Difference between revisions of "ISU Coloq 11-3-2014"
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[[File:DoubleSpinOneHalfScattering.jpeg | 200 px]] | [[File:DoubleSpinOneHalfScattering.jpeg | 200 px]] | ||
[[File:DoubleSpinThreeHalfsScattering.jpeg | 200 px]] | [[File:DoubleSpinThreeHalfsScattering.jpeg | 200 px]] | ||
+ | |||
+ | |||
+ | ==Double Spin Asymmetry== | ||
+ | |||
+ | If both the target and the incident virtual photon are polarized then one can measure the cross-section for two separate conditions; when the spins are parallel and anti-parallel. | ||
+ | |||
+ | |||
+ | Let | ||
+ | <math>\sigma_{3/2} </math> | ||
+ | |||
+ | |||
+ | :<math>\sigma_{1/2} \equiv</math> photon helicity is anti-parallel to the target spin | ||
+ | |||
+ | :<math>\sigma_{3/2} \equiv</math> photon helicity is parallel to the target spin. | ||
+ | |||
+ | |||
+ | {|border="5" | ||
+ | ![[File:TF_SigmaOneHalf_2012.png | 200 px]] || [[File:TF_Sigma3Half.png | 200 px]] | ||
+ | |- | ||
+ | |<math>\sigma_{\frac{1}{2}}</math>(Struck quark spin is <math>\parallel</math> to Nucleon spin) ||<math>\sigma_{\frac{3}{2}}</math> (Struck quark spin is <math>\not\parallel</math> to Nucleon spin) | ||
+ | |}<br> | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | ===Clebsch Gordan recoupling=== | ||
+ | The recoupling of two subsystems <math>\psi</math> with angular momenta <math>j_1</math> and <math>j_2</math> to a new system<math> \Psi</math> with total angular momentum <math>J</math> is written as | ||
+ | |||
+ | <math>\Psi^{J}_{M} = \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2} C^{j_1\;,\;j_2,\;\;J}_{m_1,m_2,M} \; \psi^{j_1}_{m_1} \psi^{j_2}_{m_2}</math> = expansion of the systems total angular momentum in terms of the uncoupled original basis states of each individual constituent | ||
+ | |||
+ | |||
+ | <math>\Psi^{3/2}_{3/2} = \sum_{m_1,m_2} C^{1,1/2,3/2}_{m_1,m_2,3/2} \psi^{1}_{m_1} \psi^{1/2}_{m_2} = C^{1,\frac{1}{2},\frac{3}{2}}_{1,\frac{1}{2},\frac{3}{2}}\psi^{1}_{1} \psi^{1/2}_{1/2} </math> : all other possible m_1 and m_2 values don't add to M | ||
+ | |||
+ | <math>\Psi^{3/2}_{1/2} = \sum_{m_1,m_2} C^{1,1/2,3/2}_{m_1,m_2,1/2} \psi^{1}_{m_1} \psi^{1/2}_{m_2} = C^{1,\frac{1}{2},\frac{3}{2}}_{0,\frac{1}{2},\frac{1}{2}}\psi^{1}_{0} \psi^{1/2}_{1/2} + C^{1,\frac{1}{2},\frac{3}{2}}_{1,\frac{-1}{2},\frac{1}{2}}\psi^{1}_{1} \psi^{1/2}_{-1/2}</math> : all other possible m_1 and m_2 values don't add to M | ||
+ | |||
+ | If I constrain the helicity of my virtual photon to be +1 by preparing electrons with spins along their direction of motion, then the state <math>\psi^{1}_{0}</math> doesn't exist so only only one state enters the sum | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | [[File:TF_FlavorTagging_Borrowed.png | 200 px]] | ||
+ | |||
+ | <math>\Psi^{3/2}_{1/2} = C^{1,\frac{1}{2},\frac{3}{2}}_{1,\frac{-1}{2},\frac{1}{2}}\psi^{1}_{1} \psi^{1/2}_{-1/2}</math> : all other possible m_1 and m_2 values don't add to M | ||
+ | |||
+ | <math>C^{j_1,j_2,J}_{m_1,m_2,M}</math> : Clebsch-Gordon Coefficient | ||
+ | <math>C^{1,\frac{1}{2},\frac{3}{2}}_{1,\frac{1}{2},\frac{3}{2}}=1</math> | ||
+ | <math>C^{1,\;\;\frac{1}{2},\frac{3}{2}}_{1,-\frac{1}{2},\frac{1}{2}}= \frac{1}{\sqrt{3}}</math> | ||
+ | |||
+ | <math>A = \frac{\sigma_{\frac{1}{2}} - \sigma_{\frac{3}{2}}}{\sigma_{\frac{1}{2}} + \sigma_{\frac{3}{2}}} = \frac{\frac{1}{3} - 1}{\frac{1}{3} + 1} = -1/2</math> | ||
+ | |||
+ | ===Quark Distributions=== | ||
+ | |||
+ | |||
+ | : | ||
+ | :<math>\sigma_p^{\pi^+ \pm \pi^-} = \frac{1}{9}[4( u + \bar{u}) \pm ( d + \bar{d})]D_u^{\pi^+ \pm \pi^-}</math> | ||
+ | |||
+ | :<math>\sigma_n^{\pi^+ \pm \pi^-} = \frac{1}{9}[4(d + \bar{d}) \pm (u + \bar{u})]D_u^{\pi^+ \pm \pi^-}</math> | ||
+ | |||
+ | :<math>\sigma_{2H}^{\pi^+ \pm \pi^-} = \frac{5}{9}[( u + \bar{u}) \pm ( d + \bar{d})]D_u^{\pi^+ \pm \pi^-}</math> | ||
+ | |||
+ | |||
+ | The polarized distributions may be expressed in a similar fashion: | ||
+ | |||
+ | |||
+ | |||
+ | {|border="5" | ||
+ | ![[File:TF_SigmaOneHalf_2012.png | 200 px]] || [[File:TF_Sigma3Half.png | 200 px]] | ||
+ | |- | ||
+ | |<math>\sigma_{\frac{1}{2}}</math>(Struck quark spin is <math>\parallel</math> to Nucleon spin) ||<math>\sigma_{\frac{3}{2}}</math> (Struck quark spin is <math>\not\parallel</math> to Nucleon spin) | ||
+ | |}<br> | ||
+ | |||
+ | |||
+ | <math>\frac{d^3 \sigma^h_{1/2}}{dxdQ^2 dz}\approx \Sigma_q e_q^2 q^{\parallel}(x,Q^2)D_q^h(z,Q^2)</math> | ||
+ | |||
+ | <math>\frac{d^3 \sigma^h_{3/2}}{dxdQ^2 dz}\approx \Sigma_q e_q^2 q^{\not\parallel}(x,Q^2)D_q^h(z,Q^2)</math> | ||
+ | |||
+ | :<math>\frac{d^3 \sigma^h_{1/2(3/2)}}{dxdQ^2 dz}\approx \Sigma_q e_q^2 q^{+(-)}(x,Q^2)D_q^h(z,Q^2)</math> | ||
+ | |||
+ | |||
+ | <math>\Delta \sigma_p^{\pi^+ \pm \pi^-} = \frac{1}{9}[4(\Delta u + \Delta \bar{u}) \pm (\Delta d + \Delta \bar{d})]D_u^{\pi^+ \pm \pi^-}</math> | ||
+ | |||
+ | :<math>\Delta \sigma_n^{\pi^+ \pm \pi^-} = \frac{1}{9}[4(\Delta d + \Delta d^-) \pm (\Delta u + \Delta u^-)]D_u^{\pi^+ \pm \pi^-}</math> | ||
+ | |||
+ | :<math>\Delta \sigma_{2H}^{\pi^+ \pm \pi^-} = \frac{5}{9}[(\Delta u + \Delta \bar{u}) \pm (\Delta d + \Delta \bar{d})]D_u^{\pi^+ \pm \pi^-}</math> | ||
+ | |||
+ | ===Polarized QUark distributions=== | ||
+ | |||
+ | === Delta d-over d=== | ||
+ | |||
+ | |||
+ | {| border="0" style="background:transparent;" align="center" | ||
+ | |- | ||
+ | | | ||
+ | [[File:statistical_estimate_of_deltad_overd_comparisontodata.jpg|400px|thumb|<math>X_{Bj}</math> vs <math>\frac{\Delta d_v}{d_v}</math>]] | ||
+ | |}<br> | ||
== References== | == References== |
Revision as of 18:31, 11 October 2014
electron scattering (collisions)
Elastic Collisions: Conserve P and E
Inelastic : Only Conserve P
Definition of Momentum Transfer
Using Ryder 4-momentum tensor convention:
- Conservation of Momentum
- Momentum Transfer
- Momentum Transfer Squared
- A space-like scattering event
Space-like interval: Two events are separated by a space like interval then there isn't enough time passing between them to allow a cause-effect relationship because a photon can't traverse the distance. This means that there is no reference frame that may be used to describe the event as happening at the same spatial location but there is a frame that describes them happening at the same time. If the spacetime interval between the two events is defined as s
then for a space-like interval
Time-like interval: Two events are separated by a time like interval if enough time passes between them to allow a cause-effect relationship. This means that there is no reference frame that may be used to describe the event as happening at the same time but there is one that describes them happening at the same spatial location. If the spacetime interval between the two events is defined as s then for a time-like interval
4-Momentum vector definition using Ryder convention
- if you define the speed of light as unity
- Note
- Other conventions used by Perkins
or Kollen
Momentum transfer is defined as
- : conservation of momentum
Definition of Missing Mass
Inelastic scattering (Energy is not conserved but absorbed from the momentum transfer)
- 4-momentum conservation
- Invariant Missing Mass = mass of the intermediate state that was created.
The Delta Resonance
Decay Modes:
- Vector current of the virtual photon exchange
Spin 1/2 and 3/2 final states
Double Spin Asymmetry
If both the target and the incident virtual photon are polarized then one can measure the cross-section for two separate conditions; when the spins are parallel and anti-parallel.
Let
- photon helicity is anti-parallel to the target spin
- photon helicity is parallel to the target spin.
![]() |
![]() |
---|---|
(Struck quark spin is to Nucleon spin) | (Struck quark spin is to Nucleon spin) |
Clebsch Gordan recoupling
The recoupling of two subsystems
with angular momenta and to a new system with total angular momentum is written as= expansion of the systems total angular momentum in terms of the uncoupled original basis states of each individual constituent
: all other possible m_1 and m_2 values don't add to M
: all other possible m_1 and m_2 values don't add to M
If I constrain the helicity of my virtual photon to be +1 by preparing electrons with spins along their direction of motion, then the state
doesn't exist so only only one state enters the sum
: all other possible m_1 and m_2 values don't add to M
: Clebsch-Gordon Coefficient
Quark Distributions
The polarized distributions may be expressed in a similar fashion:
![]() |
![]() |
---|---|
(Struck quark spin is to Nucleon spin) | (Struck quark spin is to Nucleon spin) |
Polarized QUark distributions
Delta d-over d
References
Past colloquia
Theory
Phenomenological
NP, B291(1987)793; NP, B346(1990)1;
Z. Phys. C56(1992)493; Eur. Phys. J. C44 (2005)219; hep-ph/0205123 arXiV:1310.5285
QCD inspired
NP, B483(1997)291; NP, B484(1997)265;
PRL 85(2000)3591; PRL 89(2002)162301; JHEP 0211(2002)44; NP, A720(2003)131; Eur. Phys. J. C30(2003)213; arXiV:09073534; NP, A761(2005)67; PR, C81(2010)024902
Hybrids
PYTHIA + BUU simulation
PR, C70(2004)054609; NP, A801(2008)68
Experiment
keith's 2012 talk in Italy on EG1-DVCS
Unpolarized
http://link.springer.com/article/10.1007%2FJHEP04%282014%29005
File:AndyMiller 200TalkAtDESY.pdf 2002 talk by CA Miller at DESY
File:Dueren.98.055.pdf 1998 Duren talk
July 30,2014 talk by XingLong Li at Heremes File:XingLongLi TalkOnPACIAEmdoelForSIDIS.pdf from
Xiangs proposal for CLAS12 measurement of SIDIS Xsections see Fig. 1
Semi-inclusive deep inelastic scattering at small transverse momentum
Naomi's SIDIS Hermes talk from 2011 at NNPSS11