Isospin Limit Research Articles

Final-state interactions in the CP asymmetry of D meson two-body decays

Urgent theoretical progress is needed in order to provide an estimate in the Standard Model of the recent measurement by LHCb of direct CP violation in charm-meson two-body decays. Rescattering effects must be taken into account for a meaningful theoretical description of the amplitudes involved in such category of observables, as signaled by the presence of large strong phases. We discuss the computation of the latter effects based on a two-channel coupled dispersion relation, which exploits isospin-zero phase shifts and inelasticity parameterizations of data coming from the rescattering processes [Formula: see text], [Formula: see text], and [Formula: see text]. The determination of the subtraction constants of the dispersive integrals relies on the leading contributions to the transition amplitudes from the [Formula: see text] counting, where [Formula: see text] is the number of QCD colors. Furthermore, we use the measured values of the branching ratios to help in selecting the non-perturbative inputs in the isospin limit, from which we predict values for the CP asymmetries. We find that the predicted level of CP violation is much below the experimental value.

Determination of the D→ππ ratio of penguin over tree diagrams

We study the penguin over tree ratio in D→ππ decays. This ratio can serve as a probe for rescattering effects. Assuming the Standard Model and in the isospin limit, we derive expressions that relate both the magnitude and the phase of this ratio to direct CP asymmetries and branching fractions. We find that the current data suggest that rescattering is large. A dedicated experimental analysis with current and future data will be able to significantly reduce the errors on these determinations, and enable us to check if indeed there is significant rescattering in D→ππ decays. Published by the American Physical Society 2024

On the role of isospin violation in the pion–nucleon σ-term

In recent years, a persistent tension between phenomenological and lattice QCD determinations of the pion–nucleon σ-term σπN has developed. In particular, lattice-QCD calculations have matured to the point that isospin-violating effects need to be included. Here, we point out that the standard conventions adopted in both fields are incompatible, with the data-driven extraction based on the charged-pion mass, but lattice-QCD conventions relying on the mass of the neutral pion to define the isospin limit. The corresponding correction amounts to ΔσπN=3.1(5)MeV when evaluated in chiral perturbation theory with low-energy constants determined from a Roy–Steiner analysis of pion–nucleon scattering as well as σπN itself. It reduces the tension with lattice QCD, and should be included in the comparison to phenomenological determinations. We also update the extraction from pionic atoms accounting for the latest measurement of the width of pionic hydrogen, σπN=59.0(3.5)MeV, and provide the corresponding set of scalar couplings of the nucleon.

Timelike behavior of the pion electromagnetic form factor in the functional formalism

The electromagnetic form factor of the pion is calculated within the use of functional formalism. We develop integral representation for the minimal set of Standard Model Green's functions and derive the dispersion relation for the form factor in the two-flavor QCD isospin limit ${m}_{u}={m}_{d}$. We use the dressed quark propagator as obtained from the gap equation in Minkowski space and within the Dyson-Schwinger equations formalism to derive the approximate dispersion relation for the form factor for the first time. We evaluate the form factor for the spacelike as well as for the timelike momentum in the presented formalism. A new Nakanishi-like form of integral representation is proved on the basis of the vector Bethe-Salpeter equation for the quark-photon vector with a ladder-rainbow kernel. The gauge technique turns out to be a part of the entire structure of the vertex. In the analytic approach presented here, it is shown that a large amount of the $\ensuremath{\rho}$-meson peak in the cross section ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ is governed by the gauge invariance of QED/QCD---i.e., by a gauge-technique-constructed quark-photon vertex. This approximation naturally explains the broad shape of the $\ensuremath{\rho}$-meson peak.

Compositeness of S-wave weakly-bound states from next-to-leading order Weinberg’s relations

We discuss a model-independent estimator of the likelihood of the compositeness of a shallow S-wave bound or virtual state. The approach is based on an extension of Weinberg’s relations in Weinberg (Phys Rev 137:B672, 1965) and it relies only on the proximity of the energy of the state to the two-hadron threshold to which it significantly couples. The scheme only makes use of the experimental scattering length and the effective range low energy parameters, and it is shown to be fully consistent for predominantly molecular hadrons. As explicit applications, we analyse the case of the deuteron, the ^1mathrm{S}_0 nucleon-nucleon virtual state and the exotic D^{*}_{s0}(2317)^pm , and find strong support to the molecular interpretation in all cases. Results are less conclusive for the D^{*}_{s0}(2317)^pm , since the binding energy of this state would be significantly higher than that of the deuteron, and the approach employed here is at the limit of its applicability. We also qualitatively address the case of the recently discovered T_{cc}^+ state, within the isospin limit to avoid the complexity of the very close thresholds D^0D^{*+} and D^+D^{*0}, which could mask the ingredients of the approach proposed in this work.

Scale setting the Möbius domain wall fermion on gradient-flowed HISQ action using the omega baryon mass and the gradient-flow scales t0 and w0

We report on a subpercent scale determination using the omega baryon mass and gradient-flow methods. The calculations are performed on 22 ensembles of ${N}_{f}=2+1+1$ highly improved, rooted staggered sea-quark configurations generated by the MILC and CalLat Collaborations. The valence quark action used is M\"obius domain wall fermions solved on these configurations after a gradient-flow smearing is applied with a flowtime of ${t}_{\mathrm{gf}}=1$ in lattice units. The ensembles span four lattice spacings in the range $0.06\ensuremath{\lesssim}a\ensuremath{\lesssim}0.15\text{ }\text{ }\mathrm{fm}$, six pion masses in the range $130\ensuremath{\lesssim}{m}_{\ensuremath{\pi}}\ensuremath{\lesssim}400\text{ }\text{ }\mathrm{MeV}$ and multiple lattice volumes. On each ensemble, the gradient-flow scales ${t}_{0}/{a}^{2}$ and ${w}_{0}/a$ and the omega baryon mass $a{m}_{\mathrm{\ensuremath{\Omega}}}$ are computed. The dimensionless product of these quantities is then extrapolated to the continuum and infinite volume limits and interpolated to the physical light, strange and charm quark mass point in the isospin limit, resulting in the determination of $\sqrt{{t}_{0}}=0.1422(14)\text{ }\text{ }\mathrm{fm}$ and ${w}_{0}=0.1709(11)\text{ }\text{ }\mathrm{fm}$ with all sources of statistical and systematic uncertainty accounted for. The dominant uncertainty in both results is the stochastic uncertainty, though for $\sqrt{{t}_{0}}$ there are comparable continuum extrapolation uncertainties. For ${w}_{0}$, there is a clear path for a few-per-mille uncertainty just through improved stochastic precision, as recently obtained by the Budapest-Marseille-Wuppertal Collaboration.

Aspects of model dependence of $$\eta '$$–$$\eta $$ complex treated by going beyond the isospin limit

Exploring the extent of model dependence, we study effects of certain Ansaetze for the T-dependence of the correction term in the QCD topological susceptibility. The one producing unwanted effects on results at T > 0 in the eta'-eta complex in the usual limit of isospin symmetry, is largely cured from its peculiar behavior and brought into agreement with the other, when breaking of isospin symmetry is allowed and the realistic quark mass ratio m_u/m_d = 0.50 is adopted.

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.

Temperature Dependence of the Axion Mass in a Scenario Where the Restoration of Chiral Symmetry Drives the Restoration of the UA(1) Symmetry

The temperature (T) dependence of the axion mass is predicted for T ′ s up to ∼ 2.3 × the chiral restoration temperature of QCD. The axion is related to the U A ( 1 ) anomaly. The squared axion mass m a ( T ) 2 is, modulo the presently undetermined scale of spontaneous breaking of Peccei–Quinn symmetry f a (squared), equal to QCD topological susceptibility χ ( T ) for all T. We obtain χ ( T ) by using quark condensates calculated in two effective Dyson–Schwinger models of nonperturbative QCD. They exhibit the correct chiral behavior, including the dynamical breaking of chiral symmetry and its restoration at high T. This is reflected in the U A ( 1 ) symmetry breaking and restoration through χ ( T ) . In our previous studies, such χ ( T ) yields the T-dependence of the U A ( 1 ) -anomaly-influenced masses of η ′ and η mesons consistent with experiment. This in turn supports our prediction for the T-dependence of the axion mass. Another support is a rather good agreement with the pertinent lattice results. This agreement is not spoiled by our varying u and d quark mass parameters out of the isospin limit.

Lattice QCD determination of neutron-antineutron matrix elements with physical quark masses

Matrix elements of six-quark operators are needed to extract new physics constraints from experimental searches for neutron-antineutron oscillations. This work presents in detail the first lattice quantum chromodynamics calculations of the necessary neutron-antineutron transition matrix elements including calculation methods and discussions of systematic uncertainties. Implications of isospin and chiral symmetry on the matrix elements, power counting in the isospin limit, and renormalization of a chiral basis of six-quark operators are discussed. Calculations are performed with a chiral-symmetric discretization of the quark action and physical light quark masses in order to avoid the need for chiral extrapolation. Non-perturbative renormalization is performed, including a study of lattice cutoff effects. Excited-state effects are studied using two nucleon operators and multiple values of source-sink separation. Results for the dominant matrix elements are found to be significantly larger compared to previous results from the MIT bag model. Future calculations are needed to fully account for systematic uncertainties associated with discretization and finite-volume effects but are not expected to significantly affect this conclusion.

On light vector mesons and chiral SU(3) extrapolations

A chiral extrapolation of the light vector meson masses in the up, down and strange quark masses of QCD is presented. We apply an effective chiral Lagrangian based on the hadrogenesis conjecture to QCD lattice ensembles of PACS-CS, QCDSF-UKQCD and HSC in the strict isospin limit. The leading orders low-energy constants are determined upon a global fit to the lattice data set. We use the pion and kaon masses as well as the size of the finite volume as lattice ensemble parameters only. The quark mass ratio on the various ensembles are then predicted in terms of our set of low-energy constants. An accurate reproduction of the vector meson masses and quark-mass ratios as provided by the lattice collaborations and the Particle Data Group (PDG) is achieved. Particular attention is paid to the ω−ϕ mixing phenomenon, which is demonstrated to show a strong quark mass dependence.

Variations on Photon Vacuum Polarization

I provide updates for the theoretical predictions of the muon and electron anomalous magnetic moments, for the shift in the fine structure constant α(MZ )and for the weak mixing parameter $ \mathop {\sin }\nolimits^2 \mathop \Theta \nolimits_w (\mathop M\nolimits_Z ) $. Phenomenological results for Euclidean time correlators, the key objects in the lattice QCD approach to hadronic vacuum polarization, are briefly considered. Furthermore,I present a list of isospin breaking and electromagnetic corrections for the lepton moments, which may be used to supplement lattice QCD results obtained in the isospin limit and without the e.m. corrections.

On the convergence of the chiral expansion for the baryon ground-state masses

We study the chiral expansion of the baryon octet and decuplet masses in the isospin limit. It is illustrated that a chiral expansion of the one-loop contributions is rapidly converging up to quark masses that generously encompasses the mass of the physical strange quark. We express the successive orders in terms of physical meson and baryon masses. In addition, owing to specific correlations amongst the chiral moments, we suggest a reordering of terms that make the convergence properties more manifest. Explicit expressions up to chiral order five are derived for all baryon masses at the one-loop level. The baryon masses obtained do not depend on the renormalization scale. Our scheme is tested against QCD lattice data, where the low-energy parameters are systematically correlated by large-Nc sum rules. A reproduction of the baryon masses from PACS-CS, LHPC, HSC, NPLQCD, QCDSF-UKQCD and ETMC is achieved for ensembles with pion and kaon masses smaller than 600 MeV. Predictions for baryon masses on ensembles from CLS as well as all low-energy constants that enter the baryon masses at N3LO are made.

An analytic analysis of the pion decay constant in three-flavoured chiral perturbation theory

A representation of the two-loop contribution to the pion decay constant in SU(3) chiral perturbation theory is presented. The result is analytic up to the contribution of the three (different) mass sunset integrals, for which an expansion in their external momentum has been taken. We also give an analytic expression for the two-loop contribution to the pion mass based on a renormalized representation and in terms of the physical eta mass. We find an expansion of F_{pi } and M_{pi }^2 in the strange-quark mass in the isospin limit, and we perform the matching of the chiral SU(2) and SU(3) low-energy constants. A numerical analysis demonstrates the high accuracy of our representation, and the strong dependence of the pion decay constant upon the values of the low-energy constants, especially in the chiral limit. Finally, we present a simplified representation that is particularly suitable for fitting with available lattice data.

Matrix elements ofΔB=0operators in heavy hadron chiral perturbation theory

We study the light-quark mass and spatial volume dependence of the matrix elements of $\Delta B=0$ four-quark operators relevant for the determination of $V_{ub}$ and the lifetime ratios of single-$b$ hadrons. To this end, one-loop diagrams are computed in the framework of heavy hadron chiral perturbation theory with partially quenched formalism for three light-quark flavors in the isospin limit; flavor-connected and -disconnected diagrams are carefully analyzed. These calculations include the leading light-quark flavor and heavy-quark spin symmetry breaking effects in the heavy hadron spectrum. Our results can be used in the chiral extrapolation of lattice calculations of the matrix elements to the physical light-quark masses and to infinite volume. To provide insight on such chiral extrapolation, we evaluate the one-loop contributions to the matrix elements containing external $B_d$, $B_s$ mesons and $\Lambda_b$ baryon in the QCD limit, where sea and valence quark masses become equal. In particular, we find that the matrix elements of the $\lambda_3$ flavor-octet operators with external $B_d$ meson receive the contributions solely from connected diagrams in which current lattice techniques are capable of precise determination of the matrix elements. Finite volume effects are at most a few percent for typical lattice sizes and pion masses.

K→ππΔI=3/2decay amplitude in the continuum limit

We present new results for the amplitude $A_2$ for a kaon to decay into two pions with isospin $I=2$: Re$A_2 = 1.50(4)_\mathrm{stat}(14)_\mathrm{syst}\times 10^{-8}$ GeV; Im$A_2 = -6.99(20)_\mathrm{stat}(84)_\mathrm{syst}\times 10^{-13}$ GeV. These results were obtained from two ensembles generated at physical quark masses (in the isospin limit) with inverse lattice spacings $a^{-1}=1.728(4)$ GeV and $2.358(7)$ GeV. We are therefore able to perform a continuum extrapolation and hence largely to remove the dominant systematic uncertainty from our earlier results, that due to lattice artefacts. The only previous lattice computation of $K\to\pi\pi$ decays at physical kinematics was performed using an ensemble at a single, rather coarse, value of the lattice spacing ($a^{-1}\simeq 1.37(1)$ GeV). We confirm the observation that there is a significant cancellation between the two dominant contributions to Re$A_2$ which we suggest is an important ingredient in understanding the $\Delta I=1/2$ rule, Re$A_0$/Re$A_2\simeq 22.5$, where the subscript denotes the total isospin of the two-pion final state. Our result for $A_2$ implies that the electroweak penguin contribution to $\epsilon^\prime/\epsilon$ is Re($\epsilon^\prime/\epsilon)_\textrm{EWP}=-(6.6\pm 1.0)\times 10^{-4}$.

Semileptonic kaon decay in staggered chiral perturbation theory

The determination of $|{V}_{us}|$ from kaon semileptonic decays requires the value of the form factor ${f}_{+}({q}^{2}=0)$, which can be calculated precisely on the lattice. We provide the one-loop partially quenched staggered chiral perturbation theory expressions that may be employed to analyze staggered simulations of ${f}_{+}({q}^{2})$ with three light flavors. We consider both the case of a mixed action, where the valence and sea sectors have different staggered actions, and the standard case, where these actions are the same. The momentum transfer ${q}^{2}$ of the form factor is allowed to have an arbitrary value. We give results for the generic situation where the $u$, $d$, and $s$ quark masses are all different, ${N}_{f}=1+1+1$, and for the isospin limit, ${N}_{f}=2+1$. The expression we obtain for ${f}_{+}({q}^{2})$ is independent of the mass of the (valence) spectator quark. In the limit of vanishing lattice spacing, our results reduce to the one-loop continuum partially quenched expression for ${f}_{+}({q}^{2})$, which has not previously been reported in the literature for the ${N}_{f}=1+1+1$ case. Our expressions have already been used in staggered lattice analyses of ${f}_{+}(0)$ and should prove useful in future calculations as well.

Radiative corrections to the charged pion-pair production process π − γ → π + π − π − at low energies

We calculate the one-photon loop radiative corrections to the charged pion-pair production process $\pi^-\gamma\to\pi^+\pi^-\pi^-$. In the low-energy region this reaction is governed by the chiral pion-pion interaction. The pertinent set of 42 irreducible photon-loop diagrams is calculated by using the package FeynCalc. Electromagnetic counterterms with two independent low-energy constants $\widehat k_1$ and $\widehat k_2$ are included in order to remove the ultraviolet divergences generated by the photon-loops. Infrared finiteness of the virtual radiative corrections is achieved by including soft photon radiation below an energy cut-off $\lambda$. The purely electromagnetic interaction of the charged pions mediated by one-photon exchange is also taken into account. The radiative corrections to the total cross section (in the isospin limit) vary between $+10\%$ close to threshold and about $-1\%$ at a center-of-mass energy of $7m_\pi$. The largest contribution comes from the simple one-photon exchange. Radiative corrections to the $\pi^+\pi^-$ and $\pi^-\pi^-$ mass spectra are studied as well. The Coulomb singularity of the final-state interaction produces a kink in the dipion mass spectra. The virtual radiative corrections to elastic $\pi^-\pi^-$ scattering are derived additionally.

Isospin breaking in the phases of the K e4 form factors

Isospin breaking in the K ℓ4 form factors induced by the difference between charged and neutral pion masses is studied. Starting from suitably subtracted dispersion representations, the form factors are constructed in an iterative way up to two loops in the low-energy expansion by implementing analyticity, crossing, and unitarity due to two-meson intermediate states. Analytical expressions for the phases of the two-loop form factors of the K ±→π + π − e ± ν e channel are given, allowing one to connect the difference of form-factor phase shifts measured experimentally (out of the isospin limit) and the difference of S- and P-wave ππ phase shifts studied theoretically (in the isospin limit). The isospin-breaking correction consists of the sum of a universal part, involving only ππ rescattering, and a process-dependent contribution, involving the form factors in the coupled channels. The dependence on the two S-wave scattering lengths $a_{0}^{0}$ and $a_{0}^{2}$ in the isospin limit is worked out in a general way, in contrast to previous analyses based on one-loop chiral perturbation theory. The latter is used only to assess the subtraction constants involved in the dispersive approach. The two-loop universal and process-dependent contributions are estimated and cancel partially to yield an isospin-breaking correction close to the one-loop case. The recent results on the phases of K ±→π + π − e ± ν e form factors obtained by the NA48/2 collaboration at the CERN SPS are reanalysed including this isospin-breaking correction to extract values for the scattering lengths $a_{0}^{0}$ and $a_{0}^{2}$ , as well as for low-energy constants and order parameters of two-flavour χPT.

Determining the chiral condensate from the distribution of the winding number beyond topological susceptibility

Abstract The first two non-trivial moments of the distribution of the topological charge (or gluonic winding number), i.e., the topological susceptibility and the fourth cumulant, can be computed in lattice QCD simulations and exploited to constrain the pattern of chiral symmetry breaking. We compute these two topological observables at next-to-leading order in three-flavour Chiral Perturbation Theory, and we discuss the role played by the η propagation in these expressions. For hierarchies of light-quark masses close to the physical situation, we show that the fourth cumulant has a much better sensitivity than the topological susceptibility to the three-flavour quark condensate, and thus constitutes a relevant tool to determine the pattern of chiral symmetry breaking in the limit of three massless flavours. We provide the complete formulae for the two topological observables in the isospin limit, and predict their values in the particular setting of the recent analysis of the RBC/UKQCD collaboration. We show that a combination of the topological susceptibility and the fourth cumulant is able to pin down the three flavour condensate in a particularly clean way in the case of three degenerate quarks.