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
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