Brane world unification of quark and lepton masses and its implication for the masses of the neutrinos

0.3 MB·English

by  P. Q. Hung

Brane world unification of quark and lepton masses and its implication for the masses
of the neutrinos
P.Q. Hung
∗
Dept. of Physics, University of Virginia,
382 McCormick Road, P. O. Box 400714,
Charlottesville, Virginia 22904-4714, USA
(Dated: February 7, 2008)
A TeV-scale scenario is constructed in an attempt to understandtherelationship between quark
andleptonmasses. Thisscenariocombinesamodelofearly(TeV)unificationofquarksandleptons
with the physics of large extra dimensions. It demonstrates a relationship between quark and
lepton mass scales at rather“low” (TeV) energies which will bedubbedas early quark-lepton mass
unification. It also predicts that the masses of the neutrinos are naturally light and Dirac. There
is an interesting correlation between neutrino masses and those of the unconventionally charged
5
fermions which are present in the early unification model. If these unconventional fermions were
0
to lie between 200GeV and 300GeV, the Dirac neutrino mass scale is predicted to be between
0
∼0.07eV and ∼1eV.
2

n
a I. INTRODUCTION [7] andits use in the attempt to explain the smallness of
J neutrino masses[8, 9] and the hierarchyof quark masses

1 [10, 11, 12].
Are quark and lepton masses related? This question
2
has been addressed almost thirty years ago in a famous In [3], the Standard Model (SM) with three indepen-

2 paper by [1] soonafter the concept of Grand Unification dentcouplingsismergedintoagroupGS GW withtwo
⊗
v (GUT) [2] has been put forward. From this pioneer pa- independent couplings at some scale which is supposed
2 per and subsequent works, one learns that quark-lepton to be in the TeV region. The choice of the Pati-Salam
6 unification at the GUT scale MGUT 1015 1016 GeV SU(4)PS [13] for GS was used. This scenario allowed us
22 gives rise to, for the particular case o∼f SU(5)−considered to compute sin2θW(MZ2) and to use it to constraint the
1 in [1], the equality of the τ lepton and bottom quark choicesofGW. The preferredchoiceof[3]wasthe gauge
4 masses at MGUT. After renomalization-group(RG) evo- groupSU(4)PS⊗SU(2)4withanearlyunificationscaleof
0 lution down to low energies, a remarkable “prediction” several hundreds of TeVs. Recent precise measurements
h/ for the b quark mass was made, although the complete of sin2θW(MZ2) coupled with a surge of interest in TeV
story was significantly more complicated. Despite the scale physics have prompted [4] to reexamine the petite
p
- enormous popularity of GUT, questions started to arise unificationidea. Thereitwasshownthatthepetiteunifi-
p astowhetherornotthereareactuallystructuresinstead cationscale is loweredconsiderably,to lessthan 10TeV,
he of simply a “desert” between the electroweak scale and due to the increaseofsin2θW(MZ2) as comparedwith its
M . If so, how would quark and lepton masses be value of twenty three years ago. This has the efect of
: GUT
v related if they were to have early unification? practically ruling out SU(4)PS SU(2)4 due to severe
Xi Thehopethatnewphysicsislurkingsomewhereinthe problems with the decay rate for⊗KL µe among other
→
TeV region has given rise in the past decade or two to things. Twofavoritemodelsemerged: SU(4)PS SU(2)3
ar a flurry of activities which resulted in a rich diversity of and SU(4)PS SU(3)2, both of which nicely a⊗nd natu-
⊗
topics with a variety of motivations. A common thread rally avoid the KL µe problem due pricipally to the
→
in all of these activities is the prediction of new parti- existenceofnewtypesoffermions. Adetailedanalysisof
cles of one kind or another. It goes without saying that SU(4)PS SU(2)3 wasperformedby[5],includingatwo-
⊗
discoveries of these new particles will vindicate all the loop renormalization group (RG) analysis and a discus-
efforts put into it. The present paper will rely on two of sion of the physics of the new unconventional fermions.
such scenarios with a special emphasis put into the rela- Earlyunificationinthismodeltakesplaceatamassscale
tionshipbetweenquarkandleptonmasses,including the M = (1 2 TeV).
O −
issues of neutrino mass: Is it Dirac or Majorana? Why On another front, Ref. [9] has constructed a model
is it so small? whichmadeuse ofthe mechanismofwavefunctionover-
Two TeV scenarios which form the focus of this paper lap along an extra compact dimension to “explain” the
are the following: (1) Early petite unification of quarks smallness of Dirac neutrino masses. An SU(2)R sym-
andleptons[3,4,5,6];(2)Thepossibilityoftheexistence metrywasassumedandwassubsequently spontaneously
ofextraspatialdimensions,themechanismofwavefunc- broken, giving rise to a phenomenon in which one mem-
tion overlap along an extra compact spatial dimension ber of the right-handeddoublet has a narrow wavefunc-
tion,whiletheothermemberacquiresabroadwavefunc-
tion (both localized at the same point along the extra
dimension). The overlap of the wave function of the
∗[email protected] left-handeddoubletwiththe wavefunctions ofthe right-
2
handedfieldsgivesrisetothesplittingbetweentheeffec- or pure Dirac and there are questions about the popular
tive four-dimensional Yukawa couplings (and eventually see-saw mechanism itself [15].
between the masses) of neutral and charged leptons or
of the up and down quarks. This splitting can be large
or small depending on the separation d (along the extra II. EXTRA DIMENSION, EARLY
dimension) between the wave functions for left-handed QUARK-LEPTON UNIFICATION AND MASS
RELATIONSHIP
and right-handed fields as demonstrated in [9]. It was
further noticed that there is a deep connection between
the separation d for the lepton sector and the separa- TwoTeV-scalescenariosarebrieflysummarizedbelow
l
tion d for the quark sector, giving rise to a relationship with the purpose of exposing their common threads and
q
between quark and lepton masses, a common feature in ultimately combining them in order to obtain an under-
Grand Unified Theories (GUT). standing of the possible relationship between quark and
lepton masses and the smallness of the neutrino masses.
At this point, one might ask about the distinction be-
tween the present scenario and a possible attempt to in-
corporate a GUT scenario for the masses in the context
A. Effective Yukawa couplings in models with
of Large Extra Dimensions. First, it is fair to say that,
extra dimensions
in order to achieve Grand Unification above the com-
pactificationscale,someratherstrongassumptionhasto
In its simplest version, an effective Yukawa coupling
be made about the behaviour of the running couplings,
(whichwouldbeproportionaltothemassofthefermion)
namely a power-law running. Because of this dynamical
is defined, infourdimensions, asproportionalto the size
assumption,therunningmassesusedinextrapolatingthe
of the wave function overlap between left-handed and
values at the GUT scale to low energies will also suffer
right-handedfermions along a compactfifth (spatial)di-
from large uncertainties. This is very unlike the loga-
mension [7]. Among the many applications of this idea,
rithmic behaviour used in [1]. In our case, quark-lepton
one can cite for example the attempts to give an expla-
unificationis achievedat a scale comparableto the com-
nation for the smallness of the neutrino masses [8, 9].
pactificationscaleandthepredictionsmadetherecanbe
Onecaneitherarbitrarily choosethe locations,alongthe
extrapolateddowntotheZ-massusingfamiliarrenormal-
extra dimension, of the localized wave functions for the
ization group techniques. In fact, since the quark-lepton
left-handed and right-handed neutrinos in such a way
unification scale is an order of magnitude or so larger
that the overlap is tiny, or one can try to build a model
than the Z-mass, there will not be much “running”.
in which the tiny overlap comes out more or less natu-
The plan of the paper will be as follows. First, we
rallyasRef. [9] haddone. In[9], the size ofthe neutrino
present a brief review of the essential elements that go
overlap came out small as compared with the size of the
into the wave function overlap scenario in extra dimen-
charged lepton overlap. A brief review of how this hap-
sions. We then briefly review the ideas of early quark-
pens as described in Ref. [9] will be given below. The
lepton unification with a special emphasis on the group
main point of these works is that the four-dimensional
structure and fermion representations. We then show
effective Yukawa couplings can be small even if the fun-
howonecanconnectthesetwoideasto relatethe overall
damental(four-dimensional)Yukawacouplingisoforder
mass scales in the mass matrices of the quark sector to
unity.
thoseoftheleptonsectors. Wefinishwithanumericalil-
Let us start with one extra spatial dimension y com-
lustrationofthoseresultsalongwiththeirphysicalimpli-
pactified on an orbifold S /Z and having a length L.
1 2
cations, including neutrino masses. We will present pre-
Let us, as an example, take a lepton SU(2) dou-
dictions for Dirac neutrino masses. Whether or not Ma- blet, L L (x,y), and another lepton SU(2) dLoublet,
{ } R
jorananeutrinomassesareneededisaquestionwhichde- L R (x,y), where the superscripts refer to the groups
{ }
pends onthe predictedvalues forDirac neutrinomasses.
respectively. Since a fermion in five dimensions is a
We will show a correlation between the masses of the
Diracfermion,itwillhavebothchiralities(leftandright-
neutrinosandthoseoftheunconventionalfermionswhich
handed) under four dimensions i.e. ψ = (ψ + ψ ),
L R
are present in the early unification model. If the latter
where ψ = P ψ, with P = (1 γ )/2 be-
L,R L,R L,R 5
fermions are required to have a mass between the elec- ∓
ing the usual four-dimensional chiral projection oper-
troweakscale andapproximately1TeV, it is shownthat
ator. The notations that were used in [9] and here
the Dirac neutrino masses are too small for the see-saw
will be as follows. For the SU(2) doublet, we use
L
mechanism [14] to provide the bulk of neutrino masses
if, as it is natural to assume, the Majorana scale is of L{L}(x,y)=(lL{L}+lR{L}),whilefortheSU(2)R doublet,
the order of the early unifications scale. It is also shown weuseL{R}(x,y)=(lL{R}+lR{R}). OnecanchoosetheZ2
that if the mass of the unconventional fermions is taken parity for these fields such that the only zero modes are
0, L 0, R
to lie between 200GeV and 300GeV, the range of the lL{ }(x,y) = lL(x)ξL(y) and lR{ }(x,y) = lR(x)ξR(y).
Dirac neutrino mass is found to be between 0.07eV With the introduction of the appropriate background
∼
and 1eV. In fact, there is a recentinterest in the pos- scalar fields, these zero modes can be localized at some
∼
sibility that the neutrino mass might be either mostly points along y. The effective Yukawa coupling (which
3
woulddeterminethemassofthefermion)isthenpropor- where k are normalization factors. The immediate
ν,e
tional to the overlap between ξ (y) and ξ (y). (Let us implication of Eq. 5 can be seen as follows. Using
L R
recallthatξ (y)andξ (y)aredoublets.) Themainfocus h(y)=v tanh(µy) in Eq. 5, one obtains
L R
of[9]wastheconstructionofamodelfortheSU(2) dou-
R
blet ξR(y). This constructionwillbe repeatedbelow but ξRν,e(y)=kν,ee−(CSln(cosh(µSy))±CTln(cosh(µTy))), (6)
a few words might be illuminating here. The wave func-
tions for the up and down members of ξ (y), although whereC =f /(λ /2)1/2. If the parametersofthe
R S,T S,T S,T
localized at the same point along y, have very different two scalar potentials are such that C ln(cosh(µ y))
S S
≈
shapes: one which is wide and the other which is nar- C ln(cosh(µ y)), one can immediately see that ξν(y) is
T T R
row. It is this disparity in shapes of the “right-handed” narrow while ξe(y) is broad. When these wave functions
R
wavefunctions that, whenoverlappingwiththe common overlapwith the left-handed wave function (common for
“left-handed” wave function, gives rise to the hierarchy both ν and e), one can observea large disparity between
in mass among up and down members of the doublet. the two effective Yukawa couplings. A crucial quantity
For the sake of clarity, a review of the model of [9] is which enters this hierarchy in Yukawa couplings is the
warranted here. Since the main object is the construc- separationbetweentheleft-handedwavefunctionandthe
tion of ξ (y), one will concentrate on L R . A summary two right-handed wave functions (localized at the same
R { }
ofthe mainresults of [9] cannowbe given. First,the lo- point along y) and which was denoted by ∆y(l) in [9].
calizationofξ (y)canbeachievedbyacouplingofL R The model described above has been espoused in Ref.
R { }
to a background scalar field which develops a kink solu- [9] as a mechanism for naturally small Dirac neutrino
tionalongy. Twobackgroundscalarfields areneededin masses. Furthermore,using the same wavefunction pro-
thisscenario: asingletfieldφ whosekinksolutionlocal- files for the right-handed quarks, an interesting connec-
S
izes the wave functions of both members of ξ (y) at the tion between quark and lepton mass hierarchies was no-
R
same location while keeping their shapes identical, and ticed in [9]. Basically,it was a connection between ∆y(l)
a triplet Φ = φ~ .~τ whose kink solution is responsible and ∆y(q). Possible symmetry reasons for this connec-
T T 2
fordrasticallychangingthe shapesofthe twowavefunc- tion were left open in [9]. It is the purpose of this paper
tions whilekeepingthe localizationpoints the same. (As to elucidate the relationship between quark and lepton
mentioned in [9], these backgroundscalars are chosen to mass hierarchies by considering explicitely a model of
be odd under Z so that they do not have zero modes.) TeV-scale quark-lepton unification [3, 4, 5]. To set the
2
Below is how it works. stage for that discussion, a brief summary of the early
unification model is presented below.
Theminimumenergysolutionsusedin[9]areasfollows
h (y) 0 B. Early TeV-scale quark-lepton unification
Φ = φ3 τ /2= T , (1)
h Ti h Ti 3 (cid:18) 0 hT(y)(cid:19)
−
The model that was presented in [4] and discussed in
and detail in [5] is based on the gauge group
φ =h (y), (2)
S S
h i
G =SU(4) SU(2) SU(2) SU(2) . (7)
PUT PS L R H
⊗ ⊗ ⊗
where generically h(y) = v tanh(µy), with µ = λ/2v
This group is characterized by two independent gauge
being typicallythe “thickness”of the domain walpl. Cou-
pfl(el)dL¯wRithφthLeYRuk,aownaecoobutpaliinnsgtLheY2fo=llofwT(iln)L¯g{eRq}uΦaTtioLn{sR}fo+r cSoUu(p2li)nHg,s:wghSefroerSaUp(4er)mPSuatantdiognWsfyomrSmUet(r2y)L⊗isSaUss(u2m)Re⊗d
S { } S { } amongthethreeSU(2)’s. GPUT isassumedtobebroken
ξ (y):
R down to the Standard Model in two steps, namely
∂yξRν(y)+(fS(l)hS(y)+fT(l)hT(y))ξRν(y)=0, (3) G M G M˜ G MZ SU(3) U(1) , (8)
PUT 1 2 c EM
−→ −→ −→ ⊗
where
∂ ξe(y)+(f(l)h (y) f(l)h (y))ξe(y)=0, (4)
y R S S − T T R G1 =SU(3)c U(1)S SU(2)L SU(2)R SU(2)H, (9)
⊗ ⊗ ⊗ ⊗
The solutions for the up and down members of ξ (y)
R and
(which will have the superscripts ν and e respectively)
are given as G =SU(3) SU(2) U(1) . (10)
2 c L Y
⊗ ⊗
y
ξν,e(y)=k exp( dy (f(l)h (y ) f(l)h (y )), In this scheme, quarks and leptons, which are generic
R ν,e −Z ′ S S ′ ± T T ′ termsforcolortripletsandcolorsingletsrespectively,are
0
(5) grouped into quartets of SU(4) . The scale of such a
PS
4
quark-lepton unification is denoted by M as seen above. For the sake of clarity, an explicit description of the
In contrast with GUT where such a unification occurs fermions of the model is presented below.
close to the Planck scale, it has been shown in [4], and Under SU(4) SU(2) SU(2) SU(2) , the
PS L R H
⊗ ⊗ ⊗
particularly in [5], that M 2TeV. That such a low fermions transform as
≤
scaleofunificationcanbeachievedisadistinctivefeature
(dc(1/3),U˜(4/3)) (˜l ( 1),ν(0))
obfeltohwis. model. A sketch of the arguments is presented ΨL =(4,2,1,2)L =(cid:18)(uc(−2/3),D˜(1/3)) (˜ldu(−−2),l(−1))(cid:19)L
At this point, it is worth noticing that, if the scale(s) (11)
of extra dimensions is comparable with the Petite Unifi-
cationscale,physics whichare relatedto the breakingof
(dc(1/3),U˜(4/3)) (˜l ( 1),ν(0))
Petite Unification can be extrapolated to “low energies” ΨR =(4,1,2,2)R=(cid:18)(uc( 2/3),D˜(1/3)) (˜lu(−2),l( 1))(cid:19)
with little, if any, uncertainties coming from physics be- − d − − R
(12)
yond the compactification scale. This is in contrastwith
As one can see, this model contains, besides conven-
a typical GUT scenario embedded in large extra dimen-
tionally charged fermions, unconventional fermions with
sions since its scale which would normally lie above the
chargesup to 4/3 for the quarks and downto 2 for the
compactificationscale. Asaconsequence,therearelarge
−
leptons.
uncertaintiesassociatedwiththeextrapolationof“GUT”
To understand the notations in Eqs. (11,12), one no-
physics down to the Z-mass for example.
tices the following conventions.
The mainideaofPetiteUnificationhas todo withthe
assumption that the SM, with three independent cou-
SU(2) doublets:
plings: g3, g2 and g1, is merged into the PUT group • L,R
G G which is characterized by two independent
S W
⊗
couplings: g and g . As a result, one can com-
S W dc(1/3) ν(0)
pute sin2θ (M2) as a function of the PUT unifica- , ,
W Z (cid:18)uc( 2/3)(cid:19) (cid:18)l( 1)(cid:19)
tion scale M as shown in [3, 4]. The highly precise − L,R − L,R
value of sin2θ (M2) = 0.23113(15), along with the
W Z
rreesqturiircetmtehnet cthhoaitceMs of≤G1W0,TereVsu,ltailnlogwisnutshetopsreevfeerrreeldy (cid:18)DU˜˜((41//33))(cid:19) , (cid:18)˜l˜lu((−21))(cid:19) (13)
model mentioned at the beginning of this section. (Two L,R d − L,R
other models were also found: SU(4) SU(2)4 and
PS
⊗
SU(4)PS ⊗ SU(3)2, with the former being, in some • SU(2)H doublets:
sense, ruled out due to severe problems with the process
K µe unless some exotic mechanisms are invoked,
L
for e→xample an embedding of the model into five dimen- U˜(4/3) ν(0)
, ,
sions with the gauge symmetry breaking accomplished (cid:18)dc(1/3)(cid:19)L,R (cid:18)˜lu(−1)(cid:19)L,R
by orbifold boundary conditions [16].)
Two crucial elements in the computation of
sin2θ (M2) are the group theoretical factor sin2θ0 D˜(1/3) l( 1)
(= 1/W3 forZ GW = SU(2)3) and the factor CS whicWh (cid:18)uc(−2/3)(cid:19)L,R, (cid:18)˜ld(−−2)(cid:19)L,R (14)
appears inthe expressionQ=T +T =Q +C T ,
3L Y W S 15
where QW is the ”weak” charge corresponding to SU(4)PS quartets:
the group G , and T is the unbroken diagonal •
W 15
generator of the SU(4) . The value of C de-
PS S
pends on how quarks and leptons transform under dc(1/3) D˜(1/3)
, ,
CSU(=4)PS2⊗/3SUif(2fe)rLm⊗ionSsUt(r2a)nRsf⊗orSmUa(s2)(H4.,2F,1o,r1i)nsfotarnecxe-, (cid:18) ˜lu(−1) (cid:19)L,R (cid:18) l(−1) (cid:19)L,R
S
ample,pwhile C = 8/3 if they transform as (4,2,1,2)
S
or (4,1,2,2). Thispis shown in details in [3, 4]. Since U˜(4/3) uc( 2/3)
, − (15)
sin2θW(MZ2)=(1/3)(1−0.067CS2 −logterms) (see [4]) (cid:18) ν(0) (cid:19)L,R (cid:18) ˜ld(−2) (cid:19)L,R
and coupled with the requirement that M 10TeV
≤
(which makes for little “running” between M and the Note that due the particular nature of the fermion rep-
electroweak scale, and hence small log terms), it was resentation in this model, it is
found [4] that the only acceptable fermion represen-
tations are the ones for which CS = 8/3. Using dc(1/3) =iτ u(2/3) ∗ (16)
sthinis2θva(luMe2f)o,rupCSto[t4w,o5lo],opas [d5e],tadieletdermcopinmepsutthaetiPonetitoef (cid:18)uc(−2/3)(cid:19)L,R 2(cid:18)d(−1/3)(cid:19)L,R
W Z
Unification scale to be less than 2TeV. This fermion which appears instead of the more familiar-looking
content will be the one that will be used in this paper. (u(2/3),d( 1/3)).
−
5
As emphasized in [4, 5], a nice feature of this model is of this paper, we will simply require that all unconven-
the absence oftree-levelflavor-changingneutral currents tional fermions are heavy. This will be one constraint
(FCNC) because the SU(2) and SU(4) transitions which will be used below.
H PS
only connect conventionalto unconventionalfermions as
can be seen above. A process such as K µe can only
L
occur at one loop and can easily be mad→e to obey the 1. SU(4)PS⊗SU(2)L⊗SU(2)R⊗SU(2)H in five
experimental upper bound. dimensions
Since the natural scales of the scenarios described in
Sections IIA and IIB are both in the TeV range, it is The first step one would like to do is to embed
worthwhile to see if a “marriage” of some sorts can be SU(4)PS SU(2)L SU(2)R SU(2)H in five dimen-
⊗ ⊗ ⊗
made between these two scenarios and if, as a result, sions. Letusfirstdenote,infivedimensions,thefermions
some light can be shed concerning the relationship be- presented in Section IIB by
tween quark and lepton masses and the smallness of the Ψ L (x,y)=(4,2,1,2), (17)
{ }
neutrino masses.
Ψ R (x,y)=(4,1,2,2). (18)
{ }
C. Connection between the scales of quark and
Let us recall that, in five dimensions, these are four-
lepton masses
component Dirac fields, i.e. they have both left- and
right-handedcomponents. Thesuperscripts L and R
If “quarks”and “leptons” (in the generic sense as dis- { } { }
areusedfortworeasons: (a)todenotethetransformation
cussed above) can be unified at the TeV scale, there is a
under SU(2) or SU(2) ; and (b) to show that the sur-
L R
good possibility that whatever gives rise to their masses
vivingzeromodesarerelatedtothesefields. Bychoosing
will also determine the relationship between their mass
theappropriateZ parityforthesefields,thezeromodes
2
scales. We will show below that such a possibility does of Ψ L (x,y) and Ψ R (x,y) are
{ } { }
existwithintheframeworkofthetwoscenariosdescribed
Ψ (x,y)=ψ (x)ξ (y), (19a)
above. L L L
The basic model used in this paper is SU(4)
PS
⊗
SU(2) SU(2) SU(2) . As stated above, this
groupLsp⊗ontaneouRsl⊗y breaksHdown to SU(2) U(1) ΨR(x,y)=ψR(x)ξR(y), (19b)
L Y
⊗
and then to U(1) at the scales M fewTeV s and respectively.
em ′
∼
v 250GeV respectively. Upon embedding this model Wewishtolocalizeξ (y)andξ (y)alongy. Thisisac-
L R
∼
in five dimensions, with the fifth dimension y compact- complished by coupling these fields to some background
ified on an S /Z orbifold, it is shown below that the scalar fields. To see the group representations of these
1 2
following features occur: 1) The breaking of SU(4) scalar fields, we consider the following bilinears:
PS
splits the positions, along y, of wave functions of the Ψ¯{L}(x,y)Ψ{L}(x,y)=(1+15,1+3,1,1+3), (20)
zero modes of “quarks” and “leptons”; 2) The breaking
of SU(2) gives rise to two vastly diferent profiles for
R
the wave functions of the “right-handed” zero modes; 3) Ψ¯ R (x,y)Ψ R (x,y)=(1+15,1,1+3,1+3). (21)
{ } { }
Since a SU(2) doublet groups together a conventional
H
quark(orlepton)withanunconventionalone,thebreak- From Eq. (20), one can see that some possible scalar
ing of SU(2) splits the locations, along y, of the wave fieldswhichcancoupletothesefermionswouldtransform
H
functions of the conventional fermions relative to those like (1,1,1,1), (15,1,1,1), (15,1,1,3), (15,1,3,1), etc..
of the unconventional ones; 4) And finally, the break- We will show step-by-step below how these scalar fields
ing of SU(2) U(1) provides a mass scale for all the helpestablishthe link between∆y(l) and∆y(q). We will
L Y
fermions. Asw⊗ehaveexplainedintheprevioussection,a successively invoke one scalar at a time and show how it
crucial quantity that appears in the hierarchy of masses modifies the behaviour of the fermion zero modes.
among the up and down members of an SU(2) dou- As we shall see, the scalar fields which are needed for
L
blet is the separationalong y between the wave function our scenario are the following:
of the left-handed doublet and that of the right-handed Φ =(1,1,1,1) (22a)
fields, namely ∆y(l,q). We will show below that points S1,S2
# 1, 2 and 3 help establish a relationship between ∆y(l)
and ∆y(q). Σ=(15,1,1,1), (22b)
In the construction of the model, one important point
we would like to stress is the following. The model
SU(4)PS SU(2)L SU(2)R SU(2)H contains un- ΦR =(1,1,3,1), (22c)
⊗ ⊗ ⊗
conventional quarks and leptons which were assumed to
be heavy enough to escape detection. The fate of these
fermionswerewelldescribedinRef. [4]. Forthe purpose ΦH =(15,1,1,3). (22d)
6
2. The role of the singlet scalar fields ΦS1,S2 =(1,1,1,1) 3. The roles of Σ=(15,1,1,1) and ΦH =(15,1,1,3)
The Yukawa coupling of this field with the fermions FromthegrouprepresentationsofΣ=(15,1,1,1)and
takes the form Φ = (15,1,1,3), one can see that, in principle, both
H
Ψ L and Ψ R can couple to Σ and Φ . However, for
=f (Ψ¯ L Ψ L +Ψ¯ R Ψ R )Φ , (23) { } { } H
LY1 S { } { } { } { } S1 reasons of economy, we shall see that it is sufficient to
with fS >0. We assume a kink solution for ΦS1 couple Ψ{L} to ΦH.
We now concentrate on ξ (y). As it is men-
L
hΦS1i=hS(y), (24) tioned above, one has to differentiate the unconven-
tionalfermions fromthe conventionalones as wellas the
whichlocalizesallfermionsatthesamepointy =0along
“quarks” from the “leptons”. Let us remind ourselves
y. In fact,the equationgoverningthe zero modes, atthis
that the conventional and unconventional fermions are
stage, is
grouped into SU(2) doublets as shown in Eq. (14).
Inordertobemoregeneral,wewillallowthefermions H
To differentiate the aforementionedfermions,we needto
to be localized, still at this stage, at some common ar-
break both SU(4) and SU(2) . This is accomplished
bitray point which might be different from the origin. PS H
by the use of Φ = (15,1,1,3) and of Σ = (15,1,1,1).
The most economical scenario is one in which the “left- H
The Yukawa interaction between Ψ L , Φ and Σ =
handed” fermions are localizedat that other point. This { } H
(15,1,1,1)is given by
can be accomplished by the following coupling:
LY1p =−fS′ΦS2Ψ¯{L}Ψ{L}, (25) LY2 =Ψ¯{L}(fHΦH +fΣΣ)Ψ{L}, (30)
where fS′ > 0 and the negative sign in front of it is an with f ,f >0.
H Σ
arbitrary choice. Assuming
We will assume a vacuum expectation value for Φ
H
Φ =δ, (26) and Σ as follows
h S2i
we obtain the following equations for the zero modes: 1 0 0 0
0 1 0 0
∂yξL(y)+{fShS(y)−fS′δ}ξL(y)=0. (27a) hΣi=σ 0 0 1 0  , (31a)
0 0 0 3
 − 
∂ ξ (y)+ f h (y) ξ (y)=0. (27b)
y R S S R
{ }
1 0 0 0
We will present below two scenarios.
0 1 0 0 v 0
Φ =  H , (31b)
Scenario I: h Hi 0 0 1 0 ⊗(cid:18) 0 vH (cid:19)
•  0 0 0 3 −
 − 
δ =0. (28) where the firstmatrix onthe right-handside of Eq. (31)
referstothedirectionT ofSU(4) whereasthesecond
15 PS
Scenario II: matrix in the second equation refers to the direction τ
• 3
in SU(2) , all of which refer to (15,1,1,3). Here σ and
H
v are constants.
H
δ =0. (29) When Eq. (30) is combined with Eq. (23), the equa-
6
tion for the left-handed zero modes is now given by
Atthis stage,ScenarioIimpliesthatall(leftandright-
handed) fermions are localized at the origin. Scenario II ∂yξL(y)+ fShS(y) fS′δ+fH ΦH +fΣ Σ ξL(y)=0.
implies that the right-handed ones are localized at the { − h i h i} (32)
origin while the left-handed ones are localized at a com- We now make an important assumption:
mon point away from the origin. As we shall see in the
last section, it will be Scenario II with δ = 0 that is f v =f σ. (33)
6 H H Σ
favored phenomenologically.
It is clear that this is not the end of the story because This assumptionhasa far-reachingconsequence: Allun-
the effective Yukawa couplings to an SU(2)L-doublet conventional quarks and leptons will have large wave
Higgs field, which depend on the overlap of the left and function overlaps resulting in “large” mass scales for
rightwavefunctions, wouldbe universalforallfermions, those sectors as we shall see below.
a clearly undesirable feature. We therefore have to split MakinguseexplicitelyofEq. (31)andthe assumption
the various wave functions along y. To do this, one (33), one can now rewrite (32) in terms various SU(2)
L
hastoinvokescalarswhichtransformnon-triviallyunder doublets as follows
SU(4) SU(2) SU(2) SU(2) . Thisiswhatwe
PS L R H
will proce⊗ed to do n⊗ext. ⊗ ∂yξLQ(y)+{fShS(y)+2fHvH−fS′δ}ξLQ(y)=0, (34a)
7
Eq. (38) then takes the following forms:
∂yξLQ˜(y)+{fShS(y)−fS′δ}ξLQ˜(y)=0, (34b) ∂yξRQ,up(y)+hsym(y)ξRQ,up(y)=0, (40a)
∂yξLL(y)+{fShS(y)−6fHvH −fS′δ}ξLL(y)=0, (34c) ∂yξRQ,down(y)+hasym(y)ξRQ,down(y)=0, (40b)
∂yξLL˜(y)+{fShS(y)−fS′δ}ξLL˜(y)=0, (34d) ∂ ξL,up(y)+hsym(y)ξL,up(y)=0, (40c)
y R R
where the superscripts Q,Q˜,L,L˜ refer to the normal
quark,unconventionalquark,normallepton, and uncon-
ventional lepton SU(2) doublets as shown in Eq. (15).
L ∂ ξL,down(y)+hasym(y)ξL,down(y)=0. (40d)
The above equations show the splitting between conven- y R R
tional and unconventional fermions as well as between
InEqs. (40)andaccordingtotheparticlecontentgiven
quarks and leptons. As we shall show below, this split-
inEqs. (15), “Q,up”referstodc(1/3)andU˜(4/3). Like-
ting is crucial to the success of this model.
wise, “Q,down” refers to uc( 2/3) and D˜(1/3). Simi-
From Eqs. (34), it also is easy to see that
−
larly, “L,up refers to ν(0) and ˜l ( 1) while “L,down
′′ u ′′
ξLQ˜(y)=ξLL˜(y). (35) refers to l(−1) and ˜ld(−2). It is a−lso clear, from Eqs.
(40), that
4. The role of ΦR =(1,1,3,1) ξRQ,up(y)=ξRL,up(y)=ξRup, (41a)
WehaveencounteredinSectionIIAtheSU(2) triplet
R
scalar field whose kink solution, when combined with
ξQ,down(y)=ξL,down(y)=ξdown. (41b)
a singlet kink, gives rise to very different profiles for R R R
the wave functions of the up and down members of an
A quick look at Eq. (39) reveals that u (2/3) and
SU(2) fermion doublet. In the present context, the R
R D˜ (1/3) as well as l ( 1) and ˜l ( 2) have “broad”
ttroipΨletRsccaalanrbfieewldriitstennowasΦR = (1,1,3,1). Its coupling waRve functions whileRdR−( 1/3) andd,RU˜−R(1/3) and νR(0)
{ } −
and ˜l ( 1) have “narrow” wave functions. These fea-
u,R
Y3 =fRΨ¯{R}ΦRΨ{R}, (36) tures will−be shown explicitly below.
L
From Eqs. (40), one can easily see that all right-
with f > 0. Notice that here we use f instead of the
R R handed wave functions are localized at the origin. They
notationf usedinSectionIIAinordertobeconsistent
T have,however,differentprofiles,asituationwhichisvery
with the notation used for Φ .
R similar to the scenario which is summarized in Section
The minimum energy solution for Φ can be written
R IIA. It is this difference in profiles, when combined
as
withthedifferentlocationsofleft-handedwavefunctions,
1 0 0 0 which gives rise to the disparity in mass scales.
0 1 0 0 h (y) 0 We turn next our attention to the separations along y
hΦRi= 0 0 1 0⊗(cid:18) T0 hT(y) (cid:19) , (37) betweenleft-handedandright-handedfermionswhichare
 0 0 0 1 − crucial, along with the different profiles, in determining
 
the mass scale of each sector.
where we have assumed that there is a kink solution for
Φ .
R
When one combines Eq. (23) with Eqs. (36,37), the 5. Wave function localizations in a linear approximation
equation for the zero modes looks as follows.
To see heuristically how Eqs. (34,40) help split the
locations of the “quarks” and “leptons” along the extra
∂ ξ (y)+ f h (y)+f Φ ξ (y)=0. (38)
y R { S S Rh Ri} R dimension,letusmakealinearapproximationtothekink
solutions h (y) and h (y), namely
Let us define the following effective kinks: S T
hsym(y)=fShS(y)+fRhT(y), (39a) hS(y)≈2µ2Sy. (42a)
hasym(y)=f h (y) f h (y). (39b) h (y) 2µ2y. (42b)
S S − R T T ≈ T
8
Let us recall that, with the linear approximation, out of eight locations, there are three (or four) predic-
a wave function behaves like a Gaussian ξ(y) tions. In principle, we would then obtain three (or four)
∝
exp( µ2y2). It then follows that that the overlap be- predictions for mass scalesonce the other four (or three)
−
tween two functions separated by a distance ∆y along y are fixed by the choices of the aforementioned parame-
goeslike exp( µ2(∆y)2) (where for heuristic purposeµ2 ters. We shall come back to this point below.
−
aretakentobethesameforbothwavefunctionswhichin Inordertomakesenseoutoftheabovelocations,afew
general is not the case). The effective Yukawa couplings remarksareinorderhere. TheeffectiveYukawacoupling,
in four dimensions are proportional to the overlaps be- infourdimensions,whichgovernsthefermionmassscale
tween “right-handed” and “left-handed” fermions and it depends on the overlap between left-handed and right-
can be seen that they can be “large”or “small” depend- handedfermions. Thisoverlapdependsontheseparation
ingonwhetherornot(∆y)2 µ2 or(∆y)2 µ2. What between the two fermions as well as onthe shapes of the
≫ ≪
we will set out to derive in our model is the relationship fermion wave functions. As mentioned above,the spread
between ∆y(l) and ∆y(q). of the wave functions is crucial in our scenario. This
With the approximation (42), let us apply it to Eqs. spread is rougly proportional to 1/µ. From Eqs. (43),
(39), resulting in the following definitions onecandeducethat,fortheright-handedwavefunctions,
l ( 1),˜l ( 2),u (2/3)andD˜ (1/3)havebroad wave
R d,R R R
µ2asym =fSµ2S −fRµ2T , (43a) fun−ctions sin−ce µ2asym = fSµ2S − fRµ2T. On the other
hand, d ( 1/3),U˜ (4/3),ν (0)and˜l ( 1)havenar-
R R R u,R
row wave −functions since µ2 =f µ2 +f− µ2. All the
sym S S R T
µ2 =f µ2 +f µ2 , (43b) left-handed wave functions, on the other hand, have a
sym S S R T spread of the order of 1/ f µ2. How do these facts
S S
From Eqs. (40), the locations of the Up-members and translate into the disparitipes in mass scales? To answer
the Down-members of an SU(2) doublet, for both con- this question,onehastolookatthe separationsbetween
R
ventional and unconventional quarks, are found to be at the left-handed and right-handed wave functions.
the origin. We shall denote that common point by FromEqs. (44-47),onecanreadilyderivethefollowing
left-right separations.
y =0. (44)
R
For the “quarks”:
•
For the left-handed zero modes, their locations will
f v
dependonfS′δ andfHvH. Forconvenience,letusdefine |∆yU|≡|yR−yQL|=|(fH)(µH2 )(1−r)|, (48a)
the following quantity: S S
r fS′δ . (45) fH vH
≡ 2fHvH |∆yD|≡|yR−yQL|=|(f )(µ2 )(1−r)|, (48b)
S S
Wewillassumethatr <1. FromEqs. (34),thelocations
of the SU(2)L doublets for conventional and unconven- fS′δ fH vH
tional quarks are |∆yU˜|≡|yR−yQ˜L|= 2f µ2 =(f )(µ2 )r, (48c)
S S S S
f v
H H
y =( )( )(1 r), (46a)
QL −fS µ2S − ∆y y y = fS′δ =(fH)(vH)r, (48d)
| D˜|≡| R− Q˜L| 2f µ2 f µ2
S S S S
y = fS′δ =(fH)(vH)r, (46b) For the “leptons”:
Q˜L 2f µ2 f µ2 •
S S S S
while the locations of the “lepton” doublets are f v r
H H
∆y y y =3 ( )( )(1+ ) , (49a)
| ν|≡| R− lL| | f µ2 3 |
fH vH r S S
y =(3 )( )(1+ ), (47a)
lL f µ2 3
S S
f v r
H H
∆y y y =3 ( )( )(1+ ) , (49b)
fS′δ fH vH | l(−1)|≡| R− lL| | fS µ2S 3 |
y = =( )( )r. (47b)
˜lL 2fSµ2S fS µ2S fS′δ fH vH
∆y y y = =( )( )r, (49c)
Oneimportantcommentisinorderatthispoint. From | ˜lu(−1)|≡| R− ˜lL| 2fSµ2S fS µ2S
the above equations (44,46,47) as well as (43), it is clear
fftohRuaµrt2Titn,hdeferΣepeσan,rdeaefinndvtepfaiHnradvmeHpee,tneadrlset.hnotWupgahhraatEmtqhe.tiesr(is3m:3p)flSireeµsd2Sui,scfetShs′atδto,, |∆y˜ld(−2)|≡|yR−y˜lL|= 2ffSS′µδ2S =(ffHS)(vµH2S)r. (49d)
9
Comparing Eqs. (49) with Eqs. (48), we arrive we write down the coupling between the left-handed
at the following important relationship between conven- fermions, right-handed fermions and a Higgs field whose
tional quarks and leptons: VEVgivesrise to fermionmasses. Next, we (arbitrarily)
fix the two right-handedwavefunction profiles. We then
|∆yLepton|=3(cid:16)(cid:16)(cid:16)11+rr3(cid:17)(cid:17)(cid:17)|∆yQuark|, (50) ctihoonoasleU|∆pyaQnudarkD|oswonthqautartkhesemcatossrssccaolmeseoofutthecocrornevcetlny-.
−
With the same wave function profiles, we next choose r
where r < 1. For Scenario I, one would have r = 0 and
so that, upon the use of Eq.(50), the mass scale of the
one would simply have ∆y = 3∆y . For
| Lepton| | Quark| charged lepton sector comes out correctly. We will show
Scenario II, where r =0, one obtains the above relation-
6 belowthat,asaresult,weobtainpredictionsforthemass
ship. What (50) implies is the following important fact:
scale of the Dirac neutrino sector as well as those of the
∆y 3∆y . This means that the scales of
| Lepton| ≥ | Quark| unconventional quarks and leptons. An alternative way
the lepton sector are generally a bit smaller than those
tofixtheparametersistochooseoneoftheright-handed
of the quark counterpart.
wave functions, for example the one that belongs to the
It is also useful to derive a relationship between the
charged leptons and the Up quark sector, fix r so that
separationsofconventionalandunconventionalfermions.
the mass scales come out correctly, fix the second right-
From Eqs. (46, 47, 48), it is straightforward to derive
handed wave function function so that the Down quark
the followingrelationshipbetweenthe commonleft-right
sector comes out correctly. Once this is done, the above
separationofthe unconventionalquarksandleptons and
predictions will come out the same.
that of the conventional quarks:
r
∆y = ∆y . (51) III. COMPUTATION OF MASS SCALES AND
Unconventional Quark
| | (cid:16)(cid:16)(cid:16)1−r(cid:17)(cid:17)(cid:17)| | IMPLICATIONS
Another useful form for (51) can be obtained by using
Eq. (50), namely One can now use the results of Ref. [9] and Sec-
tion IIA to estimate mass scales of the normal quark
|∆yLepton| 3 andlepton sectorsas well as those of the unconventional
∆yUnconventional = |∆yQuark| − ∆yQuark . (52) fermions. Sincethescopeofthispaperistheconstruction
| | (cid:16)(cid:16)(cid:16) 4 (cid:17)(cid:17)(cid:17)| | of a model showing a relationship between mass scales
of “quarks” and “lepton” sectors, we shall ignore issues
From (50) and (51, 52), one notices that, in Sce-
such as fermion mixings in the mass matrices. Higher
nario I where r = 0, one obtains the simple relations:
dimensionalmodelshavebeenbuilttotacklequarkmass
∆y =3∆y and ∆y =0.
Lepton Quark Unconventional
| | | | | | hierarchies,mixinganglesandCPphase(seee.g. [11]and
The above relationship is important for the following
[12]). We will therefore concentrate on the overall mass
reasons. First, it implies that the left-handedwavefunc-
scales that appear in various mass matrices.
tion for the leptons is situated much further (by a factor
of three) away from the right-handed ones than is the
caseforthequarks. Itthenmeansthatthewavefunction
A. SM Fermion-Higgs coupling
overlaps which determine the effective four-dimensional
Yukawa couplings would , in principle, be much smaller
for the lepton sector than for its quark counterpart, re- By “SM Fermion-Higgs coupling”, we mean that the
sultinginalargedisparityinmassscalesbetweenthetwo Higgs field that couples with left-handed and right-
sectors. Thisisactuallywhathappensinreality. Thede- handed fermions transforms non-triviallyunder SU(2)L.
tails of that disparity, in our scenario, will also depend Since Ψ{L} = (4,2,1,2), Ψ{R} = (4,1,2,2) and
on the difference in the wave function profiles. This will Ψ¯L Ψ R = (1+15,2,2,1+3), an appropriate Higgs
{ } { }
be discussed in the next section. field (the simplest choice) could be the following field:
Since ∆y = fS′δ for both unconventional quarks
| | 2fSµ2S H =(1,2,2,1). (53)
and leptons, this implies that unconventional quarks
and leptons have comparable mass scales which can be
The Yukawa coupling with this field can be written as
“large”. How large this might be is the subject of the
next section. Y4 =k1Ψ¯{L}HΨ{R}+k2Ψ¯{L}H˜ Ψ{R}+H.c., (54)
L
To summarize, the model contains four independent
parameters: fSµ2S, fRµ2T, fHvH, and r. From these, where H˜ = τ2H∗τ2 and where, in principle, k2 can be
we have two independent wave function profiles for the different from k1. Assuming the extra dimension to be
right-handed zero modes, one independent separation compactifiedonanorbifoldS1/Z2 andanevenZ2 parity
∆y , and one parameter r. Once r and ∆y for H, it follows that H can have a zero mode. This
Quark Quark
|are specifi|ed, all other separations can be comp|uted. | zero mode can be written as H0(x,y) = Kφ(x) where
We now present some numerical illustrations of the φ(x) is a 4-dimensional Higgs field with dimension M
above results. Our strategy will be as follows. First, (mass) and K, a constant, has a dimension √M, since
10
H has a dimension M3/2 in five dimensions. Notice that
k have a dimension M 1/2. We define the following
1,2 − L
dimensionless couplings: g =g =g dyξQ˜(y)ξup(y), (59e)
U˜1,2 ˜lu1,2 Y1,2 Z L R
0
g =k K. (55)
Y1,2 1,2
L
The VEV of φ is assumed to be g =g =g dyξQ˜(y)ξdown(y). (59f)
D˜1,2 ˜ld1,2 Y1,2 Z L R
0
v /√2 0
φ = 1 . (56) The wayv and v appearin Eqs. (58)should be clearly
h i (cid:18) 0 v /√2 (cid:19) 1 2
2
understoodthatit has to do withthe fermioncontentas
Eqs. (54) and (56) provide mass scales which appear in shown in Eqs. (15), and that is the reason why the first
mass matrices as follows two equations appear in the forms shown above.
There are two possibilities concerning Eqs. (58).
=Λ M ,
MU,D,ν,l−,U˜,D˜,˜lu,˜ld U,D,ν,l−,U˜,D˜,˜lu,˜ld U,D,ν,l−,U˜,D˜,˜lu,˜ld g =g :
(57) • Y1 Y2
where Λ are the mass scales in question This is a rather economical option, in terms of re-
and M U,D,ν,l−,U˜,D˜,˜lu,˜ldare matrices whose elements will ducing the number of parameters. From Eqs. (58)
U,D,ν,l−,U˜,D˜,˜lu,˜ld and (59), it is easy to see that, if g = g , the
depend on models of fermion masses (e.g. [11]). The Y1 Y2
ratios of scales will simply ratios of wave function
subscripts are self-explanatory.
overlaps regardless of the values of v and v as
Using the fermion representations (15) and Eqs. (54, 1 2
well as of the value of the Yukawa coupling since
56, 35, 41), the mass scales that appear in Eq. (64) now
they cancel out in the ratios. In fact, we can form
take the following forms:
six ratios (from six independent scales) which are
ΛU =gU1v2/√2+gU2v1/√2, (58a) canbetakenasΛD/ΛU,Λν/Λl−,Λν/ΛD,Λl−/ΛU,
Λ /Λ , Λ /Λ . Explicitely, one has:
U˜ D D˜ U
Λ LdyξQ(y)ξup(y)
Λ =g v /√2+g v /√2, (58b) D = 0 L R , (60a)
D D1 1 D2 2 ΛU RLdyξQ(y)ξdown(y)
0 L R
R
Λν =gν1v1/√2+gν2v2/√2, (58c) Λν = 0LdyξLL(y)ξRup(y) , (60b)
Λl− RLdyξL(y)ξdown(y)
0 L R
R
Λl− =gl−1v2/√2+gl−2v1/√2, (58d) Λ LdyξL(y)ξup(y)
ν = 0 L R , (60c)
ΛD RLdyξQ(y)ξup(y)
0 L R
R
Λ =Λ =g v /√2+g v /√2, (58e)
U˜ ˜lu U˜1 1 U˜2 2 Λl− = 0LdyξLL(y)ξRdown(y) , (60d)
ΛU RLdyξQ(y)ξdown(y)
0 L R
Λ =Λ =g v /√2+g v /√2, (58f) R
D˜ ˜ld D˜1 2 D˜2 1
where ΛU˜ = 0LdyξLQ˜(y)ξRup(y), (60e)
L ΛD RLdyξQ(y)ξup(y)
gU1,2 =gY1,2 Z dyξLQ(y)ξRdown(y), (59a) R0 L R
0
Λ LdyξQ˜(y)ξdown(y)
D˜ = 0 L R , (60f)
L ΛU RLdyξQ(y)ξdown(y)
g =g dyξQ(y)ξup(y), (59b) 0 L R
D1,2 Y1,2 Z L R R
0 Once the parameters of the wave functions and
their separations are fixed, these ratios (or any
L other combinations) can be computed unam-
gν1,2 =gY1,2 Z dyξLL(y)ξRup(y), (59c) bigously.
0
In the following, we will choose Λ and Λ as two
U D
independent inputs. From them, we can extract
L ∆y . Once the parameter r is chosen, all
gl−1,2 =gY1,2 Z dyξLL(y)ξRdown(y), (59d) |otheQrumaraks|s scales can be predicted.
0