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Publication# Chiral charge erasure via thermal fluctuations of magnetic helicity

Abstract

We consider a relativistic plasma of fermions coupled to an Abelian gauge field and carrying a chiral charge asymmetry, which might arise in the early Universe through baryogenesis. It is known that on large length scales, lambda greater than or similar to 1/(alpha mu(5)), the chiral anomaly opens an instability toward the erasure of chiral charge and growth of magnetic helicity. Here the chemical potential mu(5) parametrizes the chiral asymmetry and alpha is the fine-structure constant. We study the process of chiral charge erasure through the thermal fluctuations of magnetic helicity and contrast with the well-studied phenomenon of Chern-Simons number diffusion. Through the fluctuation-dissipation theorem we estimate the amplitude and time scale of helicity fluctuations on the length scale lambda, finding delta H similar to lambda T and tau similar to alpha lambda T-3(2) for a relativistic plasma at temperature T. We argue that the presence of a chiral asymmetry allows the helicity to grow diffusively for a time t similar to T-3/(alpha(5)mu(4)(5)) until it reaches an equilibrium value H similar to mu T-5(2)/alpha, and the chiral asymmetry is partially erased. If the chiral asymmetry is small, mu(5) < T/alpha, this avenue for chiral charge erasure is found to be slower than the chiral magnetic effect for which t similar to T/(alpha(3)mu(2)(5)). This mechanism for chiral charge erasure can be important for the hypercharge sector of the Standard Model as well as extensions including U(1) gauge interactions, such as asymmetric dark matter models.

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Solitons are stable, non-singular solutions to the classical equations of motion of non-linear field theory. Their energy is localized and finite and their shape remains unaltered during propagation. For this reason they represent particle-like states in field theory. Their mass and their size can be very large compared to those of the elementary particles in the theory. Therefore, a soliton can be viewed as a single particle-like object containing a large number of individual particles. The chiral Abelian Higgs model contains an interesting class of non-topological solitons, that carry a non-zero fermion number NF or Chern-Simons number NCS, which is the same because of the chiral anomaly. They consist of a bosonic configuration of gauge and Higgs fields characterized by NCS and are stable for sufficiently large NCS. In the first part of this thesis we study the properties of these anomalous solitons. We find that their energy-versus-fermion-number ratio is given by E ∼ NCS3/4 or E ∼ NCS2/3 depending on the structure of the scalar potential. For the former case we prove, using some inequalities from functional analysis, that there is a lower bound on the soliton energy, which reads E ≥ c NCS3/4 , where c is some parameter expressed through the masses and coupling constants of the theory. We construct the anomalous solitons numerically for two different choices for the potential accounting both for Higgs and gauge dynamics. Solutions are obtained as a function of NCS and the Higgs mass mH and we find that they are not spherically symmetric. In addition, we outline a relation between the structure of anomalous Abelian solitons and the intermediate state observed in type-I superconductors in external magnetic fields. In the limit of large NCS anomalous solitons can be described in the thin wall approximation, which allows us to remove the Higgs field from consideration. For absolute stability of anomalous solitons, it is essential that the gauge group is Abelian. If the gauge group is non-Abelian, fermions can always be converted to a gauge vacuum configuration with an arbitrary integer NCS. Therefore, if anomalous non-Abelian solitons exist, they could only be metastable. Interestingly, anomalous solitons can potentially exist in the electroweak theory, because this theory contains all necessary ingredients, namely chiral fermions and an Abelian gauge symmetry. In the second part of this thesis we investigate this possibility. Using the numerical solutions for anomalous Abelian solitons as a starting point, we construct the corresponding numerical solutions in electroweak theory. These solutions have a similar structure as the Abelian solitons with the Abelian gauge field replaced by the Z boson field. The charged boson fields W± vanish identically. However, for weak mixing angle θω > 0, the solutions have an associated magnetic field as well, that can be characterized by a magnetic dipole moment mem. Furthermore, the shape of the solutions and the structure of the gauge fields depend on θω. In the last part of this work we analyze the classical stability of the numerical solutions in the electroweak case. It is clear that the solutions are stable in the semilocal limit sin θω → 1, where the Abelian case is reproduced exactly. For arbitrary θω, we consider perturbations in the Higgs field and in the gauge fields Z and A and show that the solutions are stable with respect to these perturbations. For small θω however, the solutions are unstable with respect to the formation of a condensate of charged boson fields W± in the centre of the solution. This W-condensation instability is essentially the same, which also destabilizes the Z-string solution of electroweak theory.

An intriguing feature of the Standard Model is that the representations of the unbroken gauge symmetries are vector-like whereas those of the spontaneously broken gauge symmetries are chiral. Here we provide a toy model which shows that a natural explanation of this property could emerge in higher dimensional field theories and discuss the difficulties that arise in the attempt to construct a realistic theory. An interesting aspect of this type of models is that the 4D low energy effective theory is not generically gauge invariant. However, the non-invariant contributions to the observable quantities are very small, of the order of the square of the ratio between the light particle mass scale and the Kaluza-Klein mass scale. Remarkably, when we take the unbroken limit both the chiral asymmetry and the non-invariant terms disappear. © SISSA 2007.

2007Adrien Florio, Daniel Garcia Figueroa

We study fermion number non-conservation (or chirality breaking) in Abelian gauge theories at finite temperature. We consider the presence of a chemical potential mu for the fermionic charge, and monitor its evolution with real-time classical lattice simula- tions. This method accounts for short-scale fluctuations not included in the usual effective magneto-hydrodynamics (MHD) treatment. We observe a self-similar decay of the chemi- cal potential, accompanied by an inverse cascade process in the gauge field that leads to a production of long-range helical magnetic fields. We also study the chiral charge dynamics in the presence of an external magnetic field B, and extract its decay rate Gamma 5 equivalent to mml:mfracd mml:mspace width="0.25em"mml:mspacelog mml:mspace width="0.25em"mml:mspace mu dt mml:mfrac>. We provide in this way a new determination of the gauge coupling and magnetic field de- pendence of the chiral rate, which exhibits a best fit scaling as Gamma 5 proportional to mml:mfrac bevelled="true112mml:mfracB2. We confirm numerically the fluctuation-dissipation relation between Gamma 5 and Gamma(diff) , the Chern-Simons diffusion rate, which was obtained in a previous study. Remarkably, even though we are outside the MHD range of validity, the dynamics observed are in qualitative agreement with MHD predictions. The magnitude of the chiral/diffusion rate is however a factor similar to 10 times larger than expected in MHD, signaling that we are in reality exploring a dif- ferent regime accounting for short scale fluctuations. This discrepancy calls for a revision of the implications of fermion number and chirality non-conservation in finite tempera- ture Abelian gauge theories, though no definite conclusion can be made at this point until hard-thermal-loops are included in the lattice simulations.

2019