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QFT-II-3-2P

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The procedure we have followed is known as dimensional regularization:evaluate the integral for (for which it is convergent); analytically continue the result to arbitrary d; fix A and B by imposing our conditions; take the limit .We also could have used Pauli-Villars regularization:replacemakes the integral convergent for evaluate the integral as a function of ; fix A and B by imposing our conditions; take the limit .Would we get the same result?67We could have also calculated without explicitly calculating A and B:differentiate twice with respect to : this integral is finite for evaluate the integral; calculate by integrating it with respect to and imposing our conditions.Would we get the same result?What happened with the divergence of the original integral?68To understand this better let’s make a Taylor expansion of about : divergent for divergent for divergent for but we have only two parameters that can be fixed to get finite .Thus the whole procedure is well defined only for !For the procedure breaks down, the theory is non-renormalizable!It turns out that the theory is renormalizable only for . And it does not matter which regularization scheme we use!(due to higher order corrections; we will discuss it later)69Loop corrections to the vertexLet’s consider loop corrections to the vertex:based on S-16Exact three-point vertex function: defined as the sum of 1PI diagrams with three external lines carrying incoming momenta so that .(this definition allows to have either sign)70We will follow the same procedure as for the propagator.Feynman’s formula:71(is divergent for and finite for ) Wick rotation:for : for with the replacement we have: 72take the limit :let’s define and ; we get: 73just a number, does not depend on or we can choosefinite and independent of What condition should we impose to fix the value of ?Any condition is good!Different conditions correspond to different definitions of the coupling.E.g. we can set that corresponds to: 74E.g. for :The integral over Feynman parameters cannot be done in closed form, but it is easy to see that the magnitude of the one-loop correction to the vertex function increases logarithmically with when .the same behavior that we found for (we will discuss it


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