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5

Yes, that is the Higgs potential of the Standard Model. Note that a $\phi^3$ term is forbidden by symmetry (it would not be an $\mathrm{SU}(2)$ scalar), and $\mathcal{O}(\phi^5)$ terms would be non-renormalizable, so this is really the only potential we can write down that does not need other new physics.


3

Schwartz is simply noting that the $\beta$-function has a generic expansion in QED of the form (29) where $\beta_{0,1,2}$ are some numbers that can be computed by explicitly calculating the various Feynman diagrams. For instance the leading $\epsilon/2$ is the tree-level result in $d=4-\epsilon$ dimensions. This can be easily seen as follows. In ...


3

Consider for example, simple $\lambda \phi^3$-theory with Lagrangian $$ \mathcal{L}=-\frac{1}{2}\left|\partial_\mu\phi\right|^2-\frac{m^2}{2}\phi^2+\frac{\lambda}{3!}\phi^3. $$ One can say that the $\lambda \phi^3$ term renormalizes the mass term, because the regularization and renormalization of the divergence of the one-loop diagram will lead to a ...


3

In the lecture notes, the Feynman diagrams for the two examples of $\phi^3$ in $D=6$ and $\phi^6$ in $D=3$ are mixed up. You can easily identify this by noticing that in a scalar $\phi^n$ theory $n$ lines meet in all vertices. The example graph for $\phi^6$ theory in the lecture notes (which is wrongly listed as an example of $\phi^3$ in $D=6$) has six ...


2

Yes tensor network algorithms have been developed to describe braiding of anyons, Abelian and non-Abelian. The networks are constructed from tensors that explicitly conserve topological charge and the braiding, fusion, and recoupling data are taken as input to the algorithms. This reference: http://arxiv.org/pdf/1311.0967.pdf describes how to use the Time ...


1

For your first question: the physical mass is the measured mass. If $m$ is not the measured mass, don't call it the physical mass, because it isn't. If someone does, correct them. :-) For your second question, I think we should look at the physical interpretation of all those mathematical manipulations. The hierarchy problem (as I understand it) comes from ...


1

I recently stumbled upon a good comment about this in Jared Kaplan's AdS/CFT notes Any quantum field theory which has hope of having an UV-completion can be viewed as as effective theory at point in the RG flow from an UV complete theory. Field theories at the UV fixed point are conformal. Hence all 'well-defined' field theories are either CFTs or points ...


1

The first statement is correct to some extent, the second isn't. Take the case of vector gauge theories, like the ones in the Standard Model. These theories have a massless vector field, which can be described by two degrees of freedom (2 polarisations) while the classical field itself, $A_\mu$, is described by 4 components. Gauge invariance is related to ...


1

If a theory contains divergences, there is no need to assume it is plain wrong. It probably won't be the full theory of everything, but that's the only criticism that can be leveled at it. It can still be in perfect agreement with experiments below very-high-energy. The issue with QED is that QED is not a complete theory. In the real world, QED is just a ...


1

I think the confusion is due to a lack of mathematically precise definitions of what is quantum field theory? what is one trying to construct and how? etc. There are a lot of vague notions used in the physics literature: the partition function (which does not make much sense in infinite volume), the effective action,...but the bottom line is the collection ...


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No. Gauge invariance is not a real physical symmetry but a mathematical property of the formalism while renormalization is more deep property related to the scaling of the coupling constant. One can think interaction that breaks gauge but is still renormalizable. It is fact that QED and QCD are gauge theories but gravity is a gauge theory and not ...


1

He is writing the $\beta$ function as a perturbation series in the small parameter $\alpha$, $$\beta(\alpha) = \sum_{n=1}^\infty c_n \alpha^n.$$ By convention, it's common to write this expansion in the form he has given. The first coefficient is always $c_1 = - 4\pi\epsilon$ (where $\epsilon$ is the dimensional regularization parameter). The remaining terms ...



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