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 ′ � � 1 ( Q 2 ) + iσ µν ( p ′ − p ) ν � p ′ � � � � V q u ( p ′ ) γ µ F q F q � p 2 ( Q 2 ) = ¯ u ( p ) , µ 2 m p p where V q µ = ¯ qγ µ q . Q 2 = −( p ′ − p ) 2 Electric and magnetic form factors: Q 2 G q E ( Q 2 ) = F q ( 2 m p ) 2 F q G q M ( Q 2 ) = F q 1 ( Q 2 ) + F q 1 ( Q 2 ) − 2 ( Q 2 ) , 2 ( Q 2 ) . For a photon, weight quarks with their charges: G γ 3 G u 3 G d 3 G s E , M ≡ 2 E , M − 1 E , M − 1 E , M + . . . Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 3

Electromagnetic form factors: nonrelativistic p ′ = −� In the Breit frame ( � p = � q / 2 ): G E ( Q 2 ) ∼ ρ em , G M ( Q 2 ) ∼ � J em . Non-relativistically, this motivates the definition of a charge density: ∫ d 3 � q ( 2 π ) 3 e i � q ·� r 2 ) = r G E (� q 2 ) . ρ NR (� Integrating the density yields ∫ ∫ d 3 � r 2 ) = G E ( 0 ) , d 3 � r 2 ρ NR (� r 2 ) = − 6 G ′ r ρ NR (� r � E ( 0 ) . At Q 2 = 0 , we get the charge and magnetic moment of the proton, and the slopes define the mean-squared electric and magnetic radii: G E ( Q 2 ) = 1 − 1 6 ( r 2 E ) p Q 2 + O ( Q 4 ) G M ( Q 2 ) = µ p � 1 − 1 M ) p Q 2 + O ( Q 4 ) � 6 ( r 2 µ N 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. ct 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 ∫ d 2 � q q ·� ρ (� ( 2 π ) 2 e i � b 2 ) = b F 1 (� q 2 ) . z 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 E ′ Electromagnetic form factors are studied using θ E elastic scatering of an electron off a fixed proton target. The differential scatering cross section behaves like τ = Q 2 d Ω ∝ G E ( Q 2 ) 2 + τ dσ ϵ − 1 = 1 + 2 ( 2 + τ ) tan 2 θ ϵ G M ( Q 2 ) 2 , , 2 , 4 m 2 p so that G E and G M 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 Q 2 regions. Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 6

Proton radius How to measure r E : ◮ From scatering experiments: measure G E ( Q 2 ) , then do curve fiting to find slope at Q 2 = 0 . ◮ From atomic spectroscopy: the 2S–2P Lamb shif is ∆ E finite size ∝ r 2 E m 3 sensitive to r E . e Muonic hydrogen spectrum is much more sensitive to r E . This experiment led to proton radius puzzle. electron-proton scattering Hydrogen spectroscopy CODATA average muonic Hydrogen 0 . 84 0 . 85 0 . 86 0 . 87 0 . 88 0 . 89 r E (fm) Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 7

Proton radius: ongoing experiments electron-proton scattering Hydrogen spectroscopy 2S–4P (Garching 2017) 1S–3S (Paris 2018) CODATA average muonic Hydrogen 0 . 83 0 . 84 0 . 85 0 . 86 0 . 87 0 . 88 0 . 89 r E (fm) ◮ New ep scatering experiments at low Q 2 ◮ Mainz ISR: r E = 0 . 870 ( 28 ) fm M. Mihovilovič et al. , 1905.11182 ◮ JLab PRad: r E ∼ 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: G E / G M at high Q 2 1 . 6 1 . 4 1 . 2 Polarization transfer , proton µ G E / G M 1 . 0 ep → e � � p , gives a direct 0 . 8 measurement of G E / G M . 0 . 6 Result disagreed with 0 . 4 Polarization transfer Rosenbluth separation. 0 . 2 Rosenbluth separation 0 . 0 0 2 4 6 8 10 Q 2 (GeV 2 ) Can be explained by contributions from two-photon exchange. Explanation was tested via σ ( e + p ) σ ( e − p ) , but not at high Q 2 . Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 9

Flavour separation Elastic ep scatering gives one flavour combination: G p E , M = 2 3 G u E , M − 1 3 G d E , M − 1 3 G s 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 2 H or 3 He targets. 2. Contribution from Z exchange. Obtained from parity-violating asymmetry in elastic � ep scatering. Neutron-electron scatering length yields neutron charge radius: b ne = α 3 m n r 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. µ p = 2 . 792 847 344 62 ( 82 ) µ N (0.3 ppb) PDG; G. Schneider et al. , Science 358 , 1081–1084 (2017) µ n = − 1 . 913 042 73 ( 45 ) µ N (0.2 ppm) CODATA; G. L. Greene et al. , Phys. Rev. D 20 , 2139 (1979) e Nuclear magneton: µ N = 2 m p Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 11

Magnetic moment Good benchmark observable: experimental situation is solid. µ p = 2 . 792 847 344 62 ( 82 ) µ N (0.3 ppb) PDG; G. Schneider et al. , Science 358 , 1081–1084 (2017) µ n = − 1 . 913 042 73 ( 45 ) µ N (0.2 ppm) CODATA; G. L. Greene et al. , Phys. Rev. D 20 , 2139 (1979) µ s ≈ − 0 . 02 µ N (LQCD) e Nuclear magneton: µ N = 2 m p Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 11

Magnetic radius No independent measurements of r M : only elastic scatering data. Proton: reanalysis using z expansion G. Lee et al. , Phys. Rev. D 92 , 013013 (2015) [1505.01489] r Mp = 0 . 776 ( 34 )( 17 ) Mainz data r Mp = 0 . 914 ( 35 ) world data excluding Mainz Neutron: PDG 2019 cites two analyses r Mn = 0 . 89 ( 3 ) z expansion Z. Epstein et al. , Phys. Rev. D 90 , 074027 (2014) [1407.5687] r Mn = 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) � � m π � � 1 1 ) p − n = − ( r 2 1 + 7 д 2 A + ( 2 + 10 д 2 + 12 B r A ) log 10 ( Λ ) ( 4 πF π ) 2 Λ − д 2 A m π m N κ p − n = κ p − n 0 4 πF 2 π д 2 A m N κ p − n ( r 2 2 ) p − n = 8 πF 2 π m π Isovector r 2 1 and r 2 2 diverge as m π → 0 . Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 13

Hadron correlation functions Compute two-point and three-point functions, using interpolator χ and operator insertion O . In simplest case: C 2pt ( t ) ≡ � χ ( t ) χ † ( 0 )� � | Z n | 2 e − E n t = 0 t n → | Z 0 | 2 e − E 0 t � � 1 + O ( e − ∆ Et ) , where Z n = � Ω | χ | n � , C 3pt ( τ , T ) ≡ � χ ( T )O( τ ) χ † ( 0 )� 0 τ T � Z n ′ Z ∗ n � n ′ |O| n � e − E n τ e − E n ′ ( T − τ ) = n , n ′ → | Z 0 | 2 � 0 |O| 0 � e − E 0 T � � 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 ) ≡ C 3pt ( τ , T ) = � 0 |O| 0 � + O ( e − ∆ Eτ ) + O ( e − ∆ E ( T − τ ) ) C 2pt ( T ) Midpoint yields R ( T 2 , T ) = � 0 |O| 0 � + O ( e − ∆ ET / 2 ) . Summation method � d dT S ( T ) = � 0 |O| 0 � + O ( Te − ∆ ET ) S ( T ) ≡ R ( τ , T ) , τ Sum can be over all timeslices or from τ 0 to T − τ 0 . 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 � e − E n t � � � 2 C 2pt ( t ) = � χ ( t ) χ † ( 0 )� = � � n | χ † | 0 � n 10 − 11 10 − 12 10 − 13 C 2pt 10 − 14 10 − 15 10 − 16 10 − 17 0 2 4 6 8 10 12 14 16 18 20 t / a For a nucleon, the signal-to-noise asymptotically decays as e −( m N − 3 2 m π ) t . Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 16

Two-point function: excited states � e − E n t � � � 2 C 2pt ( t ) = � χ ( t ) χ † ( 0 )� = � � n | χ † | 0 � n 2 . 0 1 . 8 1 . 6 C 2pt / ae − m N t 1 . 4 1 . 2 1 . 0 0 . 8 0 2 4 6 8 10 12 14 16 18 20 t / a For a nucleon, the signal-to-noise asymptotically decays as e −( m N − 3 2 m π ) t . Jeremy Green | DESY, Zeuthen | July 23, 2019 | Page 16