Two-photon exchange calculations versus data l 0 l p 0 p - - PowerPoint PPT Presentation

▶
two photon exchange calculations versus data
SMART_READER_LITE
LIVE PREVIEW

Two-photon exchange calculations versus data l 0 l p 0 p - - PowerPoint PPT Presentation

Oct 19, 2022 627 likes •1.13k views

28 September, 2017 Two-photon exchange calculations versus data l 0 l p 0 p Oleksandr Tomalak Johannes Gutenberg University, Mainz, Germany 1 Scattering experiments and 2 - 2 is not among standard radiative corrections exp

slide-1
SLIDE 1

Oleksandr Tomalak

Johannes Gutenberg University, Mainz, Germany

Two-photon exchange calculations

1

versus data

28 September, 2017

γ γ l

p

p0

l0

slide-2
SLIDE 2

2

Scattering experiments and 2Ɣ

  • 2Ɣ is not among standard radiative corrections
  • J. C. Bernauer et al. (2014)
  • charge radius insensitive to 2Ɣ model
  • magnetic radius depends on 2Ɣ model

σexp ≡ σ1γ(1 + δrad + δsoft + δ2γ)

slide-3
SLIDE 3

3

magnetic form factor

  • 2 % systematic deviation

MAMI vs. world data

Scattering experiments and 2Ɣ

  • 2Ɣ is not among standard radiative corrections
  • J. C. Bernauer et al. (2014)
  • charge radius insensitive to 2Ɣ model
  • magnetic radius depends on 2Ɣ model

σexp ≡ σ1γ(1 + δrad + δsoft + δ2γ)

slide-4
SLIDE 4

4

µH hyperfine splitting and 2Ɣ

  • HFS in terms of forward lepton-proton scattering amplitudes
  • R. Pohl et al. (2016)

1S HFS in H

1 ppm accuracy

µ

PSI, J-PARC, RIKEN-RAL

  • leading theoretical uncertainty: 213 ppm from 2Ɣ, 109 ppm from 2Ɣ
  • C. Carlson, V. Nazaryan, K. Griffioen (2011)
  • Cl. Peset and A. Pineda (2017)

Zemach term recoil correction polarizability GE, GM

GE, GM F2, g1, g2

  • O. Tomalak (2017)
  • traditional decomposition:
  • A1@MAMI fit allows to quantify 2Ɣ uncertainty
  • J. C. Bernauer et al. (2014)

∆HFS = ∆Z + ∆R + ∆pol

  • proton radii, form factors and spin structure are important
slide-5
SLIDE 5

Zemach contribution

5

  • dependence on splitting: consistency check
  • 3 times more precise: 140 ppm → 49 ppm
  • Zemach correction expanding form factors

dependence on Q2 ΔZ, Q0 = 0

rE from μH

103 ×ΔZ −7.5 −7.4 −7.3 Q2

0, GeV2

0.01 0.02 0.03 0.04 0.05 0.06 dependence on Q2 ΔZ, Q0 = 0

rE from ep scattering

103 ×ΔZ −7.5 −7.4 −7.3 Q2

0, GeV2

0.01 0.02 0.03 0.04 0.05 0.06

  • O. Tomalak (2017)
  • 95 ppm change for μH and ep radii with Q0 = 0.2 GeV

✓

  • magnetic radius is important

∆Z = 8αmr π

∞

Z

Q0

dQ Q2 GM

  • Q2

GE

  • Q2

µP − 1 ! + 4αmrQ0 3π ✓ −r2

E − r2 M + r2 Er2 M

18 Q2 ◆

slide-6
SLIDE 6

6

  • dispersive evaluation and phenomenological extractions agree
  • compare with precise 1S HFS from eH

achieved accuracy in 70th: 10-12

Hyperfine splitting and 2Ɣ

RE from ep, eH RE from μH Carlson et al. Δpol, Faustov et al. ΔZ+ΔR, Bodwin et al. 1S HFS in eH eH ΔHFS, ppm −34 −33 −32 −31

slide-7
SLIDE 7

7

  • dispersive evaluation and phenomenological extractions agree
  • compare with precise 1S HFS from eH

achieved accuracy in 70th: 10-12

Hyperfine splitting and 2Ɣ

RE from ep, eH RE from μH Carlson et al. Δpol, Faustov et al. ΔZ+ΔR, Bodwin et al. 1S HFS in eH eH ΔHFS, ppm −34 −33 −32 −31

slide-8
SLIDE 8

8

  • dispersive evaluation and phenomenological extractions agree

Hyperfine splitting and 2Ɣ

  • compare with precise 1S HFS from eH

achieved accuracy in 70th: 10-12

  • exploit eH HFS measurements

scaled by a reduced mass mr

∆HFS (mµ) − mr(mµ) mr(me) ∆HFS (me)

∆(µH) = mr(mµ) mr(me) ∆ (eH)+

with 1S HFS in eH RE from ep, eH RE from μH Hagelstein et al. Peset et al. 2S HFS in μH, CREMA Carlson et al. Martynenko et al. Pachucki μH 103 ΔHFS −7.0 −6.5 −6.0 RE from ep, eH RE from μH Carlson et al. Δpol, Faustov et al. ΔZ+ΔR, Bodwin et al. 1S HFS in eH eH ΔHFS, ppm −34 −33 −32 −31

  • uncertainty: 100 ppm → 16 ppm !!!
slide-9
SLIDE 9

9

  • dispersive evaluation and phenomenological extractions agree

Hyperfine splitting and 2Ɣ

  • compare with precise 1S HFS from eH

achieved accuracy in 70th: 10-12

  • exploit eH HFS measurements

scaled by a reduced mass mr

∆HFS (mµ) − mr(mµ) mr(me) ∆HFS (me)

∆(µH) = mr(mµ) mr(me) ∆ (eH)+

with 1S HFS in eH RE from ep, eH RE from μH Hagelstein et al. Peset et al. 2S HFS in μH, CREMA Carlson et al. Martynenko et al. Pachucki μH 103 ΔHFS −7.0 −6.5 −6.0 RE from ep, eH RE from μH Carlson et al. Δpol, Faustov et al. ΔZ+ΔR, Bodwin et al. 1S HFS in eH eH ΔHFS, ppm −34 −33 −32 −31

  • uncertainty: 100 ppm → 16 ppm !!!
slide-10
SLIDE 10

Q2 = −(k − k0)2

photon polarization momentum transfer parameter forward scattering

ε → 1

ε

10

crossing-symmetric variable

Elastic lepton-proton scattering

  • 2Ɣ correction to cross section is given by amplitudes real parts

ν = (k, p + p0) 2

p

l(k)

l(k0)

p0

γ

γ γ

l

p p

l

2

  • leading 2Ɣ contribution: interference term

δ2γ = 2 P

spin

T 1γ<T 2γ P

spin

|T 1γ|2

l0

l0

p0

p0

slide-11
SLIDE 11

11

Elastic lepton-proton scattering

  • M. Gorchtein, P.A.M. Guichon and M. Vanderhaeghen (2004)

P.A.M. Guichon and M. Vanderhaeghen (2003)

p

l(k)

l(k0)

p0

K = k + k0 2 P = p + p0 2

  • electron-proton scattering: 3 structure amplitudes
  • muon-proton scattering: add helicity-flip amplitudes
  • 2Ɣ correction to cross section is given by amplitudes real parts

T non−flip = e2 Q2 ¯ lγµl · ¯ N GM(ν, Q2)γµ − F2(ν, Q2)P µ M + F3(ν, Q2) ˆ KP µ M 2 ! N

T flip = e2 Q2 m M ¯ ll · ¯ N F4(ν, Q2) + F5(ν, Q2) ˆ K M ! N + e2 Q2 m M F6(ν, Q2)¯ lγ5l · ¯ Nγ5N

slide-12
SLIDE 12

γ γ

at low momentum transfer

p

l

p0

l0

12

box diagram

assumption about the vertex

non-forward scattering

photoproduction vertex or Compton tensor

slide-13
SLIDE 13

γ γ

at low momentum transfer

p

l

p0

l0

13

dispersion relations box diagram

assumption about the vertex based on on-shell information

photoproduction vertex or Compton tensor

non-forward scattering

slide-14
SLIDE 14

non-forward scattering

γ γ

proton state

p

l

p0

l0

14

dispersion relations box diagram

assumption about the vertex based on on-shell information

Borisyuk and Kobushkin (2008), O. T. and M. Vanderhaeghen (2014) Blunden, Melnitchouk and Tjon (2003)

Dirac and Pauli form factors

slide-15
SLIDE 15

near-forward scattering account for all inelastic 2Ɣ

γ γ

l

p

γ

γ

l

p

p0

l0

15

p

l

forward scattering

slide-16
SLIDE 16

Low-Q2 inelastic 2Ɣ correction (e-p)

  • 2Ɣ at large agrees with empirical fit

unpolarized proton structure

  • 2Ɣ blob: near-forward virtual Compton scattering
  • O. T. and M. Vanderhaeghen (2016)

γ

γ p

l

rE extraction ✓

  • M. E. Christy, P. E. Bosted (2010)

l0 p0

Q2 = 0.05 GeV2

Q2 = 0.25 GeV2

16

ε

δ2γ ∼ a p Q2 + b Q2 ln Q2 + c Q2 ln2 Q2

Feshbach elastic inelastic

  • R. W. Brown (1970), M. Gorchtein (2013), O. T. and M. Vanderhaeghen (2014)

A1@MAMI Feshbach box diagram model total 2γ

δ2γ, %

0.5 1.0 1.5 ε 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 A1@MAMI Feshbach box diagram model total 2γ

δ2γ, %

0.5 1.0 1.5 ε 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

δ2γ = Z dνγdQ2(w1(νγ, Q2) · F1(νγ, Q2) + w2(νγ, Q2) · F2(νγ, Q2))

slide-17
SLIDE 17

17

  • μp elastic scattering is planned by MUSE@PSI(2018-19)
  • 2Ɣ correction in MUSE ?

μp scattering ???? ep scattering μH, μD spectroscopy eH, eD spectroscopy

  • charge radius extractions:

measure with both electron/muon charges

Scattering experiments and 2Ɣ

slide-18
SLIDE 18

MUSE@PSI (2018-19) estimates ( -p)

µ

18

  • O. T. and M. Vanderhaeghen (2014, 2016)

δ2γ, %

box diagram model, μ-p total, μ-p total, e-p 0.5 1.0 Q2, GeV2 0.010 0.020 0.005 0.015 0.025

δ2γ, %

0.5 1.0 Q2, GeV2 0.02 0.04 0.06 0.08

k = 115 MeV k = 210 MeV

  • proton box diagram model + inelastic 2Ɣ
slide-19
SLIDE 19
  • expected muon over electron ratio
  • K. Mesick talk (PAVI 2014), MUSE TDR (2016)
  • proton box diagram model + inelastic 2Ɣ
  • O. T. and M. Vanderhaeghen (2014, 2016)
  • MUSE can test rE in

small inelastic 2Ɣ small 2Ɣ uncertainty

  • ne charge channel

19

δ2γ, %

box diagram model, μ-p total, μ-p total, e-p 0.5 1.0 Q2, GeV2 0.010 0.020 0.005 0.015 0.025

δ2γ, %

0.5 1.0 Q2, GeV2 0.02 0.04 0.06 0.08

k = 115 MeV k = 210 MeV

MUSE@PSI (2018-19) estimates ( -p)

µ

slide-20
SLIDE 20

20

dispersion relations near-forward scattering p + all inelastic

γ

γ

l

p

γ

γ

l

p

X

X = p + πN

p0

l0

p0

l0

(large )

ε

(arbitrary )

ε

slide-21
SLIDE 21

2Ɣ real parts 2Ɣ imaginary parts

  • disp. rel.

cross section correction experimental data unitarity

2Ɣ prediction

Fixed-Q2 dispersion relation framework

21

<F(ν) = 2ν π P Z 1

νmin

=F(ν0 + i0) ν02 ν2 dν0

  • n-shell 1Ɣ amplitudes
slide-22
SLIDE 22

−0.2 0.2 0.4 ν, GeV2 0.2 0.4 0.6 0.8 1.0 Q2, GeV2

s = M 2 s = (M + mπ)2

22

  • proton intermediate state is outside physical region for Q2 > 0
  • πN intermediate state is outside physical region for Q2 > 0.064 GeV2

Mandelstam plot (ep)

elastic threshold inelastic threshold physical region unitarity relations work in

  • O. T. and M. Vanderhaeghen (2014)
slide-23
SLIDE 23

23

  • contour deformation method:

Analytical continuation. Elastic state

  • central value: form factor fit of A1@MAMI (2014)
  • uncertainty: difference to 2Ɣ with dipole form factors
  • O. T. and M. Vanderhaeghen (2014), Blunden and Melnitchouk (2017)

angular integration to integration on curve in complex plane keeping poles inside going to unph. region deform integration contour

  • analytical continuation

reproduces results

unphysical physical

in unphysical region

e−µ− box

=GM νph

0.01 ν, GeV2 −0.02 0.02 0.04 0.06 0.08

Q2 = 0.1 GeV2

✓

Z dΩ

slide-24
SLIDE 24

24

  • pion electroproduction amplitudes: MAID2007

2=

γ γ

Z X

γ

γ

l l p

p

N

π

l0

p0

  • analytical continuation: fit of low-Q2 expansion in physical region
  • D. Drechsel. S. Kamalov and L. Tiator (2007)

Analytical continuation. πN states

△ resonance

✓

  • uncertainty: extrapolation + large invariant masses

s = 1.6 GeV2

unphysical physical

=G2γ

2 , %

box diagram model 4-parameter fit 6-parameter fit weighted Δ Q2

ph

0.05 −0.20 −0.05 −0.10 −0.15 Q2, GeV2 0.5 1.0

=G2γ

2 , %

4-parameter fit 6-parameter fit πN Q2

ph

0.05 −0.20 −0.05 −0.10 −0.15 Q2, GeV2 0.5 1.0

πN

G1,2(s, Q2), Q2F3(s, Q2) ∼ a1Q2 ln Q2 + a2Q2 + a3Q4 ln Q2 + ...

l0

p0

slide-25
SLIDE 25

−0.2 0.2 0.4 ν, GeV2 0.2 0.4 0.6 0.8 1.0 Q2, GeV2

s = (M + mπ)2

25

  • πN intermediate state is outside physical region for Q2 > 0.064 GeV2

Mandelstam plot (ep)

inelastic threshold physical region unitarity relations work in

slide-26
SLIDE 26

−0.2 0.2 0.4 ν, GeV2 0.2 0.4 0.6 0.8 1.0 Q2, GeV2

s = 1.6 GeV2 s = (M + mπ)2

26

  • πN intermediate state is outside physical region for Q2 > 0.064 GeV2

Mandelstam plot (ep)

inelastic threshold physical region unitarity relations work in

slide-27
SLIDE 27

−0.2 0.2 0.4 ν, GeV2 0.2 0.4 0.6 0.8 1.0 Q2, GeV2

s = 1.6 GeV2 s = (M + mπ)2

27

  • πN intermediate state is outside physical region for Q2 > 0.064 GeV2

Mandelstam plot (ep)

inelastic threshold physical region unitarity relations work in

slide-28
SLIDE 28

28

  • pion electroproduction amplitudes: MAID2007

2=

γ γ

Z X

γ

γ

l l p

p

N

π

l0

p0

l0

p0

  • analytical continuation: fit of low-Q2 expansion in physical region
  • D. Drechsel. S. Kamalov and L. Tiator (2007)

Analytical continuation. πN states

△ resonance

✓

  • uncertainty: extrapolation + large invariant masses

s = 1.6 GeV2

unphysical physical

=G2γ

2 , %

box diagram model 4-parameter fit 6-parameter fit weighted Δ Q2

ph

0.05 −0.20 −0.05 −0.10 −0.15 Q2, GeV2 0.5 1.0

=G2γ

2 , %

4-parameter fit 6-parameter fit πN Q2

ph

0.05 −0.20 −0.05 −0.10 −0.15 Q2, GeV2 0.5 1.0

πN

G1,2(s, Q2), Q2F3(s, Q2) ∼ a1Q2 ln Q2 + a2Q2 + a3Q4 ln Q2 + ...

slide-29
SLIDE 29

29

πN in dispersive framework (e-p)

at large agree with near-forward

ε

  • dispersion relations

Q2 = 0.005 GeV2

δ2γ, %

πN, unsubtracted dispersion relations πN, near-forward from structure functions −0.01 0.01 0.02 ε 0.5 1.0

slide-30
SLIDE 30

30

  • πN is dominant inelastic 2Ɣ

γ

γ

p

l

N

π

  • O. T., B. Pasquini and M. Vanderhaeghen (2017)

δ2γ, %

A1@MAMI elastic elastic + πN total 2γ, near-forward 0.5 1.0 1.5 ε 0.2 0.4 0.6 0.8 1.0

Q2 = 0.05 GeV2

πN in dispersive framework (e-p)

at large agree with near-forward

ε

  • dispersion relations

Q2 = 0.005 GeV2

δ2γ, %

πN, unsubtracted dispersion relations πN, near-forward from structure functions −0.01 0.01 0.02 ε 0.5 1.0

p0

l0

slide-31
SLIDE 31

31

  • πN contribution is closer to data than △ only
  • weighted △ is similar to narrow one of Blunden et al. (2017)

Comparison with data

R2γ = σ(e+p) σ(e−p) ≈ 1 − 2δ2γ

k = 2.01 GeV

OLYMPUS (2016) elastic elastic + Δ elastic + πN

R2γ

Maximon and Tjon IR prescription

  • uncorr. + corr. uncertainties

0.98 0.99 1.00 1.01 1.02 ε 0.8 0.9 1.0

  • O. T., B. Pasquini and M. Vanderhaeghen (2017)
slide-32
SLIDE 32

32

  • near-forward 2Ɣ agree with data
  • multi-particle 2Ɣ, e.g. ππN, is important

OLYMPUS (2016) Feshbach elastic elastic + πN total 2 γ, near-forward

R2γ

Maximon and Tjon IR prescription

  • uncorr. + corr. uncertainties

0.98 0.99 1.00 1.01 1.02 ε 0.8 0.9 1.0

k = 2.01 GeV

Comparison with data

  • O. T., B. Pasquini and M. Vanderhaeghen (2017)

R2γ = σ(e+p) σ(e−p) ≈ 1 − 2δ2γ

slide-33
SLIDE 33

33

  • dispersion relations agree with CLAS data

elastic elastic + πN total 2 γ, near-forward

Comparison with data

VEPP-3 (2015) CLAS (2016) δ2γ, % 0.5 −0.5 1.0 −1.0 Q2, GeV2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

  • O. T., B. Pasquini and M. Vanderhaeghen (2017)
slide-34
SLIDE 34

34

Conclusions

largest theoretical uncertainty in low-energy proton structure

why 2Ɣ ?

slide-35
SLIDE 35

35

how to study ?

dispersion relations box diagram all scatt. angles

ep (p + πN states)

largest theoretical uncertainty in low-energy proton structure small scatt. angles

ep, µp (all states)

why 2Ɣ ?

  • multi-particle 2Ɣ, e.g. ππN, within dispersion relations is important

Conclusions

slide-36
SLIDE 36

36

Our best 2Ɣ knowledge

Q2 = 0.1 GeV2

  • small Q2: near-forward at large , all inelastic states
  • Q2≲1 GeV2: elastic+πN within dispersion relations

dispersion near-forward relations

  • intermediate range: interpolation

ε

A1@MAMI elastic + πN total 2 γ, near-forward interpolation

δ2γ, %

0.5 1.0 1.5 2.0 ε 0.2 0.4 0.6 0.8 1.0

slide-37
SLIDE 37

37

application to 1S HFS exp JLAB data MUSE data dispersive 2Ɣ evaluation

µp

Outlook

g1, g2 theoretical 2Ɣ magnetic radius extraction

ep

slide-38
SLIDE 38

Thanks for your attention !!!

38

slide-39
SLIDE 39

39

theory vs exp. 3.6 σ electron vs muon up to 7σ anomalous magnetic moment Lamb shift

µH

µH

Pohl et al., Nature (2010)

decrease hadronic uncertainties study the low energy proton structure

Our goals

hadronic uncertainty is dominant in theory discrepancy proton size

γ

µ

µ

Muon discrepancies: new physics?

39

slide-40
SLIDE 40

40

photon-proton vertex

Tool to explore the proton structure

l-p amplitude

Q2 = −(k − k0)2 momentum transfer lepton energy

Dirac and Pauli form factors

γ

u(k, h)

¯ u(k0, h0)

N(p, λ)

¯ N(p0, λ0)

Γµ(Q2) = γµFD(Q2) + iσµνqν 2M FP (Q2) T = e2 Q2 (¯ u (k0, h0) γµu (k, h)) · ¯ N (p0, λ0) Γµ(Q2)N (p, λ)

  • ω
slide-41
SLIDE 41

41

Form factors measurement

Qattan et al. (2005)

G2

M(Q2)

ε τ G2

E(Q2)

kinematical variables

τ

GE = FD − τFP GM = FD + FP

dσunpol dΩ ∼ G2

M(Q2) + ε

τ G2

E(Q2)

  • Rosenbluth separation:

dσunpol dΩ

ε

  • Sachs electric and magnetic form factors:
  • Rosenbluth slope is sensitive to corrections beyond 1Ɣ
slide-42
SLIDE 42

42

  • polarization transfer method:

PT PL ∼ GE(Q2) GM(Q2)

e + p -> e + p realized in 2000 at JLab e e p p

~ n

T

L

PT ∼ GE(Q2)GM(Q2) PL ∼ G2

M(Q2)

  • Sachs electric and magnetic form factors:

Form factors measurement

GE = FD − τFP GM = FD + FP

slide-43
SLIDE 43

43

  • V. Punjabi et al. (2015)

Gayou Jones, Punjabi Puckett Meziane

Polarization transfer

JLab (Hall A, C)

Rosenbluth separation

SLAC, JLab (Hall A, C)

Proton form factors puzzle

vs.

slide-44
SLIDE 44

44

  • V. Punjabi et al. (2015)

a possible explanation

Gayou Jones, Punjabi Puckett Meziane

vs. Polarization transfer

JLab (Hall A, C)

Rosenbluth separation

SLAC, JLab (Hall A, C)

γ

γ

l

two-photon exchange

p0

l0 2Ɣ measurements

e+p/e-p cross section ratio

p

R2γ = σ(e+p) σ(e−p) ≈ 1 − 2δ2γ

  • discrepancy motivates model-independent study of 2Ɣ

Proton form factors puzzle

slide-45
SLIDE 45

45

Proton charge radius

electric charge radius

0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 0.05 0.1 0.15 0.2 GE/Gstd.dipole (a)

Q2, GeV2

  • J. C. Bernauer et al. (2014)

< r2

E >≡ −6 dGE(Q2)

dQ2

  • Q2=0

rE = 0.879 ± 0.008 fm

ep elastic scattering

slide-46
SLIDE 46

46

Proton charge radius

rE = 0.8758 ± 0.0077 fm

rE = 0.8409 ± 0.0004 fm

CODATA 2010

rE = 0.879 ± 0.008 fm

CREMA (2010, 2013)

electric charge radius ep elastic scattering μH Lamb shift H, D spectroscopy

∆EnS ∼ m3

r < r2 E > 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 0.05 0.1 0.15 0.2 GE/Gstd.dipole (a)

Q2, GeV2

atomic spectroscopy

  • J. C. Bernauer et al. (2014)

< r2

E >≡ −6 dGE(Q2)

dQ2

  • Q2=0
slide-47
SLIDE 47

47

rE = 0.8758 ± 0.0077 fm

rE = 0.8409 ± 0.0004 fm

CODATA 2010

rE = 0.879 ± 0.008 fm

CREMA (2010, 2013)

electric charge radius ep elastic scattering μH Lamb shift H, D spectroscopy

Proton radius puzzle

4 % difference

∆EnS ∼ m3

r < r2 E > 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 0.05 0.1 0.15 0.2 GE/Gstd.dipole (a)

Q2, GeV2

atomic spectroscopy

  • J. C. Bernauer et al. (2014)

< r2

E >≡ −6 dGE(Q2)

dQ2

  • Q2=0
slide-48
SLIDE 48

48

2Ɣ hadronic correction

  • C. Carlson, M. Vanderhaeghen (2011) + M. Birse, J. McGovern (2012)

∆E2γ

2P−2S = 33 ± 2 µeV

charge radius discrepancy 310 µeV μH experimental uncertainty 2.5 µeV 2P-2S transition in μH

PSI, µH Antognini et al. H&D spectr CODATA 2010 MAMI scatt Bernauer et al JLAB scatt Zhan et al <r2

E>, fm

0.84 0.86 0.88 0.90 0.92

µH Lamb shift and 2Ɣ

important to reduce ambiguities of 2Ɣ

slide-49
SLIDE 49

DR

  • ptical theorem

Dispersion relation framework

analyticity

f(z)

<F(ν) = 2ν π P Z 1

νmin

=F(ν0 + i0) ν02 ν2 dν0

experimental cross sections energy levels correction amplitudes: imaginary parts amplitudes: real parts

49