Overview of two-photon and two-boson exchange Peter Blunden - PowerPoint PPT Presentation

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

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 
 • γ Z box contribuVons to PV electron scaXering –amenable to dispersion analysis in forward limit ( Q ²→ 0 ) –disVncVon between axial and vector hadron coupling –use of inelasVc PV data in resonance and DIS regions 2

Hadronic Approach Low to moderate Q 2 : k ! k ! k k hadronic: N + Δ + N * etc. q 1 q q 1 q 2 k k 2 k p ! p ! k p p • as Q 2 increases more and 
 more parameters • Loop integraVon using sum of monopole PGB, Melnitchouk, & Tjon, PRL 91 , 142304 (2003) transiVon form factors fit to spacelike Q 2 Nucleon (elasVc) intermediate state Feshbach limit 0.01 0 (iterated Coulomb) 0.01 2 =1 GeV 2 Q 0.001 0.1 2 ) 2 ) − 0.02 − ( ε , Q − ( ε , Q 0 2 δ 0.5 δ − 0.01 − 0.04 3 2 =1 GeV 2 Q 6 − 0.02 − 0.06 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ε ε 3

Δ and N* intermediate states e ′ ( p 3 ) e ( p 1 ) γ γ k P ′ ( p 4 ) P ( p 2 ) N, ∆ N, ∆ ( a ) ( b ) Direct loop integraVon method Kondratyuk et al., PRL 95 , 172503 (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

Dispersive method on shell k ₁ ! ! ! S = 1 + i M # ! # " S † = 1 − i M † " ! " SS † = 1 Unitarity → M � M † � = 2 ⇥ m M = M † M � � i Z � m ⇥ f |M| i ⇤ = 1 X ⇥ f |M ∗ | n ⇤⇥ n |M| i ⇤ d ρ 2 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

TPE using dispersion relaVons Generalized form factors ◆ ( p ) ✓ 2 ( Q 2 , ν ) i σ µ ν q ν 1 ( Q 2 , ν ) γ µ + F 0 M γγ → ( γ µ ) ( e ) ⊗ F 0 2 M � ( p ) + ( γ µ γ 5 ) ( e ) ⊗ a ( Q 2 , ν ) γ µ γ 5 � G 0 δ γγ = 2Re ε G E ( F 0 1 − τ F 0 2 ) + τ G M ( F 0 1 + F 0 2 ) + ν (1 − ε ) G M G 0 a ε G 2 E + τ G 2 M Dispersion relaVons Z 1 1 ( Q 2 , ν ) = 2 ν 1 ( Q 2 , ν 0 ) , Re F 0 d ν 0 ν 0 2 − ν 2 Im F 0 π P � τ Z 1 2 ( Q 2 , ν ) = 2 ν 2 ( Q 2 , ν 0 ) , Re F 0 d ν 0 ν 0 2 − ν 2 Im F 0 π P � τ Z 1 a ( Q 2 , ν ) = 2 ν 0 a ( Q 2 , ν 0 ) . Re G 0 d ν 0 ν 0 2 − ν 2 Im G 0 π P � τ Integral extends into ``unphysical region’’ down to zero energy ( cos θ < -1 ) 6

A few technical details on shell k ₁ ! ! ! 4 π Q 2 1 Im { L α µ ν H α µ ν } α Z d 4 q 1 # ! # " ( q 2 1 − λ 2 )( q 2 i π 2 2 − λ 2 ) " ! " � Q 2 1 , Q 2 � G 1 ( Q 2 1 ) G 2 ( Q 2 s − W 2 2 ) f Z 2 d Ω k 1 ( Q 2 1 + λ 2 ) ( Q 2 4 s 2 + λ 2 ) • L and H are leptonic and hadronic tensors • f is a polynomial in photon virtualiVes Q 12 and Q 22 • G i ( Q i 2 ) is a transiVon form factor with poles in the complex Q i 2 plane 0.6 θ = 30 ∘ Use numerical contour integraVon 0.5 Allows for use of arbitrary funcVonal 0.4 Q 22 ( GeV 2 ) forms for transiVon form factors G i ( Q i 2 ) θ = 90 ∘ 0.3 0.2 0.1 Contours are concentric ellipses of radial parameter r θ = 150 ∘ 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Q 12 ( GeV 2 ) 7

Nucleon (elasVc) intermediate state Q 2 = 3 GeV 2 0 ′ �� � � 0.04 �� ( ��� ) � (× �� - � ) ′ �� � � - 2 �� ( ��� ) � 0.02 � � = � ��� � ′ �� � � - 4 0.00 - 6 ( � ) - 0.02 Unphysical Physical ( � ) - 8 10 - 3 10 - 2 0.1 1 10 0 2 4 6 8 10 � ( ��� ) � ( ��� ) Logarithmic divergence No subtracVons needed at low energies 0.01 0.00 ��� - 0.01 Agrees with old loop � - 0.02 integra3on method δ � - 0.03 � - 0.04 � � = � ��� � - 0.05 0.0 0.2 0.4 0.6 0.8 1.0 8 ε

Δ intermediate state (zero width approximaVon) 8 � � = � ��� � ( � ) ( � ) 10 ′ �� � � �� ( ��� ) Δ (× �� - � ) �� ( ��� ) Δ (× �� - � ) 6 ′ �� � � 4 0 ′ �� � � 2 ′ �� � � - 10 0 ′ �� � � ′ - 2 �� � � - 20 Unphysical Physical - 4 0.5 1 5 10 0 2 4 6 8 10 12 14 � ( ��� ) � ( ��� ) •Include all 3 mulVpoles, with form factors fit to recent CLAS data • G M* x G M* dominates, but G M* x G E* interference is significant 0.015 � � = � ��� � No unphysical 0.010 divergence at ε → 1 � δ Δ 0.005 � 0.000 changes sign at Q 2 ≈ 0.6 GeV 2 ��� 0.0 0.2 0.4 0.6 0.8 1.0 9 ε

Direct measurements of Im part Target normal spin asymmetry Ee = 0.570 GeV Proton Neutron % (taken from Pasquini & Vanderhaeghen) π N (inelastic) N (elastic) total This is all in the physical region. 10

PolarizaVon data N 0.74 Q 2 = 2.50 GeV 2 N + Δ 0.72 R TL 0.70 R TL indicates mild sensiVvity 
 ● ● ● ● ● ● to G E form factor at low 𝜁 0.68 ● ● ● 0.66 ( b ) Venkat form factors 0.0 0.2 0.4 0.6 0.8 1.0 ε N 0.74 Q 2 = 2.50 GeV 2 N 1.04 N + Δ 0.72 N + Δ 1.02 ( 0 ) ● GEp2 γ R TL 0.70 ● ● ● ● P L / P L ● ● ● ● ● ● 1.00 ● ● ● 0.68 ● ● ● ( a ) 0.98 0.66 ( b ) Kelly form factors 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ε ε 11

TPE effect on raVo of e + p to e - p cross secVons TPE interference changes sign for positrons vs electrons R 2 γ = σ e + σ e − ≈ 1 − 2 δ γγ VEPP-3 (Novosibirsk) 1.04 1.04 � ���� - � ▼ ▼ 1.03 1.03 � + Δ 1.02 1.02 ▼ ▼ � � γ ▼ ▼ ▼ ▼ 1.01 1.01 1.00 1.00 ▼ ▼ ( � ) ( � ) � = ����� ��� � = ����� ��� 0.99 0.99 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ε ε 12

TPE effect on raVo of e + p to e - p cross secVons CLAS (Jefferson Lab) 1.06 1.06 < � � > = ���� ��� � < � � > = ���� ��� � � 1.04 1.04 � + Δ ���� ● ● ● 1.02 1.02 � � γ ● ● ● ● ● ● 1.00 1.00 ● ● ● ● ● ● ● ● ● ● 0.98 0.98 ( � ) ( � ) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ε ε 1.06 1.06 < ε > = ���� � < ε > = ���� ( � ) 1.04 1.04 ���� � + Δ ● ● ● 1.02 1.02 � � γ ● ● ● ● 1.00 1.00 ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.98 0.98 ( � ) 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 � � ( ��� � ) � � ( ��� � ) 13

TPE effect on raVo of e + p to e - p cross secVons OLYMPUS (Doris ring @ DESY) N OLYMPUS 1.04 ■ ■ N + Δ 1.02 ■ ■ R 2 γ ■ ■ ■ ■ ■ ■ What is going on 1.00 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ at low Q ² ? 0.98 E = 2.01 GeV 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ε 14

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

Allowing normalizaVon to float VEPP3 CLAS OLYMPUS 16

CorrecVon to proton weak charge e.w. vertex including one-loop radiaVve correcVons correcVons s 2 + ∆ 0 Q p � � W = ρ 1 − 4 κ PT (0)ˆ e + ∆ W box diagrams + ⇤ W W + ⇤ ZZ + ⇤ γ Z X Box corrections WW and ZZ box diagrams large but dominated by short distances; can be evaluated perturbaVvely γ Z box diagram sensiVve to long distance physics, has two contribuVons: O γ Z = O A γ Z + O V γ Z A( e ) x V( h ) V( e ) x A( h ) (finite at E =0 ) (inelastic vanishes at E =0 ) 17

Axial h correcVon axial h correcVon dominant γ Z correcVon in ⇤ A γ Z 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) high-energy part (above scale Λ ~ 1 GeV ) computed perturbaVvely in terms of q q scaXering from free quarks q q q q ✓ ◆ Z ∞ s 2 � 5 α dQ 2 1 − α s ( Q 2 ) � ⇤ A γ Z = 1 − 4ˆ Q 2 (1 + Q 2 /M 2 2 π Z ) π Λ 2 | {z } M 2 z Λ 2 + c ∼ log Marciano, Sirlin, PRD 29 (1984) 75; Erler et al., PRD 68 (2003) 016006 18

Forward angle dispersion method Gorchtein, Horowitz, PRL 102 (2009) 091806 on-shell states S = 1 + i M k’ k k ≈ S † = 1 − i M † q γ ∗ Z SS † = 1 p’ p p ≈ Unitarity → M � M † � = 2 ⇥ m M = M † M � � i Z � m ⇥ f |M| i ⇤ = 1 X ⇥ f |M ∗ | n ⇤⇥ n |M| i ⇤ d ρ 2 n Forward scattering amplitude: | f 〉 ≈ | i 〉 L µ ν W µ ν ⇤ m ⌅ i |M| i ⇧ = 1 Z Z | ⌅ n |M| i ⇧ | 2 ⇥ X d 3 k 1 d ρ q 2 ( q 2 � M 2 2 Z ) n vector h axial h + p µ p ν − i ε µ νλρ p λ q ρ hadronic tensor: MW µ ν γ Z = − g µ ν F γ Z p · q F γ Z 2 p · q F γ Z 1 2 3 19