UCRHEP-T394 August 2005
E?cacious Additions to the Standard Model
arXiv:hep-ph/0508030v2 22 Aug 2005
Ernest Ma Physics Department, University of California, Riverside, California 92521, USA, and Institute for Particle Physics Phenomenology, Department of Physics, University of Durham, Durham, DH1 3LE, UK
Abstract If split supersymmetry can be advocated as a means to have gauge-coupling uni?cation as well as dark matter, another plausible scenario is to enlarge judiciously the particle content of the Standard Model to achieve the same goals without supersymmetry. A simple e?cacious example is presented.
It has been proposed in the last year that supersymmetry could be split  so that scalar quarks and leptons are very heavy whereas gauginos and higgsinos remain light at the electroweak scale. This scenario ignores the hierarchy problem but has the virtue of retaining gauge-coupling uni?cation as well as dark matter. It has inspired a large number of phenomenological studies. On the other hand, if the goal is simply to preserve gaugecoupling uni?cation and the appearance of dark matter, then a judicious choice of particle content beyond that of the Standard Model (SM) works just as well. Such an alternative may have very di?erent predictions at the TeV scale and should not be overlooked in this context. A simple e?cacious example is presented below. It has been known for a long time that the particle content of the SM does not lead to uni?cation of the three gauge couplings of the standard SU(3)C × SU(2)L × U(1)Y gauge group, whereas that of the Minimal Supersymmetric Standard Model (MSSM) does . To understand how this works, consider the one-loop renormalization-group equations governing the evolution of these couplings with mass scale: 1 bi M2 1 ? = ln , αi (M1 ) αi (M2 ) 2π M1 (1)
2 where αi = gi /4π and the numbers bi are determined by the particle content of the model
between M1 and M2 . In the SM with one Higgs scalar doublet, these are given by SU(3)C : bC = ?11 + (4/3)Nf = ?7, SU(2)L : bL = ?22/3 + (4/3)Nf + 1/6 = ?19/6, U(1)Y : bY = (4/3)Nf + 1/10 = 41/10, (2) (3) (4)
where Nf = 3 is the number of quark and lepton families and bY has been normalized by the well-known factor of 3/5. In the MSSM with two Higgs super?elds, the shifts in bi from those of the SM are given by ?bC = (2/3)Nf + 2, 2 (5)
?bL = (2/3)Nf + 13/6, ?bY = (2/3)Nf + 1/2.
Since the scalar quarks and leptons are contained in the terms proportional to Nf , they do not change the relative values of bi . Hence they do not a?ect the uni?cation condition αC (MU ) = αL (MU ) = (5/3)αY (MU ) = αU , (8)
or the value of MU ; they only change the value of αU . Split supersymmetry is essentially the scenario where ?bC = 2, ?bL = 13/6, ?bY = 1/2, (9)
which is realized by adding to the SM new fermions transforming as (8,1,0), (1,3,0), (1,1,0), (1,2,±1/2), and a second scalar Higgs doublet. Suppose instead we add two complex scalar octets ζ, ζ ′ ? (8, 1, 0), one Majorana fermion triplet f ? (1, 3, 0) as well as one complex scalar triplet s ? (1, 3, 0), then ?bC = 2, ?bL = 2, ?bY = 0, (10)
which mimics Eq. (9) to a large extent. Assuming that these new particles appear at around MX , then the uni?cation condition of Eq. (8) implies 15 91 3 23 MX 1 + = ? ln , αC (MZ ) 158 αL (MZ ) αY (MZ ) 158π MZ ln Using the input  √ 2 αL (MZ ) = ( 2/π)GF MW = 0.0340, αY (MZ ) = αL (MZ ) tan2 θW = 0.0102, and 0.115 < αC (MZ ) < 0.119, 3 (15) (13) (14) 30 MX 3 30π 1 MU ? = ? ln . MZ 79 5αY (MZ ) αL (MZ ) 79 MZ (11) (12)
we ?nd 1800 GeV > MX > 500 GeV, and 5.1 × 1016 GeV < MU < 8.3 × 1016 GeV. (17) (16)
These are certainly acceptable values for new particles at the TeV scale and the proper suppression of proton decay. The new particles assumed are adjoint representations of SU(3)C or SU(2)L . They could for example be components of the adjoint 24 representation of SU(5). There is no fundamental understanding of why they are light, but the other components are heavy, just as there is no fundamental understanding of why the scalar SU(2)L doublet components of the 5 representation of SU(5) are light, but the SU(3)C triplet components are heavy in the canonical SU(5) model of grand uni?cation. We now come to the phenomenology of the new particles. The SU(3)C scalar octets ζ, ζ ′ may be thought of as scalar gluons. They decay into two vector gluons in one loop, i.e. ζ → ζζ → gg, etc. If kinematically allowed, they would be produced copiously at the Large Hadron Collider (LHC) and easily detected. As far as ?bC = 2 in Eq. (10) is concerned, η and ζ may be replaced by a single Majorana fermion octet, i.e. the gluino in the MSSM, but then the latter would be absolutely stable without scalar quarks, which is not acceptable cosmologically. [In split supersymmetry, the scalar quarks are still present, only much heavier.] Instead of the required two Higgs super?elds in the MSSM, this model has only the single Higgs boson of the SM. Hence its mass is not constrained by the MSSM and may exceed the latter’s current upper bound of about 127 GeV . The Majorana fermion triplet (f + , f 0 , f ? ) are like the three SU(2)L gauginos of the
MSSM. However, there is no U(1)Y gaugino here. After all, it contributes nothing to bC , bL , or bY , so it is not necessary for gauge-coupling uni?cation. In the MSSM, the gauginos couple to the Higgs scalars through the higgsinos, but here there are no higgsinos. However, since the left-handed leptons (ν, l) are doublets, f can couple to them through the SM Higgs doublet, which would allow f to decay and disqualify it from being considered as dark matter. The solution here is the same as in the MSSM. We simply assign a conserved quantity to f , say a multiplicative parity under which f is odd and all other particles are even, in exact analogy to R-parity in the MSSM. The complex scalar triplet (s+ , s0 , s? ) couples to the SM Higgs doublet (φ+ , φ0 ) according to √ ? ? s+ φ? φ0 + s0 (φ0 φ0 ? φ? φ+ )/ 2 ? s? φ0 φ+ , which means that s0 will acquire a nonzero vacuum expectation value. This has to be smaller than about a GeV to satisfy the precision electroweak measurements of the W and Z masses. It is nevertheless important for the phenomenology of dark matter because it allows f ± to be heavier than f 0 from the coupling s+ f 0 f ? + s0 f ? f + + s? f + f 0 , so that f ± may decay weakly into f 0 plus a virtual W ± which becomes a quark-antiquark pair or lepton-antilepton pair. Thus f 0 (the analog of the neutral wino in the MSSM) is the candidate for dark matter in this model. The idea that particles beyond those of the SM can restore gauge-coupling uni?cation and suppress proton decay is not new , but with the possible abandonment of low-energy supersymmetry as the only acceptable scenario below the TeV scale, new alternatives should be explored. One such simple example has been presented in this paper, with experimentally veri?able predictions very di?erent from those of the MSSM or split supersymmetry. 5
This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG03-94ER40837.
 N. Arkani-Hamed and S. Dimopoulos, hep-th/0405159; G. F. Giudice and A. Romanino, Nucl. Phys. B699, 65 (2004) [Erratum: ibid. B706, 65 (2005)]; N. Arkani-Hamed, S. Dimopoulos, G. F. Giudice, and A. Romanino, ibid. B709, 3 (2005). See also J. D. Wells, Phys. Rev. D71, 015013 (2005).  See for example W. de Boer and C. Sander, Phys. Lett. B585, 276 (2004).  S. Eidelman et al., Particle Data Group, Phys. Lett. B592, 1 (2004).  See for example R. Kinnunen et al., J. Phys. G31, 71 (2005).  See for example K. S. Babu and E. Ma, Phys. Lett. 144B, 381 (1984); also W. Grimus and L. Lavoura, Eur. Phys. J. C28, 123 (2003).