-sea mesons, thus it does not depend upon the parameter, ${C}_{\mathrm{Mix}}$, that appears in the chiral Lagrangian of the mixed theory. In addition, the contributions which nominally arise from operators appearing in the mixed action $\mathcal{O}({a}^{2}{m}_{q})$ Lagrangian exactly cancel when the scattering length is written in this form. This is in contrast to the scattering length expressed in terms of the bare parameters of the chiral Lagrangian, which explicitly exhibits all the sicknesses and lattice spacing dependence allowed by a partially quenched mixed action theory. These results hold for both 2 and $2+1$ flavors of sea quarks.

Nucleon and delta masses in twisted mass chiral perturbation theory

We calculate the masses of the nucleons and deltas in twisted mass heavy baryon chiral perturbation theory. We work to quadratic order in a power counting scheme in which we treat the lattice spacing and the quark masses to be of the same order. We give expressions for the mass and the mass splitting of the nucleons and deltas both in and away from the isospin limit. We give an argument using the chiral Lagrangian treatment that, in the strong isospin limit, the nucleons remain degenerate and the delta multiplet breaks into two degenerate pairs to all orders in chiral perturbation theory. We show that the mass splitting between the degenerate pairs of the deltas first appears at quadratic order in in the lattice spacing. We discuss the subtleties in the effective chiral theory that arise from the inclusion of isospin breaking.

Baryon electromagnetic properties in partially quenched heavy hadron chiral perturbation theory

The electromagnetic properties of baryons containing a heavy quark are calculated at next-to-leading order in partially quenched heavy hadron chiral perturbation theory. Calculations are performed for three light flavors in the isospin limit and additionally for two light nondegenerate flavors. We use partially quenched charge matrices that are easy to implement on the lattice. The results presented are necessary for the light-quark mass extrapolation and zero-momentum extrapolation of lattice QCD and partially quenched lattice QCD calculations of heavy hadron electromagnetic properties. Additionally relations between the sextet electromagnetic form factors and transition form factors are derived.

The role of strange sea quarks in chiral extrapolations on the lattice

Since the strange quark has a light mass of order Lambda_QCD, fluctuations of sea s-s bar pairs may play a special role in the low-energy dynamics of QCD by inducing significantly different patterns of chiral symmetry breaking in the chiral limits N_f=2 (m_u=m_d=0, m_s physical) and N_f=3 (m_u=m_d=m_s=0). This effect of vacuum fluctuations of s-s bar pairs is related to the violation of the Zweig rule in the scalar sector, described through the two O(p^4) low-energy constants L_4 and L_6 of the three-flavour strong chiral lagrangian. In the case of significant vacuum fluctuations, three-flavour chiral expansions might exhibit a numerical competition between leading- and next-to-leading-order terms according to the chiral counting, and chiral extrapolations should be handled with a special care. We investigate the impact of the fluctuations of s-s bar pairs on chiral extrapolations in the case of lattice simulations with three dynamical flavours in the isospin limit. Information on the size of the vacuum fluctuations can be obtained from the dependence of the masses and decay constants of pions and kaons on the light quark masses. Even in the case of large fluctuations, corrections due to the finite size of spatial dimensions can be kept under control for large enough boxes (L around 2.5 fm).

Baryon masses in partially quenched heavy hadron chiral perturbation theory

The masses of baryons containing a heavy quark are calculated to next-to-leading order in partially quenched heavy hadron chiral perturbation theory. Calculations are performed for three light flavors in the isospin limit and additionally for two light non-degenerate flavors. The results presented are necessary for extrapolating lattice QCD and partially quenched lattice QCD calculations of the heavy hadron masses.

Isospin breaking in $K\to3\pi$ decays II: radiative corrections

The five different CP conserving amplitudes for the decays $K\to3\pi$ are calculated using Chiral Perturbation Theory. The calculation is made to next-to-leading order and includes full isospin breaking. The squared amplitudes are compared with the corresponding ones in the isospin limit to estimate the size of the isospin breaking effects. In this paper we add the radiative corrections to the earlier calculated $m_u-m_d$ and local electromagnetic effects. We find corrections of order 5-10 percent.

Decuplet baryon masses in partially quenched chiral perturbation theory

The masses of the spin- 3 2 baryons are calculated to next-to-next-to-leading order in heavy baryon chiral perturbation theory and partially quenched heavy baryon chiral perturbation theory. The calculations are performed for three light flavors in the isospin limit. These results are necessary for extrapolating QCD and partially quenched QCD lattice calculations of the decuplet baryon masses.

Octet baryon masses in partially quenched chiral perturbation theory

The mass spectrum of the lowest lying octet baryons is calculated to next-to-next-to-leading order in heavy baryon chiral perturbation theory and partially quenched heavy baryon chiral perturbation theory. We work in the isospin limit and treat the decuplet baryons as dynamical fields. These results are necessary for extrapolating QCD and partially quenched QCD lattice simulations of the octet baryon masses.

Hadronic electromagnetic properties at finite lattice spacing

Electromagnetic properties of the octet mesons as well as the octet and decuplet baryons are augmented in quenched and partially quenched chiral perturbation theory to include $O(a)$ corrections due to lattice discretization. We present the results for the $\mathrm{SU}(3)$ flavor group in the isospin limit as well as the results for $\mathrm{SU}(2)$ flavor with nondegenerate quarks. These corrections will be useful for extrapolation of lattice calculations using Wilson valence and sea quarks, as well as calculations using Wilson sea quarks and Ginsparg-Wilson valence quarks.

Erratum: Electromagnetic properties of the baryon decuplet in quenched and partially quenched chiral perturbation theory [Phys. Rev. D68, 114503 (2003)

We calculate electromagnetic properties of the decuplet baryons in quenched and partially quenched chiral perturbation theory. We work at next-to-leading order in the chiral expansion, leading order in the heavy baryon expansion, and obtain expressions for the magnetic moments, charge radii, and electric quadrupole moments. The quenched calculation is shown to be pathological since only quenched chiral singularities are present at this order. We present the partially quenched results for both the SU(2) and SU(3) flavor groups and use the isospin limit in the latter. These results are necessary for proper extrapolation of lattice calculations of decuplet electromagnetic properties.

Isospin breaking in $K \rightarrow \pi\pi$ decays

We perform a complete analysis of isospin breaking in K -> Pi Pi amplitudes in chiral perturbation theory, including both strong isospin violation and electromagnetic corrections to next-to-leading order in the low-energy expansion. The unknown chiral couplings are estimated at leading order in the 1/N_c expansion. We study the impact of isospin breaking on CP conserving amplitudes and rescattering phases. In particular, we extract the effective couplings g_8 and g_27 from a fit to K -> Pi Pi branching ratios, finding small deviations from the isospin-limit case. The ratio Re(A_0)/Re(A_2) measuring the Delta I=1/2 enhancement is found to decrese from 22.1 +/- 0.1 in the isospin limit to 20.3 +/- 0.5 in the presence of isospin breaking. We also analyse the effect of isospin violation on the CP violation parameter epsilon-prime, finding a destructive interference between three different sources of isospin violation. Within the uncertainties of large-N_c estimates for the low-energy constants, the isospin violating correction for epsilon-prime is below 15%.

Baryon decuplet to octet electromagnetic transitions in quenched and partially quenched chiral perturbation theory

We calculate baryon decuplet to octet electromagnetic transition form factors in quenched and partially quenched chiral perturbation theory. We work in the isospin limit of SU(3) flavor, up to next-to-leading order in the chiral expansion, and to leading order in the heavy baryon expansion. Our results are necessary for proper extrapolation of lattice calculations of these transitions. We also derive expressions for the case of SU(2) flavor away from the isospin limit.