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Physics
Beyond the Standard Models
Vacuum energy in the Wess-Zumino model
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[QUOTE="nrqed, post: 2215941, member: 15416"] :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. [b] SUM = -4 I [/b] Now, can anyone check any of this?? [/QUOTE]
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Beyond the Standard Models
Vacuum energy in the Wess-Zumino model
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