Difference between revisions of "Limits based on Mandelstam Variables"

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<center><math>x = \arccos{3}</math></center>
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<center><math>\theta = \arccos{3}</math></center>
  
  
<center><math>cos(x) = \cos{\arccos{3}}</math></center>
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<center><math>\cos{\theta} = \cos{\arccos{3}}</math></center>
  
  
<center><math>\cos{x} = 3</math></center>
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<center><math>\cos{\theta} = 3</math></center>
  
  
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<center><math>\cos{x} = \frac{e^{ix} + e^{-ix}}{2}</math></center>
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<center><math>\cos{x} = \frac{e^{i \theta} + e^{-i \theta}}{2}</math></center>
  
  
<center><math> \frac{e^{ix} + e^{-ix}}{2} = 3</math></center>
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<center><math> \frac{e^{i \theta} + e^{-i \theta}}{2} = 3</math></center>
  
  
<center><math> e^{ix} + e^{-ix} = 6</math></center>
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<center><math> e^{i \theta} + e^{-i \theta} = 6</math></center>
  
  
Multiply with <math>e^{ix}</math>
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Multiply with <math>e^{i \theta}</math>
  
  
<center><math>e^{2ix} + 1 = 6e^{ix}</math></center>
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<center><math>e^{2i \theta} + 1 = 6e^{i \theta}</math></center>
  
  
<center>Letting <math>y = e^{ix}</math></center>
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<center>Letting <math>y = e^{i \theta}</math></center>
  
  
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<center><math>y_1 = 5.828427 = e^{ix}</math></center>
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<center><math>y_1 = 5.828427 = e^{i \theta}</math></center>
  
  
  
<center><math>y_2 = 0.171573 = e^{ix}</math></center>
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<center><math>y_2 = 0.171573 = e^{i \theta}</math></center>
  
  
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<center><math>x_1 = \frac{ln(5.828427) }{ i}=-1.76275i</math></center>
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<center><math>\theta_1 = \frac{ln(5.828427) }{ i}=-1.76275i</math></center>
  
  
  
<center><math>x_2 = \frac{ln(0.171573) }{ i}=1.76275i</math></center>
+
<center><math> \theta_2 = \frac{ln(0.171573) }{ i}=1.76275i</math></center>

Revision as of 16:37, 12 June 2017

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Limits based on Mandelstam Variables

Since the Mandelstam variables are the scalar product of 4-momenta, which are invariants, they are invariants as well. The sum of these invariant variables must also be invariant as well. Find the sum of the 3 Mandelstam variables when the two particles have equal mass in the center of mass frame gives:


[math]s+t+u=(4(m^2+ p \ ^{*2}))+(-2 p \ ^{*2}(1-cos\ \theta))+(-2 p \ ^{*2}(1+cos\ \theta))[/math]


[math]s+t+u \equiv 4m^2[/math]


Since

[math]s \equiv 4(m^2+\vec p \ ^{*2})[/math]


This implies

[math]s \ge 4m^2[/math]


In turn, this implies


[math] t \le 0 \qquad u \le 0[/math]


At the condition both t and u are equal to zero, we find


[math] t = 0 \qquad u = 0[/math]


[math]-2 p \ ^{*2}(1-cos\ \theta) = 0 \qquad -2 p \ ^{*2}(1+cos\ \theta) = 0[/math]


[math](-2 p \ ^{*2}+2 p \ ^{*2}cos\ \theta) = 0 \qquad (-2 p \ ^{*2}-2 p \ ^{*2}cos\ \theta) = 0[/math]


[math]2 p \ ^{*2}cos\ \theta = 2 p \ ^{*2} \qquad -2 p \ ^{*2}cos\ \theta = 2 p \ ^{*2}[/math]


[math]\cos\ \theta = 1 \qquad \cos\ \theta = -1[/math]


[math]\Rightarrow \theta_{t=0} = \arccos \ 1=0^{\circ} \qquad \theta_{u=0} = \arccos \ -1=180^{\circ}[/math]

Holding u constant at zero we can find the maximum of t


[math]s+t \equiv 4m^2[/math]


[math]\Rightarrow t=4m^2-s[/math]


[math]t=4m^2-4m^2+ 4p \ ^{*2})[/math]


[math]t=4p \ ^{*2}[/math]


[math]-2 p \ ^{*2}(1-cos\ \theta)=4p \ ^{*2}[/math]


[math](1-cos\ \theta)=-2[/math]


[math]-cos\ \theta=-3[/math]


[math] \theta_{max} \equiv \arccos 3[/math]


The domain of the arccos function is from −1 to +1 inclusive and the range is from 0 to π radians inclusive (or from 0° to 180°). This implies for arccos 3, the range will include imaginary numbers. Knowing that the range of the cosine function is -1 to +1 inclusive and the domain to be any angle


[math]\theta = \arccos{3}[/math]


[math]\cos{\theta} = \cos{\arccos{3}}[/math]


[math]\cos{\theta} = 3[/math]


From Euler's formula


[math]\cos{x} = \frac{e^{i \theta} + e^{-i \theta}}{2}[/math]


[math] \frac{e^{i \theta} + e^{-i \theta}}{2} = 3[/math]


[math] e^{i \theta} + e^{-i \theta} = 6[/math]


Multiply with [math]e^{i \theta}[/math]


[math]e^{2i \theta} + 1 = 6e^{i \theta}[/math]


Letting [math]y = e^{i \theta}[/math]


We get an quadratic equation:


[math]y^2 - 6y + 1 = 0[/math]


[math]y = (6 ± √32)/2[/math]


[math]y_1 = 5.828427 = e^{i \theta}[/math]


[math]y_2 = 0.171573 = e^{i \theta}[/math]


Apply ln on both sides gives the solution for arccos 3:


[math]\theta_1 = \frac{ln(5.828427) }{ i}=-1.76275i[/math]


[math] \theta_2 = \frac{ln(0.171573) }{ i}=1.76275i[/math]