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Register a new userLeading isospin-breaking corrections to meson masses on the lattice
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We present a study of the isospin-breaking (IB) corrections to pseudoscalar (PS) meson masses using the gauge configurations produced by the ETM Collaboration with $N_f=2+1+1$ dynamical quarks at three lattice spacings varying from 0.089 to 0.062 fm. Our method is based on a combined expansion of the path integral in powers of the small parameters $(widehat{m}_d - widehat{m}_u)/Lambda_{QCD}$ and $alpha_{em}$, where $widehat{m}_f$ is the renormalized quark mass and $alpha_{em}$ the renormalized fine structure constant. We obtain results for the pion, kaon and $D$-meson mass splitting; for the Dashens theorem violation parameters $epsilon_gamma(overline{mathrm{MS}}, 2~mbox{GeV})$, $epsilon_{pi^0}$, $epsilon_{K^0}(overline{mathrm{MS}}, 2~mbox{GeV})$; for the light quark masses $(widehat{m}_d - widehat{m}_u)(overline{mathrm{MS}}, 2~mbox{GeV})$, $(widehat{m}_u / widehat{m}_d)(overline{mathrm{MS}}, 2~mbox{GeV})$; for the flavour symmetry breaking parameters $R(overline{mathrm{MS}}, 2~mbox{GeV})$ and $Q(overline{mathrm{MS}}, 2~mbox{GeV})$ and for the strong IB effects on the kaon decay constants.
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We calculate the strong isospin breaking and QED corrections to meson masses and the hadronic vacuum polarization in an exploratory study on a $64times24^3$ lattice with an inverse lattice spacing of $a^{-1}=1.78$ GeV and an isospin symmetric pion mass of $m_pi=340$ MeV. We include QED in an electro-quenched setup using two different methods, a stochastic and a perturbative approach. We find that the electromagnetic correction to the leading hadronic contribution to the anomalous magnetic moment of the muon is smaller than $1%$ for the up quark and $0.1%$ for the strange quark, although it should be noted that this is obtained using unphysical light quark masses. In addition to the results themselves, we compare the precision which can be reached for the same computational cost using each method. Such a comparison is also made for the meson electromagnetic mass-splittings.
Lattice QCD simulations are now reaching a precision where isospin breaking effects become important. Previously, we have developed a program to systematically investigate the pattern of flavor symmetry beaking within QCD and successfully applied it to meson and baryon masses involving up, down and strange quarks. In this Letter we extend the calculations to QCD + QED and present our first results on isospin splittings in the pseudoscalar meson and baryon octets. In particular, we obtain the nucleon mass difference of $M_n-M_p=1.35(18)(8),mbox{MeV}$ and the electromagnetic contribution to the pion splitting $M_{pi^+}-M_{pi^0}=4.60(20),mbox{MeV}$. Further we report first determination of the separation between strong and electromagnetic contributions in the $bar{MS}$ scheme.
Isospin symmetry is not exact and the corrections to the isosymmetric limit are, in general, at the percent level. For gold plated quantities, such as pseudoscalar meson masses or the kaon leptonic and semileptonic decay rates, these effects are of the same order of magnitude of the errors quoted in nowadays lattice calculations and cannot be neglected any longer. In this talk I discuss the methods that have been developed in the last few years to calculate isospin breaking corrections by starting from first principles lattice simulations. In particular, I discuss how to perform a combined QCD+QED lattice simulation and a renormalization prescription to be used in order to separate QCD from QED isospin breaking effects. A brief review of recent lattice results of isospin breaking effects on the hadron spectrum is also included.
Isospin symmetry is explicitly broken in the Standard Model by the non-zero differences of mass and electric charge between the up and down quarks. Both of these corrections are expected to have a comparable size of the order of one percent relatively to hadronic energies. Although these contributions are small, they play a crucial role in hadronic and nuclear physics. In this review we explain how to properly define QCD and QED on a finite and discrete space-time so that isospin corrections to hadronic observables can be computed ab-initio. We then consider the different approaches to compute lattice correlation functions of QCD and QED observables. Finally we summarise the actual lattice results concerning the isospin corrections to the light hadron spectrum.
We summarize our lattice QCD determinations of the pion-pion, pion-kaon and kaon-kaon s-wave scattering lengths at maximal isospin with a particular focus on the extrapolation to the physical point and the usage of next-to-leading order chiral perturbation theory to do so. We employ data at three values of the lattice spacing and pion masses ranging from around 230 MeV to around 450 MeV, applying Lueschers finite volume method to compute the scattering lengths. We find that leading order chiral perturbation theory is surprisingly close to our data even in the kaon-kaon case for our entire range of pion masses.
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