View source for ISU Coloq 11-3-2014

==Abstract==

ISU's intermediate energy nuclear physics group is presently involved in several fundamental physics measurements. This talk will describe a  program to measure the fractional polarization of down quarks in a nucleon using polarized electrons to probe polarized nucleon targets.  Quantum chromodynamcs (QCD) is a theory of the strong interaction; one of the four fundamental forces in nature.  QCD predicts that the down quark will carry all of the nucleon's spin and result in a fractional polarization of unity when the probe interacts with down quarks that carry all of the nucleon's momentum.  This theory contradicts the leading constituent quark model of the nucleon.  The world's current data set is unable to discriminate between QCD's prediction and the constituent quark model.    A description of this experimental program and the roles of ISU graduate students will be described.

==electron scattering (collisions) ==

Elastic Collisions: Conserve P and E

Inelastic : Only Conserve P

==Four Momentum==

Using Ryder 4-momentum tensor convention:  <math>P_{\mu} \equiv (E/c, -\vec{p}) \;\;\;\;\;  P^{\mu} \equiv (E/c, \vec{p})</math>

If I take the inner product of a particles four momentum with itself I get
<math>P_{\mu} P^{\mu} \equiv \left | \vec P \right |^2 = \frac{E}{c} - \vec{p} \cdot \vec{p} </math>

==Definition of Momentum Transfer==



[[File:TF_Elastic_e-p_Feynman.png | 200 px]]


<math>q_\mu \equiv (\omega, \vec{q})</math>
<math>k^i_\mu \equiv (E_i, \vec{k_i})</math>
<math>k^f_\mu \equiv (E_f, \vec{k_f})</math>

;Conservation of Momentum:<math> k^i_\mu =q_\mu +  k^f_\mu </math>

;Momentum Transfer :<math> q_\mu =k^i_\mu - k^f_\mu </math>

;Momentum Transfer Squared:<math> -Q^2 = q^2 = q_\mu q^{\mu} =(E_i-E_f)^2 - (\vec{k}_i - \vec{k}_f) \cdot (\vec{k}_i - \vec{k}_f) </math>
:<math>= m_i^2 + m_f^2 -2E_iE_f + 2\left | \vec{k}_i\right | \left | \vec{k}_f\right | cos(\theta)</math>
:<math>=-4E_iE_f \sin^2(\theta/2) \;\;\; m_i = m_f \ll E_i \mbox{ and } E \approx \left | \vec{k}_i\right |</math>

:<math>Q^2 =4E_iE_f \sin^2(\theta/2) > 0</math> 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<math>^2 = (\Delta r)^2 - (c\Delta t)^2</math> then <math>s^2 > 0</math> 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<math>^2 = (\Delta r)^2 - (c\Delta t)^2</math> then <math>s^2 < 0</math> for a time-like interval


4-Momentum vector definition using Ryder convention

:<math>P_{\mu} \equiv \left ( \frac{E}{c} , - \vec p \right )</math>
:<math>P^{\mu} \equiv \left ( \frac{E}{c} , \vec p \right )</math>

:<math>P_{\mu} P^{\mu} =  \left ( \frac{E}{c}\right )^2 - \vec p^2 = E^2-p^2 = m^2</math>  if you define the speed of light as unity


;Note: Other conventions used by Perkins 

:<math>P_{\mu} \equiv \left (  \vec p, -E  \right )</math>
:<math>P^{\mu} \equiv \left ( \vec p , E\right )</math>

or Kollen

:<math>P_{\mu} \equiv \left (  \vec p, iE  \right )</math>
:<math>P^{\mu} \equiv \left ( \vec p , iE\right )</math>


Momentum transfer is defined as 

:<math>q_{\mu} \equiv ( \omega, \vec q) = P^i_{\mu} - P^f_{\mu}</math>  : conservation of momentum 

: <math>q_{\mu}q^{\mu} = (E_i - E_f)^2 - (\vec {P}_i - \vec{P}_f) \cdot  (\vec {P}_i - \vec{P}_f)</math> 
:: <math>= m_i^2 +m_f^2 - 2E_iE_f + 2 \left | \vec {P}_i  \right | \left | \vec {P}_f  \right |</math>

== Definition of Missing Mass==


Inelastic scattering (Energy is not conserved but absorbed from the momentum transfer)


<math>P_e^{\mu} \equiv (E_i,\vec{k}_i)</math>
<math>\left(P_e^{\mu}\right)^{\prime} \equiv (E_f,\vec{k}_f)</math>
<math>P_p^{\mu} \equiv (M_p,0)</math>
<math>\left(P_p^{\mu}\right)^{\prime} \equiv (E_X,\vec{k}_X)</math>

;4-momentum conservation:<math>P_e^{\mu} + P_p^{\mu}  = \left(P_e^{\mu}\right)^{\prime}  + \left(P_p^{\mu}\right)^{\prime}</math>


<math>\left(P_p^{\mu}\right)^{\prime}  \left(P_{\mu}^p\right)^{\prime} = [P_e^{\mu} +  P_p^{\mu}  - \left(P_e^{\mu}\right)^{\prime}  ][P^e_{\mu} +  P^p_{\mu}  - \left(P^e_{\mu}\right)^{\prime}  ]</math>

<math>E_X^2 - P_X^2 \equiv W^2 = (E_i^2 - k_i^2) + (E_f^2-k_f^2) + M_p^2 + 2M_p(E_i - E_f) -2(E_iE_f-\vec{k_i}\cdot \vec{k_f})</math>

:<math>W^2=M_p^2 + 2M_p(E_i-E_f) -Q^2 \equiv </math>Invariant Missing Mass = mass of the intermediate state that was created.


[[File:CLAS_InelasticSpectrum_2012.png | 200 px]]

===The Delta Resonance===

[[File:epDeltaFeynman2012.png | 200 px]]

Decay Modes:  <math>ep \rightarrow \;\; \left \{ {N \pi \;\;\; 99.5% \atop p \gamma \;\;\; 0.5% } \right .</math>


<math>\sigma = <\Delta^+ | V_{\mu}^3| p></math>

:<math>V_{\mu}^3 =</math> Vector current of the virtual photon exchange

[[File:01092012_epnipiTrack.png| 200 px]][[File:Cal01092012_epnipiCal.png| 200 px]]

== Spin 1/2 and 3/2 final states==

[[File:DoubleSpinOneHalfScattering.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==
Past colloquia

[[TF_01092012_Colloq]]

===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===


[http://pos.sissa.it/archive/conferences/160/044/Bormio2012_044.pdf  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

[http://meetings.aps.org/Meeting/APR13/Event/193112 2013 APS talk on EG1-DVCS]

July 30,2014 talk by XingLong Li at Heremes
[[File:XingLongLi_TalkOnPACIAEmdoelForSIDIS.pdf]]
from 

http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=31&ved=0CB0QFjAAOB4&url=http%3A%2F%2Fhadron2014.csp.escience.cn%2Fdct%2Fattach%2FY2xiOmNsYjpwZGY6MzAwMjA%3D&ei=UPw1VNiQAdHlsATA7IHwDw&usg=AFQjCNH5yyek_zTJ22TZq0Dw_YLUeAxe2Q


[https://www.jlab.org/exp_prog/proposals/07/LOI12-07-103.pdf Xiangs proposal for CLAS12 measurement of SIDIS Xsections  see Fig. 1]



[http://arxiv.org/abs/hep-ph/0611265  Semi-inclusive deep inelastic scattering at small transverse momentum]


[http://www.tunl.duke.edu/nnpss/lectures/11/NNPSS11-2-nobuilds.pdf  Naomi's SIDIS Hermes talk from 2011 at NNPSS11]


[http://link.springer.com/article/10.1007%2FBF01552496 1986 article ob QCD effects in semi-inclusive deep inelastic scattering from a polarized target by P. Chiappetta J. -Ph. Guillet ]

==theses==

[https://www.ideals.illinois.edu/handle/2142/46764 2014 Ph.D thesis of  Naomi's student Sylvester Joosten]

http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=7&cad=rja&uact=8&ved=0CFYQFjAG&url=http%3A%2F%2Fwww.nikhef.nl%2Fpub%2Fservices%2Fbiblio%2Ftheses_pdf%2Fthesis_E_Garutti.pdf&ei=Osw3VJbSAY27ogSx3IDwDQ&usg=AFQjCNENlNdgPr6dgfq9OanlL8cUnYHmMQ&bvm=bv.77161500,d.cGU


[[TF_SIDIS_Physics]]