Table Of ContentSU(3) Family Gauge Symmetry and the Axion
Thomas Appelquist,1 Yang Bai,1 and Maurizio Piai2
1Department of Physics, Sloane Laboratory, Yale University, New Haven, CT 06520
2Department of Physics, University of Washington, Seattle, WA 98195
(Dated: February 2, 2008)
Weanalyzethestructureofarecentlyproposedeffectivefieldtheory(EFT)forthegeneration of
quark and lepton mass ratios and mixing angles, based on the spontaneous breaking of an SU(3)
family gauge symmetry at a high scale F. We classify the Yukawa operators necessary to seed the
masses, making use of the continuous global symmetries that they preserve. One global U(1), in
addition to baryon number and electroweak hypercharge, remains unbroken after the inclusion of
alloperatorsrequiredbystandard-model-fermionphenomenology. Anassociatedvacuumsymmetry
insuresthevanishingofthefirst-familyquarkandcharged-leptonmassesintheabsenceofthefamily
gaugeinteraction. IfthisU(1)symmetryistakentobeexactintheEFT,brokenexplicitlybyonly
7
theQCD-inducedanomaly,and if thebreakingscale F istaken toliein therange109−1012 GeV,
0
0 then theassociated Nambu-Goldstoneboson is a potential QCD axion.
2
PACSnumbers: 12.15.Ff,14.60.Pq,14.80.Mz
n
a
J
INTRODUCTION setofglobalU(1)symmetriesassociatedwitheachofthe
4 complexfieldsofthemodel. Onecombinationisrendered
2
anomalous by the SU(3) family gauge interaction. We
In a recent set of papers [1] [2], we developed an F
showthatthedominantYukawaoperatorsrequiredtode-
2 effective-field-theory (EFT) framework for the computa-
v scribe(with the family gaugeinteractions)mostfeatures
tionofquarkandleptonmassesandmixinganglesbased
1 ofthe quarkandcharged-leptonmassmatrices thenpre-
on an SU(3) family gauge symmetry. The largest el-
6 F servetwoU(1)symmetriesinadditiontothoseassociated
3 ements of the quark and charged-lepton mass matrices
with baryon number and electroweak hypercharge.
2 are seeded phenomenologically through a set of Yukawa
1 operators,bilinear in the quark and lepton fields and in- In order to fit precisely the quark and charged-lepton
6 massmatrices,andtogeneratetheneutrinomassmatrix,
cluding the Higgs doublet. They also include standard-
0 it is necessary to include some additional, smaller oper-
model (SM)-singlet scalars transforming as sextets un-
/
h der the SU(3) family group. The family symmetry is atorsthat explicitly break these two U(1) symmetries to
F
p one. Thisfinalsymmetry,U(1) ,isbrokenspontaneously
broken spontaneously at a high scale F by vacuum ex- a
- atthe scaleF. Ifitis takentobe exactintheEFT,bro-
p pectation values (VEV’s) of these scalars. The SU(3)F
e family symmetry is realized nonlinearly among the SM- ken explicitly by only QCD anomalies, it could play the
h role of a Peccei-Quinn symmetry to address the strong
singlet scalars, so that only Nambu-Goldstone (NGB)
: CP problem [3].
v and pseudo-Nambu-Goldstone (PNGB) degrees of free-
i dom remain in the EFT. We first discuss the model and the Yukawa operators
X
necessary to seed the quark and lepton mass matrices.
The small charged-lepton mass ratios, and the small
r
a up-type and down-type quark mass ratios and Cabibbo- We then describe the approximate global symmetries of
the EFT, broken explicitly by the family gauge interac-
Kobayashi-Maskawa(CKM)mixinganglesarethencom-
tions and the Yukawa operators. We discuss the vac-
putedperturbativelyinthefamilygaugecoupling. Small,
uum structure of the EFT, enumerating the NGB’s and
hierarchical neutrino masses, and large leptonic mixing
PNGB’s, and then classify the fermion mass matrices
angles are naturally accommodated at zeroth order in
that emerge from the Yukawa operators. We conclude
the family gauge coupling, although the specific values
with a discussion of the U(1) global symmetry of the
of the mixing angles are not predicted [2]. Imposing a
EFT, broken explicitly by QCD anomalies, and leading
theconstraintsfromthemeasuredsolarandatmospheric
to a potential axion [4].
mass differences and mixing angles restricts the parame-
ters describing the vacuum symmetry structure and can
relatesomeoftheotherwisefreeparametersinthequark
and charged-lepton mass-matrix estimates. One small THE MODEL
leptonic mixing angle then emerges, and is predicted to
lie within the reach of planned experiments. The model of Refs.[1] [2] consists ofthe three families
ToclassifytheYukawaoperatorsoftheEFT,wefound of SM fermions, together with two additional fermions,
it helpful in Refs. [1] [2] to make use of a discrete, Z3, χ and χc, also coming in three families, required to ex-
symmetry. Here we dispense with the Z3 and show that plain the up-type quark mass ratios. Each of the (left-
a complete classification scheme is provided through the handed, chiral) fermion fields, q,uc,dc,χ,χc,l,ec, trans-
2
forms as a 3 under a family SU(3)1 symmetry. Two SU(3)1 SU(3)2 SU(3)c SU(2)L U(1)Y
complex, symmetric-tensor fields S and Σ (¯6’s) are em- q 3 1 3 2 1
6
ployed to seed the spontaneous breaking of the SU(3)1. uc 3 1 ¯3 1 −2
These fields constitute the “visible” sector of the model. 3
dc 3 1 ¯3 1 1
Withelectroweaksymmetrybreakingdescribedbyasin- 3
χ 3 1 3 1 2
gleHiggs-doubletfield,someadditionalmechanismisre- 3
quiredtostabilizetheHiggsmass. Thisproblemwasnot χc 3 1 ¯3 1 −32
addressedinRefs.[1][2], andwillnotbe addressedhere. ℓ 3 1 1 2 −21
c
e 3 1 1 1 1
In order to compute the small quark mass ratios h 1 1 1 2 −1
2
md/mb, ms/mb, mu/mt, mc/mt, and the CKM mix- S ¯6 1 1 1 0
ing angles radiatively in the family gauge interaction, Σ ¯6 1 1 1 0
these quantities must vanish in its absence. To this end,
H 1 ¯6 1 1 0
a “hidden sector” is introduced transforming according
to its own SU(3)2. The SU(3)F family gauge interac-
TABLE I: Field content and symmetries of the model. All
tion then arises from gauging the diagonal subgroup of
fermionsareLHchiralfields. ThesymbolsS,Σ,andHdenote
SU(3)1×SU(3)2. SM-singlet scalar fields.
ThefamilybreakingscaleF istakentobelargeenough
to suppress flavor-changing neutral currents, and the YUKAWA OPERATORS OF THE EFT, AND U(1)
familygaugecouplinggisweakenoughsothatthegauge- SYMMETRIES
boson masses, of order gF, are small compared to the
cut-off MF = 4πF of the EFT. Their effects can there- Dominant Yukawa Operators
fore be computed perturbatively within the EFT. The
vanishing of the above mass ratios and the CKM angles
In the absence of the Yukawa operators,there exists a
in the absence of the family gauge interaction follows
U(1) global symmetry for each of the 11 complex fields
from the symmetries and vacuum structure in the vis-
of Table I. A minimal set of Yukawa operators required
ible sector. These symmetries are then broken in the
to seed most features of the quark- and charged-lepton
hidden sector, with the breaking communicated to the
mass matrices is given by
visible sector through the gauge interactions, leading to
nonzero, calculable values for the mass ratios and CKM qhSdc qh˜Sχc
angles. −LY = yd +y1 +y2χSuc+y3χΣχc
F F
ℓhSec
The matter-field content of the EFT is summarized in +ye +h.c.. (1)
F
Table I. The hidden sector is described by a single com-
plex,symmetrictensorfieldH,transformingasa¯6under The dimensionless coupling constants, fit to experiment,
SU(3)2. Note that no SM-singlet neutrinos are included range in size from O(10−3) to O(1), with electroweak
in the EFT. If they exist, they are taken to have masses symmetry breaking arising from the Higgs VEV v ≃250
abovethe cutoffM , andhavebeenintegratedout. The GeV. The VEV’s of S and Σ are of order F. (The H
F
EFT includes the fermion fields, the NGB and PNGB field,sofarnotdirectlycoupledtothevisiblesector,also
components of S, Σ and H, the family gauge fields, and developsaVEVoforderF.) Thefirstandlasttermsseed
SM gauge fields. thelargestelementsofthedown-typeandcharged-lepton
massmatrices. Theotherthreeterms arerequiredtoset
TheSU(3) familygaugeinteractionis,sofar,anoma- up a (“see-saw”) mass-generating mechanism in the up-
F
lous,requiringtheexistenceofadditionalheavyfermions type sector [1]. All these operators are dimension-3 or 4
to remove the anomalies. An example is a set of three in the fields with SM quantum numbers.
SM-singlet fermions, each transforming as a ¯6 under Thephenomenologicalconsequencesoftheseoperators
SU(3)2. With these “hidden-sector” fermions coupled were analyzed in Refs. [1] [2]. There are many other
to H, they all become massive when H develops its Yukawa operators allowed by the SM symmetries and
symmetry-breaking VEV of order F. If their Yukawa theSU(3)F gaugesymmetry,especiallysincetheSU(3)F
couplings are strong (O(4π)), then the masses will be symmetryis realizednonlinearlyinthe scalar(S, Σ,and
O(M = 4πF), and they will not be part of the EFT. H) sectors. In order to justify using only these opera-
F
Whenintegratedout,theygenerateanappropriateWess- tors we will make use of the U(1) symmetries that are
Zumino-Witten (WZW) term at energies below M [5]. naturally part of the model.
F
It must be included in the EFT, but it does not affect The 5 operators of L break 5 of the 10 U(1) symme-
Y
the mass estimates of Refs. [1] [2] to leading order. triesassociatedwiththevisible-sectorfieldsofTableI. In
3
q uc dc χ χc ℓ ec h S Σ H the correct order of magnitude for the neutrino masses.
′ ′ ′
Thus y is of order y and y or smaller, providing F is
U(1)a 0 0 2 1 0 -11 13 -1 -1 -1 20 ν u e
oforder1011 GeVorsmaller. (The chargeassignmentof
U(1)b 1 0 0 0 -2 -1 2 -1 0 2 0
H under U(1) is chosen so that the third operator pre-
b
serves this symmetry, even though it is already broken
TABLE II: Two linearly independent U(1)’s, in addition to
U(1)B and U(1)Y, left unbrokenby the operators of LY and by the the first two operators.)
byanomaliesgeneratedbySU(3)F familygaugeinteractions. With H charged under U(1)a, the additional, heavy,
Thesmall operators of L′Y breakU(1)b leaving U(1)a unbro- hidden-sectorfermionsrequiredtoremoveSU(3)F gauge
ken. anomalies may also carry U(1) charge. An example is
a
the set of 3 heavy ¯6’s coupled to H, discussed above. In
orderthatthe globalU(1) notbe anomalousdue to the
addition,onecombination,whichcanbetakentobe lep- a
SU(3) gaugeinteraction,theU(1) chargeassignments
tonnumber,U(1) ,isrenderedanomalousbytheSU(3) F a
ℓ F of all the fermions will then have to be adjusted relative
family gauge interaction. Of the remaining 4 U(1)’s, 2
tothevaluesinTableII,butnothinginthepresentpaper
are U(1) corresponding to baryon number and U(1)
B Y depends on these specific values.
corresponding electroweak hypercharge. The final 2 are
TheU(1) symmetry,unbrokenbytheoperatorsofL
denoted U(1)a and U(1)b. We exhibit in Table II one and L′ oraby SU(3) -generated anomalies, is spontaY-
possible choice for the charge assignments of each of the Y F
neouslybrokenatthe scaleF andisrenderedanomalous
complex fields under U(1) ×U(1) . The reason for the
a b by QCD interactions. If it is respected by all the oper-
charge assignments of H will be made clear shortly.
ators of the EFT, it is a candidate for a Peccei-Quinn
We will show using the vacuum structure of the EFT
symmetry. We return to this topic after discussing the
that the operators of L provide the required dominant
Y vacuum structure of the EFT and its consequences for
seeding of the quark and charged-lepton mass matri-
the fermion mass matrices.
ces, that is, that other Yukawa operators respecting the
U(1) × U(1) symmetry provide no new mass-matrix
a b
structure. VACUUM STRUCTURE
InRefs.[1][2],weassumedthattheglobalsymmetries
Smaller, Symmetry-Breaking Yukawa Operators
SU(3)1×SU(3)2 are broken spontaneously at the scale
F by VEV’s of the scalar fields S, Σ and H. The VEV’s
The 5 operators of L allow us to fit the quark mass
Y were taken to be
ratios and CKM mixing angles, except for the smallest
CKM angle, θq . Also, there is nothing in the model 0 0 0
13
so far to generate charged-lepton mass ratios that differ hSi = F 0 0 0
from the down-type quark mass ratios. Finally, there is
0 0 s
no mechanism so far to provide the very small neutrino
masses and leptonic mixing angles. 0 0 0
Each of these problems can be addressed by including hΣi = F 0 σ 0
a set of “smaller” operators that explicitly break one or 0 0 0
more of the symmetries preservedso far. A minimal set,
employed in Ref. [2], is given by b21 b2 b3
hHi = F b2 a1 a2 , (3)
−L′ = y′ qh˜Σuc +y′ℓhΣec + yν′ ℓh˜Hh˜ℓ +h.c..(2) b3 a2 a3
Y u F e F 2 F2
where |s|, |σ| and the |a | are O(1), while the |b | are of
i i
The first two operators each break U(1) but preserve orderthe Cabibbo angleθq . This pattern,which wasat
b 12
U(1) . The phenomenological use of these operators re- the coreofthe phenomenologyofRefs.[1][2],is adopted
a
quires that y′ and y′ be of order 10−4. here. We next discuss the broken symmetries associated
u e
The thirdoperator,dimension-5 in the SM fields, cou- with this VEV pattern, and the associated NGB’s and
ples the hidden and visible sectors directly. With the PNGB’s.
chargeassignmentforH underU(1) ,showninTableII, We first neglect the family gauge coupling and the
a
′
this operator preserves this symmetry. It breaks U(1) small operators of L . The visible-sector scalars S and
H Y
to a combination of U(1) and U(1) , which is anoma- Σ are taken to couple strongly in the underlying theory,
H ℓ
lous due to the SU(3)F family gauge interactions. It transformingaccordingtoasingleSU(3)1 symmetry,to-
also breaks SU(3)1×SU(3)2 to the diagonal subgroup, gether with U(1)a×U(1)b. The underlying dynamics is
asdoes the family gaugeinteraction. With v ≃250Gev, assumedtotriggerthespontaneousbreakingofthissym-
we have y′ ≃ F/(1015 GeV) if the operator is to give metry in the above pattern, with hSi and hΣi together
ν
4
leaving two unbroken U(1) symmetries of the vacuum to make massive the associated NGB.
and producing 8 NGB’s. The vacuum-symmetry gener-
ators are linear combinations of those of U(1) , U(1) , Consider now the bilinear part of the effective la-
a b
andthe two diagonalgeneratorsof SU(3)1. The hidden- grangian involving the (P)NGB’s associated with the
sectorVEV,hHi,produces9NGB’s. Therearetherefore spontaneous breaking of U(1)a × U(1)b × U(1)H. Af-
a total of 17 NGB’s, with the 9 arising from the hidden ter proper diagonalization and normalization of the ki-
sector decoupled so-far from the visible sector. netic part, and neglecting QCD anomalies, there will
In Fig. 1, we show the symmetry breaking pattern of exist a rank-two mass matrix generated at the multi-
the model, with the first line corresponding to the limit loop level. Its entries are proportional to F2 with co-
inwhichthegaugecouplingsandtheoperatorsofL′Y are efficients determined by the small parametersin L′Y and
set to zero. The two unbroken U(1) symmetries of the the family gauge coupling g. The 2 resulting PNGB’s
visible-sector vacuum are designated U(1) and U(1) . (as well as the 6 PNGB’s with masses of order g2F/4π)
va vb
In the next section (Eq. 4), we exhibit these symmetries have non-diagonal couplings to the SM fermions (they
explicitly andusethem tostudy the allowedYukawaop- arefamilons), while the masslessNGB has diagonalcou-
erators. plings. In addition, since they all couple through the
′
thirdoperatorofL totheMajoranamassmatrixofthe
The underlying physics in the visible and hidden sec- Y
neutrinos; they are Majorons. Clearly, F must be taken
tors, leading to these patterns, produces a set of nonlin-
large enough so that the 8 PNGB’s lie beyond current
ear constraints in the EFT, reducing the 24 degrees of
experimental reach.
freedom in S and Σ to the 8 NGB’s of the visible sector,
and the 12 degrees of freedom in H to the 9 NGB’s of
The weak coupling of the visible and hidden sectors
the hidden sector. They are described in the Appendix.
bythe familygaugeinteractionandthethirdinteraction
We next include the family gauge coupling and the ′
of L means that hHi cannot in generalbe diagonalized
small operators of L′ . As described above, the fam- Y
Y in the frame in which hSi and hΣi are diagonal. Its ori-
ily gaugeinteractionmakes anomalous one visible-sector
entation, described by 3 mixing angles, is a dynamical,
U(1), taken to be U(1) . Since the family gauge in-
ℓ vacuumalignmentquestion. Themixinganglesenterthe
′
teraction and the third operator of L explicitly break
Y neutrinomassmatrixdirectlythroughthethirdoperator
SU(3)1×SU(3)2 →SU(3)F, since the third operator of ofL′ andthey enterthequarkandcharged-leptonmass
L′ explicitlybreaksU(1) ,andsincethefirsttwobreak Y
Y H matrices through the SU(3) radiative corrections since
U(1) , the full symmetry of the EFT, excluding the SM F
b the gauge-bosonmass matrix depends on hHi.
interactions, is SU(3) ×U(1) , along with U(1) and
F a B
U(1) . With the visible and hidden sectors now cou-
Y The effective potential determining this orientation is
′
pled by the family-gauge and L interactions, the spon-
Y generatedfromthe weakcouplingsofthe EFTaswellas
taneous breaking of the SU(3) × U(1) symmetry is
F a other possible weak interactions linking the two sectors
complete, producing9 NGB’s, ofwhich8are eaten. The
in the underlying theory. To account for the CKM and
remaining NGB is the candidate axion. This count is
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing an-
described in the last line of Fig. 1.
gles, all the off-diagonalentries of hHi must be non-zero
Of the original 17 NGB’s, therefore, 8 have become in the basis in which hSi and hΣi are diagonal. Equiva-
PNGB’s. We discuss their masses by first noting that if
lently, the breaking pattern must not leave any residual
onlythefamilygaugeinteractionisincluded(secondline Z2 symmetries. (The local operators corresponding to
of Fig. 1), 2 of the PNGB’s, corresponding to the spon- the masses of the PNGB’s described above are a part of
taneous breaking of U(1)b and U(1)H, remain massless. the full effective potential.)
The other 6 develop masses at one-loop in the family
gauge interaction, of order g2F/4π. Finally, we note that CP-violating phases are natu-
ThereremaintwoPNGB’sassociatedwiththeexplicit rally present in the model. They can emerge from the
′
breakingof U(1) and U(1) by the operatorsofL . In underlying theory and are then present directly in the
b H Y
the case of U(1) , a combination of U(1) and U(1) is Yukawa couplings of the EFT. It can be seen that they
H H ℓ
′
still preserved by the operators of L (and L ). How- cannot in general be removed from all allowed opera-
Y Y
ever, this (SU(3) -anomalous) symmetry is not an es- tors by phase rotations of the fields. Phases can also
F
sential ingredient in the quark and charged-lepton phe- arise spontaneously through the weak effective potential
nomenology,and may be brokenby additional operators coupling the visible and hidden sectors. The combina-
which preserve U(1) , but are small enough not to dis- tion ofall these phases will determine the measuredCP-
a
turbsignificantlytheneutrinophenomenologyofRef.[2]. violating phase in the CKM matrix, and the predicted
We take these operators to be present generically. The Dirac and Majoranaphases in the leptonic (PMNS) ma-
explicit breaking of U(1) by the first two operators of trix. If the spontaneously generated phases and those
b
′
L mustbeaccompaniedbyacoupling(thefamilygauge present in the operators of the EFT are O(1), the same
Y
coupling)betweenthevisibleandhiddensectorsinorder will be true of the measured and predicted phases.
5
FIG. 1: The symmetry breaking pattern of the model. Horizontal arrows represent the spontaneous symmetry breaking of
Eq.3. ThenumbersovereacharrowcountthenumberofNGB’sgenerated. Thefirstrowcorrespondstothelimitinwhichthe
gauge couplings and thesmall symmetry breakingoperators of L′Y are set tozero, in which case 17NGB’s are produced. The
vacuum symmetries U(1)va and U(1)vb are shown in Eq. 4. In the second row, the SU(3)F gauge coupling is turned on, and
as a result only the diagonal combination of SU(3)1×SU(3)2 is preserved in the EFT. At the same time, SU(3)F anomalies
explicitly break lepton number U(1)ℓ. There are 11 NGB’s, and 6 PNGB’s with masses ∼ g2F/4π. In the third row, the
operators in L′Y are turned on, explicitly breaking U(1)b (by yu′ and ye′) and U(1)H (by yν′). This gives mass to two of the11
NGB’s. Ofthese,8disappearthroughtheHiggsmechanism, andoneisthecandidateaxion. Fiveadditional global U(1)’s are
explicitly broken by LY, U(1)B is unbroken,and U(1)Y is broken spontaneously at themuch lower electroweak scale v. They
are not included here.
YUKAWA OPERATORS AND FERMION MASS develop VEV’s. If attention is restricted to operators
MATRICES with only one power of S or Σ, as in L , there is no
Y
such quantity. But it easy to write down operators of
In this section we examine the phenomenological ef- this type if more powers of S and Σ are admitted. A
fects of all admissible Yukawa operators,including those simple example is qh˜(S ×S)×Σ∗uc/ F3, where S ×S
not in L and L′ , making use of the EFT symmetries represents the 6 in the product of the two ¯6’s. Clearly
Y Y
and the vacuum symmetries. We show that other al- this operator vanishes in the vacuum of Eq. 3, but what
lowedoperatorswith the U(1) ×U(1) symmetryofL about the general class of such operators?
a b Y
(or which break U(1) by small amounts as in L′ ), give To answer this question, we note that under the
b Y
no qualitatively new contributions to the mass matrices U(1)va ×U(1)vb vacuum symmetry of the visible sector
at zeroth order in the family gauge interaction. (Fig. 1), the fields transform as
Theeffectoftheadditionaladmissibleoperatorswedid
not include is therefore at most a redefinition of some of q → diag{eiθa,e−21iθa+2iθb,e−21iθa+iθb}q
the couplings that seed the mass matrices, and hence, uc → diag{eiθa−iθb,e−21iθa+iθb,e−12iθa}uc
evenatloop-orderinthefamilygaugeinteractions,leave
′ dc → diag{e3iθa−iθb,e23iθa+iθb,e23iθa}dc
thephenomenologicalsuccessofL andL undisturbed.
Y Y
χ → diag{e2iθa−iθb,e21iθa+iθb,e21iθa}χ
χc → diag{eiθa−3iθb,e−21iθa−iθb,e−21iθa−2iθb}χc
Dominant Yukawa Operators
ℓ → diag{e−10iθa−2iθb,e−223iθa,e−223iθa−iθb}ℓ
We firstdiscussYukawaoperatorsrespecting the sym- ec → diag{e14iθa+iθb,e225iθa+3iθb,e225iθa+2iθb}ec
metries of LY: U(1)a × U(1)b × SU(3)1. The family h → e−iθa−iθbh
gauge interaction is initially neglected, it’s effects to be S → diag{e−32iθa+iθb,e−iθb,1}Sdiag{e−32iθa+iθb,e−iθb,1}
includedperturbatively. Theoperatorsofinterestarebi-
linear in the fermion fields and include up to one power Σ → diag{e−32iθa+2iθb,1,eiθb}Σdiag{e−32iθa+2iθb,1,eiθb},
of the Higgs-doublet field h. Any number of S and Σ (4)
fields may be included since they aresubject to the non-
linearconstraintsthatfreezeoutallbutNGBandPNGB where θ and θ are the arbitrary parameters associated
a b
degrees of freedom. with the symmetries U(1) and U(1) .
a b
We begin with operatorswith S’s and Σ’s sandwiched The most general mass operator involving q and uc,
betweenqh˜ anduc,thatis,operatorspotentiallycapable emerging from the VEV’s of S and Σ, is of the form
ofdirectly giving up-type quark masseswhen the scalars qh˜diag{y ,y ,y }uc. Inorderthatitbeinvariantun-
u1 u2 u3
6
der U(1)va ×U(1)vb, we must have together break U(1)b, U(1)H, and SU(3)1 ×SU(3)2 →
SU(3) , leaving the global U(1) symmetry.
F a
diag{yu1,yu2,yu3} = diag{e3iθa,e4iθb,e2iθb} With U(1)a as the only global U(1) symmetry, many
×diag{y ,y ,y }. (5) other Yukawa operators, comparably small compared to
u1 u2 u3 ′
those of L and breaking U(1) , are allowed. Note that
Y b
The only solution is y = y = y = 0. Thus there there is no distinction between S and Σ at this level
u1 u2 u3
is no Yukawa operator involving q and uc giving a non- since they have the same U(1) charges. The question
a
vanishing mass matrix. is whether any of these operators can give rise in the
One can show more generally that the mass matrices vacuum of Eq. 3 to fermion mass matrices that disturb
generated by the operators of L are the most general the successful phenomenology based on the operators of
Y
′
fermionmassmatricesallowedbyU(1) ×U(1) . Con- L .
va vb Y
sider, for example, a down-type operator with VEV’s of Toseethatthisdoesnothappen,notethattheresidual
S and Σ sandwiched between qh and dc. It must be of symmetry of the vacuum (present before gauge interac-
the form qhdiag{y ,y ,y }dc. In order that it be in- tionslinkthevisibleandhiddensectors),withU(1) now
d1 d2 d3 b
variant, we must have explicitly broken, is just U(1) . It can be read off from
va
Eq.4bysettingθ =0. ThissingleU(1)vacuumsymme-
b
diag{yd1,yd2,yd3} = diag{e3iθa−2iθb,e2iθb,1} try allows nonzero values for only the 22 and 33 entries
×diag{y ,y ,y }, (6) of both hSi and hΣi. In the absence of the family gauge
d1 d2 d3
interaction, there can therefore be no masses present for
Thus the only possible non-vanishing entry is the 33 ele- the first-family quarks and charged leptons. This is an
ment, which is generated by the operator y qhSdc/F of essential role of the U(1) symmetry. Its spontaneous
d a
L . Other operators may be written down that do the breakingin the hidden sectorand transmittalto the vis-
Y
same thing, for example qh(S×Σ)×Σ∗dc/ F3 with its iblesectorbythefamilygaugeinteractionthenproduces
own(complex) coefficient. It, too, has only a 33 entry in the small, first-family masses.
the vacuum of Eq. 3. The contributions arising from L′ to the 22 and 33
Y
AsimilarargumentappliestoallYukawaoperatorsre- entriesofthequarkandcharged-leptonmassmatricesare
spectingthesymmetryU(1)a×U(1)b×SU(3)1ofthevis- O(10−4). They produce small but important corrections
iblesector. Alloperatorsthathavenon-vanishingVEV’s in the quark and lepton phenomenology [2].
in the vacuum of Eq. 3, with it’s symmetry (Eq. 4), give
rise to the same mass matrices as those arising from the
operators of LY. For the charged-lepton sector there is U(1)a AND THE QCD AXION
onlya33entry. Fortheup-typesector,theentrieslaythe
groundwork for the see-saw explanation of the masses. We have shown that a minimal EFT, capable of ac-
To summarize, we have included in LY a minimal set counting for the quark and lepton masses, mixing an-
ofYukawaoperatorsnecessarytoexplain,alongwiththe gles, and phases [1] [2], naturally includes one global
SU(3)F gaugeinteraction,mostfeaturesofthequarkand U(1)symmetry,U(1)a ofTableII. Thebreakingpattern
charged-lepton mass matrices. The two U(1) vacuum leaves an associated vacuum symmetry U(1) (Eq. 4)
va
symmetries imply that the quark and charged-lepton in the visible sector, protecting the first-family quarks
massmatricesgeneratedbytheoperatorsofLY arecom- and chargedleptons fromgaining mass in the absence of
pletely general. Perturbation theory in the family gauge the SU(3) family gauge interaction. The breaking of
F
interaction then couples the visible and hidden sectors, U(1) inthe hiddensectoratscaleF, communicatedto
va
communicating the breaking of the two U(1) vacuum thequarksandleptonsbytheSU(3) gaugeinteraction,
F
symmetries to the visible sector, and leading to non- leads to finite first-family masses, and produces a so-far
vanishing values for up-type and down-type quark mass massless NGB.
ratios,CKMmixing angles,andcharged-leptonmass ra- Suppose next that the U(1) symmetry is classically
a
tios. They are finite and calculable within the EFT. exact,respectedbyalloperatorsoftheEFT.Then,since
it is anomalous due to QCD interactions, the NGB is
a candidate for a QCD axion [6]. The axion field is a
Smaller, Symmetry-Violating Yukawa Operators linear combination of NGB fields in S, Σ, and H, the
combination that remains massless and survives below
To incorporate necessary small corrections to the thescaleswherethefamilygaugebosonsandthePNGB’s
quark- and charged-lepton mass matrices, and to gen- decouple. (Thisisalsobelowthescalewheretheχandχc
erate the entire, small mass matrix of the neutrinos, the fieldshavebeenintegratedout,havinggeneratedtheup-
′
additionalsmalloperatorsof L , arerequired. The cou- type quark masses.) The linear combination is dictated
Y
plingsy′ andy′ areO(10−4),andy′ isnolargerthanthis by ratios of the dimensionless parametersthat appear in
e u ν
if F is no larger than about 1011 GeV. These operators the VEV’s of Eq. 3.
7
The axion couples to visible matter through the oper- symmetriesoftheEFTtoclassifythequarkandcharged-
′
ators of L and L . With the family gauge corrections lepton masses that emerge. CP-violating phases, which
Y Y
included, these operators lead to the observed masses of lead to the CKM phase as well as Dirac and Majorana
all the quarks and leptons. Thus, in the effective theory phasesintheleptonicPMNSmatrix,arisespontaneously
at scales low enough so that only the SM fields and the within the EFT, and also enter the parameters of the
axion survive, the axion couples to all the quarks and EFT directly from the underlying physics.
leptons with coupling strength given by m /F , where The U(1) symmetry is unbroken by any of the phe-
f a a
m is the fermion mass and F is related to F by ratios nomenologically necessary interactions of the EFT, ex-
f a
the dimensionless parameters that appear in the VEV’s ceptforQCDanomalies. Thespontaneousbreakingpat-
ofEq.3. Since theyareallexpectedtobe roughlyofthe tern preserves an associated vacuum symmetry, U(1) ,
va
sameorder,F isofthesameorderasF. Sincetheaxion in the visible sector, enforcing the masslessness of the
a
couplestoneutrinomassthroughtheMajoranaoperator first-family quarksandchargedleptons in the absenceof
′
in L , it is also a Majoron. the family gauge interaction. The U(1) symmetry is
Y va
It is not our purpose to discuss the phenomenology of broken in the hidden sector at the family-breaking scale
this axion candidate here, except to observe that with F, with the breaking communicated to the standard-
F in the allowed window 109 <∼ F <∼ 1012 GeV [7, 8], model fields by the family gauge interaction.
corresponding to a mass range 10−3 >∼ma >∼10−6 eV, it IftheU(1)a symmetryistakentobeexactintheEFT
evades all axion and Majoron searches to date. except for QCD anomalies (a Peccei-Quinn symmetry),
The U(1) symmetry is a natural feature of the EFT andif F istakentolie in the allowedwindow 109 <F <
a
operators required to compute quark and lepton mass 1012 GeV, then the associated PNGB is a viable axion,
matrices,andiftakentobeexactitleadstoaviableQCD coupling to all the particles of the standardmodel. This
axion. But the imbedding ofthis EFT in a largerframe- conclusion relies on the large hierarchy between F and
workcouldingeneralleadtohigher-dimensionoperators the electroweak scale v. Also, it is not clear whether the
thatexplicitly breakU(1)a andgivecontributionsto the U(1)a symmetrysurvivesthe imbedding ofthe EFT ina
axion potential that swamp the QCD contribution [9]. larger framework.
SUMMARY APPENDIX - NONLINEAR CONSTRAINTS
We have explored an effective field theory (EFT) We summarize here the nonlinear constraints that
frameworkproposedrecently for the generationof quark must emerge from the underlying dynamics in the vis-
andleptonmassmatrices [1][2]. AnSU(3)familygauge ible sector and the hidden sector, corresponding to the
symmetry,brokenspontaneouslyat a highscaleF, com- VEVpatternofEq.3andreducingthedegree-of-freedom
municates symmetry breaking from a hidden sector to count to only the NGB’s. The nonlinear constraints for
the visible-sector standard model fields. S are
To classify the Yukawa operators that seed the mass Tr[SS∗] = s2F2 (7)
matrices, we have employed the set of global U(1) sym-
∗
Tr[(S×S)(S×S) ] = 0. (8)
metries that are naturally part of the EFT. The dom-
inant required operators preserve two such symmetries,
With S written in the form
U(1) andU(1) ,inadditiontobaryonnumberandelec-
a b
troweak hypercharge. A set of smaller operators, neces- s11 s12 s13
sarytogeneratetheneutrinomassmatrixandtoprovide S = s12 s22 s23 , (9)
small corrections to the quark and charged lepton mass s13 s23 s33
matrices,preserveonlyU(1) alongwithbaryonnumber
a
and electroweak hypercharge. wherethe the sij arecomplexfields, Eq.7givesonecon-
We have described the vacuum structure of the EFT, straint for these 12 real fields.
enumerating the Nambu-Goldstone bosons (NGB’s) and Eq. 8 can be written in the form
pseudo-Nambu Goldstone bosons (PNGB’s), as deter- ∗
Tr[(S×S)(S×S) ]
mined by the symmetry-breaking interactions that link
the visible and hidden sectors. The PNGB’s that gain = |s11s22−s212|2+|s11s33−s213|2+|s22s33−s223|2
mass because of the family gauge coupling and the +2|s12s13−s11s23|2+2|s12s23−s22s13|2
small symmetry-breaking Yukawa operators, couple off- +2|s13s23−s33s12|2
diagonallyinfamilyspace(theyarefamilons),andcouple
= 0. (10)
to the Majorana mass matrix of the neutrinos (they are
Majorons). Eachof the absolute values must vanish, leading to a set
We have used the vacuum structure together with the of three, independent complex equations (6 constraints
8
in all) . They can be taken to be Tr[(H ×H)(H ×H)∗] = O(1)F4
|detH| = O(b2)F3, (15)
s11s22 =s212 (11)
s11s33 =s213 (12)
s22s33 =s223 (13) reducing the 12 degrees of freedom in H to the 9 NGB’s
of the hidden sector. There are now a total of 17 NGB’s
The same constraints apply to the elements of Σ.
Once the visible and hidden sectors are linked by the
The nonlinear constraint coupling S and Σ is:
′
family gauge interaction and the third operator of L ,
Y
Tr[SΣ∗S∗Σ]=0. all but 9 of these become PNGB’s.
It can be written in the form
|s12σ12s23σ23+s12σ12s13σ13+s23σ23s13σ13|2 This work was partially supported by Department of
×|s12σ1∗2s23σ2∗3+s12σ1∗2s13σ1∗3+s23σ2∗3s13σ1∗3|2 Energy grants DE-FG02-92ER-40704 (T.A. and Y.B.)
/|s12s13s23σ12σ13σ23|2 and DE-FG02-96ER40956 (M.P.). We thank Michele
Frigerio, Walter Goldberger, Adam Martin, Robert
= 0, (14)
Shrock, and Witold Skiba for useful discussions.
where σ are the elements of Σ. This form, which is a
ij
setof2constraints,iswrittenmakinguseoftheseparate
nonlinearconstraintsonS andΣ. Thusthetotalnumber
of constraints is 16, reducing the 24 degrees of freedom
[1] T. Appelquist, Y. Bai and M. Piai, Phys. Lett. B 637,
in S and Σ to the 8 NGB’s of the visible sector.
245 (2006) [arXiv:hep-ph/0603104].
These constraints also lead to the VEV’s of Eq. 3. If [2] T. Appelquist, Y. Bai and M. Piai, Phys. Rev. D 74,
werotatehSiintodiagonalform,thenthenonlinearcon- 076001 (2006) [arXiv:hep-ph/0607174].
straintsonS allowonlyonenon-vanishingelementwhich [3] R.D.PecceiandH.R.Quinn,Phys.Rev.Lett.38,1440
we take to be the 33 element. The constraint coupling (1977).
[4] F.Wilczek,Phys.Rev.Lett.40,279(1978);S.Weinberg,
S and Σ then gives hσ33i = 0. But then the above con-
Phys. Rev.Lett. 40, 223 (1978).
straintequations onS, with s replacedby σ , demand
ij ij [5] J. Wess and B. Zumino, Phys. Lett. B 37, 95 (1971);
that hσ13i = hσ23i = 0. Then hΣi can be put into di- E. Witten,Nucl. Phys. B 223, 422 (1983);
agonalformby anSU(2)transformationleavinghSiun- [6] The possible existence of a Peccei-Quinn symmetry
touched. The constraints on Σ demand that only one and associated axion in the context of models with an
element be nonzero,which we take to be the 22element. SU(3)familysymmetrywasnotedbyZ.G.Berezhianiand
We note that each of the above constraints can M.Yu.Khlopov,YadernayaFizika(1990)V.51,PP.1157-
1170. [English translation: Sov.J.Nucl.Phys. (1990) V.
be derived from an appropriate potential providing a
51, PP. 739-746]; Z.Phys.C- Particles and Fields (1991),
phenomenological description of the underlying dynam-
V. 49, PP. 73-78.
ics [10]. We assume here that they emerge from the true
[7] W. M. Yao et al. [Particle Data Group], J. Phys. G 33,
underlying theory, the UV completion of our EFT. 1 (2006).
In a similar manner, the set of 3 nonlinear constraints [8] P. Sikivie, hep-ph/0509198, J. E. Kim, hep-ph/0612141
on the hidden-sector H is [9] M. Dine, arXiv:hep-ph/0011376.
[10] For the analysis of a potential with a single sextet field,
Tr[HH∗] = O(1)F2 see: L.F. Li, Phys.Rev.D 9, 1723 (1974).
