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Schwinger trick and following change of variables

  1. Mar 20, 2012 #1
    1. The problem statement, all variables and given/known data

    We have the evaluation of the integral associated with the electron self-energy diagram and I am following Brian Hatfield's book "Quantum Field Theory of point particles and strings", and I am having problems with the integration limits after changing variables from [itex]z_2[/itex] to [itex]1-z_2[/itex], at p.388.

    2. Relevant equations

    [itex]\int_0^\infty \frac{d\beta}{\beta} \int_0^\infty dz_2 \left[2m_o - (1-z_2)\gamma^\mu p_\mu)\right] e^{-\beta[(1-z_2)z_2 p^2 - m^2_o z_2 - \lambda^2(1-z_2)]}[/itex]

    Since it may be relevant, some additional information: before this integral, we had an integration over [itex]z_1[/itex], which involved a Dirac delta function, [itex]\delta(1-z_1-z_2)[/itex].

    3. The attempt at a solution

    I expected the change of variables to be as straightfoward as it seemed to be, resulting in:

    [itex]\int_0^\infty \frac{d\beta}{\beta} \int_{-\infty}^1 dz \left(2m_o - z\gamma^\mu p_\mu)\right) e^{-\beta[z(1-z) p^2 - m^2_o(1-z) - \lambda^2z]}[/itex]

    But, in Hatfield's book, the lower limit of the integral in [itex]z[/itex] becomes [itex]0[/itex] instead of [itex]-\infty[/itex]. I wonder if the integration from [itex]-\infty[/itex] to [itex]0[/itex] may vanish somehow, but I just can't figure it.

    In advance, thank you all for the help!

    (English is not my native language. I apologize any mistake.)
     
  2. jcsd
  3. Mar 20, 2012 #2
    If think I found my mistake: evaluating the previous [itex]z_1[/itex] integration with the delta function will automatically change the integration limits of the [itex]z_2[/itex] integration.

    Looking at the step before the above integral, where we have the integral over [itex]z_1[/itex], from [itex]z_1=0[/itex] to [itex]z_1=\infty[/itex], we note that:

    - There is the delta, [itex]\delta(1-z_1-z_2)[/itex], demanding [itex]z_1=1-z_2[/itex].

    - But [itex]z_2[/itex] is integrated from [itex]z_2=0[/itex] to [itex]z_2=\infty[/itex], saying that [itex]z_1[/itex] may assume values between [itex]-\infty[/itex] and [itex]1[/itex].

    - On the other hand the integration over [itex]z_1[/itex] puts its values between [itex]0[/itex] and [itex]\infty[/itex].

    - The range of values [itex]z_2[/itex] may assume so that the delta function argument is satisfied goes from [itex]0[/itex] to [itex]1[/itex].

    Therefore, this should be the range of integration for the [itex]z_2[/itex] integration after the evaluation of the delta function: from [itex]z_2=0[/itex] to [itex]z_2=1[/itex]. In such a way, the change of variables from [itex]z_2[/itex] to [itex]z=1-z_2[/itex] is indeed as straightfoward as it seems, and also the integration limits are as expected.
     
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