We present numerical evidence for the existence of stationary spinning generalizations for the static sphaleron in the Weinberg-Salam theory. Our results suggest that, for any value of the mixing angle $\thetw$ and for any Higgs mass, the spinning sphalerons comprise a family labeled by their angular momentum $J$. For $\thetw\neq 0$ they possess an electric charge $Q=eJ$ where $e$ is the electron charge. Inside they contain a monopole-antimonopole pair and a spinning loop of electric current, and for large $J$ they show a Regge-type behavior. It is likely that these sphalerons mediate the topological transitions in sectors with $J\neq 0$, thus enlarging the number of transition channels. Their action {\it decreases} with $J$, which may considerably affect the total transition rate.
We correct an error in our treatment of the tadpole contribution to the fluctuation determinant of the sphaleron, and also a minor mistake in a previous estimate. Thereby the overall agreement between the two existing exact computations and their consistency with the estimate is improved considerably.
We present the relation between the sphaleron energy and the gravitational wave signals from a first order electroweak phase transition. The crucial ingredient is the scaling law between the sphaleron energy at the temperature of the phase transition and that at zero temperature. We estimate the baryon number preservation criterion, and observe that for a sufficiently strong phase transition, it is possible to probe the electroweak sphaleron using measurements of future space-based gravitational wave detectors.
We perform a systematic study of stationary sphaleron-antisphaleron systems of Weinberg-Salam theory at the physical value of the weak mixing angle. These systems include rotating sphaleron-antisphaleron pairs, chains and vortex rings. We show that the angular momentum of these solutions is proportional to their electric charge. We study the dependence of their energy and magnetic moment on their angular momentum. We also investigate the influence of their angular momentum on their local properties, in particular on their energy density and on the node structure of their Higgs field configurat…
A numerical study of static, spherically symmetric sphaleron solutions in the standard model coupled to the dilaton field is presented. We show that sphaleron is surrounded by strong dilaton cloud which vanishes inside the sphaleron.
In this work we investigate the sphaleron solution in a $SU(2)\times U(1)_X$ gauge theory, which also encompasses the Standard Model, with higher scalar representation(s) ($J^{(i)},X^{(i)}$). We show that the field profiles describing the sphaleron in higher scalar multiplet, have similar trends like the doublet case with respect to the radial distance. We compute the sphaleron energy and find that it scales linearly with the vacuum expectation value of the scalar field and its slope depends on the representation. We also investigate the effect of $U(1)$ gauge field and find that it is small f…
At finite weak mixing angle the sphaleron solution of Weinberg-Salam theory can be endowed with angular momentum proportional to the electric charge. Here we show, that this holds also for sphaleron-antisphaleron systems such as pairs, chains and vortex rings. We also address the equilibrium conditions for these solutions.
After the discovery of the Higgs boson and the rather precise measurement of all electroweak boson's masses the local structure of the electroweak symmetry breaking potential is already quite well established. However, despite being a key ingredient to a fundamental understanding of the underlying mechanism of electroweak symmetry breaking, the global structure of the electroweak potential remains entirely unknown. The existence of sphalerons, unstable solutions of the classical action of motion that are interpolating between topologically distinct vacua, is a direct consequence of the Standar…
In an external magnetic field, the energy of the electroweak sphaleron---representing the energy barrier to baryon and lepton number violation---decreases but remains nonzero until the upper Ambjorn-Olesen critical field strength set by the Higgs mass and the electric charge, where it vanishes. We demonstrate this by numerically computing the sphaleron configuration in the presence of an external magnetic field, over the full range of field strengths until the energy barrier vanishes. We discuss the implications for baryogenesis in the early universe, and the possibility of observing of baryon…
New physics at the TeV scale can affect the dynamics of the electroweak phase transition in many ways. In this note, we evaluate its impact on the rate of baryon-number violation via sphaleron transitions. We parameterize the effect of new physics with dimension-6 operators, and we use the Newton-Kantorovich method to numerically solve the resulting equations of motion. Depending on the sign of the coefficient of the dimension-6 operators, their presence can either increase or decrease the sphaleron energy at the level of a few percent, parametrically of order $m_W^2 / Λ^2$ where $Λ$ is the sc…
We elucidate a magnetic mass effect on a sphaleron energy that is crucial for baryon number preservation needed for successful electroweak baryogenesis. It is found that the sphaleron energy increases in response to the magnetic mass. As an application, we study the sphaleron energy and electroweak phase transition with the magnetic mass in a two-Higgs-doublet model. Although the magnetic mass can screen the gauge boson loops, it relaxes a baryon number preservation criterion more effectively, broadening the baryogenesis-possible region. Our findings would be universal in any new physics model…
3D electroweak sphalerons on the lattice are used as test configurations for definitions of various topological defects. In the maximally Abelian gauge they are shown to contain a symmetric array of Abelian monopoles and anti-monopoles connected by two kinds of Abelian vortex strings. Gauge-invariant lattice definitions of the Nambu monopole and the Z-vortex string are formulated which correspond to Abelian projection from the unitary gauge. The sphalerons contain in their core just one (non-Abelian) Nambu monopole-anti-monopole pair (connected by a Z-string) in an unstable saddle point bound …
Nonlinear electroweak sphaleron clouds around Kerr black holes
Universidade de Aveiro（阿威罗大学）；Centre for Research and Development in Mathematics and Applications (CIDMA)（数学与应用数学研究与发展中心）；Universität Oldenburg（奥尔登堡大学）；U. Federal do Pará（帕拉联邦大学）
We propose a formalism to obtain the electroweak sphaleron, which is one of the static classical solutions, using the gradient flow method. By adding a modification term to the gradient flow equation, we can obtain the sphaleron configuration as a stable fixed point of the flow in the large flow time. Applying the method to the $SU(2)$-Higgs model (the Weinberg angle $θ_W$ is $0$) in four dimensions, we obtain the sphaleron configuration whose energy coincides with previous studies numerically. We can also show that the Chern-Simons number of the solution has a half-integer.
Spectral flow of chiral fermions in the background of electroweak sphalerons is studied. A fermion field configuration that interpolates between the sphaleron zero mode and the asymptotic fermion modes is proposed for the level crossing left handed fermion fields. It is shown that the fermionic electromagnetic currents with non-trivial helicity are produced during the level crossing. Cosmic magnetic field generation due to these currents is briefly discussed.
For the periodic sphaleron potential in the electroweak theory, we find the one-dimensional time-independent Schrödinger equation with the Chern-Simons number as the coordinate, construct the Bloch wave function and determine the corresponding conducting (pass) band structure. We show that the baryon-lepton number violating processes can take place without the exponential tunneling suppression (at zero temperature) at energies around and above the barrier height (sphaleron energy) at 9.0 TeV. Phenomenologically, probable detection of such processes at LHC is discussed.
In the Randall-Sundrum model where the radion is stabilized by a Goldberger-Wise (GW) potential there is a supercooled transition from a deconfined to a confined phase at temperatures orders of magnitude below the typical Standard Model critical temperature. When the Higgs is localized at the IR brane the electroweak phase transition is delayed and becomes a strong first-order one where the Universe expands by a few e-folds. This generates the possibility of having the out-of-equilibrium condition required by electroweak baryogenesis in the electroweak phase transition. We have studied numeric…
We make use of a semiclassical method for calculating the suppression exponent for topology changing transitions in high-energy electroweak collisions. In the Standard Model these processes are accompanied by violation of baryon and lepton number. By using a suitable computational technique we obtain results for s-wave scattering in a large region of initial data. Our results show that baryon and lepton number violation remains exponentially suppressed up to very high energies of at least 30 sphaleron masses (250 TeV). We also conclude that the known analytic approaches inferred from low energ…
We present a new scenario for the baryon number violation that may take place in models with large extra dimensions. Our idea is interesting because leptogenesis with a low reheating temperature requires an alternative source of the B+L violation to convert the produced leptons into baryons. If one considers a model with an intermediate compactification radius, the reheating temperature may be allowed to become higher than the critical temperature of the conventional electroweak phase transition. Our mechanism is still important in this case, because it provides the phase boundary, which is an…
The vortex solution (Z-string) of the electroweak interactions can be interpreted as the 2-dimensional sphaleron at the top of a non-contractible sphere. The same holds for another type of solution, the W-string.
Many models of baryogenesis rely on anomalous particle physics processes to give baryon number violation. By numerically evolving the electroweak equations on a lattice, we show that baryogenesis in these models creates helical cosmic magnetic fields. After a transitory period, electroweak dynamics is found to conserve the Chern-Simons number and the total electromagnetic helicity. We argue that baryogenesis could lead to magnetic fields of nano-Gauss strength today on astrophysical length scales. In addition to being astrophysically relevant, such helical magnetic fields can provide an indepe…
We study the SU(2) electroweak model in which the standard Yang-Mills coupling is supplemented by a Born-Infeld term. The deformation of the sphaleron and bisphaleron solutions due to the Born-Infeld term is investigated and new branches of solutions are exhibited. Especially, we find a new branch of solutions connecting the Born-Infeld sphaleron to the first solution of the Kerner-Gal'tsov series.
Results of a large scale numerical simulation show that the high temperature Chern-Simons number diffusion rate in the electroweak theory has a classical limit $Γ= κ(α_w T)^4$, where $κ= 1.09\pm 0.04$ and $α_w$ is the weak fine structure constant.
We reanalyse the sphaleron bound of electroweak baryogenesis when allowing deviations to the Friedmann equation. These modifications are well motivated in the context of brane cosmology where they appear without being in conflict with major experimental constraints on four-dimensional gravity. While suppressed at the time of nucleosynthesis, these corrections can dominate at the time of the electroweak phase transition and in certain cases provide the amount of expansion needed to freeze out the baryon asymmetry without requiring a strongly first order phase transition. The sphaleron bound is …
The standard requirement for the production of baryons at the electroweak phase transition, that the phase transition be first order and the sphaleron bound be satisfied, is predicated on the assumption of a radiation dominated universe at that epoch. One simple alternative - domination by the energy in a kinetic mode of a scalar field which scales as $1/a^6$ - gives a significantly weakened sphaleron bound for the preservation of a baryon asymmetry produced at a first-order phase transition, and allows the possibility that the observed baryon asymmetry be produced when the phase transition is…
We initiate a study of cosmological implications of sphaleron-mediated CP-violation arising from the electroweak vacuum angle under the reasonable assumption that the semiclassical suppression is lifted at finite temperature. In this article, we explore the implications for existing scenarios of baryogenesis. Many compelling models of baryogenesis rely on electroweak sphalerons to relax a $(B+L)$ charge asymmetry. Depending on the sign of the CP-violating parameter, it is shown that the erasure of positive $(B+L)$ will proceed more or less quickly than the relaxation of negative $(B+L)$. This …
The sphaleron type solution in the electroweak theory, generalized to include the dilaton field, is examined. The solutions describe both the variarions of Higgs and Gauge fields inside the sphaleron and the shape of the dilaton cloud surrounding the sphaleron. Such a cloud is large and extends far outside. These phenomena may play an important role during the baryogenesis which probably took place in the Early Universe.
It is widely believed that the existence of singlet scalars in some Standard Model extensions can easily make the electroweak phase transition strongly first order, which is needed for the electroweak baryogenesis scenario. In this paper, we will examine the strength of the electroweak phase transition in the simplest extension of the Standard Model with a real singlet using the sphaleron energy at the critical temperature. We find that the phase transition is stronger by adding a singlet; and also that the criterion for a strong phase transition Omega (Tc)/Tc>1, where Omega =(v^2+(x-x0)^2)^(1…
In this paper we describe a new type of topological defect, called a homilia string, which is stabilized via interactions with the string network. Using analytical and numerical techniques, we investigate the stability and dynamics of homilia strings, and show that they can form stable electroweak strings. In SU(2)xU(1) models of symmetry breaking the intersection of two homilia strings is identified with a sphaleron. Due to repulsive forces, the homilia strings seperate, resulting in sphaleron annihilation. It is shown that electroweak homilia string loops cannot stabilize as vortons, which c…
