Experimental and theoretical aspects of nucleon form factors

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1 Experimental and theoretical aspects of nucleon form factors Rinaldo Baldini Ferroli and Simone Pacetti Scattering and annihilation electromagnetic processes 8-22 February 23 ECT* - Trento

2 Outline Last News on Baryon FF near threshold The Neutral Baryon Puzzle Spacelike - Timelike Relationship Conclusions and Perspectives 2

3 Cross sections and analyticity Space-like region eb eb FF s are real Im[q 2 ] Time-like region Unphysical region Data region No data e + e BB FF s are complex Time-like: had. helicity = s th = 4Mπ 2 { GE G M s phy = 4M 2 B Re[q 2 ] G E (4M 2 B ) = G M (4M 2 B ) e B B e θ Elastic scattering dσ dω = α2 E e cos 2 θ 2 4Ee 3 sin 4 θ 2 ( [G 2E τ +2( τ ) tan 2 θ ) ] GM 2 2 τ τ = q2 4MB 2 B e e + θ B Annihilation Coulomb correction dσ dω = α2 βc [(+cos 2 4q 2 θ) G M 2 + τ ] β = sin2 θ G E 2 τ 3

4 The Coulomb Factor γ B pp Coulomb interaction as FSI [Sommerfeld, Sakharov, Schwinger, Fadin, Khoze] B Distorted wave approximation C = Ψ Coul () 2 πα S-wave: C = β exp D-wave: C = ( πα β πα ) β β Coulomb factor C No Coulomb factor for boson pairs (P-wave) q 2 (GeV/c) 4

5 Sommerfeld Enhancement and Resummation Factors Coulomb Factor C for S-wave only: Partial wave FF: Cross section: G S = 2G M q 2 /4M 2 + G E 3 G D = G M q 2 /4M 2 G E 3 σ(q 2 ) = 2πα 2 β 4M2 (q 2 ) 2 [ C G S (q 2 ) G D (q 2 ) 2] C = E R Enhancement factor: E = πα/β Step at threshold: σ(4m 2 ) = π2 α 3 β 2M 2 β G S(4M 2 ) 2 =.85 G S (4M 2 ) 2 nb Resummation factor: R = /[ exp( πα/β)] Few MeV above threshold: C σ(q 2 ) β G S (q 2 ) 2 5

6 The e + e τ + τ case σ ττ (nb).75 KEDR BES With Coulomb corr. Without Coulomb corr. With only enhancement factor W ττ (GeV) 6

7 BABAR : G p eff with and without resummation [PRD73, 25] G p eff.5 G p eff, no resum..5 G p eff with resum W pp (GeV) 7

8 Pointlike Baryons? R. Baldini Ferroli, S. Pacetti, A. Zallo and A. Zichichi 8

9 I.S.R. versus c.m. Advantages All q at the same time = Better control on systematics c.m. boost = at threshold efficiency + σ W MeV Detected ISR γ = full pp angular coverage Drawbacks L invariant mass bin w More background 9

10 Mass resolution Mass resolution (MeV/c 2 ) M pp (GeV/c 2 ) Incredibly good at threshold ( MeV/c 2 ), as e + e c.m. p T /p T.5% at GeV

11 BABAR : e + e pp [PRD73, 25] σ pp (nb) W pp (GeV)

12 BABAR 23: e + e pp PRD73-25, arxiv:32.55 σ pp (nb).5 L = 232 fb, W pp = 23, 25 MeV L = 469 fb, W pp = 23, 25 MeV L = 469 fb, W pp = 3.5, 5 MeV first bin W pp (GeV) 2

13 e + e pp: efficiency arxiv:32.55 Detection efficiency BABAR W pp (GeV) 3

14 Proton form factor at q 2 = 4M 2 p σ(e + e pp)(4m 2 p ) =.83 ±.5 nb σ(e + e pp)(4mp) 2 = π2 α 3 β 2Mp 2 β Gp (4Mp) 2 2 =.85 G p (4Mp) 2 2 nb G p (4M 2 p ) G p (4M 2 p) =.99 ±.4(stat) ±.3(syst) 4

15 Proton form factor at q 2 = 4M 2 p G p (4M 2 p ) At q 2 = 4MP 2 protons behave as pointlike fermions! 4

16 Sommerfeld Resummation Factor Needed? 5

17 Resummation Factor Needed? At threshold: G E /G M = σ(q 2 ), G E /G M G S, G D G S = exp( πα/β) No need of Resummation Factor { GS R G D = R For a wide energy range ( 2 MeV): Proton behaves as a pointlike particle e.m. dominance, no strong interaction? Mild sensitivity to BB invariant mass resolution 6

18 BABAR : G p E /Gp M and σ(e+ e pp) [PRD73, 25] G E p /GM p σ pp (nb) W pp (GeV) 7

19 BABAR : G p E /Gp M and σ(e+ e pp) [PRD73, 25] G S p, GD p exp( πα/β) G S p G D p W pp (GeV) 8

20 BABAR : G S, G D G p eff [PRD73, 25] G S, G D.5 G p eff.5 G p eff, no resum. G S.5 G D W pp (GeV).5 G p eff with resum W pp (GeV) 9

21 Integrated Sommerfeld factor W W [ exp( πα/β)]dw W (MeV) 2

22 Other charged baryon FF s at threshold 2

23 e + e Λ + c Λ c and e+ e pn(44)+c.c. [Belle PRL, 72] [BABAR PRD73, 25] σ ΛcΛc (nb).6 σ pn* (nb) W ΛcΛc (GeV) W pn* (GeV) 22

24 e + e pn(44)+c.c. BABAR PRD73, 25 σ Coulomb = 6π2 α 3 M 3/2 p M 3/2 N(44) (M p + M N(44) ) 5 G pn(44) 2 = G pn(44) 2.49 nb Events/. GeV σ pn(44) (nb).6.4 Coulomb W pπ (GeV) q 2 (GeV/c) G pn(44) =.4 ±.9 23

25 The neutral baryons puzzle 24 February 8th, 23

26 Neutral Baryons puzzle (BABAR) [PRD76, 926] σ(e + e B B )= 4πα2 βc 3q 2 [ G B M 2 + 2M2 B q 2 G B E 2 No Coulomb correction at hadron level: C = ] πα 2 β q 2 2M 2M 2 G B 2 B B 3 σ(e + e ΛΛ) (pb) BABAR.4 σ(e + e Σ Σ ) (pb).8 σ(e + e ΛΣ ) (pb) 2 DM2 σ th = 2±5 pb.3.2. σ th = 3±3 pb σ th = 47±23 pb q 2 (GeV/c) q 2 (GeV/c) q 2 (GeV/c) Like a remnant of Coulomb interactions C β at quark level? as q 2 2M B For any neutral baryon σb B GB M B 25

27 Baryon octet and U-spin arxiv: n Y p Σ Σ Σ + - Λ Ξ Ξ - I 3 (Y, I 3 ) (Y U, U 3 ) U 3 = 2 I Y Y U = Q Ξ Ξ Y U Σ Λ = 3Λ+Σ 2 Σ + - Σ = 3Σ Λ 2 p n U 3 U-spin relation: G Σ G Λ G ΛΣ = M Σ σσ M Σ Λ σλλ + 2 M ΛΣ σλσ = (.6 ± 6.)

28 Baryon octet and U-spin arxiv: n Y p Σ Σ Σ + - Λ Ξ Ξ - I 3 (Y, I 3 ) (Y U, U 3 ) U 3 = 2 I Y Y U = Q Ξ Ξ Y U Σ Λ = 3Λ+Σ 2 Σ + - Σ = 3Σ Λ 2 p n U 3 U-spin relation: G Σ G Λ G ΛΣ = M Σ σσ M Σ Λ σλλ + 2 M ΛΣ σλσ = (.6 ± 6.)

29 BABAR : e + e ΛΛ [PRD76, 926] σ ΛΛ (nb).3 BESIII collected data at threshold and above W ΛΛ (GeV) 27

30 Time-like G n M measurements G n M (q2 ).8.6 No Coulomb correction FENICE DM2 DM2 extr. from G Λ G p M (q2 ) Q d /Q u G n M /Gp M Data.5 Naively Q d /Q u.4 pqcd < q 2 (GeV) Soliton models VMD (Dubnicka) Only SND, CMD2(?) and BESIII can measure this cross section No other experiments at present and in near future will be able to perform such a measurement 28

31 e + e nn: preliminary result from SND σ nn (nb) e + e nn SND (2) FENICE (988) Scan 2 Maximum energy: 2 GeV Efficiency 3% Above nn threshold: σ nn =.8 ±.2 nb SND preliminary q 2 (GeV) 29

32 e + e nn 3

33 e + e nn (FENICE) 3

34 Dispersive analysis of the ratio R = µ p G p E G p M Eur. Phys. J. A32, 42 R. Baldini, S. Pacetti and A. Zallo space-like unphysical region time-like Re(q 2 ) 32

35 Space-like G p E /Gp M measurements µpg p E (q2 )/G p M (q2 ) Space-like G p E = F p + q2 F p 4Mp 2 2 G p M = F p + F p 2 F / q 2 4Mp 2 F 2 cancellation G p E (q2 ) G p M (q2 ) < PRD5 549 q 2 (GeV 2 /c 2 ) Time-like F / q 2 4Mp 2 F 2 enhancement G p E (q2 ) G p M (q2 ) > Radiative corrections of polarization technique < Radiative corrections in Rosenbluth method 33

36 Space-like G p E /Gp M measurements µpg p E (q2 )/G p M (q2 ) Space-like G p E = F p + q2 F p 4Mp 2 2 G p M = F p + F p 2 F / q 2 4Mp 2 F 2 cancellation G p E (q2 ) G p M (q2 ) < PRL PRL PRD5 549 q 2 (GeV 2 /c 2 ) Time-like F / q 2 4Mp 2 F 2 enhancement G p E (q2 ) G p M (q2 ) > Radiative corrections of polarization technique < Radiative corrections in Rosenbluth method 33

37 Time-like G p E /Gp M measurements [ ] dσ d cos θ = πα2 βc 2q 2 G p M 2 (+cos 2 θ)+ 4M2 p q 2 µ 2 sin 2 θ R 2 p R(q 2 ) = µ p G p E (q2 ) G p M (q2 ) R(q 2 ) /µp BABAR (ISR) PRD73, 25 LEAR (pp e + e ) NPB4, 3 FENICE+DM2 E835 EPJC46, 42 Scaling γγ exchange e e γ C = + γ γγ exchange interferes with the Born term p p q 2 (GeV/c) Asymmetry in angular distributions [PLB659, 97] 34

38 γγ exchange from e + e ppγ BABAR data E. Tomasi-Gustafsson, E. A. Kuraev, S. Bakmaev, SP PLB659, 97 dσ A(cos θ, q 2 ) = dω (cos θ, q2 ) dσ dω ( cos θ, q2 ) dσ dω (cos θ, q2 ) + dσ dω ( cos θ, q2 ). A cos θ -. A cos θ,q 2 =. ± q2 (GeV/c) 35

39 γγ exchange from e + e ppγ BABAR 23 data PLB659-97, arxiv:32.55 (dσ/dcos θ p ) meas /(dσ/dcos θ p ) fit...9 2M p W 3 GeV Integrated over the energy from threshold up to 3 GeV The MC-fit assumes one-photon exchange Slope =.4±.26± A cos θp = cos θ p Integral asymmetry σ(cos θp > ) σ(cos θp < ) =.25 ±.4 ±.3 σ(cos θ p > ) + σ(cos θ p < ) σ(cos θ p ) is the cross section integrated with W 3 GeV and cos θ p 36

40 R(q 2 ) in the complex plane G E, G M and also R, if G M has no zeros, are analytic on the q 2 plane with a cut (s th = 4M 2 π, ) [see e. g.: Eur. Phys. J. C, 79 (999)] R(q 2 ) experimental sheet Im(q 2 ) s th s phy Re(q 2 ) physical sheet unphysical sheet 37

41 R(q 2 ) in the complex plane R(q 2 ) experimental sheet Dispersion relation for the imaginary part (q 2 s th ) G(q 2 ) = G(z)dz lim R 2πi C z q 2 = π s th ImG(s)ds s q 2 Im(q 2 ) s th s phy path C R physical sheet Re(q 2 ) unphysical sheet 37

42 R(q 2 ) in the complex plane R(q 2 ) Dispersion relation for R with subtraction at q 2 = experimental sheet R(q 2 ) = R() + q2 π s th ImR(s)ds s(s q 2 ) Im(q 2 ) s th s phy path C R physical sheet Re(q 2 ) unphysical sheet 37

43 R(q 2 ) EPJA32 42 R(q2 ) = R() + q2 π 4M 2 π ImR(s) s(s q 2 ) ds R(q 2 ) space-like R(q 2 ) time-like Req 2 JLab+MIT-Bates BABAR +DM2/FENICE+E q 2 (GeV 2 /c 2 ) q 2 (GeV 2 /c 2 ) 38

44 R(q 2 ) EPJA32 42 R(q2 ) = R() + q2 π 4M 2 π ImR(s) s(s q 2 ) ds R(q 2 ) space-like R(q 2 ) time-like Req 2 JLab+MIT-Bates BABAR +DM2/FENICE+E q 2 (GeV 2 /c 2 ) q 2 (GeV 2 /c 2 ) 38

45 R(q 2 ) EPJA32 42 R(q2 ) = R() + q2 π 4M 2 π ImR(s) s(s q 2 ) ds R(q 2 ) space-like R(q 2 ) time-like Req 2 JLab+MIT-Bates BABAR +DM2/FENICE+E835.5 DR Approach /Q log 2 Q 2 /Q 2 Impr. log 2 Q 2 /Q 2 IJL q 2 (GeV 2 /c 2 ) q 2 (GeV 2 /c 2 ) 38

46 R(q 2 ) EPJA32 42 R(q2 ) = R() + q2 π 4M 2 π ImR(s) s(s q 2 ) ds R(q 2 ) space-like R(q 2 ) time-like Req 2 JLab+MIT-Bates BABAR +DM2/FENICE+E835 JLab preliminary V. Punjabi DSPIN-9 Dubna, Russia.5 DR Approach /Q log 2 Q 2 /Q 2 Impr. log 2 Q 2 /Q 2 IJL q 2 (GeV 2 /c 2 ) q 2 (GeV 2 /c 2 ) 38

47 Asymptotic G P E (q2 )/G p M (q2 ) and phase G P E (q2 )/G p M (q2 ) 2 Space Time Phase of G P E (q2 )/G p M (q2 ) 2 Phragmèn Lindelöf phase limit zeros q 2 (GeV 2 /c 2 ) q 2 (GeV/c) pqcd prediction G p E (q2 ) G p M (q2 ) q 2 Phase from DR φ(q 2 ) = q 2 s π ln R(s) ds Pr s s s (s q 2 ) 39

48 Time-like magnetic proton form factor.6 (ppγ)babar PRD73,25(6) (pp)adone NC4A,(73) (e + e )SPKR PLB64,475(76) (pp)dm PLB86,395(79) - (ppγ)babar PRD73,25(6) (pp)bes PLB63,4(5) (e + e )CALO PRL7,22(93) (e + e )CALO PRD6,322(99) G p eff (q2 ).4.2 (pp)dm2 NPB224,379(83) (pp)dm2 ZPC48,23(9) (pp)fenice PLB334,43(94) (e + e )WIRE NPB4,3(94) (pp)bes PLB63,4(5) -2 (e + e )CALO PLB559,2(3) q 2 (GeV 2 ) 5 2 q 2 (GeV 2 ) Data obtained assuming G p M = Gp E Gp eff (true only at threshold) G p eff 2 = 6πα 2 C e 3 σ pp (q 2 ) ( ) /τ + 4q 2 2τ 4

49 The integral equation for G M EPJC 79 Dispersion relation subtracted at t = ln G(t) = t s th t ln G(s) ds π s s s th (s t) s th Less dependent on the asymptotic behavior of the FF ln G() = no further terms have to be considered Splitting the integral s th into s phy s th Data and Theory {}}{ ln G(t) Iphy (t) = t s th t π + s phy we obtain the integral equation s phy s th Unknown {}}{ ln G(s) ds s s s th (s t) To avoid instabilities around s phy = 4MN 2, the upper boundary has been shifted to s phy = s phy +, with.5 GeV 2 We impose continuity of the FF at s phy and s th, in addition, at the upper boundary s phy, continuity of the first derivative is also required A regularization, depending on a free parameter τ, is introduced by requiring the FF total curvature in the unphysical region to be limited 4

50 Solving procedure and test EPJC 79 Solving procedure Fπ(q 2 ) Minimize: χ 2 = χ 2 data + χ2 theory + τ 6 χ 2 regu s [ χ 2 regu = phy d 2 ] ln G(s) 2 ds 2 ds s th [ total curvature ] in [s th, s phy ] Pion FF to fix the regularization parameter τ Space-like (DR) and time-like data (yellow bands) have been used as input in the integral equation to retrieve the time-like FF in the nucleon unphysical region. q 2 (GeV 2 ) 42

51 Solving procedure and test EPJC 79 Solving procedure Fπ(q 2 ) τ m π Minimize: χ 2 = χ 2 data + χ2 theory + τ 6 χ 2 regu s [ χ 2 regu = phy d 2 ] ln G(s) 2 ds 2 ds s th [ total curvature ] in [s th, s phy ] Pion FF to fix the regularization parameter τ Space-like (DR) and time-like data (yellow bands) have been used as input in the integral equation to retrieve the time-like FF in the nucleon unphysical region (gray band). q 2 (GeV 2 ) 42

52 Nucleon magnetic form factors EPJC 79 G p M (q2 )/µp Steep behavior near by the threshold G n M (q2 )/µn input data outcome q 2 (GeV 2 ) q 2 (GeV 2 ) M 77 MeV M 2 6 MeV Γ 35 MeV Γ 2 35 MeV 43

53 Conclusions Pointlike Behavior at and well above threshold No Sommerfeld Resummation Factor Neutral baryon non zero cross section at threshold? G p E space-like asymptotically? Perspectives BESIII: ISR and scan Data from SND and CMD2 up to 2 GeV PANDA could explore FFs below threshold through pp π l + l SuperTauCharm? 44