Dynamical Symmetry Breaking of Extended Gauge Symmetries 1
Thomas Appelquist1 and Robert Shrock2
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)P S × 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 why GLR and G422 break to SU(3)c × SU(2)L × U(1)Y , and why this takes place at a scale (∼ 103 TeV) which is large compared to the electroweak scale. 12.60.Cn, 12.60.Nz, 14.60.Pq
and hence (gU /gP S )2 = 3/2 at ΛP S , where ΛP S is the breaking scale of the G422 group. This model also has the appealp that it quantizes electric charge, since Q = T3L + T3R + 2/3TP S,15 = T3L + T3R + (1/6)diag(1, 1, 1, −3).
The standard model (SM) gauge group GSM = SU(3)c × SU(2)L × U(1)Y has provided a successful description of both strong and electroweak interactions. Although the standard model itself predicts zero neutrino masses, its fermion content can be augmented to accomodate the current evidence for neutrino masses and lepton mixing. But the origin of the electroweak symmetry breaking (EWSB) is still not understood. It might occur via the Higgs mechanism, as in the SM. An alternative is dynamical symmetry breaking (DSB) of the electroweak symmetry, driven by a strongly coupled, asymptoticallyfree, vectorial gauge interaction associated with an unbroken gauge symmetry, denoted generically as technicolor (TC) [1]- [8]. There has also long been interest in models with gauge groups larger than GSM . One such model has the gauge group [9] GLR = SU(3)c × SU(2)L × SU(2)R × U(1)B−L
The conventional approach to the gauge symmetry breaking of these models employs elementary Higgs fields and arranges for a hierarchy of breaking scales by making the vacuum expectation values (vev’s) of the Higgs that break GLR or G422 to GSM much larger than the Higgs vev’s that break SU(2)L × U(1)Y → U(1)em [9,11]. This hierarchy is necessitated by the experimental lower limits on the masses of a possible WR or Z 0 [12]. An interesting question is whether one can construct asymptotically free gauge theories containing the group GLR and/or G422 that exhibit dynamical breaking of all the gauge symmetries other than SU (3)c and U (1)em , that naturally explain the hierarchy of breaking scales, and that yield requisite light neutrino masses. In this letter, we present such models.
(1)
in which the fermions of each generation transform as (3, 2, 1)1/3,L , (3, 1, 2)1/3,R , (1, 2, 1)−1,L, and (1, 1, 2)−1,R . The gauge couplings are defined via the covariant derivative Dµ = ∂µ − ig3 Tc · Ac,µ − ig2L TL · AL,µ − ig2R TR · AR,µ − i(gU /2)(B − L)Uµ . In this model the electric charge is given by the elegant relation Q = T3L + T3R + (B − L)/2, where B and L denote baryon and (total) lepton number. GLR would break at a scale ΛLR well above the electroweak scale. The model based on GLR may be further embedded in a model with gauge group [10] G422 = SU(4)P S × SU(2)L × SU(2)R .
Technicolor itself cannot provide a mechanism for all the breaking, because it is too weak at the the scale ΛLR or ΛP S and because the technifermion condensate hF¯ F i = hF¯L FR i + hF¯R FL i would break both SU(2)L and SU(2)R at the same scale (to the diagonal (vector) group SU(2)V ). Of course, to explain quark and lepton mass generation and incorporate the three families, technicolor has to be enlarged to an extended technicolor (ETC) theory [3]. Our models are ETC-type theories, with the breaking of GLR and G422 to GSM being driven by the same interactions that break the ETC group and generate quark and lepton masses.
(2)
Taking the technicolor gauge group to be SU(NT C ), the technifermions comprise an additional family, viz., U N , L = QL = D L E L , UR , DR , NR , ER transL forming according to the fundamental representation of SU(NT C ) and the usual representations of GSM (where color and TC indices are suppressed). Vacuum alignment considerations yield the desired color- and charge-
This model provides a higher degree of unification since it combines U(1)B−L and SU(3)c (in a maximal subgroup) in the Pati-Salam group SU(4)P S and hence relates gU and g3 . Denoting the generators √ of SU(4)P S as TP S,i , 1 ≤ i ≤ 15, with TP S,15 = (2 6)−1 diag(1, 1, p1, −3) and setting Uµ = AP S,15,µ , one has (B − L)/2 = 2/3TP S,15 , 1
Thus the fermions include a vectorlike set of quarks and techniquarks in the representations (5, 1, 3, 2, 1)1/3,L, (5, 1, 3, 1, 2)1/3,R and leptons and technileptons in (5, 1, 1, 2, 1)−1,L, (5, 1, 1, 1, 2)−1,R, together with a set of GLR -singlet fermions in (¯5, 1, 1, 1, 1)0,R, (10, 1, 1, 1, 1)0,R, and (10, 2, 1, 1, 1)0,R [20]. The leptons and technileptons are denoted Li,p χ , where χ = L, R, 1 ≤ i ≤ 5, and p = 1, 2. The GLR -singlets are denoted respectively Ni,R , ψij,R , ij,α , where 1 ≤ i, j ≤ 5 are ETC indices and α, β and ζR are SU(2)HC indices. The models with GLR and G422 share several features with the ETC model in [7]. The SU(5)ET C theory is an anomaly-free, chiral gauge theory and, like the ETC and HC theories, is asymptotically free. There are no bilinear fermion operators invariant under G, and hence there are no bare fermion mass terms. The SU(2)HC and SU(2)T C subsectors of SU(5)T C are vectorial. To analyze the stages of symmetry breaking, we identify plausible preferred condensation channels using a generalized-most-attractive-channel (GMAC) approach that takes account of one or more strong gauge interactions at each breaking scale, as well as the energy cost involved in producing gauge boson masses when gauge symmetries are broken. In this framework, an approximate measure of the attractiveness of a channel R1 × R2 → Rcond. is ∆C2 = C2 (R1 ) + C2 (R2 ) − C2 (Rcond. ), where Rj denotes the representation under a relevant gauge interaction and C2 (R) is the quadratic Casimir. As the energy decreases from some high value, the SU(5)ET C and SU(2)HC couplings increase. We envi3 sion that at E ∼ ΛLR > ∼ 10 TeV, αET C is sufficiently strong [16] to produce condensation in the channel
conserving TC condensates [14]. To satisfy constraints from flavor-changing neutral-current processes, the ETC vector bosons that can mediate generation-changing transitions must have large masses. We envision that these arise from self-breaking of an ETC gauge symmetry, which requires that ETC be a strongly coupled, chiral gauge theory. The self-breaking occurs in stages, for example at the three stages Λ1 ∼ 103 TeV, Λ2 ∼ 50 TeV, and Λ3 ∼ 3 TeV, corresponding to the 3 standard-model fermion generations. Hence NET C = NT C + 3. A particularly attractive choice for the technicolor group, used in the models studied here, is SU(2)T C , which thus entails NET C = 5. With Nf = 8 vectorially coupled technifermions in the fundamental representation, studies suggest that this SU(2)T C theory could have an (approximate) infrared fixed point (IRFP) in the confining phase with spontaneous chiral symmetry breaking [15,16]. This approximate IRFP produces a slowly running (“walking”) TC gauge coupling, which can yield realistically large quark and charged lepton masses [5]. The choice NT C = 2 and the walking can strongly reduce TC contributions to the S parameter [8,17]. Further ingredients may be needed to account for the top-quark mass. In Ref. [18], we studied the generation of neutrino masses in an ETC model of this sort and showed that light neutrino masses and lepton mixing can be produced via a seesaw without any superheavy mass scales. Here we extend this model to the groups GLR and G422 . We recall that ΛT C is determined by using the relation 2 + fL2 ) ' (g 2 /4)(Nc + 1)fF2 , where m2W = (g 2 /4)(Nc fQ for our purposes we take fL ' fQ ≡ fF . This gives fF ' 130 GeV. In QCD, fπ = 93 MeV and ΛQCD ∼ 170 MeV, so that ΛQCD /fπ ∼ 2; using this as a guide to technicolor, we infer ΛT C ∼ 260 GeV. The induced fermion masses in the i’th generation are given by mfi ∼ 2 η NT C Λ3T C /(4π 2 Mi2 ), where Mi ∼ gET C Λi is the gET C i mass of the ETC gauge bosons that gain mass at scale Λi and gET C is the running ETC gauge coupling evaluated at this scale. The quantity ηi is a possible enhancement factor incorporating walking, for which ηi ∼ Λi /fF [5,19]. We first consider the standard-model extension based on GLR . Our model for the DSB utilizes the gauge group
(5, 1, 1, 1, 2)−1,R × (¯5, 1, 1, 1, 1)0,R → (1, 1, 1, 1, 2)−1 (5)
with ∆C2 = 24/5, breaking GLR to SU(3)c × SU(2)L × T CNi,R i, U(1)Y . The associated condensate is hLi,p R where 1 ≤ i ≤ 5 is an SU(5)ET C index and p ∈ {1, 2} is an SU(2)R index. With no loss of generality, we use the initial SU(2)R invariance to rotate the condensate to ≡ niR , which is electrically the p = 1 component, Li,p=1 R neutral and has weak hypercharge Y = 0; the condensate is thus hniRT CNi,R i so that the niR and Ni,R gain dynamical masses ∼ ΛLR . (3) G = SU(5)ET C × SU(2)HC × GLR There exists a more attractive channel than (5) in a simple MAC analysis: (10, 1, 1, 1, )0,R ×(10, 2, 1, 1, )0,R → where HC denotes hypercolor, a second strong gauge in(1, 2, 1, 1, )0, with ∆C2 = 36/5. But with the coupling teraction which, together with ETC, triggers the requialso large at ΛLR , a sizeable energy price would g HC site sequential breaking pattern. The fermion content be incurred in this channel to generate the vector boof this model is listed below; the numbers indicate the son masses associated with the breaking of the SU(2)HC . representations under SU(5)ET C × SU(2)HC × SU(3)c × We assume here that this price is higher than the enSU(2)L × SU(2)R and the subscript gives B − L: 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 g2R g2u ΛLR ΛLR , mZ 0 = (6) mWR = (¯5, 1, 1, 1, 1)0,R , (10, 1, 1, 1, 1)0,R , (10, 2, 1, 1, 1)0,R . (4) 2 2 2
q ± 2 + g 2 , for the W ± where g2u ≡ g2R R,µ = AR,µ gauge U bosons and the linear combination Zµ0 =
g2R A3,R,µ − gU Uµ . g2u
it is plausible that Λ1 < ∼ ΛLR , since an energy price (∼ gET C Λ1 ) is incurred by the breaking of SU (5)ET C . The SU(5)ET C → SU(4)ET C breaking entails the separation of the first generation of quarks and leptons from the components of SU(5)ET C fermion fields with indices 2 ≤ i ≤ 5. The further ETC gauge symmetry breaking occurs in stages, leading eventually to the SU(2)T C subgroup of the original SU(5)ET C group. We have identified two plausible sequences for this breaking [7,18]. Both sequences yield a strongly coupled SU(2)T C gauge interaction that produces a TC condensate, breaking SU(2)L × U(1)Y → U(1)em [22]. Dirac mass terms for the neutrinos are formed dynamically, involving the left-handed neutrinos in the (5, 1, 1, 2, 1)−1,L, but not their respective right-handed counterparts in the (5, 1, 1, 1, 2)−1,R. Instead, the righthanded partners emerge from the (10, 1, 1, 1, 1)0,R (as ψ1j,R , j = 2, 3). Thus there are only two right-handed neutrinos. In a model in which L is not gauged, it is a convention how one assigns the lepton number L to the SM-singlet fields. Here, L = 0 for the fields that are singlets under GLR or G422 , since they are singlets under U(1)B−L and have B = 0. Hence, the neutrino Dirac mass terms violate L by 1 unit. There are also larger, Majorana masses generated for the ψij,R fields themselves; the seesaw mechanism then leads to left-handed ∆L = 2 Majorana neutrino bilinears [23]. We next consider the extension of the standard model gauge group to G422 . In this case, our full model is based on the gauge group G = SU(5)ET C × SU(2)HC × G422 with fermion content
(7)
This leaves the orthogonal combination Bµ =
gU A3,R,µ + g2R Uµ g2u
(8)
as the weak hypercharge U(1)Y gauge boson, which is massless at this stage. The hypercharge coupling is then g0 =
g2R gU . g2u
(9)
−2 −2 −2 + (g 0 )−2 = g2L + g2R + gU−2 , the so that, with e−2 = g2L weak mixing angle is given by 2
sin θW = 1 +
g
2L
g2R
2
+
g
2L
2 −1
gU
(10)
at the scale ΛLR . The experimental value of sin2 θW at MZ can be accommodated naturally, for example with all couplings in (10) of the same order ( even with g2R = g2L ) and with modest RG running from ΛLR to MZ . For E < ΛLR , the fermion content of the effective theory is (5, 1, 3, 2)1/3,L ,
(5, 1, 3, 1)4/3,R ,
(5, 1, 1, 2)−1,L ,
(5, 1, 1, 1)−2,R ,
(10, 1, 1, 1)0,R ,
(10, 2, 1, 1)0,R ,
(5, 1, 3, 1)−2/3,R
where the entries refer to SU(5)ET C ×SU(2)HC ×SU(3)c × SU(2)L and Y is a subscript. This is precisely the gauge group and fermion content of the ETC model that we analyzed in Ref. [18] with a focus on the formation of neutrino masses. We therefore summarize the subsequent stages of breaking only briefly, drawing on results of [18]. At a value E ∼ Λ1 ∼ 103 TeV comparable to ΛLR , a GMAC analysis suggests that there is condensation in the channel (10, 1, 1, 1)0,R × (10, 1, 1, 1)0,R → (5, 1, 1, 1)0 .
(5, 1, 4, 2, 1)L ,
(5, 1, 4, 1, 2)R ,
(¯5, 1, 1, 1, 1)R ,
(10, 1, 1, 1, 1)R ,
(11) (10, 2, 1, 1, 1)R . (13)
Again, as E decreases from high values, the SU(5)ET C and SU(2)HC couplings increase. At a scale ΛP S , the SU(5)ET C coupling will be large enough to produce condensation in the channel (5, 1, 4, 1, 2)R × (¯5, 1, 1, 1, 1)R → (1, 1, 4, 1, 2) .
(14)
This breaks SU(4)P S × SU(2)R directly to SU(3)c × U(1)Y . The value ΛP S ∼ 103 TeV satisfies phenomenological constraints, e.g. from the upper limit on BR(KL → µ± e∓ ). The associated condensate is again hniRT CNi,R i, and the niR and Ni,R gain masses ∼ ΛP S . The results (6)-(10) apply with the condition (gU /gP S )2 = 3/2 at ΛP S . Further breaking at lower scales proceeds as in the GLR model and as described in Ref. [18]. Dirac mass terms for the neutrinos are formed from the (5, 1, 4, 2, 1)L and the (10, 1, 1, 1, 1)R, leading to the same type of seesaw as in [18] and the GLR model. The experimental value of sin2 θW can again be accommodated by (10), although this now necessarily requires g2R < g2L at ΛP S . To see this, we evolve the
(12)
Thus, SU(5)ET C self-breaks to SU(4)ET C , producing masses ∼ gET C Λ1 for the nine gauge bosons in the coset SU(5)ET C /SU(4)ET C . As at ΛLR , we assume that a GMAC analysis favors this channel over the 10 × 10 channel in which SU (2)HC -breaking gauge boson masses ∼ gHC Λ1 would have to be formed. Although the latter channel is more attractive, a very large energy price would have to be paid for the associated vector boson mass generation for sufficiently large αHC > αET C . Also, although (12) has the same ∆C2 -value (= 24/5) as (5), 3
SM gauge couplings from µ = mZ to the EWSB scale −1/2 = 174 GeV and then from ΛEW up ΛEW = 2−3/4 GF to ΛP S using dαj /dt = −b0 α2j /(2π) + O(α3j ) + ... where t = ln µ, α1 ≡ (g 0 )2 /(4π), and ... denotes theoretical uncertainties associated with mass thresholds. In the interval ΛEW ≤ µ ≤ ΛP S we include the contributions from the t quark and relevant technifermions, so that (3) (2) (1) b0 = 13/3, b0 = 2/3, and b0 = −10. The initial values at mZ are α3 (mZ ) = 0.118, αem (mZ )−1 = 129, and (sin2 θW )MS (mZ ) = 0.231 [13,17]. With ΛP S = 106 GeV and the calculated values α3 = 0.064, α2L = 0.032, α1 = 0.012 at ΛP S , we find α2R (ΛP S ) ' 0.013 so that g2R /g2L ' 0.64 at this scale. It may be possible to allow g2R = g2L at ΛP S , and still match (sin2 θW )exp. , by further expanding the (4D) gauge theory to one with, e.g., SU(4)P S × SU(2)4 as in [24] but with DSB; we are currently studying this [25]. To summarize, we have constructed asymptotically free models with dynamical symmetry breaking of the extended gauge groups GLR and G422 . These models involve higher unification, and G422 has the appeal of quantizing electric charge. Our models naturally explain why (i) GLR and G422 break to GSM and (ii) this breaking occurs at the scales ΛLR , ΛP S >> mW,Z . The models incorporate technicolor for electroweak symmetry breaking, and extended technicolor for fermion mass generation including a seesaw mechanism for the generation of realistic neutrino masses. A different approach appears to be needed to construct a theory with dynamical breaking of the grand unified groups GGUT = SU(5) or SO(10) because, among other things, if the ETC group commuted with GGUT , then, with the standard fermion assignments in these GUT groups, the quarks and charged leptons would not transform in a vectorial manner under GET C , so that the usual ETC mechanism for the corresponding fermion mass generation would not apply. This research was partially supported by the grants DE-FG02-92ER-4074 (T.A.), NSF-PHY-00-98527 (R.S.).
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[17]
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4
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