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Vacuum energy in the Wess-Zumino model

  1. May 28, 2009 #1

    nrqed

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    :cry:I am at my wit's end so I hope someone can help.

    I am trying to do what is (almost) the simplest SUSY calculation one can think of: the calculation of the vacuum energy in the Wess-Zumino model. The result shoudl be zero but I don't get that.

    Since SUSY is "beyond the standard model" physics, I decided to cross-post my question here.

    Let me start by asking a simpler question and see if anyone can help.

    Let me break up the Wess-Zumino interactions into four terms

    [tex] L_1 = - \frac{1}{2} g^2 (A^2 + B^2)^2 [/tex]

    [tex] L_2 = - M g (A^3 + A B^2) [/tex]

    [tex] L_3 = - g A \overline{\Psi} \Psi[/tex]

    [tex] L_4 = - ig B \overline{\Psi} \gamma_5 \Psi [/tex]

    where A and B are scalar fields and Psi is a Majorana spinor, although for the terms I want to check in this post, this makes no difference, they can be treated as Dirac spinors.

    I consider the contributions to the vacuum energy of order g^2 so I calculate the time
    ordered expectation values of

    [tex] i L_1 - \frac{1}{2} ( L_2^2 + L_3^2 + L_4^2 + 2 L_2 L_3 + 2 L_2 L_4 + 2 L_3 L_4 ) [/tex]


    There are two main classes of diagrams: diagrams in which three lines connect two distinct points.
    (i.e. "sunset" diagrams). These are a bit more tricky.

    Simpler diagrams are those that contain two loops over independent variables (so the
    diagrams contain two independent loops). These should be simple to double check.
    I will only give my results for these diagrams, not the sunset diagrams for now. These diagrams should
    cancel independently of the sunset diagrams (a fact confirmed by a paper of Zumino).
    But I can't get it to work!

    These diagrams either have the form of two circles touching each other or two circles connected by
    line (forming a dumbell).

    These diagrams are all proportional to the integral

    [tex] I \equiv i g^2 \int D(z-z) D(w-w) [/tex]

    where D is just the usual boson propagator. I will quote all my results in terms of I .


    CONTRIBUTION FROM L1

    Diagram with two A loops: -3I/2

    Diagram with one A loop and one B loop: -I

    Diagram with two A loops: -3I/2

    CONTRIBUTION FROM L2^2

    Dumbell diagram with two A loops: 9 I/2

    Dumbell diagram with one A loop and one B loop: 3 I

    Dumbell diagram with two B loops: I/2

    CONTRIBUTION FROM L3^2

    Dumbell diagram, with two fermion loops: 8 I

    CONTRIBUTION FROM L4^2

    Because there is a gamma 5, there are no dumbell type contribution

    CONTRIBUTION FROM L2 L3

    Contribution with one A loop and one fermion loop: - 12 I

    Contribution with one B loop and one fermion loop: -4 I

    CONTRIBUTION FROM L3 L4 is identically zero.


    SUM = -4 I

    Now, can anyone check any of this??
     
  2. jcsd
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