Overview of two-photon and two-boson exchange

Transcription

1 Overview of two-photon and two-boson exchange Peter Blunden University of Manitoba Electroweak Box Workshop, September 28, 2017 in collaboration with Wally Melnitchouk and AJM Collaboration

2 2 Outline Recent advances in TPE theory (2008-present) Review: Afanasev, PGB, Hassell, Raue, Prog. Nucl. Part. Phys. (2017) improved hadronic model parameters (fit to data) use of dispersion relavons and connecvon to data new experimental results γz box contribuvons to PV electron scaxering amenable to dispersion analysis in forward limit (Q² 0) disvncvon between axial and vector hadron coupling use of inelasvc PV data in resonance and DIS regions

3 Hadronic Approach Low to moderate Q 2 : hadronic: N + Δ + N* etc. as Q 2 increases more and more parameters Loop integravon using sum of monopole transivon form factors fit to spacelike Q 2 k k! k k! q 1 q 2 p p! Nucleon (elasvc) intermediate state q 1 k k k k q 2 p p! PGB, Melnitchouk, & Tjon, PRL 91, (2003) δ (ε,q 2 ) Q 2 =1 GeV 2 Feshbach limit (iterated Coulomb) ε 2 δ (ε,q ) Q 2 =1 GeV ε 3

4 Δ and N* intermediate states e(p 1 ) e (p 3 ) γ γ k P (p 2 ) N, P (p 4 ) (a) N, (b) Direct loop integravon method Kondratyuk et al., PRL 95, (2005) Zhou & Yang, Eur. Phys. J. A. 51, 105 (2015) Unphysical divergence Include all 3 N Δ mulvpoles, with form factors fit to CLAS data Opposite sign to nucleon contribuvon QualitaVvely correct, BUT diverges as ε 1, implying a violavon of unitarity (Froissart bound) 4

5 Dispersive method on shell S =1+iM S =1 im SS =1 k₁!!! #! # " " "! Unitarity i M M =2 mm = M M m f M i = 1 Z d f M n n M i 2 X n Imaginary part determined by unitarity Uses only on-shell form factors Use form factors directly fit to data, not reparametrized by sum of monopoles Real part determined from dispersion relavons 5

6 TPE using dispersion relavons Generalized form factors M! ( µ ) (e) F1(Q 0 2, ) µ + F2(Q 0 2, ) i µ q 2M (p) +( µ 5 ) (e) G 0 a(q 2, ) µ 5 (p) = 2Re "G E(F 0 1 F 0 2)+ G M (F F 0 2)+ (1 ")G M G 0 a "G 2 E + G2 M Dispersion relavons Re F1(Q 0 2, ) = 2 Z 1 P Re F2(Q 0 2, ) = 2 Z 1 P Re G 0 a(q 2, ) = 2 Z 1 P d 0 d Im F 0 1(Q 2, 0 ), 02 2 Im F 0 2(Q 2, 0 ), d Im G0 a(q 2, 0 ). Integral extends into ``unphysical region down to zero energy (cos θ < -1) 6

7 1 4 Q2 i 2 s W 2 4s Z Z A few technical details d 4 q 1 Im {L µ H µ } (q )(q ) d k1 f Q 2 1,Q 2 2 G 1 (Q 2 1) G 2 (Q 2 2) (Q )(Q ) k₁ on shell!!! #! # " " "! L and H are leptonic and hadronic tensors f is a polynomial in photon virtualives Q1 2 and Q2 2 Gi(Qi 2 ) is a transivon form factor with poles in the complex Qi 2 plane Use numerical contour integravon Allows for use of arbitrary funcvonal forms for transivon form factors Gi(Qi 2 ) Q2 2 (GeV 2 ) θ=30 θ=90 Contours are concentric ellipses of radial parameter r Q 1 2 (GeV 2 ) θ=150 7

8 Nucleon (elasvc) intermediate state Q 2 = 3 GeV 2 ( ) Unphysical ( ) Physical ( ) ( - ) = ( ) ( ) Logarithmic divergence at low energies Agrees with old loop integra3on method δ = ( ) No subtracvons needed

9 Δ intermediate state (zero width approximavon) Unphysical δδ = Physical ( ) = ( ) ( ) Δ ( - ) ( ) ( ) Include all 3 mulvpoles, with form factors fit to recent CLAS data GM * x GM * dominates, but GM * x GE * interference is significant No unphysical divergence at ε ε changes sign at Q GeV 2 9

10 Direct measurements of Im part Target normal spin asymmetry Ee = GeV Proton Neutron % π N (inelastic) (taken from Pasquini & Vanderhaeghen) N (elastic) total This is all in the physical region. 10

11 PolarizaVon data 0.74 Q 2 = 2.50 GeV 2 N 0.72 N+Δ RTL indicates mild sensivvity to GE form factor at low ε RTL (b) Venkat form factors ε 1.04 N N+Δ Q 2 = 2.50 GeV 2 N N+Δ PL/PL (0) GEp2γ RTL (a) ε 0.66 (b) Kelly form factors ε 11

12 TPE effect on ravo of e + p to e - p cross secvons TPE interference changes sign for positrons vs electrons R 2 = e + e 1 2 VEPP-3 (Novosibirsk) γ Δ ( ) = ε ( ) = ε 12

13 TPE effect on ravo of e + p to e - p cross secvons CLAS (Jefferson Lab) 1.06 < >= 1.06 < >= Δ 1.04 γ ( ) ( ) ε ε Δ <ε> = ( ) <ε>= γ ( ) ( ) ( ) 13

14 TPE effect on ravo of e + p to e - p cross secvons OLYMPUS (Doris DESY) 1.04 OLYMPUS N R2 γ N+Δ What is going on 0.98 E = 2.01 GeV at low Q²? ε 14

15 Comparing theory and experiment VEPP3 CLAS OLYMPUS About 1% below theory over all ε 15

16 Allowing normalizavon to float VEPP3 CLAS OLYMPUS 16

17 CorrecVon to proton weak charge including one-loop radiavve correcvons Q p W = 1 4apple PT(0)ŝ e + W + WW + ZZ + Z e.w. vertex correcvons box diagrams Box corrections X WW and ZZ box diagrams large but dominated by short distances; can be evaluated perturbavvely γz box diagram sensivve to long distance physics, has two contribuvons: O Z = O A Z + O V Z V(e) x A(h) (finite at E=0) A(e) x V(h) (inelastic vanishes at E=0) 17

18 Axial h correcvon A Z axial h correcvon dominant γz correcvon in atomic parity violavon at very low (zero) energy computed by Marciano & Sirlin in 1983 as sum of two parts: low-energy part approximated by Born contribuvon (elasvc intermediate state) q q q A Z = 1 4ŝ q q q high-energy part (above scale Λ ~ 1 GeV) computed perturbavvely in terms of scaxering from free quarks Z 1 dq 2 s (Q 2 ) Q 2 (1 + Q 2 /M 2 2 Z ) 1 {z } log M2 z 2 +c Marciano, Sirlin, PRD 29 (1984) 75; Erler et al., PRD 68 (2003)

19 Forward angle dispersion method Gorchtein, Horowitz, PRL 102 (2009) S =1+iM S =1 im SS =1 k p on-shell states q Z k k p p Unitarity i M M =2 mm = M M m f M i = 1 Z d f M n n M i 2 X n Forward scattering amplitude: f i Z m i M i = 1 X d 2 n n M i 2 Z d 3 k 1 L µ W µ q 2 (q 2 M 2 Z ) vector h axial h hadronic tensor: MW µ Z = gµ F Z 1 + pµ p p q F Z 2 i µ p q 2p q F Z 3 19

20 At low energy, dominant Axial h correcvon V e A h using forward dispersion relavons correcvon evaluated <e A Z(E) = 2 Z 1 0 de 0 E 0 E 02 E 2 =m A Z(E 0 ) imaginary part given by F Z 3 structure funcvon k p Im A Z(E) = =m A Z (E) = q V A Z 1 (2ME) 2 k k p p Z s Z Q 2 max dw 2 dq 2 v e(q 2 ) (Q 2 ) M Q 2 /MZ 2 2ME 1 W 2 M 2 + Q 2 2 F Z 3 with v e (Q 2 )=1 4apple(Q 2 )ŝ 2 20

21 21 Axial h correcvon DIS part (dominant contribuvon) DIS part dominated by leading twist PDFs at small x (MSTW, CTEQ, Alekhin parametrizations) F Z(DIS) 3 (x, Q 2 )= X q in DIS region ( Q 2 & 1 GeV 2 2 e q g q A q(x, Q2 ) q(x, Q 2 ) ), expand integrand in powers of x <e A(DIS) Z (E) = 3 2 Z 1 dq 2 v e (Q 2 ) (Q 2 ) Q 2 Q 2 (1 + Q 2 /M 2 0 Z apple ) M (1) 3 (Q2 )+ 2M 2 9Q 4 (5E2 3Q 2 )M (3) 3 (Q2 )+... with moments M Z(n) 3 = Z 1 0 dx x n 1 F Z 3 (x, Q 2 )

22 structure funcvon moments n = 1 Re n = 3 M Z(1) 3 (Q 2 )= 5 3 Axial h correcvon 1 γz analog of Gross-Llewellyn Smith sum rule A(DIS) Z (1 4ŝ 2 ) 5 2 1R Q 2 0 s(q 2 ) dq 2 s(q Q 2 (1+Q 2 /MZ 2 ) 1 2 ) precisely result from Marciano & Sirlin! (works because result depends on lowest moment of valence PDF, with model-independent normalizavon!) M Z(3) 3 (Q 2 )= 1 3 2hx2 i u + hx 2 i d 1+ 5 s(q 2 ) 12 related to x 2 -weighted moment of valence quarks log M 2 Z Q

23 Axial h elasvc + resonance correcvon elasvc part: F Z(el) 3 (Q 2 )= Q 2 G p M (Q2 )G Z A(Q 2 ) (W 2 M 2 ) resonance part from parametrizavon of ν scaxering data; includes lowest four spin 1/2 and 3/2 states (Lalakulich-Paschos) Least well-constrained part of the calculavon A -4 Re γz (E) (x 10 ) elastic resonance E (GeV) 23

24 Vector h correcvon vector h correcvon O V Z vanishes at E = 0, but experiment has E ~ 1 GeV - what is energy dependence? forward dispersion relavon e O V Z (E) = 2E R 1 0 de 0 1 E 02 E 2 m O V Z (E0 ) k p imaginary part given by m O V Z(E) = (s M 2 ) 2 s q A V Z k k p p W 2 dw 2 F Z 1 + F Z 2 0 Q 2 max dq 2 1+Q 2 /M 2 Z s (Q 2 max Q 2 ) Q 2 (W 2 M 2 + Q 2 ) 24

25 3 groups doing independent analyses Hall et al. PRD 88, (2013) Carlson and Rislow PRD 83, (2011) Gorchtein et al. PRC 84, (2011) Qweak energy: 8% correction! Mainly different treatments of low Q², low W region background contributions Agree on overall magnitude of 8% correction, but disagree on errors and details 25

26 AJM structure funcvon model Accurate knowledge of γγ and γz structure funcvons (at all kinemavcs) vital for determinavon of radiavve correcvons Wealth of data on F i structure funcvons over large range of kinemavcs in Q 2 and W (or x) with some gaps RelaVvely lixle known about i interference structure funcvons below HERA measurements, with Q GeV 2 F Z F i Fit over all kinemavcs in Q 2 and W, then rotate to using available theorevcal/phenomenological constraints e.g. isospin symmetry hn J µ Z pi =(1 4sin2 W )hn J µ pi hn J µ ni F Z i 26

27 IntegraVon region (structure funcvon map) elasvc DIS Leading twist PDFs Q 2 GeV 2 10 (quark-parton model) III Regge resonance Bosted-Christy γγ fit 2.5 I II Pomeron and Reggeon exchange + isospin rotavon to γz W 2 GeV 2 27

28 Basic issue: how to relate F Z to F? 1,2 1,2 Scaling region III F 2 = X q F Z 2 = X q e 2 qx(q + q) 2e q g q V x(q + q) x = Q 2 W 2 M 2 + Q 2 F Z 3 Resonance region I largest contribuvon (unlike ) For γγ use Christy-Bosted (CB) fit to e-p cross secvons T,L = T,L (res) + T,L (bg) T,L(res) Includes 7 most prominent N* resonances below 2 GeV. Generally agrees with data to ~ 5% For γz modify fit by ravo of weak to e.m. transivon amplitudes. 28

29 Background T,L(bg) Use Vector Meson Dominance (VMD) models fit to high energy data, plus isospin rotavons p Z p = p V V Z p V = ρ, ω, φ + continuum Z V = apple V V Z = + R + R + C R C 1+R + R + R C Isospin rotavon: =2 4 sin 2 W, = 4 sin 2 W, apple =3 4 sin 2 W convnuum parameter κc not constrained in VMD GHRM: assign 100% uncertainty on convnuum contribuvon (dominates errors) AJM model: constrain convnuum (higher Q²) contribuvon by matching with PDF ravos (γz to γγ) across boundaries of Regions I, II and III. 29

30 ContribuVon from different regions to O V Z (relavve to weak charge of ) Re É V gz Total I II III E HGeVL 30

31 êq 2 Hppm GeV -2 L p A PV Parity-violaVng Deep InelasVc ScaXering (PVDIS) asymmetry allows a direct measurement of the γz structure funcvons A PV = g e A Ê -120 Ê GF Q 2 2 p 2 AJM γz model direct test xy 2 F Z 1 +(1 y)f Z 2 + ge V g e A (y GHRM W HGeVL W HGeVL AJM Q²=0.34 GeV², E=0.69 GeV xy 2 F 1 +(1 y)f 2 AJM y 2 /2)xF Z 3 ABM11 From PDFs W HGeVL Q² = 2.5 GeV², E = 6 GeV Androic et al. (G0 collaboration), arxiv:

32 Potential impact of constraints from deuteron PV inelastic asymmetries 100% uncertainty on continuum background 32

33 Potential impact of constraints from deuteron PV inelastic asymmetries 50% uncertainty on continuum background 33

34 Potential impact of constraints from deuteron PV inelastic asymmetries 25% uncertainty on continuum background 34

35 Constraints from PV inelasvc asymmetries êq 2 Hppm GeV -2 L d A PV E constrained Ê Q 2 = 0.95 GeV 2 Q 2 = 0.76 GeV W HGeVL Ú Q 2 = 0.83 GeV 2 Wang et al. PRL 111, (2013) AJM model asymmetries and uncertainves for PV deuteron asymmetry constrained by fit to E data Hall et al. (2013) 35

36 Ê Ú PredicVons for PV deuteron asymmetry in DIS kinemavcs êq 2 Hppm GeV -2 L d A PV PDF constrained Q 2 = 1.09 GeV 2 Q 2 = 1.90 GeV W HGeVL Prediction: Hall et al. (2013) E08-011: Wang et al. Nature 506, 67 (2014) A PV = A PV = 92.4 ± 6.8 ppm ± 12.2 ppm 36

37 Parity-violaVng inelasvc asymmetries Expected inelasvc asymmetry data from Qweak -3-4 Q 2 = 0.09 GeV 2-5 HppmL p A PV Ï 100% AJM W HGeVL AJM model uncertainves compared with 100% on convnuum contribuvon Hall et al. (2013) 37

38 Duality in electron-nucleon scattering deep inelastic function average over (strongly Q 2dependent) 2 resonances Q independent scaling function Nachtmann scaling variable 2x = M 2 x 2 /Q 2 Niculescu et al., PRL 85, 1182 (2000) WM, Ent, Keppel, PRep. 406, 127 (2005) Separates higher twist (HT) effects from target mass corrections to leading twist (LT) 38

39 γγ Leading Twist (LT) F1,2 moments vs. Nachtmann moments Proton Neutron Res DIS Elastic m 1 H2L Total LT HCNL m 2 H2L 0.10 m 2 H2L lnhq 2 êgev 2 L lnhq 2 êgev 2 L Compare total empirical moments of structure funcvons to leading twist (LT) contribuvons down to low Q 2 Difference indicavve of highter twist (HT) contribuvons Sum is approximately independent of Q 2 Note isospin independence Apply to γz structure functions? 39

40 γz Leading Twist (LT) moments vs. Nachmann Proton Neutron Res DIS Elastic m 1 H2L gz Total LT HCNL m 2 H2L gz 0.10 m 2 H2L gz lnhq 2 L lnhq 2 L Allows us to extend PDF region down to Q²=1 GeV² (from Q²=2.5) V Z@1.165 GeV : (5.6 ± 0.4) 10 3, 2013 (5.4 ± 0.4) 10 3,

41 Summary Lots of interesvng new theorevcal work movvated by new experimental results Dispersive method only feasible approach for TPE, with connecvon to data in forward angle limit Efforts underway to incorporate electroproducvon data throughout the resonance region, including background Clear need for definivve e + p measurements at high Q 2, low ε Dispersion approach significant improvement over old methods PDF region provides constraints on model-dependence of isospin rotavon Direct comparison with PV inelasvc data in resonance and DIS regions e-d PVDIS asymmetry strongly constrains the uncertainty checking Δ region at Mainz or JLab would be useful quark-hadron duality approach allows further constraints on uncertainves 41