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I will present the simplest example of beta functions arising in string theory, specifically within bosonic string theory. The states transform in the $24 \otimes 24$ representation of $SO(24)$, equivalent to three irreducible representations; schematically, $$(\mathrm{traceless \;symmetric} )\otimes (\mathrm{antisymmetric}) \otimes (\mathrm{singlet})$$ To ...


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Unfortunately, Bardeen seems to misunderstand the naturalness problem that has nothing to do with quadratic divergences per se. In the strict SM, there is no naturalness problem because the running Higgs mass squared is proportional to itself. But this is not the setup that people care about when talking about the actual naturalness problem that emerges ...


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This is a special case of a more general phenomena. Conserved currents never acquire anomalous dimensions, they are protected by the symmetry. If you have a conserved current, you have a symmetry algebra for the theory \begin{equation} [Q^a,Q^b]=if^{abc}Q_c \end{equation} For this to hold, the charges need to be dimensionless. But the charges are given by ...


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The main point that you have to always keep in mind is that relevant/irrelevant coupling constants are defined with respect to a fixed point. The standard/naive power counting is done assuming that the fixed point controlling the RG flow is the gaussian. This is true for massless QED and $\phi^4$ theories in four dimensions in the infrared, and for QCD in ...


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The renormalization factor $1/Z_S$ that Manohar and Wise discuss in their book is not the renormalization of the coupling constant. It is there to remove divergences that come from the fact that we have inserted an operator that is a product of fields. As I understand their line of reasoning, this is before we discuss a coupling constant at all, so if we add ...


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Taking your integral as an example, i.e. $$\int \mathrm{d}^{4-\epsilon}k \, \, F(k,m,s) = \frac{2}{\epsilon} + \frac{m^2}{3}\left( \gamma + \log (4\pi) -\frac{1}{\epsilon} \right)$$ Renormalization does not simply 'delete' the $1/\epsilon$ divergences. It is a well-defined procedure which expresses the amplitudes in terms of renormalized, measurable and ...


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In general, derivative couplings lead to momentum-dependencies in scattering amplitudes. This can be seen from the fact that the Fourier transform of a derivative operator corresponds to a multiplication by the relevant momentum. A mass dependence is implicit through by having a momentum, since the momentum of a fermion depends on its mass. In this case, ...


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Not an expert on this, but I would think about from a top-down approach rather than the other way round: So we start from a "fundamental" theory (FT), valid at all energy scales (high energy included) and we then derive the low energy effective field theory (LEFT), which is sufficient to describe physics at the low energies. There are two logical ...


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The definite answer to your question is: There is no mathematicaly precise, commonly accepted definition of the term "regularization procedure" in perturbative quantum field theory. Instead, there are various regularization schemes with their advantages and disatvantages. Maybe you'll find Chapter B5: Divergences and renormalization of my theoretical ...



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