Twowo--Ph t Ph iPhoton Physics: Theory - uni-mainz.de Dienstag/08 Pasquini.pdfTwowo--Ph t Ph iPhoton...
Transcript of Twowo--Ph t Ph iPhoton Physics: Theory - uni-mainz.de Dienstag/08 Pasquini.pdfTwowo--Ph t Ph iPhoton...
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]