Numerical investigation of QED finite-volume effects for meson mass and HVP James Harrison University of Southampton RBC/UKQCD 22 nd June 2017 1/23
The RBC & UKQCD collaborations Jiqun Tu BNL and RBRC KEK Bigeng Wang Mattia Bruno Julien Frison Tianle Wang Tomomi Ishikawa Taku Izubuchi University of Liverpool University of Connecticut Luchang Jin Nicolas Garron Tom Blum Chulwoo Jung Dan Hoying Christoph Lehner Peking University Cheng Tu Meifeng Lin Xu Feng Hiroshi Ohki Edinburgh University Shigemi Ohta (KEK) University of Southampton Peter Boyle Amarjit Soni Guido Cossu Jonathan Flynn Sergey Syritsyn Vera Guelpers Luigi Del Debbio Richard Kenway James Harrison Columbia University Andreas Juettner Julia Kettle Ziyuan Bai Ava Khamseh Andrew Lawson Norman Christ Edwin Lizarazo Brian Pendleton Duo Guo Antonin Portelli Chris Sachrajda Christopher Kelly Tobias Tsang Bob Mawhinney York University (Toronto) Oliver Witzel David Murphy Azusa Yamaguchi Renwick Hudspith Masaaki Tomii 2/23
Introduction • We have made an exploratory calculation of isospin-breaking corrections to the HVP [V. Guelpers, Thursday, 17:30; arXiv:1706.05293] . • QED finite volume effects (FVE) must be taken into account for a physical calculation of the HVP . • 2-loop analytical calculation of HVP FVE has not yet been carried out. • We use lattice scalar QED, as a quicker numerical calculation of FVE and as a cross-check for the analytical result. • Cross-checked method against known results for meson mass FVE. • Preliminary results for FV behaviour of HVP . 3/23
Isospin breaking � � α , m u − m d • Isospin-breaking effects enter at O ∼ 1% Λ QCD • Precision in QCD now approaching 1% for several quantities. – e.g. hadronic contribution to muon g − 2: 1% precision or better required to compete with determination from e + e − → hadrons, and to prepare for upcoming experiments at Fermilab and J-PARC. • Isospin is significant systematic uncertainty • Electromagnetism and light quark mass difference need to be included. 4/23
QED in finite volume • For sufficiently large volume, QCD FVE are exponentially suppressed. • QED is long-range due to massless photon, so QED FVE are much larger than for QCD. • Typically scale with inverse powers of L , not exponential. • QED FVE can be comparable in magnitude to QED corrections [Borsanyi et al., arXiv:1406.4088] . • FVE must be accounted for in any lattice calculation involving QED. 5/23
QED finite volume effects • FVE are not very sensitive to higher modes (discretisation, hadron structure). • Can study these analytically using an effective theory, with point-like hadrons – Scalar QED (pseudoscalar mesons) – Spinor QED (spin-1/2 baryons) • In momentum space, integrals become discrete sums in finite volume. • Derive relations between finite-volume quantites and their infinite-volume counterparts. • FVE have been calculated for hadron masses [Davoudi & Savage, arXiv:1402.6741; Borsanyi et al., arXiv:1406.4088] and leptonic decay amplitudes [Lubicz et al., arXiv:1611.08497] . 6/23
QED finite volume effects Alternative: lattice scalar QED. • Generate U ( 1 ) gauge configurations • Calculate scalar propagators and expectation values • Repeat for several volumes • Fit polynomial in 1 / L to extract coefficients Offers a quick numerical method for obtaining FVE, which is generally applicable to a wide range of observables. 7/23
Scalar QED on the lattice Discretised scalar QED action: a 4 φ ∗ ( x ) ∆ φ ( x ) + S γ � S � φ , A µ � � A µ � = 2 x � D ∗ µ D µ + m 2 − ∆ = µ a − 1 � e ieaA µ ( x ) f ( x + a ˆ � D µ f ( x ) µ ) − f ( x ) = Quenched theory: set scalar determinant = 1 in path integral. 8/23
Sampling gauge field configurations Non-compact photon action: a 4 � � � 2 � A µ � � ∂ µ A ν − ∂ ν A µ S γ = 4 x µ , ν a − 1 [ f ( x + a ˆ ∂ µ f ( x ) µ ) − f ( x )] = • Feynman gauge. • In momentum space, ˜ A µ ( k ) is Gaussian - cheap to sample gauge configurations [Duncan, Eichten & Thacker, arXiv:hep-lat/9602005] . • Subtract zero mode using QED L scheme [Uno & Hayakawa, arXiv:0804.2044] . 9/23
Calculating the scalar propagator Alternative to CG, making use of FFT. Expand scalar propagator around e = 0: � µ D µ + m 2 ∆ 0 + e ∆ 1 + e 2 ∆ 2 + O � e 3 � D ∗ ∆ = − = µ ∆ − 1 ∆ − 1 0 − e ∆ − 1 0 ∆ 1 ∆ − 1 = 0 + e 2 � ∆ − 1 0 ∆ 1 ∆ − 1 0 ∆ 1 ∆ − 1 0 − ∆ − 1 0 ∆ 2 ∆ − 1 � � e 3 � + O 0 + + + 10/23
Calculating the scalar propagator ∆ − 1 F − 1 1 k 2 + m 2 F 0 = ˆ e iak µ � 1 ∆ − 1 0 ∆ 1 ∆ − 1 − ia − 1 � µ F − 1 k 2 + m 2 F A µ F − 1 0 = ˆ ˆ k 2 + m 2 − e − iak µ 1 � k 2 + m 2 F A µ F − 1 F ˆ ˆ k 2 + m 2 e iak µ � 1 ∆ − 1 0 ∆ 2 ∆ − 1 1 k 2 + m 2 F A 2 µ F − 1 µ F − 1 � 0 = 2 ˆ ˆ k 2 + m 2 − e − iak µ � 1 k 2 + m 2 F A 2 µ F − 1 F ˆ ˆ k 2 + m 2 where F represents the Fourier transform. 11/23
Finite volume corrections - meson mass Scalar mass FV corrections are known: � � m 0 + 2 ∞ − α κ m 2 ( L ) ∼ m 2 L L κ = 2 . 837297 ( 1 ) in QED L , up to terms exponentially suppressed in m 0 L [Borsanyi et al., arXiv:1406.4088] . m 0 is bare mass, m ∞ is infinite-volume mass. We can use this as a validity check of our method. 12/23
Finite volume corrections - meson mass We define an effective mass difference from ratio of charged and free propagators (neglect backward propagating states for simplicity of presentation): ∆ − 1 � t , � � ∆ − 1 � t + 1 , � � 0 0 δ m eff ( t ) = � − � t , � � t + 1 , � � ∆ − 1 ∆ − 1 0 0 0 0 Can compare to perturbation theory: ∆ − 1 � t , � � 0 � T + Σ � � = c + te 2 � t , � 2 m 0 ∆ − 1 0 0 13/23
Finite volume corrections - meson mass Effective mass difference, 24 3 × 128, m 0 = 0.2 0.1075 0.1074 0.1073 0.1072 m eff ( t ) 0.1071 0.1070 0.1069 0.1068 0.1067 0 10 20 30 40 50 60 t 14/23
Finite volume corrections - meson mass • Eight volumes, from L = 4 to L = 64 • T = 128 for all L • 100 configurations per volume • Computed on a single KNL • Largest volume took ∼ 1 day 15/23
Finite volume corrections - meson mass L 3 × 128, m 0 = 0.2 m 2 L ( m 0 + 2 L ) 0.0832 Lattice 0.0831 0.0830 0.0829 m 2 0.0828 0.0827 0.0826 0.0825 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1/L Good agreement between lattice and analytical: χ 2 / d . o . f . = 0 . 44 (note: this is not a fit). 16/23
Finite volume corrections - meson mass L 3 × 128, m 0 = 0.2 L ( m 0 + 2 L ) m 2 0.0830 Lattice 0.0825 0.0820 0.0815 m 2 0.0810 0.0805 0.0800 0.0795 0.0790 0.00 0.05 0.10 0.15 0.20 0.25 1/L 17/23
Finite volume corrections - HVP Leading effective field theory contribution to HVP is scalar bubble diagram. Vacuum polarisation tensor: C µ ν ( x ) = � V µ ( x ) V ν ( 0 ) � Π µ ν ( Q ) = a 4 � e − iQ · x C µ ν ( x ) − a 4 � C µ ν ( x ) x x � ˆ Q 2 − ˆ � δ µ ν ˆ Q µ ˆ � Q 2 � Π µ ν ( Q ) = Q ν Π 18/23
Finite volume corrections - HVP Conserved vector current: φ ∗ ( x ) e ieaA µ ( x ) φ ( x + a ˆ µ ) − φ ∗ ( x + a ˆ V µ ( x ) = a 2 � µ ) e − ieaA µ ( x ) φ ( x ) � Contact term at x = 0 due to non-local currents: T µ ( x ) = φ ∗ ( x ) e ieaA µ ( x ) φ ( x + a ˆ µ ) + φ ∗ ( x + a ˆ µ ) e − ieaA µ ( x ) φ ( x ) Contact term is removed by subtracting zero-mode ˜ Π µ ν ( 0 ) . 19/23
Finite volume corrections - HVP O ( α ) contributions to scalar VP: 20/23
Finite volume corrections - HVP Free scalar VP, L 3 × 128, Q 0 = 2 / T , m 0 = 0.2 PRELIMINARY c 1 e c 2 L Lattice 10 3 10 4 0 ( L ) 10 5 c 0 10 6 10 7 10 20 30 40 50 60 L 21/23
Finite volume corrections - HVP Scalar VP ( ) correction, L 3 × 128, Q 0 = 2 / T , m 0 = 0.2 PRELIMINARY 0.0078 0.0079 0.0080 c 0 + c 1 / L 3 , 2 /d. o. f. = 8.50 0 ( L ) c 0 + c 1 / L 4 , 2 /d. o. f. = 0.27 0.0081 c 0 + c 1 / L 5 , 2 /d. o. f. = 1.94 ( L ) Lattice 0.0082 0.0083 0.0084 0.015 0.020 0.025 0.030 0.035 0.040 0.045 1/ L 22/23
Summary • We want to calculate QED FVE for the HVP . • As a first step before analytical calculation, we use lattice scalar QED to calculate FVE • Simulations are very cheap ( ∼ 1 day on a single KNL) • We successfully reproduce known results for meson mass FVE • Preliminary data suggest leading QED FV behaviour of HVP may be O � 1 / L 4 � • This technique is more generally applicable to a wide range of quantities 23/23
Acknowledgements We acknowledge financial support from the EPSRC Centre for Doctoral Training in Next Generation Computational Modelling grant EP/L015382/1. 24/23
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