We complete an existing result for how the baryon asymmetry left over after a
period of full thermal equilibrium depends on different lepton asymmetries.Comment: 5 pages; small clarifications and a reference added; to appear in PR

A. D. Dolgov

A. E. Nelson

H. Dreiner

J. A. Harvey

J. March-Russell

M. Laine

M. Shaposhnikov

S. Davidson

S. Yu. Khlebnikov

V. A. Kuzmin

English

arXiv

Laine M, Shaposhnikov M. Remark on sphaleron erasure of baryon asymmetry. PHYSICAL REVIEW D. 2000;61(11): 117302.We complete an existing result for how the baryon asymmetry left over after a period of full thermal equilibrium depends on different lepton asymmetries

Laine, Mikko 

Shaposhnikov, M

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Remark on sphaleron erasure of baryon asymmetry

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Laine, M

Serveur académique lausannois

A remark on sphaleron erasure of baryon asymmetry.

We complete an existing result for how the baryon asymmetry left over after a period of full thermal equilibrium depends on different lepton asymmetries

Laine, Mikko

Shaposhnikov, Mikhail E

CERN Document Server

CERN-TH/99-353UNIL-IPT/99-6hep-ph/9911473November 1999A REMARK ON SPHALERON ERASUREOF BARYON ASYMMETRYM. Lainea,b,1, M. Shaposhnikovc,2aTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandbDept. of Physics, P.O.Box 9, FIN-00014 Univ. of Helsinki, FinlandcInstitute for Theoretical Physics, University of Lausanne,BSP-Dorigny, CH-1015 Lausanne, SwitzerlandAbstractWe complete an existing result for how the baryon asymmetry left over after a periodof full thermal equilibrium depends on dierent lepton asymmetries.Introduction. Baryon plus lepton number (B + L) is violated by anomalous elec-troweak processes, and at the high temperatures appearing in the Early Universe theseprocesses are fast enough to be in thermal equilibrium [1]. This leaves only threeconserved global charges,Xi =1nFB − Li; (1)where i = 1; :::; nF , and nF denotes the number of generations3.For cosmological applications it is important to know what is the value of the baryonnumber B in an equilibrium system with given Xi (see, e.g., a recent work [2]). Thisproblem has been addressed in a number of papers [3]{[8], and dierent answers tothe same questions have been given. The confusing points were related to accountingfor the scalar degrees of freedom and to xing the boundary conditions for the gaugecharges (electric charge versus hypercharge in the Higgs phase of the theory).1mikko.laine@cern.ch2mikhail.shaposhnikov@ipt.unil.ch3In extensions of the Minimal Standard Model even these charges may be violated, for instancedue to flavour non-diagonal slepton masses or Majorana-type right-handed neutrino masses, but weassume here that the Xi are strictly conserved.1In the limit of small Yukawa couplings the baryon number can be written in the formB = f0∑iXi +∑iXi(f1m2iT 2+ f2h2i); (2)where mi = hi=p2 are the lepton masses, hi are the lepton Yukawa couplings,  isthe expectation value of the Higgs eld, T is the temperature, and∑iXi = B−L. Thefi are some functions of T , , the number of fermionic flavours nF , and the number ofHiggs scalars nS. The function f0 was computed correctly in the symmetric phase ofthe EW theory in [4], and in the Higgs phase in [9]; the limit   T of f0 coincideswith the results of refs. [5, 6, 7, 8].The situation with f1 and f2 is more obscure. In ref. [4] it was claimed that f1 =(4=132)A, f2 = (1=132)A with A ’ 1 for both symmetric and the Higgs phasesprovided the temperature is much larger than the vector boson masses (however, noexplicit expression for A was presented). This result was stated to be wrong in [8]based on a reasoning related to the boundary conditions, and another form for f1 wasderived. In [8] it was said, moreover, that in the symmetric phase B = 0 if B −L = 0,meaning eectively that f2 = 0.The question of how to really carry out the computation was nally claried in [9],but no estimates were presented for f1 and f2, as only f0 was computed. We feel thatthe correct results for f1 and f2, valid both in the symmetric and the Higgs phases,should appear in the literature at last, and this is the aim of the present note.Method. Let us briefly recall the method of computing the baryon number [9]. Wehave nF conserved global charges Xi, with which we can associate chemical potentials.However, it is more convenient to introduce rst the nF +1 chemical potentials B; Li,and impose the constraint following from the sphaleron processes,nFB +nF∑i=1Li = 0 (3)only later on. We now have to compute the eective potential V (;Aa0; B0;B; Li; T ),where  is the Higgs expectation value, and Aa0; B0 are the temporal components ofthe SU(2) and U(1) gauge elds. The elds Aa0; B0 have to be included, since thechemical potentials break Lorentz invariance and induce expectation values for them.The minimization @V =@Aa0 = @V =@B0 = 0 corresponds to the neutrality of the systemwith respect to gauge charges. From the gauge-invariant value of the eective potentialat the minimum, we get B = −@V =@B; Li = −@V =@Li (these are really the baryonand lepton numbers per unit volume). Utilising Eq. (3), we can nally eliminate B; Lito obtain an expression of the desired form B = f(Xi).We will work at high temperatures, assuming B; Li  T and <(a few)T . Theexpectation values of Aa0; B0 are proportional to the ’s, so it is sucient to keep terms2up to quadratic order in Aa0; B0; B; Li. It is sucient to choose only A30 non-zero. Thebosonic degrees of freedom (Higgses, gauge elds) only contribute to terms quadratic inA30; B0, while B; Li come from the fermionic contributions. We denote the fermionicterms involving Aa0; B0; B; Li by=  γ0[− i2~AaL +i2~B(YLaL + YRaR)] ; (4)where  is either B=3 (for quarks) or Li (for leptons), aL; aR are the left and rightprojectors, YL,R are the corresponding hypercharges, and ~A  gA30; ~B  g′B0.Baryon asymmetry. In order to account for flavour dependent contributions to thebaryon asymmetry, we have to supplement the eective potential computed in [9] withthe dominant terms dierentiating between the Li’s. Such terms must involve Yukawacouplings, and can arise either as mass corrections in the 1-loop eective potential, oras 2-loop corrections directly proportional to the Yukawa couplings. For   T bothtypes of terms ( m2i =T 2; h2i ) are of the same order of magnitude and must be includedsimultaneously.The dominant fermionic 1-loop mass corrections come from the graphOn the other hand, denoting the Higgs with a dashed line, the 2-loop graphs potentiallycontributing are (we need only terms with at least one power of Li)However, it is easy to see that the rst 2-loop graph does in fact not contribute. Thereason is that viewed as a self-energy correction for the Higgs eld, the fermionic loopdoes not have a term linear in , because the insertions of γ0 to the two dierentfermion lines cancel each other; thus the result is / ~A2; ~B2.Furthermore, it turns out the second 2-loop graph does not contribute either. It isproportional to the divergent integralI =∑∫P,Qpf (pf + qf )(P +Q)2Q2(P 2)2; (5)where P = (pf ;p) and pf are the fermionic Matsubara frequencies. It turns out thatthe coecient of this integral gets contributions also from the last graph, and altogetherthe result is again just a higher order correction of the form ~A2; ~B2.The remaining nite contribution from the last 2-loop graph, together with the 1-loopcontribution as well as all the bosonic contributions, give the eective potentialV =12~A2[142 +(23+nS6+nF3)T 2]+12~B2[142 +(nS6+5nF9)T 2]+14~A ~B23+i3~BT 2(nF3B −∑iLi)− nF92BT2 − 14∑i2LiT2− i82~A∑iLim2i +T 282∑i(3i ~BLi + 22Li)(m2iT 2+h2i4): (6)Working as outlined above, we obtain for the physical case (nF = 3; nS = 1)B = 477T 2 + 272869T 2 + 3332∑iXi+112247T 2 + 182869T 2 + 3332∑iXim2iT 2+11621034T 2 + 4052869T 2 + 3332∑iXih2i : (7)The rst term coincides with the one found in [9]. The latter line is the dominantone if∑iXi = 0. Using the notation of Eq. (2) we have f1 = (4=132)(143=148) (4=132)  0:9662 and f2 = (1=132)(585=592)  (1=132)  0:9882 for   T , andf1 = (4=132)(611=632)  (4=132)  0:9668 and f2 = f1=4 for   T , in numericalagreement with the estimates in [4].When higher order corrections are taken into account, one expects fi ! fi[1 +O(W=; S=; h2=2; m2=(T )2)], where h is a general Yukawa coupling and m ageneral mass.In summary, we have derived the leading order expressions for the flavour depen-dent contributions to the baryon asymmetry remaining after a period of full thermalequilibrium, Eq. (7). The result should be evaluated at   T if the sphaleron pro-cesses fall out of thermal equilibrium after the system has smoothly passed from thesymmetric to the Higgs phase. If, on the contrary, there is a strong rst order elec-troweak phase transition such that the sphaleron processes are always switched o inthe Higgs phase (but at the same time no new B + L asymmetry is generated duringthe transition due to, say, too little CP-violation), it should be evaluated at  = 0.The work of M.L. was partly supported by the TMR network Finite TemperaturePhase Transitions in Particle Physics, EU contract no. FMRX-CT97-0122. The work ofM.S. was partly supported by the Swiss Science Foundation, contract no. 21-55560.98.References[1] V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. B 155 (1985)36.[2] J. March-Russell, H. Murayama and A. Riotto, hep-ph/9908396.4[3] V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. B 191 (1987)171.[4] S.Yu. Khlebnikov and M.E. Shaposhnikov, Nucl. Phys. B 308 (1988) 885.[5] J.A.Harvey and M.S. Turner, Phys. Rev. D 42 (1990) 3344.[6] A.E. Nelson and S.M. Barr, Phys. Lett. B 246 (1990) 141.[7] A.D. Dolgov, Phys. Rept. 222 (1992) 309.[8] H. Dreiner and G.G. Ross, Nucl. Phys. B 410 (1993) 188 [hep-ph/9207221].[9] S.Yu. Khlebnikov and M.E. Shaposhnikov, Phys. Lett. B 387 (1996) 817 [hep-ph/9607386].5

A remark on sphaleron erasure of baryon asymmetry

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A remark on sphaleron erasure of baryon asymmetry

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