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Electroweak Baryogenesis with Cosmic Strings ?

I report on a critical analysis of the scenario of electroweak baryogenesis mediated by nonsuperconducting cosmic strings. This mechanism relies upon electroweak symmetry restoration in a region around cosmic strings, where sphalerons would be unsuppressed. I discuss the various problems this scenario has to face, presenting a careful computation of the sphaleron rates inside the strings, of the chemical potential for chiral number and of the efficiency of baryogenesis in different regimes of string networks. The conclusion is that the asymmetry in baryon number generated by this scenario is smaller than the observed value by at least 10 orders of magnitude.


Introduction
Electroweak baryogenesis 1 is a beautiful idea which fails (or is about to fail) in the best motivated models we have for physics at the Fermi scale (∼ 100 GeV ). In the Standard Model, LEP II experiments set a lower bound on the mass of the Higgs boson of about 97 GeV, implying that the electroweak phase transition in that model is not first order but rather a crossover 2 . In the Minimal Supersymmetric Standard Model the electroweak phase transition can be first order and sufficiently strong to allow for electroweak baryogenesis, but this occurs in a very small region of parameter space 3 which presumably will be ruled out by LEP II in a couple of years.
One may take the previous negative results as indication that the asymmetry in baryon number was not created at the electroweak epoch, but rather related to the physics of B − L violation and neutrino masses. To stick to electroweak baryogenesis one can consider extensions of the particle content of the model to get a stronger electroweak phase transition (e.g. extensions which include singlets). In this talk I will consider another possibility: how the remnants of physics at energy scales higher than the electroweak scale (cosmic strings in this case) can be useful to overcome the problems of having a weak electroweak phase transition.
Electroweak baryogenesis requires the co-existence of regions of large and small ϕ /T , where T is the temperature and ϕ the (T -dependent) Higgs vacuum expectation value. At small or zero ϕ /T sphalerons are unsuppressed and mediate baryon number violation, while large ϕ /T is needed to store the created baryon number (for ϕ /T ≥ 1 sphaleron transitions are ineffective and baryon number is conserved). Below the critical temperature T EW c of the electroweak phase transition and irrespective of whether it is first or second order, ϕ /T grows until sphaleron transitions are shut-off. For baryogenesis to be possible at those times, we need some region where ϕ is forced to remain zero or small. The idea we examine in this talk is that this can be the case along topological defects (like cosmic strings) left over from some other cosmological phase transition that took place before the electroweak epoch 4 . If the electroweak symmetry is restored in some region around the strings, sphalerons could be unsuppressed in the string cores while they would be ineffective in the bulk of space, away from the strings. The motion of the string network, in a similar way as the motion of bubble walls in the usual first-order phase-transition scenario, will leave a trail of net baryon number behind.
Some problems with this scenario come immediately to mind. First, it is clear that the space swept by the defects is much smaller than the total volume, so there will be a geometrical suppression factor with respect to the usual bubble-mediated scenario 4 . Another suppression factor arises from the fact that there is a partial cancellation between front and back walls of the string, which tend to produce asymmetries of opposite signs 4 . Another problem comes from the condition that the symmetry restoration region (which naively would be of size R rest ∼ 1/ √ λ ϕ , where λ is the quartic Higgs coupling) should be large enough to contain sphalerons (which in the symmetric phase have size R sph ∼ 1/g 2 T ), while outside the strings, sphalerons should be suppressed ( ϕ /T ≥ 1). Combining both conditions one obtains λ ≤ g 4 , which means the scenario would require small values of the Higgs mass, in conflict with experimental bounds. LEP II tells us that λ is at least of order g 2 , so that sphalerons won't fit in the restoration region. In other words, for realistic values of the Higgs mass sphalerons are not going to be fully unsuppressed. We will measure how effective they are by writing the rate of sphaleron transitions per unit time and unit of string length as Γ l = κ l α 2 w T 2 . For a string with R rest = R sph , one has Γ l R 2 rest equal to the rate in the symmetric phase, corresponding to κ l ∼ 1. Values of κ l much smaller than 1 would mean that sphalerons are not really unsuppressed inside the strings.
In the rest of the talk I review the careful analysis of this mechanism contained in ref. 5 , to which I refer the interested reader for further details.
admits global strings: configurations with S = 0 along some line (say the zaxis) and S(r) = f (r)S 0 e iθ , with f (∞) → 1, where r is the distance to the z-axis and θ the azimuthal angle. The radius of these strings (where most of the energy is trapped) is set by the scale 1/m S ≡ 1/ √ λ S S 0 . If the U (1) is made local, in addition to the S field, a non-zero gauge field is also present, A µ = −a(r)∂ µ θ/q S , with a(∞) = 1, where q S is the U (1) charge of the S field. This gauge field is such that the covariant derivative D µ S goes to zero for large r resulting in a finite energy per unit length of string.
We assume that S-strings (global or local) form at some temperature T S c > T EW c and are present at the time of the electroweak phase transition. To force ϕ → 0 in the cores of the strings, the Higgs field must interact either with the S field or the A µ field (if the strings are local):

S − ϕ interaction
Suppose the scalar potential has the form with γ > 0. The mass squared of the Higgs field in the string background is m 2 ϕ (r) ∼ γ(S 2 0 − |S(r)| 2 ) − 2λϕ 2 0 , which is negative outside the string core but can be positive inside, so that electroweak symmetry tends to be restored along the strings. Exploring the (S 0 , λ S , γ, λ) parameter space, the typical case, with λ S S 2 0 ≫ λϕ 2 0 leads to R rest ∼ 1/m ϕ (∞). The best posible case to get a large restoration region has λ S ≪ γ ≪ λ and S 0 ≫ ϕ 0 and gives R rest ∼ γ/λ S /m ϕ (∞).

S − A µ interaction
In this case we assume that the Higgs field carries a charge q ϕ under the extra U (1) responsible for the strings, so that its covariant derivative has an extra piece. As we saw, the A µ field in the string goes like −1/q S r at large r to cancel the azimuthal derivative of S, give vanishing D µ S and minimize energy. In D µ ϕ, the A µ contribution is now proportional to q ϕ /q S and the azimuthal derivative of ϕ can cancel D µ ϕ only if q ϕ /q S is an integer. If that is not the case, a Z µ boson condensate is induced until the covariant derivative is cancelled 7 . In any case, a non-zero winding of ϕ forces ϕ → 0 in the string core (r = 0). The restoration region around r = 0 is larger in the presence of a non-zero Z µ string (case of non-integer q ϕ /q S ).

Sphaleron rates and CP asymmetry in the string cores
In general, with no tuning of potential parameters nor a Z µ condensate, ϕ is zero only at the string core (r = 0) and rises inmediately away from that line. As the symmetry is never really restored in a wide region, the energy of the sphaleron in such background (it can be computed in the lattice looking for a saddle point of the energy functional) is only about a factor 0.7 smaller than the sphaleron energy in the broken phase (alternatively κ l ∼ 10 −6 : that is, sphalerons are not really unsuppressed in this type of strings).
The situation is better when a Z µ -field is induced, in which case κ l ∼ 1/30 for ϕ /T ∼ 1 (this number can be obtained in the lattice using a fully non-perturbative approach and tracking Chern-Simons number in real time evolution). However this number is very sensitive to T and drops significantly when T decreases.
Fully unsuppressed sphalerons can only be obtained in the global U (1) case for large enough γ/λ S . In fact, to obtain an asymmetry of the order of the observed one, one would need γ/λ S ∼ 10 14 . On the other hand, stability of the potential requires 4λ/γ > γ/λ S , so that λ/λ S ∼ 10 28 . Such an ad-hoc and wild fine-tuning of the parameters prevents us from taking this particular case seriously.
Unsuppressed sphaleron transitions inside the string cores are not sufficient to generate the baryon asymmetry: they must occur in a background with CP asymmetric particle distributions so that the sign of the B-violation is biased. This asymmetry comes about if the interactions between the particles in the plasma and the string walls violate CP. In that case the walls of a moving string act as sources of chiral-number flux (which would be zero if the string velocity v S were zero). This asymmetry diffuses away from the walls and only that inside the string is useful to create baryons (for geometrical reasons it is also clear that this diffusion effect is less efficient for strings than for bubbles). In conclusion, we have to compute the chemical potential µ for chiral number inside the strings. General arguments (confirmed by detailed analysis of particular models) give the result µ = Kv 2 S T for small v S , with K < ∼ 0.01 and µ = K ′ T for v S ∼ 1 with K ′ of order 1.

Evolution of string networks and efficiency of baryogenesis
To get a final number for the asymmetry generated by this mechanism, we need to know how many strings there are and how quickly they are moving (the best case being that of a dense network of fast moving strings). We can describe the string network by a mean average separation between strings R(t) and a mean average velocity v S (t). The evolution of these quantities with time t is governed by Hubble expansion (H ∼ 1/2t); energy loss by loop formation; and friction with the plasma. The friction force goes like F ∼ v S T 3 : it is important at early times when it dominates the dynamics of the evolution. This is the friction dominated or Kibble regime, with R(t) ∼ t 5/4 and v S (t) ∼ t 1/4 ∼ HR(t).
Eventually, friction will no longer be important and a scaling regime is reached with R(t) ∼ 1/H and v S ∼ 1.
In conclusion, to get the final number for the baryon asymmetry we start with the equation for the rate of change of baryon number N B per unit time and unit length of string: If we use the results for κ l and µ previously discussed, and integrate eq.(3) in one Hubble time (this is because κ l is shut-off quickly with decreasing T ) using the network evolution results just presented we end up with the result that That is, the mechanism just studied is uncapable of generating a sufficiently large matter-antimatter asymmetry.