UCRHEP-T138 November 1994 LEFT-RIGHT GAUGE SYMMETRY AT THE TEV ENERGY SCALE?
arXiv:hep-ph/9411408v1 28 Nov 1994
ERNEST MA Department of Physics, University of California, Riverside, California 92521, USA
ABSTRACT Two ?rst examples beyond the standard model are given which exhibit left-right symmetry (gL = gR ) and supersymmetry at a few TeV, together with gaugecoupling uni?cation at around 1016 GeV.
1. Introduction What lies beyond the standard model at or below the TeV energy scale? One very well-motivated possibility is supersymmetry. In particular, the minimal supersymmetric standard model (MSSM) is being studied by very many people. Another possibility is left-right gauge symmetry, but there are a lot fewer advocates here and for good reason, as I will explain in this talk. I will also discuss how these problems may be overcome, assuming both supersymmetry and left-right gauge symmetry at the TeV energy scale. There are two problems with the conventional left-right gauge model at the TeV energy scale with or without supersymmetry. One is the unavoidable occurrence of ?avor-changing neutral currents (FCNC) at tree level. The other is the lack of gaugecoupling uni?cation which is known to be well satis?ed by the MSSM.1 In this talk, I will o?er two new models.2,3 Both allow the gauge couplings to be uni?ed at around 1016 GeV. The second has the added virtue of being free of FCNC at tree level. Hence left-right gauge symmetry at a few TeV should be considered a much more attractive possibility than was previously recognized. 2. Origin of FCNC in Left-Right Models Consider the gauge symmetry SU(3)C × SU(2)L × SU(2)R × U(1)B?L which breaks down to the standard SU(3)C × SU(2)L × U(1)Y at MR ? few TeV with particle
To appear in the Proceedings of the 7th Adriatic Meeting on Particle Physics, Brijuni, Croatia (September 1994)
content given by Q≡ L≡ ν l u d ? (3, 2, 1, 1/6),
Qc ≡ Lc ≡
dc uc lc νc
? (3, 1, 2, ?1/6),
? (1, 2, 1, ?1/2),
? (1, 1, 2, 1/2).
Note that each generation of quarks and leptons (i.e. Q + Qc + L + Lc ) ?ts naturally into a 16 representation of SO(10). In order for the quarks and leptons to obtain nonzero masses, a scalar bidoublet η≡
0 η1 ? η1 + η2 0 η2
? (1, 2, 2, 0)
is required. Consider the interaction of η with the quarks:
0 ? 0 + QQc η = ddc η1 ? udc η1 + uuc η2 ? duc η2 .
If there is just one η, then the mass matrices for the u and d quarks are related by
0 Md η1 ?1 0 = Mu η2 ?1
which means that there can be no mixing among generations and the ratio mu /md is the same for each generation. This is certainly not realistic and two η’s will be required.
0 ′0 Md = f η1 + f ′ η1 , 0 ′0 Mu = f η2 + f ′ η2 .
As a result, the diagonalizations of Mu and Md do not also diagonalize the respective Yukawa couplings, hence FCNC are unavoidable. To suppress these contributions to processes such as K 0 ? K 0 mixing, the ?ne tuning of couplings is required if MR ? few TeV. In the nonsupersymmetric case, η ′ can be simply taken to be σ2 η ? σ2 , but that will not alleviate the FCNC problem. Similarly, if the f ′ terms were radiative corrections from, say, soft supersymmetry breaking, FCNC would still be present. 3. Evolution of Gauge Couplings Consider now the evolution of the gauge couplings to one-loop order.
?1 ?1 αi (MU ) = αi (MR ) ?
MU bi ln , 2π MR
2 where αi ≡ gi /4π and bi are constants determined by the particle content contributing to αi . Using the standard model to evolve αi from their experimentally determined
values at MZ to MR ? few TeV and requiring that they converge to a single value at around 1016 GeV, the constraints b2 ? b3 ? 4, b1 ? b2 ? 14, (9)
are obtained. It is easily seen that these constraints are not satis?ed by the conventional left-right gauge model with or without supersymmetry. Note that b2 ? b3 = 4 in the MSSM, corresponding to two SU(2)L doublets, whereas in the supersymmetric left-right model with two bidoublets (four SU(2)L doublets), b2 ? b3 = 5. 4. First Example with Uni?cation Suppose the FCNC problem is disregarded, then the conventional left-right model with particle assignments given by Eqs. (1) and (2) can be made to have gaugecoupling uni?cation if new particles are added at the TeV energy scale.2 Supersymmetry is also assumed so that MR and MU can be separated naturally. Now bS = ?9 + 2(3) + nD = ?1, bLR = ?6 + 2(3) + n22 + nH = 3, (3/2)bX = 2(3) + 3nH + nD + 3nE = 17, (10) (11) (12)
and the constraints of Eq. (9) are satis?ed. The gauge couplings do meet at one point as shown in Fig. 1, based on a full two-loop numerical analysis. In this model n22 = 2 is the number of bidoublets, nH = 1 is the number of an anomaly-free set of Higgs doublets needed to break the SU(2)R symmetry independent of SU(2)L : ΦL ? (1, 2, 1, ?1/2), Φc ? (1, 2, 1, 1/2), L ΦR ? (1, 1, 2, 1/2), Φc ? (1, 1, 2, ?1/2), R (13) (14)
nD = 2 is the number of exotic singlet quarks of charge ?1/3: D ? (3, 1, 1, ?1/3), D c ? (3, 1, 1, 1/3), (15)
and nE = 2 is the number of exotic singlet leptons of charge ?1: E ? (1, 1, 1, ?1), E c ? (1, 1, 1, 1). (16)
Note that n22 = 2 and nH = 1 are required for fermion masses and SU(2)R breaking respectively. To obtain bLR ? bS = 4, nD = 2 is then assumed. At this stage, (3/2)bX ? bLR = 8. To increase that to 14, nE = 2 is just right. This should not be considered ?ne tuning because the contribution of each new set of particles comes in large chunks, 3 in the case of the E’s for example; so if 6 did not happen to be the desired number, it would not have been possible to achieve uni?cation with the addition of new particles this way. 3
5. Left-Right Model without FCNC Consider the E6 superstring-inspired left-right model proposed some years ago.4,5 In the fundamental 27 representation of E6 , there is an additional quark singlet of charge ?1/3. An alternative to the conventional left-right assignment is then possible: Q≡ Qc ≡ hc uc u d ? (3, 2, 1, 1/6),
dc ? (3, 1, 1, 1/3), L hc ? (3, 1, 1, ?1/3), L
? (3, 1, 2, ?1/6),
where the switch hc for dc has been made. The doublets ΦL,R and the bidoublet η are also in the 27. Hence the following terms are allowed:
0 ? 0 + QQc η = dhc η1 ? uhc η1 + uuc η2 ? duc η2 , Qdc ΦL = ddc φ0 ? udc φ? , L L hQc ΦR = hhc φ0 ? huc φ+ . R R
(19) (20) (21) (22)
As a result,
0 Mu ∝ η2 ,
M d ∝ φ0 , L
M h ∝ φ0 . R
Since each quark type has its own source of mass generation, FCNC are now guaranteed to be absent at tree level. This is the only example of a left-right model without FCNC. 6. Extended De?nition of Lepton Number Since the (1,2,1,?1/2) component of the 27 is now identi?ed as the Higgs super?eld ΦL , where are the leptons of this model? One lepton doublet is in fact contained in the bidoublet, i.e. (ν, l)L should be identi?ed with the spinor components ? 0 c of (η1 , η1 ), and one lepton singlet lL with that of φ+ . Since R
0 ? 0 + ΦL ΦR η = φ? φ+ η1 ? φ0 φ+ η1 + φ0 φ0 η2 ? φ? φ0 η2 , L R L R L R L R
the lepton l gets a mass from φ0 . Furthermore, from Eq. (19), it is seen that the L exotic quark h must have lepton number L = 1 and since uc and hc are linked by ? SU(2)R , the WR gauge boson must also have L = 1. This extended de?nition of lepton number is consistent with all the interactions of this model and is conserved. The production of WR in this model6,7 is very di?erent from that of the conventional left-right model. Because of lepton-number conservation, the best scenario is + to have u + g → h + WR , where g is a gluon. The decay of WR must end up with a lepton as well as a particle with odd R parity. Note also that WL ? WR mixing is strictly forbidden and WR does not contribute to ?mK or ? decay. Since the absence of FCNC allows only one bidoublet, only one lepton generation is accounted for in the above. Let it be the τ lepton. The e and ? generations are 4
then accommodated in the ΦL,R components of the other two 27’s, but they must not couple to Qdc or hQc . This can be accomplished by extending the discrete symmetry necessary for maintaining the conservation of lepton number as de?ned above.3 7. Precision Measurements at the Z Because of the Higgs structure of this model, there is in general some Z ?Z ′ mixing 0 which depends on the ratio of the WL to WR masses. Let η2 = v, φ0 L,R = vL,R , 2 2 2 2 r = v /(v + vL ), x = sin θW , then 1 2 2 MWL,R = g 2 (v 2 + vL,R ), 2 and
2 MZ 2 MWL x 1? r? ? 1?x 1?x 2
(24) 1?x 2 M , 1 ? 2x WR
2 MZ ′ ?
2 2 where ξ = MWL /MWR . Deviations from the standard model can now be expressed in terms of the three oblique parameters ?1,2,3 or S, T, U. Using the precision experimental inputs α, GF , MZ , and the Z → e? e+ , ?? ?+ (but not τ ? τ + ) rates and forward-backward asymmetries, they are given by
2 ? 3x x ξ, ?r r? 1?x 1?x αU x ?2 = ? ξ, = ? r? 4x 1?x x αS 1 ? 2x ?3 = r? ξ. = ? 4x 2x 1?x ?1 = αT = ?
(26) (27) (28)
Note that the ratio S/T must be positive and of order unity here. Experimentally, S, T, U are all consistent with being zero within about 1σ, but the central S and T values are ?0.42 and ?0.35 respectively.8 These imply that r ? 0.8 and ξ ? 6 × 10?3, hence the WR mass should be about 1 TeV whcih is exactly consistent with this model’s assumed SU(2)R breaking scale. In this model, the τ generation transforms di?erently under SU(2)R , hence there is a predicted di?erence in the ρl and sin2 θl parameters governing Z → l? l+ decay. Speci?cally, x ρτ ? ρe,? = 2 r ? ξ ? 6 × 10?3 , (29) 1?x compared with the experimental value of 0.0064 ± 0.0048, and sin2 θτ ? sin2 θe,? = ?x r ? x ξ ? ?7 × 10?4 , 1?x (30)
compared with the experimental value of ?0.0043 ± 0.0022. The standard model’s prediction for either quantity is of course zero. 5
8. Second Example with Uni?cation Fig. 2 shows the two-loop evolution of gauge couplings corresponding to the following situation. Let the particle content of the proposed left-right model be restricted to only components of the 27 and 27* representations of E6 , then uni?cation is achieved with3 bS = ?9 + 2(3) + nh = 0, bLR = ?6 + 2(3) + n22 + nφ = 4, (3/2)bX = 2(3) + nh + 3nφ = 18, (31) (32) (33)
where nh = 3 and n22 = 1 are required as already discussed, and nφ = 3 is the number of extra sets of ΦL + ΦR + Φc + Φc . Note that at least one such set is needed for L R SU(2)R breaking and that the two constraints of Eq. (9) are simultaneously satis?ed with the one choice of nφ = 3. To complete the model, six singlets N ? (1, 1, 1, 0) are also assumed. At the uni?cation scale MU , there are presumably six 27’s and three 27*’s of E6 , which is then broken down to supersymmetric SU(3)C × SU(2)L × SU(2)R × U(1) supplemented by a discrete Z4 × Z2 symmetry3 . Of the three 27’s and three 27*’s, only the combinations ΦL + ΦR + Φc + Φc survive. Of the other three 27’s, only two L R bidoublets do not survive. At MR ? few TeV, ΦR and Φc break SU(2)R × U(1) down R to U(1)Y . Supersymmetry is also broken softly at MR . The surviving model at the electroweak energy scale is the standard model with two Higgs doublets but not those of the MSSM, as already explained in my ?rst talk9 at this meeting. 9. Lepton Masses The τ gets its mass from the ΦL ΦR η term, but there can be no such term for the e and ?. Hence the latter two are massless at tree level. However, the soft supersymmetry-breaking term ΦL ΦR η (where η = σ2 η ? σ2 and all three ?elds are ? ? scalars) is allowed, hence me and m? are generated radiatively from the mass of the U(1) gauge fermion.10 The neutrinos obtain small seesaw masses from their couplings with the three NL ’s which are assumed to have large Majorana masses. The ντ NL mass comes from the ηηNL term, and the νe NL , ν? NL masses come from the ΦL Φc NL L terms. 10. Conclusion New physics in the framework of left-right gauge symmetry is possible at the TeV energy scale even if grand uni?cation is required. Two examples have been given, the second of which is particularly attractive: it is free of FCNC at tree level and has negative contributions to the oblique parameters S and T consistent with present experimental central values. 6
11. Acknowledgements I thank Profs. D. Tadic and I. Picek and the other organizers of the 7th Adriatic Meeting on Particle Physics for their great hospitality and a very stimulating program. This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG03-94ER40837. 12. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. U. Amaldi, W. de Boer, and H. Furstenau, Phys. Lett. B260, 447 (1991). E. Ma, UCRHEP-T124, Phys. Rev. D, to be published. E. Ma, UCRHEP-T129, Phys. Lett. B, to be published. E. Ma, Phys. Rev. D36, 274 (1987). K. S. Babu, X.-G. He, and E. Ma, Phys. Rev. D36, 878 (1987). J. F. Gunion, J. L. Hewett, E. Ma, and T. G. Rizzo, Int. J. Mod. Phys. A2, 1199 (1987). P. Chiappetta and M. Deliyannis, Phys. Lett. B276, 67 (1992). Particle Data Group, Phys. Rev. D50, 1173 (1994). E. Ma, ”Two-Doublet Higgs Structure at the Electroweak Energy Scale,” these Proceedings. E. Ma, Phys. Rev. D39, 1922 (1989).