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Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 769240 9 pageshttpdxdoiorg1011552013769240

Research ArticleTowards Reviving Electroweak Baryogenesis witha Fourth Generation

Wei-Shu Hou and Masaya Kohda

Department of Physics National Taiwan University Taipei 10617 Taiwan

Correspondence should be addressed to Masaya Kohda mkohdahep1physntuedutw

Received 22 June 2012 Accepted 27 December 2012

Academic Editor Tao Han

Copyright copy 2013 W-S Hou and M Kohda This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Electroweak baryogenesis is an attractive scenario for explaining the baryon asymmetry of the universe However it does not workwithin the standardmodel due to two reasons (1) the strength of CP violation from the Kobayashi-Maskawamechanismwith threegenerations is too small (2) the electroweak phase transition is not first order for the experimentally allowed Higgs boson massWe discuss possibilities to solve these problems by introducing a fourth generation of fermions and how electroweak baryogenesismight be revived We also discuss briefly the recent observation of a Higgs-like boson with mass around 125GeV which puts thefourth generation in a difficult situation and the possible way out

1 Introduction

The origin of the baryon asymmetry of the universe (BAU)is not only a big mystery in particle physics and cosmologybut it is a core problem related to our very existence Theobservational evidence is very compelling There is no indi-cation for any macroscopic objects made from antiparticlesThe antiparticles observed in cosmic rays are thought to be ofsecondary origin Thus it is accepted that the universe con-tains negligible amount of antibaryons as compared to thebaryons that constitute usual matter (including ourselves)or 119899119887119899119887 ≪ 1 The standard big bang cosmology requires the

baryon-to-entropy ratio of the universe to be [1]

119899119861

119904

=

119899119887 minus 119899119887

119904

≃ 09 times 10minus10 (1)

If one does not take the view that this is merely an initialcondition for the expanding universe then the challenge isnot so much the dominance of radiation but that whetherthere is anymatter left from the primordial matter-antimatterannihilation

Many scenarios have been attempted to explain theorigins of the minute amount of leftover matter based on adynamical mechanism (baryogenesis) A common basis for

such mechanisms is provided by the three Sakharov condi-tions [2 3] which consist of (i) baryon number (B) violation(ii) 119862 and CP violation and (iii) departure from equilibriumIn order to produce the BAU these three conditions have tobe satisfied simultaneously at some stage in the early universe

Among many proposals electroweak baryogenesis(EWBG) [4] is one of the well-motivated and appealingmechanisms to explain BAU especially since its ingredientscan be possibly probed by the Large Hadron Collider (LHC)This mechanism could operate during the electroweak phasetransition (EWPT) occurring when the temperature ofthe universe is around 119879119888 sim 100GeV and it is a possiblebaryogenesis mechanism within the standard model (SM)of particle physics One should recall the truly remarkablefact that the SM carries all the necessary ingredients for theSakharov conditions

(1) B is violated due to the chiral anomaly inducedby the SU(2) times U(1) gauge interaction B-violatingprocesses are nonperturbative phenomena mediatedby topologically nontrivial SU(2) gauge field configu-rations Although the instanton-mediated B violationat zero temperature is highly suppressed the rate of Bviolation at finite temperature is not negligible via the

2 Advances in High Energy Physics

so-called sphaleron transition especially when thetemperature is around or above 119879119888

(2) 119862 is violated by the 119881 minus 119860 structure of the weakinteraction while CP is violated by the Kobayashi-Maskawa (KM) mechanism [5] with three genera-tions of quarks A complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix provides thesource of CP violation which nicely describes allterrestrially observed CP-violating phenomena thatis CP violation in the 119870- and 119861-meson (even the 119863-meson) systems

(3) SU(2) times U(1) gauge symmetry is expected to berestored at high temperature above 119879119888 Therefore asthe universe cools down to a temperature of around119879119888 the universe undergoes a phase transition (theEWPT) from symmetric phase of SU(2) times U(1) tothe broken phase If the EWPT is first order thephase transition proceeds through bubble nucleationof the broken phase where expansion of the bubblesfills up the entire universe Thus temporal departurefrom equilibrium is achieved near the surface of theexpanding bubble

Unfortunately themeasured parameters of the SMappearto be insufficient to bring about EWBG The SM fails toexplain the BAU due to two reasons The first reason is thattheKMmechanismwith three quark generations cannot offerlarge enough CP violation (CPV) to produce the observedbaryon-to-entropy ratio during the EWPT The second rea-son is that the EWPT is not first order for a Higgs boson with119898ℎ gt 114GeV required already by direct search at the LEP[6] and now indicated to be around 125GeV at the LHC [7 8]Therefore physics beyond the SM is needed for EWBG andvarious extensions of the SMhave been investigated from thispoint of view In particular there are extensive studies (for arecent review on EWBG see [9] and references therein) forthe twoHiggs doublet model and supersymmetric extensionsof the SM such as the MSSM and the NMSSM It shouldbe noted however that MSSM itself is under stress from theLHC and its case for EWBG is only marginal [9] One of thebiggest means of attraction to have EWBG is that the impliednew physics should be accessible at the LHC at least partially

In this contribution we consider a four-generation (4G)extension of the SM and discuss a possibility to revive EWBG(beside a possible revival of EWBG the 4G has anotherinteresting implication on baryogenesis if it is long-livedsee [10]) fully within the SM framework as 4G does notintroduce anything that was not already contained in SMWefocus on the quark sector which includes fourth generationquarks 1199051015840 and 1198871015840 Compared to the three-generation casethe 4G quarks bring three additional mixing angles andtwo additional complex phases in the 4 times 4 CKM matrixthus providing new sources of CP violation One can argueintuitively for a possible effect of new CP violation on BAUby studying basis-independent invariants constructed fromthe quark mass matrices (or the Yukawa coupling matrices)which are extensions of the Jarlskog determinant in the SMOne of us [11] pointed out that such an invariant quantitycan be highly enhanced compared to the three-generation

case mainly due to the large masses (or the large Yukawacouplings) of 1199051015840 and 1198871015840(see [12 13] for an earlier discussionon 4G-enhanced CP violation for BAU) As for the issue ofthe EWPT a one-loop analysis shows that the 4G quarks donot play a positive role towards a first-order phase transition[14] However given that bounds on the 4G quark massesfrom direct search at the LHC have entered 600GeV level[15 16] which is beyond the tree-level perturbative unitaritybound (UB) of 500ndash550GeV [17] we have to reconsider theproblemof EWPTbeyondperturbation theory In this regardthe nature of the EWPT with 4G quarks is still quite an openproblem (for earlier works which address the issue of theEWPTwithin the four-generation framework (but not withinthe SM framework) see [14 18] (supersymmetric model withthe 4G fermions) and [19] (dynamical electroweak symmetrybreaking model due to strong four-fermion interactionsamong the 4G fermions))

The recent observation of a Higgs-like object with massaround 125GeV poses a special impasse situation since anaive application of heavy 1199051015840 and 119887

1015840 quarks in the loopwould enhance the gluon-gluon fusion production of theHiggs boson by about an order of magnitude which is notsupported by 119892119892 rarr 119881119881 data [7 8] where 119881 is the 119882 or119885 boson Furthermore quadratic corrections to the Higgsmass from heavy 1199051015840 and 1198871015840 quarks make the light Higgs massrather difficult to sustain With the deep conflict between alight Higgs and having 4G quarks beyond UB it may seemcavalier to relegate this again to nonperturbative treatmentsHowever the aim of this work is very modest which is justto show the nontrivial nature of 4G towards EWBG Sincethe source of CPV arises from off-diagonal couplings theHiggs boson does not directly enter EWBG computationWewill discuss towards the end the possibility that the observed125GeV boson could be a dilaton rather than a bona fide SMHiggs boson or it may be a pseudo-Goldstone Higgs boson

This paper is organized as follows In Section 2 we discussthe effect of CP violation from the fourth generation quarksector on BAU Extending the EWBGmechanism in the caseof the SM to the fourth generation case we estimate theamount of the BAU produced during a first-order EWPT InSection 3 we summarize the EWPT in the SM briefly anddiscuss possible mechanisms to induce a first-order phasetransition within the four-generation framework Finally inSection 4 we offer some discussions and give our summary

2 CP Violation

One of the reasons for the failure of EWBG based on SM isthe insufficiency of CP violation from the KM phase Thissituation can be naively understood as follows

In the SM with three generations a basis-independentinvariant for CP violation is given by the well-known Jarlskogdeterminant [20]

det [119872119906119872dagger

119906119872119889119872

dagger

119889]

100381610038161003816100381610038163minusgen

equiv 119894119869SM (2)

Advances in High Energy Physics 3

where119872119906119889 are the up- and down-type quark mass matricesIn terms of quarkmasses and CKMmatrix elements one gets

119869SM = 2JSM (1198982

119905minus 1198982

119888) (1198982

119905minus 1198982

119906) (1198982

119888minus 1198982

119906)

times (1198982

119887minus 1198982

119904) (1198982

119887minus 1198982

119889) (1198982

119904minus 1198982

119889)

(3)

where JSM is twice the area of any triangle defined by theunitarity condition of the CKM matrix 119881dagger119881 = 1 withthe experimentally determined value of JSM ≃ 3 times 10

minus5One can estimate a naive strength of CP violation duringthe EWPT by constructing a dimensionless quantity from119869SM and the critical temperature 119879119888 sim 100GeV This gives119869SM119879

12

119888sim 10minus19 which is too small to account for BAU in

(1)It is readily observed that the smallness of 119869SM originates

mainly from the powers of light quarkmasses11989821198881198984

1198871198982

1199041198798

119888sim

10minus15 with milder suppression from the CKM factor JS119872

Motivated by this observation and a possible hint of newphysics in 119887 rarr 119904 transition it was suggested in [11] tointroduce the 4G quarks and replace (3) by an analogousquantity involving second to fourth generation quarks (orequivalently treating the first two generations as degenerate)

119869119904119887

(234)= 2J119904 (119898

2

1199051015840minus 1198982

119905) (1198982

1199051015840minus 1198982

119888) (1198982

119905minus 1198982

119888)

times (1198982

1198871015840minus 1198982

119887) (1198982

1198871015840minus 1198982

119904) (1198982

119887minus 1198982

119904)

sim

J119904

JSM(

1198982

1199051015840

1198982119905

minus 1)

1198982

1199051015840

1198982119888

1198984

1198871015840

1198982

1198871198982119904

119869SM

(4)

whereJ119904 = Im(119881lowast119905119904119881119905119887119881lowast

11990510158401198871198811199051015840119904) In fact 119869

119904119887

(234)is a leading term

of a basis-independent set of invariants for CP violation inthe four-generation case and is given by Tr[119872119906119872

dagger

119906119872119889119872

dagger

119889]3

up to an overall factor [20 21] Due to heaviness of 1199051015840 and 1198871015840119869119904119887

(234)can be highly enhanced compared to 119869SM in (3) For

1198981199051015840 sim 1198981198871015840 sim 600GeV for instance the enhancement factorreaches sim1018 solely from the mass factors thus seeminglypossible to overcome the smallness of 119869SM

In the following after we briefly review EWBG in theSM case we take one step further and examine the previousexpectation of 4G-enhanced CP violation for the EWBGscenario

A concrete mechanism of EWBG within SM was pro-posed by Farrar and Shaposhnikov (FS) [22ndash25] In the FSmechanism a baryon asymmetry is produced through CPviolating scattering of quarks at the surface of an expandingbubble that is the bubble wall generated during a first-orderEWPT CP asymmetry in reflection of the quarks off thebubble wall is induced by thermal effects that is interactionsof the quarks with the 119882 and charged-Higgs bosons inthe cosmic plasma FS included these effects employing aquasiparticle picture and treated the scattering problem ina quantum mechanical manner Solving an effective Diracequation for the quasiparticles they found that CP violationfrom the KM phase is sufficient to explain the BAU underoptimal conditions contrary to the naive argument givenprevious

The very attractive SM explanation for BAU by FS washowever refuted by subsequent works by Gavela et al [26ndash28] These authors pointed out that the width of the quasi-particle (damping rate) was not included in the study of FSand they found that inclusion of the width reduces the baryonasymmetry to a negligible amount This result was alsoconfirmed by Huet and Sather (HS) [29] who interpreted thedamping as quantum decoherence phenomenon induced byplasma effects leading to reduction of the CP asymmetry inwhich quantummechanical coherence plays an essential roleAfter the demonstration of these results it was (re)acceptedthat CP violation from the KM phase is not sufficient toexplain BAU This conclusion is in accordance with thenaive dimensional argument of 119869SM given previous We nowestimate by extending thework ofHS the baryon asymmetrygenerated by the FS mechanism in the four-generation case

We assume the existence of a first-order phase transitionThe baryon asymmetry generated during the first-orderEWPT is given by [29]

119899119861

119904

sim minus

10minus2

119879

int

119889120596

2120587

1198990 (120596) [1 minus 1198990 (120596)]

Δp sdot v119882119879

Δ (120596) (5)

where 1198990(120596) = 1[exp(120596119879) + 1] is the Fermi-Diracdistribution Δp equiv p119871 minus p119877 is the difference between left-handed and right-handed quasiparticle momenta for a givenenergy120596 and v119882 is the velocity of the expanding bubble wallWe neglect O(v2

119882) contribution by assuming |v119882| sim 01 We

take 119879 sim 119879119888 sim 100GeV in the previous formula Δ(120596) is areflection asymmetry defined by

Δ (120596) equiv Tr [119877dagger119871119877119877119871119877 minus 119877

dagger

119871119877119877119871119877] (6)

where reflection coefficients 119877119871119877 and 119877119871119877 are matrices inflavor space that is 119877119891119894

119871119877is the reflection coefficient for

119902119894

119871rarr 119902

119891

119877 where 119894 and 119891 refer to quark flavors and

119877

119891119894

119871119877corresponds to the CP-conjugate process Δ(120596) provides

the CP asymmetry for the reflection rate of a left-handedquasiparticle incident from the symmetric phase with anenergy 120596 summed over all flavors We further assume thatthe EWPT is strongly first order so that the generated baryonasymmetry is not washed out by the sphaleron processes inthe broke phase (see next section) We return to discuss theissue of order of phase transition in 4G context later

Δ(120596) is obtained by solving an effectiveDirac equation forthe quasiparticles in the presence of space-dependent quarkmass terms Assuming planar wall with zero thickness theeffective Dirac equation is given by

(

2(120596 minus Ω119871 + 119894120574 +

1

3

119894120590 sdot 120597) 119872119906119889 120579 (119911)

119872dagger

119906119889120579 (119911) 2 (120596 minus Ω119877 + 119894120574 minus

1

3

119894120590 sdot 120597)

)

times Ψ119906119889 (119911) = 0

(7)

where Ψ119906 = (120595119906 120595119888 120595119905 1205951199051015840)119879 Ψ119889 = (120595119889 120595119904 120595119887 1205951198871015840)

119879 andeach 120595119894 are four-component spinor wavefunctions In (7)


Ω119871(119877) is a thermal mass matrix for the left- (right-) handedquasiparticles 120574 is the quasiparticle width andwe use the onecalculated in QCD [30] 120574 ≃ 015119892

2

119904119879 neglecting possible

flavor-dependent corrections from the Yukawa interactionsFollowing HS we solve the effective Dirac equations

analytically based on the Greenrsquos function method whichgives perturbative expansions of the reflection coefficients interms of the quark mass matrices 119872119906 119889 The leading-ordercontribution to Δ(120596) from the scattering of the 119889-type quarksis given by

Δ 119889 (120596) =

4

3

(

271205871205721198821198792

64Ω01198722

119882

)

3

[1 + (

120596 minus Ω0

120574

)

2

]

minus6

times (

1

6120574

)

9

Im Tr [119872119906119872dagger

119906 119872119889119872

dagger

119889]

3

(8)

where Ω0 ≃ 119892119904119879radic6 is the dominant part in the thermal

masses of quarks coming from QCD The 4G effects entersolely through the last factor in (8) which arises as onefollows the scattering of an incoming 119889-type quark againstthe bubble wall In the three-generation case this factor isnothing but the Jarlskog determinant discussed previous

Im Tr [119872119906119872dagger

119906 119872119889119872

dagger

119889]

31003816100381610038161003816100381610038163minusgen

= 3 Im det [119872119906119872dagger

119906 119872119889119872

dagger

119889]

100381610038161003816100381610038163minusgen

= 3 119869SM

(9)

where 119869SM is defined in (2) and (3) On the other hand thefour-generation counterpart is more complicated but it canbe simplified [21] by exploiting the hierarchical structure ofthe quark masses and the CKMmatrix elements with a mildassumption on newmixings |119881119905119889119881

lowast

1199051015840119889| ≪ |119881119905119894119881

lowast

1199051015840119894| (119894 = 119904 119887 1198871015840)

Then the last factor in (8) is approximated as

Im Tr [119872119906119872dagger

119906 119872119889119872

dagger

119889]

3

≃ minus6J119904 (1198982

1199051015840minus 1198982

119905)1198982

11990510158401198982

1199051198984

11988710158401198982

119887

(10)

where J119904 = Im(119881lowast

119905119904119881lowast

11990510158401198871198811199051198871198811199051015840119904) One notes that

Im Tr [119872119906119872dagger

119906119872119889119872

dagger

119889]

3

≃ minus3 119869119904119887

(234) where 119869

119904119887

(234)is

defined in (4) Therefore Δ 119889(120596) can be actually enhancedby large masses of 1199051015840 and 1198871015840 compared to the SM case asanticipated in [11]

As discussed by HS dimensionless perturbative expan-sion parameters in the Greenrsquos function method are given by119872119906 119889(6120574) ≃ 119872119906 119889(14119879) which are of order unity or largerfor the 119905 1199051015840 and 1198871015840 quarks given 119879 sim 100GeV during theEWPT Hence the perturbative expansion breaks down forthese heavy quarks and the previous result for Δ 119889(120596) couldbe reduced due to large 1198981198871015840 (equation (8) is obtained by theexpansion in the 119889-type quark mass matrix119872119889 based on theGreenrsquos function method The 119906-type quark mass matrix119872119906in (8) originates from the thermal mass matrix for the 119889-type quasiparticles that is the Yukawa interaction with theplasma Thus the heaviness of only the 1198871015840 quark matters to

Δ119889(120596)) if the perturbation is not used Thus the previousresult might overestimate the CP asymmetry from the 119889-typequarksThe 119906-type quark contribution at leading order is thesame as (8) except for the difference of the overall sign henceleading to a complete cancelation at this level The previousmentioned remark is however also applied for the 119906-typequark contribution and a possible amount of reductionwould be even larger as it contains the contributions fromthe two heavy quarks 119905 and 1199051015840 We naively accept (8) asa dominant contribution to the reflection asymmetry Δ(120596)simply neglecting the 119906-type quark contribution (there isanother type of major contribution to Δ(120596) called Δ 7 in HSwhich arises when the finite quarkmasses in the broken phaseare taken into account in the self-energy of quarks While Δ 7gives larger contribution to the BAU than (8) in the three-generation case we confirmed that the contribution from(8) dominates in the four-generation case for experimentallyallowed masses of 4G quarks)

Eventually the baryon asymmetry generated during theEWPT is given by

119899119861

119904

sim 09 times 10minus10(

J119904

10minus4)(

1198981199051015840

650GeV)

4

(

1198981198871015840

650GeV)

4

(11)

where 119879 = 100GeV and119872119882(119879) = 50GeV are adopted J119904includes newCKMparameter1198811199051015840119904119881

lowast

1199051015840119887which can bemeasured

via flavor observables related with 119887 rarr 119904 transition Wenote that measurements for the CP violating phase 120601119904 in119861119904 minus 119861119904 mixing the forward-backward asymmetry for 119861 rarr

119870lowast120583+120583minus and the 119861119904 rarr 120583

+120583minus rate are making rapid progress

at the LHC Theoretical predictions for these observables arenot subject to hadronic uncertainties somuch hence they areuseful to constrain1198811199051015840119904119881

lowast

1199051015840119887[31] From the recent LHCdataJ119904

of around 10minus4 with1198981199051015840 = 650GeV seems to be at the borderof the experimentally allowed region [32] However withrising 1198981199051015840 (and 1198981198871015840) there is a tendency that J119904 drops butthe high powers of 11989811990510158401198981198871015840 in (11) should easily compensatefor it

With reasonable 4G parameters therefore the FS mech-anism with 4G quarks seems to generate the correct orderof magnitude of the baryon-to-entropy ratio and hence canin principle explain BAU We note that the previous estimateof BAU relies on the perturbative treatments of the Yukawacouplingsmasses of the quarks including 1199051015840 and 1198871015840 in a fewsteps while the LHC bounds on the 4G quark masses sug-gest a nonperturbative nature of the corresponding Yukawacouplings For instance the Greenrsquos functionmethod invokesthe perturbation in the quark mass matrices as we alreadydiscussed Besides this point the thermal properties for thequasiparticles are obtained via perturbative calculations forexample the thermal mass matrices are evaluated at one-loop level Therefore our result would be semiquantitativeat most Nevertheless given remarkable enhancement of theCP asymmetry due to the 4G quarks and the agreementwith the observed BAU as shown in (11) our finding shoulddeserve further investigationWe further remark that the CP-violating Jarlskog-like invariants of (2) and (10) are of purelyalgebraic nature Thus we suspect that a fully dynamicalcalculation should still reflect this fact


3 Electroweak Phase Transition

Besides CP violation the other essential issue for EWBG isthe nature of the EWPT EWBG requires first-order EWPTwhich proceeds through nucleation and growth of broken-phase bubbles In addition there is a stronger requirement[12 13] on the EWPT as explained later

In EWBG the baryon asymmetry is generated at theEWPT utilizing the B-violating sphaleron transition How-ever the sphaleron transition must decouple just after thephase transition otherwise the generated baryon asymmetrywould be washed out In the broken phase the sphalerontransition rate is proportional to the Boltzmann factorexp(minus119864sph119879) where 119864sph is the energy of a sphaleronconfiguration and is given by 119864sph sim 4120587120601119892 where 120601 is thethermal average of the Higgs field ⟨1198670⟩119879 = 120601radic2 and 119892 isthe SU(2) gauge coupling In order to ensure the decouplingof the sphaleron transition in the broken phase 120601 should belarge enough to suppress the previous rate leading to thecondition 120601119888119879119888 ≳ 1 where 120601119888 is given by ⟨1198670⟩119879

119888

= 120601119888radic2

Therefore the first-order EWPT has to be strong enough toavoid a washout of the generated baryon asymmetry

A basic tool to analyze the EWPT is the finite temperatureeffective potential (FTEP) ([33ndash35] and for a review onEWPT and FTEP see [36] and references therein) In SM theone-loop FTEP is given by

119881(0)(120601) = 1198810 (120601) + 119881

(0)

1(120601) + 119881

(119879)

1(120601 119879) (12)

where 1198810 is the tree-level potential for the Higgs field 119881(0)

1

is the temperature independent one-loop contribution and119881(119879)

1(120601 119879) represents the finite temperature correction to the

zero-temperature potential 119881(0)1

in Landau gauge and M119878scheme is given by the Coleman-Weinberg potential

119881(0)

1(120601) = sum

119894=ℎ120594119882119885119905

119899119894

1198984

119894(120601)

641205872[ln

1198982

119894(120601)

1205832

minus 119862119894] (13)

where 1198982119894(120601) is the 120601-dependent squared mass for particle

119894 with 119899119894 the corresponding number of degrees of freedomSee Table 1 for the definitions where we note that fermionscontribute with opposite sign

The one-loop finite temperature contribution 119881(119879)1

isgiven by

119881(119879)

1(120601 119879)

=

1198794

21205872[

[

sum

119894=ℎ120594119882119885

119899119894119869119861 [

1198982

119894(120601)

1198792] + 119899119905119869119865 [

1198982

119905(120601)

1198792]]

]

(14)

where 119869119861 and 119869119865 are defined by

119869119861 (119886) = int

infin

0

119889119909 1199092 ln [1 minus 119890minusradic119909

2+119886]

119869119865 (119886) = int

infin

0

119889119909 1199092 ln [1 + 119890minusradic119909

2+119886]

(15)

Table 1 The 120601-dependent squared masses 1198982119894(120601) and the number

of degrees of freedom 119899119894 for particle 119894 in SM 1198982119894(120601 = 119907) gives the

physical masses for 119907 = 246GeV which is the vacuum expectationvalue of the Higgs field 119862

119894are given for MS scheme We include

the contributions coming from the Higgs boson ℎ the would-beNambu-Goldstone bosons 120594 the gauge bosons 119882 and 119885 and thetop quark 119905

Particle 1198982

119894(120601) 119898

2

119894(119907) 119899119894 119862119894

ℎ minus1198982+ 3120582120601

221205821199072 1 32

120594 minus1198982+ 1205821206012 0 3 32

119882

1198922

4

1206012

1198922

4

1199072 6 56

119885

1198922+ 11989210158402

4

1206012

1198922+ 11989210158402

4

1199072 3 56

119905

1205822

119905

2

1206012

1205822

119905

2

1199072

minus12 32

In the high temperature limit where 119898(120601) ≪ 119879 119869119861119865 can beexpanded in terms of119898(120601)119879 (high-temperature expansion)as

119869119861 (

1198982

1198792) = minus

1205874

45

+

1205872

12

1198982

1198792minus

120587

6

(

1198982

1198792)

32

minus

1

32

1198984

1198794ln 1198982

1198861198871198792+ O(

1198986

1198796)

119869119865 (

1198982

1198792) =

71205874

360

minus

1205872

24

1198982

1198792

minus

1

32

1198984

1198794ln 119898

2

1198861198911198792+ O(

1198986

1198796)

(16)

where 119886119887 = 161205872 exp(32 minus 2120574119864) and 119886119891 = 120587

2 exp(32 minus 2120574119864)that is ln 119886119887 asymp 54076 and ln 119886119891 asymp 26351

Using high temperature expansion the one-loop FTEPcan be written as

119881 (120601 119879) ≃ 119863 (1198792minus 1198792

119900) 1206012minus 119864119879120601

3+

120582 (119879)

4

1206014 (17)

where

119863 =

21198982

119882+ 1198982

119885+ 21198982

119905

81199072

1198792

119900=

1198982

ℎ

4119863

119864 =

21198983

119882+ 1198983

119885

41205871199073

120582 (119879) = 120582 minus

3

1612058721199074

times (21198984

119882ln

1205832

1198601198611198792+1198984

119885ln

1205832

1198601198611198792minus41198984

119905ln

1205832

1198601198651198792)

(18)


Veff T gt Tc T = Tc

T = 0

φφc

Figure 1 Typical forms of the effective potential in a case of the first-order phase transition for 119879 = 0 119879 = 119879119888 and 119879 gt 1198791

Here 1198982119894equiv 1198982

119894(119907) with 119907 = 246GeV are physical masses

for particle 119894 and ln119860119861 = ln 119886119887 minus 32 as well as ln119860119865 =ln 119886119891minus32 In (17) termswhich do not have 120601 dependence areomitted For sake of illustration we also neglect contributionsfrom ℎ and 120594 by assuming that ℎ is lighter than 119882 and 119885though this is phenomenologically not correct One sees thatonly weak bosons contribute to the coefficient 119864 of the cubicterm in (17) hence 119864 is rather small in SM

With this FTEP one can examine the behaviorof the EWPT analytically For 119879 gt 1198791 equiv

radic8120582(1198791)1198631198792119900(8120582(1198791)119863 minus 9119864

2) the only minimum of

the effective potential (17) is 120601 = 0 Hence the electroweaksymmetry is restored As the electroweak symmetry is brokenat 119879 = 0 the phase transition must occur at a temperature119879 that satisfies 1198791 gt 119879 gt 0 Inspection of (17) one clearlysees that the phase transition is first order for 119864 = 0 takingthe form depicted in Figure 1 and is second order for 119864 = 0When 119864 = 0 the critical temperature 119879119888 and 120601119888 are given by

1198792

119888=

120582 (119879119888)1198631198792

0

120582 (119879119888)119863 minus 1198642 120601119888 =

2119864119879119888

120582 (119879119888)

(19)

Then the strength of the first-order phase transition is givenby

120601119888

119879119888

=

2119864

120582 (119879119888)

sim

41198641199072

1198982

ℎ

(20)

One notes that a lighter Higgs boson is preferred to maintaina stronger first-order phase transition

At high temperature a certain class of higher-orderdiagrams the so-called ring diagrams (or daisy diagrams)[36] give significant contributions to the FTEP Dominantcontributions from the ring-diagrams can be resummed andit amounts to shifting the masses 1198982

119894(120601) for bosons in the

one-loop contributions given in (13) and (14) to thermalmasses given by M2

119894(120601 119879) equiv 119898

2

119894(120601) + Π119894(119879) where Π119894(119879)

is the one-loop self-energy of particle 119894 in the infrared limitAs for gauge bosons only the longitudinal modes receivecorrections to the masses when one-loop self-energies areadopted for Π119894(119879) Basically this forbids the longitudinalmodes of 119882 and 119885 to contribute to the cubic term in (17)

As a consequence the coefficient of the cubic term is reducedas

119864 =

21198983

119882+ 1198983

119885

41205871199073

997888rarr

2

3

21198983

119882+ 1198983

119885

41205871199073

sim 95 times 10minus3 (21)

leading to the reduction of the strength of the EWPTWith these results the sphaleron decoupling condition

120601119888119879119888 ≳ 1 reads

119898ℎ ≲radic4119864119907 sim 42GeV (22)

However this condition conflicts with the mass bound fromdirect search at the LEP 119898ℎ gt 114GeV [6] Furthermorerecent results from the LHC indicate a Higgs-like particle at125GeV So the EWPT is not strongly first order in the SMand the baryon asymmetry generated during EWPT cannotbe retained in the broken phase

For a heavierHiggs boson119898ℎ ≳ 119898119882 the approximationsused in the previous analysis are no longer correct in theSM In particular higher-order diagrams beyond the ringdiagrams also become nonnegligible at critical temperatureand a nonperturbative analysis is required Lattice studies[37ndash40] suggest however that there is an endpoint of first-order EWPT around 119898ℎ sim 70GeV above which thetransition turns into a continuous crossover Therefore thereis no EWPT for the experimentally preferred Higgs bosonmass

For successful EWBG then physics beyond the SM is alsorequired It is clear that the existence of the cubic term in theFTEP of (17) is essential for a first-order phase transitionand the cubic term arises from (119898

2)32 term in the high

temperature expansion for the bosonic loop function 119869119861given in (16) while there is no such term from fermioniccontributions So if a new boson with strong coupling to theHiggs boson is introduced their thermal loop can enhancethe cubic term and the EWPT can be strengthened On theother hand introducing a new fermion does not affect thecubic term and hence does not improve the strength of theEWPT at one-loop level

Based on the one-loop result therefore the introductionof 4G fermions seems useless from the viewpoint of makinga first-order EWPT However given that bounds on the 4Gquark masses from direct search at the LHC have reachedbeyond the 600GeV level that is beyond the perturbativeunitarity bound of 500ndash550GeV we have to reconsider theproblem of the EWPT beyond perturbative level Interest-ingly several studies (for zero temperature case) suggestthat strong Yukawa couplings of 4G quarks can induce newbound states of 4G quarks [41ndash43] If such Yukawa boundstates are bound tightly enough so that they do not dissolvearound the critical temperature of the EWPT bosonic boundstates may contribute to the FTEP via loop effect and mayinduce the cubic term leading to strongly first-order phasetransition Beside this possibility if the 4G quarks 119876 form apair condensate ⟨119876119876⟩ = 0 due to strong Yukawa couplingsan effective description of the theory would be given by athreeHiggs doubletmodel (1 elementary + 2 compositeHiggsdoublet) [41] (the EWPT in multicomposite Higgs doubletmodel with the 4G fermions was studied in [19] where


the composite Higgs fields appear as bound states of the4G fermions formed by strong four-fermion interactions)Therefore the potential for the Higgs field would bemodifiedeven at tree level and dynamics of the EWPT could bedrastically changed

4 Discussion and Summary

Let us continue the discussion from the previous sectionBosonization is only one aspect of very heavy 4G quarksthat could change the landscape for EWBG After all thebosonization described previous is due to strong Yukawacoupling A special feature of the SM is that fermion in factall masses reflect a dynamical coupling But fermions arespecial in that there is no theory of these Yukawa couplingsand neglecting neutrinos they span a range of six orders ofmagnitude If a fourth generation exists above the unitaritybound the strong Yukawa coupling could induce ⟨119876119876⟩condensation which in principle could replace the usualcondensation of the Higgs field as the electroweak symmetrybreaking mechanism A ldquobootstraprdquo dynamical symmetrybreaking (DSB) equation was recently formulated [44] andstudied [45] and it was foundnumerically thatDSB can occurfor Yukawa coupling 120582119876 ≳ 4120587 For such strong Yukawacoupling our traditional notions for EWBG may have to bereconsidered

The difficulty for the bootstrap DSB scenario is again thenewly observed [7 8] 125GeV boson at the LHC Howeverthe observed state could be a dilaton of scale invariance viola-tion rather than the genuine SMHiggs bosonThe couplingsof the dilaton to vector bosons and fermions are suppressedby 119907119891 compared to the SM Higgs boson case where 119891is the dilaton decay constant while 120574120574 and 119892119892 couplingsof the dilaton are essentially free parameters depending onthe details of scale invariance violation [46] Therefore ifthe observed signal arises mostly from the gluon fusion thedilaton could mimic the SM Higgs boson Discrimination isprovided by the detection of the Higgs production throughthe vector boson fusion (VBF) or the bremsstrahlung offa vector boson (VH) In particular important modes arethe VBF-produced 120574120574 mode and the 119887119887 and 120591120591 modes inthis regard As VBF and VH production are subdominantcompared with gluon fusion these modes are not yet firmlyestablished by the LHC experiments It is interesting that thebootstrap DSB equation is scale invariant by constructionand at the present level of study the scale is introducedheuristically as a physical condition for the bootstrap [45]hence a dilaton is in principle allowed Whether the 125GeVstate is the SM-like Higgs or a dilaton with rather modifiedcouplings can be checked by ATLAS and CMS

Even if the 125GeV object is verified as SM-like Higgsboson the strong Yukawa coupling of 4G quarks may still berelevantThe lightness of theHiggs bosonmay be because it isa pseudo-Goldstone boson from a TeV scale strongly coupledtheory which indeed the previous 120582119876 ≳ 4120587 situation seemsto qualify it as a candidate Pseudo-Goldstone Baryogenesis(PGBG) has been advocated [47] as a possible mechanismwhere strong coupling brings about parametric enhancement

of effective dimension-six interactions that loosen the rela-tion between Higgs self-coupling and the Higgs mass it is120582(119879119888) that really appears in (20) and Higgs mass enters onlythrough the standard relation of 1198982

ℎ= 2120582119907

2 Thus whetherone could have PGBG at work or not has to be tested bychecking the Higgs boson self-coupling which likely canbe done only at an 119890+119890minus Linear Collider with energy above500GeV which will take several decades

Of course the formulation of the bootstrap DSB equationis not yet at the level to demonstrate a possible PGBG andmuch more work needs to be done The formulation ofthe bootstrap DSB equation itself may offer a different pathtowards the study of order of phase transition The equationis a coupled set of two integral equations of loop momentumin the ladder approximation [45] At finite temperature thesewould become four equations since the temporal integrationwould be replaced by a summation One could check thetemperature dependence of DSB both in finding the critical119879119888 when symmetry is restored but also check what is ldquo120601119888rdquothe critical Higgs field expectation value and whether itsstrength allows a stronglyfirst-order transitionOf course thisis not yet done but it may offer further insight that is alonga different path than the usual approach of [36] discussed inthe previous section

To summarize EWBG is an attractive scenario to addressthe BAU puzzle especially because this scenario is basedon particle physics models that can be tested at the LHCAlthough the EWBG scenario fails within the minimal SMwith three quark generations introduction of the fourthgeneration may revive the scenario offering hope to solvethe problems in the SM case CP violation coming from the4 times 4 CKM matrix would be highly enhanced comparedto the SM mainly through the large masses (or the largeYukawa couplings) of 1199051015840 and 1198871015840 quarks Naive extension ofthe EWBG mechanism of SM to the 4G case shows that thisCP violating effect can be large enough to explain BAU withreasonable 4G parameters The issue of whether the EWPTbecomes strongly first order with 4G quarks is still an openproblem especially since the bounds on 4G quark massesfrom direct search at the LHC have reached the 600GeVlevel which is beyond the perturbative unitarity bound of500ndash550GeV A mechanism of the strongly first-order phasetransitionmay be accommodated by new bound states andor119876119876 condensation of 4G quarks both induced by strongYukawa couplings which couldmodify the FTEP at looptreelevel Higgs boson search at the LHC has uncovered a newboson with SMHiggs featuresThe study is still ongoing andits nature is important to obtain better understanding for theEWPT All in all the possibility of providing both sufficientCP violation as well as perhaps a strongly first-order phasetransition (which we did not demonstrate) all rooted in largeYukawa couplings of fourth generation quarks makes this aworthy pursuit

Acknowledgments

The authors are grateful to Y Kikukawa for collaboration onthe FS mechanism in the four-generation case presented in


Section 2 M Kohda is supported by the NTU Grant no10R40044 and the Laurel Program and W-S Hou by theAcademic Summit Grant NSC 100-2745-M-002-002-ASP ofthe National Science Council of Taiwan and various NTUgrants under the Excellence Program of the Ministry ofEducation of Taiwan

References

[1] J Beringer J F Arguin R M Barnett et al ldquoReview of particlephysicsrdquo Physical Review D vol 86 Article ID 010001 2012

[2] A D Sakharov ldquoCP symmetry violation C-asymmetry andbaryonic asymmetry of the universerdquoZhurnal Eksperimentalnoii Teoreticheskoi Fiziki Pisma vol 5 p 32 1967

[3] A D Sakharov ldquoCP symmetry violation C-asymmetry andbaryonic asymmetry of the universerdquo Journal of Experimentaland Theoretical Physics Letters vol 5 p 24 1967

[4] V A Kuzmin V A Rubakov and M E Shaposhnikov ldquoOnanomalous electroweak baryon-number non-conservation inthe early universerdquo Physics Letters B vol 155 no 1-2 pp 36ndash421985

[5] M Kobayashi and T Maskawa ldquoCP-Violation in the renor-malizable theory of weak interactionrdquo Progress of TheoreticalPhysics vol 49 no 2 pp 652ndash657 1973

[6] R Barate R Bruneliere I de Bonis et al ldquoSearch for thestandard model higgs boson at LEPrdquo Physics Letters B vol 565pp 61ndash75 2003

[7] G Aad T Abajyan B Abbott et al ldquoObservation of a newparticle in the search for the Standard Model Higgs boson withthe ATLAS detector at the LHCrdquo Physics Letters B vol 716 no1 pp 1ndash29 2012

[8] S Chatrchyan V Khachatryan A M Sirunyan et al ldquoObser-vation of a new boson at a mass of 125GeV with the CMSexperiment at the LHCrdquo Physics Letters B vol 716 no 1 pp30ndash61 2012

[9] D E Morrissey andM J Ramsey-Musolf ldquoElectroweak baryo-genesisrdquoNew Journal of Physics vol 14 Article ID 125003 2012

[10] H Murayama V Rentala J Shu and T T Yanagida ldquoSavingfourth generation and baryon number by living longrdquo PhysicsLetters B vol 705 no 3 pp 208ndash211 2011

[11] W S Hou ldquoSource of CP violation for baryon asymmetry of theuniverserdquo Chinese Journal of Physics vol 47 p 134 2009

[12] M E Shaposhnikov ldquoPossible appearance of the baryon asym-metry of the universe in an electroweak theoryrdquo Jounal ofExperimental and Theoretical Physics Letters vol 44 no 8 pp465ndash468 1986

[13] ME Shaposhnikov ldquoBaryon asymmetry of the universe instandard electroweak theoryrdquo Nuclear Physics B vol 287 pp757ndash775 1987

[14] R Fok and G D Kribs ldquoFour generations the electroweakphase transition and supersymmetryrdquo Physical Review D vol78 no 7 Article ID 075023 2008

[15] S Chatrchyan V Khachatryan A M Sirunyan et al ldquoSearchfor heavy top-like quarkpair production in the dilepton finalstate in pp collisions at sqrt(s) = 7 TeVrdquo Physics Letters B vol716 pp 103ndash121 2012

[16] S ChatrchyanVKhachatryanAM Sirunyan et al ldquoSearch forheavy bottom-like quarks in 49fbminus1 of pp collisions at radics = 7TeVrdquo Journal of High Energy Physics vol 1205 p 123 2012

[17] M S Chanowitz M A Furman and I Hinchliffe ldquoWeakinteractions of ultra heavy fermionsrdquo Physics Letters B vol 78no 2-3 pp 285ndash289 1978

[18] S W Ham S K Oh and D Son ldquoElectroweak phase transitionin the minimal supersymmetric standard model with fourgenerationsrdquo Physical Review D vol 71 Article ID 015001 6pages 2005

[19] Y Kikukawa M Kohda and J Yasuda ldquoThe strongly coupledfourth family and a first-order electroweak phase transition Imdashquark sectorrdquo Progress of Theoretical Physics vol 122 no 2 pp401ndash426 2009

[20] C Jarlskog ldquoFlavor projection operators and applications to CPviolation with any number of familiesrdquo Physical Review D vol36 no 7 pp 2128ndash2136 1987

[21] W S Hou Y Y Mao and C H Shen ldquoLeading effect of CPviolation with four generationsrdquo Physical Review D vol 82Article ID 036005 10 pages 2010

[22] G R Farrar and M E Shaposhnikov ldquoBaryon asymmetry ofthe universe in the minimal standard modelrdquo Physical ReviewLetters vol 70 no 19 pp 2833ndash2836 1993

[23] G R Farrar and M E Shaposhnikov ldquoErratum lsquoBaryonasymmetry of the universe in the minimal standard modelrsquordquoPhysical Review Letters vol 71 p 210 1993

[24] G R Farrar andM E Shaposhnikov ldquoBaryon asymmetry of theuniverse in the standard electroweak theoryrdquo Physical ReviewDvol 50 no 2 pp 774ndash818 1994

[25] G R Farrar and M E Shaposhnikov ldquoNote added tolsquoBaryon asymmetry of the universe in the standard modelrsquordquohttparxivorgabshep-ph9406387

[26] M B Gavela P Hernandez J Orloff and O Pene ldquoStandardmodel Cp-violation and baryon asymmetryrdquo Modern PhysicsLetters A vol 9 no 9 p 795 1994

[27] M B Gavela M Lozano J Orloff and O Pene ldquoStandardmodel CP-violation and baryon asymmetry (I) Zero temper-aturerdquo Nuclear Physics B vol 430 no 2 pp 345ndash381 1994

[28] M B Gavela P Hernandez J Orloff O Pene and C QuimbayldquoStandard model CP-violation and baryon asymmetry (II)Finite temperaturerdquo Nuclear Physics B vol 430 no 2 pp 382ndash426 1994

[29] P Huet and E Sather ldquoElectroweak baryogenesis and standardmodel CP violationrdquo Physical Review D vol 51 no 2 pp 379ndash394 1995

[30] E Braaten andRD Pisarski ldquoCalculation of the quark dampingrate in hot QCDrdquo Physical Review D vol 46 no 4 pp 1829ndash1834 1992

[31] W S Hou M Kohda and F Xu ldquoMeasuring the fourth-generation brarr s quadrangle at the LHCrdquo Physical Review Dvol 84 no 9 Article ID 094027 7 pages 2011

[32] W S Hou M Kohda and F Xu ldquoHints for a low 119861119904 rarr 120583+120583minus

rate and the fourth generationrdquo Physical Review D vol 85 no9 Article ID 097502 5 pages 2012

[33] L Dolan and R Jackiw ldquoSymmetry behavior at finite tempera-turerdquo Physical Review D vol 9 no 12 pp 3320ndash3341 1974

[34] S Weinberg ldquoGauge and global symmetries at high tempera-turerdquo Physical Review D vol 9 no 12 pp 3357ndash3378 1974

[35] G W Anderson and L J Hall ldquoElectroweak phase transitionand baryogenesisrdquo Physical Review D vol 45 no 8 pp 2685ndash2698 1992

[36] M Quiros ldquoFinite temperature fieldtheory and phase transi-tionsrdquo httparxivorgabshep-ph9901312


[37] M Gurtler E M Ilgenfritz and A Schiller ldquoWhere theelectroweak phase transition endsrdquo Physical Review D vol 56no 7 pp 3888ndash3895 1997

[38] M Laine and K Rummukainen ldquoWhatrsquos new with the elec-troweak phase transitionrdquo Nuclear Physics B vol 73 no 1ndash3pp 180ndash185 1999

[39] F Csikor Z Fodor and J Heitger ldquoEnd point of the hotelectroweak phase transitionrdquo Physical Review Letters vol 82no 1 pp 21ndash24 1999

[40] Y Aoki F Csikor Z Fodor and A Ukawa ldquoThe end point ofthe first-order phase transition of the SU(2) gauge-Higgs modelon a 4-dimensional isotropic latticerdquo Physical Review D vol 60no 1 Article ID 013001 pp 1ndash8 1999

[41] P Q Hung and C Xiong ldquoDynamical electroweak symmetrybreakingwith a heavy fourth generationrdquoNuclear Physics B vol848 no 2 pp 288ndash302 2011

[42] K Ishiwata and M B Wise ldquoFourth generation bound statesrdquoPhysical ReviewD vol 83 no 7 Article ID 074015 8 pages 2011

[43] T Enkhbat W S Hou and H Yokoya ldquoEarly LHC phe-nomenology of Yukawa-bound heavy 119876119876 mesonsrdquo PhysicalReview D vol 84 no 9 Article ID 094013 14 pages 2011

[44] W S Hou ldquoSome unfinished thoughts on strong yukawacouplingsrdquo Chinese Journal of Physics vol 50 p 375 2012

[45] Y Mimura W S Hou and H Kohyama ldquoBootstrapdynamical symmetrybreaking with new heavy chiral quarksrdquohttparxivorgabs12066063

[46] D Elander and M Piai ldquoThe decay constant of the holographictechni-dilaton and the 125 GeV bosonrdquo Nuclear Physics B vol867 no 3 pp 779ndash809 2013

[47] B Grinstein and M Trott ldquoElectroweak baryogenesis with apseudo-Goldstone Higgs bosonrdquo Physical Review D vol 78 no7 Article ID 075022 28 pages 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


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Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


so-called sphaleron transition especially when thetemperature is around or above 119879119888

(2) 119862 is violated by the 119881 minus 119860 structure of the weakinteraction while CP is violated by the Kobayashi-Maskawa (KM) mechanism [5] with three genera-tions of quarks A complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix provides thesource of CP violation which nicely describes allterrestrially observed CP-violating phenomena thatis CP violation in the 119870- and 119861-meson (even the 119863-meson) systems

(3) SU(2) times U(1) gauge symmetry is expected to berestored at high temperature above 119879119888 Therefore asthe universe cools down to a temperature of around119879119888 the universe undergoes a phase transition (theEWPT) from symmetric phase of SU(2) times U(1) tothe broken phase If the EWPT is first order thephase transition proceeds through bubble nucleationof the broken phase where expansion of the bubblesfills up the entire universe Thus temporal departurefrom equilibrium is achieved near the surface of theexpanding bubble

Unfortunately themeasured parameters of the SMappearto be insufficient to bring about EWBG The SM fails toexplain the BAU due to two reasons The first reason is thattheKMmechanismwith three quark generations cannot offerlarge enough CP violation (CPV) to produce the observedbaryon-to-entropy ratio during the EWPT The second rea-son is that the EWPT is not first order for a Higgs boson with119898ℎ gt 114GeV required already by direct search at the LEP[6] and now indicated to be around 125GeV at the LHC [7 8]Therefore physics beyond the SM is needed for EWBG andvarious extensions of the SMhave been investigated from thispoint of view In particular there are extensive studies (for arecent review on EWBG see [9] and references therein) forthe twoHiggs doublet model and supersymmetric extensionsof the SM such as the MSSM and the NMSSM It shouldbe noted however that MSSM itself is under stress from theLHC and its case for EWBG is only marginal [9] One of thebiggest means of attraction to have EWBG is that the impliednew physics should be accessible at the LHC at least partially

In this contribution we consider a four-generation (4G)extension of the SM and discuss a possibility to revive EWBG(beside a possible revival of EWBG the 4G has anotherinteresting implication on baryogenesis if it is long-livedsee [10]) fully within the SM framework as 4G does notintroduce anything that was not already contained in SMWefocus on the quark sector which includes fourth generationquarks 1199051015840 and 1198871015840 Compared to the three-generation casethe 4G quarks bring three additional mixing angles andtwo additional complex phases in the 4 times 4 CKM matrixthus providing new sources of CP violation One can argueintuitively for a possible effect of new CP violation on BAUby studying basis-independent invariants constructed fromthe quark mass matrices (or the Yukawa coupling matrices)which are extensions of the Jarlskog determinant in the SMOne of us [11] pointed out that such an invariant quantitycan be highly enhanced compared to the three-generation

case mainly due to the large masses (or the large Yukawacouplings) of 1199051015840 and 1198871015840(see [12 13] for an earlier discussionon 4G-enhanced CP violation for BAU) As for the issue ofthe EWPT a one-loop analysis shows that the 4G quarks donot play a positive role towards a first-order phase transition[14] However given that bounds on the 4G quark massesfrom direct search at the LHC have entered 600GeV level[15 16] which is beyond the tree-level perturbative unitaritybound (UB) of 500ndash550GeV [17] we have to reconsider theproblemof EWPTbeyondperturbation theory In this regardthe nature of the EWPT with 4G quarks is still quite an openproblem (for earlier works which address the issue of theEWPTwithin the four-generation framework (but not withinthe SM framework) see [14 18] (supersymmetric model withthe 4G fermions) and [19] (dynamical electroweak symmetrybreaking model due to strong four-fermion interactionsamong the 4G fermions))

The recent observation of a Higgs-like object with massaround 125GeV poses a special impasse situation since anaive application of heavy 1199051015840 and 119887

1015840 quarks in the loopwould enhance the gluon-gluon fusion production of theHiggs boson by about an order of magnitude which is notsupported by 119892119892 rarr 119881119881 data [7 8] where 119881 is the 119882 or119885 boson Furthermore quadratic corrections to the Higgsmass from heavy 1199051015840 and 1198871015840 quarks make the light Higgs massrather difficult to sustain With the deep conflict between alight Higgs and having 4G quarks beyond UB it may seemcavalier to relegate this again to nonperturbative treatmentsHowever the aim of this work is very modest which is justto show the nontrivial nature of 4G towards EWBG Sincethe source of CPV arises from off-diagonal couplings theHiggs boson does not directly enter EWBG computationWewill discuss towards the end the possibility that the observed125GeV boson could be a dilaton rather than a bona fide SMHiggs boson or it may be a pseudo-Goldstone Higgs boson

This paper is organized as follows In Section 2 we discussthe effect of CP violation from the fourth generation quarksector on BAU Extending the EWBGmechanism in the caseof the SM to the fourth generation case we estimate theamount of the BAU produced during a first-order EWPT InSection 3 we summarize the EWPT in the SM briefly anddiscuss possible mechanisms to induce a first-order phasetransition within the four-generation framework Finally inSection 4 we offer some discussions and give our summary

2 CP Violation

One of the reasons for the failure of EWBG based on SM isthe insufficiency of CP violation from the KM phase Thissituation can be naively understood as follows

In the SM with three generations a basis-independentinvariant for CP violation is given by the well-known Jarlskogdeterminant [20]

det [119872119906119872dagger

119906119872119889119872

dagger

119889]

100381610038161003816100381610038163minusgen

equiv 119894119869SM (2)



119869SM = 2JSM (1198982

119905minus 1198982

119888) (1198982

119905minus 1198982

119906) (1198982

119888minus 1198982

119906)

times (1198982

119887minus 1198982

119904) (1198982

119887minus 1198982

119889) (1198982

119904minus 1198982

119889)

(3)



12




1198871198982

1199041198798

119888sim



119869119904119887

(234)= 2J119904 (119898

2

1199051015840minus 1198982

119905) (1198982

1199051015840minus 1198982

119888) (1198982

119905minus 1198982

119888)

times (1198982

1198871015840minus 1198982

119887) (1198982

1198871015840minus 1198982

119904) (1198982

119887minus 1198982

119904)

sim

J119904

JSM(

1198982

1199051015840

1198982119905

minus 1)

1198982

1199051015840

1198982119888

1198984

1198871015840

1198982

1198871198982119904

119869SM

(4)


11990510158401198871198811199051015840119904) In fact 119869

119904119887



dagger

119906119872119889119872

dagger

119889]3








119899119861

119904

sim minus

10minus2

119879

int

119889120596

2120587

1198990 (120596) [1 minus 1198990 (120596)]

Δp sdot v119882119879

Δ (120596) (5)





dagger

119871119877119877119871119877] (6)



119902119894

119871rarr 119902

119891


119877

119891119894




(

2(120596 minus Ω119871 + 119894120574 +

1

3

119894120590 sdot 120597) 119872119906119889 120579 (119911)

119872dagger

119906119889120579 (119911) 2 (120596 minus Ω119877 + 119894120574 minus

1

3

119894120590 sdot 120597)

)

times Ψ119906119889 (119911) = 0

(7)

where Ψ119906 = (120595119906 120595119888 120595119905 1205951199051015840)119879 Ψ119889 = (120595119889 120595119904 120595119887 1205951198871015840)




2




Δ 119889 (120596) =

4

3

(

271205871205721198821198792

64Ω01198722

119882

)

3

[1 + (

120596 minus Ω0

120574

)

2

]

minus6

times (

1

6120574

)

9

Im Tr [119872119906119872dagger

119906 119872119889119872

dagger

119889]

3

(8)



Im Tr [119872119906119872dagger

119906 119872119889119872

dagger

119889]

31003816100381610038161003816100381610038163minusgen

= 3 Im det [119872119906119872dagger

119906 119872119889119872

dagger

119889]

100381610038161003816100381610038163minusgen

= 3 119869SM

(9)


lowast

1199051015840119889| ≪ |119881119905119894119881

lowast

1199051015840119894| (119894 = 119904 119887 1198871015840)


Im Tr [119872119906119872dagger

119906 119872119889119872

dagger

119889]

3

≃ minus6J119904 (1198982

1199051015840minus 1198982

119905)1198982

11990510158401198982

1199051198984

11988710158401198982

119887

(10)


119905119904119881lowast

11990510158401198871198811199051198871198811199051015840119904) One notes that

Im Tr [119872119906119872dagger

119906119872119889119872

dagger

119889]

3

≃ minus3 119869119904119887

(234) where 119869

119904119887

(234)is





119899119861

119904


J119904

10minus4)(

1198981199051015840

650GeV)

4

(

1198981198871015840

650GeV)

4

(11)


lowast






lowast








119888

= 120601119888radic2



119881(0)(120601) = 1198810 (120601) + 119881

(0)

1(120601) + 119881

(119879)

1(120601 119879) (12)


1





119881(0)

1(120601) = sum

119894=ℎ120594119882119885119905

119899119894

1198984

119894(120601)

641205872[ln

1198982

119894(120601)

1205832

minus 119862119894] (13)




isgiven by

119881(119879)

1(120601 119879)

=

1198794

21205872[

[

sum

119894=ℎ120594119882119885

119899119894119869119861 [

1198982

119894(120601)

1198792] + 119899119905119869119865 [

1198982

119905(120601)

1198792]]

]

(14)


119869119861 (119886) = int

infin

0


2+119886]

119869119865 (119886) = int

infin

0

119889119909 1199092 ln [1 + 119890minusradic119909

2+119886]

(15)






Particle 1198982

119894(120601) 119898

2

119894(119907) 119899119894 119862119894

ℎ minus1198982+ 3120582120601

221205821199072 1 32

120594 minus1198982+ 1205821206012 0 3 32

119882

1198922

4

1206012

1198922

4

1199072 6 56

119885

1198922+ 11989210158402

4

1206012

1198922+ 11989210158402

4

1199072 3 56

119905

1205822

119905

2

1206012

1205822

119905

2

1199072

minus12 32


119869119861 (

1198982

1198792) = minus

1205874

45

+

1205872

12

1198982

1198792minus

120587

6

(

1198982

1198792)

32

minus

1

32

1198984

1198794ln 1198982

1198861198871198792+ O(

1198986

1198796)

119869119865 (

1198982

1198792) =

71205874

360

minus

1205872

24

1198982

1198792

minus

1

32

1198984

1198794ln 119898

2

1198861198911198792+ O(

1198986

1198796)

(16)




119881 (120601 119879) ≃ 119863 (1198792minus 1198792

119900) 1206012minus 119864119879120601

3+

120582 (119879)

4

1206014 (17)

where

119863 =

21198982

119882+ 1198982

119885+ 21198982

119905

81199072

1198792

119900=

1198982

ℎ

4119863

119864 =

21198983

119882+ 1198983

119885

41205871199073

120582 (119879) = 120582 minus

3

1612058721199074

times (21198984

119882ln

1205832

1198601198611198792+1198984

119885ln

1205832

1198601198611198792minus41198984

119905ln

1205832

1198601198651198792)

(18)


Veff T gt Tc T = Tc

T = 0

φφc


Here 1198982119894equiv 1198982




radic8120582(1198791)1198631198792119900(8120582(1198791)119863 minus 9119864



1198792

119888=

120582 (119879119888)1198631198792

0

120582 (119879119888)119863 minus 1198642 120601119888 =

2119864119879119888

120582 (119879119888)

(19)


120601119888

119879119888

=

2119864

120582 (119879119888)

sim

41198641199072

1198982

ℎ

(20)





119894(120601 119879) equiv 119898

2

119894(120601) + Π119894(119879) where Π119894(119879)



119864 =

21198983

119882+ 1198983

119885

41205871199073

997888rarr

2

3

21198983

119882+ 1198983

119885

41205871199073



120601119888119879119888 ≳ 1 reads

119898ℎ ≲radic4119864119907 sim 42GeV (22)














ℎ= 2120582119907




Acknowledgments




References























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119869SM = 2JSM (1198982

119905minus 1198982

119888) (1198982

119905minus 1198982

119906) (1198982

119888minus 1198982

119906)

times (1198982

119887minus 1198982

119904) (1198982

119887minus 1198982

119889) (1198982

119904minus 1198982

119889)

(3)



12




1198871198982

1199041198798

119888sim



119869119904119887

(234)= 2J119904 (119898

2

1199051015840minus 1198982

119905) (1198982

1199051015840minus 1198982

119888) (1198982

119905minus 1198982

119888)

times (1198982

1198871015840minus 1198982

119887) (1198982

1198871015840minus 1198982

119904) (1198982

119887minus 1198982

119904)

sim

J119904

JSM(

1198982

1199051015840

1198982119905

minus 1)

1198982

1199051015840

1198982119888

1198984

1198871015840

1198982

1198871198982119904

119869SM

(4)


11990510158401198871198811199051015840119904) In fact 119869

119904119887



dagger

119906119872119889119872

dagger

119889]3








119899119861

119904

sim minus

10minus2

119879

int

119889120596

2120587

1198990 (120596) [1 minus 1198990 (120596)]

Δp sdot v119882119879

Δ (120596) (5)





dagger

119871119877119877119871119877] (6)



119902119894

119871rarr 119902

119891


119877

119891119894




(

2(120596 minus Ω119871 + 119894120574 +

1

3

119894120590 sdot 120597) 119872119906119889 120579 (119911)

119872dagger

119906119889120579 (119911) 2 (120596 minus Ω119877 + 119894120574 minus

1

3

119894120590 sdot 120597)

)

times Ψ119906119889 (119911) = 0

(7)

where Ψ119906 = (120595119906 120595119888 120595119905 1205951199051015840)119879 Ψ119889 = (120595119889 120595119904 120595119887 1205951198871015840)




2




Δ 119889 (120596) =

4

3

(

271205871205721198821198792

64Ω01198722

119882

)

3

[1 + (

120596 minus Ω0

120574

)

2

]

minus6

times (

1

6120574

)

9

Im Tr [119872119906119872dagger

119906 119872119889119872

dagger

119889]

3

(8)



Im Tr [119872119906119872dagger

119906 119872119889119872

dagger

119889]

31003816100381610038161003816100381610038163minusgen

= 3 Im det [119872119906119872dagger

119906 119872119889119872

dagger

119889]

100381610038161003816100381610038163minusgen

= 3 119869SM

(9)


lowast

1199051015840119889| ≪ |119881119905119894119881

lowast

1199051015840119894| (119894 = 119904 119887 1198871015840)


Im Tr [119872119906119872dagger

119906 119872119889119872

dagger

119889]

3

≃ minus6J119904 (1198982

1199051015840minus 1198982

119905)1198982

11990510158401198982

1199051198984

11988710158401198982

119887

(10)


119905119904119881lowast

11990510158401198871198811199051198871198811199051015840119904) One notes that

Im Tr [119872119906119872dagger

119906119872119889119872

dagger

119889]

3

≃ minus3 119869119904119887

(234) where 119869

119904119887

(234)is





119899119861

119904


J119904

10minus4)(

1198981199051015840

650GeV)

4

(

1198981198871015840

650GeV)

4

(11)


lowast






lowast








119888

= 120601119888radic2



119881(0)(120601) = 1198810 (120601) + 119881

(0)

1(120601) + 119881

(119879)

1(120601 119879) (12)


1





119881(0)

1(120601) = sum

119894=ℎ120594119882119885119905

119899119894

1198984

119894(120601)

641205872[ln

1198982

119894(120601)

1205832

minus 119862119894] (13)




isgiven by

119881(119879)

1(120601 119879)

=

1198794

21205872[

[

sum

119894=ℎ120594119882119885

119899119894119869119861 [

1198982

119894(120601)

1198792] + 119899119905119869119865 [

1198982

119905(120601)

1198792]]

]

(14)


119869119861 (119886) = int

infin

0


2+119886]

119869119865 (119886) = int

infin

0

119889119909 1199092 ln [1 + 119890minusradic119909

2+119886]

(15)






Particle 1198982

119894(120601) 119898

2

119894(119907) 119899119894 119862119894

ℎ minus1198982+ 3120582120601

221205821199072 1 32

120594 minus1198982+ 1205821206012 0 3 32

119882

1198922

4

1206012

1198922

4

1199072 6 56

119885

1198922+ 11989210158402

4

1206012

1198922+ 11989210158402

4

1199072 3 56

119905

1205822

119905

2

1206012

1205822

119905

2

1199072

minus12 32


119869119861 (

1198982

1198792) = minus

1205874

45

+

1205872

12

1198982

1198792minus

120587

6

(

1198982

1198792)

32

minus

1

32

1198984

1198794ln 1198982

1198861198871198792+ O(

1198986

1198796)

119869119865 (

1198982

1198792) =

71205874

360

minus

1205872

24

1198982

1198792

minus

1

32

1198984

1198794ln 119898

2

1198861198911198792+ O(

1198986

1198796)

(16)




119881 (120601 119879) ≃ 119863 (1198792minus 1198792

119900) 1206012minus 119864119879120601

3+

120582 (119879)

4

1206014 (17)

where

119863 =

21198982

119882+ 1198982

119885+ 21198982

119905

81199072

1198792

119900=

1198982

ℎ

4119863

119864 =

21198983

119882+ 1198983

119885

41205871199073

120582 (119879) = 120582 minus

3

1612058721199074

times (21198984

119882ln

1205832

1198601198611198792+1198984

119885ln

1205832

1198601198611198792minus41198984

119905ln

1205832

1198601198651198792)

(18)


Veff T gt Tc T = Tc

T = 0

φφc


Here 1198982119894equiv 1198982




radic8120582(1198791)1198631198792119900(8120582(1198791)119863 minus 9119864



1198792

119888=

120582 (119879119888)1198631198792

0

120582 (119879119888)119863 minus 1198642 120601119888 =

2119864119879119888

120582 (119879119888)

(19)


120601119888

119879119888

=

2119864

120582 (119879119888)

sim

41198641199072

1198982

ℎ

(20)





119894(120601 119879) equiv 119898

2

119894(120601) + Π119894(119879) where Π119894(119879)



119864 =

21198983

119882+ 1198983

119885

41205871199073

997888rarr

2

3

21198983

119882+ 1198983

119885

41205871199073



120601119888119879119888 ≳ 1 reads

119898ℎ ≲radic4119864119907 sim 42GeV (22)














ℎ= 2120582119907




Acknowledgments




References























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





2




Δ 119889 (120596) =

4

3

(

271205871205721198821198792

64Ω01198722

119882

)

3

[1 + (

120596 minus Ω0

120574

)

2

]

minus6

times (

1

6120574

)

9

Im Tr [119872119906119872dagger

119906 119872119889119872

dagger

119889]

3

(8)



Im Tr [119872119906119872dagger

119906 119872119889119872

dagger

119889]

31003816100381610038161003816100381610038163minusgen

= 3 Im det [119872119906119872dagger

119906 119872119889119872

dagger

119889]

100381610038161003816100381610038163minusgen

= 3 119869SM

(9)


lowast

1199051015840119889| ≪ |119881119905119894119881

lowast

1199051015840119894| (119894 = 119904 119887 1198871015840)


Im Tr [119872119906119872dagger

119906 119872119889119872

dagger

119889]

3

≃ minus6J119904 (1198982

1199051015840minus 1198982

119905)1198982

11990510158401198982

1199051198984

11988710158401198982

119887

(10)


119905119904119881lowast

11990510158401198871198811199051198871198811199051015840119904) One notes that

Im Tr [119872119906119872dagger

119906119872119889119872

dagger

119889]

3

≃ minus3 119869119904119887

(234) where 119869

119904119887

(234)is





119899119861

119904


J119904

10minus4)(

1198981199051015840

650GeV)

4

(

1198981198871015840

650GeV)

4

(11)


lowast






lowast








119888

= 120601119888radic2



119881(0)(120601) = 1198810 (120601) + 119881

(0)

1(120601) + 119881

(119879)

1(120601 119879) (12)


1





119881(0)

1(120601) = sum

119894=ℎ120594119882119885119905

119899119894

1198984

119894(120601)

641205872[ln

1198982

119894(120601)

1205832

minus 119862119894] (13)




isgiven by

119881(119879)

1(120601 119879)

=

1198794

21205872[

[

sum

119894=ℎ120594119882119885

119899119894119869119861 [

1198982

119894(120601)

1198792] + 119899119905119869119865 [

1198982

119905(120601)

1198792]]

]

(14)


119869119861 (119886) = int

infin

0


2+119886]

119869119865 (119886) = int

infin

0

119889119909 1199092 ln [1 + 119890minusradic119909

2+119886]

(15)






Particle 1198982

119894(120601) 119898

2

119894(119907) 119899119894 119862119894

ℎ minus1198982+ 3120582120601

221205821199072 1 32

120594 minus1198982+ 1205821206012 0 3 32

119882

1198922

4

1206012

1198922

4

1199072 6 56

119885

1198922+ 11989210158402

4

1206012

1198922+ 11989210158402

4

1199072 3 56

119905

1205822

119905

2

1206012

1205822

119905

2

1199072

minus12 32


119869119861 (

1198982

1198792) = minus

1205874

45

+

1205872

12

1198982

1198792minus

120587

6

(

1198982

1198792)

32

minus

1

32

1198984

1198794ln 1198982

1198861198871198792+ O(

1198986

1198796)

119869119865 (

1198982

1198792) =

71205874

360

minus

1205872

24

1198982

1198792

minus

1

32

1198984

1198794ln 119898

2

1198861198911198792+ O(

1198986

1198796)

(16)




119881 (120601 119879) ≃ 119863 (1198792minus 1198792

119900) 1206012minus 119864119879120601

3+

120582 (119879)

4

1206014 (17)

where

119863 =

21198982

119882+ 1198982

119885+ 21198982

119905

81199072

1198792

119900=

1198982

ℎ

4119863

119864 =

21198983

119882+ 1198983

119885

41205871199073

120582 (119879) = 120582 minus

3

1612058721199074

times (21198984

119882ln

1205832

1198601198611198792+1198984

119885ln

1205832

1198601198611198792minus41198984

119905ln

1205832

1198601198651198792)

(18)


Veff T gt Tc T = Tc

T = 0

φφc


Here 1198982119894equiv 1198982




radic8120582(1198791)1198631198792119900(8120582(1198791)119863 minus 9119864



1198792

119888=

120582 (119879119888)1198631198792

0

120582 (119879119888)119863 minus 1198642 120601119888 =

2119864119879119888

120582 (119879119888)

(19)


120601119888

119879119888

=

2119864

120582 (119879119888)

sim

41198641199072

1198982

ℎ

(20)





119894(120601 119879) equiv 119898

2

119894(120601) + Π119894(119879) where Π119894(119879)



119864 =

21198983

119882+ 1198983

119885

41205871199073

997888rarr

2

3

21198983

119882+ 1198983

119885

41205871199073



120601119888119879119888 ≳ 1 reads

119898ℎ ≲radic4119864119907 sim 42GeV (22)














ℎ= 2120582119907




Acknowledgments




References























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119888

= 120601119888radic2



119881(0)(120601) = 1198810 (120601) + 119881

(0)

1(120601) + 119881

(119879)

1(120601 119879) (12)


1





119881(0)

1(120601) = sum

119894=ℎ120594119882119885119905

119899119894

1198984

119894(120601)

641205872[ln

1198982

119894(120601)

1205832

minus 119862119894] (13)




isgiven by

119881(119879)

1(120601 119879)

=

1198794

21205872[

[

sum

119894=ℎ120594119882119885

119899119894119869119861 [

1198982

119894(120601)

1198792] + 119899119905119869119865 [

1198982

119905(120601)

1198792]]

]

(14)


119869119861 (119886) = int

infin

0


2+119886]

119869119865 (119886) = int

infin

0

119889119909 1199092 ln [1 + 119890minusradic119909

2+119886]

(15)






Particle 1198982

119894(120601) 119898

2

119894(119907) 119899119894 119862119894

ℎ minus1198982+ 3120582120601

221205821199072 1 32

120594 minus1198982+ 1205821206012 0 3 32

119882

1198922

4

1206012

1198922

4

1199072 6 56

119885

1198922+ 11989210158402

4

1206012

1198922+ 11989210158402

4

1199072 3 56

119905

1205822

119905

2

1206012

1205822

119905

2

1199072

minus12 32


119869119861 (

1198982

1198792) = minus

1205874

45

+

1205872

12

1198982

1198792minus

120587

6

(

1198982

1198792)

32

minus

1

32

1198984

1198794ln 1198982

1198861198871198792+ O(

1198986

1198796)

119869119865 (

1198982

1198792) =

71205874

360

minus

1205872

24

1198982

1198792

minus

1

32

1198984

1198794ln 119898

2

1198861198911198792+ O(

1198986

1198796)

(16)




119881 (120601 119879) ≃ 119863 (1198792minus 1198792

119900) 1206012minus 119864119879120601

3+

120582 (119879)

4

1206014 (17)

where

119863 =

21198982

119882+ 1198982

119885+ 21198982

119905

81199072

1198792

119900=

1198982

ℎ

4119863

119864 =

21198983

119882+ 1198983

119885

41205871199073

120582 (119879) = 120582 minus

3

1612058721199074

times (21198984

119882ln

1205832

1198601198611198792+1198984

119885ln

1205832

1198601198611198792minus41198984

119905ln

1205832

1198601198651198792)

(18)


Veff T gt Tc T = Tc

T = 0

φφc


Here 1198982119894equiv 1198982




radic8120582(1198791)1198631198792119900(8120582(1198791)119863 minus 9119864



1198792

119888=

120582 (119879119888)1198631198792

0

120582 (119879119888)119863 minus 1198642 120601119888 =

2119864119879119888

120582 (119879119888)

(19)


120601119888

119879119888

=

2119864

120582 (119879119888)

sim

41198641199072

1198982

ℎ

(20)





119894(120601 119879) equiv 119898

2

119894(120601) + Π119894(119879) where Π119894(119879)



119864 =

21198983

119882+ 1198983

119885

41205871199073

997888rarr

2

3

21198983

119882+ 1198983

119885

41205871199073



120601119888119879119888 ≳ 1 reads

119898ℎ ≲radic4119864119907 sim 42GeV (22)














ℎ= 2120582119907




Acknowledgments




References























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Veff T gt Tc T = Tc

T = 0

φφc


Here 1198982119894equiv 1198982




radic8120582(1198791)1198631198792119900(8120582(1198791)119863 minus 9119864



1198792

119888=

120582 (119879119888)1198631198792

0

120582 (119879119888)119863 minus 1198642 120601119888 =

2119864119879119888

120582 (119879119888)

(19)


120601119888

119879119888

=

2119864

120582 (119879119888)

sim

41198641199072

1198982

ℎ

(20)





119894(120601 119879) equiv 119898

2

119894(120601) + Π119894(119879) where Π119894(119879)



119864 =

21198983

119882+ 1198983

119885

41205871199073

997888rarr

2

3

21198983

119882+ 1198983

119885

41205871199073



120601119888119879119888 ≳ 1 reads

119898ℎ ≲radic4119864119907 sim 42GeV (22)














ℎ= 2120582119907




Acknowledgments




References























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










ℎ= 2120582119907




Acknowledgments




References























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





References























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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Research Article Towards Reviving Electroweak Baryogenesis ...downloads.hindawi.com/journals/ahep/2013/769240.pdf · Research Article Towards Reviving Electroweak Baryogenesis with