The scalar pion form factor in two-flavor lattice QCD

We calculate the scalar form factor of the pion using two dynamical flavors of non-perturbatively $\mathcal{O}(a)$-improved Wilson fermions, including both the connected and the disconnected contribution to the relevant correlation functions. We employ the calculation of all-to-all propagators using stochastic sources and a generalized hopping parameter expansion. From the form factor data at vanishing momentum transfer, $Q^2=0$, and two non-vanishing $Q^2$ we obtain an estimate for the scalar radius \left^\pi_{_{\rm S}} of the pion at one value of the lattice spacing and for five different pion masses. Using Chiral Perturbation Theory at next-to-leading order, we find \left^\pi_{_{\rm S}}=0.635\pm0.016 fm$^2$ at the physical pion mass (statistical error only). This is in good agreement with the phenomenological estimate from $\pi\pi$-scattering. The inclusion of the disconnected contribution is essential for achieving this level of agreement.Comment: 15 pages, 10 pdf figures, uses revtex4-1; version to appear in PR

Isospin Breaking Corrections to the HVP with Domain Wall Fermions

We present results for the QED and strong isospin breaking corrections to the hadronic vacuum polarization using $N_f=2+1$ Domain Wall fermions. QED is included in an electro-quenched setup using two different methods, a stochastic and a perturbative approach. Results and statistical errors from both methods are directly compared with each other.Comment: 8 pages, 6 figures, presented at the 35th International Symposium on Lattice Field Theory (Lattice 2017), Granada, Spain, June 18-24, 201

Prospects for a lattice calculation of the rare decay Σ+→pℓ+ℓ−\Sigma^+\to p\ell^+\ell^-

We present a strategy for calculating the rare decay of a $\Sigma^+ (uus)$ baryon to a proton $(uud)$ and di-lepton pair using lattice QCD. To determine this observable one needs to numerically evaluate baryonic two-, three-, and four-point correlation functions related to the target process. In particular, the four-point function arises from the insertion of incoming and outgoing baryons, together with a weak Hamiltonian mediating the $s \to d$ transition and an electromagnetic current creating the outgoing leptons. As is described in previous work in other contexts, this four-point function has a highly non-trivial relation to the physical observable, due to nucleon and nucleon-pion intermediate states. These lead to growing Euclidean time dependence and, in the case of the nucleon-pion states, to power-like volume effects. We discuss how to treat these issues in the context of the $\Sigma^+\rightarrow p\ell^+\ell^-$ decay and, in particular, detail the relation between the finite-volume estimator and the physical, complex-valued amplitude. In doing so, we also make connections between various approaches in the literature

The scalar radius of the pion from Lattice QCD in the continuum limit

We extend our study of the pion scalar radius in two-flavour lattice QCD to include two additional lattice spacings as well as lighter pion masses, enabling us to perform a combined chiral and continuum extrapolation. We find discretisation artefacts to be small for the radius, and confirm the importance of the disconnected diagrams in reproducing the correct chiral behaviour. Our final result for the scalar radius of the pion at the physical point is $\left\langle r^2\right\rangle^\pi_{\rm S}=0.600\pm0.052$ fm$^2$, corresponding to a value of $\overline{\ell}_4=4.54\pm0.30$ for the low-energy constant $\overline{\ell}_4$ of NLO chiral perturbation theory.Comment: 4 pages, 4 figures, uses svjour.cl

Isospin-breaking corrections to light leptonic decays in lattice QCD+QED at the physical point

We report on the physical-point RBC/UKQCD calculation of the leading isospin-breaking corrections to light-meson leptonic decays. This is highly relevant for future precision tests in the flavour physics sector, in particular the first-row unitarity of the Cabibbo-Kobayashi-Maskawa matrix containing the elements $V_{us}$ and $V_{ud}$. The simulations were performed using Domain-Wall fermions for $2+1$ flavours, and with isospin-breaking effects included perturbatively in the path integral through order $\alpha$ and $(m_u - m_d)/\Lambda _{\mathrm{QCD}}$. We use QED$_{\mathrm{L}}$ for the inclusion of electromagnetism, and discuss here the non-locality of this prescription which has significant impact on the infinite-volume extrapolation.Comment: Proceedings for The 39th International Symposium on Lattice Field Theory, 8th-13th August, 2022, Rheinische Friedrich-Wilhelms-Universit\"at Bonn, Bonn, German

Hadronic correlation functions with quark-disconnected contributions in lattice QCD

One of the fundamental interactions in the Standard Model of particle physicsrnis the strong force, which can be formulated as a non-abelian gauge theoryrncalled Quantum Chromodynamics (QCD). rnIn the low-energy regime, where the QCD coupling becomes strong and quarksrnand gluons are confined to hadrons, a perturbativernexpansion in the coupling constant is not possible.rnHowever, the introduction of a four-dimensional Euclidean space-timernlattice allows for an textit{ab initio} treatment of QCD and provides arnpowerful tool to study the low-energy dynamics of hadrons.rnSome hadronic matrix elements of interest receive contributionsrnfrom diagrams including quark-disconnected loops, i.e. disconnected quarkrnlines from one lattice point back to the same point. The calculation of suchrnquark loops is computationally very demanding, because it requires knowledge ofrnthe all-to-all propagator. In this thesis we use stochastic sources and arnhopping parameter expansion to estimate such propagators.rnWe apply this technique to study two problems which relay crucially on therncalculation of quark-disconnected diagrams, namely the scalar form factor ofrnthe pion and the hadronic vacuum polarization contribution to the anomalousrnmagnet moment of the muon.rnThe scalar form factor of the pion describes the coupling of a charged pion torna scalar particle. We calculate the connected and the disconnected contributionrnto the scalar form factor for three different momentum transfers. The scalarrnradius of the pion is extracted from the momentum dependence of the form factor.rnThe use ofrnseveral different pion masses and lattice spacings allows for an extrapolationrnto the physical point. The chiral extrapolation is done using chiralrnperturbation theory ($chi$PT). We find that our pion mass dependence of thernscalar radius is consistent with $chi$PT at next-to-leading order.rnAdditionally, we are able to extract the low energy constant $ell_4$ from thernextrapolation, and ourrnresult is in agreement with results from other lattice determinations.rnFurthermore, our result for the scalar pion radius at the physical point isrnconsistent with a value that was extracted from $pipi$-scattering data. rnThe hadronic vacuum polarization (HVP) is the leading-order hadronicrncontribution to the anomalous magnetic moment $a_mu$ of the muon. The HVP canrnbe estimated from the correlation of two vector currents in the time-momentumrnrepresentation. We explicitly calculate the corresponding disconnectedrncontribution to the vector correlator. We find that the disconnectedrncontribution is consistent with zero within its statistical errors. This resultrncan be converted into an upper limit for the maximum contribution of therndisconnected diagram to $a_mu$ by using the expected time-dependence of therncorrelator and comparing it to the corresponding connected contribution. Wernfind the disconnected contribution to be smaller than $approx5$ of thernconnected one. This value can be used as an estimate for a systematic errorrnthat arises from neglecting the disconnected contribution.rnDie starke Kraft ist eine der fundamentalen Wechselwirkungen imrnStandardmodellrnder Teilchenphysik. Sie kann als nicht-abelsche Eichtheorie -- diernQuantenchromodynamik -- formuliert werden. Im Niederenergiebereich, in dem diernQCD Kopplung groß wird und Quarks und Gluonen in Hadronen gebunden sind, istrnes nicht möglich die QCD in einer Störungsreihe in der Kopplungskonstantenrnzu entwickeln. Die Formulierung der QCD auf einem vierdimensionalen Raum-ZeitrnGitter erlaubt eine theoretische Beschreibung der starken Wechselwirkung undrnermöglicht es, die Dynamik der Hadronen im Niederenergiebereich zu studieren.rnEinige hadronische Matrixelemente enthalten Beiträge von Diagrammen mitrnquark-unverbundenen Schleifen, d.h. Quarklinien von einemrnGitterpunkt zurück zum selben Gitterpunkt. Die Berechnung solcherrnQuarkschleifen ist sehr aufwendig, da man hierfür den Propagator von jedemrnGitterpunkt zu allen Gitterpunkten (all-to-all propagator) benötigt.rnIn der vorliegenden Arbeitrnbenutzen wir stochastische Quellen und eine Hopping Parameter-Entwicklung umrnsolche Propagatoren abzuschätzen. Diese Methode wird verwendet, um zweirnProbleme zu erfoschen, die wesentlich von der Berechnung unverbundenerrnBeiträge abhängen, nämlich der skalare Formfaktor des Pions und derrnBeitrag der hadronische Vakuumpolarisation zum anomalen magnetischen Moment desrnMyons.rnDer skalare Formfaktor des Pions beschreibt die Kopplung eines geladenen Pionsrnan ein skalares Teilchen. Wir berechnen den verbundenen und den unverbundenenrnBeitrag zum skalaren Formfaktor für drei verschiedene Impulsüberträge.rnAus der Impulsabhängigkeit des Formfaktors kann der skalare Radius des Pionsrnbestimmt werden. Da wir verschiedene Pionmassen und Gitterabständernverwenden, können wir unsere Ergebnisse zum physikalischen Punktrnextrapolieren. Für die chirale Extrapolation verwenden wir chiralernStörungstheorie ($chi$PT). Unsere Werte sind konsistent mit $chi$PT zurrntextit{next-to-leading} Ordung. Zusätzlich können wir ausrnder Extrapolation diernNiederenergiekonstante $ell_4$ bestimmen. Unser Ergebnis ist konsistent mitrnResultaten anderer Gitterrechnungen. Das entsprechende Resultat für denrnskalaren Pionradius am physikalischenrnPunkt stimmt mit einem Wert überein, der aus $pipi$-Streuung bestimmt wurde.rnDie hadronische Vakuumpolarisation (HVP) ist der führende hadronischernBeitrag zum anomalen magnetischen Moment $a_mu$ des Myons. Die HVP kann ausrnder Korrelation zweier Vektorströme bestimmt werden. Wir berechnen explizitrnden unverbundenen Anteil des Vektorkorrelators. Wir erhalten einen rnunverbundene Beitrag, der innerhalb der statistischen Fehler mit nullrnverträglichrnist. Daraus lässt sich eine obere Schranke für den maximalen Beitrag desrnunverbundenen Diagramms zu $a_mu$ ermitteln, indem wir die erwarteternZeitabhängigkeit des Korrelators verwenden und ihn mit dem entsprechendenrnverbundenen Anteil vergleichen. Der unverbundenernBeitrag ist kleiner als $approx5$ des verbundenen Anteils. Dieser Wertrnkann als systematischer Fehler verwendet werden, der aus der Vernachlässigungrndes unverbundenen Diagramms resultiert.r

Efficiently unquenching QCD+QED at O(α)

We outline a strategy to efficiently include the electromagnetic interactions of the sea quarks in QCD+QED. When computing iso-spin breaking corrections to hadronic quantities at leading order in the electromagnetic coupling, the sea-quark charges result in quark-line disconnected diagrams which are challenging to compute precisely. An analysis of the variance of stochastic estimators for the relevant traces of quark propagators helps us to improve the situation for certain flavour combinations and space-time decompositions. We present preliminary numerical results for the variances of the corresponding contributions using an ensemble of Nf=2+1 domain-wall fermions generated by the RBC/UKQCD collaboration.ISSN:1824-803