Input tracer package

\(\(\(Needs["Tracer`", 
      ToString[ToFileName[{$TopDirectory, "Addons", "Applications", "dirac"}, "Tracer.m"]]]\)\(
\)\(		\)\)\)

VectorDimension[4]

Electron Tensor

electron  =  
      G[le, k + me U]  **  G[le, {mu}]  **  G[le, kp  +  me U] **  
        G[le, {nu}]

Spur[1]

1/2 comes from averaging over incoming electron spins

etr  =  Simplify[ 1/2 electron /. le -> 1]

General Hadron tensor

hadron  =  -W1 S[{mu}, {nu}]  +  
        W2 /mp^2 S[P, {mu}] S[P, {nu}]  + 
        W3/mp^2 S[q, {mu}] S[q, {nu}]  +  
        W4/mp^2 (S[P, {mu}] S[q, {nu}]  + 
              S[P, {nu}] S[q, {mu}])  +  
        W5/mp^2 (S[P, {mu}] S[q, {nu}]  - 
              S[P, {nu}] S[q, {mu}])

Current Conservation

The hadron tensor must conserve current.  Therefore, q.W=0.  This implies relationships between the V's.  In particular, V5=0 and:

a1 = 
      Collect[ContractEpsGamma[hadron S[q, {mu}] ], {S[P, {nu}], 
          S[q, {nu}]}]

a2 = 
      Collect[Evaluate[
          ContractEpsGamma[hadron S[q, {nu}]] /. mu -> nu ], {S[P, {nu}], 
          S[q, {nu}]}]

aa = 
      Collect[Simplify[(a1 + a2)/2], {S[P, {nu}], S[q, {nu}]}]

eq1 = aa[[1, 3]] == 0

eq2 =  aa[[2, 2]] == 0

solution1  =  Solve[{eq1, eq2}, {W4, W3}][[1]]

Substitute V's into hadron tensor:

\(hadron1\  = \ 
      Collect[Simplify[\((hadron /. solution1)\) /. 
					W5 -> 
              0], {W1, W2}]\)

Matrix Element Squared

Contract lepton and hadron tensors:

LH  =  
      Collect[Simplify[ContractEpsGamma[etr hadron1]], {W1, W2}]

Remove kp:

LH1 = 
      Collect[Simplify[LH /. kp -> k - q], {W1, W2}] /. {k . k -> me^2, 
          P . P -> mp^2}

Evaluate in Laboratory Frame:

k1 = {k0, 0, 0, k0};
kp1 = {kp0, kp0 Sin[theta], 0, kp0 Cos[theta]};
P1  =  {mp, 0, 0, 0};
metric = {{1, 0, 0, 0}, {0, -1, 0, 0}, {0, 0, -1, 0}, {0, 0, 
            0, -1}};
q1 = k1 - kp1;
LH1a = 
      Collect[Simplify[
          LH1 /. {k . q -> k1 . metric . q1, me -> 0, 
              k . P -> k1 . metric . P1, P . q -> P1 . metric . q1, 
              q . q -> q1 . metric . q1}], {W1, W2}]

LH1b = 
      Simplify[LH1a[[2]] /. Cos[theta] -> 1 - 2 Sin[theta/2]^2, 
          Trig -> False] + 
        Expand[LH1a[[1]] /. 
            Cos[theta] -> 2  Cos[theta/2]^2 - 1]

qq  = 
      Simplify[q1 . metric . q1] /. 
        Cos[theta] -> 1 - 2 Sin[theta/2]^2

alpha = e^2/(4 PI)

msq  =  
      Simplify[(4 Pi)^2 alpha^2/qq^2 4 Pi mp LH1b, 
        Trig -> False]

xsect  =  msq/4/mp kp0/k0/(2 Pi)^3/2

Equation 8.33 in Griffiths.

Change to DIS variables:

\(LH2 = 
      Collect[Simplify[
          LH1 /. {P . q -> mp\ nu, P . k -> mp\ nu\ /y, \ 
              q . q -> \(-2\)\ x\ nu\ mp, 
		k . 
                  q -> \(-x\)\ nu\ mp	}], {W1, W2}]\)

Change to standard values of structure functions:

LH3 = 
      Collect[Simplify[LH2 /. {W1 -> F1/mp, W2 ->  F2/nu}] , {F1, 
          F2}]

msqdis  =  (4 Pi)^2 alpha^2 /qsq 4  Pi mp LH3

Get cross section:
	Notes:  Let me->0
		y=nu/k0 in lab
		kp0/(2 k0 2 mp (2Pi)^3 2) is phase space factor
		2 Pi mp nu/kp0 is conversion from dkp0 dOmenga to dx dy

factor = 8 Pi alpha^2 k0 mp/qsq;
\(xsectdis\  = 
      Collect[Simplify[
            Expand[\ \(\((\((msqdis /. me -> 0)\) y^2\ *k0^2/nu^2)\)/4\)/
                    mp\ \(\(kp0/k0\)/\((2\ Pi)\)^3\)/2*
		2\ Pi\ mp\ nu/
                    kp0]/factor], {F1, F2}]*factor\)

To get from in Particle data book:
	2 k0 mp = s-mp^2
	y/2nu = mp x y/(2 mp x nu)=mp x y/Q^2=mp/(s-mp^2)