Forest Bhabha Scattering - Revision history

Foretony: /* Inserting all terms */

2012-04-22T18:13:06Z

Inserting all terms

Foretony at 03:02, 19 April 2012

2012-04-19T03:02:55Z

Foretony: /* Positron absorbs photn Vertex */

2012-04-19T02:57:33Z

Positron absorbs photn Vertex

Foretony: /* Step 2 Label 4-Momentum */

2012-04-19T02:55:56Z

Step 2 Label 4-Momentum

Foretony: /* Inserting all terms */

2012-04-17T19:25:41Z

Inserting all terms

Foretony: /* Inserting all terms */

2012-04-17T19:24:13Z

Inserting all terms

Foretony: /* Inserting all terms */

2012-04-17T19:23:50Z

Inserting all terms

Foretony: /* Inserting all terms */

2012-04-17T19:21:31Z

Inserting all terms

Foretony: /* Spin averaged matrix element */

2012-04-17T19:13:36Z

Spin averaged matrix element

Foretony: /* Spin averaged matrix element */

2012-04-17T19:09:49Z

Spin averaged matrix element

@@ Line 197: / Line 197: @@
 Mandelstam variables are defined as
-:<math>s^2 = (p_1 + p_3)^2 = (p_2 + p_4)^2\approx 2 p_1 \cdot p_3 \approx 2 p_2 \cdot p_4</math>
+:<math>s = (p_1 + p_3)^2 = (p_2 + p_4)^2\approx 2 p_1 \cdot p_3 \approx 2 p_2 \cdot p_4</math>
-:<math>t^2 = (p_1 - p_2)^2 = (p_3-p_4)^2 \approx -2 p_1 \cdot p_2 \approx -2 p_3 \cdot p_4</math>
+:<math>t = (p_1 - p_2)^2 = (p_3-p_4)^2 \approx -2 p_1 \cdot p_2 \approx -2 p_3 \cdot p_4</math>
-:<math>u^2 = (p_1 - p_4)^2 = (p_3 - p_2)^2\approx -2 p_1 \cdot p_4 \approx -2 p_3 \cdot p_2</math>
+:<math>u = (p_1 - p_4)^2 = (p_3 - p_2)^2\approx -2 p_1 \cdot p_4 \approx -2 p_3 \cdot p_2</math>
 **Approximations only valid in high relativistic limit

@@ Line 93: / Line 93: @@
 The Matrix element is a product of the vertex and propagator terms
-:<math>\mathcal{M}_t = (2\pi)^4 \int \left [ \bar{u}_2 (ig_e\gamma^{\mu}) u_1 \right ] \left ( \frac{-ig_{\mu \nu}}{(q)^2} \right) \left [ \bar{u}_3 (ig_e \gamma^{\nu}) u_4 \right ] \delta^4(p_1 -p_2 -q)\delta^4(p_3 +q -p_4) </math>
+:<math>\mathcal{M}_t = (2\pi)^4 \int \left [ \bar{u}_2 (ig_e\gamma^{\mu}) u_1 \right ] \left ( \frac{-ig_{\mu \nu}}{(q)^2} \right) \left [ \bar{v}_3 (ig_e \gamma^{\nu}) v_4 \right ] \delta^4(p_1 -p_2 -q)\delta^4(p_3 +q -p_4) </math>
 :::= t-channel scattering matrix amplitude
-::<math> = \left [ \bar{u}_2 (ig_e\gamma^{\mu}) u_1 \right ] \left ( \frac{1}{(p_1-p_2)^2} \right) \left [ \bar{u}_3 (ig_e \gamma_{\mu}) u_4\right ]</math> did the integral over the momentum delta functions which cancels out the <math>(2\pi)^4</math>
+::<math> = \left [ \bar{u}_2 (ig_e\gamma^{\mu}) u_1 \right ] \left ( \frac{1}{(p_1-p_2)^2} \right) \left [ \bar{v}_3 (ig_e \gamma_{\mu}) v_4\right ]</math> did the integral over the momentum delta functions which cancels out the <math>(2\pi)^4</math>
-::<math> =  \left ( \frac{-g_e^2}{(p_1-p_2)^2} \right) [\bar{u}_2 \gamma^{\mu} u_1] [\bar{u}_3 \gamma_{\mu} u_4]</math>
+::<math> =  \left ( \frac{-g_e^2}{(p_1-p_2)^2} \right) [\bar{u}_2 \gamma^{\mu} u_1] [\bar{v}_3 \gamma_{\mu} v_4]</math>
-<math>|\mathcal{M}_t |^2 = \frac{e^4}{(p_1-p_2)^4} \Big( [\bar{u}_2 \gamma^{\mu} u_1] [\bar{u}_3 \gamma_{\mu} u_4]\Big)^* \Big( ([\bar{u}_2 \gamma^{\mu} u_1] [\bar{u}_3 \gamma_{\mu} u_4]\Big) </math>
+<math>|\mathcal{M}_t |^2 = \frac{e^4}{(p_1-p_2)^4} \Big( [\bar{u}_2 \gamma^{\mu} u_1] [\bar{v}_3 \gamma_{\mu} v_4]\Big)^* \Big( ([\bar{u}_2 \gamma^{\mu} u_1] [\bar{v}_3 \gamma_{\mu} v_4]\Big) </math>
-:<math>= \frac{e^4}{(p_1-p_2)^4} \Big( (\bar{u}_1 \gamma^\mu u_2 ) ( \bar{u}_4 \gamma_\mu u_3) \Big) \Big( (\bar{u}_2 \gamma^\nu u_1)( \bar{u}_3 \gamma_\nu u_4 \Big )</math>(complex conjugate will flip order)
+:<math>= \frac{e^4}{(p_1-p_2)^4} \Big( (\bar{u}_1 \gamma^\mu u_2 ) ( \bar{v}_4 \gamma_\mu v_3) \Big) \Big( (\bar{u}_2 \gamma^\nu u_1)( \bar{v}_3 \gamma_\nu v_4 \Big )</math>(complex conjugate will flip order)
-:<math>= \frac{e^4}{(p_1-p_2)^4} \Big( (\bar{u}_1 \gamma^\mu u_2 ) (\bar{u}_2 \gamma^\nu u_1)( \bar{u}_4 \gamma_\mu u_3)( \bar{u}_3 \gamma_\nu u_4 \Big)</math> (move terms that depend on same momentum to be next to each other)
+:<math>= \frac{e^4}{(p_1-p_2)^4} \Big( (\bar{u}_1 \gamma^\mu u_2 ) (\bar{u}_2 \gamma^\nu u_1)( \bar{v}_4 \gamma_\mu v_3)( \bar{v}_3 \gamma_\nu v_4 \Big)</math> (move terms that depend on same momentum to be next to each other)
@@ Line 113: / Line 113: @@
 <math>< \left | \mathcal{M}^2 \right | > \equiv</math> average over all intial spin states and sum over all the final spin states for <math>\left | \mathcal{M} \right |^2</math>
-:<math>= \frac{e^4}{(p_1-p_2)^4}  \sum_{s_1} \left ( \bar{u}_1 \gamma^\mu \left [  \sum_{s_2} u_2 \bar{u}_2 \right ]\gamma^\nu u_1 \right) \sum_{s_4} \left ( \bar{u}_4 \gamma_\mu \left [ \sum_{s_3} u_3\bar{u}_3 \right ] \gamma_\nu u_4 \right )</math>
+:<math>= \frac{e^4}{(p_1-p_2)^4}  \sum_{s_1} \left ( \bar{u}_1 \gamma^\mu \left [  \sum_{s_2} u_2 \bar{u}_2 \right ]\gamma^\nu u_1 \right) \sum_{s_4} \left ( \bar{v}_4 \gamma_\mu \left [ \sum_{s_3} v_3\bar{v}_3 \right ] \gamma_\nu v_4 \right )</math>
@@ Line 122: / Line 122: @@
 ;The completeness relation
 :<math>\sum_{s_2} u_2 \bar{u}_2  = p\!\!\!/_2 + m_2 </math>
@@ Line 141: / Line 142: @@
 Using the above relation for the matrix element
-:<math>\mathcal{M}_t= \frac{e^4}{(p_1-p_2)^4}  \sum_{s_1} \left ( \bar{u}_1 \gamma^\mu \left [  \sum_{s_2} u_2 \bar{u}_2 \right ]\gamma^\nu u_1 \right) \sum_{s_4} \left ( \bar{u}_4 \gamma_\mu \left [ \sum_{s_3} u_3\bar{u}_3 \right ] \gamma_\nu u_4 \right )</math>
+:<math>\mathcal{M}_t= \frac{e^4}{(p_1-p_2)^4}  \sum_{s_1} \left ( \bar{u}_1 \gamma^\mu \left [  \sum_{s_2} u_2 \bar{u}_2 \right ]\gamma^\nu u_1 \right) \sum_{s_4} \left ( \bar{v}_4 \gamma_\mu \left [ \sum_{s_3} v_3\bar{v}_3 \right ] \gamma_\nu v_4 \right )</math>
 ::<math>= \frac{e^4}{(p_1-p_2)^4}  Tr\left [  \gamma^\mu \left ( p\!\!\!/_2 + m_2 \right ) \gamma^\nu ( p\!\!\!/_1 + m_1) \right] Tr\left [  \gamma_\mu \left ( p\!\!\!/_3 + m_3 \right ) \gamma_\nu ( p\!\!\!/_4 + m_4) \right]</math>

@@ Line 85: / Line 85: @@
 The positron absorbs the photon emitted by the electron
-:<math>\bar{u}_3 (ig_e\gamma^{\nu} ) u_4</math>
+:<math>\bar{v}_3 (ig_e\gamma^{\nu} ) v_4</math>
 ;because of the arrow for anti-particles the final state u_4 appears on the most left hand side as the initial state.

@@ Line 35: / Line 35: @@
 :<math>p_2=p_{out}(e^-) \equiv</math> final electron 4-momentum
-:<math>u_2 \equiv</math> final electron spinor
+:<math>\bar{u}_2 \equiv</math> final electron spinor
 :<math>p_3= p_{in}(e^+) \equiv</math> initial positron 4-momentum
-:<math>\bar{u}_3 \equiv</math> initial positron spinor
+:<math>\bar{v}_3 \equiv</math> initial positron spinor
 :<math>p_4 = p_{out}(e^+) \equiv</math> final positron 4-momentum
-:<math>\bar{u}_4 \equiv</math> final positron spinor
+:<math>v_4 \equiv</math> final positron spinor

@@ Line 199: / Line 199: @@
 :<math>t^2 = (p_1 - p_2)^2 = (p_3-p_4)^2 \approx -2 p_1 \cdot p_2 \approx -2 p_3 \cdot p_4</math>
 :<math>u^2 = (p_1 - p_4)^2 = (p_3 - p_2)^2\approx -2 p_1 \cdot p_4 \approx -2 p_3 \cdot p_2</math>
 =e+e- Annihilation (s-channel) (time-like)=

@@ Line 196: / Line 196: @@
 Mandelstam variables are defined as
-:<math>s^2 = (p_1 + p_3)^2 = (p_2 + p_4)^2</math>
+:<math>s^2 = (p_1 + p_3)^2 = (p_2 + p_4)^2\aprox 2 p_1 \cdot p_3 \aprox 2 p_2 \cdot p_4</math>
-:<math>t^2 = (p_1 - p_2)^2 = (p_3-p_4)^2 \approx -2 p_1 \cdot p_2 \aprox -2 p_3 \cdot p_4</math>
+:<math>t^2 = (p_1 - p_2)^2 = (p_3-p_4)^2 \approx -2 p_1 \cdot p_2 \approx -2 p_3 \cdot p_4</math>
 =e+e- Annihilation (s-channel) (time-like)=

@@ Line 193: / Line 193: @@
 :<math>\mathcal{M}_t = \frac{32e^4}{(p_1-p_2)^4} \left [  (p_2\cdot p_3) (p_1 \cdot p_4) + (p_2 \cdot p_4 ) (p_1 \cdot p_3) \right ]</math>
 ::<math>= 2e^4\frac{s^2+u^2}{t^2}</math>
 =e+e- Annihilation (s-channel) (time-like)=

@@ Line 109: / Line 109: @@
 ;or you can average over all the initial spin states and sum over all the final spin states to determine the cross section.
-;Note: there is a factor of 1/2 which is left out for now
+;Note: there is a factor of 1/2 if only one of the states is the initial-state particle, the factor is 1/4 is both particles are in the initial state, the factor is 1 if neither particle is in the initial state.
 <math>< \left | \mathcal{M}^2 \right | > \equiv</math> average over all intial spin states and sum over all the final spin states for <math>\left | \mathcal{M} \right |^2</math>

@@ Line 108: / Line 108: @@
 To determine actual cross-sections and lifetimes you need to insert the specific spinors for the physics process
-;or you can overage over all the initial spin states and sum over all the final spin states to determine the cross section.
+;or you can average over all the initial spin states and sum over all the final spin states to determine the cross section.
 <math>< \left | \mathcal{M}^2 \right | > \equiv</math> average over all intial spin states and sum over all the final spin states for <math>\left | \mathcal{M} \right |^2</math>
@@ Line 135: / Line 136: @@
 ::<math>= \left ( Q ( p\!\!\!/_1 + m_1)  \right )_{ii}</math>
 ::<math>= Tr\left [ Q ( p\!\!\!/_1 + m_1) \right]</math>: Def of trace is the sum of diagonal elements in a matrix
 ::<math>= Tr\left [  \gamma^\mu \left ( p\!\!\!/_2 + m_2 \right ) \gamma^\nu ( p\!\!\!/_1 + m_1) \right]</math>:
 ====Evaluating the Matrix element in terms of Traces====