Leading electromagnetic corrections to meson masses and the HVP Vera G¨ ulpers James Harrison, Andreas J¨ uttner, Antonin Portelli, Christopher Sachrajda School of Physics and Astronomy University of Southampton July 26, 2016 LATTICE 2016
RBC/UKQCD Collaboration BNL and RBRC Greg McGlynn Peking University David Murphy Mattia Bruno Xu Feng Jiqun Tu Tomomi Ishikawa Taku Izubuchi Plymouth University Chulwoo Jung University of Connecticut Christoph Lehner Nicolas Garron Tom Blum Meifeng Lin Taichi Kawanai University of Southampton Edinburgh University Hiroshi Ohki Shigemi Ohta (KEK) Jonathan Flynn Peter Boyle Amarjit Soni Vera G¨ ulpers Guido Cossu Sergey Syritsyn James Harrison Luigi Del Debbio Andreas J¨ uttner Richard Kenway CERN Andrew Lawson Julia Kettle Edwin Lizarazo Ava Khamseh Marina Marinkovic Chris Sachrajda Brian Pendleton Francesco Sanfilippo Antonin Portelli Columbia University Matthew Spraggs Oliver Witzel Tobias Tsang Azusa Yamaguchi Ziyuan Bai Norman Christ York University (Toronto) Luchang Jin KEK Christopher Kelly Renwick Hudspith Julien Frison Bob Mawhinney
Outline Introduction QED correction to meson masses QED correction to the HVP
Introduction Introduction ◮ Isospin breaking corrections − different masses of u and d quark − QED corrections ◮ expected to be of order of 1% ◮ e.g. a µ , isospin breaking effects crucial to be competitive with determination from e + e − → hadrons ◮ QED effects ◮ stochastic QED using U(1) gauge configurations [J. Harrison, Tue 15:00] ◮ expansion of the path integral in α [RM123 Collaboration, Phys.Rev. D87 , 114505 (2013)] �O� = 1 � D [U] D [A] D [Ψ , Ψ] O e − S F [Ψ , Ψ , A , U] e − S A [A] e − S G [U] Z → compute the leading order QED corrections Vera G¨ ulpers (University of Southampton) Lattice 2016 July 26, 2016 1 / 12
Introduction Diagrams at O ( α ) ◮ two insertions of the conserved vector current or one insertion of the tadpole operator at O ( α ) ◮ three different types of (connected) diagrams photon exchange self energy tadpole x x y x z 0 z z 0 0 y ◮ e.g. photon exchange diagram for a charged Kaon � � � � S s (z , x) Γ c ν S s (x , 0) γ 5 S u (0 , y) Γ c µ S u (y , z) γ 5 C(z 0 ) = Tr ∆ µν (x − y) # » x , y z Vera G¨ ulpers (University of Southampton) Lattice 2016 July 26, 2016 2 / 12
Introduction photon propagator ◮ photon propagator (Feynman gauge) e ik · (x − y) 1 � ∆ µν (x − y) = δ µν sin 2 k ρ V 4 � k , # » 2 k � =0 ρ ◮ subtract all spatial zero modes → QED L [Borsanyi et al., Science 347 (2015) 1452-1455] ◮ rewrite photon propagator � ∆ µν (x − u) η (u) η † (y) = ˜ ∆ µν (x) η † (y) ∆ µν (x − y) ≈ u with a stochastic source (e.g. Z 2 ) N 1 � η i (u) η † i (y) ≈ δ u , y N i=1 ◮ calculate ˜ ∆ µν (x) = � ∆ µν (x − u) η (u) using Fast Fourier Transform u Vera G¨ ulpers (University of Southampton) Lattice 2016 July 26, 2016 3 / 12
Introduction construction of the correlators ◮ photon exchange for a charged Kaon � � � � S s (z , x) Γ c ν S s (x , 0) γ 5 S u (0 , y) Γ c µ S u (y , z) γ 5 ∆ µν (x) η † (y) ˜ C(z 0 ) = Tr # z » x , y ◮ sequential propagators ˜ η † (y) ∆ µν (x) Γ c Γ c ν µ ◮ contraction x z 0 y ◮ similar for the self energy using a double sequential propagator Vera G¨ ulpers (University of Southampton) Lattice 2016 July 26, 2016 4 / 12
Introduction setup of the run ◮ N f = 2 + 1 Domain Wall Fermions ◮ 64 × 24 3 lattice with a − 1 = 1 . 78 GeV ◮ L s = 16 , M 5 = 1 . 8 ◮ 87 gauge configurations ◮ pion mass m π = 340 Mev ◮ different masses for valence u and d quarks ≈ physical mass difference [BMW Collaboration, 1604.07112] ◮ physical valence strange quark mass [T. Blum et al , Phys. Rev. D93, 074505 (2016)] ◮ one Z 2 noise for the stochastic insertion of the photon propagator per gauge configuration and source position ◮ 3 source positions ◮ computational cost: 17 inversions per valence quark and source position Vera G¨ ulpers (University of Southampton) Lattice 2016 July 26, 2016 5 / 12
QED correction to meson masses results - correlators ◮ two quarks with m u 10000 ◮ photon exchange t w opt � x µ self energy � µ photon ex hange 0 z � µ tadp ole 1000 y ◮ self energy PRELIMINAR Y C ( t ) x y 100 z 0 ◮ tadpole 10 x z 1 0 0 8 16 24 32 t Vera G¨ ulpers (University of Southampton) Lattice 2016 July 26, 2016 6 / 12
QED correction to meson masses corrections to meson masses ◮ extract mass correction from an O ( α ) diagram by [RM123 Collaboration, Phys.Rev. D87 , 114505 (2013)] C O ( α ) (t) C(t) = C 2 pt (t) + C O ( α ) (t) = A e − (m+ δ m) · t ⇒ δ m = − ∂ t C 2 pt (t) photon exchange self energy tadpole ◮ x y x x 0 z 0 z 0 z y ◮ example: charged Kaon 1 0 . 4 C h /C 2 C self /C 2 pt C tad /C 2 ex pt pt linear �t linear �t linear �t 0 . 8 0 . 2 0 . 3 pt ( t ) pt ( t ) pt ( t ) 0 . 6 h ( t ) /C 2 self,u ( t ) /C 2 tad,u ( t ) /C 2 0 . 2 0 . 4 0 . 1 ex PRELIMINAR Y PRELIMINAR Y PRELIMINAR Y C C C 0 . 1 0 . 2 0 0 0 0 8 16 24 32 0 8 16 24 32 0 8 16 24 32 t t t Vera G¨ ulpers (University of Southampton) Lattice 2016 July 26, 2016 7 / 12
QED correction to meson masses results QED corrections to meson masses ◮ some (very preliminary) results for QED corrections to meson masses (w/o finite volume correction) Quantity this work stochastic QED [Tue, 15:00] M γ 2 . 70 ± 0 . 02 MeV 3 . 42 ± 0 . 02 MeV π + M γ 0 . 70 ± 0 . 02 MeV 1 . 52 ± 0 . 01 MeV π 0 M π + − M π 0 2 . 00 ± 0 . 03 MeV 1 . 90 ± 0 . 02 MeV M γ 2 . 12 ± 0 . 02 MeV 2 . 70 ± 0 . 02 MeV K + M γ 0 . 28 ± 0 . 01 MeV 0 . 55 ± 0 . 01 MeV K 0 Vera G¨ ulpers (University of Southampton) Lattice 2016 July 26, 2016 8 / 12
QED correction to meson masses results QED corrections to meson masses ◮ some (very preliminary) results for QED corrections to meson masses (w/o finite volume correction) Quantity this work stochastic QED [Tue, 15:00] M γ 2 . 70 ± 0 . 02 MeV 3 . 42 ± 0 . 02 MeV π + M γ 0 . 70 ± 0 . 02 MeV 1 . 52 ± 0 . 01 MeV π 0 M π + − M π 0 2 . 00 ± 0 . 03 MeV 1 . 90 ± 0 . 02 MeV M γ 2 . 12 ± 0 . 02 MeV 2 . 70 ± 0 . 02 MeV K + M γ 0 . 28 ± 0 . 01 MeV 0 . 55 ± 0 . 01 MeV K 0 ◮ pion mass splitting is a special case [RM123 Collaboration, Phys.Rev. D87 , 114505 (2013)] → depends only on photon exchange diagram x M π + − M π 0 = (q u − q d ) 2 C exch (t) 0 z e 2 ∂ t 2 C 2pt (t) y ◮ problem in the self energy and/or the tadpole diagram? → needs to be resolved Vera G¨ ulpers (University of Southampton) Lattice 2016 July 26, 2016 8 / 12
QED correction to meson masses Comparison of statistical precision ◮ computational cost perturbative method 17 inversions per quark flavor • stochastic method 3 inversions per quark flavor • ◮ statistical error ∆ of QED contribution to effective Kaon mass ◮ scaled by √ # inversions 3 2 . 5 2 h sto ert / ∆ 1 . 5 p ∆ 1 0 . 5 PRELIMINAR Y 0 0 8 16 24 32 t Vera G¨ ulpers (University of Southampton) Lattice 2016 July 26, 2016 9 / 12
QED correction to the HVP The hadronic vacuum polarisation ◮ vacuum polarisation tensor e i Q · x � � � = (Q µ Q ν − δ µν Q 2 ) Π(Q 2 ) j γ µ (x) j γ Π µν (Q) = ν (0) x ◮ correlator � � C µν (x) = Z v q 2 V c µ (x)V ℓ ν (0) f ◮ construction of the HVP tensor, see eg. [RBC/UKQCD, JHEP 1604 (2016) 063], [M. Spraggs, Tue 17:10] � e − iQ · x C µν (x) − � Π µν (Q) = C µν (x) x x (with zero mode subtraction) ◮ vacuum polarisation Q 2 ) = 1 Π jj (Q) Π(ˆ � ˆ 3 Q 2 j Q 2 ) ≡ 4 Q 2 ) + 1 Q 2 ) + 1 Π(ˆ 9Π u (ˆ 9Π d (ˆ 9Π s (ˆ Q 2 ) Vera G¨ ulpers (University of Southampton) Lattice 2016 July 26, 2016 10 / 12
QED correction to the HVP First look at HVP ◮ hadronic vacuum polarisation for the u quark 0 (ˆ Q 2 ) , right: QED corrections to HVP δ Π u (ˆ ◮ left: HVP without QED Π u Q 2 ) Π u (ˆ Q 2 ) = Π u 0 (ˆ Q 2 ) + δ Π u (ˆ Q 2 ) 0 0 . 0006 Π u ( Q 2 ) w/o QED self energy ex hange 0 . 0004 tadp ole total QED orre tion − 0 . 05 0 . 0002 δ Π u ( Q 2 ) Π u ( Q 2 ) − 0 . 1 0 − 0 . 0002 − 0 . 15 − 0 . 0004 PRELIMINAR Y PRELIMINAR Y − 0 . 2 − 0 . 0006 0 2 4 6 8 0 2 4 6 8 ˆ ˆ Q 2 / Ge Q 2 / V 2 V Ge Vera G¨ ulpers (University of Southampton) Lattice 2016 July 26, 2016 11 / 12
QED correction to the HVP Summary ◮ Leading order QED corrections by expansion of the path integral ◮ corrections to meson masses and HVP ◮ exploratory study ◮ currently, discrepancy between results from stochastic and perturbative approach → needs to be resolved Outlook ◮ Coulomb gauge for the photon propagator ◮ more gauge ensembles ◮ matrix elements [N. Carrasco et al , Phys. Rev. D91 (2015) 074506], [N. Tantalo, Wed 11:50], [S. Simula, Wed 12:10] Vera G¨ ulpers (University of Southampton) Lattice 2016 July 26, 2016 12 / 12
Backup Backup Vera G¨ ulpers (University of Southampton) Lattice 2016 July 26, 2016 13 / 12
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