QED in muon g 2 , hadron spectroscopy, and beyond The RBC & - - PowerPoint PPT Presentation

QED∞ in muon g − 2, hadron spectroscopy, and beyond

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

The RBC & UKQCD collaborations

BNL and BNL/RBRC Ryan Abbot Norman Christ Duo Guo Christopher Kelly Bob Mawhinney Masaaki T

• mii

Jiqun Tu University of Connecticut Peter Boyle Luigi Del Debbio Felix Erben Vera Gülpers T adeusz Janowski Julia Kettle Michael Marshall Fionn Ó hÓgáin Antonin Portelli T

• bias T

sang Andrew Yong Azusa Yamaguchi Nicolas Garron Nils Asmussen Jonathan Flynn Ryan Hill Andreas Jüttner James Richings Chris Sachrajda Julien Frison Xu Feng Bigeng Wang Tianle Wang Yidi Zhao UC Boulder Yasumichi Aoki (KEK) T aku Izubuchi Yong-Chull Jang Chulwoo Jung Meifeng Lin Aaron Meyer Hiroshi Ohki Shigemi Ohta (KEK) Amarjit Soni Oliver Witzel Columbia University T

• m Blum

Dan Hoying (BNL) Luchang Jin (RBRC) Cheng Tu Edinburgh University University of Southampton Peking University University of Liverpool KEK Stony Brook University Jun-Sik Yoo Sergey Syritsyn (RBRC) MIT David Murphy Mattia Bruno CERN Christoph Lehner (BNL) University of Regensburg Masashi Hayakawa (Nagoya)

Outline 1 / 22

• 1. Introduction
• 2. QED∞ in Muon g − 2

Long-distance contribution to the HLbL in position space from the π0 pole

• 3. QED∞ in hadron spectroscopy (X. Feng and L. Jin 2018)

Pion mass splitting with the infjnite volume reconstruction method

• 4. Conclusion

Muon g − 2: experiments 2 / 22

⃗ µ = −g e 2m⃗ s World Average dominated by BNL aµ = (11659208.9 ± 6.3) × 10−10 In comparison, for electron ae = (11596521.8073 ± 0.0028) × 10−10

• Fermilab E989 (0.14 ppm) Almost 4 times more accurate then the previous experiment.
• J-PARC E34 also plans to measure muon g − 2 with similar precision.

Muon g − 2: theory 3 / 22

aµ × 1010 QED incl. 5-loops 11658471.9 ± 0.0 Aoyama, et al, 2012 Weak incl. 2-loops 15.4 ± 0.1 Gnendiger et al, 2013 HVP 692.5 ± 2.7 Talk: C. Lehner (Mon 14:20)

• C. Lehner et al (RBC-UKQCD), 2018

HVP NLO&NNLO −8.7 ± 0.1 FJ17 HLbL 10.3 ± 2.9 FJ17 Hadronic Models, “Consensus” Standard Model 11659181.3 ± 4.0 Experiment 11659208.9 ± 6.3 E821, The g − 2 Collab. 2006 Difgerence (Exp-SM) 27.6 ± 7.5

• More than 3 standard deviations due to mistake in the highly sophisticated perturbative

calculation, inaccuracy of e+e− → hadrons experiments, incorrect hadronic model, or new physics?

Hadronic contribution to Muon g − 2 4 / 22

q = p′ − p, ν p p′

HVP: Hadronic Vacuum Polarization

q = p′ − p, ν p p′

HLbL: Hadronic Light by Light

• Dispersive approach with e+e− → hadrons experimental data is very successful for HVP.
• HLbL is much more complicated. Anyway, lots of works have been done in this direction.

Pion pole, pion box, rescattering

• G. Colangelo, M. Hoferichter, B. Kubis,
• M. Procura, P. Stofger.

Light by light scattering sum rule

• I. Danilkin, O. Deineka, M. Vanderhaeghen.

Light by light scattering sum rule

• L. Dai, M. Pennington.

Schwinger’s sum rule

• F. Hagelstein, V. Pascalutsa.

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.

Muon g − 2: theory 6 / 22

aµ × 1010 QED incl. 5-loops 11658471.9 ± 0.0 Aoyama, et al, 2012 Weak incl. 2-loops 15.4 ± 0.1 Gnendiger et al, 2013 HVP 692.5 ± 2.7 Talk: C. Lehner (Mon 14:20)

• C. Lehner et al (RBC-UKQCD), 2018

HVP NLO&NNLO −8.7 ± 0.1 FJ17 HLbL 7.4 ± 6.6 Talk: T. Blum (Tue 16:30) Lattice QCD, RBC-UKQCD Standard Model 11659178.5 ± 7.1 Experiment 11659208.9 ± 6.3 E821, The g − 2 Collab. 2006 Difgerence (Exp-SM) 30.4 ± 9.5

• More than 3 standard deviations due to mistake in the highly sophisticated perturbative

calculation, inaccuracy of e+e− → hadrons experiments, under estimating sys/stat error in lattice calculation, or new physics?

Outline 7 / 22

• 1. Introduction
• 2. QED∞ in Muon g − 2

Long-distance contribution to the HLbL in position space from the π0 pole

• 3. QED∞ in hadron spectroscopy (X. Feng and L. Jin 2018)

Pion mass splitting with the infjnite volume reconstruction method

• 4. Conclusion

HLbL: QED∞ to eliminate power-law FV efgects 8 / 22

• We start the calculation by simulating both

the QCD part and the QED part on the lattice within a fjnite volume using the QEDL

• formalism. [T. Blum et al, 2014,2016,2017]
• Mainz’s group fjrst demonstrated that it is

possible to effjciently calculate the QED part semi-analytically in the infjnite volume. [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]

QCD Box QED Box z′ y′ x′ y x z xop

At physical point, π0 is very light, therefore:

• Large volume is needed.
• Large statistical error from the long

distance region, especially for the disconnected diagrams. [L. Jin et al, 2016] [J. Bijnens and J. Relefors, 2016]

Long-distance part of HLbL from the π0 pole 9 / 22

• Norman’s initial idea for

calculating the long-distance part of HLbL from π0 pole. (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 HLbL: further tweaks 10 / 22

x x′ y y′ π0 π0

• We can rotate the modest size

lattice so its time direction is aligned with the pion propagating direction.

• Only need to measure the

π0 → γγ three-point function with zero momentum pion.

• We only need the following

three-point function for ⃗ p = 0 pion. Hµ,ν(x − y) = ⟨0|Jµ(x)Jν(y)|π0⟩

Long-distance HLbL: numerical results (prelim) 11 / 22

• UCONN graduate student Cheng Tu did the calculation.

1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 aµ × 1010 Rmax (fm) 32ID

• 32ID: 323 × 64, a−1 = 1.015 GeV,

Mπ = 142 MeV.

• Rmax = max(|x−y|, |x−y ′|, |y−y ′|).

Reverse partial sum plotted.

x x′ y y′ π0 π0

The three-point function: π0 → γγ 12 / 22

Hµ,ν(x − y) = ⟨0|Jµ(x)Jν(y)|π0⟩

• We calculated it using one point source propagator

from y and wall source propagators seperated by enough distance from both x and y to create the pion.

• We averaged it within a confjguration, saved it to disk,

then performed furthur contractions.

y x π0

• π0 → γγ decay width from a

coordinate-space method. Poster: X. Feng (Tue 17:50).

• Yidi Zhao and Norman Christ’s

very noval calculation π0 → e+e−, which serves as the starting point for a more adventurous calculation KL → µ+µ−. Talk: N. Christ (Tue 17:10). Talk: Y. Zhao (Tue 17:30).

• Likely more applications.

Outline 13 / 22

• 1. Introduction
• 2. QED∞ in Muon g − 2

Long-distance contribution to the HLbL in position space from the π0 pole

• 3. QED∞ in hadron spectroscopy (X. Feng and L. Jin 2018)

Pion mass splitting with the infjnite volume reconstruction method

• 4. Conclusion

QED correction to hadron masses 14 / 22

∆M = I = 1 2 ∫ d4x Hµ,ν(x)Sγ

µ,ν(x)

Hµ,ν(x) = 1 2M ⟨N|TJµ(x)Jν(0)|N⟩ Sγ

µ,ν(x)

= δµ,ν 4π2x2

0, ν x, µ

• We can evaluate the QED part, the photon propagator, in infjnite volume.
• The hadronic function do not always fall exponentially in the long distance region.

∆M = I = I(s) + I(l) I(s) = 1 2 ∫ ts

−ts

dt ∫ d3x Hµ,ν(x)Sγ

µ,ν(x)

I(l) = ∫ ∞

ts

dt ∫ d3x Hµ,ν(x)Sγ

µ,ν(x)

• To eliminate all power-law suppressed fjnite volume efgects, a difgerent treatment for the

long distance part is required. (ts ≲ L)

The infjnite volume reconstruction method 15 / 22

• For the short distance part, I(s) can be directly calculated on a fjnite volume lattice:

I(s) ≈ I(s,L) = 1 2 ∫ ts

−ts

dt ∫ L/2

L/2

d3x Hµ,ν(x)Sγ

µ,ν(x)

• For the long distance part, we need to evaluate Hµ,ν(x) indirectly.

Note that when t is large, the intermediate states between the two currents are mostly ground states (possibly with small momentum). Therefore: Hµ,ν(x) ≈ ∫ d3p (2π)3 [ 1 2E⃗

p

1 2M ⟨N|Jµ(0)|N(⃗ p)⟩⟨N(⃗ p)|Jν(0)|N⟩ ] ei⃗

p·⃗ x−(E⃗

p−M)t

– We only need to calculate the form factors: ⟨N(⃗ p)|Jν(0)|N⟩! – Values for all ⃗ p are needed. Simply inverse Fouier transform the above relation! ∫ d3xHµ,ν(ts,⃗ x)e−i⃗

p·⃗ x+(E⃗

p−M)ts ≈

1 2E⃗

p

1 2M ⟨N|Jµ(0)|N(⃗ p)⟩⟨N(⃗ p)|Jν(0)|N⟩

Master formula for QED self-energy 16 / 22

• The fjnal expression for QED correction to hadron mass is split into two parts:

∆M = I ≈ I(s,L) + I(l,L)

• For the short distance part:
• For the long distance part:
• For Feynman gauge:

I(s) ≈ I(s,L) = 1 2 ∫ ts

−ts

dt ∫ L/2

L/2

d3x Hµ,ν(x)Sγ

µ,ν(x)

I(l) ≈ I(l,L) = ∫ L/2

−L/2

d3x HL

µ,ν(ts,⃗

x)Lµ,ν(ts,⃗ x) Sγ

µ,ν(x)

= δµ,ν 4π2x2 Lµ,ν(ts,⃗ x) = δµ,ν 2π2 ∫ ∞ dp sin(p|⃗ x|) 2(p + Ep − M)|⃗ x|e−pts

• Only exponentially suppressed fjnite volume errors.

Power-law suppressed fjnite volume errors are removed to all orders.

Pion mass splitting: Mπ+ − Mπ0 17 / 22

γ π0 π0

Type 1

γ π0 π0

Type 2

• Diagrams are very similar to the π− → π+ee 0ν2β decay calculation.

(D. Murphy and W. Detmold, 2018.) Poster: X. Tuo (Tue 17:50).

• All UV divergence and other disconnected diagrams are cancelled. (RM123 2013)
• RM123 2013 (type 2 only): M2

π+ − M2 π0 = 1.44(13)stat(16)chiral × 103 MeV2

• A. Risch and H. Wittig, 2018.
• Talk: J. Richings (Wed 11:50).

Pion mass splitting: Mπ+ − Mπ0 (prelim) 18 / 22

Type 1 and Type 2

3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 1 1.5 2 2.5 3 3.5 4 4.5 5 Mπ+ − Mπ0(MeV) ts (fm) 24ID short 24ID 32ID short 32ID

Type 1 only

0.01 0.02 0.03 0.04 0.05 0.06 1 2 3 4 5 Mπ+ − Mπ0(MeV) ts (fm)

• Mπ = 142 MeV.
• a−1 = 1.015 GeV,
• 24ID: 243 × 64,
• 32ID: 323 × 64,

Pion mass splitting: Mπ+ − Mπ0 (prelim) 19 / 22

∆M2

π(a, Mπ) = 2Mπ∆Mπ = ∆M2 π(0, Mphys π+ ) + c1a2 + c2(M2 π − (Mphys π+ )2)

900 1000 1100 1200 1300 1400 1500 1600 0.2 0.4 0.6 0.8 1 1.2 ∆M 2

π(MeV2)

a2(GeV−2) 2M phys

π+ ∆M phys π

2M phys

π0

∆M phys

π

Extrapolation 48I 139 MeV 24ID 142 MeV 32ID 142 MeV 32IDF 144 MeV 24ID 340 MeV

∆M2

π(0, Mphys π+ )

= 1.275(15) × 103 MeV ∆Mπ(0, Mphys

π+ )

= 4.57(6) MeV

• Both type 1 and type 2 diagrams included.
• 10 times more accurate than previous.

QED correction for meson leptonic decay 20 / 22

The calculation of QED correction to QCD

• bservables, and the appropriate treatment of

the problems arising due to the fjnite size of the simulated lattice, is a hot topic in the fjeld; nonetheless the main aspects debated at the moment concern actually the infrared singular- ities appearing in the matrix elements due to the presence of a fjnite box, which is not the subject of the present paper. – Anonymous Talk: X. Feng (Wed 11:10).

• Infrared divergence shall cancel analytically as it always does in infjnite volume perturbative

calculations.

Outline 21 / 22

• 1. Introduction
• 2. QED∞ in Muon g − 2

Long-distance contribution to the HLbL in position space from the π0 pole

• 3. QED∞ in hadron spectroscopy (X. Feng and L. Jin 2018)

Pion mass splitting with the infjnite volume reconstruction method

• 4. Conclusion

Conclusion 22 / 22

• In many cases, it is very benefjcial to view the problem from a position space perspective.
• For a class of problems, we can split the contribution into two difgerent regions and adopt

difgerent treatment. – Muon g − 2: * Window method and GEVP for HVP C. Lehner et al, 2018. * Long-distance π0 pole contribution for HLbL. – Hadron Spectroscopy: * The infjnite volume reconstruction method. X. Feng and L. Jin 2018. Applied in calculating Mπ+ − Mπ0. Work in progress for leptonic decay.

• More applications?

ThankYou!