Luchang Jin University of Connecticut / RIKEN BNL Research Center Thomas Blum (UConn / RBRC), Norman Christ (Columbia), Masashi Hayakawa (Nagoya), Xu Feng (Peking U), Taku Izubuchi (BNL / RBRC), Chulwoo Jung (BNL), Christoph Lehner (Regensburg), Cheng Tu (UConn) and RBC-UKQCD collaborations June 20, 2019 37th international conference on lattice fjeld theory (LATTICE 2019) Hilton Hotel Wuhan Riverside, Wuhan, China QED ∞ in muon g − 2 , hadron spectroscopy, and beyond
The RBC & UKQCD collaborations Bigeng Wang KEK BNL and BNL/RBRC Tianle Wang Julien Frison Yasumichi Aoki (KEK) Yidi Zhao T aku Izubuchi University of Liverpool Yong-Chull Jang University of Connecticut Nicolas Garron Chulwoo Jung T om Blum Meifeng Lin MIT Dan Hoying (BNL) Aaron Meyer Luchang Jin (RBRC) David Murphy Hiroshi Ohki Cheng Tu Shigemi Ohta (KEK) Peking University Amarjit Soni Edinburgh University Xu Feng Peter Boyle UC Boulder University of Regensburg Luigi Del Debbio Oliver Witzel Christoph Lehner (BNL) Felix Erben CERN Vera Gülpers University of Southampton T adeusz Janowski Mattia Bruno Nils Asmussen Julia Kettle Jonathan Flynn Columbia University Michael Marshall Ryan Hill Fionn Ó hÓgáin Ryan Abbot Andreas Jüttner Antonin Portelli Norman Christ James Richings T obias T sang Duo Guo Chris Sachrajda Andrew Yong Christopher Kelly Azusa Yamaguchi Bob Mawhinney Stony Brook University Masaaki T omii Jun-Sik Yoo Jiqun Tu Masashi Hayakawa (Nagoya) Sergey Syritsyn (RBRC)
Outline 1 / 22 1. Introduction Pion mass splitting with the infjnite volume reconstruction method 4. Conclusion 2. QED ∞ in Muon g − 2 Long-distance contribution to the HLbL in position space from the π 0 pole 3. QED ∞ in hadron spectroscopy (X. Feng and L. Jin 2018)
World Average dominated by BNL 2 / 22 In comparison, for electron • Fermilab E989 (0.14 ppm) Almost 4 times more accurate then the previous experiment. Muon g − 2 : experiments µ = − g e ⃗ 2 m⃗ s a µ = (11659208 . 9 ± 6 . 3) × 10 − 10 a e = (11596521 . 8073 ± 0 . 0028) × 10 − 10 • J-PARC E34 also plans to measure muon g − 2 with similar precision.
new physics ? Weak incl. 2-loops HVP NLO&NNLO HVP Gnendiger et al, 2013 3 / 22 FJ17 HLbL Aoyama, et al, 2012 Talk: C. Lehner (Mon 14:20) FJ17 Hadronic Models, “Consensus” Standard Model QED incl. 5-loops Experiment Difgerence (Exp-SM) • More than 3 standard deviations due to mistake in the highly sophisticated perturbative C. Lehner et al (RBC-UKQCD), 2018 Muon g − 2 : theory 10 10 a µ × 11658471 . 9 ± 0 . 0 15 . 4 ± 0 . 1 692 . 5 ± 2 . 7 − 8 . 7 ± 0 . 1 10 . 3 ± 2 . 9 11659181 . 3 ± 4 . 0 11659208 . 9 ± 6 . 3 E821, The g − 2 Collab. 2006 27 . 6 ± 7 . 5 calculation, inaccuracy of e + e − → hadrons experiments, incorrect hadronic model, or
F. Hagelstein, V. Pascalutsa. G. Colangelo, M. Hoferichter, B. Kubis, Schwinger ’ s sum rule L. Dai, M. Pennington. Light by light scattering sum rule HVP: Hadronic Vacuum Polarization I. Danilkin, O. Deineka, M. Vanderhaeghen. Light by light scattering sum rule M. Procura, P. Stofger. HLbL: Hadronic Light by Light 4 / 22 • HLbL is much more complicated. Anyway, lots of works have been done in this direction. Pion pole, pion box, rescattering Hadronic contribution to Muon g − 2 q = p ′ − p, ν q = p ′ − p, ν p ′ p ′ p p • Dispersive approach with e + e − → hadrons experimental data is very successful for HVP.
Lattice QCD 5 / 22 • Discrete lattice usually corresponds to hard cut ofg in momentum space. • Hard cut ofg regularization is said to break gauge invariance, while lattice does not. • Position space formulation can provide new insight.
in lattice calculation, or new physics ? Weak incl. 2-loops HVP NLO&NNLO HVP Gnendiger et al, 2013 6 / 22 FJ17 HLbL Aoyama, et al, 2012 Talk: C. Lehner (Mon 14:20) Talk: T. Blum (Tue 16:30) Lattice QCD, RBC-UKQCD Standard Model QED incl. 5-loops Experiment Difgerence (Exp-SM) • More than 3 standard deviations due to mistake in the highly sophisticated perturbative C. Lehner et al (RBC-UKQCD), 2018 Muon g − 2 : theory 10 10 a µ × 11658471 . 9 ± 0 . 0 15 . 4 ± 0 . 1 692 . 5 ± 2 . 7 − 8 . 7 ± 0 . 1 7 . 4 ± 6 . 6 11659178 . 5 ± 7 . 1 11659208 . 9 ± 6 . 3 E821, The g − 2 Collab. 2006 30 . 4 ± 9 . 5 calculation, inaccuracy of e + e − → hadrons experiments, under estimating sys/stat error
Outline 7 / 22 1. Introduction Pion mass splitting with the infjnite volume reconstruction method 4. Conclusion 2. QED ∞ in Muon g − 2 Long-distance contribution to the HLbL in position space from the π 0 pole 3. QED ∞ in hadron spectroscopy (X. Feng and L. Jin 2018)
2016] [J. Bijnens and J. Relefors, 2016] • Large statistical error from the long • We start the calculation by simulating both the QCD part and the QED part on the disconnected diagrams. [L. Jin et al, formalism. [T. Blum et al, 2014,2016,2017] • Mainz ’s group fjrst demonstrated that it is possible to effjciently calculate the QED part distance region, especially for the [N. Asmussen et al, LATTICE2016] • We can improve the infjnite volume QED part to reduce the discretization error signifjcantly with a subtracted QED kernel . [T. Blum et al, 2017] 8 / 22 HLbL: QED ∞ to eliminate power-law FV efgects QED Box QCD Box lattice within a fjnite volume using the QED L z x x op y z ′ y ′ x ′ At physical point, π 0 is very light, therefore: semi-analytically in the infjnite volume . • Large volume is needed.
9 / 22 • Norman’s initial idea for calculating the long-distance (Norman’s talk at HLbL workshop in UCONN, 2018/03/13) • The hadronic 4-point function in the long-distance region can be calculated with two three-point functions, which can be directly calculated in a modest size lattice. Long-distance part of HLbL from the π 0 pole part of HLbL from π 0 pole.
Long-distance HLbL: further tweaks lattice so its time direction is pion. • We only need the following with zero momentum pion. • Only need to measure the propagating direction. 10 / 22 aligned with the pion • We can rotate the modest size y y ′ π 0 π 0 → γγ three-point function π 0 x ′ x p = 0 three-point function for ⃗ H µ,ν ( x − y ) = ⟨ 0 | J µ ( x ) J ν ( y ) | π 0 ⟩
Long-distance HLbL: numerical results (prelim) 11 / 22 Reverse partial sum plotted. • UCONN graduate student Cheng Tu did the calculation. • 32ID: 32 3 × 64 , a − 1 = 1 . 015 GeV, 6 32ID M π = 142 MeV. 5 • R max = max( | x − y | , | x − y ′ | , | y − y ′ | ) . 4 a µ × 10 10 3 y y ′ 2 π 0 1 π 0 x ′ 0 x 0 1 2 3 4 5 6 7 8 9 R max (fm)
• Likely more applications. Poster: X. Feng (Tue 17:50). Talk: Y. Zhao (Tue 17:30). • We calculated it using one point source propagator Talk: N. Christ (Tue 17:10). adventurous calculation • We averaged it within a confjguration, saved it to disk, then performed furthur contractions. starting point for a more very noval calculation • Yidi Zhao and Norman Christ’s 12 / 22 coordinate-space method. The three-point function: π 0 → γγ • π 0 → γγ decay width from a H µ,ν ( x − y ) = ⟨ 0 | J µ ( x ) J ν ( y ) | π 0 ⟩ from y and wall source propagators seperated by enough distance from both x and y to create the pion. π 0 → e + e − , which serves as the K L → µ + µ − . x π 0 y
Outline 13 / 22 1. Introduction Pion mass splitting with the infjnite volume reconstruction method 4. Conclusion 2. QED ∞ in Muon g − 2 Long-distance contribution to the HLbL in position space from the π 0 pole 3. QED ∞ in hadron spectroscopy (X. Feng and L. Jin 2018)
QED correction to hadron masses • We can evaluate the QED part, the photon propagator, in infjnite volume. • To eliminate all power-law suppressed fjnite volume efgects, a difgerent treatment for the 14 / 22 • The hadronic function do not always fall exponentially in the long distance region. 1 ∫ d 4 x H µ,ν ( x ) S γ ∆ M = I = µ,ν ( x ) x, µ 2 1 H µ,ν ( x ) = 2 M ⟨ N | TJ µ ( x ) J ν (0) | N ⟩ δ µ,ν S γ µ,ν ( x ) = 4 π 2 x 2 0 , ν ∫ t s 1 ∫ I ( s ) d 3 x H µ,ν ( x ) S γ = dt µ,ν ( x ) 2 ∆ M = I = I ( s ) + I ( l ) − t s ∫ ∞ ∫ I ( l ) d 3 x H µ,ν ( x ) S γ = dt µ,ν ( x ) t s long distance part is required . ( t s ≲ L )
The infjnite volume reconstruction method ground states (possibly with small momentum). Therefore: 15 / 22 • For the short distance part, I ( s ) can be directly calculated on a fjnite volume lattice: ∫ t s ∫ L/ 2 I ( s ) ≈ I ( s,L ) = 1 d 3 x H µ,ν ( x ) S γ dt µ,ν ( x ) 2 − t s L/ 2 • For the long distance part, we need to evaluate H µ,ν ( x ) indirectly. Note that when t is large, the intermediate states between the two currents are mostly [ 1 d 3 p ∫ 1 ] e i⃗ p · ⃗ x − ( E ⃗ p − M ) t H µ,ν ( x ) ≈ 2 M ⟨ N | J µ (0) | N ( ⃗ p ) ⟩⟨ N ( ⃗ p ) | J ν (0) | N ⟩ (2 π ) 3 2 E ⃗ p – We only need to calculate the form factors: ⟨ N ( ⃗ p ) | J ν (0) | N ⟩ ! – Values for all ⃗ p are needed. Simply inverse Fouier transform the above relation! ∫ 1 1 p − M ) t s ≈ d 3 x H µ,ν ( t s ,⃗ x ) e − i⃗ p · ⃗ x +( E ⃗ 2 M ⟨ N | J µ (0) | N ( ⃗ p ) ⟩⟨ N ( ⃗ p ) | J ν (0) | N ⟩ 2 E ⃗ p
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