Kaon form factor and decay constant from lattice QCD Aida X. - PowerPoint PPT Presentation

Kaon form factor and decay constant from lattice QCD Aida X. El-Khadra (University of Illinois) Current and Future Status of the First-Row CKM Unitarity, 16-18 May 2019 HC2NP workshop Puerto de la Cruz, Tenerife, 26-30 Sep 2016

Fermilab Lattice and MILC collaboration Fermilab Lattice Collaboration: 
 AXK, E. Freeland, E. Gámiz, S. Gottlieb, A. Kronfeld, J. Laiho, P . Mackenzie, E. Neil, J. Simone, R. Van de Water 
 Z. Gelzer, W. Jay MILC: 
 A. Bazavov, C. Bernard, C. DeTar, S. Gottlieb, U. Heller, J. Osborn, R. Sugar, D. Toussaint, 
 J. Komijani, R. Li, Y. Liu, A. Vaquero 
 Computing done at: 
 Mira, Theta (Argonne under DOE INCITE, ALCC) 
 Blue Waters (NCSA, UIUC under NSF PRAC and BW Illinois) 
 BigRed 2+ (Indiana U), TACC, NCAR (XSEDE); NICS, ORNL (Teragrid); FNAL, BNL (USQCD), … A. El-Khadra FirstRow workshop 2019, 16-18 May 2019 2 �

Outline Introduction and Motivation Lattice QCD introduction Leptonic and semileptonic K decay amplitudes Set-up and analysis ( K ℓ 3 ) Systematic error analysis Results in comparison Implications for … ✦ | V us | ✦ first row unitarity Summary and Outlook A. El-Khadra FirstRow workshop 2019, 16-18 May 2019 � 3

<latexit sha1_base64="+TQolo0IdUiTNLaeameq5z6UEUg=">ACMnicbVBLSwMxGMzWV62vqkcvwSJUqstuFfRY9KDipYJ9QHdbsmnahibZJckKpfQ3efGXCB70oIhXf4Rpu4JWhwSGmflIvgkiRpV2nGcrNTe/sLiUXs6srK6tb2Q3t6oqjCUmFRyUNYDpAijglQ01YzUI0kQDxipBf3zsV+7I1LRUNzqQUR8jrqCdihG2kit7JV3gThH+eumAz0dQi+izUPo8bhZgJ6IW4btH0DPnO9cYZybDdi23crmHNuZAP4lbkJyIEG5lX302iGOREaM6RUw3Ui7Q+R1BQzMsp4sSIRwn3UJQ1DBeJE+cPJyiO4Z5Q27ITSXKHhRP05MURcqQEPTJIj3VOz3lj8z2vEunPqD6mIYk0Enj7UiRk03Yz7g20qCdZsYAjCkpq/QtxDEmFtWs6YEtzZlf+SatF2j+zizXGudJbUkQY7YBfkgQtOQAlcgjKoAzuwRN4BW/Wg/VivVsf02jKSma2wS9Yn18VYqZk</latexit> ⬆ ⬆ Introduction and Motivation µ + example: K ℓ 3 decay ν µ W ¯ s ¯ u K 0 π − d Experiment vs. SM theory: (experiment) = (known) x ( CKM factors ) x (had. matrix element) Lattice QCD Γ ( K 0 → π − µ + ν µ ) , Γ ( K + → µ + ν µ ) , ... , d Γ ( B → K ` + ` − ) d Γ ( B → ⇡`⌫ ) parameterize the MEs in , . . . dq 2 dq 2 terms of form factors, d Γ ( B → D `⌫ ) , d Γ ( B → D ⌧⌫ ) ∆ m d ( s ) decay constants, bag , . . . d ! d ! parameters, ... … A. El-Khadra FirstRow workshop 2019, 16-18 May 2019 � 4

⬆ <latexit sha1_base64="+TQolo0IdUiTNLaeameq5z6UEUg=">ACMnicbVBLSwMxGMzWV62vqkcvwSJUqstuFfRY9KDipYJ9QHdbsmnahibZJckKpfQ3efGXCB70oIhXf4Rpu4JWhwSGmflIvgkiRpV2nGcrNTe/sLiUXs6srK6tb2Q3t6oqjCUmFRyUNYDpAijglQ01YzUI0kQDxipBf3zsV+7I1LRUNzqQUR8jrqCdihG2kit7JV3gThH+eumAz0dQi+izUPo8bhZgJ6IW4btH0DPnO9cYZybDdi23crmHNuZAP4lbkJyIEG5lX302iGOREaM6RUw3Ui7Q+R1BQzMsp4sSIRwn3UJQ1DBeJE+cPJyiO4Z5Q27ITSXKHhRP05MURcqQEPTJIj3VOz3lj8z2vEunPqD6mIYk0Enj7UiRk03Yz7g20qCdZsYAjCkpq/QtxDEmFtWs6YEtzZlf+SatF2j+zizXGudJbUkQY7YBfkgQtOQAlcgjKoAzuwRN4BW/Wg/VivVsf02jKSma2wS9Yn18VYqZk</latexit> ⬆ Introduction and Motivation µ + example: K ℓ 3 decay ν µ W ¯ s ¯ u K 0 π − d Experiment vs. SM theory: (experiment) = (known) x ( CKM factors ) x (had. matrix element) Two main purposes: combine experimental measurements with Lattice QCD Γ ( K 0 → π − µ + ν µ ) , Γ ( K + → µ + ν µ ) , ... LQCD results to determine CKM parameters. , d Γ ( B → K ` + ` − ) d Γ ( B → ⇡`⌫ ) confront experimental measurements of rare parameterize the MEs in , . . . dq 2 dq 2 processes or lepton flavor (universality) terms of form factors, d Γ ( B → D `⌫ ) , d Γ ( B → D ⌧⌫ ) violating observables with SM theory using ∆ m d ( s ) decay constants, bag , . . . d ! d ! LQCD inputs. parameters, ... … A. El-Khadra FirstRow workshop 2019, 16-18 May 2019 � 4

Introduction V us K + → µ + ν µ example: µ + ¯ s W K + u ν µ K + → ` + ⌫ ` ( � ) = (known) × (1 + � ` EM ) × | V us | 2 × f 2 � � Γ K + Needed to relate pure QCD decay constant to experiment. Davide Giusti ➠ Currently estimated phenomenologically. [Cirigliano et al, arXiv: talk 1107.6001, RMP 2012] use experiment + LQCD input for determination of CKM element ratios such as : reduced statistical and systematic errors. f K + /f π + A. El-Khadra FirstRow workshop 2019, 16-18 May 2019 5 �

Introduction µ + V us example: K 0 → ⇡ − ` + ⌫ ` ν µ W ¯ s ¯ u K 0 π − d ✓ ◆ phase SU(2) ) × | V us | 2 × | f K 0 ⇡ − (0) | 2 × (1 + δ K ` EM + δ K ⇡ Γ K ` 3 = (known) × + space Needed to include charged kaon decay Needed to relate pure QCD form factor in the experimental average. to experiment. Mode dependent. Both are currently estimated phenomenologically. [Cirigliano et al, arXiv:1107.6001, RMP 2012] . A. El-Khadra FirstRow workshop 2019, 16-18 May 2019 � 6

Lattice QCD Introduction / + m f ) ψ f + 1 ¯ X 4tr F µ ν F µ ν L QCD = ψ f ( D f discrete Euclidean space-time (spacing a ) 
 derivatives ➙ difference operators, etc… 
 L x finite spatial volume ( L ) a finite time extent ( T ) adjustable parameters lattice spacing: a ➙ 0 finite volume, time: L ➙ ∞ , T > L quark masses ( m f ): 
 M H, lat = M H, exp m ud m s m c m b tune using hadron masses 
 m f ➙ m f, phys extrapolations/interpolations A. El-Khadra FirstRow workshop 2019, 16-18 May 2019 7 �

Lattice QCD Introduction / + m f ) ψ f + 1 ¯ X 4tr F µ ν F µ ν L QCD = ψ f ( D f discrete Euclidean space-time (spacing a ) 
 derivatives ➙ difference operators, etc… 
 L x finite spatial volume ( L ) a finite time extent ( T ) Integrals are evaluated numerically using monte adjustable parameters carlo methods. lattice spacing: a ➙ 0 finite volume, time: L ➙ ∞ , T > L quark masses ( m f ): 
 M H, lat = M H, exp m ud m s m c m b tune using hadron masses 
 m f ➙ m f, phys extrapolations/interpolations A. El-Khadra FirstRow workshop 2019, 16-18 May 2019 7 �

Lattice QCD Introduction L x a use monte carlo methods (importance sampling) to evaluate the integral. Note: Integrating over the fermion fields leaves det( D +m ) in the integrand. The / correlation functions, O , are then written in terms of ( D+m ) - 1 and gluon fields. / steps of a lattice QCD calculation: 1. generate gluon field configurations according to det( D+m ) e -S / 2. calculate quark propagators, ( D+m q ) -1 , for each valence quark flavor and source point / 3. tie together quark propagators into hadronic correlation functions (usually 2 or 3-pt functions) 4. statistical analysis to extract hadron masses, energies, hadronic matrix elements, …. from correlation functions 5. systematic error analysis A. El-Khadra FirstRow workshop 2019, 16-18 May 2019 � 8

Lattice QCD Introduction L x a systematic error analysis ...of lattice spacing, chiral, heavy quark, and finite volume effects is based on EFT (Effective Field Theory) descriptions of QCD ➙ ab initio • finite a : Symanzik EFT • light quark masses: Chiral Perturbation Theory • heavy quarks: HQET • finite L : finite volume EFT 
 • need large enough L and small enough a and simulations with several a, L, … A. El-Khadra FirstRow workshop 2019, 16-18 May 2019 9 �

Lattice QCD Introduction L x a discretization effects — continuum extrapolation • typical momentum scale of quarks gluons inside hadrons: ~ Λ QCD • make a small to separate the scales: Λ QCD ≪ 1/ a h O i lat = h O i cont + O ( a Λ ) n • Symanzik EFT: , n ≥ 2 
 provides functional form for extrapolation (depends on the details of the lattice action) can be used to build improved lattice actions can be used to anticipate the size of discretization effects • to control and reliably estimate 
 L the error, repeat … a (fm) A. El-Khadra FirstRow workshop 2019, 16-18 May 2019 10 �

Lattice QCD Introduction L x a finite volume effects One stable hadron (meson) in initial/final state: If L is large enough, FV error ∼ e − m π L keep m π L & 4 To quantify residual error: include FV effects in χ PT compare results at several L s (with other parameters fixed) The story changes completely with two or more hadrons in initial/final state or if there are two or more intermediate state hadrons. ➠ “simple quantities”: 
 no more than one stable hadron in initial/final state 
 If QED is included, FV effects also become more complicated… A. El-Khadra FirstRow workshop 2019, 16-18 May 2019 11 �

Lattice QCD Introduction L x a combined chiral-continuum interpolation/extrapolation When m light ≶ 1/2 ( m u + m d ) phys 
 𝜓 PT guides the interpolation/extrapolation to the physical point. combined chiral-continuum interpolation/extrapolation include (light quark) discretization effects 
 (for example, staggered 𝜓 PT ) A. El-Khadra FirstRow workshop 2019, 16-18 May 2019 � 12

Lattice QCD Introduction L x a combined chiral-continuum interpolation/extrapolation MILC n f = 2+1+1 Growing number of collaborations have generated sets of ensembles that include sea quarks with physical light-quark masses and use improved lattice actions : 
 PACS-CS, BMW, MILC, RBC/UKQCD, ETM,… A. El-Khadra FirstRow workshop 2019, 16-18 May 2019 � 13