UT-766 January ’96
A natural solution to the ?-problem
arXiv:hep-ph/9701232v4 29 Apr 1998
in dynamical supergravity model
Department of Physics, University of Tokyo Bunkyo-ku, Tokyo 113,Japan
Abstract The Higgs mixing term coe?cient ? is calculated in the supersymmetric theory which possesses a non-anomalous U (1)R symmetry in the limit of global supersymmetry. In this model, supersymmetry is assumed to be broken by gaugino condensation in the hidden sector when the supergravity e?ects are turned on. The soft breaking terms in the visible sector and the ? term of order the weak scale are produced in a simple manner.
The Standard Model(SM) of particle interactions enjoys overwhelming phenomeno-
logical successes, but it does not account for the gravitational interactions nor does it explain the origin or naturalness of the electroweak scale MW ? Mpl . These theoretical problems result in a brief that the high-energy physics should be described by supergravity. The hierarchy of mass scales can be naturally explained if supersymmetry is exact at high energies but becomes spontaneously broken, above MW , by a non-perturbative mechanism. At low energies, this mechanism should decouple from the observable physics and supersymmetry would appear to be broken by explicit soft terms in the e?ective low-energy Lagrangian. ?From the low-energy point of view, these soft terms - which include the masses of the super-partners of all known particles - are simply independent input parameters, just like the gauge and the Yukawa couplings of the Standard Model, but from the high-energy point of view, they are calculable in terms of the supergravity couplings. Because of its non-renormalizability, supergravity itself has to be thought of as an e?ective theory, valid below the Planck scale Mpl . Currently, the best candidate for a consistent theory governing the physics of energies near the Planck scale is the superstring theory. However, in general, we have no reliable stringy mechanisms that lead to non-perturbative spontaneous breaking of supersymmetry. Instead, one generally assumes that the dominant non-perturbative e?ects emerges at energies well below Mpl . Gaugino condensation in an asymptotically free hidden sector of the e?ective supergravity is a prime example of this type of mechanisms. These models, however, face another problem of naturalness which we call the ?-problem where ? is the coe?cient of the H1 H2 term in the low energy superpotential and H1 and H2 denote the usual Higgs SU(2) doublet chiral super?elds. In the minimal supersymmetric standard model(MSSM) the matter content consists of three generations of quarks and lepton super?elds plus two higgs doublets H1 and H2 of opposite hypercharge. The most general e?ective observable superpotential has the form: W M SSM = ? W 0 + W ??term ? ? W0 =
generations (hu QL H2 uR
W ??term = ?H1 H2 2
+ hd QL H1 dR + he LL H1 eR ) (1.1)
where hi presents dimensionless Yukawa coupling constants. This includes the usual Yukawa couplings plus a possible supersymmetric mass term for the Higgses. In addition to these supersymmetric terms, we should include soft breaking terms of order 1Tev. For this model to work well, it is known that ? should not be too large and its value is estimated to be the same order as the soft breaking terms. Once it is accepted that the presence of the ?-term is essential, immediately a question arises. Is there any dynamical reason why ? should be so small of the order of the electroweak scale? We should note that, to this respect, the ?-term is di?erent from the supersymmetry breaking terms so its origin should be di?erent from the supersymmetry breaking mechanism. In principle the natural scale of ? would be Mpl , but this would re-introduce the hierarchy problem since the Higgs scalars get a contribution ?2 to their squared mass. Thus, any complete explanation of the electroweak breaking scale must justify the origin of ?. This is the so-called ?-problem and this has been considered by several authors. In this letter we suggest a natural solution to the ?-problem without introducing complicated non-renormalizable terms and any additional mechanisms other than dynamical supersymmetry breaking in the hidden sector.
A natural solution to the ?-problem
The model of supersymmetry which we study is based on the continuous U (1)R
symmetry that is extended from R-parity. We de?ne this R-symmetry by giving the coordinate of superspace θ charge +1/2, all matter ?elds charge +1/2, and all Higgs super?elds charge 0. Expansions of the super?elds in terms of the component ?elds then show that all ordinary particles are R-neutral while all superpartners carry non-zero Rcharge. For gauginos in the hidden sector, there is an R-symmetry which is spontaneously broken if a gaugino condensate forms and leads to a Goldstone mode. (Here we should note that this R-symmetry should be explicitly broken when we ?ne-tune the cosmological constant by adding a constant to the superpotential and turn on the supergravity e?ects. See also ?? for more detailed discussions on this point.) In this case the auxiliary ?eld φ describing this would-be mode must be embedded in a chiral super?eld Φ which is coupled in a supersymmetric way. (Here we can also consider the auxiliary ?eld Φ as 3
the compensator super?eld for the anomaly of the R-symmetry.) In this respect, we consider an auxiliary ?eld Φ which also couples to the ordinary components of MSSM. This ?eld Φ has R-symmetry +1, and its scalar component is an order parameter of gaugino condensation. (The construction of the Lagrangian is motivated by ref.) The general supersymmetric lagrangian is generally characterized by three functions; K¨ ahler
? potential K (zi , zi ), superpotential W (zi ), and kinetic function f (zi ) for vector multiplets. ? ? K¨ ahler potential K (zi , zi ) is a function of scalar ?elds zi and zi , while superpotential
W (zi ) and the kinetic function f (zi ) depend scalar ?elds with de?nite chirality. Using these functions, the general form of the supersymmetry lagrangian can be written in the super?eld formalism, L = + d4 θK d2 θ W + 1 fHS WW 4 +
1 fV S WW 4
where fHS and fV S denote the kinetic functions for the hidden and the visible sector. Here we simply assume that the hidden sector consists of SU (Nc ) supersymmetric pure Yang-Mills. Demanding that the e?ective theory has non-anomalous U (1)R invariance in the limit of global supersymmetry, which is compensated by the auxiliary ?eld Φ, the form of the W and fHS are determined: W = W (Φ) + W 0 W (Φ) = (m2 + λH1 H2 )Φ fHS = S ? ξln(Φ/?) (2.2)
where m and ? are the mass parameters, ξ is a dimensionless constant and S is a dilaton super?eld. W is the total superpotential which can be written in the sum of the hidden, observable and mixing terms. One can determine ξ by demanding that the low-energy effective Lagrangian is anomaly free under the non-anomalous R-symmetry transformation. Here, for simplicity, we neglect the gauge interactions in the observable sector because such interactions are not important when we consider the strong dynamics in the hidden sector. The Higgs ?elds Hi does not have any U (1)R charge. In general, the explicit mass parameter m should be the order of the Planck mass Mpl . Other contributions to the superpotential, such as the non-renormalizable couplings suppressed by the Planck 4
mass, are not important in this model. The classical equation of motion for the auxiliary component of Φ (φ, χ, h) yields: 1 ?f ?W + λλ = 0 ?φ 4 ?φ This gives the relation: ξ (m2 + λH1 H2 )φ = λλ. 4 (2.4) (2.3)
Once gaugino condensate in the hidden sector, the right hand side of (2.4) becomes about
12 Λ3 HS where ΛHS is the dynamical scale of the hidden sector (ΛHS ? 10 Gev). For H1
and H2 , it is obvious that their vacuum expectation values cannot become large once φ develops non-zero vacuum expectation value. As a result, from the equation (2.4) we obtain a solution φ ?< λλ > /m2 ? m 3 . For the super?eld Hi , this induces a
supersymmetric mass term of the electroweak scale. W ??term ? m 3 H1 H2
This explains the scale of the ?-term in a natural way. The scalar component of Φ is determined mainly from the strong coupling e?ect which is relevant to the gaugino condensation in the hidden sector, and its non-zero value induces a supersymmetric mass term to the Higgs super?elds which appears as the ?-term in the observable sector.
The ?-problem is the necessity of introducing by hand a small mass term(?) of order
the soft breaking mass scale in the observable sector which (in general) is not correlated to the breaking of supergravity. We have shown that this problem can be avoided if we consider an e?ective Lagrangian which possesses U (1)R symmetry which is compensated by an auxiliary super?eld Φ.
We thank K.Fujikawa and K.Tobe for many helpful discussions.
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