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 models as long as the gauge sector is common to the standard model.
We develop an analytical technique to evaluate the magnetic helicity in the debris from sphaleron decay. We show that baryon number production leads to left-handed magnetic fields, and that the magnetic helicity is conserved at late times. Our analysis explicitly demonstrates the connection between sphaleron-mediated cosmic baryogenesis and cosmic magnetogenesis.
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…
The recently discovered inverse magnetic catalysis around the critical temperature indicates that some important information is missing in our current understanding of conventional chiral dynamics of QCD, which is enhanced by the magnetic field. In this work, we provide a mechanism to explain that the inverse magnetic catalysis around the critical temperature is induced by sphalerons. At high temperatures, sphaleron transitions between distinct classical vacua cause an asymmetry between the number of right- and left-handed quarks due to the axial anomaly of QCD. In the presence of a strong mag…
The presence of a primordial magnetic field in the early universe affects the dynamic of the electroweak phase transition enhancing its strength. This effect may enlarge the window for electroweak baryogenesis in the minimal supersymmetric extension of the standard model or even resurrect the electroweak baryogenesis scenario in the standard model. We compute the sphaleron energy in the background of the magnetic field and show that, due to the sphaleron dipole moment, the barrier between topologically inequivalent vacua is lowered. Therefore, the preservation of the baryon asymmetry calls for…
We describe electroweak strings and their ability to carry Chern-Simons number. Certain string configurations, for any $θ_W$, carry Chern-Simons number equal to that of the sphaleron and we conjecture that such strings are ``extended sphalerons''. The production of electroweak strings during the phase transition is discussed and it is suggested that electroweak strings could survive long enough for them to play a role in baryogenesis. Finally the issue of magnetic field generation during the electroweak phase transition is addressed and several points not discussed in the existing literature a…
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…
The chiral magnetic effect is a novel quantum phenomenon proposed for high-energy nuclear collisions but which has yet to be observed. We quantify the axial charge relaxation time, due to sphalerons, which enters in simulations of this effect. An extrapolation of weak coupling calculations of the sphaleron rate yields rather different relaxation times than strong coupling AdS/CFT calculations. The AdS/CFT relaxation time is the larger one of the two by an order of magnitude, but the weak coupling relaxation time may not be reliable because it is only marginally bigger than the microscopic ther…
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…
This work is based on a paper with Margaret James, to appear in Phys Rev D. In it we showed that the dipole moment of the sphaleron has its origin in two components: a ring of electric current circulating around the edge of the sphaleron; and also two regions of opposite magnetic charge above and below the ring. This magnetic charge has a partly topological explanation, arising from the fact that the sphaleron is axisymmetric and parity invariant. Here, I discuss the definition of the electromagnetic field and its sources in the Standard Model, comparing the one we use with the better-known on…
Euclidean d=3 SU(2) Yang-Mills-Chern-Simons (YMCS) theory, including Georgi-Glashow (GGCS) theory, may have solitons in the presence of appropriate mass terms. For integral CS level k and for solitons carrying integral CS number, YMCS is gauge-invariant and consistent. However, individual solitons such as sphalerons and linked center vortices with CS number of 1/2 and writhing center vortices with arbitrary CS number are non-compact; a condensate of them threatens compactness of the theory. We study various forms of the non-compact theory in the dilute-gas approximation, treating the parameter…
Standard decoupling of heavy fermions may fail when there are non-perturbative variations in a scalar field which gives masses to the fermions. One situation of phenomenological relevance is the case of sphalerons in the presence of fermions at finite temperatures. The free energy of a simple model is determined using a non-perturbative technique to study the effect of fermions on the scalar field. The effects of quantum and thermal fermionic fluctuations on the free energy of the thermal sphaleron are calculated, including contributions from the gradients of the scalar field to all orders.
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 topo…
Theoretical arguments suggest that the Higgs sector of the standard model is an effective theory. We parametrize the new physics by means of an effective Lagrangian technique and study its effect on the energy of the electroweak sphaleron. We found that in the presence of a certain class of higher dimension operators the sphaleron energy becomes arbitrarily large as the Higgs boson mass increases. The physical meaning of this result and its implications to the electroweak baryogenesis in the dynamical models are discussed.
We measure the sphaleron rate with the physical parameters of the Standard Model. In particular, we plug into the calculations the recently found Higgs mass m_H = 125 GeV. The sphaleron rate tells us about the efficiency of baryon number violation through sphaleron transitions. These occur above the electroweak scale of 100 GeV and get exponentially suppressed at temperatures substantially below the electroweak crossover. The sphaleron rate enters computations of Baryogenesis via Leptogenesis, where it converts non-zero lepton number into non-zero baryon number. We simulate the effective elect…
We calculate the dissipation of the baryon number after the electroweak phase transition due to thermal fluctuations above the sphaleron barrier. We consider not only the classical Boltzmann factor but also fermionic and bosonic one-loop contributions. We find that both bosonic and especially fermionic fluctuations can considerably suppress the transition rate. Assuming the Langer--Affleck formalism for this rate, the condition that an initial baryon asymmetry must not be washed out by sphaleron transitions leads, in the Minimal Standard Model ($\sinθ_W=0$), to an upper bound for the Higgs mas…
We apply the gradient approach to obtain a path over the sphaleron barrier and to demonstrate the fermionic level crossing phenomenon. Neglecting the mixing angle dependence and assuming that the fermions of a doublet are degenerate in mass we employ spherically symmetric ansätze for the fields. The gradient path over the barrier is smooth, even for large values of the Higgs boson mass or of the fermion mass, where the extremal energy path bifurcates.
We investigate the effect of dynamical fermions on the sphaleron transition rate at finite temperature for the Abelian Higgs model in one spatial dimension. The fermion degrees of freedom are included through bosonization. Using a numerical simulation, we find that massless fermions do not change the rate within the measurement accuracy. Surprisingly, the exponential dependence of the sphaleron energy on the Yukawa coupling is not borne out by the transition rate, which shows a very weak dependence on the fermion mass.
In the electroweak sector of the standard model topologically inequivalent vacua are separated by finite energy barriers, whose height is given by the sphale\-ron. For large values of the Higgs mass there exist several sphaleron solutions and the barriers are no longer symmetric. We construct paths of classical configurations from one vacuum to a neighbouring one and solve the fermion equations in the background field configurations along such paths, choosing the fermions of a doublet degenerate in mass. As in the case of light Higgs masses we observe the level crossing phenomenon also for lar…
We study the process of thermal activation mediated by sphaleron transitions by analyzing the real-time dynamics of the decay out of equilibrium in a $1+1$ dimensional field theory with a metastable state. The situation considered is that of a rapid supercooling in which the system is trapped in a metastable state at a temperature larger than the mass of the quanta, but smaller than the energy to create a critical droplet. The initial density matrix is evolved in time and the nucleation rate (probability current at the saddle point) is computed. The nucleation rate is {\it time dependent}, van…
Since the TRIVIALITY argument of the Higgs sector requires the existence of new physics beyond the standard model, there should exist a cutoff $Λ$ beyond which the standard model will breakdown. The cutoff can be determined from the position of the Landau pole. We study the effects of this cutoff on the energy of the electroweak sphaleron, $E^{spha}$, in the large Higgs boson mass limit. We found that $E^{spha}$ becomes arbitrarily large as the Higgs boson mass increases. This is in contrast to the well-known result, which is held in the standard model and a wide class of its extensions, that …
We construct the sphaleron for several temperature dependent effective potentials. We determine the sphaleron energy as a function of temperature and demonstrate that the sphaleron energy at a given temperature $T$ is well approximated by the sphaleron energy at temperature zero scaled by the ratio of the vacuum expectation values of the Higgs field at temperatures $T$ and zero. We address the cosmological upper bound on the Higgs mass.
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…
The classical transitions between topologically distinct vacua in a SU(2)-Higgs model, using a Higgs field of mass approximately 120 GeV, is examined to probe the crossover region between the symmetric and broken phase. Assuming the Higgs mass is constant, we find the width of this crossover region is approximately 20% of the average temperature. We suggest that this observable is a better parameter to explore this region of phase space than equal-time correlation functions.
I show that a cosmological baryon asymmetry generated at the GUT scale is in general safe against washout due to sphalerons and generic $B$- or $L$-violating effects. This result is mainly due to the (almost) conserved number of right-handed electrons at high temperatures $T \gsim {\cal O}(10) $ TeV, but also the mass corrections, in particular the thermal masses of leptons act as the protector of the primordial baryon asymmetry.
We discuss one-loop radiative corrections to the sphaleron-induced baryon number-violating transition rate near the electroweak phase transition in the standard model. We emphasize that in the case of a first-order transition a rearrangement of the loop expansion is required close to the transition temperature. The corresponding expansion parameter, the effective 3-dimensional gauge coupling approaches a finite $λ$ dependent value at the critical temperature. The $λ$ (Higgs mass) dependence of the 1-loop radiative corrections is discussed in the framework of the heat kernel method. Radiative c…
The baryon number dissipation rate due to sphaleron transitions at high temperatures in the minimal standard model is evaluated. We find that this rate can be considerably suppressed by one loop contributions of bosonic and fermionic fluctuations which are particularly important for a small mass of the Higgs boson and a large top quark mass. Fixing the latter to its recently stated value of 174 GeV the complete erasure of the baryon asymmetry is prevented within the framework of the minimal standard model if the Higgs mass is less than about 66 GeV.
We study the problem of existence of finite energy monopole solutions in the Weinberg-Salam model starting with a most general ansatz for static axially-symmetric electroweak magnetic fields. The ansatz includes an explicit construction of field configurations with various topologies described by the monopole and Hopf charges. We introduce a unique SU(2) gauge invariant definition for the electromagnetic field. It has been proved that the magnetic charge of any finite energy monopole solution must be screened at far distance. This implies non-existence of finite energy monopole solutions with …
We start by considering the production rates of sphalerons with different size $ρ$ in the symmetric phase, $T>T_{EW}$. At small $ρ$, the distribution is cut off by the growing mass $M\sim 1/ρ$, and at large $ρ$ by the magnetic screening mass. In the broken phase, $T<T_{EW}$ the scale is set by the Higgs VEV $v(T)$. We introduce the concept of "Sphaleron freezeout" whereby the sphaleron production rate matches the Hubble Universe expansion rate. At freezeout the sphalerons are out of equilibrium. Sphaleron explosions generate sound and even gravity waves, when nonzero Weinberg angle make them n…
We investigate the effect on the sphaleron in the two Higgs doublet electroweak theory of including CP violation in the Higgs potential. To have better control over the relation between the sphaleron energy and the physical quantities in the theory, we show how to parametrize the Higgs potential in terms of physical masses and mixing angles, one of which causes CP violation. By altering this CP violating angle (and keeping the other physical quantities fixed) the sphaleron energy increases by up to 10%. We also calculate the static minimum energy path between adjacent vacua as a function of Ch…
The electroweak phase transition and the sphaleron decoupling condition in the MSSM are revisited taking the latest experimental data into account. The light Higgs boson scenario and the ordinary decoupling limit which are classified by the relative size between the CP-odd Higgs boson mass and Z boson mass are considered within the context of electroweak baryogenesis. We investigate v/T at not only the critical temperature at which the effective potential has two degenerate minima but also the nucleation temperature of the critical bubbles, where v is a vacuum expectation value of the Higgs bo…
In 2+1 dimensional gauge thories with $SU(N_c)$ color and $2N_f$ flavors of quarks in the fundamental representation, sphalerons, if present may lead to quark condensation and chiral symmetry breaking. The effect is similar to instanton induced chiral symmetry breaking in 3+1 dimensions and is due to the interaction of quark propagators with sphalerons. The existence of sphalerons requires that color symmetries be broken by a Higgs in the fundamental representation. We show that the sphaleron effect may persist for arbitrarily large $N_f$ but vanishes along with the mass of the gauge fields co…
We discuss baryogenesis in scenarios where the Universe is reheated to temperatures $\lesssim 100\,$GeV by the decay of long-lived massive particles into energetic SM particles. Before its thermalization, the center-of-mass energy in collisions between such a particle and a particle from the ambient plasma can be higher than the typical sphaleron mass, even if the temperature of the plasma itself is much lower. Optimistic estimates for the high energy enhancement of the sphaleron cross section suggest that successful baryogenesis is possible for reheating temperatures as low as $0.1\text{-}1\,…
We construct the sphaleron solution, at zero and finite temperature, in the Minimal Supersymmetric Standard Model as a function of the supersymmetric parameters, including the leading one-loop corrections to the effective potential in the presence of the sphaleron. At zero temperature we have included the one-loop radiative corrections, dominated by the top/stop sector. The sphaleron energy $E_{\rm MSSM}$ mainly depends on an effective Higgs mass ${\displaystyle m_h^{\rm eff}=\lim_{m_A\gg m_W} m_h}$, where $m_h$ is the lightest CP-even Higgs mass and $m_A$ the pseudoscalar mass. We have compar…
We report on work studying the properties of the sphaleron in models of the electroweak interactions with two Higgs doublets in as model-independent a way as possible: by exploring the physical parameter space described by the masses and mixing angles of the Higgs particles. If one of the Higgs particles is heavy, there can be several sphaleron solutions, distinguished by their properties under parity and the behaviour of the Higgs field at the origin. In general, these solutions are not spherically symmetric, although the departure from spherical symmetry is small.
We construct multisphaleron solutions in the Weinberg-Salam theory. The multisphaleron solutions carry Chern-Simons charge $n/2$, where $n$ is an integer, counting the winding of the fields in the azimuthal angle. The well-known sphaleron has $n=1$. The multisphalerons possess axial symmetry and parity reflection symmetry. We vary the Higgs mass and the mixing angle. For small $n$ the energies of the multisphalerons are on the order of $n$ times the energy of the sphaleron and their magnetic dipole moments are on the order of $n$ times the magnetic dipole moment of the sphaleron.
Sphalerons of a new gauge interaction can convert a primordial asymmetry in B or L into a dark matter asymmetry. From the equilibrium conditions for the sphalerons of both the electroweak and the new interactions, one can compute the ratios of B, L, and X, where X is the dark matter number, thus determining the mass of the dark matter particle fairly precisely. Such a scenario can arise naturally in the context of unification with larger groups. An illustrative model embeddable in $SU(6) \times SU(2) \subset E_6$ is described as well as an equally simple model based on SU(7).
As a result of the Standard Model chiral anomalies, baryon number is violated in the early universe in the presence of a hypermagnetic field with varying helicity. We investigate whether the matter / anti-matter asymmetry of the universe can be created from the decaying helicity of a primordial (hyper)magnetic field before and after the electroweak phase transition. In this model, baryogenesis occurs without $(B-L)$-violation, since the $(B+L)$ asymmetry generated by the hypermagnetic field counteracts the washout by electroweak sphalerons. At the electroweak crossover, the hypermagnetic field…
We present a new mechanism for generating the baryon asymmetry of the universe directly in the decay of a singlet scalar field $S_r$ with a weak scale mass and a high dimensional baryon number violating coupling. Unlike most currently popular models, this mechanism, which becomes effective after the electroweak phase transition, does not rely on the sphalerons for inducing a nonzero baryon number. CP asymmetry in $S_r$ decay arises through loop diagrams involving the exchange of $W^\pm$ gauge bosons, and is suppressed by light quark masses, leading naturally to a value of $η_B \sim 10^{-10}$. …
We investigate the $B+L$ violation process by performing three-dimensional lattice simulations in an electroweak theory with first-order phase transition and the electroweak sphaleron decay. The simulation results indicate that the Chern-Simons number changes along with the helical magnetic field production when the sphaleron decay occurs. Our study suggests that, for the electroweak phase transition with nucleation rate being smaller than $\sim \mathcal{O}(10)$, the helical magnetic field with the fractional magnetic helicity $ε_M\leq 0.2$ can be probed by Cherenkov Telescope Array through th…
We present details for a method to compute the broken phase sphaleron rate (rate of hot baryon number violation below the electroweak phase transition) nonperturbatively, using a combination of multicanonical and real time lattice techniques. The calculation includes the ``dynamical prefactor,'' which accounts for prompt recrossings of the sphaleron barrier. The prefactor depends on the hard thermal loops, getting smaller with increasing Debye mass; but for realistic Debye masses the effect is not large. The baryon number erasure rate in the broken phase is slower than a perturbative estimate …
Using lattice simulations, we measure the sphaleron rate in the Standard Model as a function of temperature through the electroweak cross-over, for the Higgs masses m_H=115 and m_H=160 GeV. We pay special attention to the shutting off of the baryon rate as the temperature is lowered. This quantity enters computations of Baryogenesis via Leptogenesis, where non-zero lepton number is converted into non-zero baryon number by equilibrium sphaleron transitions. Combining existing numerical methods applicable in the symmetric and broken electroweak phases, we find the temperature dependence of the s…
We consider the top-bottom doublet in the background of the sphaleron for the realistic case of large non-degeneracy of fermion masses, in particular $m_b=5$ GeV and $m_t=175$ GeV. We propose an axially symmetric $(r,θ)$-dependent ansatz for fermion fields and investigate the effects of the non-degeneracy on them. The exact solution is described, with an error less than 0.01\%, by a set of ten radial functions. We also propose an approximate solution, in the $m_b/m_t\rightarrow 0$ limit, with an error ${\cal O}(m_b/m_t)$. We have found that the effects of non-degeneracy provide a $θ$-dependenc…
We discuss the effects of various processes that can be active during the leptogenesis era, and present the Boltzmann equations that take them into account appropriately. A non-vanishing Higgs number asymmetry is always present, enhancing the washout of the lepton asymmetry. This is the main new effect when leptogenesis takes place at $T>10^{12}$ GeV, reducing the final baryon asymmetry and tightening the leptogenesis bound on the neutrino masses. If leptogenesis occurs at lower temperatures, electroweak sphalerons partially transfer the lepton asymmetry to a baryonic one, while Yukawa interac…
We discuss aidnogenesis, the generation of a dark matter asymmetry via new sphaleron processes associated to an extra non-abelian gauge symmetry common to both the visible and the dark sectors. Such a theory can naturally produce an abundance of asymmetric dark matter which is of the same size as the lepton and baryon asymmetries, as suggested by the similar sizes of the observed baryonic and dark matter energy content, and provide a definite prediction for the mass of the dark matter particle. We discuss in detail a minimal realization in which the Standard Model is only extended by dark matt…
New defects (Z-vortices and Nambu monopoles) are found to become thermodynamically relevant for the broken phase near to the (weakly first order) electroweak phase transition, and below the crossover for higher Higgs mass. The symmetric phase is characterized by vortex condensation (percolation). The percolation transition persists in the crossover region. The quasiclassical nature of the vortices is supported by correlations with Higgs field and gauge field energy. Sphalerons are shown to be related to monopolium bound states.
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.
We elaborate upon the model of baryogenesis from decaying magnetic helicity by focusing on the evolution of the baryon number and magnetic field through the Standard Model electroweak crossover. The baryon asymmetry is determined by a competition between the helical hypermagnetic field, which sources baryon number, and the electroweak sphaleron, which tends to wash out baryon number. At the electroweak crossover both of these processes become inactive; the hypermagnetic field is converted into an electromagnetic field, which does not source baryon number, and the weak gauge boson masses grow, …
We construct multisphaleron solutions in the weak interactions. The multisphaleron solutions carry Chern-Simons charge $n/2$, where $n$ is an integer. The well-known sphaleron has $n=1$ and is spherically symmetric for vanishing mixing angle. In contrast the multisphalerons with $n>1$ are only axially symmetric, even for vanishing mixing angle. The greater $n$, the stronger the energy density deforms. While for small Higgs masses the energy of the multisphalerons is smaller than $n$ times the energy of the sphaleron, the energy of the multisphalerons is larger than $n$ times the energy of the …
We investigate the phenomenological aspects of non-perturbative baryon- and lepton-number-violating processes at hadron colliders. Such processes, induced by instanton/sphaleron configurations of the electroweak gauge fields, are believed to play a crucial role in the generation of baryon asymmetry in the early Universe at finite temperature. On the other hand, at colliders (that represent the zero-temperature high-energy regime) the rate and observability of such processes are still under debate. Motivated by current theoretical considerations, we construct a modern event generator within the…
The generation of the observed baryon asymmetry may have taken place during the electroweak phase transition, thus involving physics testable at LHC, a scenario dubbed electroweak baryogenesis. In this paper we point out that the magnetic field which is produced in the bubbles of a first order phase transition endangers the baryon asymmetry produced in the bubble walls. The reason being that the produced magnetic field couples to the sphaleron magnetic moment and lowers the sphaleron energy; this strengthens the sphaleron transitions inside the bubbles and triggers a more effective wash out of…
We construct the multi-charge generalizations for the electroweak magnetic monopole solution of Cho and Maison within a wide range of values of the magnetic charge. We use the same ansatz for the axially symmetric fields as the one previously employed to construct the electroweak sphalerons and compare the internal structure of monopoles with that of sphalerons. The monopoles have zero dipole moment but a finite quadrupole momentum that rapidly increases with growing magnetic charge. For large charges, the monopole configurations are strongly squashed and show inside a bubble of symmetric phas…
If the present baryon-asymmetry is due to a Planck or GUT-scale matter asymmetry then baryon- or lepton-number violating processes are constrained by the condition that they do not subsequently erase this asymmetry. We present a revision of the analysis of sphaleron baryon-number violating processes in the standard model including lepton-mass effects. We find the surprising result that a GUT-scale matter-asymmetry can survive the $B$ and $L$ violating sphaleron interactions even though ($B- L$) is conserved and equals zero for all temperatures. We extend the analysis to cover the minimal super…
We present a semiclassical study of the suppression of topology changing, baryon number violating transitions induced by particle collisions in the electroweak theory. We find that below the sphaleron energy the suppression exponent is remarkably close to the analytic estimates based on a low energy expansion about the instanton. Above the sphaleron energy, the relevant semiclassical solutions have qualitatively different properties from those below the sphaleron: they correspond to jumps on top of the barrier. Yet these processes remain exponentially suppressed, and, furthermore, the tunnelin…
I calculate the rate of $B+L$ violating processes in the ``symmetric" phase of the Standard Model by means of semiclassical methods, which are based on an appropriately resummed loop expansion within the dimensionally reduced theory. The rate is found to reach its asymptotic form $Γ/V=c(\a_W T)^4$ at a temperature about three times the critical temperature of the electroweak phase transition, with a coefficient $c\sim 0.01$ at the one-loop level. The order of magnitude of $c$ is sensitive to higher order corrections to the dynamically generated vector boson mass if these are greater than 20\%.…
