beam single spin asymmetries in electron-proton scattering · p 2 p 1 p 2+q 1 q 1 p 4 q 2 k p 3...
TRANSCRIPT
Beam Single Spin Asymmetries in Electron-ProtonScattering
Pratik SachdevaWashington University in St. Louis
Mentors:Wally Melnitchouk
Jefferson Laboratory Theory Center
Peter BlundenUniversity of Manitoba
July 23, 2014
Outline
1 IntroductionBeam Single Spin Asymmetries (BSSA)Transverse and Normal PolarizationsMotivation: Q-weak Collaboration
2 Elastic ScatteringBeam Normal SSABeam Transverse Single Spin AsymmetryCombination of Asymmetries
3 Inelastic Hadronic Intermediate StateM ∗
γMγγ InterferenceLeptonic and Hadronic TensorsNormal BSSA
4 Conclusion
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 2 / 26
Introduction Beam Single Spin Asymmetries (BSSA)
IntroductionBeam Single Spin Asymmetries (BSSA)
Polarized electron beam (longitudinal, transverse, normal), butunpolarized target.
In general the beam asymmetry, B, is
B = σ↑ − σ↓σ↑ + σ↓
.
BSSA disappear for one-photon exchange.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 3 / 26
Introduction Beam Single Spin Asymmetries (BSSA)
IntroductionBeam Single Spin Asymmetries (BSSA)
Polarized electron beam (longitudinal, transverse, normal), butunpolarized target.
In general the beam asymmetry, B, is
B = σ↑ − σ↓σ↑ + σ↓
.
BSSA disappear for one-photon exchange.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 3 / 26
Introduction Beam Single Spin Asymmetries (BSSA)
IntroductionBeam Single Spin Asymmetries (BSSA)
Polarized electron beam (longitudinal, transverse, normal), butunpolarized target.
In general the beam asymmetry, B, is
B = σ↑ − σ↓σ↑ + σ↓
.
BSSA disappear for one-photon exchange.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 3 / 26
Introduction Beam Single Spin Asymmetries (BSSA)
What exchanges can cause BSSA?
Consider the scattering amplitude:
|M |2 = |Mγ + MZ + Mγγ + · · · |2
= |Mγ |2 + M ∗γMZ + M ∗
γMγγ + · · ·
At the Born level, the spin orientation has no effect on |Mγ |2, so theasymmetry vanishes.Higher order effects result in asymmetries dependent on whether thespin is transversely, normally, or longitudinally polarized.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 4 / 26
Introduction Beam Single Spin Asymmetries (BSSA)
What exchanges can cause BSSA?
Consider the scattering amplitude:
|M |2 = |Mγ + MZ + Mγγ + · · · |2
= |Mγ |2 + M ∗γMZ + M ∗
γMγγ + · · ·
At the Born level, the spin orientation has no effect on |Mγ |2, so theasymmetry vanishes.Higher order effects result in asymmetries dependent on whether thespin is transversely, normally, or longitudinally polarized.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 4 / 26
Introduction Beam Single Spin Asymmetries (BSSA)
What exchanges can cause BSSA?
Consider the scattering amplitude:
|M |2 = |Mγ + MZ + Mγγ + · · · |2
= |Mγ |2 + M ∗γMZ + M ∗
γMγγ + · · ·
At the Born level, the spin orientation has no effect on |Mγ |2, so theasymmetry vanishes.
Higher order effects result in asymmetries dependent on whether thespin is transversely, normally, or longitudinally polarized.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 4 / 26
Introduction Beam Single Spin Asymmetries (BSSA)
What exchanges can cause BSSA?
Consider the scattering amplitude:
|M |2 = |Mγ + MZ + Mγγ + · · · |2
= |Mγ |2 + M ∗γMZ + M ∗
γMγγ + · · ·
At the Born level, the spin orientation has no effect on |Mγ |2, so theasymmetry vanishes.Higher order effects result in asymmetries dependent on whether thespin is transversely, normally, or longitudinally polarized.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 4 / 26
Introduction Transverse and Normal Polarizations
Transverse and Normal Polarization
Transverse asymmetry Bt behaves like cosφs and normal asymmetryBn behaves like sinφs .
Figure 2. Electron-Proton Scattering
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 5 / 26
Introduction Motivation: Q-weak Collaboration
Motivation: Q-weak Collaboration
Precise measurement of the weak charge of the proton, QPW , using
asymmetry produced by longitudinally polarized electrons.
Transversely polarized electrons: BSSA has azimuthal (φ) dependence(Buddhini Waidyawansa, CIPANP 2012 Conference).
Concerned about possible phase offsets due to unconsidered BSSAeffects.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 6 / 26
Introduction Motivation: Q-weak Collaboration
Motivation: Q-weak Collaboration
Precise measurement of the weak charge of the proton, QPW , using
asymmetry produced by longitudinally polarized electrons.Transversely polarized electrons: BSSA has azimuthal (φ) dependence(Buddhini Waidyawansa, CIPANP 2012 Conference).
Concerned about possible phase offsets due to unconsidered BSSAeffects.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 6 / 26
Introduction Motivation: Q-weak Collaboration
Motivation: Q-weak Collaboration
Precise measurement of the weak charge of the proton, QPW , using
asymmetry produced by longitudinally polarized electrons.Transversely polarized electrons: BSSA has azimuthal (φ) dependence(Buddhini Waidyawansa, CIPANP 2012 Conference).
Concerned about possible phase offsets due to unconsidered BSSAeffects.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 6 / 26
Introduction Motivation: Q-weak Collaboration
Approach
Elastic Scattering Contribution
Interference of one- and two-photon exchange.Interference of one-photon and Z exchange.
Hadronic Inelastic Intermediate StatePasquini and Vanderhaeghen (PRC 70, 2004) showed that inelastichadronic intermediate states have a larger contribution to theasymmetry than the elastic case.Consider the case of near forward limit.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 7 / 26
Introduction Motivation: Q-weak Collaboration
Approach
Elastic Scattering ContributionInterference of one- and two-photon exchange.Interference of one-photon and Z exchange.
Hadronic Inelastic Intermediate StatePasquini and Vanderhaeghen (PRC 70, 2004) showed that inelastichadronic intermediate states have a larger contribution to theasymmetry than the elastic case.Consider the case of near forward limit.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 7 / 26
Introduction Motivation: Q-weak Collaboration
Approach
Elastic Scattering ContributionInterference of one- and two-photon exchange.Interference of one-photon and Z exchange.
Hadronic Inelastic Intermediate State
Pasquini and Vanderhaeghen (PRC 70, 2004) showed that inelastichadronic intermediate states have a larger contribution to theasymmetry than the elastic case.Consider the case of near forward limit.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 7 / 26
Introduction Motivation: Q-weak Collaboration
Approach
Elastic Scattering ContributionInterference of one- and two-photon exchange.Interference of one-photon and Z exchange.
Hadronic Inelastic Intermediate StatePasquini and Vanderhaeghen (PRC 70, 2004) showed that inelastichadronic intermediate states have a larger contribution to theasymmetry than the elastic case.Consider the case of near forward limit.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 7 / 26
Elastic Scattering
Elastic Scattering
Elastic scattering for electron-proton scattering,
e(p1) + N(p2)→ e(p3) + N(p4).
�p2p1
p2 + q1
q1
p4
q2
k
p3
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 8 / 26
Elastic Scattering
Kinematic Variables
Elastic scattering process:
e(p1) + N(p2)→ e(p3) + N(p4)
Using the following notation for kinematic variables:
P = p2 + p42 , K = p1 + p3
2 , q = p1 − p3, Q2 = −q2;
s = (p1 + p2)2, τ = Q2
4M2 , ν = P · K , ε = ν2 −M4τ(1 + τ)ν2 + M4τ(1 + τ) ;
W 2 = (p2 + q1)2, Q21 = −q2
1 , Q22 = −q2
2 ,
for electron mass me and hadron mass M.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 9 / 26
Elastic Scattering
Kinematic Variables
Elastic scattering process:
e(p1) + N(p2)→ e(p3) + N(p4)
Using the following notation for kinematic variables:
P = p2 + p42 , K = p1 + p3
2 , q = p1 − p3, Q2 = −q2;
s = (p1 + p2)2, τ = Q2
4M2 , ν = P · K , ε = ν2 −M4τ(1 + τ)ν2 + M4τ(1 + τ) ;
W 2 = (p2 + q1)2, Q21 = −q2
1 , Q22 = −q2
2 ,
for electron mass me and hadron mass M.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 9 / 26
Elastic Scattering Beam Normal SSA
Beam Normal Single Spin Asymmetry: SetupUsing six invariant amplitudes given by Goldberger et al. (Ann. Phys. 2,1957), we can construct a general elastic lepton-nucleon scatteringamplitude:
T = T non-flip + T flip,
where
T non-flip = e2
Q2 ue(p3)γµue(p1) · uN(p4)(
GMγµ − F2
Pµ
M + F3/KPµ
M2
)uN(p2)
and
T flip = e2
Q2meM
[ue(p3)ue(p1) · uN(p4)
(F4 + F5
/KM
)uN(p2)
+ F6ue(p3)γ5ue(p1) · uN(p4)γ5uN(p2)].
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 10 / 26
Elastic Scattering Beam Normal SSA
Beam Normal Single Spin Asymmetry: SetupUsing six invariant amplitudes given by Goldberger et al. (Ann. Phys. 2,1957), we can construct a general elastic lepton-nucleon scatteringamplitude:
T = T non-flip + T flip,
where
T non-flip = e2
Q2 ue(p3)γµue(p1) · uN(p4)(
GMγµ − F2
Pµ
M + F3/KPµ
M2
)uN(p2)
and
T flip = e2
Q2meM
[ue(p3)ue(p1) · uN(p4)
(F4 + F5
/KM
)uN(p2)
+ F6ue(p3)γ5ue(p1) · uN(p4)γ5uN(p2)].
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 10 / 26
Elastic Scattering Beam Normal SSA
Beam Normal Single Spin Asymmetry: SetupUsing six invariant amplitudes given by Goldberger et al. (Ann. Phys. 2,1957), we can construct a general elastic lepton-nucleon scatteringamplitude:
T = T non-flip + T flip,
where
T non-flip = e2
Q2 ue(p3)γµue(p1) · uN(p4)(
GMγµ − F2
Pµ
M + F3/KPµ
M2
)uN(p2)
and
T flip = e2
Q2meM
[ue(p3)ue(p1) · uN(p4)
(F4 + F5
/KM
)uN(p2)
+ F6ue(p3)γ5ue(p1) · uN(p4)γ5uN(p2)].
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 10 / 26
Elastic Scattering Beam Normal SSA
Beam Normal Single Spin Asymmetry: Setup
In the two previous equations,
GM , F2, F3, F4, F5, F6
are complex functions of ν and Q2. In the Born approximation, thesereduce to
GBornM (ν,Q2) = GM(Q2)
F Born2 (ν,Q2) = F2(Q2)
F Born3,4,5,6(ν,Q2) = 0.
We can also writeGE = GM − (1 + τ)F2.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 11 / 26
Elastic Scattering Beam Normal SSA
Beam Normal Single Spin Asymmetry: Setup
In the two previous equations,
GM , F2, F3, F4, F5, F6
are complex functions of ν and Q2. In the Born approximation, thesereduce to
GBornM (ν,Q2) = GM(Q2)
F Born2 (ν,Q2) = F2(Q2)
F Born3,4,5,6(ν,Q2) = 0.
We can also writeGE = GM − (1 + τ)F2.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 11 / 26
Elastic Scattering Beam Normal SSA
Beam Normal Single Spin Asymmetry: Setup
In the two previous equations,
GM , F2, F3, F4, F5, F6
are complex functions of ν and Q2. In the Born approximation, thesereduce to
GBornM (ν,Q2) = GM(Q2)
F Born2 (ν,Q2) = F2(Q2)
F Born3,4,5,6(ν,Q2) = 0.
We can also writeGE = GM − (1 + τ)F2.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 11 / 26
Elastic Scattering Beam Normal SSA
Beam Normal SSA
Gorchtein et al. found that for a spin parallel (anti-parallel) to the normalpolarization vector
Sµ = (0, ~Sn), ~Sn = (~p1 × ~p3) / |~p1 × ~p3|
there is a beam normal SSA Bn due to M ∗γMγγ . It is equal to
Bn = 2meQ
√2ε(1− ε)
√1 + 1
τ
(G2
M + G2E
)−1
×{−τGM Im
(F3 + 1
1 + τ
ν
M2 F5
)− GE Im
(F4 + 1
1 + τ
ν
M2 F5
)}.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 12 / 26
Elastic Scattering Beam Normal SSA
Beam Normal SSA
Gorchtein et al. found that for a spin parallel (anti-parallel) to the normalpolarization vector
Sµ = (0, ~Sn), ~Sn = (~p1 × ~p3) / |~p1 × ~p3|
there is a beam normal SSA Bn due to M ∗γMγγ . It is equal to
Bn = 2meQ
√2ε(1− ε)
√1 + 1
τ
(G2
M + G2E
)−1
×{−τGM Im
(F3 + 1
1 + τ
ν
M2 F5
)− GE Im
(F4 + 1
1 + τ
ν
M2 F5
)}.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 12 / 26
Elastic Scattering Beam Normal SSA
Beam Normal SSA
Figure 3. Transverse Spin Polarization
If we consider a general transverse spin
~s = cosφs x + sinφs y ,
the general beam normal SSA can be written
Bn, gen = Bn sinφs .
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 13 / 26
Elastic Scattering Beam Normal SSA
Beam Normal SSA
Figure 3. Transverse Spin Polarization
If we consider a general transverse spin
~s = cosφs x + sinφs y ,
the general beam normal SSA can be written
Bn, gen = Bn sinφs .
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 13 / 26
Elastic Scattering Beam Normal SSA
Beam Normal SSA
Figure 3. Transverse Spin Polarization
If we consider a general transverse spin
~s = cosφs x + sinφs y ,
the general beam normal SSA can be written
Bn, gen = Bn sinφs .
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 13 / 26
Elastic Scattering Beam Transverse Single Spin Asymmetry
Beam Transverse SSA
We found that a beam transverse SSA exists at the Born level due toM ∗
γMZ . It is equal to
Bt = GF me2πα
Q2
(s −M2)
√ε(1− ε)τ2(τ + 1)
(G2
M + ε
τG2
E
)−1
×{
geV GMGZ
Mτ(τ + 1) + geA
(GMGZ
A τ(1 + τ − ν) + GE GZE (1− τ − ν)
)},
where GZM and GZ
E are the weak form factors and GF is Fermi’s couplingconstant.
For a general transverse spin,
Bt, gen = Bt cosφs .
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 14 / 26
Elastic Scattering Beam Transverse Single Spin Asymmetry
Beam Transverse SSA
We found that a beam transverse SSA exists at the Born level due toM ∗
γMZ . It is equal to
Bt = GF me2πα
Q2
(s −M2)
√ε(1− ε)τ2(τ + 1)
(G2
M + ε
τG2
E
)−1
×{
geV GMGZ
Mτ(τ + 1) + geA
(GMGZ
A τ(1 + τ − ν) + GE GZE (1− τ − ν)
)},
where GZM and GZ
E are the weak form factors and GF is Fermi’s couplingconstant.
For a general transverse spin,
Bt, gen = Bt cosφs .
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 14 / 26
Elastic Scattering Beam Transverse Single Spin Asymmetry
Beam Transverse SSA
We found that a beam transverse SSA exists at the Born level due toM ∗
γMZ . It is equal to
Bt = GF me2πα
Q2
(s −M2)
√ε(1− ε)τ2(τ + 1)
(G2
M + ε
τG2
E
)−1
×{
geV GMGZ
Mτ(τ + 1) + geA
(GMGZ
A τ(1 + τ − ν) + GE GZE (1− τ − ν)
)},
where GZM and GZ
E are the weak form factors and GF is Fermi’s couplingconstant.
For a general transverse spin,
Bt, gen = Bt cosφs .
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 14 / 26
Elastic Scattering Combination of Asymmetries
Combination of Asymmetries
The interference between one- and two-photon exchange onlyproduces a normal BSSA.The interference between one-photon and Z exchange only producesa transverse BSSA.
The combination of these two asymmetries gives
B = Bt cosφs + Bn sinφs
=√
B2t + B2
n sin (φs + δ)
where δ = tan−1(
BtBn
).
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 15 / 26
Elastic Scattering Combination of Asymmetries
Combination of Asymmetries
The interference between one- and two-photon exchange onlyproduces a normal BSSA.The interference between one-photon and Z exchange only producesa transverse BSSA.
The combination of these two asymmetries gives
B = Bt cosφs + Bn sinφs
=√
B2t + B2
n sin (φs + δ)
where δ = tan−1(
BtBn
).
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 15 / 26
Elastic Scattering Combination of Asymmetries
Combination of Asymmetries
The interference between one- and two-photon exchange onlyproduces a normal BSSA.The interference between one-photon and Z exchange only producesa transverse BSSA.
The combination of these two asymmetries gives
B = Bt cosφs + Bn sinφs
=√
B2t + B2
n sin (φs + δ)
where δ = tan−1(
BtBn
).
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 15 / 26
Elastic Scattering Combination of Asymmetries
Combination of Asymmetries: Q-weak Kinematics
In Q-weak, Bn was measured as
Bn ≈ −5.350 ppm.
Using Q-weak kinematics, Q2 = 0.025 GeV, E1 = 1.125 GeV (electronbeam energy), we find
|δ| =∣∣∣∣tan−1
(BtBn
)∣∣∣∣ ≈ ∣∣∣∣BtBn
∣∣∣∣ ≈ 1.116× 10−11
5.350× 10−6 = 2.086× 10−6.
→ This is too small to affect the Q-weak measurements.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 16 / 26
Elastic Scattering Combination of Asymmetries
Combination of Asymmetries: Q-weak Kinematics
In Q-weak, Bn was measured as
Bn ≈ −5.350 ppm.
Using Q-weak kinematics, Q2 = 0.025 GeV, E1 = 1.125 GeV (electronbeam energy), we find
|δ| =∣∣∣∣tan−1
(BtBn
)∣∣∣∣ ≈ ∣∣∣∣BtBn
∣∣∣∣ ≈ 1.116× 10−11
5.350× 10−6 = 2.086× 10−6.
→ This is too small to affect the Q-weak measurements.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 16 / 26
Elastic Scattering Combination of Asymmetries
Combination of Asymmetries: Q-weak Kinematics
In Q-weak, Bn was measured as
Bn ≈ −5.350 ppm.
Using Q-weak kinematics, Q2 = 0.025 GeV, E1 = 1.125 GeV (electronbeam energy), we find
|δ| =∣∣∣∣tan−1
(BtBn
)∣∣∣∣ ≈ ∣∣∣∣BtBn
∣∣∣∣ ≈ 1.116× 10−11
5.350× 10−6 = 2.086× 10−6.
→ This is too small to affect the Q-weak measurements.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 16 / 26
Elastic Scattering Combination of Asymmetries
Combination of Asymmetries: Q-weak Kinematics
In Q-weak, Bn was measured as
Bn ≈ −5.350 ppm.
Using Q-weak kinematics, Q2 = 0.025 GeV, E1 = 1.125 GeV (electronbeam energy), we find
|δ| =∣∣∣∣tan−1
(BtBn
)∣∣∣∣ ≈ ∣∣∣∣BtBn
∣∣∣∣ ≈ 1.116× 10−11
5.350× 10−6 = 2.086× 10−6.
→ This is too small to affect the Q-weak measurements.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 16 / 26
Elastic Scattering Combination of Asymmetries
Order of Magnitude Estimates
Beam Normal SSA:
Bn ∼αmeM ∼ 5× 10−6 → 5 ppm
APV (from longitudinal polarization):
APV ∼Q2
M2Z∼ Q2 × 10−4
Beam Transverse SSA:
Bt ∼Q2
M2Z
meM ∼ Q2 ×
(5× 10−8
)
What factors contribute to Bn?
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 17 / 26
Elastic Scattering Combination of Asymmetries
Order of Magnitude Estimates
Beam Normal SSA:
Bn ∼αmeM ∼ 5× 10−6 → 5 ppm
APV (from longitudinal polarization):
APV ∼Q2
M2Z∼ Q2 × 10−4
Beam Transverse SSA:
Bt ∼Q2
M2Z
meM ∼ Q2 ×
(5× 10−8
)
What factors contribute to Bn?
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 17 / 26
Elastic Scattering Combination of Asymmetries
Order of Magnitude Estimates
Beam Normal SSA:
Bn ∼αmeM ∼ 5× 10−6 → 5 ppm
APV (from longitudinal polarization):
APV ∼Q2
M2Z∼ Q2 × 10−4
Beam Transverse SSA:
Bt ∼Q2
M2Z
meM ∼ Q2 ×
(5× 10−8
)
What factors contribute to Bn?
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 17 / 26
Elastic Scattering Combination of Asymmetries
Order of Magnitude Estimates
Beam Normal SSA:
Bn ∼αmeM ∼ 5× 10−6 → 5 ppm
APV (from longitudinal polarization):
APV ∼Q2
M2Z∼ Q2 × 10−4
Beam Transverse SSA:
Bt ∼Q2
M2Z
meM ∼ Q2 ×
(5× 10−8
)
What factors contribute to Bn?Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 17 / 26
Inelastic Hadronic Intermediate State
Inelastic Hadronic Intermediate StateScattering process:
e(p1) + N(p2)→ e(p3) + N(p4)
with intermediate leptonic momentum k and intermediate hadronicmomentum W = p2 + q1.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 18 / 26
Inelastic Hadronic Intermediate State M∗γ Mγγ Interference
One- and Two- Photon Interference
We consider the contribution to the normal BSSA from M ∗γMγγ .
For polarized beam, spinor relation becomes
ue(k)ue(k) = (1 + γ5/s)(/k + me).
The BSSA comes from the absorptive part of the two-photonexchange amplitude (Pasquini & Vanderhaeghen PRC 70, 2004):
Bn =
2 Im
∑spins
M ∗γ · Abs Mγγ
∑spins|Mγ |2
.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 19 / 26
Inelastic Hadronic Intermediate State M∗γ Mγγ Interference
One- and Two- Photon Interference
We consider the contribution to the normal BSSA from M ∗γMγγ .
For polarized beam, spinor relation becomes
ue(k)ue(k) = (1 + γ5/s)(/k + me).
The BSSA comes from the absorptive part of the two-photonexchange amplitude (Pasquini & Vanderhaeghen PRC 70, 2004):
Bn =
2 Im
∑spins
M ∗γ · Abs Mγγ
∑spins|Mγ |2
.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 19 / 26
Inelastic Hadronic Intermediate State M∗γ Mγγ Interference
One- and Two- Photon Interference
We consider the contribution to the normal BSSA from M ∗γMγγ .
For polarized beam, spinor relation becomes
ue(k)ue(k) = (1 + γ5/s)(/k + me).
The BSSA comes from the absorptive part of the two-photonexchange amplitude (Pasquini & Vanderhaeghen PRC 70, 2004):
Bn =
2 Im
∑spins
M ∗γ · Abs Mγγ
∑spins|Mγ |2
.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 19 / 26
Inelastic Hadronic Intermediate State Leptonic and Hadronic Tensors
Leptonic and Hadronic TensorsWe can write this as
Bn = e6
(2π)3Q2
(∑spins|Mγ |2
)−1 ∫ d3~kEk
1Q2
1Q22
Im {LαµνHαµν} ,
where
Lαµν = Tr[1
2(1 + γ5/s)( /p1 + me)γα( /p1 − /q + me)γµ( /p1 − /q1 + me)γν].
and we take the forward limit on the hadronic tensor:
Hαµν = 2pα2 W µν
where (de Rujula et al. Nucl. Phys. B35, 1971)
W µν =(−gµν + qµ1 qν1
q21
)W1+ 1
M2
(pµ2 −
p2 · q1q2
1qµ1
)(pν2 −
p2 · q1q2
1qν1
)W2.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 20 / 26
Inelastic Hadronic Intermediate State Leptonic and Hadronic Tensors
Leptonic and Hadronic TensorsWe can write this as
Bn = e6
(2π)3Q2
(∑spins|Mγ |2
)−1 ∫ d3~kEk
1Q2
1Q22
Im {LαµνHαµν} ,
where
Lαµν = Tr[1
2(1 + γ5/s)( /p1 + me)γα( /p1 − /q + me)γµ( /p1 − /q1 + me)γν].
and we take the forward limit on the hadronic tensor:
Hαµν = 2pα2 W µν
where (de Rujula et al. Nucl. Phys. B35, 1971)
W µν =(−gµν + qµ1 qν1
q21
)W1+ 1
M2
(pµ2 −
p2 · q1q2
1qµ1
)(pν2 −
p2 · q1q2
1qν1
)W2.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 20 / 26
Inelastic Hadronic Intermediate State Leptonic and Hadronic Tensors
Leptonic and Hadronic TensorsWe can write this as
Bn = e6
(2π)3Q2
(∑spins|Mγ |2
)−1 ∫ d3~kEk
1Q2
1Q22
Im {LαµνHαµν} ,
where
Lαµν = Tr[1
2(1 + γ5/s)( /p1 + me)γα( /p1 − /q + me)γµ( /p1 − /q1 + me)γν].
and we take the forward limit on the hadronic tensor:
Hαµν = 2pα2 W µν
where (de Rujula et al. Nucl. Phys. B35, 1971)
W µν =(−gµν + qµ1 qν1
q21
)W1+ 1
M2
(pµ2 −
p2 · q1q2
1qµ1
)(pν2 −
p2 · q1q2
1qν1
)W2.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 20 / 26
Inelastic Hadronic Intermediate State Leptonic and Hadronic Tensors
Leptonic and Hadronic TensorsWe can write this as
Bn = e6
(2π)3Q2
(∑spins|Mγ |2
)−1 ∫ d3~kEk
1Q2
1Q22
Im {LαµνHαµν} ,
where
Lαµν = Tr[1
2(1 + γ5/s)( /p1 + me)γα( /p1 − /q + me)γµ( /p1 − /q1 + me)γν].
and we take the forward limit on the hadronic tensor:
Hαµν = 2pα2 W µν
where (de Rujula et al. Nucl. Phys. B35, 1971)
W µν =(−gµν + qµ1 qν1
q21
)W1+ 1
M2
(pµ2 −
p2 · q1q2
1qµ1
)(pν2 −
p2 · q1q2
1qν1
)W2.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 20 / 26
Inelastic Hadronic Intermediate State Normal BSSA
Finally, we can write the asymmetry as
Bn = − 1(2π)3
e2Q2
D(s,Q2)1
8E1E3M
∫ 2π
0dφk
∫ s
M2dW 2
∫ Q21,max
0dQ2
1
× Im {LαµνHαµν}Q2
1 |~k| (1− cos θ cos θk − sin θ sin θk cosφk)
where E1 and E3 are the energies of the incoming and outgoing leptons, θkis the angle between ~p1 and ~k, φk is the azimuthal angle, and
D(s,Q2) = 8Q4
1− ε
{G2
M + ε
τG2
E
},
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 21 / 26
Inelastic Hadronic Intermediate State Normal BSSA
Finally, we can write the asymmetry as
Bn = − 1(2π)3
e2Q2
D(s,Q2)1
8E1E3M
∫ 2π
0dφk
∫ s
M2dW 2
∫ Q21,max
0dQ2
1
× Im {LαµνHαµν}Q2
1 |~k| (1− cos θ cos θk − sin θ sin θk cosφk)
where E1 and E3 are the energies of the incoming and outgoing leptons, θkis the angle between ~p1 and ~k, φk is the azimuthal angle, and
D(s,Q2) = 8Q4
1− ε
{G2
M + ε
τG2
E
},
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 21 / 26
Inelastic Hadronic Intermediate State Normal BSSA
Finally, we can write the asymmetry as
Bn = − 1(2π)3
e2Q2
D(s,Q2)1
8E1E3M
∫ 2π
0dφk
∫ s
M2dW 2
∫ Q21,max
0dQ2
1
× Im {LαµνHαµν}Q2
1 |~k| (1− cos θ cos θk − sin θ sin θk cosφk)
where E1 and E3 are the energies of the incoming and outgoing leptons, θkis the angle between ~p1 and ~k, φk is the azimuthal angle, and
D(s,Q2) = 8Q4
1− ε
{G2
M + ε
τG2
E
},
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 21 / 26
Inelastic Hadronic Intermediate State Normal BSSA
Once again,
Bn = − 1(2π)3
e2Q2
D(s,Q2)1
8E1E3M
∫ 2π
0dφk
∫ s
M2dW 2
∫ Q21,max
0dQ2
1
× Im {LαµνHαµν}Q2
1 |~k| (1− cos θ cos θk − sin θ sin θk cosφk)
The tensor contraction is
Im LαµνHαµν = 8meM2Q2
1
[2W1M2Q2
1 (εp1p2qs + εp2qq1s)
+W2εp2qq1s
((M2 − Q2
1 −W 2)(p1 · p2)−M2Q21
)].
→ Currently evaluating for Q-weak kinematics.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 22 / 26
Inelastic Hadronic Intermediate State Normal BSSA
Once again,
Bn = − 1(2π)3
e2Q2
D(s,Q2)1
8E1E3M
∫ 2π
0dφk
∫ s
M2dW 2
∫ Q21,max
0dQ2
1
× Im {LαµνHαµν}Q2
1 |~k| (1− cos θ cos θk − sin θ sin θk cosφk)
The tensor contraction is
Im LαµνHαµν = 8meM2Q2
1
[2W1M2Q2
1 (εp1p2qs + εp2qq1s)
+W2εp2qq1s
((M2 − Q2
1 −W 2)(p1 · p2)−M2Q21
)].
→ Currently evaluating for Q-weak kinematics.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 22 / 26
Inelastic Hadronic Intermediate State Normal BSSA
Once again,
Bn = − 1(2π)3
e2Q2
D(s,Q2)1
8E1E3M
∫ 2π
0dφk
∫ s
M2dW 2
∫ Q21,max
0dQ2
1
× Im {LαµνHαµν}Q2
1 |~k| (1− cos θ cos θk − sin θ sin θk cosφk)
The tensor contraction is
Im LαµνHαµν = 8meM2Q2
1
[2W1M2Q2
1 (εp1p2qs + εp2qq1s)
+W2εp2qq1s
((M2 − Q2
1 −W 2)(p1 · p2)−M2Q21
)].
→ Currently evaluating for Q-weak kinematics.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 22 / 26
Conclusion
Conclusion
1 Beam single spin asymmetries result from the difference in crosssections produced by the beam polarization in electron-protonscattering experiments.
2 The interference between one- and two-photon exchange amplitudesproduces a beam normal single spin asymmetry.
3 The interference between one-photon and Z exchange amplitudesproduces a beam transverse single spin asymmetry.
4 The combination of these two asymmetries produces a small phaseshift that may be detectable in the future.
5 The inelastic hadronic intermediate state produces an importantcontribution to the BSSA (soon to be quantified).
6 The framework developed here could potentially be used to computeother beam or target SSA at near-forward angles.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 23 / 26
Conclusion
Conclusion
1 Beam single spin asymmetries result from the difference in crosssections produced by the beam polarization in electron-protonscattering experiments.
2 The interference between one- and two-photon exchange amplitudesproduces a beam normal single spin asymmetry.
3 The interference between one-photon and Z exchange amplitudesproduces a beam transverse single spin asymmetry.
4 The combination of these two asymmetries produces a small phaseshift that may be detectable in the future.
5 The inelastic hadronic intermediate state produces an importantcontribution to the BSSA (soon to be quantified).
6 The framework developed here could potentially be used to computeother beam or target SSA at near-forward angles.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 23 / 26
Conclusion
Conclusion
1 Beam single spin asymmetries result from the difference in crosssections produced by the beam polarization in electron-protonscattering experiments.
2 The interference between one- and two-photon exchange amplitudesproduces a beam normal single spin asymmetry.
3 The interference between one-photon and Z exchange amplitudesproduces a beam transverse single spin asymmetry.
4 The combination of these two asymmetries produces a small phaseshift that may be detectable in the future.
5 The inelastic hadronic intermediate state produces an importantcontribution to the BSSA (soon to be quantified).
6 The framework developed here could potentially be used to computeother beam or target SSA at near-forward angles.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 23 / 26
Conclusion
Conclusion
1 Beam single spin asymmetries result from the difference in crosssections produced by the beam polarization in electron-protonscattering experiments.
2 The interference between one- and two-photon exchange amplitudesproduces a beam normal single spin asymmetry.
3 The interference between one-photon and Z exchange amplitudesproduces a beam transverse single spin asymmetry.
4 The combination of these two asymmetries produces a small phaseshift that may be detectable in the future.
5 The inelastic hadronic intermediate state produces an importantcontribution to the BSSA (soon to be quantified).
6 The framework developed here could potentially be used to computeother beam or target SSA at near-forward angles.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 23 / 26
Conclusion
Conclusion
1 Beam single spin asymmetries result from the difference in crosssections produced by the beam polarization in electron-protonscattering experiments.
2 The interference between one- and two-photon exchange amplitudesproduces a beam normal single spin asymmetry.
3 The interference between one-photon and Z exchange amplitudesproduces a beam transverse single spin asymmetry.
4 The combination of these two asymmetries produces a small phaseshift that may be detectable in the future.
5 The inelastic hadronic intermediate state produces an importantcontribution to the BSSA (soon to be quantified).
6 The framework developed here could potentially be used to computeother beam or target SSA at near-forward angles.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 23 / 26
Conclusion
Conclusion
1 Beam single spin asymmetries result from the difference in crosssections produced by the beam polarization in electron-protonscattering experiments.
2 The interference between one- and two-photon exchange amplitudesproduces a beam normal single spin asymmetry.
3 The interference between one-photon and Z exchange amplitudesproduces a beam transverse single spin asymmetry.
4 The combination of these two asymmetries produces a small phaseshift that may be detectable in the future.
5 The inelastic hadronic intermediate state produces an importantcontribution to the BSSA (soon to be quantified).
6 The framework developed here could potentially be used to computeother beam or target SSA at near-forward angles.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 23 / 26
Conclusion
References
1 Waidyawansa, B. P. (2013). A 3% Measurement of the Beam NormalSingle Spin Asymmetry in Forward Angle Elastic Electron-ProtonScattering using the Qweak Setup. Ohio University: U.S.A.
2 B. Pasquini and M. Vanderhaeghen, Physical Review C 70, 045206(2004).
3 M. Gorchtein et al., Nucl. Phys. A741, 234 (2004).4 M.L. Goldberger, Y. Nambu, R. Oehme, Ann. Phys. 2 (1957) 226.5 A. De Rujula, J. M. Kaplan, and E. de Rafael, Nucl. Phys. B35, 365
(1971).
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 24 / 26
Conclusion
Acknowledgements
I would like to thank my mentor, Dr. Wally Melnitchouk, for his guidance,instruction, and patience. I would also like to thank Dr. Peter Blunden forhis aid in completing the calculations for this project.
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 25 / 26
Conclusion
Questions?
Pratik Sachdeva Beam Single Spin Asymmetries in Electron-Proton Scattering 26 / 26