Electroweak Baryogenesis with a Second Order Phase Transition
이전에 제안된 전자약리력 baryogenesis의 대부분은 첫 번째 단계 전환을 필요로했다. 하지만, 이 연구에서는 두 번째 단계 전환에서도 stable electroweak strings를 이용한 baryogenesis를 가능하게 한다.
electroweak strings는 SU(2) × U(1) 이론에서 Nielsen-Olesen strings의 아벨 subgroup에 의해 형성된다. Strings은 안정적일 수 있으며, 안정적인 strings network가 formation될 것이다. Inside the string, anomalous baryon number violating processes가 발생하며, strings이 이동하면 thermal equilibrium를 벗어날 것이라 예상한다.
이 논문에서는 two Higgs model의 prototype로 사용되는 Lagrangian에 explicit CP violation을 추가하여 이 현상을 설명한다. Two Higgs model은 electroweak phase transition에서 stable electroweak strings가 존재할 수 있다고 시사한다.
연구에서는 electroweak string contraction으로 인해 net baryon symmetry를 생성하는 것을 보여주고, 이는 전자약리력 phase transition의 second order condition과 compatible하다고 주장한다.
이 연구는 전자약리력 baryogenesis scenario에 대한 새로운 접근법을 제공하며, stable electroweak strings가 존재할 경우 baryon to entropy ratio를 생성하는 것으로 나타난다.
한글 요약 끝:
영어 요약 시작:
This paper proposes a possibility of implementing electroweak baryogenesis using stable electroweak strings during the second order phase transition.
Most previously suggested mechanisms for electroweak baryogenesis relied on having a first-order phase transition, but this work shows that it is possible to generate baryon asymmetry in the universe without such a transition.
Electroweak strings are non-topological solitons arising in the standard electroweak theory (and its extensions). If stable, they can form a network during the phase transition. Inside the string, anomalous baryon number violating processes occur, and if the string moves out of thermal equilibrium is expected.
The paper uses the two-Higgs model as an example to describe this phenomenon with explicit CP violation added to the Lagrangian. The two-Higgs model suggests that stable electroweak strings could exist during the electroweak phase transition.
The research shows how net baryon symmetry can be generated through string contraction and argues that it is compatible with the second-order condition of the electroweak phase transition.
영어 요약 끝:
Electroweak Baryogenesis with a Second Order Phase Transition
arXiv:hep-ph/9206260v1 29 Jun 1992BROWN-HET-865Electroweak Baryogenesis with a Second Order Phase Transition⋆Robert H. Brandenberger1)andAnne-Christine Davis2)1) Physics Department, Brown UniversityProvidence, RI 02912, USA2) Department of Applied Mathematicsand Theoretical Physics & Kings CollegeUniversity of Cambridge, Cambridge, CB39EW, U.K.ABSTRACTIf stable electroweak strings are copiously produced during the electroweakphase transition, they may contribute significantly to the presently observed baryonto entropy ratio of the universe. This analysis establishes the feasibility of imple-menting an electroweak baryogenesis scenario without a first order phase transi-tion.⋆To be published in the proceedings of the March 1992 Texas-Yale Workshop on ElectroweakBaryogenesis, ed.
by L. Krauss and S.-J. Rey (World Scientific, Singapore, 1992)1
1. IntroductionThe mechanisms suggested so far1−4) for electroweak baryogenesis all rely onhaving a first order phase transition.
The resulting bubble walls were required inorder to obtain a region of unsuppressed baryon number violation occurring outof thermal equilibrium. Our work5) is based on the observation that topologicaldefects forming in a second order phase transition may play a similar role to thebubble walls.
We propose a specific mechanism in which electroweak strings areresponsible for baryogenesis.Electroweak strings6) are nontopological solitons which arise in the standardelectroweak theory (and extensions thereof). They are essentially Nielsen-Olesenstrings of U(1)Z embedded in the SU(2) × U(1) theory (U(1)Z is the Abeliansubgroup which is broken during the electroweak phase transition).
For certainranges of the parameters of the standard model, electroweak strings are energeti-cally stable7) (They are not topologically stable).If, however, we are in a region of parameter space in which electroweak stringsare stable, a network of such strings will form during the electroweak phase tran-sition, even if it is second order. Inside the strings, anomalous baryon numberviolating processes are unsuppressed.
If the strings move, the out of thermal equi-librium condition will be satisfied. Finally, the standard model contains CP vio-lation.
Hence all of Sakharov’s criteria are satisfied. As we shall demonstrate, it isin fact possible to generate a substantial nB/s using electroweak strings.In order to obtain a sufficiently large baryon to entropy ratio, the standardelectroweak model must be extended by adding new terms in the Lagrangian whichcontain explicit CP violation.
An often used prototype theory is the two Higgsmodel.2−4)The construction of nontopological vortex solutions in theories which do notsatisfy the topological criterion for strings is not specific to the minimal standardmodel.Thus, we expect electroweak strings to exist also in extensions of the2
Weinberg-Salam model (This has recently been demonstrated in the two Higgsmodel8)). It is possible that these strings could be stable even for experimentallyallowed values for the model parameters.
In the following we shall assume thatelectroweak strings exist and are stable.In models admitting stable electroweak strings, a network of such strings willform during the electroweak phase transition. If we consider a theory with HiggspotentialV (φ) = λ(φ+φ −η2/2)2,(1)then the initial correlation length (mean separation of strings) will be9)ξ(tG) ≃λ−1η−1,(2)where tG is the time corresponding to the Ginsburg temperature of the phasetransition.The initial network of electroweak strings will be quite different from that ofcosmic strings, the reason being that electroweak strings can end on local monopoleand antimonopole configurations.
From thermodynamic considerations10), we ex-pect most of the strings to be short, i.e., of length l ≃ξ(tG), since this maximizesthe entropy of the network for fixed energy.After the phase transition, the vortices will contract along their axes and decayafter a time interval∆tS ≃1v(λη)−1(3)where v is the velocity of contraction (expected to be ≃1). In the following, weshall demonstrate that the string contractions will produce a net baryon symmetry.We are using units in which c = ¯h = kB = 1.3
2. The Baryogenesis MechanismWe shall consider an extension of the standard electroweak theory in whichthere is additional CP violation in the Higgs sector.
An example is the two Higgsmodel used in Refs. 2-4.
We assume that electroweak strings can be embedded inthis model8), and we choose the values of the parameters in the Lagrangian forwhich these strings are stable. Furthermore, the phase transition is taken to besecond order.A key issue is the formation probability of electroweak strings.
In the following,we make the rather optimistic assumption that both the mean length and averageseparation of electroweak strings at tG will equal the correlation length ξ(tG).For topological defects, this result follows from the Kibble mechanism9). Whenapplied to electroweak strings, the Kibble mechanism implies that the vortex fieldsφ and Z have the correlation length ξ(tG).
However, to form an electroweak string,the other fields must be sufficiently small such that the configuration relaxes tothe exact electroweak string configuration. Obviously, the restriction this imposes(and the consequent increase in the mean separation of electroweak strings) isparameter dependent - the more stable the strings, the smaller the increase in themean separation.
Pieces of string are bounded by monopole-antimonopole pairs.Energetic arguments tell us that the string will shrink. We now argue that themoving string ends will have the same effects on baryogenesis as the expandingbubble walls in Refs.
2&3.We can phrase our argument either in terms of the language of Ref. 2 or of Ref.3.
The phase of the extra CP violation is nonvanishing in the region in which theHiggs fields φ are changing in magnitude, i.e., at the edge of the string. Since |φ|increases in magnitude, CP violation has a definite sign.
Hence, in the languageof Ref. 3, a chemical potential with definite sign for baryon number is induced atthe tips of the string (where |φ| is increasing).
This chemical potential induces anonvanishing baryon number.In the language of Ref. 2, the CP violation with definite sign at the tips of the4
string leads to preferential decay of local texture configurations with a definite netchange in Chern-Simons (i.e., baryon) number.Let us now estimate the magnitude of this effect. The rate of baryon numberviolating events inside the string (in the unbroken phase) isΓB ∼α4wT 4.
(4)The volume in which CP violation is effective changes at a rate (g is the gaugecoupling constant)dVdt = g2w2V,(5)where w ≃λ−1/2η−1 is the width of the string and v is its contraction velocity.The factor g comes from the observation that baryon number violating processesare unsuppressed only if |φ| < gη.11) The rate of baryon number generation perstring isdNBdt∼w2vΓBǫ∆tc,(6)where ǫ is a dimensionless constant measuring the strength of CP violation and∆tc = gwv1γ(v)(7)is the time a fixed point in space is in the transition region. Here, γ(v) is the usualrelativistic γ factor.
Since there is one string per correlation volume ξ(tG)3, theresulting rate of increase in the baryon number density nB isdnBdt∼λ−3/2η−3g3α4wT 41γ(v)ǫξ(tG)−3. (8)The net baryon number density is obtained by integrating (8) from tG, the time5
corresponding to the Ginsburg temperature, and tG + ∆tS (see (3)). The result isnB ∼λvγ(v)g3α4wT 3Gǫ.
(9)Our result (9) must be compared to the entropy density at tG:s(tG) = π245g∗T 3G,(10)where g∗is the number of relativistic spin degrees of freedom. From (9) and (10)we obtainnBs ∼45π2g∗λγ(v)vǫg3α4w.
(11)For λ ∼v ∼1 and ǫ ∼1, the ratio obtained is only slightly smaller than theobservational value.In order for our mechanism to work, the core radius of the string (|φ| < gη)must be large enough to contain the nonperturbative configurations which medi-ate baryon number violating processes. This leads to the condition λ < g4, i.e.small Higgs mass.
In addition, the sphaleron must be sufficiently heavy such thatsphaleron transitions in the broken symmetry phase are suppressed for T = TG.For small values of λ, this condition will automatically be satisfied. Finally, themodel parameters must be such that the phase transition is of second order.
Inthe standard electroweak theory, this condition is incompatible with λ ≪g4. Inany extended electroweak theory, the consistency of the above conditions must besatisfied in order for our baryogenesis mechanism to be effective.6
3. DiscussionWe have presented a counterexample to the “folk theorem” stating that elec-troweak baryogenesis requires a first order electroweak phase transition.
We pro-pose a mechanism in which finite length electroweak strings during their contrac-tion generate a nonvanishing net baryon number. The strings play a similar roleto the expanding bubble walls in a first order phase transition: they provide outof equilibrium processes, and also a region where CP violation occurs.The mechanism presented here requires stable electroweak strings and an extrasource of CP violation (which is present in the two Higgs models used in Refs.
2-4).Based on the stability analysis of electroweak strings in the standard model7), it isunlikely that these strings will be stable for experimentally allowed values of theparameters in the Lagrangian.AcknowledgementsFor interesting discussions we are grateful to M. Einhorn, R. Holman andL. McLerran.
We also thank L. Krauss and S.-J. Rey for organizing a stimulat-ing and timely workshop.
This work was supported in part (at Brown) by DOEgrant DE-AC02-76ER03130 Task A, by an Alfred P. Sloan Fellowship (R.B. ), andby an NSF-SERC Collaborative Research Award NSF-INT-9022895 and SERCGR/G37149.
One of us (R.B.) acknowledges the hospitality of the Institue forTheoretical Physics at the University of California in Santa Barbara, where thiswork was completed with support from NSF Grant No.
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