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Register a new userRenormalizable Expansion for Nonrenormalizable Theories: II. Gauge Higher Dimensional Theories
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The previously developed renormalizable perturbative 1/N-expansion in higher dimensional scalar field theories is extended to gauge theories with fermions. It is based on the $1/N_f$-expansion and results in a logarithmically divergent perturbation theory in arbitrary high odd space-time dimension. Due to the self-interaction of non-Abelian fields the proposed recipe requires some modification which, however, does not change the main results. The new effective coupling is dimensionless and is running in accordance with the usual RG equations. The corresponding beta function is calculated in the leading order and is nonpolynomial in effective coupling. The original dimensionful gauge coupling plays a role of mass and is also logarithmically renormalized. Comments on the unitarity of the resulting theory are given.
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We demonstrate how one can construct renormalizable perturbative expansion in formally nonrenormalizable higher dimensional scalar theories. It is based on 1/N-expansion and results in a logarithmically divergent perturbation theory in arbitrary high odd space-time dimension. The resulting effective coupling is dimensionless and is running in accordance with the usual RG equations. The corresponding beta function is calculated in the leading order and is nonpolynomial in effective coupling. It exhibits either UV asymptotically free or IR free behaviour depending on the dimension of space-time.
We demonstrate how one can construct renormalizable perturbative expansion in formally nonrenormalizable higher dimensional field theories. It is based on $1/N_f$-expansion and results in a logarithmically divergent perturbation theory in arbitrary high space-time dimension. First, we consider a simple example of $N$-component scalar filed theory and then extend this approach to Abelian and non-Abelian gauge theories with $N_f$ fermions. In the latter case, due to self-interaction of non-Abelian fields the proposed recipe requires some modification which, however, does not change the main results. The resulting effective coupling is dimensionless and is running in accordance with the usual RG equations. The corresponding beta function is calculated in the leading order and is nonpolynomial in effective coupling. It exhibits either UV asymptotically free or IR free behaviour depending on the dimension of space-time. The original dimensionful coupling plays a role of a mass and is also logarithmically renormalized. We analyze also the analytical properties of a resulting theory and demonstrate that in general it acquires several ghost states with negative and/or complex masses. In the former case, the ghost state can be removed by a proper choice of the coupling. As for the states with complex conjugated masses, their contribution to physical amplitudes cancels so that the theory appears to be unitary.
We discuss a natural scenario to solve the strong CP problem in the framework of the higher dimensional gauge theory. An axion-like field $A_y$ has been built-in as the extra-space component of the higher dimensional gauge field. The coupling of $A_y$ with gluons is attributed to the radiatively induced Chern-Simons (CS) term. We adopt a toy model with some unknown gauge symmetry U(1)$_X$. The CS term is obtained in two ways: first by a concrete 1-loop calculation and next by use of the Fujikawas method to deal with the chiral anomaly in 4D space-time. The obtained results are identical, which implies that the radiative correction to the CS term is 1-loop exact and is also free from UV-divergence even though the theory itself is non-renormalizable. As a novel feature of this scenario, such obtained CS term is no longer linear in the field $A_y$ as in the usually discussed CS term in 5D space-time but a periodic function of $A_y$, since $A_y$ has a physical meaning as the Wilson-loop phase. We argue how such novel feature of this scenario causes the modification of the ordinary solutions of the strong CP problem based on the axion fields.
Some nonrenormalizable theories are less singular than all renormalizable theories, and one can use lattice simulations to extract physical information from them. This paper discusses four nonrenormalizable theories that have finite euclidian and minkowskian Greens functions. Two of them have finite euclidian action densities and describe scalar bosons of finite mass. The space of nonsingular nonrenormalizable theories is vast.
We show that the subleading soft photon theorem in a $(d+2)$-dimensional massless abelian gauge theory gives rise to a Ward identity corresponding to divergent large gauge transformations acting on the celestial sphere at null infinity. We further generalize our analysis to $(d+2)$-dimensional non-abelian gauge theories and show that the leading and subleading soft gluon theorem give rise to Ward identities corresponding to asymptotic symmetries of the theory.
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