## Standard Model

We prove the Adler-Bardeen theorem in a large class of general gauge theories, including nonrenormalizable ones. We assume that the gauge symmetries are general covariance, local Lorentz symmetry and Abelian and non-Abelian Yang-Mills symmetries, and that the local functionals of vanishing ghost numbers satisfy a variant of the Kluberg-Stern–Zuber conjecture. We show that if the gauge anomalies are trivial at one loop, for every truncation of the theory there exists a subtraction scheme where they manifestly vanish to all orders, within the truncation. Outside the truncation the cancellation of gauge anomalies can be enforced by fine-tuning local counterterms. The framework of the proof is worked out by combining a recently formulated chiral dimensional regularization with a gauge invariant higher-derivative regularization. If the higher-derivative regularizing terms are placed well beyond the truncation, and the energy scale $\Lambda$ associated with them is kept fixed, the theory is super-renormalizable and has the property that, once the gauge anomalies are canceled at one loop, they manifestly vanish from two loops onwards by simple power counting. When the $\Lambda$ divergences are subtracted away and $\Lambda$ is sent to infinity, the anomaly cancellation survives in a manifest form within the truncation and in a nonmanifest form outside. The standard model coupled to quantum gravity satisfies all the assumptions, so it is free of gauge anomalies to all orders.

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Phys. Rev. D 91 (2015) 105016 | DOI: 10.1103/PhysRevD.91.105016

arXiv: 1501.07014 [hep-th]

We consider renormalizable Standard-Model extensions that violate Lorentz symmetry at high energies, but preserve CPT, and do not contain elementary scalar fields. A Nambu–Jona-Lasinio mechanism gives masses to fermions and gauge bosons, and generates composite Higgs fields at low energies. We study the effective potential at the leading order of the large-$N_{c}$ expansion, prove that there exists a broken phase and study the phase space. In general, the minimum may break invariance under boosts, rotations and CPT, but we give evidence that there exists a Lorentz invariant phase. We study the spectrum of composite bosons and the low-energy theory in the Lorentz phase. Our approach predicts relations among the parameters of the low-energy theory. We find that such relations are compatible with the experimental data, within theoretical errors. We also study the mixing among generations, the emergence of the CKM matrix and neutrino oscillations.

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Phys. Rev. D83 (2011) 056005 | DOI: 10.1103/PhysRevD.83.056005

arXiv:1101.2014 [hep-th]

If Lorentz symmetry is violated at high energies, interactions that are usually non-renormalizable can become renormalizable by weighted power counting. Recently, a CPT invariant, Lorentz violating extension of the Standard Model containing two scalar-two fermion interactions (which can explain neutrino masses) and four fermion interactions (which can explain proton decay) was proposed. In this paper we consider a variant of this model, obtained suppressing the elementary scalar fields, and argue that it can reproduce the known low energy physics. In the Nambu$-$Jona-Lasinio spirit, we show, using a large $N_c$ expansion, that a dynamical symmetry breaking takes place. The effective potential has a Lorentz invariant minimum and the Lorentz violation does not reverberate down to low energies. The mechanism generates fermion masses, gauge-boson masses and scalar bound states, to be identified with composite Higgs bosons. Our approach is not plagued by the ambiguities of approaches based on non-renormalizable vertices. The low-energy effective action is uniquely determined and predicts relations among parameters of the Standard Model.

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Eur.Phys.J. C65 (2010) 523-536 | DOI: 10.1140/epjc/s10052-009-1211-z

arXiv:0904.1849 [hep-ph]

We study the Standard-Model extensions that have the following features: they violate Lorentz invariance explicitly at high energies; they are unitary, local, polynomial and renormalizable by weighted power counting; they contain the vertex $(LH)^2$, which gives Majorana masses to the neutrinos after symmetry breaking, and possibly four fermion interactions; they do not contain right-handed neutrinos, nor other extra fields. We study the simplest CPT invariant Standard-Model extension of this type in detail and prove the cancellation of gauge anomalies. We investigate the low-energy recovery of Lorentz invariance and comment on other types of extensions.

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Phys.Rev. D79 (2009) 025017 | DOI: 10.1103/PhysRevD.79.025017

arXiv:0808.3475 [hep-ph]

In flat space, $\gamma_5$ and the epsilon tensor break the dimensionally continued Lorentz symmetry, but propagators have fully Lorentz invariant denominators. When the Standard Model is coupled with quantum gravity $\gamma_5$ breaks the continued local Lorentz symmetry. I show how to deform the Einstein lagrangian and gauge-fix the residual local Lorentz symmetry so that the propagators of the graviton, the ghosts and the BRST auxiliary fields have fully Lorentz invariant denominators. This makes the calculation of Feynman diagrams more efficient.

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Phys.Lett. B596 (2004) 90-95 | DOI: 10.1016/j.physletb.2004.06.089

arXiv:hep-th/0404032

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14B1 D. Anselmi
Renormalization

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Last update: May 9th 2015, 230 pages

Contents: Preface | 1. Functional integral | 2. Renormalization | 3. Renormalization group | 4. Gauge symmetry | 5. Canonical formalism | 6. Quantum electrodynamics | 7. Non-Abelian gauge field theories | Notation and useful formulas | References

Course on renormalization, taught in Pisa in 2015. (More chapters will be added later.)