Dynamical Symmetry Breaking of Extended Gauge Symmetries

4 Pages·0.14 MB·English

by  Thomas Appelquist

Dynamical Symmetry Breaking of Extended Gauge Symmetries
Thomas Appelquist1 and Robert Shrock2
1 Department of Physics, Sloane Laboratory, Yale University, New Haven, CT 06520
C. N. Yang Institute for Theoretical Physics, State University of New York, Stony Brook, NY 11794
We construct asymptotically free gauge theories exhibiting dynamical breaking of the left-right,
strong-electroweak gauge group GLR = SU(3)c ×SU(2)L×SU(2)R×U(1)B−L, and its extension
to the Pati-Salam gauge group G422 = SU(4)PS × SU(2)L × SU(2)R. The models incorporate
technicolor for electroweak breaking, and extended technicolor for the breaking of GLR and G422
and the generation of fermion masses, including a seesaw mechanism for neutrino masses. These
models explain whyGLR and G422 break toSU(3)c×SU(2)L×U(1)Y, andwhythistakesplace at
a scale (∼103 TeV) which is large compared to theelectroweak scale.
3
12.60.Cn, 12.60.Nz, 14.60.Pq
0
0
2

n The standard model (SM) gauge group G = and hence (g /g )2 =3/2 at Λ , where Λ is the
a SM U PS PS PS
J SU(3)c SU(2)L U(1)Y has provided a successful de- breakingscaleoftheG422 group. Thismodelalsohasthe
× ×
scriptionofbothstrongandelectroweakinteractions. Al- appealthat it quantizes electric charge,since Q=T +
5 3L
1 though the standard model itself predicts zero neutrino T3R+ 2/3TPS,15 =T3L+T3R+(1/6)diag(1,1,1, 3).
−
masses, its fermion content can be augmented to acco- p
The conventional approach to the gauge symmetry
1 modatethecurrentevidenceforneutrinomassesandlep-
breakingofthesemodelsemployselementaryHiggsfields
v
ton mixing. But the origin of the electroweak symmetry
8 and arranges for a hierarchy of breaking scales by mak-
breaking(EWSB)is stillnotunderstood. Itmightoccur
0 ing the vacuum expectation values (vev’s) of the Higgs
1 viatheHiggsmechanism,asintheSM.Analternativeis that break G or G to G much larger than the
LR 422 SM
1 dynamical symmetry breaking (DSB) of the electroweak
Higgsvev’sthatbreakSU(2) U(1) U(1) [9,11].
0 symmetry, driven by a stronglycoupled, asymptotically- L× Y → em
This hierarchy is necessitated by the experimental lower
3
0 free, vectorial gauge interaction associated with an un- limits on the masses of a possible WR or Z′ [12]. An
broken gauge symmetry, denoted generically as techni-
/ interestingquestioniswhetheronecanconstructasymp-
h color (TC) [1]- [8].
totically free gauge theories containing the group G
p LR
Therehasalsolongbeeninterestinmodelswithgauge
- and/or G422 that exhibit dynamical breaking of all the
p groups larger than G . One such model has the gauge
SM gauge symmetries other than SU(3) and U(1) , that
e group [9] c em
naturally explain the hierarchy of breaking scales, and
h
: that yield requisite light neutrino masses. In this letter,
v GLR =SU(3)c SU(2)L SU(2)R U(1)B−L (1)
× × × we present such models.
i
X in which the fermions of each generation transform as Technicolor itself cannot provide a mechanism for all
ar (3,2,1)1/3,L, (3,1,2)1/3,R, (1,2,1)−1,L, and (1,1,2)−1,R. the breaking, because it is too weak at the the scale
Thegaugecouplingsaredefinedviathecovariantderiva- Λ or Λ and because the technifermion condensate
LR PS
tAivRe,µD−µ =i(g∂Uµ/−2)(iBg3T−cL·)AUc,µµ.−Iing2tLhTisLm·AodLe,lµt−heige2lReTctRric· hSFU¯F(2i)R=ahtF¯LthFeRsia+mheF¯sRcaFlLei(twootuhldedbiraegaoknbaolt(hveScUto(r2))gLraonudp
charge is given by the elegant relation Q=T3L+T3R+ SU(2)V). Of course, to explain quark and lepton mass
(B L)/2,whereB andLdenotebaryonand(total)lep- generationandincorporatethethreefamilies,technicolor
−
tonnumber. GLR wouldbreakatascaleΛLR wellabove hastobeenlargedtoanextendedtechnicolor(ETC)the-
the electroweak scale. ory [3]. Our models are ETC-type theories, with the
ThemodelbasedonGLR maybe furtherembeddedin breaking of GLR and G422 to GSM being driven by the
a model with gauge group [10] same interactions that break the ETC group and gener-
ate quark and lepton masses.
G =SU(4) SU(2) SU(2) . (2)
422 PS L R
× × Taking the technicolor gauge group to be SU(N ),
TC
Thismodelprovidesahigherdegreeofunificationsinceit the technifermions comprise an additional family, viz.,
combines U(1)B−L and SU(3)c (in a maximal subgroup) QL = DU L, LL = NE L, UR, DR, NR, ER trans-
in the Pati-Salam group SU(4)PS and hence relates gU forming(cid:0)ac(cid:1)cording to t(cid:0)he(cid:1)fundamental representation of
and g3. Denoting the generators of SU(4)PS as TPS,i, SU(NTC) and the usual representations of GSM (where
1 i 15, with TPS,15 = (2√6)−1diag(1,1,1, 3) and color and TC indices are suppressed). Vacuum align-
≤ ≤ −
settingU =A ,onehas(B L)/2= 2/3T , ment considerations yield the desired color- and charge-
µ PS,15,µ PS,15
−
p
1
conserving TC condensates [14]. To satisfy constraints Thus the fermions include a vectorlike set of quarks
from flavor-changingneutral-currentprocesses,the ETC andtechniquarksintherepresentations(5,1,3,2,1) ,
1/3,L
vector bosons that can mediate generation-changing (5,1,3,1,2) and leptons and technileptons in
1/3,R
transitions must have large masses. We envision that (5,1,1,2,1) , (5,1,1,1,2) , together with a set of
−1,L −1,R
these arise from self-breaking of an ETC gauge symme- G -singletfermionsin(¯5,1,1,1,1) ,(10,1,1,1,1) ,
LR 0,R 0,R
try,whichrequiresthatETCbeastronglycoupled,chiral and (10,2,1,1,1) [20]. The leptons and technileptons
0,R
gauge theory. The self-breaking occurs in stages, for ex- aredenotedLi,p,whereχ=L,R,1 i 5,andp=1,2.
χ ≤ ≤
ample at the three stages Λ 103 TeV, Λ 50 TeV, The G -singlets are denoted respectively , ψ ,
1 2 LR i,R ij,R
∼ ∼ N
and Λ3 ∼3 TeV, correspondingto the 3 standard-model and ζRij,α, where 1 ≤ i,j ≤ 5 are ETC indices and α,β
fermion generations. Hence NETC =NTC +3. are SU(2)HC indices. The models with GLR and G422
A particularly attractive choice for the technicolor share several features with the ETC model in [7].
group, used in the models studied here, is SU(2)TC, TheSU(5)ETC theoryisananomaly-free,chiralgauge
which thus entails NETC = 5. With Nf = 8 vectorially theory and, like the ETC and HC theories, is asymp-
coupled technifermions in the fundamental representa- totically free. There are no bilinear fermion operators
tion,studiessuggestthatthisSU(2)TC theorycouldhave invariant under G, and hence there are no bare fermion
an(approximate)infraredfixedpoint(IRFP)inthecon- mass terms. The SU(2) and SU(2) subsectors of
HC TC
fining phase with spontaneous chiralsymmetry breaking SU(5) are vectorial.
TC
[15,16]. This approximate IRFP produces a slowly run- To analyze the stages of symmetry breaking, we iden-
ning (“walking”)TC gaugecoupling,whichcanyieldre- tify plausible preferred condensation channels using a
alisticallylargequarkandchargedleptonmasses[5]. The generalized-most-attractive-channel (GMAC) approach
choiceNTC =2andthe walkingcanstronglyreduceTC that takes account of one or more strong gauge inter-
contributionstotheS parameter[8,17]. Furtheringredi- actions at each breaking scale, as well as the energy cost
ents may be needed to account for the top-quark mass. involved in producing gauge boson masses when gauge
In Ref. [18], we studied the generation of neutrino symmetries are broken. In this framework, an approx-
masses in an ETC model of this sort and showed that imate measure of the attractiveness of a channel R
1
×
lightneutrinomassesandleptonmixingcanbeproduced R R is ∆C = C (R )+C (R ) C (R ),
2 cond. 2 2 1 2 2 2 cond.
→ −
via a seesaw without any superheavy mass scales. Here where R denotes the representation under a relevant
j
we extend this model to the groups GLR and G422. gauge interaction and C2(R) is the quadratic Casimir.
WerecallthatΛTC isdeterminedbyusingtherelation As the energy decreases from some high value, the
m2W = (g2/4)(NcfQ2 + fL2) ≃ (g2/4)(Nc +1)fF2, where SU(5)ETC and SU(2)HC couplings increase. We envi-
ffoFr our13p0urGpeoVse.sInweQtCaDke, ffLπ =≃ 9f3QM≡eVfFa.ndTΛhiQsCgDives sstiornontgh[a1t6]attoEpr∼odΛucLeRco>∼nd1e0n3saTteioVn, iαnEtThCeicshasunffinecliently
≃ ∼
170 MeV, so that Λ /f 2; using this as a guide
QCD π ∼ (5,1,1,1,2) (¯5,1,1,1,1) (1,1,1,1,2) (5)
to technicolor, we infer ΛTC 260 GeV. The induced −1,R× 0,R→ −1
∼
fermionmassesinthe i’thgenerationaregivenbym with ∆C = 24/5, breaking G to SU(3) SU(2)
fi ∼ 2 LR c× L ×
gE2TCηiNTCΛ3TC/(4π2Mi2), where Mi ∼ gETCΛi is the U(1)Y. The associated condensate is hLiR,p TCNi,Ri,
massoftheETCgaugebosonsthatgainmassatscaleΛi where 1 i 5 is an SU(5)ETC index and p 1,2
≤ ≤ ∈ { }
and g is the running ETC gauge coupling evaluated is an SU(2) index. With no loss of generality, we use
ETC R
at this scale. The quantity ηi is a possible enhancement the initialSU(2)R invarianceto rotate the condensateto
factorincorporatingwalking,forwhichηi ∼Λi/fF [5,19]. the p = 1 component, LiR,p=1 ≡ niR, which is electrically
We first consider the standard-model extension based neutral and has weak hypercharge Y = 0; the conden-
onGLR. OurmodelfortheDSButilizesthegaugegroup sate is thus hniRTCNi,Ri so that the niR and Ni,R gain
dynamical masses Λ .
LR
G=SU(5) SU(2) G (3) ∼
ETC × HC × LR There exists a more attractive channel than (5) in a
where HC denotes hypercolor,a second strong gauge in- simpleMACanalysis: (10,1,1,1,)0,R (10,2,1,1,)0,R
× →
teraction which, together with ETC, triggers the requi- (1,2,1,1,)0, with ∆C2 = 36/5. But with the coupling
site sequential breaking pattern. The fermion content gHC also large at ΛLR, a sizeable energy price would
of this model is listed below; the numbers indicate the be incurred in this channel to generate the vector bo-
representations under SU(5)ETC SU(2)HC SU(3)c sonmassesassociatedwiththebreakingoftheSU(2)HC.
SU(2) SU(2) and the subscri×pt gives B ×L: × We assume here that this price is higher than the en-
L R
× − ergy advantage due to the greater attractiveness of the
(5,1,3,2,1)1/3,L , (5,1,3,1,2)1/3,R , channel (10,1,1,1,)0,R (10,2,1,1,)0,R (1,2,1,1,)0
× →
[21].
(5,1,1,2,1)−1,L , (5,1,1,1,2)−1,R , The condensation (5) generates masses
g g
2R 2u
(¯5,1,1,1,1)0,R , (10,1,1,1,1)0,R , (10,2,1,1,1)0,R . (4) mWR = 2 ΛLR mZ′ = 2 ΛLR , (6)
2
where g2u ≡ qg22R+gU2, for the WR±,µ = A±R,µ gauge i(t igs plaΛusi)bilseitnhcautrrΛed1b<∼y tΛheLRb,resaiknicnegaonf SeUne(r5g)y pr.ice
bosons and the linear combination ∼ ETC 1 ETC
The SU(5) SU(4) breaking entails the sep-
ETC ETC
→
Z′ = g2RA3,R,µ−gUUµ. (7) aration of the first generation of quarks and leptons
µ g from the components of SU(5) fermion fields with
2u ETC
indices 2 i 5. The further ETC gauge symme-
This leaves the orthogonalcombination ≤ ≤
try breaking occurs in stages, leading eventually to the
SU(2) subgroup of the original SU(5) group. We
g A +g U TC ETC
U 3,R,µ 2R µ
Bµ = (8) have identified two plausible sequences for this breaking
g
2u [7,18]. Both sequences yield a strongly coupled SU(2)
TC
as the weak hypercharge U(1) gauge boson, which is gaugeinteractionthatproducesaTCcondensate,break-
Y
massless at this stage. The hyperchargecoupling is then ing SU(2)L U(1)Y U(1)em [22].
× →
Dirac mass terms for the neutrinos are formed dy-
g g
g′ = 2R U . (9) namically, involving the left-handed neutrinos in the
g2u (5,1,1,2,1)−1,L, but not their respective right-handed
counterparts in the (5,1,1,1,2) . Instead, the right-
so that, with e−2 =g−2+(g′)−2 =g−2+g−2+g−2, the −1,R
2L 2L 2R U handed partners emerge from the (10,1,1,1,1)0,R (as
weak mixing angle is given by
ψ , j = 2,3). Thus there are only two right-handed
1j,R
neutrinos. In a model in which L is not gauged, it is a
g 2 g 2 −1
sin2θ = 1+ 2L + 2L (10) convention how one assigns the lepton number L to the
W
(cid:20) (cid:16)g2R(cid:17) (cid:16)gU (cid:17) (cid:21) SM-singlet fields. Here, L = 0 for the fields that are
singlets under G or G , since they are singlets un-
at the scale Λ . The experimental value of sin2θ at LR 422
LR W derU(1) andhaveB =0. Hence,theneutrinoDirac
B−L
M canbeaccommodatednaturally,forexamplewithall
Z masstermsviolateLby1unit. Therearealsolarger,Ma-
couplingsin(10)ofthesameorder(evenwithg =g )
2R 2L jorana masses generated for the ψ fields themselves;
ij,R
and with modest RG running from Λ to M .
LR Z the seesaw mechanism then leads to left-handed ∆L=2
For E <Λ , the fermioncontent ofthe effective the-
LR Majorana neutrino bilinears [23].
ory is
We next consider the extension of the standard model
gaugegrouptoG . Inthiscase,ourfullmodelisbased
(5,1,3,2) , (5,1,3,1) , (5,1,3,1) 422
1/3,L 4/3,R −2/3,R on the gauge group G = SU(5) SU(2) G
ETC HC 422
× ×
with fermion content
(5,1,1,2) , (5,1,1,1) ,
−1,L −2,R
(5,1,4,2,1) , (5,1,4,1,2) ,
L R
(10,1,1,1) , (10,2,1,1) , (11)
0,R 0,R
(¯5,1,1,1,1) , (10,1,1,1,1) , (10,2,1,1,1) . (13)
R R R
wheretheentriesrefertoSU(5) SU(2) SU(3)
ETC HC c
× × × Again, as E decreases from high values, the SU(5)
SU(2) and Y is a subscript. This is precisely the gauge ETC
L
and SU(2) couplings increase. At a scale Λ , the
group and fermion content of the ETC model that we HC PS
SU(5) coupling will be large enoughto produce con-
analyzed in Ref. [18] with a focus on the formation of ETC
densation in the channel
neutrinomasses. Wethereforesummarizethesubsequent
stagesofbreakingonlybriefly,drawingonresultsof[18]. (5,1,4,1,2) (¯5,1,1,1,1) (1,1,4,1,2) . (14)
R R
At a value E Λ 103 TeV comparable to Λ , × →
1 LR
∼ ∼ This breaks SU(4) SU(2) directly to SU(3)
a GMAC analysis suggests that there is condensation in PS × R c ×
U(1) . The value Λ 103 TeV satisfies phe-
the channel Y PS ∼
nomenological constraints, e.g. from the upper limit
(10,1,1,1)0,R×(10,1,1,1)0,R →(5,1,1,1)0 . (12) aognaiBnR(nKi LTC→ µ±,e∓an).d tTheheniassaoncdiated cognadinenmsaatesseiss
h R Ni,Ri R Ni,R
Thus, SU(5)ETC self-breaks to SU(4)ETC, producing ∼ ΛPS. The results (6)-(10) apply with the condition
masses g Λ for the nine gauge bosons in the coset (g /g )2 =3/2 at Λ .
∼ ETC 1 U PS PS
SU(5) /SU(4) . As at Λ , we assume that a FurtherbreakingatlowerscalesproceedsasintheG
ETC ETC LR LR
GMAC analysis favors this channel over the 10 10 model and as described in Ref. [18]. Dirac mass terms
×
channelinwhichSU(2) -breakinggaugebosonmasses for the neutrinos are formed from the (5,1,4,2,1) and
HC L
g Λ would have to be formed. Although the lat- the (10,1,1,1,1) , leadingto the sametypeofseesawas
∼ HC 1 R
ter channel is more attractive, a very large energy price in [18] and the G model.
LR
would have to be paid for the associated vector boson The experimental value of sin2θ can again be ac-
W
massgenerationfor sufficiently largeα >α . Also, commodated by (10), although this now necessarily re-
HC ETC
although (12) has the same ∆C -value (= 24/5) as (5), quires g < g at Λ . To see this, we evolve the
2 2R 2L PS
3
SM gauge couplings from µ = mZ to the EWSB scale andL.C.R.Wijewardhana,Phys.Rev.D35,774(1987);
Λ = 2−3/4G−1/2 = 174 GeV and then from Λ up Phys. Rev.D 36, 568 (1987).
toEΛW using dαF /dt = b α2/(2π)+O(α3)+...EwWhere [6] T. Appelquist and J. Terning, Phys. Lett. B315, 139
PS j − 0 j j (1993);T.Appelquist,J.Terning,L.C.R.Wijewardhana,
t = lnµ, α (g′)2/(4π), and ... denotes theoretical
1 ≡ Phys. Rev.Lett. 77, 1214 (1996); ibid.79, 2767 (1997).
uncertainties associatedwith mass thresholds. In the in-
[7] T.Appelquist,J.Terning,Phys.Rev.D50,2116(1994).
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from the t quark and relevant technifermions, so that (1999); ibid.60, 116007 (1999).
b(3) = 13/3, b(2) = 2/3, and b(1) = 10. The initial [9] R. N. Mohapatra and J. C. Pati, Phys. Rev. D 11, 566
v0alues at m a0re α (m ) = 0.1018, α−(m )−1 = 129, (1975); ibid. 11, 2558 (1975); R. N. Mohapatra and G.
Z 3 Z em Z
and (sin2θ ) (m )= 0.231 [13,17]. With Λ = 106 Senjanovi´c, ibid.,12, 1502 (1975); ibid.,23, 165 (1981).
GeV and tWheMcaSlculaZted values α =0.064, α P=S 0.032, [10] J. C. Pati and A.Salam, Phys. Rev.D 10, 275 (1974).
3 2L
[11] Supersymmetric versions are K. Babu, R. Mohapatra,
α = 0.012 at Λ , we find α (Λ ) 0.013 so that
1 PS 2R PS ≃ Phys.Lett.B518,269(2001); J.Pati,hep-ph/0106082.
g /g 0.64 at this scale.
2RIt m2Lay≃be possible to allow g = g at Λ , and [12] Current data implies that, for g2R ≃ g2L, mWR >∼ 800
still match (sin2θ ) , by furth2Rer expa2Lnding tPhSe (4D) GeV, with a similar lower bound on an mZ′ [13].
W exp. [13] http://pdg.lbl.gov.
gauge theory to one with, e.g., SU(4)PS SU(2)4 as in [14] M. Peskin, Nucl. Phys. B175, 197 (1980); J. Preskill,
×
[24] but with DSB; we are currently studying this [25]. ibid. 177, 21 (1981). The TC theory forms condensates
To summarize, we have constructed asymptotically hF¯Fi, where F = Ua,Da,E,N, but not, e.g., hU¯aEi,
free models with dynamical symmetry breaking of the hD¯aEi,hU¯aNi,hD¯aNi,hU¯aDai,hE¯Ni,or,forNTC =2,
extended gaugegroups G andG . These models in- hǫijFχi TCFχ′ji, χ = L,R. The excluded condensates
LR 422 would incur an energy price due to gauge boson mass
volvehigherunification,andG hastheappealofquan-
422 generationwhenthe(weaker)gaugesymmetriesarebro-
tizing electric charge. Our models naturally explain why
ken.
(i) G and G break to G and (ii) this breaking
LR 422 SM [15] A vectorial SU(N) theory with Nf massless fermions in
occurs at the scales ΛLR, ΛPS >> mW,Z. The models the fundamental representation is expected to exist in
incorporatetechnicolorfor electroweaksymmetry break- a confining phase with SχSB if N < N , where
f f,cond.
ing, and extended technicolor for fermion mass genera- N ≃(2/5)N(50N2−33)/(5N2−3) and in a non-
f,cond.
tion including a seesaw mechanism for the generation of abelian Coulomb phase if Nf,cond. < Nf < 11N/2. For
realistic neutrino masses. N =2, we haveNf,cond. ≃8.
[16] Intheapproximationofsingle-gauge-bosonexchange,the
Adifferentapproachappearstobeneededtoconstruct
critical coupling for condensation R1 ×R2 → Rcond. is
agrothueposryGwith=dySnUa(m5i)caolrbSrOea(1k0in)gbeocfatuhsee,garmanodngunoitfiheedr given by 32απ∆C2=1, where ∆C2 =[C2(R1)+C2(R2)−
GUT C2(Rcond.)]and C2(R) is the quadraticCasimir.
things, if the ETC group commuted with GGUT, then, [17] WenotethatglobalelectroweakfitsyieldingS andT are
with the standard fermion assignments in these GUT complicatedbytheNuTeVanomalyreportedinG.Zeller
groups,the quarksandchargedleptons wouldnottrans- et al., Phys. Rev.Lett. 88, 091802 (2002).
forminavectorialmannerunderG ,sothattheusual [18] T. Appelquist and R. Shrock, Phys. Lett. B 548, 204
ETC
ETCmechanismforthecorrespondingfermionmassgen- (2002) and to appear.
eration would not apply. [19] Here ηa = exp[ fΛFa(dµ/µ)γ(α(µ))], and in walking TC
This research was partially supported by the grants theoriestheanoRmalousdimensionγ ≃1soηa ≃Λa/fF.
[20] We write SM-singlet fields as right-handed.
DE-FG02-92ER-4074(T.A.),NSF-PHY-00-98527(R.S.).
[21] This problem is currently under study. The analysis is
more challenging than perturbative vacuum alignment
[14] since all the relevant couplings are strong.
[22] For a different approach to DSB of GLR using Nambu
Jona-Lasino-type four-fermion couplings, see E. Akhme-
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