Electromagnetic form factors and the proton radius Jeremy Green - - PowerPoint PPT Presentation

Electromagnetic form factors and the proton radius

Jeremy Green

NIC, DESY, Zeuthen

Advances in Latice Gauge Theory 2019 CERN, July 23, 2019

Outline

• 1. Nucleon form factors
• 2. Latice QCD: standard methods
• 3. Direct methods for radii and magnetic moment
• 4. Outlook

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 2

Electromagnetic (vector) form factors

Elastic ep scatering cross section depends on the strength of the coupling of a proton to a current.

• p′

V q

µ

• p
• = ¯

u(p′)

• γµFq

1 (Q2) + iσµν(p′ − p)ν

2mp Fq

2 (Q2)

• u(p),

where V q

µ = ¯

qγµq.

p′ p

Q2 = −(p′ − p)2

Electric and magnetic form factors: Gq

E(Q2) = Fq 1 (Q2) − Q2 (2mp)2 Fq 2 (Q2),

Gq

M(Q2) = Fq 1 (Q2) + Fq 2 (Q2).

For a photon, weight quarks with their charges: Gγ

E,M ≡ 2 3Gu E,M − 1 3Gd E,M − 1 3Gs E,M + . . .

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 3

Electromagnetic form factors: nonrelativistic

In the Breit frame ( p′ = − p = q/2): GE(Q2) ∼ ρem, GM(Q2) ∼ Jem. Non-relativistically, this motivates the definition of a charge density: ρNR( r 2) = ∫ d3 q (2π)3ei

q· rGE(

q2). Integrating the density yields ∫ d3 r ρNR( r 2) = GE(0), ∫ d3 r r 2ρNR( r 2) = −6G′

E(0).

At Q2 = 0, we get the charge and magnetic moment of the proton, and the slopes define the mean-squared electric and magnetic radii: GE(Q2) = 1 − 1

6(r 2 E)pQ2 + O(Q4)

GM(Q2) = µp µN 1 − 1

6(r 2 M)pQ2 + O(Q4)

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 4

EM form factors: relativistic interpretation

“Earthrise” photo taken by Apollo 8 astronauts. ◮ Photo has 4 ms exposure time. ◮ Image of Earth taken ∼ 1 s before Moon. ◮ Farthest point of Earth viewed ∼ 20 ms before nearest point. We actually see things along the light cone rather than at fixed time. Apply same light-front approach to the proton. Then we get a 2d transverse charge density ρ( b2) = ∫ d2 q (2π)2ei

q· bF1(

q2).

z ct

• M. Diehl, Eur. Phys. J. C 25, 223 [hep-ph/0205208],
• M. Burkardt, Int. J. Mod. Phys. A 18, 173 [hep-ph/0207047]

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 5

Electron-proton scatering

Electromagnetic form factors are studied using elastic scatering of an electron off a fixed proton target.

E θ E′

The differential scatering cross section behaves like dσ dΩ ∝ GE(Q2)2 + τ ϵ GM(Q2)2, τ = Q2 4m2

p

, ϵ−1 = 1 + 2(2 + τ) tan2 θ

2,

so that GE and GM can be measured in experiments (Rosenbluth separation). ◮ First experiments in 1950s (R. Hofstadter). ◮ Recent years: experiments in Mainz and JLab. ◮ Ongoing work studying low and high Q2 regions.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 6

Proton radius

How to measure rE: ◮ From scatering experiments: measure GE(Q2), then do curve fiting to find slope at Q2 = 0. ◮ From atomic spectroscopy: the 2S–2P Lamb shif is sensitive to rE. ∆Efinite size ∝ r 2

Em3 e

Muonic hydrogen spectrum is much more sensitive to rE. This experiment led to proton radius puzzle.

0.84 0.85 0.86 0.87 0.88 0.89 rE (fm) muonic Hydrogen CODATA average Hydrogen spectroscopy electron-proton scattering

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 7

Proton radius: ongoing experiments

0.83 0.84 0.85 0.86 0.87 0.88 0.89 rE (fm) muonic Hydrogen CODATA average Hydrogen spectroscopy electron-proton scattering 2S–4P (Garching 2017) 1S–3S (Paris 2018)

◮ New ep scatering experiments at low Q2

◮ Mainz ISR: rE = 0.870(28) fm M. Mihovilovič et al., 1905.11182 ◮ JLab PRad: rE ∼ 0.830(20) fm (preliminary: CERN Courier) ◮ Tohoku: planned ULQ2 experiment using low beam energy (20–60 MeV).

◮ Muon-proton scatering

◮ MUSE: µ±p, e±p scatering at PSI ◮ COMPASS: proposed µ±p experiment

◮ New hydrogen spectroscopy experiments: Garching, Paris, Toronto Also note: analyses of scatering data based on dispersion relations yield small radius. M. A. Belushkin et al., Phys. Rev. C 75, 035202 (2007) [hep-ph/0608337]

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 8

An older puzzle: GE/GM at high Q2

2 4 6 8 10 Q2 (GeV2) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 proton µGE/GM Polarization transfer Rosenbluth separation

Polarization transfer,

• ep → e

p, gives a direct measurement of GE/GM. Result disagreed with Rosenbluth separation. Can be explained by contributions from two-photon exchange. Explanation was tested via σ(e+p)

σ(e−p), but not at high Q2.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 9

Flavour separation

Elastic ep scatering gives one flavour combination: Gp

E,M = 2 3Gu E,M − 1 3Gd E,M − 1 3Gs E,M − · · ·

Separating out u, d, and s contributions requires two more independent combinations

• 1. Neutron electromagnetic form factors (assuming isospin): swap role of

u and d. Obtained using 2H or 3He targets.

• 2. Contribution from Z exchange. Obtained from parity-violating

asymmetry in elastic ep scatering. Neutron-electron scatering length yields neutron charge radius: bne = α 3mnr 2

En.

PDG average: r 2

En = −0.1161(22).

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 10

Magnetic moment

Good benchmark observable: experimental situation is solid. µp = 2.792 847 344 62(82)µN (0.3 ppb) PDG; G. Schneider et al., Science 358, 1081–1084 (2017) µn = −1.913 042 73(45)µN (0.2 ppm) CODATA; G. L. Greene et al., Phys. Rev. D 20, 2139 (1979) Nuclear magneton: µN = e 2mp

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 11

Magnetic moment

Good benchmark observable: experimental situation is solid. µp = 2.792 847 344 62(82)µN (0.3 ppb) PDG; G. Schneider et al., Science 358, 1081–1084 (2017) µn = −1.913 042 73(45)µN (0.2 ppm) CODATA; G. L. Greene et al., Phys. Rev. D 20, 2139 (1979) µs ≈ −0.02µN (LQCD) Nuclear magneton: µN = e 2mp

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 11

Magnetic radius

No independent measurements of rM: only elastic scatering data. Proton: reanalysis using z expansion

• G. Lee et al., Phys. Rev. D 92, 013013 (2015) [1505.01489]

rMp = 0.776(34)(17) Mainz data rMp = 0.914(35) world data excluding Mainz Neutron: PDG 2019 cites two analyses rMn = 0.89(3) z expansion Z. Epstein et al., Phys. Rev. D 90, 074027 (2014) [1407.5687] rMn = 0.862+9

−8 disp. rel. M. A. Belushkin et al., Phys. Rev. C 75, 035202 (2007) [hep-ph/0608337]

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 12

Need for physical pion mass

Pion cloud makes large contribution to isovector radii. Heavy baryon ChPT: V. Bernard et al., Nucl. Phys. B 388, 315–345 (1992) (r 2

1)p−n = −

1 (4πFπ )2

• 1 + 7д2

A + (2 + 10д2 A) log

mπ Λ

• + 12Br

10(Λ)

• κp−n = κp−n

− д2

AmπmN

4πF 2

π

κp−n(r 2

2)p−n =

д2

AmN

8πF 2

πmπ

Isovector r 2

1 and r 2 2 diverge as mπ → 0.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 13

Hadron correlation functions

Compute two-point and three-point functions, using interpolator χ and

• perator insertion O. In simplest case:

t τ T

C2pt(t) ≡ χ(t)χ†(0) =

• n

|Zn|2e−Ent → |Z0|2e−E0t 1 + O(e−∆Et)

• ,

where Zn = Ω|χ|n, C3pt(τ,T) ≡ χ(T)O(τ)χ†(0) =

• n,n′

Zn′Z ∗

nn′|O|ne−Enτe−En′(T −τ )

→ |Z0|20|O|0e−E0T 1 + O(e−∆Eτ ) + O(e−∆E(T −τ ))

• Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 14

Hadron matrix elements

Ratio method R(τ,T) ≡ C3pt(τ,T) C2pt(T) = 0|O|0 + O(e−∆Eτ ) + O(e−∆E(T −τ )) Midpoint yields R(T

2 ,T) = 0|O|0 + O(e−∆ET /2).

Summation method S(T) ≡

• τ

R(τ,T), d dT S(T) = 0|O|0 + O(Te−∆ET ) Sum can be over all timeslices or from τ0 to T − τ0. Improved asymptotic behaviour noted in talks at Latice 2010.

• S. Capitani et al., PoS LATTICE2010 147 [1011.1358]; J. Bulava et al., ibid. 303 [1011.4393]

In practice noisier than ratio method at same T.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 15

Two-point function: excited states

C2pt(t) = χ(t)χ†(0) =

• n

e−Ent n|χ†|0

• 2

2 4 6 8 10 12 14 16 18 20 t/a 10−17 10−16 10−15 10−14 10−13 10−12 10−11 C2pt

For a nucleon, the signal-to-noise asymptotically decays as e−(mN − 3

2mπ )t. Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 16

Two-point function: excited states

C2pt(t) = χ(t)χ†(0) =

• n

e−Ent n|χ†|0

• 2

2 4 6 8 10 12 14 16 18 20 t/a 0.8 1.0 1.2 1.4 1.6 1.8 2.0 C2pt/ae−mNt

For a nucleon, the signal-to-noise asymptotically decays as e−(mN − 3

2mπ )t. Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 16

Three-point function: excited states

ratio: R(T,τ) = C3pt(T,τ) C2pt(T) → N |O|N τ T

χ χ J −7.5 −5.0 −2.5 0.0 2.5 5.0 7.5

(τ − T/2)/a

1.10 1.15 1.20 1.25 1.30

RA(τ, T) coarse

T/a = 3 4 5 6 7 8 10 12

• N. Hasan, JG, et al., Phys. Rev. D 99, 114505 (2019) [1903.06487]

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 17

Three-point function: excited states

ratio: R(T,τ) = C3pt(T,τ) C2pt(T) → N |O|N τ T

χ χ J −7.5 −5.0 −2.5 0.0 2.5 5.0 7.5

(τ − T/2)/a

1.10 1.15 1.20 1.25 1.30

RA(τ, T) coarse

T/a = 3 4 5 6 7 8 10 12

2 4 6 8

(T/2 or Tmin)/a

1.10 1.15 1.20 1.25 1.30 1.35

gbare

A

coarse

Plateau Summation

• N. Hasan, JG, et al., Phys. Rev. D 99, 114505 (2019) [1903.06487]

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 17

Off-forward matrix elements

C O

3pt(τ,T;

p, p′) = ∫ d3 xd3 y e−i

p· xei( p′− p)· y Tr

• Γpol χ(

x,T)O( y,τ) ¯ χ(0)

• → Z(

p)Z( p′) 4E( p)E( p′)e−E(

p)τe−E( p′)(T −τ ) s,s′

¯ u(p,s)Γpolu(p′,s′) p′,s′|O|p,s . Cancel overlaps and exponents using ratio: RO(τ,T; p, p′) = C O

3pt(τ,T;

p, p′)

• C2pt(T,

p)C2pt(T, p′)

• C2pt(

p,T − τ)C2pt( p′,τ) C2pt( p′,T − τ)C2pt( p,τ)

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 18

Extracting form factors

Matrix elements are linear combinations of form factors:

• p′,s′

Vµ

• p,s
• = ¯

u(p′,s′)

• γµF1(Q2) + iσµν(p′ − p)ν

2mp F2(Q2)

• u(p,s).

Gather matrix elements with the same Q2: ◮ different µ ◮ different p, p′ ◮ different polarization (i.e. s, s′). This gives a linear system of equations with two unknowns: F1(Q2) and F2(Q2).

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 19

Extracting form factors: rest frame

Consider states with p′ = 0 and spin in the +ˆ z direction. ℜ(RVi) ∼ ϵij3pjGM(Q2) ℑ(RVi) ∼ −piGE(Q2) ℜ(RV4) ∼ (mN + EN ( p))GE(Q2) Most common approach: take ℜ(RVi) and ℜ(RV4).

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 20

Latice results

Largest volume: L = 10.8 fm → Q2

min = 0.013 GeV2

• E. Shintani et al. (PACS), Phys. Rev. D 99, 014510 (2019) [1811.07292]

0.05 0.1 q2 [GeV2] 0.7 0.75 0.8 0.85 0.9 0.95 1 GE

v(q2) Dipole Kelly Isovector, tsep/a={14,16} Isovector, tsep/a={12,14,16} PACS’18, tsep/a=15

0.05 0.1 q2 [GeV2] 2 2.5 3 3.5 4 4.5 5 5.5 6 GM

v (q2) Dipole Kelly Isovector, tsep/a={14,16} Isovector, tsep/a={12,14,16} PACS’18, tsep/a=15

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 21

Latice results

Largest volume: L = 10.8 fm → Q2

min = 0.013 GeV2

• E. Shintani et al. (PACS), Phys. Rev. D 99, 014510 (2019) [1811.07292]

0.05 0.1 q2 [GeV2] 0.7 0.75 0.8 0.85 0.9 0.95 1 GE

v(q2) Dipole Kelly Isovector, tsep/a={14,16} Isovector, tsep/a={12,14,16} PACS’18, tsep/a=15

0.05 0.1 q2 [GeV2] 2 2.5 3 3.5 4 4.5 5 5.5 6 GM

v (q2) Dipole Kelly Isovector, tsep/a={14,16} Isovector, tsep/a={12,14,16} PACS’18, tsep/a=15

Minimum momentum transfer Q2

min ≈ (2π L )2.

Latice L (fm) Q2

min(GeV2)

LHPC 2010 2.7 0.203 LHPC 2014 5.6 0.048 PACS 2019 10.8 0.013

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 21

Latice results

Largest volume: L = 10.8 fm → Q2

min = 0.013 GeV2

• E. Shintani et al. (PACS), Phys. Rev. D 99, 014510 (2019) [1811.07292]

0.05 0.1 q2 [GeV2] 0.7 0.75 0.8 0.85 0.9 0.95 1 GE

v(q2) Dipole Kelly Isovector, tsep/a={14,16} Isovector, tsep/a={12,14,16} PACS’18, tsep/a=15

0.05 0.1 q2 [GeV2] 2 2.5 3 3.5 4 4.5 5 5.5 6 GM

v (q2) Dipole Kelly Isovector, tsep/a={14,16} Isovector, tsep/a={12,14,16} PACS’18, tsep/a=15

Minimum momentum transfer Q2

min ≈ (2π L )2.

Latice L (fm) Q2

min(GeV2)

LHPC 2010 2.7 0.203 LHPC 2014 5.6 0.048 PACS 2019 10.8 0.013 Experiment Q2

min(GeV2)

Mainz 2014 0.003 Mainz ISR 0.001 JLab PRad 0.0002

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 21

Background field methods

Alternative approach with different systematics. E.g. neutral hadron in uniform magnetic field: E = m − µ · B + O( B2).

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 22

Background field methods

Alternative approach with different systematics. E.g. neutral hadron in uniform magnetic field: E = m − µ · B + O( B2). Implement by modifying gauge links: Uµ(x) → eieqaAµ(x)Uµ(x). For constant B = Bˆ z, can choose Aµ(x) such that with periodic boundary conditions, eqB = 2π L2 n, n ∈ Z. Can study radii using spatially-varying E fields.

• Z. Davoudi and W. Detmold, Phys. Rev. D 92, 074506 (2015) [1507.01908]
• Z. Davoudi and W. Detmold, Phys. Rev. D 93, 014509 (2016) [1510.02444]

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 22

Direct calculation of µ (naïve atempt)

For spin +ˆ z, we have ℜ[RVx (p ˆ y, 0)] ∼ pGM(Q2). Therefore µ µN = GM(0) ∼ lim

p→0

1 p ℜ[RVx (p ˆ y, 0)] = ∂ ∂py ℜ[RVx ( p, 0)]

• p=0

= 1 C2pt(T, 0) ∂ ∂py ℜ

• CVx

3pt(τ,T;

p, 0)

• p=0

, where ∂ ∂py CVx

3pt(τ,T;

p, 0)

• p=0

= ∫ d3 x1d3 x2(−ix2y) Tr

• Γpol χ(

x1,T)Vx( x2,τ) ¯ χ(0)

• .

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 23

Failure of naïve approach

• W. Wilcox, “Continuum moment equations on the latice”, Phys. Rev. D 66, 017502 (2002)

[hep-lat/0204024]

Need to choose finite-volume expression for the moment in y: y → fL(y) =          y y < L/2 y = L/2 y − L y > L/2

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 24

Failure of naïve approach

• W. Wilcox, “Continuum moment equations on the latice”, Phys. Rev. D 66, 017502 (2002)

[hep-lat/0204024]

Need to choose finite-volume expression for the moment in y: y → fL(y) =          y y < L/2 y = L/2 y − L y > L/2 In finite volume, this can always be decomposed into Fourier modes: fL(y) =

• n∈Z

cne−ipny, pn = 2π L n =⇒ cn =

• n = 0

(−1)n 2i L πn

n 0

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 24

Failure of naïve approach

• W. Wilcox, “Continuum moment equations on the latice”, Phys. Rev. D 66, 017502 (2002)

[hep-lat/0204024]

Need to choose finite-volume expression for the moment in y: y → fL(y) =          y y < L/2 y = L/2 y − L y > L/2 In finite volume, this can always be decomposed into Fourier modes: fL(y) =

• n∈Z

cne−ipny, pn = 2π L n =⇒ cn =

• n = 0

(−1)n 2i L πn

n 0 At large τ, C3pt will be dominated by the lowest-lying initial states with energies EN ( p = pn ˆ y), i.e. with n = ±1.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 24

Failure of naïve approach

• W. Wilcox, “Continuum moment equations on the latice”, Phys. Rev. D 66, 017502 (2002)

[hep-lat/0204024]

Need to choose finite-volume expression for the moment in y: y → fL(y) =          y y < L/2 y = L/2 y − L y > L/2 In finite volume, this can always be decomposed into Fourier modes: fL(y) =

• n∈Z

cne−ipny, pn = 2π L n =⇒ cn =

• n = 0

(−1)n 2i L πn

n 0 At large τ, C3pt will be dominated by the lowest-lying initial states with energies EN ( p = pn ˆ y), i.e. with n = ±1. Two problems:

• 1. The denominator C2pt(T,

0) doesn’t cancel Z( p)e−EN (

p)τ .

• 2. We get matrix elements proportional to GM(Q2

min).

Nothing new over standard approach.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 24

Order of limits

Need to take L → ∞ before p → 0. Correct order of limits for standard calculation: µ µN ∝ lim

p→0 lim L→∞

lim

τ ,T −τ →∞

1 p ℜ

• RVx (τ,T;p ˆ

y, 0)

• .

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 25

Order of limits

Need to take L → ∞ before p → 0. Correct order of limits for standard calculation: µ µN ∝ lim

p→0 lim L→∞

lim

τ ,T −τ →∞

1 p ℜ

• RVx (τ,T;p ˆ

y, 0)

• .

Can postpone ground-state isolation: µ µN ∝ lim

τ ,T −τ →∞ lim p→0 lim L→∞

1 p ℜ

• RVx (τ,T;p ˆ

y, 0)

• .

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 25

Order of limits

Need to take L → ∞ before p → 0. Correct order of limits for standard calculation: µ µN ∝ lim

p→0 lim L→∞

lim

τ ,T −τ →∞

1 p ℜ

• RVx (τ,T;p ˆ

y, 0)

• .

Can postpone ground-state isolation: µ µN ∝ lim

τ ,T −τ →∞ lim p→0 lim L→∞

1 p ℜ

• RVx (τ,T;p ˆ

y, 0)

• .

We can take finite-volume moments of a correlator: µ µN ∝ lim

τ ,T −τ →∞ lim L→∞

∂ ∂py ℜ

• RVx (τ,T;

p, 0)

• p=0

. This approach not currently being taken, but discussed in

• C. Alexandrou et al. (ETMC), Phys. Rev. D 94, 074508 (2016) [1605.07327].

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 25

Twisted boundary conditions

For pµ = 2π

Lµ nµ, nµ ∈ Z, can absorb momentum into quark propagator:

e−ip·(x−y)D−1(x,y;U ) = D−1(x,y;U (p)), U (p)

µ (x) = eiapµUµ(x).

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 26

Twisted boundary conditions

For pµ = 2π

Lµ nµ, nµ ∈ Z, can absorb momentum into quark propagator:

e−ip·(x−y)D−1(x,y;U ) = D−1(x,y;U (p)), U (p)

µ (x) = eiapµUµ(x).

For non-Fourier modes pµ =

1 Lµ (2πnµ + θµ), using U (p) implies twisted

boundary conditions D−1(x + Lµ ˆ µ,y;U ) = eiθµD−1(x,y;U ).

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 26

Twisted boundary conditions

For pµ = 2π

Lµ nµ, nµ ∈ Z, can absorb momentum into quark propagator:

e−ip·(x−y)D−1(x,y;U ) = D−1(x,y;U (p)), U (p)

µ (x) = eiapµUµ(x).

For non-Fourier modes pµ =

1 Lµ (2πnµ + θµ), using U (p) implies twisted

boundary conditions D−1(x + Lµ ˆ µ,y;U ) = eiθµD−1(x,y;U ). Derivatives with respect to p can be evaluated exactly

• G. M. de Divitiis, R. Petronzio, N. Tantalo, Phys. Let. B 718, 589–596 (2012) [1208.5914]

∂D−1 ∂pµ = −D−1 ∂D ∂pµ D−1 For Wilson-type fermions, ∂D

∂pµ amounts to an insertion of the conserved

vector current.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 26

Twisted boundary conditions: form factors

Need to absorb momentum into D−1(x,y) with x y =⇒ can’t use for disconnected diagrams. Equivalently, need flavor-changing vector current ¯ qθ ′γµqθ with different twist angles θ ′ θ.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 27

Twisted boundary conditions: form factors

Need to absorb momentum into D−1(x,y) with x y =⇒ can’t use for disconnected diagrams. Equivalently, need flavor-changing vector current ¯ qθ ′γµqθ with different twist angles θ ′ θ. Use partially twisted boundary conditions, i.e. only on valence quarks connected to current. Nonunitarity vanishes as L → ∞.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 27

Twisted boundary conditions: form factors

Need to absorb momentum into D−1(x,y) with x y =⇒ can’t use for disconnected diagrams. Equivalently, need flavor-changing vector current ¯ qθ ′γµqθ with different twist angles θ ′ θ. Use partially twisted boundary conditions, i.e. only on valence quarks connected to current. Nonunitarity vanishes as L → ∞. Two applications:

• 1. Studying finite-volume effects. Twisted BC allows for the same

momentum on different volumes.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 27

Twisted boundary conditions: form factors

Need to absorb momentum into D−1(x,y) with x y =⇒ can’t use for disconnected diagrams. Equivalently, need flavor-changing vector current ¯ qθ ′γµqθ with different twist angles θ ′ θ. Use partially twisted boundary conditions, i.e. only on valence quarks connected to current. Nonunitarity vanishes as L → ∞. Two applications:

• 1. Studying finite-volume effects. Twisted BC allows for the same

momentum on different volumes.

• 2. Momentum derivatives at

p = 0. ∂ ∂pj → µ, ∂2 ∂p2

j

• r

∂2 ∂p′

j∂pj

→ r 2

E.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 27

Latice study: derivative method

Calculations at physical mπ using BMW 2HEX-clover action. First version used

∂ ∂pj and ∂2 ∂p2

j

→ extra propagators shared for all source-sink separations.

• N. Hasan, JG, et al., Phys. Rev. D 97, 034504 (2018) [1711.11385]

✵✳✵ ✵✳✷ ✵✳✹ ✵✳✻ ✵✳✽ ✶✳✵ ✵✳✵ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✹ ✵✳✺ ✵✳✻ ✵✳✼ ✵✳✽ ✵✳✾ F v

1 (Q2)

Q2 (GeV2) P❘❊▲■▼■◆❆❘❨ st❛♥❞❛r❞ ♠❡t❤♦❞ ③✲❡①♣❛♥s✐♦♥ ✜t ❙❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ✇✳r✳t p ▼✐①❡❞ ❞❡r✐✈❛t✐✈❡s ✇✳r✳t p, p′ ❑❡❧❧② ✵✳✺ ✶✳✵ ✶✳✺ ✷✳✵ ✷✳✺ ✸✳✵ ✸✳✺ ✹✳✵ ✵✳✵ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✹ ✵✳✺ ✵✳✻ ✵✳✼ ✵✳✽ ✵✳✾ F v

2 (Q2)

Q2 (GeV2) P❘❊▲■▼■◆❆❘❨ st❛♥❞❛r❞ ♠❡t❤♦❞ ③✲❡①♣❛♥s✐♦♥ ✜t ❋✐rst ❞❡r✐✈❛t✐✈❡ ✇✳r✳t p ❑❡❧❧②

Revised approach also uses

∂2 ∂p′

j ∂pj

→ reduced statistical errors for r 2

E.

• N. Hasan, talk at Latice 2017

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 28

Finite-volume effects: ChPT

F.-J. Jiang and B. C. Tiburzi, Flavor twisted boundary conditions and the nucleon vector current,

• Phys. Rev. D 78, 114505 (2008) [0810.1495]

◮ Heavy baryon partially quenched ChPT. ◮ Two sea quarks plus additional valence and ghost quarks with twisted BC. ◮ New unphysical baryon LEC д1: equals 2дd,conn

A

in chiral limit. E.g. for p = 0, p′ = p ˆ y: δL[Gu−d

M

(Q2)] = −2mN f 2p (д2

A + дAд1)K2(mπ,

p, 0) + 3mN f 2 ∫ 1 dx

• д2

AL33(mπPπ,

0, p,x p, 0) + 2 9д2

∆N L33(mπPπ,

0, p,x p, ∆)

• ,

where Pπ =

• 1 + x(1 − x)p2/m2

π and Ki, Lij are finite-volume functions.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 29

Finite-volume effects from ChPT: isovector GM

−0.14 −0.12 −0.10 −0.08 −0.06 −0.04 −0.02 0.00 ✵✳✵ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✹ ✵✳✺ δL[GM(Q2)] Q2 ✭●❡❱2✮ t✇✐st❡❞ ❇❈ ♣❡r✐♦❞✐❝ ❇❈

physical mπ , mπL = 4 Derivative method: δL[µ/µN ] → −mN

πf 2 (2д2 A + дAд1)mπe−mπ L.

Compare with δL[дA] ∼ m2

πe−mπ L/√mπL.

−0.45 −0.40 −0.35 −0.30 −0.25 −0.20 −0.15 −0.10 −0.05 0.00 ✸✳✵ ✸✳✺ ✹✳✵ ✹✳✺ ✺✳✵ ✺✳✺ ✻✳✵ δL[µ/µN] mπL ❢✉❧❧ ❡①♣r❡ss✐♦♥ ❛s②♠♣t♦t✐❝ Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 30

Finite-volume effects from ChPT: isovector GE

−0.001 0.000 0.001 0.002 0.003 0.004 0.005 ✵✳✵ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✹ ✵✳✺ δL[GE(Q2)] Q2 ✭●❡❱2✮ t✇✐st❡❞ ❇❈ ♣❡r✐♦❞✐❝ ❇❈

physical mπ , mπL = 4 Derivative method: δL[r 2

E] ∼ (mπL)3/2e−mπ L.

Slow approach to asymptote; beter approx.: (2/f 2)e−mπ L. Compare with δL[дA] ∼ m2

πe−mπ L/√mπL.

0.00 0.05 0.10 0.15 0.20 0.25 ✸✳✵ ✸✳✺ ✹✳✵ ✹✳✺ ✺✳✵ ✺✳✺ ✻✳✵ δL[r2

E] ✭❢♠2✮

mπL ❢✉❧❧ ❡①♣r❡ss✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥ Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 31

Latice study: finite volume

Two ensembles with mπ ≈ 250 MeV, a = 0.116 fm:

• 1. 323 × 48, mπL = 4.8
• 2. 243 × 48, mπL = 3.6, plus twisted BC to match momenta

Results are PRELIMINARY.

✵✳✵ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✹ ✵✳✺ ✭●❡❱ ✮ ❞✐✛❡r❡♥❝❡ ✵✳✵ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✹ ✵✳✺ ✭●❡❱ ✮ ❞✐✛❡r❡♥❝❡

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 32

Latice study: finite volume

Two ensembles with mπ ≈ 250 MeV, a = 0.116 fm:

• 1. 323 × 48, mπL = 4.8
• 2. 243 × 48, mπL = 3.6, plus twisted BC to match momenta

Results are PRELIMINARY.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 GE (isovector) Q2 (GeV2) periodic 323 twisted 243 periodic 243 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.2 0.4 0.6 0.8 1.0 GM (isovector) Q2 (GeV2) periodic 323 twisted 243 periodic 243 ✵✳✵ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✹ ✵✳✺ ✭●❡❱ ✮ ❞✐✛❡r❡♥❝❡ ✵✳✵ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✹ ✵✳✺ ✭●❡❱ ✮ ❞✐✛❡r❡♥❝❡

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 32

Latice study: finite volume

Two ensembles with mπ ≈ 250 MeV, a = 0.116 fm:

• 1. 323 × 48, mπL = 4.8
• 2. 243 × 48, mπL = 3.6, plus twisted BC to match momenta

Results are PRELIMINARY.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 GE (isovector) Q2 (GeV2) periodic 323 twisted 243 periodic 243 δ24 − δ32 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.2 0.4 0.6 0.8 1.0 GM (isovector) Q2 (GeV2) periodic 323 twisted 243 periodic 243 δ24 − δ32 −0.01 0.00 0.01 0.02 ✵✳✵ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✹ ✵✳✺ δL[GE(Q2)] Q2 ✭●❡❱2✮ 323 243 ❞✐✛❡r❡♥❝❡ −0.4 −0.3 −0.2 −0.1 0.0 ✵✳✵ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✹ ✵✳✺ δL[GM(Q2)] Q2 ✭●❡❱2✮ 323 243 ❞✐✛❡r❡♥❝❡

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 32

Latice study: finite volume derivative method

0.30 0.35 0.40 0.45 0.50 0.55

• 6
• 4
• 2

2 4 6 r2

E (fm2)

(τ − T/2)/a T/a = 6 7 8 10 12 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4

• 6
• 4
• 2

2 4 6 µ/µN (τ − T/2)/a T/a = 6 7 8 10 12

• pen symbols: 243

filled symbols: 323 ∼ −2% effect for r 2

E; ChPT: (r 2 E)24 − (r 2 E)32 = +0.076 fm2

∼ −5% effect for µ; ChPT: µ24 − µ32 = −0.28

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 33

Latice study: finite volume derivative method

0.30 0.35 0.40 0.45 0.50 0.55

• 6
• 4
• 2

2 4 6 r2

E (fm2)

(τ − T/2)/a T/a = 6 7 8 10 12 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4

• 6
• 4
• 2

2 4 6 µ/µN (τ − T/2)/a T/a = 6 7 8 10 12

• pen symbols: 243

filled symbols: 323 ∼ −2% effect for r 2

E; ChPT: (r 2 E)24 − (r 2 E)32 = +0.076 fm2

∼ −5% effect for µ; ChPT: µ24 − µ32 = −0.28 Excited-state effects are significant!

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 33

Disconnected diagrams?

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Q2 (GeV2) −6 −4 −2 2 4 6 8 10 GP

u+d connected u+d u+d disconnected 2s

mπ = 317 MeV.

JG et al., Phys. Rev. D 95, 114502 (2017) [1703.06703]

Extreme case: isoscalar induced pseudoscalar form factor. Connected diagrams have pole from partially quenched pion. Disconnected diagrams must cancel it.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 34

Disconnected diagrams?

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Q2 (GeV2) −6 −4 −2 2 4 6 8 10 GP

u+d connected u+d u+d disconnected 2s

mπ = 317 MeV.

JG et al., Phys. Rev. D 95, 114502 (2017) [1703.06703]

Extreme case: isoscalar induced pseudoscalar form factor. Connected diagrams have pole from partially quenched pion. Disconnected diagrams must cancel it. Likewise, disconnected diagrams must cancel log(mπ ) term in isoscalar r 2

E.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 34

Disconnected diagrams?

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Q2 (GeV2) −6 −4 −2 2 4 6 8 10 GP

u+d connected u+d u+d disconnected 2s

mπ = 317 MeV.

JG et al., Phys. Rev. D 95, 114502 (2017) [1703.06703]

Extreme case: isoscalar induced pseudoscalar form factor. Connected diagrams have pole from partially quenched pion. Disconnected diagrams must cancel it. Likewise, disconnected diagrams must cancel log(mπ ) term in isoscalar r 2

E.

In practice: contribution is less than 5%. e.g. χQCD, PRD 96, 114504 [1705.05849] Large fiting uncertainty is acceptable.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 34

Understanding excited-state effects

Two main models used:

• 1. “N ∗ model”: well-separated states. Produces OK fits but not very

predictive.

• 2. ChPT: nucleon-pion states. Predictive at leading order but not

consistent with data.

• Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 35

Understanding excited-state effects

Two main models used:

• 1. “N ∗ model”: well-separated states. Produces OK fits but not very

predictive.

• 2. ChPT: nucleon-pion states. Predictive at leading order but not

consistent with data.

˜ GP(Q2)

• exp. (muon capture)

• exp. (π electroprod.)


pion pole dominance model
 PACS data
 PACS data, Nπ removed
 ppd model, Nπ added
 pole ansatz (PACS)

Q2 [(GeV)2]

• O. Bär, Phys. Rev. D 99, 054506 (2019) [1812.09191]

Special case: ChPT provides good description for (induced) pseudoscalar form factor. Tree-level diagram dominates.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 35

Understanding excited-state effects

Two main models used:

• 1. “N ∗ model”: well-separated states. Produces OK fits but not very

predictive.

• 2. ChPT: nucleon-pion states. Predictive at leading order but not

consistent with data.

˜ GP(Q2)

• exp. (muon capture)

• exp. (π electroprod.)


pion pole dominance model
 PACS data
 PACS data, Nπ removed
 ppd model, Nπ added
 pole ansatz (PACS)

Q2 [(GeV)2]

• O. Bär, Phys. Rev. D 99, 054506 (2019) [1812.09191]

Special case: ChPT provides good description for (induced) pseudoscalar form factor. Tree-level diagram dominates. Speculation: Could resonance models provide good description of excited-state effects? Are Nρ-type states the dominant contribution for GE and GM?

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 35

Outlook

Derivative method makes it possible to avoid uncertainty in rE from fiting GE(Q2). ◮ Applies to connected diagrams: largest contribution to rE. ◮ Exponentially suppressed finite-volume effects.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 36

Outlook

Derivative method makes it possible to avoid uncertainty in rE from fiting GE(Q2). ◮ Applies to connected diagrams: largest contribution to rE. ◮ Exponentially suppressed finite-volume effects. Can compare observed volume dependence with ChPT. ◮ Magnetic moment: same sign, similar size. ◮ Charge radius: opposite sign, observed effect much smaller. ChPT: should aim for mπL 5 for useful results.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 36

Outlook

Derivative method makes it possible to avoid uncertainty in rE from fiting GE(Q2). ◮ Applies to connected diagrams: largest contribution to rE. ◮ Exponentially suppressed finite-volume effects. Can compare observed volume dependence with ChPT. ◮ Magnetic moment: same sign, similar size. ◮ Charge radius: opposite sign, observed effect much smaller. ChPT: should aim for mπL 5 for useful results. Excited-state effects remain a significant challenge.

Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 36