TT Ph t Ph iPh t Ph iTwoTwo--Photon Physics:Photon Physics:TheoryTheoryyy

Barbara PasquiniBarbara Pasquini

U i i f P i d INFN P iU i i f P i d INFN P i I lI lUniversity of Pavia and INFN PaviaUniversity of Pavia and INFN Pavia--ItalyItaly

MAMI and Beyond 30 MarchMAMI and Beyond 30 March--3 April 2009 Budenheim (Mainz)3 April 2009 Budenheim (Mainz)MAMI and Beyond,30 MarchMAMI and Beyond,30 March 3 April 2009, Budenheim (Mainz)3 April 2009, Budenheim (Mainz)

OutlineOutline

Real Compton scattering and polarizabilities Theoretical predictionsAnalysis of RCS with Dispersion Relation FormalismExtraction of spin polarizabilities from double polarization experimentsExtraction of spin polarizabilities from double polarization experiments(MAMI, HIGS,….)

B m N m l Spin As mm t in l sti l t n n l n s tt inBeam Normal Spin Asymmetry in elastic electron nucleon scatteringAbsorptive part of two-photon exchange amplitudeMAMI experiment in the resonance region: new tool to study resonance MAMI experiment in the resonance region: new tool to study resonance transition form factors

Beam Normal Spin Asymmetry with inelastic electron nucleon scatteringΔ-resonance in the final state → unique tool to extract γ*ΔΔ form factors f ibilit t MAMI feasibility at MAMI

Static polarizabilities in Real Compton ScatteringStatic polarizabilities in Real Compton Scattering

P ll h ff Powell cross section: photon scattering off a pointlike nucleon with anomalous magnetic moment

Static polarizabilities: response of the internal Static polarizabilities: response of the internal nucleon degrees of freedom to a static electric and magnetic field

spin-independent dipole

spin-dependent dipole

spin d p nd nt spin-dependent dipole-quadrupole

Spin independent dipole polarizabilitiesSpin independent dipole polarizabilities

600

b )

Baldin Sum Rule (1960)

300

μcr

oss

sect

ion(

b

Tot

al c

ro

Compton scattering

0 210

0.5 1.5ν (GeV)Olmos de Leon et la., EPJ A10 (2001)

Spin independent dipole PolarizabilitiesSpin independent dipole Polarizabilities

HB3 HB4 SSE3+ LECs LC3 LC4 O(p4/Δ M) EXP

α 12.2 12.1± 1.2 11.5± 2.4 6.8 12.2 10.8 11.9± 0.5

β 1 2 3 4± 1 1 3 4± 1 7 1 8 1 6 2 9 1 2 0 7β 1.2 3.4± 1.1 3.4± 1.7 -1.8 1.6 2.9 1.2 ± 0.7

EXP: global fit to RCS data below pion-threshold [Olmos de Leon et al., 2001]

HB3 (Heavy Baryon ChPT at O(p3)): no free parameters, good agreement with exp. polarizabilities, but poor description of RCS cross section [Bernard, Kaiser, Meissner, 2004]

HB4 LEC t th t ll t fit th l i biliti t d t b l th h ld HB4: LECs enter that allow to fit the polarizabilities to data below threshold [McGovern, 2002, Beane et al, 2004]

SSE3 + LECs (HB3 + Δ-resonance + LECs): large paramagnetic contribution to βSSE LECs (HB Δ resonance LECs): large paramagnetic contribution to β→ higher order LECs are promoted to lower order [Hildebrandt et al., 2003]

LC3 (Lorentz covariant ChPT at O(p3)): relativistic effects are important[Bernard Kaiser Meissner 1992][Bernard, Kaiser, Meissner, 1992]

LC4: LECs enter that allow to fit the polarizabilities to data below threshold [Djukanovic, PhD Thesis, Mainz, 2008]j

O(p4/Δ M) (LC3 + Δ-resonance): expansion in δ ∼ mπ/Δ M ∼ Δ M/M; no free-parameters;good description of RCS cross section [Lensky, Pascalutsa, 2008]

Spin polarizabilitiesSpin polarizabilities

GDH Sum Rule

GDH Coll. (MAMI & ELSA)

Ahrens et al., PRL87 (2001)Dutz et al. PRL91 (2003)

Unpolarized Compton scattering

TAPS, LARA, SENECASchumacher, Prog. Part. Nucl. Phys. 55 (2005)

Spin PolarizabilitiesSpin PolarizabilitiesHB3 HB4 SSE LC3 LC4 DRs HB HB SSE LC LC DRs

γE1E1 -5.7 -1.4 -5.4 -3.2 -2.8 -4.3γM1M1 -1.1 3.3 1.4 -1.4 -3.1 2.9γM1M1

γE1M2 1.1 0.2 1.0 0.7 0.8 0.0γM1E2 1.1 1.8 1.0 0.7 0.3 2.1γM1E2

γ0 4.6 -3.9 2.0 3.1 4.8 -0.7

γ 4 6 6 3 6 8 1 8 -0 8 9 3γπ 4.6 6.3 6.8 1.8 0.8 9.3

HB3: Heavy Baryon ChPT at O(p3) [Hemmert et al, 1998]

HB4: Heavy Baryon ChPT at O(p4) [Kumar et al, 2000]H H a y aryon h at O(p ) [Kumar t a , ]

SSE: Heavy Baryon ChPT with Δ at O(p3) [Hemmert et al, 1998]

LC: Lorentz covariant ChPT [Djukanovic, PhD Thesis, Mainz, 2008]

DRs: Dispersion Relations [Drechsel et al., 2003]

How to extract the RCS polarizabilitiesHow to extract the RCS polarizabilities

Born (anomalous magnetic moment)

1

LEX(polarizabilities at leading order)

2

( g )

Dispersion relations (full calculation)

3

RCS below pion threshold peffects of the polarizabilities rather small (∼ 10 %) below threshold

limitation in energy in order to apply low energy expansion (LEX)

Dispersion relation formalism allows

to check validity of LEX

to extract polarizabilities with a minimum of model dependence

to go to higher energies to enhance the sensitivity to polarizabilities

Dispersion Relations at fixed Dispersion Relations at fixed t t

: : 6 Lorentz invariant functions of ν=Eγ+t/4M and t=-2E’γ Eγ(1-cosθ)

Im(ν)analytical functions in the complex ν plane with cuts and poles on the real axis

ν + i ε

Re(ν)νB νB

Cauchy integral formula

Crossing symmetry and analyticityCrossing symmetry and analyticity

Unsubtracted Dispersion RelationsUnsubtracted Dispersion Relations

h li d A i h i 3 4 5 6 d f h hi h i f b d the amplitudes Ai with i=3,4,5,6 drop fast enough at high energy to satisfy unsubtracted dispersion relations

the amplitudes A1 and A2 do not satisfy unsubtracted DRs due to high energybehaviour

modeled by energy independent poles in the t channel with parameters fitted to experimental data

fitted to data

L’vov, Petrun’kin, Schumacher, PRC55 (1997)

Subtracted Dispersion RelationsSubtracted Dispersion Relations

= const. , subtraction point at

c n nc f

subtraction functions are determined from subtracted DRs

convergence for all 6 amplitudes

subtraction functions are determined from subtracted DRsin at fixed

subtraction constants are directly related to linearcombinations of static polarizabilities

Subtracted Dispersion Relations Subtracted Dispersion Relations

Drechsel, Gorchtein, BP, Vanderhaeghen, PRC61 (1999)

s-channel dispersion integral

s ≥ (M + m )2 s ≥ (M + 2 m )2s ≥ (M + mπ )2 s ≥ (M + 2 mπ)2

one-pion intermediate states: 1π photoproduction multipoles from MAID analysis

resonance contribution from multipion intermediate states

t-channel dispersion integral

p siti t t: t ≥ 4 m 2positive – t cut: t ≥ 4 mπ2

t ≥ 4 mπ2

γγ→ππ: unitarized S and D waves amplitudesππ→ N N: extrapolation of the crossed π N → π N helicity amplitudes [Hoehler, 1983]

negative – t cut: t ≤ - 2mπ (M+mπ)extrapolation of the s-channel amplitudes (Δ(1232) and non-resonant π N exchange)in the unphysical region at ν =0 and t ≤ 0

and : input from available experimental information of different p ss s ( → N → nd → NN)processes (γ π→ π N, γγ→ ππ and ππ→ NN)

: subtraction constants given by the polarizabilities

free parameters to be fitted to RCS data

g y p

RCS below pion threshold: fit with fixedRCS below pion threshold: fit with fixed--t DRst DRs

Baldin sum rule:Baldin sum rule:

Unsubtracted DRsOlmos de Leon et al, EPJA100,2001

Subtracted DRsD h l t l Ph R 378 (2003)Drechsel et al., Phys. Rep. 378 (2003)

Strategy to extract the Spin Polarizabilities

f b d DR 6 f ( l b l )

Use the available experimental values for α, β, γ0 and γπ

Start from Subtracted DRs → 6 free parameters (polarizabilities)

αE1 + βM1 = (13.8 ± 0.4) ⋅10-4 fm3 Baldin sum ruleOlmos de Leon et la., EPJ A10 (2001)

γ0 = γE1E1 - γM1M1 - γM1E2 - γE1M2 = (-1 00 ± 0 08 ± 0 10) ⋅10-4 fm4 GDH coll

αE1 −βM1 = (10.5 ± 0.9 ± 0.7) ⋅10-4 fm3 World average value,Olmos de Leon et al., EPJ A10 (2001)

γ0 = γE1E1 γM1M1 γM1E2 γE1M2 = ( 1.00 ± 0.08 ± 0.10) ⋅10 fm GDH coll.PRL87 (2001) and PRL91 (2003)

γπ = γE1E1 + γM1M1 + γM1E2 - γE1M2 = (-38.7 ± 1.8) ⋅10-4 fm4 Unpolarized Compton data γπ γE1E1 γM1M1 γM1E2 γE1M2 ( )(TAPS, LARA, SENECA)

Schumacher, Prog. Part. Nucl. Phys. 55(2005)

Eliminate γE1M2 and γM1E2 through γ0 and γEliminate γE1M2 and γM1E2 through γ0 and γπ

Fit γE1E1 and γM1M1 to double polarized Compton scattering experiments

Check also dependence on γπ ! BP, Drechsel, Vanderhaeghen, PRC76 (2007)

Circularly polarized photonsCircularly polarized photons

Longitudinally polarized target

Transversely polarized target

Circularly pol. Photon Circularly pol. Photon -- Proton Target pol. along zProton Target pol. along z-2.3 4.9 9.8-4.3-6.3

γE1E1 2.90.9

γM1M1 8.07.2

γπ

θlab = 30o

effects of15 20% MAMI

θlab = 90o

15-20% MAMIand

HIGSproposalsproposals

θlab = 150o

Circularly pol. Photon Circularly pol. Photon -- Proton Target pol. along xProton Target pol. along x

-2.3 4.9 9.8-4.3-6.3

γE1E1 2.90.9

γM1M1 8.07.2

γπ

θlab = 30o

MAMI

θlab = 90o15%

40%

MAMIand

HIGSproposalsproposals

θlab = 150o

Linearly polarized PhotonLinearly polarized Photon

azimuthal angle between the scattering plane and the photon polarization vector

φ = 0 and unpolarized target → Σ3φ 0 and unpolarized target → Σ3

φ = 0 and transversely polarized target in the y direction → Σ3y

φ = 45o and longitudinally pol. target → Σ1z

φ = 45o and transv. pol. target in the x direction → Σ1xφ p g 1x

Linearly pol. Photon with Linearly pol. Photon with φφ=0=0oo –– Unpol. TargetUnpol. Target-2.34 3

4.92 9

9.88 0-4.3

-6.3γE1E1 2.9

0.9γM1M1 8.0

7.2γπ

θcm = 65o

θcm = 90o

θcm = 135o Blanpied, et al., PRC64 (2001)

Linearly pol. Photon with Linearly pol. Photon with φφ = 0= 0oo

-2.3-4.3-6.3

γE1E1

4.92.90.9

γM1M1

Blanpied, et al., PRC64 (2001)( )

Unpol. target

MAMIproposal

Pol. target in the y direction in the y direction

Beam Normal Spin Asymmetry inelastic eN scattering

directly proportional to the Imaginary part of 2-photon exchange amplitudes

spin of beam NORMAL

to scattering plane

on-shell intermediate state

order of magnitude:

De Rujula et al. (1971) to

1γ exchange

function of elastic nucleon form factors

2γ exchange

factors

absorptive part of non-forward

double virtual Compton scatteringdouble virtual Compton scattering

Hadronic Tensor: Absorptive part of Doubly Virtual Compton Tensor

λnλn

sum over intermediate states with M2 ≤ W2 ≤ s

on-shell intermediate state (MX2 = W2)

Transverse spin asymmetries

sum over intermediate states with M ≤ W ≤ s

lepton

hadronhadron

Beam normal spin asymmetry :Beam normal spin asymmetry : experimentsexperiments

Expt E(GeV) θ Q2 (GeV2) B (ppm)

Beam normal spin asymmetry :Beam normal spin asymmetry : experimentsexperiments

Expt. E(GeV) θe Q2 (GeV2) Bn(ppm)SAMPLE 0.192 146 0.10 -16.4±5.9

A4 0.570 35 0.11 -8.59±0.89

A4 0 855 35 0 23 8 52±2 31A4 0.855 35 0.23 -8.52±2.31

HAPPEX 3.0 16 0.11 -6.7 ± 1.5

G0 3.0 20.2 0.15 -4.06 ±0.99

G0 3.0 25.9 0.25 -4.84 ±1.87

E-158 46.0 ~3.0 0.06 -3.5 -> -2.5E 158 46.0 3.0 0.06 3.5 2.5

Resonance region (WResonance region (W≤≤ 2 GeV)2 GeV)

elastic contribution

on-shell nucleon intermediate nucleon

inelastic contribution

X= π N

resonant and non-resonant π N intermediate states calculated with MAID2003 : unitary isobar model

X= π N

yall 13 **** resonances below 2 GeV included

Drechsel, Hanstein, Kamalov, Tiator (1999)

Beam normal spin asymmetry

N (elastic)

π N (inelastic)

total (N + π N)

B.P. & Vanderhaeghen (2004)

Ee = 0.570 GeV and Ee= 0.855 GeV Ee = 0.315 GeV

MAMI data at Ee=0.424 GeV under analysis

MAMI dataF. Maas et al., PRL 94 (2005)

Preliminary MAMI dataS. Baunack, L. Capozza and A4 Coll., Proc. PAVI06

Beam Spin Asymmetry ininelastic eN scattering with Δ in the final stateinelastic eN scattering with Δ in the final state

1γ exchange e- e-1γ exchange

N Δ

2γ exchange

N Δ

γ*NΔ form factors2γ exchange

e- e- e- e-

for s� M2Δ

N N Δ Δ ΔNγ*ΔΔ form factorsγ*NΔ form factors

for s M Δ

γ ΔΔ form factors

unique tool to learn about the γ*ΔΔ form factors

γ NΔ form factors

Beam asymmetry in inelastic electron scattering

B.P. & Vanderhaeghenin preparation

l i i h f d ilarge asymmetries in the forward region

sensitive to γ*ΔΔ form factors

l (N )

N intermediate stateΔ intermediate state

total (N + Δ)

γ*NΔ form factors from MAID07 parametrization

γ*Δ Δ form factors from LATTICE QCDAlexandrou et al., arXiv:0901.3457 [hep-ph]

Energy spectrum of the final electron in MAMI experimentCourtesy of S BaunackCourtesy of S. Baunack

Ee=0.855 GeV θlab=35o

elastic peakE’e=730 MeVΔ peak

E’e=440 MeV

promising to extract the beam asymmetry in the inelastic channel

SummaryMany recent RCS data both below and above threshold Many recent RCS data both below and above threshold

αE1 and βM1 are known at 5-10% levelthe small γ0 is known to 10% and the larger γπ

disp to about 25%

Proposals to measure the spin polarizabilities with double polarizationexperiments (MAMI,HIGS)

i l l l h t d l it di ll l t t circularly pol. photons and longitudinally pol. target → γM1M1

circularly pol. photons and transversely pol. target → γE1E1

beam asymmetry → γM1M1beam asymmetry → γM1M1

linearly pol. photons and transversely pol. target → γE1E1

Transverse beam asymmetry in elastic electron scattering to access the imaginaryTransverse beam asymmetry in elastic electron scattering to access the imaginarypart of 2γ exchange amplitude absorptive part of non-forward doubly VCS

MAMI experiment in the resonance region: new tool to study the resonance transition form factors

Transverse beam asymmetry in inelastic electron scattering with Δ-resonance in the final statein the final state

unique tool to study the γ*ΔΔ form factors data available from MAMI measurements

Integrand : beam normal spin asymmetryg p y yEe = 0.855 GeV

π0 p

π+ n

tot Quasi-RCS peak

Integrand : beam normal spin asymmetryg p y yEe = 3 GeV

Δ (1232) D13 (1520)F15 (1680)

Δ (1232) 13 ( )

π0 p

π+ n

totWmax = 2.55 GeV,

but with MAID we can integrate only up to W = 2 GeV

(near) collinear singularities

Q21 � 0, Q2

2 ≠ 0Quasi - VCS�

k // k1

Q21 ≠ 0, Q2

2 � 0 Quasi - VCS�

k 1 // k2

Q21 � 0, Q2

2 � 0 Q 1 0, Q 2 0

Quasi - RCSk1 = 0 W2 � sk1 = 0, W � s

�

Kinematical bounds for Q2 and Q2Kinematical bounds for Q 1 and Q 2

Elastic contribution

Inelastic contribution

Kinematical bounds for Q21 and Q2

2

Δ in the final state

Phase space integrationPhase space integration• 2-dim integration (Q1

2, Q22) for the elastic intermediate state

• 3-dim integration (Q12, Q2

2,W2) for inelastic excitations

‘Soft’ intermediate electron;Both photons are hard collinear

1.5

p

1�M2 � W2

MAMI A4E = 855 MeV Θcm= 57 deg

0 75

1

0

0.5One photon is Hard collinear

Θcm 57 deg

00.25

0.5

0.75Q120.25

0.5

0.75

Q22

000.25

0.5

0.75Q12

1Q

01Q

Beam normal spin asymmetryBeam normal spin asymmetryEe = 0.570 GeV

P t N tProton Neutron

π N (inelastic)

N (elastic)

totaltotal

Target normal spin asymmetryTarget normal spin asymmetry

%%%

π N (inelastic)N (elastic)

total

Target normal spin asymmetryTarget normal spin asymmetryEe = 0.570 GeV

P t N tProton Neutron

%

N (elastic)

π N (inelastic)

totaltotal

Linearly pol. Photon with Linearly pol. Photon with φφ = 45= 45oo –– pol. Target along z/xpol. Target along z/x-2.3 4.9 9.8-4.3-6.3

γE1E1 2.90.9

γM1M1 8.07.2

γπ

Long. pol. targetl

θ=90o

along z

Transv. pol. targetalong x

θ=150o

along x

Circularly pol. Photon and Proton Target pol. along z or xCircularly pol. Photon and Proton Target pol. along z or x-2.3 4.9.-4.3-6.3

γE1E1

.92.90.9

γM1M1

longitudinal asymmetry

(proton pol. along z)

transverse asymmetry

(proton pol. along x)

Spin PolarizabilitiesSpin PolarizabilitiesHB3 HB4 SSE LC3 LC4 DRs EXPHB HB SSE LC LC DRs EXP

γE1E1 -5.7 -1.4 -5.4 -3.2 -2.8 4.3γM1M1 -1.1 3.3 1.4 -1.4 -3.1 2.9γM1M1 . 3.3 . . 3. .9γE1M2 1.1 0.2 1.0 0.7 0.8 0.0

γM1E2 1.1 1.8 1.0 0.7 0.3 2.1γM1E2

γ0 4.6 -3.9 2.0 3.1 4.8 -0.7 -1.0±0.08

γ 4 6 6 3 6 8 1 8 -0 8 9 3 8 0± 1 8γπ 4.6 6.3 6.8 1.8 0.8 9.3 8.0± 1.8

HB3: Heavy Baryon ChPT at O(p3) [Hemmert et al, 1998]

HB4: Heavy Baryon ChPT at O(p4) [Kumar et al, 2000]H H a y aryon h at O(p ) [Kumar t a , ]

SSE: Heavy Baryon ChPT with Δ at O(p3) [Hemmert et al, 1998]

LC: Lorentz covariant ChPT [Djukanovic, PhD Thesis, Mainz, 2008]

DRs: Dispersion Relations [Drechsel et al., 2003]

Beam asymmetry in inelastic electron scattering

B.P. & Vanderhaeghenin preparation

large asymmetries in the forward region dominated by quasi-VCS kinematics where dominated by quasi VCS kinematics where one exchanged photon becomes quasi-real

l (N )

N intermediate stateΔ intermediate state

total (N + Δ)

γ*NΔ form factors from MAID07 parametrization

γ*Δ Δ form factors from LATTICE QCDAlexandrou et al., arXiv:0901.3457 [hep-ph]

Beam asymmetry in inelastic electron scattering

B.P. & Vanderhaeghenin preparation

l i i h f d ilarge asymmetries in the forward region

sensitive to γ*ΔΔ form factors

l (N )

N intermediate stateΔ intermediate state

total (N + Δ)

γ*NΔ form factors from MAID07 parametrization

γ*Δ Δ form factors from LATTICE QCDAlexandrou et al., arXiv:0901.3457 [hep-ph]