SPHALERONS WITH TWO HIGGS DOUBLETS ∗ MARKHINDMARSHAND JACKIEGRANT Centre for Theoretical Physics University of Sussex Falmer, Brighton BN1 9QJ 1 U.K. 0 E-mail: [email protected], [email protected] 0 2 Wereportonworkstudyingthepropertiesofthesphaleroninmodelsoftheelec- n troweak interactions with two Higgs doublets in as model-independent a way as a possible: by exploring the physical parameter space described by the masses and J mixinganglesoftheHiggsparticles. IfoneoftheHiggsparticlesisheavy,therecan beseveral sphaleronsolutions, distinguishedbytheir properties under parityand 0 the behaviour of the Higgs field at the origin. In general, these solutions are not 1 spherically symmetric, although the departure from spherical symmetry is small. SUSX-TH-01-001 1 v 6 1 Introduction 9 0 One of the major unsolved problems in particle cosmology is accounting for 1 the baryon asymmetry of the Universe. This asymmetry is usually expressed 0 1 in terms of the parameter η, defined as the ratio between the baryon num- 0 ber density n and the entropy density s: η = n /s ∼ 10−10. Sakharov 1 B B h/ laid down the framework for any explanation: the theory of baryogenesis p must contain B violation; C and CP violation; and a departure from ther- - mal equilibrium. All these conditions are met by the Standard Model 2 and p its extensions, and so there is considerable optimism that the origin of the e h baryonasymmetry can be found in physics accessible at current and planned : accelerators (see 3 for reviews). v i Current attention is focused on the Minimal Supersymmetric Standard X Model, where there are many sources of CP violation over and above the r CKM matrix 4, and the phase transition can be first order for Higgs masses a up to 120 GeV, providing the right-handed stop is very light and the left- handed stop very massive 5. B violation is provided by sphalerons 6, at a rate Γ ≃ exp(−E (T)/T), s s whereE (T)is the energyofthe sphaleronattemperatureT. This ratemust s notbesolargethatthebaryonasymmetryisremovedbehindthebubblewall, andthisconditioncanbetranslatedintoalowerboundonthesphaleronmass ∗TalkgivenatStrong and Electroweak Matter,Marseille,14-17June2000 procs4xxx: submitted to World Scientific on February 1, 2008 1 Es(Tc)/Tc >∼45.Thusitisclearthatsuccessfulbaryogenesisrequiresacareful calculation of the sphaleron properties. Here we report on work on sphalerons in the two-doublet Higgs model (2DHM) in which we study the properties of sphalerons in as generala set of realistic models as possible. In doing so we try to express parameter space in terms of physical quantities: Higgs masses and mixing angles, which helps us avoid regions of parameter space which have already been ruled out by LEP. Previous work on sphalerons in 2DHMs 7,8,9,10 has restricted either the Higgspotentialortheansatzinsomeway. Ourpotentialisrestrictedonlybya softlybrokendiscretesymmetryimposedtominimizeflavour-changingneutral currents(FCNCs). Ouransatzisthemostgeneralsphericallysymmetricone, including possible C and P violating field configurations11. We firstly check our results against the existing literature, principally BTT8,whofoundanewP-violating“relativewinding”(RW)sphaleron,spe- cifictomulti-doubletmodels,albeitatMA =MH± =0. Thisisdistinguished from Yaffe’s P-violating deformed sphaleron 11 or “bisphaleron” by a differ- ence in the behaviour of each of the two Higgs fields at the origin. We then reexamine the sphaleron in a more realistic part of parameter space, where MA and MH± are above their experimental bounds. We reiterate the point made in 12 that introducing Higgs sector CP violation makes a significant difference to the sphaleron mass (between ten and fifteen percent), and may significantlychangeboundsontheHiggsmassfromelectroweakbaryogenesis. 2 Two Higgs doublet electroweak theory The most general quartic potential for 2DHMs has 14 parameters, only one of which, the Higgs vacuum expectation value, υ, is known. However, we are aided by the observation 13 that FCNCs can be suppressed by imposing a softly-brokendiscretesymmetryφ →+(−)φ ,andresultsinapotential 1(2) 1(2) with10realparameters. Oneofthesemayberemovedbyaphaseredefinition of the fields; the vacuum configurationis then entirely real, and CP violation iscontainedinoneterm2χ Re(φ†φ )− υ1υ2 Im(φ†φ ). Ignoringcouplings 2 1 2 2 1 2 tootherfields,whenχ =0t(cid:16)hereisadiscretes(cid:17)ymmetryφ →−iσ φ∗,which 2 α 2 α sends Im(φ†φ )→−Im(φ†φ ). This can be identified as C invariance. 1 2 1 2 Following 12 we determine as many as possible of the nine parameters in the potential from physical ones. The physical parameters at hand are the four masses of the Higgs particles, the three mixing angles of the neutral Higgses,oneofwhich,θ ,istheonlyCPviolatingphysicalparameter,(θ CP CP mixes the CP even and CP odd neutral Higgs sector), and υ. procs4xxx: submitted to World Scientific on February 1, 2008 2 We further check that the potential for these sets of physical parameters is always bounded from below. 3 Sphaleron ansatz and numerical methods The most general static spherically symmetric ansatz is, in the radial gauge 0 (1+β) α φ =(F +iG xˆaσa) , Wa = ε xˆ + (δ −xˆ xˆ ) .(1) α α α (cid:18)1(cid:19) i (cid:20) r aim m r ai a i (cid:21) Here, the subscript α = 1,2, and F = a +ib and G = c +id are α α α α α α complex functions. The boundary conditions can be most easily expressedin terms of the functions χ, Ψ, H , h , and Θ , defined by α α α −β+iα=χeiΨ, a +ic =H eiΘα, b +id =h eiΘα, (2) α α α α α α and one can show that as r → 0, either H2 +h2 → 0 or Θ → Ψ/2+n π α α 1 1 and Θ → Ψ/2+n π, (n ,n ∈ Z). These boundary conditions distinguish 2 2 1 2 between the various types of sphaleron solution: the ordinary sphaleron has H2 +h2 → 0 as r → 0, the bisphaleron has non-vanishing Higgs fields with α α n = n , and the RW sphaleron non-vanishing Higgs fields with n 6= n . 1 2 1 2 These integers represent the winding of the Higgs field around spheres of constant|φ |,althoughonlytheirdifferencehasanygaugeinvariantmeaning. α Notethattheansatzispotentiallyinconsistent,asIm(φ†φ )∝xˆ ,apoint 1 2 3 which has not been noted before. In practice, the non-spherically symmetric parts of the static energy functional, E[f ], contribute less than 1% of the A total, and so we are justified in assuming the fields of the ansatz, f , are a A function only of the radial co-ordinate r and then integrating over xˆ . 3 We look for solutions to E[f ] using a Newton method 11 which is an A efficient way of finding extrema (and not just minima). The method can be brieflycharacterisedasupdatingthefieldsf byδf ,givenbythesolutionof A A ′′ ′ E δf =−E , where the primes denote functional differentialion with respect to f . A particular advantage to using this method is that because we calcu- A ′′ late E , it is straightforward to get the curvature eigenvalues, and therefore to check that the solution really is the lowest energy unstable solution. 4 Results We first checked our method and code against the results of BTT 8, and Yaffe11 findingagreementintheenergyofbetterthan1partin103 forawide range of parameters. We also measured the Chern-Simons numbers, n , of CS the solutions that they discovered and determined that they were near 1/2, procs4xxx: submitted to World Scientific on February 1, 2008 3 Energy of sphaleron and RWS Chern−Simons number of RWS 800 800 M (Gev)H123456700000000000000 3.40633.54343.68063.81773.9549 4.092 4.2292 4.3663 4.5035 4.6407 M (Gev)H123456700000000000000 0.494730.496204.497740.49849 0.49925 0 0 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 M (Gev) M (Gev) h h Figure1. Contours ofsphaleronenergy(MW/αW)andChern-Simonsnumber. Most negative eigenvalue of sphaleron and RWS Second most negative eigenvalue of sphaleron 800 800 M (Gev)H123456700000000000000 −2.4498 −3.181 −3.9121 −4.6433 −5.3744 −6.1056 −6.8367 −7.5678 −8.299 −9.0301−9.7613−10.4924 M (Gev)H123456700000000000000 −1.541−80−.230.1853865 −0.92507 0 0 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 M (Gev) M (Gev) h h Figure 2. Eigenvalues (Mw) of the sphaleron solutions as a function of the CP even Higgs masses Mh and MH. Therewas no mixing, and MA =241 Gev, MH± =161 Gev, tanβ =6, λ3 =−0.05. The solidlines inthe most negative eigenvalue plot represent the relative winding sphaleron, while the dashed lines are for the ordinary sphaleron, for the dottedregionthepotential wasunbounded frombelow. but not exactly 1/2 as with the sphaleronsolution. Further they appearedin Pconjugatepairswithn ofthepairaddingtoexactlyone. Wethenlooked CS at more realistic values of MA and MH±, with results that are displayed in Figs.1and2. Notefirstofallthewell-knownfeaturethatthesphaleronmass depends mainlyonM . Secondly,forincreasingM ,the curvaturematrixof h H the sphalerondevelopsa secondnegativeeigenvalue(Fig. 2, right),signalling procs4xxx: submitted to World Scientific on February 1, 2008 4 the appearance of a pair of RW sphalerons. The lower of the two n is CS plotted on the right Fig. 1. The departure from n =1/2 is small, as is the CS difference in energy between the RW and ordinary sphalerons for the Higgs masses we examined, however the most negative curvature eigenvalue of the RWsphaleroncanbe double thatofthe sphaleron. Moredetailedresultsand discussion of their significance are reserved for a future publication 14. Acknowledgments MH and JG are supported by PPARC. This work was conducted on the SGI Origin platform using COSMOS Consortium facilities, funded by HEFCE, PPARC and SGI. We also acknowledge computing support from the Sussex High Performance Computing Initiative. References 1. A.D. Sakharov,JETP Lett. 6 24 (1967). 2. V.A. Kuzmin, V.A. Rubakov, and M.E. Shaposhnikov, Phys. Lett. B 155 36 (1985). 3. V.A Rubakov and M.E. Shaposhnikov, Phys. Usp. 39, 461 (1996) [hep- ph/9603208]; 4. P.HuetandA.E.Nelson,Phys. Rev. D534578(1996)[hep-ph/9506477]; M. Carena, M. Quiros and C.E.M. Wagner, Phys. Lett. B390 6919 (1997) [hep-ph/9603420]; J. Cline, M.Joyce and K. 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