The strongly coupled fourth family and a first-order electroweak phase transition (I) quark sector

0.96 MB·English

by  Y. Kikukawa

UT-Komaba/09-1
The strongly coupled fourth family and a first-order electroweak phase transition
(I) quark sector
Yoshio Kikukawa,1, Masaya Kohda,2, and Junichiro Yasuda3,
∗ † ‡
1Institute of Physics, University of Tokyo Tokyo 153-8092, Japan
2Department of Physics, Nagoya University Nagoya 464-8602, Japan
3Center for the Studies of Higher Education, Nagoya University Nagoya 464-8601, Japan
(Dated: January 27, 2009)
In models of dynamical electroweak symmetry breaking due to strongly coupled fourth-family
quarks and leptons, their low-energy effective descriptions may involve multiple composite Higgs
fields, leading to a possibility that the electroweak phase transition at finite temperature is first
orderduetotheColeman-Weinbergmechanism. Weexaminethebehavioroftheelectroweakphase
9 transition based on the effective renormalizable Yukawa theory which consists of the fourth-family
0 quarks and two SU(2)-doublet Higgs fields corresponding to the bilinear operators of the fourth-
0 family quarks with/without imposing the compositeness condition. The strength of the first-order
2 phase transition is estimated by using the finite-temperature effective potential at one-loop with
the ring-improvement. In the Yukawa theory without the compositeness condition, it is found
n
that there is a parameter region where the first-order phase transition is strong enough for the
a
J electroweak baryogenesis with the experimentally acceptable Higgs boson and fourth-family quark
masses. On the other hand, when the compositeness condition is imposed, the phase transition
7
turns out to be weakly first order, or possibly second order, although the result is rather sensitive
2
to the details of the compositeness condition. Combining with the result of the Yukawa theory

without the compositeness condition, it is argued that with the fourth-family quark masses in the
]
h range of 330-480 GeV, corresponding to the compositeness scale in the range of 1.0-2.3 TeV, the
p four-fermion interaction among the fourth-family quarks does not lead to the strongly first-order
- electroweak phasetransition.
p
e
PACSnumbers: 11.10.Wx, 12.60.Fr,98.80.Cq.
h
[

2 I. INTRODUCTION possibility beyond the SM. The constraint from the in-
v visible Z width is insignificant for the fourth-family neu-
2 trino being heavier than m /2. Though the electroweak
The standard model (SM) can in principle fulfill all Z
6
precisiondata[19]givestringentconstraintonthefourth
9 three Sakharov conditions [1] for generating a baryon
family[20, 21, 22, 23], it is known that the data do not
1 asymmetryintheuniverse[2,3,4]. Themodelfails,how-
exclude their existence [24, 25, 26, 27]. It was shown
. ever, for two reasons, to explain the value of the asym-
1
that there still remain a parameter region being consis-
metry required for the primordial nucleosynthesis [5], or
0
tent with all current experimental bounds[28].
9 the value measured through the cosmic microwaveback-
0 ground[6]. ThefirstreasonisthattheCPviolationfrom The existence of the fourth-family quarks accommo-
v: the Kobayashi-Maskawamechanism [7], which nicely ex- date the extra mixings and CP violating phases within
i plains CP violation in K- and B-systems, is highly sup- the Cabibbo-Kobayashi-Maskawa scheme. It has been
X
pressed [8, 9, 10, 11, 12, 13]. The second reason is that arguedthat the observedanomaly in B-CP asymmetries
r theelectroweakphasetransition(EWPT)isnotstrongly maybeexplainedbytheeffectofthefourth-familyquarks
a
first order. The experimental lower bound on the Higgs [29,30,31,32]. Recently,itwaspointedoutthatCPvio-
mass,mh >114GeV[14],impliesthatthereisnoEWPT lationfromthesenewphasescouldbelargeenoughtoex-
in the SM [15, 16, 17, 18]. Consequently, sphaleron- plain the baryonasymmetry in the universe on the basis
induced (B+L)-violating interactions are not sufficiently of the dimensional analysis using the Jarlskog invariants
suppressedinthe brokenphaseandwashoutthe baryon extended to four families[33].
asymmetry. Therefore, if the physics at the electroweak
The question is then whether the EWPT can be
scaleistoexplainthebaryonasymmetryintheuniverse,
strongly first order with the fourth family: the mere ad-
the better understanding of the structure of the Higgs
dition of the fourth family to the SM is of no help in
sector and of the source of CP violation would be re-
this respect, as long as the standard Higgs sector with
quired.
the single SU(2) doublet is considered. (See, for exam-
The fourth family is still a viable phenomenological
ple, a recent study by Fok and Kribs [34].) Carena et
al. [35] has first discussed the possibility of a first-order
EWPT due to new heavy fermions coupled strongly to
Higgsbosons. Theyfoundthatsomeheavyandstrongly-
∗[email protected]
†[email protected] interacting bosonic fields are required both to stabilize
‡[email protected] theeffectivepotentialagainstthelargeeffectoftheheavy
2
fermionsandtocauseafirst-orderEWPT[130]. Thisled 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96]. The analysis of
the authors to consider a supersymmetric model. The the effect of the heavy charged lepton and neutrino will
EWPT in the supersymmetric model with the fourth be reported in a subsequent paper [97]. We also neglect,
family has recently been examined in [34]. (See [36] for in this paper, the SU(3)×SU(2)×U(1) gauge interaction
earlier works.) and consider the global symmetry limit because we do
If the masses of the fourth-family quarks and lep- notexpectalargeeffectoftheelectroweakinteractionto
tons are quite large and are comparable to the uni- the dynamics of the first-order phase transition in this
tarity bounds, the fourth family must couple strongly model [65].
to the Higgs sector [37]. In this case, the masses of This paper is organized as follows. In section II, we
the fourth-family quarks and leptons (or their vacuum formulate the effective Yukawa theory and introduce the
condensates) may be regarded as the order parameters cutoff scale Λ by considering the vacuum instability and
of the electroweak symmetry breaking (EWSB) [131] the triviality bounds. We then specify the composite-
[38, 45, 46, 47, 48, 49, 50]. The effective description ness condition for our model. In section III, we derive
of the fluctuations of the order parameters may involve thefinite-temperatureeffectivepotentialatone-loopwith
multiple Higgs scalar fields. This leads to the possibility the ring-improvement. In section IV, based on the nu-
thattheEWPTwouldbefirstorderduetotheColeman- mericalanalysisoftheeffectivepotential,weexaminethe
Weinbergmechanism(thefluctuationinducedfirst-order strengthofthefirst-orderphasetransitionintheYukawa
phase transition)[51, 52, 53, 54, 55, 56, 57, 58, 59] [132]. theory with/without the compositeness condition. Sec-
tion V is devoted to conclusion and discussion.
The goalofthis paperis to explorethe abovepossibil-
ityofafirst-orderEWPTduetotheheavyfourthfamily.
Westartfromamodelofdynamicalelectroweaksymme-
II. FOURTH FAMILY AND ELECTROWEAK
try breaking due to the effective four-fermion interac-
SYMMETRY BREAKING
tionsofthefourth-familyquarksandleptonsatthescale
Λ around a few TeV. We adopt the four-fermion in-
4f
A. Fourth family and Four-fermion interactions
teractions considered by Holdom [72]. This four-fermion
theory may be rewritten into a Yukawa theory by intro-
ducing auxiliary scalar fields which corresponds to the We assume the existence of the fourth-family quarks
bilinear operators of the fourth-family quarks and lep- and leptons, which we denote by q′ = (t′,b′)T, ℓ′L =
tons. These scalar fields consist of three SU(2) doublets (ντ′L,τL′)T, τR′ . The right-handed neutrino ντ′R is as-
and one SU(2) triplet. (It is assumed that the right- sumedtoacquireitsmassattheflavorscalearound1000
handed neutrino is extra heavy, acquiring its mass at TeV and to be absent below the flavor scale. To be con-
the flavor scale around 1000 TeV. ) The renormaliza- sistent with the electroweak precision date, the masses
tiongroupevolutionfromthe scaleΛ downtothe elec- of the fourth-family quarks should be almost degenerate
4f
troweakscalev(=246GeV)maygenerateoperatorssuch with a small mass splitting. For simplicity, we assume
as the kinetic and interaction terms of the scalar fields mt′ =mb′.
andotherhigherdimensionaloperators. Wethenextend Following Holdom[72], we introduce the four-fermion
this model by including the kinetic, cubic and quartic interactions of the fourth-family fermions as follows:
terms of the scalar fields so that it becomes renormaliz-
able,neglectingtheeffectofthehigherdimensionaloper- L4f =Gq′(q¯′LiqR′ j)(q¯′RjqL′i)+Gτ′(ℓ¯′LiτR′ )(τ¯′Rℓ′Li)
awteoresx.aImtiinsethEisWePffeTctitvheroruenghormthaelizfianbitlee-mteomdpeelrfaotruwrehiecfh- −Gντ′L(ℓ′LTiC†ℓ′Lj)(ℓ¯′LjCℓ¯′TLi), (1)
fective potential at one-loop with the ring-improvement where C is the charge conjugation matrix and color in-
[73, 74, 75, 76, 77, 78, 79, 80, 81, 82]. Strictly speaking, dexes are contracted within a bracket. The scale of
in our case, the renormalization group equations must theseinteractionsisassumedtobeΛ4f: Gq′,Gτ′,Gντ′L ≃
be subjecttothe compositenessconditionasaboundary 1/Λ2 . The interaction term among the quarks
4f
condition at the scale µ=Λ4f [44, 83]. Accordingly, the has SU(2)L×SU(2)R×U(1)V×U(1)A symmetry, where
values of the renormalized couplings at the lower scale U(1) corresponds to the baryon number. On the
V
µ=v are restricted in a certain region of the parameter other hand, the interaction terms among the leptons
space. In our analysis, however, we will first explore the have SU(2) ×U(1) ×U(1) symmetry which includes
L L R
parameterspaceoftherenormalizabletheorywithoutthe the vector like U(1) symmetry corresponding to the lep-
constraintduetothecompositenesscondition,inorderto ton number. The above four-fermioninteractions, there-
locate the parameter region where a strongly first-order fore, have the extra symmetries compared with the SM
EWPTisrealized. Wethenexaminethepossibleoverlap Higgs sector which has O(4) symmetry. We then as-
of these two regions. sume the existence of sub-leading multi-fermion opera-
In this paper (I), we concentrate on the effect of the tors,whicharesuppressedcomparedwithEq.(1),sothat
heavy quarks and consider two SU(2) doublets out of the extra symmetries are broken explicitly and, hence,
four scalar fields. The bosonic sector of our model then the possible pseudo Nambu-Goldstone (NG) bosons ac-
reducestothe twoHiggsdoubletmodel(2HDM)[84,85, quire non-zero masses.
3
The four-fermion interactions may be rewritten into We include the last term which breaks the U(1) sym-
A
the form of Yukawa interactions by introducing the aux- metry and induces the mass of the pseudo NG boson.
iliary scalar fields Φ, Hτ′ and χa (a = 1,2,3), which Then,thesymmetryofthetheoryisthechiralsymmetry
correspondto the bilinear operatorsof the fourth-family SU(2) ×SU(2) plus the U(1) symmetry correspond-
L R V
quarks and leptons as follows: ing to the baryon number. We do not include the terms
whichconsistofǫΦ ǫotherthaninthedeterminantterm.
Φij ∼q¯′RjqL′i , Hτ′ ∼τ¯′Rℓ′Li , χa ∼ℓ¯′LτaǫCℓ¯′TL. (2) ∗
Then one obtains
C. Electroweak Symmetry Breaking
L′4f =−m2Φ0tr(Φ†Φ)−m2Hτ′0Hτ†′Hτ′ −m2χ0χa∗χa+LY,
We assume that the chiral symmetry SU(2) ×SU(2)
(3) L R
breaks down to the diagonal subgroup SU(2) by the
V
where LY is the Yukawa interaction terms given by vacuum expectation value (VEV) of Φ(x):
LY =−yq′0(q¯′LΦqR′ +c.c.)−yτ′(ℓ¯′LHτ′τR′ +c.c.) hΦi= φ I, (7)
−f(ℓ′LTC†ǫτaχaℓ′L+c.c.). (4) 2Nf
where N (=2) is the numbper of the fourth-family quark
f
B. Effective Renormalizable Theory flavors,Iis the Nf×Nf unit matrixandφ≥0. Attree-
level, the VEV is determined by the effective potential:
Throughtherenormalizationgroupevolutionfromthe 1 1 λ
scale µ = Λ4f down to the electroweak scale µ = v V0(φ)= 2(m2Φ−c)φ2+ 8 λ1+ N2 φ4. (8)
(=246 GeV), the kinetic and interaction terms of the (cid:18) f(cid:19)
scalarfields andother higher dimensionaloperatorsmay For (m2 −c)<0, the VEV is given by
Φ
be generated. We then extend this model by including
the kinetic, cubic and quartic terms of the scalar fields −2(m2 −c)
sothatitbecomesrenormalizable,neglectingthe effectof φ0 = Φ . (9)
thehigherdimensionaloperators. Theeffectiverenormal- sλ1+λ2/Nf
izable theory is then given by the following Lagrangian:
For the effective potential to be stable in this channel,
the following conditions must be satisfied:
L=L +L −V, (5)
k Y
where Lk consists of the kinetic terms for fourth-family λ1+λ2/Nf ≥0, λ2 ≥0. (10)
fermionsandscalarbosonsandV is the scalarpotential.
Aroundthe VEV, we may parametrize the fluctuation
The explicit form of V is given in appendix A [133].
of Φ(x) as follows:
Strictlyspeaking,therenormalizationgroupequations
are subject to the compositeness condition as a bound- φ+h+iη 3 σα
ary condition at the scale µ=Λ4f [44]. Accordingly, the Φ(x)= I+ (ξα+iπα) , (11)
values of the renormalized couplings at the lower scale 2Nf α=1 2
X
µ = v are restricted in a certain region of the param-
where σα (α=1p,2,3) are the Pauli matrices. The fields
eter space. In the following analysis, however, we will
h,η,ξα,πα andq acquiremassesattreelevelassumma-
first explore the parameter space of the renormalizable ′
rized in Table I, where, for notational simplicity, we use
theory without the constraint due to the compositeness
the following abbreviations:
condition,inordertolocatetheparameterregionwherea
strongly first-order EWPT is realized. We then examine 3 1
the possible overlap of these two regions. ah = (λ1+λ2/Nf), aξ = (λ1+3λ2/Nf), (12)
2 2
In this paper (I), we concentrate on the effect of the
1 1
fourth-family quarks and consider two SU(2) doublets aη = aπ = (λ1+λ2/Nf), aq′ = y2, (13)
2 2N
out of four scalar fields. We also neglect, in this paper, f
the SU(3)×SU(2)×U(1) gauge interaction and consider and
the globalsymmetrylimit, simply because we do notex-
pect a largeeffect of the color andthe electroweakinter- b = a −a =(λ +λ /N ), (14)
h h π 1 2 f
actionsonthedynamicsofthefirst-orderphasetransition b = a −a =(λ /N ). (15)
ξ ξ π 2 f
in this model. Then the LagrangianEq. (5) reduces to
The bosonic sector of this model is just the 2HDM. h is
L = q¯i∂/q −y(q¯ Φq +c.c.)
′ ′ ′L R′ the singlet of SU(2)V and corresponds to the SM Higgs
+ tr(∂µΦ†∂µΦ)−m2ΦtrΦ†Φ boson. The adjoint πα are the NG bosons of the break-
− λ1(trΦ†Φ)2− λ2 tr(Φ†Φ)2+c(detΦ+c.c.).(6) idnogscoaflaSrUH(2ig)gLs×bSoUs(o2n)Ra,ndwhisilealtshoetshiengplseetuηdoisNtGhebposseoun-
2 2
4
particle m2i(φ) m2i(φ0) ni one-loop:
h m2 −c+a φ2 b φ2 1
Φ h h 0 ∂ 1
ξ m2 +c+a φ2 2c+b φ2 3 µ λ¯ = [(N2+4)λ¯2+4N λ¯ λ¯ +3λ¯2+2N y2λ¯ ],
Φ ξ ξ 0 ∂µ 1 8π2 f 1 f 1 2 2 c 1
η m2Φ+c+aηφ2 2c 1 ∂ 1
π m2Φ−c+aπφ2 0 3 µ∂µλ¯2 = 8π2(6λ¯1λ¯2+2Nfλ¯22+2Ncy2λ¯2−2Ncy¯4),
q′ aq′φ2 aq′φ20 −24
∂ 1
µ y¯= (N +N )y¯3, (16)
TABLEI:Theeffectivemassesandthenumbersofthedegrees ∂µ 16π2 f c
of freedom in themodel for thefourth family quarks
with the initial conditions λ¯ (v) = λ , λ¯ (v) = λ and
1 1 2 2
y¯(v)=ygivenattheelectroweakscale. Asonecanseein
FIG. 1, the Yukawa coupling, which is large at the elec-
troweak scale for the heavy fourth-family quarks, would
associated with the breaking of the U(1)A symmetry. blow up to infinity (due to the Landau pole) at a cer-
The adjoint ξα consist of the extra neutral Higgs bo- tainenergyscalenotfarfromtheelectroweakscale[134].
son and the charged Higgs bosons. Three NG bosons The ultraviolet behaviors of the scalar quartic couplings
πα are eaten by W and Z bosons when the electroweak then take two types of patterndepending onthe relative
interactions are introduced. As for the fourth-family size of the scalar quartic couplings and Yukawa coupling
quarks, the experimental lower bound from the direct at the electroweak scale: (i) the scalar quartic coupling
search mq′ &256 GeV [109] implies y &2.1 at tree-level λ¯1(µ)+λ¯2(µ)/Nf and/orλ¯2(µ)aredriventonegativeat
by taking φ0 =v(= 246GeV). someenergyscale,implyingthattheelectroweakvacuum
is unstable; (ii) the scalar quartic couplings encounter
theLandaupoleatsomeenergyscale. Inbothcases,one
should introduce a cutoff before these problems happen.
WeestimatethecutoffΛ,forgiveninitialvaluesofthe
couplings at the electroweak scale, as the scale at which
D. Cutoff Scale of the Effective Theory
one of the following conditions is first hit:
The applicabilityofthe effective theorydefinedby the λ¯1(Λ)+λ¯2(Λ)/Nf =0, λ¯2(Λ)=0, (17)
Lagrangian Eq. (6) would break down at some energy
scaleandoneneedstointroduceacutoffΛ. Therunning whichcorrespondtothecase(i)(thevacuuminstability)
coupling constants in this model, λ¯ (µ), λ¯ (µ) and y¯(µ), and
1 2
obey the following renormalization group equations at
16π2
y¯(Λ)2 = , (18)
N
c
16π2 16π2
λ¯ (Λ)+λ¯ (Λ)/N = , λ¯ (Λ)= ,(19)
1 2 f N2 2 N
f f
10 which correspond to the case (ii) (the Landau pole).
Here, we adopt the upper limits of the perturbativity
8 bounds,
(cid:3)(cid:3)(cid:3)(cid:4)Μy 6 y¯(Λ)2 ≤ 1N6π2, (20)
c
4 16π2 16π2
λ¯ (Λ)+λ¯ (Λ)/N ≤ , λ¯ (Λ)≤ ,(21)
1 2 f N2 2 N
f f
2
200 500 1000 2000 5000 1(cid:1)104 as a criterion for the Landau pole.
Μ(cid:1)GeV(cid:2)
In FIG. 2, we show the contours of the estimated cut-
off Λ for y = 2.0 in λ –λ /N plane. We see that in the
FIG. 1: The renormalization group flows of the Yukawacou- 1 2 f
most region, the cutoff scale is around 1 TeV or lower.
pling for various initial values at the electroweak scale. The
Forthefixedλ ,Λtendstoincreasewithλ /N forsmall
valuesofyare2.0,2.5,3.0frombottomtotop. Thegraysolid 1 2 f
lineindicatesthevaluey=2.1, which correspondstotheex- λ2/Nf andtendstodecreasewithλ2/Nf forlargeλ2/Nf.
perimental lower bound from thedirect search of the fourth- The former (latter) behavior is due to the fact that Λ
family quarks, mq′ & 256 GeV. The blue dashed line indi- is determined via the vacuum instability (the Landau
cates the upper limit of the perturbativity condition, which pole) conditions in that region. FIG. 3 is the similar
is adopted in our analysis as a criterion for theLandau pole. plot for y = 2.5. We see that for the larger value of the
Yukawa coupling, the relatively larger values of λ /N
2 f
5
A
1TeV
1TeV
BB
C
1TeV
FIG. 2: An estimate of the cutoff scale Λ for y = 2.0 in FIG.4: AnestimateofthecutoffΛforλ +λ /N =0.05in
1 2 f
λ –λ /N plane. The dashed, dot-dashed, solid and dotted y–λ /N plane. Thedashed,solidanddottedcontourscorre-
1 2 f 2 f
contourscorrespondtoΛ=5.0,1.5,1.0,0.5TeV,respectively. spond to Λ = 5.0,1.0,0.5 TeV, respectively. The blue filled-
In the shaded region, the effective potential at tree-level is circle (indicated with ”A”), which is located at the ”cusp”
unstable and we consider the region where λ +λ /N ≥ 0 on the boundary of the allowed region with Λ ≥ 1.0 TeV,
1 2 f
only. indicatestheelectroweak-scalevaluesoftherunningcoupling
constants subject to the compositeness condition (Criterion
A) at Λ =1.0 TeV.
4f
This requirement leads to the constraint on the quark
masses, mq′ . 370 GeV at tree-level, corresponding to
1TeV
the Yukawa coupling y ≤3.0.
E. Compositeness condition
1TeV
Just below the scale of the four-fermion interaction
µ.Λ ,thefour-fermiontheoryEq.(3)withonlyquark
4f
fields q ,
′
L′4f =q¯′i∂/q′−yq′0(q¯′LΦqR′ +c.c.)−m2Φ0tr(Φ†Φ), (22)
is renormalizedto the Yukawa theory Eq. (6), where the
renormalized couplings are given by
FIG. 3: An estimate of the cutoff scale Λ for y = 2.5 in
λ1–λ2/Nf plane. The dot-dashed, solid and dotted contours λ¯1(µ) = 0, (23)
correspond to Λ=1.5,1.0,0.5 TeV, respectively.
32π2 1
λ¯ (µ) = , (24)
2 N ln(Λ2 /µ2)
c 4f
16π2 1
y¯(µ)2 = . (25)
N ln(Λ2 /µ2)
are required to fulfill the vacuum stability condition. In c 4f
FIG. 4, we show the contours of the estimated cutoff Λ
(See appendix B for detail.) In the limit µ → Λ , one
for λ +λ /N =0.05 in y–λ /N plane. 4f
1 2 f 2 f finds
In order to ensure the applicability of the effective
renormalizable theory, the cutoff Λ should be taken to λ¯1(µ) → 0,
be large enough compared with other mass scales in the λ¯ (µ) → ∞, (26)
2
theory: Λ ≫ mi(φ0),φ0. In the following analysis of y¯(µ)2 → ∞, λ¯ (µ)/y¯(µ)2 →2.
2
the first-order EWPT,it turns out that the largestmass
scale is given by m (φ ) around 400–700 GeV. Then, Thisprovidesthecompositenessconditionintermsofthe
ξ 0
we require Λ ≥ 1 TeV and exclude the region of the renormalizedcouplings as the boundary condition of the
parameter space where this condition is not fulfilled. renormalizationgroup equations at µ=Λ [44].
4f
6
Criterion:A(cid:2)dashed(cid:3),B(cid:2)dotted(cid:3),C(cid:2)dot(cid:1)dashed(cid:3) fromtoplefttobottomright,respectively. Thecasewith
40
Λ = 1.0 TeV comes close to the stability boundary at
4f
theelectroweakscale,takingthevalueλ +λ /N =0.05.
1 2 f
Then,inFIG.4,thevaluesy,λ ofthiscaseareindicated
2
30
by the blue filled-circle, which is located at the ”cusp”
ontheboundaryoftheallowedregionwithΛ≥1.0TeV.
In fact, the electroweak-scale values of the couplings
Λ(cid:1)2Nf20 are rather sensitive to the choice of their values at the
compositeness scale Λ . To see this, let us refer the
4f
above criterion for the compositeness condition as ”A”,
10
and introduce slightly modified criteria ”B” and ”C” as
follows:
0(cid:1)12 (cid:1)10 (cid:1)8 (cid:1)Λ16 (cid:1)4 (cid:1)2 0 λ¯1(Λ4f)=0, λ¯2(Λ4f)= 8π2, y¯(Λ4f)2 = 8π2,
N N
f c
λ¯ (Λ )/y¯(Λ )2 =N /N [Criterion B], (28)
2 4f 4f c f
0.5
and
4π2 4π2
λ¯ (Λ )=0, λ¯ (Λ )= , y¯(Λ )2 = ,
1 4f 2 4f 4f
0.5 Nf Nc
1
λ¯ (Λ )/y¯(Λ )2 =N /N [Criterion C]. (29)
2 4f 4f c f
1.5
11 00.55
In FIG. 5, the red filled-circles and green filled-circles
1.5
5 1 shows the electroweak-scale values λ , λ of the running
1 2
5 1.5 coupling constants subject to the compositeness condi-
5
tions Eqs. (28) and (29), respectively. We will discuss
thispointfurtherinsectionIVinrelationtotheanalysis
of EWPT.
III. EFFECTIVE POTENTIAL
FIG.5: Theelectroweak-scalevaluesoftherunningcoupling
constants, which are subject to thecompositeness conditions
at various Λ , are shown in λ –λ /N plane. The blue, red, A. Zero-temperature effective potential
4f 1 2 f
green data sets correspond to the criterion A, B, C, respec-
tively. Thebluefilled-circlesindicatethevaluesλ ,λ /N for
1 2 f At zero temperature, the one-loop effective potential
Λ (y) equal to 0.5 TeV (4.0), 1.0 TeV (3.0), 1.5 TeV (2.7),
4f is given by
5.0 TeV (2.2) from top left to bottom right, respectively.
V(0)(φ)=V (φ)+V(0)(φ), (30)
0 1
In our effective theory formulated as above, however, where V0 is the tree-level effective potential, V1(0) is the
thecompositenessconditionshouldbemodified. Onecan one-loop contributions at zero temperature.
notimposetheaboveconditionEq.(26)literallybecause V0, the tree-level effective potential, is given by
the values of the couplings λ¯ (µ) and y¯(µ) must exceed
2
tahnedpy¯e(rµtu)rbisatdivuietytobotuhnedsL.anBduatu, thpeolde.iveTrgheennc,eiotfsλ¯e2e(mµ)s V0(φ)= 21(m2Φ−c)φ2+ 81(cid:18)λ1+ Nλ2f(cid:19)φ4. (31)
reasonableinourcasetosubstitutetheupperlimitofthe
perturbativity bounds for the compositeness condition: V1(0),theone-loopcontributionatzerotemperature,is
given by
16π2 16π2
λ¯1(Λ4f)=0, λ¯2(Λ4f)= Nf , y¯(Λ4f)2 = Nc , V1(0)(φ)=641π2 nim4i(φ) lnmµ2i(2φ) − 32 + 21Aφ2.
λ¯2(Λ4f)/y¯(Λ4f)2 =Nc/Nf [Criterion A]. (27) i=h,Xξ,η,π,q′ (cid:20) (cid:21)
(32)
InFIG.5,weplottheelectroweak-scalevaluesλ ,λ of
1 2
the running coupling constants which are subject to the mi(φ)andniaretheeffectivemassesdependingonφand
compositenessconditionsEqs.(27)atvariousscales. The thenumberofdegreesoffreedom,respectively,whichare
blue filled-circles correspond to the values of Λ (y) = given in Table I. In the calculation of the loop integral
4f
0.5TeV(4.0),1.0TeV(3.0),1.5TeV(2.7),5.0TeV(2.2) in V(0), we have taken the limit Λ → ∞ and have used
1
7
the MS scheme with a slightmodification to renormalize where J and J are defined by
B F
theultravioletdivergences. Thefirsttermistheone-loop
contribution in ordinary MS scheme with the renormal-
ization scale µ. The modification is the existence of the
sVeEcoVndφter=m w−hi2c(hma2re−acd)d/(eλd t+o λpr/eNserv)eatnhde, ttrheeen-l,evweel JB(a)= ∞dx x2ln 1−e−√x2+a ,
set φ =0 v(=246GeVΦ). The p1aram2eterfA is determined Z0 (cid:16) (cid:17)
0
through 0=p∂V(0)(φ)/∂φ at φ=v and is given by JF(a)= ∞dx x2ln 1+e−√x2+a . (37)
1
Z0 (cid:16) (cid:17)
1 m2(v)
A=− n a m2(v) ln i −1 . (33)
16π2 i i i µ2
i=h,ξ,η,q′ (cid:18) (cid:19)
X
InthefollowinganalysisoftheEWPT,wecarryoutanu-
At one-loop, the Higgs boson mass mh is shifted from mericalintegralforJB andJF withouthightemperature
the tree-level value (mh)tree = λ1+λ2/Nfv. In this expansion.
paper, we adopt the following definition for the Higgs
p
boson mass mh at one-loop: In the ordinary perturbation theory at finite temper-
ature, the perturbative expansion breaks down near the
λ v2 m2(v)
m2 ≡ λ + 2 v2+ n a2ln i . (34) critical temperature due to the existence of the higher-
h (cid:18) 1 Nf(cid:19) 8π2 i=h,ξ,q′ i i µ2 loop IR divergent diagrams in the massless limit. To
X improve the reliability of the perturbative expansion, we
This is the curvature of the effective potential at φ=v : include the contributions from the ring diagrams which
V(0) (φ = v) with neglecting the contributions from the are the most dominant IR contributions at eachorder of
′′
light or massless scalar bosons η and πs [135]. the perturbative expansion [76, 77, 78, 79, 80, 81, 82]
As for the mass of the extra scalar bosons ξ, η and [136].
the fourth-family quarksq , we adopt the formula at the
′
tree-level: One can include the contribution of ring diagrams,
V (φ,T),byreplacingm2(φ)(i=h,ξ,η,π)inV(0)and
λ y2 ring i 1
m2ξ ≡2c+ N2fv2 , m2η ≡2c , m2q′ ≡ 2Nfv2. (35) mV1(2T()φ)w+ithΠth,eweffheecrteivΠe T-idsetpheendseelnft-emnearsgsyesoMf t2ih(eφ,sTca)la≡r
i Φ Φ
In the following analysis,we use these definitions for the bosons in the IR limit where the Matsubara frequency
masses. and the momentum of the external fields go to zero and
in the leading order of m (φ)/T and is given by
i
B. Finite-temperature effective potential
Theone-loopcontributionatfinitetemperature,V(T), Π = 1 [(N2+1)λ +2N λ +N y2]T2, (38)
1 Φ 12 f 1 f 2 c
is given by
V(T)(φ,T)
1
at one-loop order.
T4
= 2π2  niJB[m2i(φ)/T2]+nq′JF[m2q′(φ)/T2],
i=h,ξ,η,π Afterall,theone-loopring-improvedeffectivepotential
X
 (36) is given by
V(φ,T)=V (φ)+V(0)(φ)+V(T)(φ,T)+V (φ,T)
0 1 1 ring
1 1 M2(φ,T) 3 T4
=V (φ)+ Aφ2+ n M4(φ,T) ln i − + J [M2(φ,T)β2]
0 2 i 64π2 i µ2 2 2π2 B i
i=h,ξ,η,π (cid:20) (cid:18) (cid:19) (cid:21)
X
+nq′ 641π2m4q′(φ) lnm2qµ′2(φ) − 23 + 2Tπ42JF[m2q′(φ)β2] . (39)
" ! #
Inthe following,we study the finite-temperature EWPT by numerically evaluating this effective potential.
8
IV. NUMERICAL ANALYSIS OF
ELECTROWEAK PHASE TRANSITION
(cid:47)=0.5TeV
The EWPT should be strongly first order in order
0.4
to avoid the washout of the generated baryon asymme-
0.6 (cid:47)=1TeV
try in the broken phase. How strongly first order the
0.8
phase transition must be depends on the energy of the
1.0
sphaleron solution [100, 101] in the model considered.
As long as the classical (static) solution of the equation
of motions is concerned, one may neglect the effect of
thefourth-familyquarksevenwhentheyareheavy[137].
(cid:47)=1TeV
Then, one may use the condition
(cid:47)=0.5TeV
φ /T & 1, (40)
c c
as the criterion for a strongly first-order EWPT, as dis-
cussed in our previous work [65].
In the following analysis, assuming a first-order phase FIG.6: Contourplotsforvariousφc/Tcinλ1–λ2/Nf planeat
transition, we solve the conditions y=2.0, c=0. The dashed lines are thecontours of φc/Tc =
1.0,0.8,0.6,0.4 from bottom left to top right as indicated.
∂V(φ,Tc) The solid lines indicate the boundary of the allowed region
=0, V(φ ,T )=V(0,T ), (41)
∂φ c c c withΛ≥1TeV.(Theregionwith1TeV>Λ≥0.5TeVisalso
(cid:12)φ=φc shown.) In the shaded region, the effective potential at tree-
(cid:12)
(cid:12) level is unstable. The black filled-circle indicates the values
numericallyfor(cid:12)variousparameters,mh,mξ,mq′ andmη
ofthecouplingsλ ,λ /N whenthecompositeness condition
(see Eqs. (34), (35) for definitions). Then we evaluate 1 2 f
is satisfied at Λ = 9 TeV so that y=2.0.
φ /T in order to estimate the strength of the first-order 4f
c c
phase transition.
We first explore the parameter space of the renormal-
izable theory without the constraint due to the compos-
iteness condition, in order to locate the parameter re- (cid:47)=0.5TeV
gion where a strongly first-order EWPT is realized. We
0.8 0.6
consider only the region where the stability condition at 0.4
(cid:47)=1TeV
tree-level,λ +λ /N ≥0,issatisfied,otherwisethemass
1 2 f
parameter squares in the one-loopeffective potential be-
come negative. We also consider only the region where
the perturbation theory is reliable: (cid:47)=1TeV
(4π)2 λ (4π)2 (4π)2
λ1+λ2/Nf ≪ N2 , N2 ≪ N2 , y2 ≪ N . (42) (cid:47)=0.5TeV
f f f c
In order to fulfill the requirement Λ ≥ 1 TeV, we also
concentrate ona regionwhere y ≤3.0. For anestimated
Λ, we require that φ = v is the global minimum of the
one-loopzero-temperatureeffective potential V(0)(φ) for
0 < φ < Λ. The renormalization scale of the effective
FIG. 7: Contour plots for various φc/Tc in λ1–λ2/Nf plane
potential is set to the electroweak scale, µ=v. We next at y = 2.5, c = 0. The dashed lines are the contours of
examine the possible overlap of the parameter region of φc/Tc =0.8,0.6,0.4frombottomlefttotoprightasindicated.
the strongly first-orderEWPT and the regionsubject to The black filled-circle indicates the values of the couplings
the compositeness condition. λ ,λ /N when the compositeness condition is satisfied at
1 2 f
Λ = 2.3 TeV so that y=2.5.
4f
A. Yukawa theory without compositeness condition
We first discuss the numerical results for the Yukawa TeV. (The contours with 1 TeV≥ Λ ≥0.5 TeV are also
theory without the constraint due to the compositeness shown for reference.) We clearly see that the region of
condition. In this analysis, we neglect the effect of the the strongly first-order transition lies above the stability
U(1) symmetry breaking term, setting c=0. boundaryλ +λ /N =0. Whenλ +λ /N getslarger,
A 1 2 f 1 2 f
In FIG. 6, we show the contours of φ /T in the λ – φ /T tends to become smaller. For a fixed small λ +
c c 1 c c 1
λ /N planefory =2.0intheallowedregionwithΛ≥1 λ /N ,thereisatendencythatφ /T decreasesasλ /N
2 f 2 f c c 2 f
9
increases.
FIG. 7 is a similar contour diagram for y = 2.5. We A
500
notice that the region with φ /T >1 disappears in this
c c
case. Thesubstantialregionclosetothestabilitybound- (cid:47)=1TeV
400
ary is excluded by the condition Λ ≥ 1 TeV. This is
because one encounters the instability at a lower scale
300
for the larger Yukawa coupling. But, even when the al- (cid:47)=0.5TeV
lowed region is extended to Λ ≥ 0.5 TeV, the transition B
200
is weakly first order with φ /T <1. For larger values of
c c
y, as far as we explored, the transition gets more weakly
114 C
first order, or possibly second order.
In FIG. 8, we show the contours of φ /T in y–λ /N mq’=256GeV
c c 2 f
plane for λ + λ /N = 0.05. We see clearly that a
1 2 f
strongly first-order phase transition is realized even for
y & 2.1, which corresponds to the experimental lower
boundfromthe directsearchofthe fourth-family quarks
FIG. 9: Contour plots for various m in y–λ /N plane at
mq′ & 256 GeV, in the range 3 . λ2/Nf . 4 (corre- λ +λ /N =0.05, c=0. The dashedhlines are2 thfe contours
1 2 f
sponding to 430GeV.m .500GeV). For largervalues
ξ of m =114,200,300,400,500 GeV as indicated.
h
ofy,however,thestabilityoftheelectroweakvacuumre-
quireslagervaluesofλ /N andthenthe strengthofthe
2 f
first-order phase transition becomes weaker, or possibly
second order.
In FIG. 9, we show the contoursof the one-loopHiggs
mass m in y–λ /N plane for λ + λ /N = 0.05,
h 2 f 1 2 f mq’(cid:2)260GeV,mΗ(cid:2)0
corresponding to (m ) = 55 GeV. In this case, for
h tree 1.2
λ /N & 1 or y & 1 we can neglect the one-loop con-
2 f
tributions of η and πs to m and the use of the formula 1.0
h
EHqig.g(s3m4)asiss rveacleidivaetsedra.thIetrilsarcgleeaornteh-laotopfocroyrre&cti2o.n1atnhde (cid:3)cTcΦ0.8
0.6
isheavierthanexperimentallowerbound,m >114GeV,
h
0.4
in the whole region.
In FIG. 10, φc/Tc is plotted as a function of mh for 0.2150 200 250 300 350 400
several values of mξ with mq′ = 260 GeV fixed. We can mh(cid:1)GeV(cid:2)
see that with m fixed, φ /T decreases as m increases,
ξ c c h FIG. 10: φc/Tc as a function of mh for mq′ =260 GeV and
while, with m fixed, φ /T increases as m increases.
h c c ξ mξ =400,450,500,550 GeV from bottom left to top right.
Inthe whole range,m exceedsthe experimentalbound,
h
mΞ(cid:3)450GeV,mΗ(cid:3)0
1.2
A
0.2 1.0
0.3 (cid:47)=1TeV (cid:3)cTcΦ0.8
0.5 0.6
0.4
(cid:47)=0.5TeV
0.2
150 200 250 300 350
1.0 B mh(cid:1)GeV(cid:2)
1.5 FIG. 11: φc/Tc as a function of mh for mξ =450 GeV and
C mq′ =200,260,280,300 GeV from top to bottom.
2.5 mq’=256GeV
1.5 1.0
FIG. 8: Contour plots for various φc/Tc in y–λ2/Nf plane mh > 114GeV. In FIG. 11, on the other hand, φc/Tc is
at λ1 +λ2/Nf = 0.05, c = 0. The value of φc/Tc for each plottedasafunctionofmh forseveralvaluesofmq′ with
contour is shown in the diagram. m = 450GeV fixed. In this case, we note that with m
ξ h
fixed, φc/Tc decrease as mq′ increases.
10
B. Effect of explicit symmetry breaking term of the electroweak-scale values of the running coupling
constants which are subject to the compositeness condi-
We next examine the effect of the explicit U(1) sym- tions Eqs. (27) (Criterion A) at various scales: the val-
A
metry breakingterm by taking a value ofmη (or c) non- ues λ1, λ2 for Λ4f = 1.0 TeV come close to the stabil-
zero. The effect can be read from FIG. 12. For a fixed ity boundary at the electroweak scale, taking the value
value of mh, φc/Tc decreases as mη increases. So we see λ1 +λ2/Nf = 0.05 and it is located at the ”cusp” on
that the explicit symmetry breaking term reduces the the boundary of the allowed region with Λ ≥ 1 TeV.
strength of the first-order phase transition. The experi- In this case, corresponding to the blue filled-circle (in-
mental bound on the mass of the pseudo NG boson de- dicated with ”A”) in FIG. 8, though we could not find
pends on how η couples to the other particles, which is thesolutionofEqs.(41),wehavecheckedthatthe phase
not specified in our model. If we adopt the bound for transition is weakly first order with φc/Tc < 0.001, or
the pseudoscalar Higgs boson in supersymmetric models possibly second order.
(withtanβ >0.4),theallowedvaluesaremη >93.4GeV The above conclusion deserves discussions. In case if
[109]. For such a value of mη, we see that the region of oneadoptstheothercriterionforthecompositenesscon-
φc/Tc ≥1 becomes quite narrow [138]. dition,thephasetransitioncanbefirstorder. Inthecase
ofthecriterionB,forexample,theelectroweak-scaleval-
mΞ(cid:3)450GeV,mq’(cid:3)260GeV ues of the running coupling constants which are subject
to the compositeness condition at Λ = 3.7 TeV come
1.2 4f
close to the stability boundary at the electroweak scale,
1.0 taking the value λ +λ /N =0.05. In FIG. 8, we show
1 2 f
(cid:3)cTc0.8 the electroweak-scalevalues y, λ2 of this case by the red
Φ filled-circle (indicated with ”B”). One immediately see
0.6
that the phase transition is rather strongly first order
0.4
with φ /T ≃ 1. Thus the critical behavior of the phase
c c
0.2150 200 250 300 350 transition is rather sensitive to the choice of the values
mh(cid:1)GeV(cid:2) ofthe coupling constantsatthe compositeness scaleΛ .
4f
However, combining with the results of the Yukawa the-
FIG. 12: φc/Tc as a function of mh for mξ = 450 GeV,
ory without the compositeness condition, in particular,
mq′ = 260 GeV and mη = 0,50,100,150 GeV from top to
bottom. with the result for y =2.5 shownin FIG. 7, it seems fair
to say that the four-fermion interaction of the fourth-
family quarks, which causes EWSB at zero-temperature
Typical values of φc and Tc are shown separately in and produces the mass of the heavy quarks larger than
FIG.13asafunctionofmhformq′ =260GeV,mξ =450 mq′ ≃ 310 GeV (y & 2.5, Λ4f . 2.3 TeV), does not lead
GeV and mη =100 GeV fixed. to the strongly first-order EWPT at finite temperature
[139]. We note that a region with Λ <1 TeV (y >3.0,
4f
mΞ(cid:3)450GeV,mq’(cid:3)260GeV,mΗ(cid:3)100GeV mq′ >370GeV)isbeyondthescopeofouranalysisusing
180 the effective renormalizable theory.
160
(cid:2)V
(cid:3)(cid:4)(cid:1)TcgrayGe112400 1000
(cid:3)(cid:4)cblack,100 800
Φ 6800 180 200 220 240 260 280 300 (cid:3)(cid:3)(cid:3)(cid:3)(cid:4)(cid:1)(cid:2)ΜmGeV’q 600
mh(cid:1)GeV(cid:2) 400
FIG. 13: Plots of φc (black) and Tc (gray) as a function of 200
mh with mq′ =260 GeV,mξ =450 GeVand mη =100 GeV 200 500 1000 2000 5000 1(cid:1)104
fixed. Μ(cid:1)GeV(cid:2)
FIG. 14: Plots of the running mass of the fourth-family
quarks, m¯q′(µ) = y¯(µ)v/p2Nf, as a function of the renor-
malization scale µ. The value of ”on-shell” mass is defined
C. Compositeness condition by mq′,phys =y¯(mq′,phys)v/p2Nf. The dot-dashed line (or-
ange) indicates m¯q′(µ)=µ.
Finally, we discuss the numerical results for the
Yukawa theory with the compositeness condition im-
posed. Let us recall the plots in FIG. 4 and FIG. 5