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Electroweak symmetry breaking and s-spectrum in M-theory
Nuclear Physics B 580 (2000) 3–28 www.elsevier.nl/locate/npe
Electroweak symmetry breaking and s-spectrum in M-theory Tatsuo Kobayashi a,b , Jisuke Kubo c , Hitoshi Shimabukuro c a Department of Physics, High Energy Physics Division, University of Helsinki, Helsinki, Finland b Helsinki Institute of Physics, P.O. Box 9, FIN-00014 Helsinki, Finland c Institute for Theoretical Physics, Kanazawa University, Kanazawa, 920-1192, Japan
Received 31 March 1999; revised 19 October 1999; accepted 29 February 2000
Abstract We study soft SUSY breaking parameters in M-theory with and without 5-brane moduli fields. We investigate successful electroweak symmetry breaking and the positivity of stau mass squared. A value of α(T + T¯ ) is phenomenologically constrained. Also we have phenomenological constraints on magnitudes of derivatives of the Kähler potential for the 5-brane moduli field. We study mass spectra in the allowed regions. 2000 Elsevier Science B.V. All rights reserved. PACS: 04.65.+e; 11.25.Mj; 12.60.Jv Keywords: SUSY GUT; Coupling reduction; Soft SUSY breaking parameters; Radiative electroweak breaking
1. Introduction Supersymmetry (SUSY) breaking is very important to connect superstring theory to low-energy physics. Induced soft SUSY breaking terms are significant to currently running and future experiment, and they also have cosmological implications. Actually, soft SUSY breaking terms have been derived, e.g., within the framework of weakly coupled heterotic string theory under the assumption that only F -terms of the dilaton S and moduli fields T i contribute to SUSY breaking [1–5]. Also various phenomenological aspects of these string-inspired soft terms have been studied. Recently strongly coupled superstring theories as well as strongly coupled SUSY gauge theories have been studied. Strongly coupled E8 × E8 heterotic string theory has been considered as M-theory compactified on S 1 /Z2 [6–8]. In particular, the 4-dimensional effective theory has been obtained and SUSY breaking has been studied [9–23]. Soft parameter formulae between the weakly and strongly coupled regions are connected by the parameter α(T + T¯ ). Actually, several phenomenological aspects of soft SUSY breaking 0550-3213/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 3 8 - 3
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parameters have been investigated, e.g., radiative electroweak symmetry breaking and s-spectrum [24–29], although the large tan β region has not been studied extensively. The standard embedding leads to the gauge group E6 × E80 , where the top, bottom and tau Yukawa couplings are unified, that is, we have a large value of tan β. In this sense, the large tan β scenario is one natural target to study. When E6 breaks field-theoretically into the standard model (SM) gauge group GSM = SU (3)C × SU (2)L × U (1)Y , integrating-out of heavy modes induces additional soft scalar terms, i.e., D-term contributions. On the other hand, non-standard embedding including Wilson line breaking is also possible [30–33] and that can lead to smaller gauge groups, e.g., SU (3) × SU (2) × U (1) or SU (5). Thus, in non-standard embedding a small value of tan β can also be realized. Furthermore, non-standard embedding, in general, has 5-branes, that is, in 4-dimensional effective field theory we have a new type of moduli fields Z n whose vacuum expectation values (VEVs) correspond to positions of 5-branes zn within [0, 1] along the orbifold S 1 /Z2 . It is possible that F -terms of the 5-brane moduli fields Z n also contribute to SUSY breaking. Recently 4-dimensional effective theory with Z n has been studied [34,35] and SUSY breaking terms have also investigated [36]. The purpose of this paper is to take into account several effects which are mentioned above and to investigate successful electroweak breaking conditions and the lightest superparticle (LSP) systematically. M-inspired soft SUSY breaking parameters have a certain difference from those derived from weakly coupled string models. Such difference depends on the parameter α(T + T¯ ) and the limit α(T + T¯ ) → 0 corresponds to the weakly coupled model without threshold corrections. First of all, the gauge kinetic function has a α(T + T¯ )-dependent term [9,11,14,15] and that implies we have non-vanishing gaugino masses even in the moduli-dominant SUSY breaking. Threshold corrections in the weakly coupled models have similar effects, that is, even for the moduli-dominant SUSY breaking non-vanishing gaugino masses can appear in the weakly coupled models with threshold corrections. However, in M-theory α(T + T¯ )-dependent terms appear in the soft scalar masses and the A-terms, too. Thus, since the parameter α(T + T¯ ) is important, we investigate α(T + T¯ ) dependence of phenomenological aspects, e.g., the mass spectrum. As we shall show, sfermion mass squared can be negative in certain parameter spaces and radiative electroweak symmetry breaking is not realized. Thus, we have phenomenological constraints on the value of α(T + T¯ ) and other free parameters. In particular, in the large tan β scenario, the stau (mass)2 becomes small and negative because of large and negative radiative corrections from the large Yukawa coupling. Thus the stau mass leads to a significant constraint for large tan β. We cover the whole realistic region of tan β including large tan β. We investigate α(T + T¯ )-dependence of electroweak symmetry breaking and s-spectrum. Furthermore, we study the effect due to F -terms of the moduli fields Z n . Their Kähler potential have significant effects on soft SUSY breaking terms. Even though we lack in knowledge about an explicit form of the Kähler potential of Z n , we can derive phenomenological constraints on magnitudes of its derivatives. This paper is organized as follows. In Section 2, after we describe the structure of the soft parameters in M-theory without 5-brane, we investigate the electroweak symmetry
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breaking conditions and the positivity of stau mass squared under the assumption of top-bottom-tau Yukawa coupling unification, that is, the large tan β scenario. D-term contributions, which appear in E6 breaking into GSM , are also taken into account. We also consider the case with small tan β. In Section 3, we consider soft parameters in M-theory that the 5-brane moduli field also contribute to SUSY breaking. In this case, electroweak symmetry breaking and mass spectra are studied and allowed regions are compared with the case without 5-brane effects. Section 4 is devoted to conclusions.
2. SUSY breaking in M-theory without 5-branes In this section we study SUSY breaking in M-theory without 5-branes. For example, standard embedding corresponds to this class. We discuss the soft breaking parameters derived within this framework and study their phenomenological implications. 2.1. Soft parameters The 4-dimensional effective supergravity action of M-theory can be analyzed by expanding it in powers of the two dimensionless variables: ε1 = κ 2/3πρ/V 2/3 and ε2 = κ 2/3/(πρV 1/3 ), likewise the weakly coupled heterotic string theory expanded by the string coupling εs = e2φ /(2π)5 and the world sheet sigma model coupling εσ = 4πα 0 /V 1/3 , where κ 2 , ρ and V denote the 11-dimensional gravitational coupling, the 11-dimensional length and the Calabi–Yau volume respectively. Here we consider only the overall moduli field T . Thus, the 4-dimensional effective supergravity includes the dilaton S and the moduli field T in addition to chiral multiplets Φ p , gauge multiplets and the graviton multiplet. Scalar components of S and T can be identified as 1 (4πκ 2 )−2/3 V , 2π 61/3 (4πκ 2 )−1/3 πρV 1/3 . (1) Re(T ) = 8π The Kählar potential is computed by M-theory expansion for matter fields up to order of κ 4/3 [18], α 3 ¯ − 3 log(T + T¯ ) + + |Φ|2 . (2) K = − log(S + S) T + T¯ S + S¯ Re(S) =
Similarly the gauge kinetic functions of the observable and hidden sectors are calculated [9,11,14,15], fE6 = S + αT , where 1 (8π )α = 64π 3 α 0 2
fE8 = S − αT , Z
1 ω ∧ tr(F ∧ F ) − tr(R ∧ R) . 2
(3)
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In addition, the tree-level superpotential is obtained, W = Ypqr Φ p Φ q Φ r .
(4)
Here α(T + T¯ ) is of the order one at the “physical” point where gauge coupling unification can be realized [9]. Note that the superpotential W and gauge kinetic functions fE6 and fE8 are exact up to nonperturbative corrections, while there are small additional perturbative corrections to the Kählar potential which are of order 1/[4π 2Re(S)] or 1/[4π 2Re(T )]3 ∼ O(αGUT /π) in the strong coupling limit εs 1. Now we know the forms of the Kähler potential K, the gauge kinetic function fE6 and the tree-level superpotential W . Thus, we can derive the formula of soft SUSY breaking parameters following Refs. [3,37]. We assume that a nonperturbative superpotential of S and T is induced and F -terms of S and T contribute to SUSY breaking. The gravitino mass m3/2 is obtained 3m23/2 =
|F S |2 |F T |2 + − V0 , ¯ 2 (T + T¯ )2 3(S + S)
(5)
where V0 is the vacuum energy. Hence, following Ref. [4], we parameterize F -terms, √ ¯ sin θ e−iγS , (6) F S = 3 m3/2C(S + S) T −iγ T , (7) F = m3/2 C(T + T¯ ) cos θ e where C 2 = 1 + V0 /(3m3/2). Using these parameters, we can write the soft parameters, i.e., the gaugino mass M1/2 , the soft scalar mass m and the A-parameter as follows [19] √ 3 Cm3/2 M1/2 = ¯ (S + S) + α(T + T¯ ) ¯ ¯ sin θ e−iγS + α(T√+ T ) cos θ e−iγT , (8) × (S + S) 3 3C 2 m23/2 m2 = V0 + m23/2 − ¯ + α(T + T¯ ) 3(S + S) α(T + T¯ ) sin2 θ × α(T + T¯ ) 2 − ¯ + α(T + T¯ ) 3(S + S) ¯ 3(S + S) ¯ cos2 θ + (S + S) 2 − ¯ + α(T + T¯ ) 3(S + S) √ ¯ 2 3 α(T + T¯ )(S + S) sin θ cos θ cos(γS − γT ) , (9) − ¯ + α(T + T¯ ) 3(S + S) √ 3α(T + T¯ ) sin θ e−iγS A = 3 Cm3/2 −1 + ¯ + α(T + T¯ ) 3(S + S) ¯ √ 3(S + S) −iγT cos θ e + 3 −1 + . (10) ¯ + α(T + T¯ ) 3(S + S) In following analyses we concentrate to the case with the vanishing vacuum energy, V0 = 0 (C = 1) and no CP phase γS = γT = 0.
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Since the gauge kinetic function fE6 provides the gauge coupling constant, that is, 2 , the phenomenological requirement g −2 ' 2 leads to Re fE6 = 1/gE GUT 6 ¯ + α(T + T¯ ) ' 4. (S + S)
(11)
¯ > α(T + T¯ ). Therefore, we have Moreover, the constraint for Re fE8 > 0 leads to (S + S) 0 < α(T + T¯ ) . 2.
(12)
Note that there are three free parameters m3/2 , θ and α(T + T¯ ), and in particular, α(T + T¯ ) is the characteristic parameter of M-theory. The limit α(T + T¯ ) → 0 corresponds to weakly coupled theory. 2.2. Electroweak symmetry breaking and mass spectrum We now analyze phenomenology of M-theory by using Eqs. (8)–(10) as boundary conditions of renormalization group equations (RGEs) at the GUT scale. We investigate successful radiative electroweak symmetry breaking and positive masses squared for any sfermion. That gives constraints for the parameters m3/2 , α(T + T¯ ) and θ . If we assume a certain type of µ-term generation mechanism, we could fix magnitudes of the supersymmetric Higgs mixing mass µ and the corresponding SUSY breaking parameter B. However, we do not take such a procedure here. Because we would like to study generic case. We determine these magnitudes by using the following minimization conditions of the MSSM Higgs potential, m21 + m22 = −
2µB , sin β
m21 − m22 = − MZ2 + m21 + m22 cos 2β,
(13)
where m21,2 = m2H1 ,H2 + µ2 , and m2H1 ,H2 are soft scalar masses of Higgs fields. The Higgs fields for the down sector and the up sector are denoted by H1 and H2 . Using these equations, µ and B are written in terms of the soft scalar masses and tan β. Realization of successful electroweak symmetry breaking leads to the condition, m21 m22 < |µB|2 ,
(14)
and the bounded-from-below condition along the D-flat directions in the Higgs potential, that is, m21 + m22 > 2|µB|,
(15)
should be satisfied in order for stability of the potential along the D-flat direction. In the large tan β scenario, these conditions lead to m2H1 − m2H2 & MZ2 /2.
(16)
Also, we require that a physical mass squared should be positive for any sfermion. Since the sleptons have no large positive-definite radiative corrections from loops involving the gluino, the sleptons become lighter than squarks at low energy. In particular, in the case
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of large tan β, the stau mass squared has a sizable and negative radiative correction due to the large Yukawa coupling. The lighter stau mass becomes significant for large tan β. 1 Furthermore, what is the LSP is an important issue. One candidate is the lightest neutralino and another candidate is the lighter stau for large tan β from the above reason. 2 The stau mass matrix is 2 mτ˜L + (− 12 + sin2 θW )MZ2 cos 2β vYτ (Aτ cos β − µ sin β) , (17) m2τ˜ = m2τ˜R − MZ2 sin2 θW cos 2β vYτ (Aτ cos β − µ sin β) where v 2 ≡ hH1 i2 + hH2 i2 , and we have neglected the tau (mass)2 Mτ2 term here. The lighter stau mass squared is obtained as 1 1 2 m2τ˜L + m2τ˜R − MZ2 cos 2β mτ˜1 = 2 2 2 1 − 2 sin2 θW MZ2 cos 2β − m2τ˜L − m2τ˜R − 2 1/2 2 + 4vYτ (Aτ cos β − µ sin β) . (18) We take the following input parameters, Mτ = 1.777 GeV,
MZ = 91.188 GeV, Mt 8 −1 log (MZ ) = 127.9 + , αEM 9π MZ sin2 θW = 0.2319 − 3.03 × 10−5 ∆t − 8.4 × 10−8 ∆2t ,
(19)
where 1t = Mt [GeV] − 165.0. Here Mτ and Mt are physical tau lepton and top quark masses. We define the SUSY scale as the arithmetic average of the stop squared mass eigenvalues [44,45], 2 MSUSY
=
m2t˜ + m2t˜ 1
2
2
.
(20)
2.2.1. Standard embedding First of all, we consider the case of standard embedding, where the gauge group E6 × E80 is obtained and the top, bottom and tau Yukawa couplings are unified at the GUT scale. From the experimental value of Mτ , we obtain tan β = 53 and Mt = 175 GeV. We assume at the GUT scale E6 breaks into GSM field-theoretically and the MSSM matter content remains light. On top of that, we assume this symmetry breaking induces no extra contributions to soft parameters. Then we use Eqs. (8)–(10) as the initial conditions of RGEs at the GUT scale. The latter assumption will be relaxed later. Before studying the constraints, it is useful to show how the soft parameters, i.e., the gaugino mass M1/2 , A-parameter and the soft scalar mass m at the GUT scale, depend 1 For example, in Refs. [38,39] the constraint m2 > 0 has been considered in details for (finite) SU(5) GUTs τ˜1
and SO(10) GUTs. 2 In Refs. [40–43] cosmological implications of the stau LSP have been discussed. The stau LSP could not be a candidate for cold dark matter. In the case with the stau LSP we need another candidate for cold dark matter.
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Fig. 1. The ratios −A/M1/2 and m/M1/2 against 2α Re(T ). The three solid lines correspond to −A/M1/2 for θ = 0, π/5 and π/2. The two dotted ones correspond to m/M1/2 for θ = π/5 and π/2.
on the parameters, 2α Re(T ) and θ . Note that the A-parameter is always negative. Fig. 1 shows the ratios −A/M1/2 and m/M1/2 against 2α Re(T ) for various values of θ , where the three solid lines and the two dotted lines stand for −A/M1/2 and and m/M1/2 . The uppermost, middle and lowermost solid lines are for −A/M1/2 with θ = 0, π/5 and π/2, respectively. Similarly, the upper and lower dotted lines are for m/M1/2 with θ = π/5 and π/2, respectively. In the weak coupling limit we obtain the universal SSB parameters for for any value of θ 3 : A = −1, M1/2
m 1 =√ . M1/2 3
(21)
For any value of 2α Re(T ) in the region 0.4 < tan θ < 1.0, we find A/M1/2 ≈ −1. In the dilaton-dominant (moduli-dominant) case, however, −A/M1/2 becomes smaller (larger). √ The ratio m/M1/2 = 1/ 3 holds approximately for any value of 2α Re(T ) in the case with 2 decreases as 2α Re(T ) 0.01 . tan θ . 1.0. For most of the other values of tan θ , m2 /M1/2 increases. Also the soft scalar mass squared m2 itself decreases except 0.0 . tan θ . 0.5 as 2α Re(T ) increases. Furthermore, the soft scalar mass squared m2 can become negative. For example, for 2α Re(T ) = 0.5, 1.0 and 1.5 m2 is negative in the regions, −0.2 . tan θ . −1/70, −0.7 . tan θ . −1/30 and −6 . tan θ . −1/20, respectively. Thus, the value of 2α Re(T ) is constrained phenomenologically for fixed tan θ , in particular for negative tan θ . 3 These relations have an implication for finiteness and RG invariance [38,46–52].
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Moreover, negative radiative corrections due to Yukawa couplings make such constraint more severe for the third family of sfermions, stop, sbottom and stau. Now we consider the conditions for the electroweak symmetry breaking (14) and (15) and look for the parameter region in which these conditions are satisfied, where we investigate the cases for θ = π/5, π/4, π/2, and 3π/4. The region I with lines slanting from upper right downwards in Figs. 2(a)–(d) is the region in which the conditions (14) and (15) are NOT satisfied. As one can see from Figs. 2(a)–(d), there are two regions, leftand right-hand sides of the figures. In the region II (with lines slanting from upper left downwards) m2τ˜1 is negative, and so it is not desirable, because the electromagnetic U (1) will be broken in this region. The shaded region III denotes the parameter space in which we find m2τ˜ < m2 0 . Consequently, the parameter space in the region III which does not 1
χ˜ 1
overlap with I and II leads to an MSSM with a stau as the LSP. The constraints coming from a successful electroweak symmetry breaking (14) and (15) allow a wider space with increasing 2α Re(T ), and in particular, the larger θ is, the wider is the allowed space. But too large θ in the case of a large 2α Re(T ) is not desirable, because we would enter in the region II (mτ2˜1 < 0), as we can see from Figs. 2(c) and 2(d). This results from the fact that due to Eq. (9) we have m2 < 0 at the GUT scale. Furthermore, the negative radiative correction due to the Yukawa coupling contributes to widen the region with m2τ˜1 < 0. Thus, we have the phenomenological upper bound of 2α Re(T ) for fixed values of θ and m3/2 . In any parameter space that is not excluded by I and II, in the range of 2α Re(T ), M3/2 and θ we have considered, the LSP is a stau. That is a unique property of this model. 2.2.2. D-term contributions We have assumed that the gauge symmetry E6 breaks down to GSM field-theoretically and the symmetry breaking has no effect on soft parameters. However, integrating-out of heavy modes, in general, affects soft scalar mass terms. In particular, in the case that a gauge group symmetry breaks reducing its rank, the integrating-out of heavy modes induces the so-called D-term contribution to soft scalar mass terms [53–60]. Here we take into effects of such D-term contributions to soft scalar masses. In general, these D-term contributions are proportional to quantum numbers of broken diagonal generators. In the present case, we have two independent universal factors of D-term contributions, which we denote as m2D1 and m2D2 , because the rank of E6 is larger than GSM by two. We regard these as free parameters and then analyze their effects on the parameter space. The soft scalar masses at the GUT scale including D-term contributions are written as m2Q˜ = m2t˜ = m2τ˜R = m227 − m2D1 + m2D2 , m2b˜ = m2τ˜L = m227 + 3m2D1 + m2D2 , m2H1 = m227 − 2m2D1 − 2m2D2 , m2H2 = m227 + 2m2D1 − 2m2D2 , where m27 is the soft scalar mass m in Eq. (9).
(22)
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(a)
(b)
Fig. 2. The excluded regions by the electroweak breaking conditions (14) and (15) (region I), the constraint m2τ˜ > 0 (region II) and the region with m2τ˜ > m2 0 (region III). (a) and (b) correspond to 1
the cases with θ = π/5, π/4.
1
χ˜ 1
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(c)
(d)
Fig. 2. (Continued.) The excluded regions by the electroweak breaking conditions (14) and (15) (region I), the constraint m2τ˜ > 0 (region II) and the region with m2τ˜ > m2 0 (region III). (c) and (d) 1
correspond to the cases with θ = π/2, 3π/4.
1
χ˜ 1
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
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We now analyze D-term contribution effects on the parameter space at tan β = 53, which has been obtained under the assumption of the top-bottom-tau Yukawa coupling unification. The term m2D1 with a negative and sizable value is helpful to create a positive gap m2H 1 − m2H 2 , which is favorable to successful electroweak symmetry breaking. Also, a negative value of m2D1 increases m2τ˜R , but it reduces m2τ˜L . The (mass)2 m2τ˜R is dominant to m2τ˜1 around the universal case with m2τ˜R = m2τ˜L . Therefore negative m2D1 may increase m2τ˜ and seems to be favorable to the constraint of m2τ˜ . However, as m2D1 decreases, m2τ˜L is 1 1 decreased and below a critical value of m2D1 m2τ˜L becomes dominant to mτ2˜1 . A similar effect of the D-term contribution has been discussed within the framework of SO(10) GUTs [39]. On the other hand, a positive value of m2D2 increases both mτ2˜R and m2τ˜L . Thus, a positive value of m2D2 is favorable to increase m2τ˜1 . The effects of m2D2 to m2H 1 and m2H 2 are same. Hence, m2D2 has little effect on electroweak symmetry breaking. These results are shown in Fig. 3(a), (b) and Fig. 4(a), (b) for θ = π/2. Fig. 3 show the m2D2 effects, where we have assumed that m2D1 = 0 and have chosen m2D2 = 0.5 and 1.0 (TeV)2 for Figs. 3(a) and 3(b), respectively. A large value of m2D2 reduces the region excluded by the constraint m2τ˜1 > 0, while it has little effect on electroweak symmetry breaking. The region with the neutralino LSP appears and it becomes wider as m2D2 increases. For example, in the case of m2D2 = 1.0 (TeV)2 , the region with the neutralino LSP has the largest value m3/2 = 3.1 TeV when α(T + T¯ ) = 0.8. In this case the lightest neutral Higgs boson mass mh0 is 151 GeV. Here we use the two loop Higgs mass mh0 [44, 45,61,62]. Similarly, we obtain mh0 = 129 GeV when m3/2 = 1.0 TeV and α(T + T¯ ) = 0.8. This value is stable between m3/2 = 0.4 TeV and 1.0 TeV. Fig. 4(a), (b) show the m2D1 effects, where we have assumed that m2D2 = 0. They show that the region excluded by the electroweak symmetry breaking conditions becomes more narrow and the region with m2τ˜1 < 0 becomes wider as m2D1 decreases. In suitable values of m2D1 , we have the region with the neutralino LSP. The critical value is around m2D1 = −2.0 (TeV)2 , which is shown in Fig. 4(b), where the neutralino LSP region is widest. This should be compared with Fig. 4(a) for which we have used m2D1 = −0.05 (TeV)2 .
2.2.3. Small tan β scenario It is also possible that we obtain a smaller gauge group, e.g., GSM or SU (5) at the GUT scale by non-standard embedding including Wilson line breaking. In this case, the assumption of the top-bottom-tau Yukawa coupling unification can be relaxed and a smaller value of tan β can be realized. We assume that below the GUT scale the matter content is same as the MSSM. Here, we vary tan β holding Eqs. (8)–(10) as the initial conditions of RGEs at the GUT scale and fixing Mt = 175 GeV. Fig. 5(a), (b) show that the allowed parameter region for tan β and α(T + T¯ ). Fig. 5(a) shows that for a small value of | sin θ |; there is no excluded region at tan β < 50. However, for large values of | sin θ | and α(T + T¯ ), the excluded regions I and II appear even in the case of small tan β, which is shown in Fig. 5(b).
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T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
(a)
(b)
Fig. 3. Effects of the D-term contribution m2D2 on the excluded region for θ = π/2. (a) and (b) correspond to the cases with (m2D1 , m2D2 ) = (0, 0.5) and (0, 1.0) [(TeV)2 ], respectively.
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
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(a)
(b)
Fig. 4. Effects of the D-term contribution m2D1 on the excluded region for θ = π/2. (a) and (b) correspond to the cases with (m2D1 , m2D2 ) = (−0.05, 0) and (−2.0, 0) [(TeV)2 ], respectively.
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(a)
(b)
Fig. 5. The excluded region in the tan β − 2α Re(T ) space. (a) corresponds to the case with θ = π/5 and m3/2 = 1.0 TeV for Mtop = 175.0 GeV. Similarly, (b) corresponds to the case with θ = π/2 and m3/2 = 2.0 TeV for Mtop = 175.0 GeV.
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
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3. M-theory with 5-brane 3.1. Soft parameters In general, non-standard embedding includes 5-branes, that is, 4-dimensional effective theory includes 5-brane moduli fields Z n . It is possible that Z n also contribute to SUSY breaking. Thus, we consider such effects for generic case. If the 5-branes exist between the boundaries in M-theory, the Kähler potential and the gauge kinetic functions are modified. The relevant part of the moduli Kähler potential Kmod and the gauge kinetic functions of the observable and hidden sectors f (1) and f (2) with N 5-branes take the following form [34–36], ¯ − 3 ln(T + T¯ ) + K5 , Kmod = − ln(S + S) f (1) = S + εT β (0) +
N X
!
(23)
(1 − Z n )2 β (n) ,
n=1
f
(2)
= S + εT β
(N+1)
! N X n 2 (n) + (Z ) β ,
(24)
n=1
where K5 is the Kähler potential for the 5-brane position moduli Z n and the expansion parameter ε = (κ/4π)2/3 2π 2 ρ/V 2/3 . In addition, β (0) , β (N+1) are the instanton numbers on the observable sector boundary and the hidden sector boundary respectively, and β (n) (n = 1, . . . , N) is a magnetic charge on the each 5-brane. They should satisfy the cohomology constraint, N+1 X
(n)
βi
= 0,
(25)
n=0
which means physically that the net charges must be zero because the “flux” cannot get away anywhere in compact space. The part of the Kähler potential for the matter fields is obtained εζ 3 + |Φ|2 , (26) Kmat = ¯ (T + T¯ ) (S + S) where ζ = β (0) +
N X
(1 − zn )2 β (n) .
(27)
n=1
Following the generic formulae of Refs. [3,37] again, we can derive the soft parameters. We assume V0 = 0 again. The soft parameters are obtained, S 1 F + εζ F T + εT F n ζn , ¯ ¯ (S + S) + εζ (T + T ) S 2 9 1 2 2 − m = m3/2 − F ¯ 2 [3(S + S) ¯ + εζ (T + T¯ )]2 (S + S)
M1/2 =
(28)
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(εζ )2 1 − ¯ + εζ (T + T¯ )]2 (T + T¯ )2 [3(S + S) ¯ εζnm¯ (T + T ) ε2 ζn ζm¯ (T + T¯ )2 n ¯m ¯ − −F F ¯ + εζ (T + T¯ ) [3(S + S) ¯ + εζ (T + T¯ )]2 3(S + S) εζ εζm¯ (T + T¯ ) ¯ + 3F S F¯ m¯ + 3F S F¯ T 2 ¯ + εζ (T + T¯ )] ¯ + εζ (T + T¯ )]2 [3(S + S) [3(S + S) ε2 ζ ζm¯ (T + T¯ ) εζm¯ − − F T F¯ m¯ + (c.c.), ¯ + εζ (T + T¯ ) [3(S + S) ¯ + εζ (T + T¯ )]2 3(S + S) (29) 9 2 − A = FS ¯ + εζ (T + T¯ ) S + S¯ 3(S + S) εζ + FT ¯ + εζ(T + T¯ ) 3(S + S) 3εζn (T + T¯ ) n , (30) + F K5,n − ¯ + εζ (T + T¯ ) 3(S + S) 2 − F T
where K5,n = ∂K5 /∂Z n , ζn = ∂ζ /∂zn and ζnm = ∂ 2 ζ /(∂zn ∂zm ). We have normalized the soft scalar mass m as, m02 = m2
∂ 2 Kmat , |∂Φ|2
(31)
using the total Kähler metric, where m0 is the mass for the field Φ with non-canonical kinetic factor, ∂ 2 Kmat /|∂Φ|2 . That is different from Ref. [36], where the soft scalar mass is normalized as 3 . (32) m02 = m2 (T + T¯ )2 Here we ignore CP phases, i.e., F = F¯ . For simplicity, we assume that there is only one ¯ i.e., relevant 5-brane moduli Z and its Kähler potential is a function of only (Z + Z), ¯ K5 = K5 (Z + Z).
(33)
The gravitino mass is obtained, m23/2 =
2 |F S |2 |F T |2 1 + + K5,Z Z¯ F Z . ¯ 2 (T + T¯ )2 3 3(S + S)
Now we can parameterize each F -term as follows √ ¯ sin θ sin φ, F S = 3 m3/2 (S + S) F T = m3/2(T + T¯ ) cos θ sin φ, s 3 Z m3/2 cos φ. F = K5,Z Z¯
(34)
(35)
Still now we have several unknown parameters and most of them are due to detailed information of the 5-brane. Thus, we fix z and β (1) , e.g.,
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
1 z= , 2
4 β (1) = β (0). 3
19
(36)
Note that both the instanton number β (0) and the intersection number β (1) are integers. Eq. (36) leads to the following simple relations with Eq. (27) as 1 (37) ζ |z=1/2 = −ζ,z |z=1/2 = ζ,zz |z=1/2 = β (1) . 2 In this case we can rewrite the soft parameters, ( √ m3/2 ¯ sin θ sin φ 3(S + S) M1/2 = ¯ + εζ (T + T¯ ) (S + S) ) s ¯) 3 εζ (T + T cos φ , + εζ (T + T¯ ) cos θ sin φ − 2 K5,Z Z¯ ( ¯ 2 9(S + S) 2 2 sin2 θ sin2 φ m = m3/2 1 − 3 1 − ¯ + εζ (T + T¯ )]2 [3(S + S) " 2 # εζ (T + T¯ ) cos2 θ sin2 φ − 1− ¯ + εζ (T + T¯ ) 3(S + S) εζ (T + T¯ ) 3εζ (T + T¯ ) 2− cos2 φ − ¯ + εζ (T + T¯ )] ¯ + εζ (T + T¯ ) K5,Z Z¯ [3(S + S) 3(S + S) √ ¯ (T + T¯ ) 6 3(S + S)εζ sin θ cos θ sin2 φ + ¯ + εζ (T + T¯ )]2 [3(S + S) s √ ¯ (T + T¯ ) 3 6 3(S + S)εζ sin θ sin φ cos φ − 2 ¯ ¯ K [3(S + S) + εζ (T + T )] 5,Z Z¯ εζ (T + T¯ ) 2εζ (T + T¯ ) 1− + ¯ + εζ (T + T¯ ) ¯ + εζ (T + T¯ ) 3(S + S) 3(S + S) ) s 3 cos θ sin φ cos φ , × K5,Z Z¯ ( ¯ √ 9(S + S) sin θ sin φ 3 2− A = m3/2 ¯ + εζ (T + T¯ ) 3(S + S) 3εζ (T + T¯ ) cos θ sin φ ¯ + εζ (T + T¯ ) 3(S + S) ) s 3εζ (T + T¯ ) 3 cos φ , + K5,Z + ¯ + εζ (T + T¯ ) K5,Z Z¯ 3(S + S)
−
(38)
where we have neglected the phase of T . In addition, from the gauge kinetic functions (24) we obtain constraints for values of moduli S and T as, ¯ + εζ (T + T¯ ) = 2g −2 ' 4, (39) 2 Re f (1) = (S + S) GUT
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and ¯ − 3 εζ (T + T¯ ) > 0. (40) 2 Re f (2) = (S + S) 2 Here we use Eqs. (36), (37), (25), (39) and (40), so that we find the parameter region as 0 < εζ (T + T¯ ) < 8/5. That is almost comparable to the region 0 < α(T + T¯ ) < 2 in the case without 5-branes. 3.2. Electroweak symmetry breaking and mass spectrum Using Eq. (38), we analyze the 5-brane effects on electroweak symmetry breaking and the mass spectrum. Eq. (38) has total 6 free parameters, that is, m3/2 , εζ (T + T¯ ), θ , φ, and K5,Z Z¯ , K5,Z . Here we use the same notation for εζ (T + T¯ ) as the case without 5-brane, i.e., α(T + T¯ ) ≡ εζ (T + T¯ ). Note that K5,Z Z¯ is positive definite value due to Eq. (34), (33), but K5,Z is any real number. The parameter K5,Z appears only in the A-parameter and K5,Z -dependence of A is easy to understand, that is, A/ cos φ increases linearly as K5,Z increases. As we shall see, we can derive phenomenological constraints on the magnitudes of K5,Z Z¯ and K5,Z although we lack in their explicit forms. Now, let us concentrate ourselves to the region with |F S | |F T |, that is, sin θ ≈ 1. In this region, Eq. (38) is reduced to ( ) s ¯) m3/2 √ α(T + T 3 3 4 − α(T + T¯ ) sin φ − cos φ , M1/2 = 4 2 K5,Z Z¯ ( 9[4 − α(T + T¯ )]2 2 2 sin2 φ m = m3/2 1 − 3 1 − ¯ + α(T + T¯ )]2 [3(S + S) α(T + T¯ ) 3α(T + T¯ ) 2− cos2 φ − ¯ + α(T + T¯ )] ¯ + α(T + T¯ ) K5,Z Z¯ [3(S + S) 3(S + S) ) s √ 3 6 3[4 − α(T + T¯ )]α(T + T¯ ) sin φ cos φ , (41) − ¯ + α(T + T¯ )]2 K5,Z Z¯ [3(S + S) ( √ 9[4 − α(T + T¯ )] sin φ 3 2− A = m3/2 ¯ + α(T + T¯ ) 3(S + S) ) s 3α(T + T¯ ) 3 cos φ . (42) + K5,Z + ¯ + α(T + T¯ ) K5,Z Z¯ 3(S + S) For example, in the case with α(T + T¯ ) = 1 K5,Z Z¯ = 0.5 and 0.2 lead to m2 < 0 for −0.17 6 tan φ 6 1.9 and −1.1 6 tan φ 6 3.9, respectively. Thus, we have the constraint on the value of K5,Z Z¯ and the region with K5,Z Z¯ > O(1) is favorable. The parameter K5,Z in the A-term also contribute sfermion masses. Because the radiative corrections of the A-terms to the soft scalar masses squared is negative. Furthermore, after the electroweak symmetry breaking the A-terms contribute to the off-diagonal elements of sfermion mass matrices and that reduces the lighter eigenvalues of the masses. Hence, a large value of |K5,Z | makes the constraint m2τ˜ > 0 more severe. 1
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Fig. 6. The soft parameters M1/2 , |A| and m with fixed values of m3/2 = 2.0 TeV, θ = π/2, φ = π/4 and K5,Z Z¯ = K5,Z = 1 as functions of 2α Re(T ) when the 5-brane effects are taken into account.
To compare with the results in Section 2 without 5-brane, here we take tan β = 53. Actually, in the case with such large tan β we have stronger constraints. At first, we consider the case with K5,Z Z¯ = K5,Z = 1. Fig. 6 shows M1/2 , m and A for φ = π/4 as well as m3/2 = 2.0 TeV, θ = π/2, and K5,Z Z¯ = K5,Z = 1. Obviously, the ratios, A/M1/2 and m/M1/2 , are different from Eq. (21). The A-parameter is always positive when φ = π/4. For small α(T + T¯ ), the dominant terms are the first term of M1/2 , and the second term of m2 . The 5-brane SUSY breaking effect F Z within 0 < sin φ < 1 reduces M1/2 , but increases m2 . That is obvious from the comparison between Fig. 6 and Eq. (21). Therefore we obtain m2τ˜1 > m2 0 ∼ M12 at small α(T + T¯ ) and the lightest neutralino is almost Binoχ˜ 1
like. When α(T + T¯ ) become larger, the third and forth term in the right-hand side of (41) for m2 become more sizable and m2 itself is decreased. Thus the excluded region by the constraint m2τ˜1 > 0 is wider than the case without the 5-brane effect. The analyses are shown in Fig. 7(a), (b), where as before the region I is the excluded region by the electroweak symmetry breaking conditions (14) and (15), the region II stands for m2τ˜1 < 0, and the region III for m2τ˜1 < m2 0 . Here we have used θ = π/2, K5,Z Z¯ = K5,Z = 1, and χ˜ 1
φ = π/4 and φ = π/5 for Fig. 7(a) and Fig. 7(b). In the allowed region (white), the lightest neutralino is almost Bino-like. In the 5-brane dominant case, cos φ → 1, most of the region is excluded. Figs. 8(a) and 8(b) show the α(T + T¯ )-dependence and φ-dependence of representative superparticle masses, e.g., the lightest neutralino, chargino, stop and stau. The φdependence is slightly larger than α(T + T¯ )-dependence. When we vary φ from 0.25 to
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(a)
(b)
Fig. 7. 5-brane effects on the electroweak symmetry breaking, the constraint m2τ˜ > 0 and the LSP 1 for θ = π/2 and K5,Z Z¯ = K5,Z = 1. (a) and (b) correspond to φ = π/4 and φ = π/5.
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
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(a)
(b)
Fig. 8. The superparticle spectra against 2α Re(T ). (a) shows 2α Re(T )-dependence of the lightest neutralino, chargino, stau and stop masses for m3/2 = 1.0 TeV, θ = π/2, π/4 and K5,Z Z¯ = K5,Z = 1. Similarly, (b) shows φ-dependence of the superparticle masses for m3/2 = 1.0 TeV, θ = π/2, 2α Re(T ) = 0.5 and K5,Z Z¯ = K5,Z = 1.
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0.5, the lightest Higgs mass mh0 change from 132 GeV to 138 GeV in the allowed region for m3/2 = 1 TeV and θ = π/2. The lightest Higgs mass is rather stable for change of α(T + T¯ ). Next, we calculate K5,Z Z¯ contribution at K5,Z = 1. Fig. 9 shows M1/2 , m and |A| for m3/2 = 2.0 TeV, θ = π/2, φ = π/4, 2α Re(T ) = 1.0 and K5,Z = 1. Obviously, the A/M1/2 and m/M1/2 , are different from Eq. (21), again. The A-parameter changes from positive to negative around K5,Z Z¯ = 2.5 for φ = π/4 and 2α Re(T ) = 1.0. The terms p of the 5-brane contribution of M1/2 and m2 are proportional to 3/K5,Z Z¯ or 3/K5,Z Z¯ . When K5,Z Z¯ increases, then these terms are suppressed and these soft parameters become close to those without 5-brane except for the existence of the “reducing” factor sin φ. Therefore the effects of sin φ is favorable for the m2τ˜1 > 0 condition and it leads to mτ2˜1 > m2 0 . Fig. 10 shows such K5,Z Z¯ -dependence of the regions excluded by the electroweak χ˜ 1
symmetry breaking conditions (region I) and the constraint m2τ˜1 > 0 (region II) as well as the stau LSP condition (region III). The larger K5,Z Z¯ is, the higher is mτ2˜ , but it is not 1 favorable for a successful electroweak symmetry breaking, as we can see from Fig. 10. Finally, we show K5,Z -dependence, which has the linear effect only on A but affects the mass spectrum through A as mentioned above. Fig. 11 show the K5,Z effect on the electroweak symmetry breaking and m2τ˜1 for m3/2 = 2.0 TeV, φ = π/4 and K5,Z Z¯ = 1. As K5,Z increases, the electroweak symmetry breaking conditions exclude only smaller values of α(T + T¯ ) (region I). Consequently, there exists a white region which has no overlap with I, II and III. The region I (unsuccessful electroweak symmetry breaking) disappears at K5,Z > 1.4, but at K5,Z > 3.2 all the region is excluded because of m2τ˜1 < 0. Therefore, K5,Z is constrained by 1 . K5,Z . 2. As a final investigation, we compute representative superparticle masses (i.e., mχ˜ 0 , mτ˜1 , 1 mχ˜1c and mt˜1 ) to see their K5,Z -dependence. This is shown in Fig. 12, and we observe that the masses except for mτ˜1 are rather stable against change of K5,Z .
4. Conclusions We have studied soft SUSY breaking parameters in M-theory with and without 5brane moduli effects. We investigate successful electroweak symmetry breaking and the constraint m2τ˜1 > 0. We have derived the phenomenological constraints on the values of α(T + T¯ ), K5,Z and K5,Z Z¯ , which are characteristic parameters in M-theory. In the allowed regions, we have shown mass spectra. We have considered the large tan β scenario intensively, because these constraints are severe for large tan β. In the case without 5-brane effects, the electroweak symmetry breaking conditions are relaxed in the region with large α(T + T¯ ). In such region, however, the constraint m2τ˜1 > 0 becomes strong in particular for large θ . Then we have the phenomenological upper bound of α(T + T¯ ). In any allowed region, the LSP is the stau.
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
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Fig. 9. The soft parameters M1/2 , |A| and m when we vary K5,Z Z¯ . The other parameters are fixed as m3/2 = 2.0 TeV, θ = π/2, φ = π/4, 2α Re(T ) = 1.0 and K5,Z = 1.
Fig. 10. K5,Z Z¯ -dependence of the electroweak symmetry breaking, the constraint m2τ˜ > 0 and the 1 LSP for m3/2 = 2.0 TeV, θ = π/2, φ = π/4 and K5,Z = 1.
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Fig. 11. K5,Z -dependence of the electroweak symmetry breaking, the constraint m2τ˜ > 0 and the 1 LSP for m3/2 = 2.0 TeV, θ = π/2, φ = π/4 and K5,Z Z¯ = 1. Fig. 11 shows the regions with −1.0 6 K5,Z 6 1.0 .
Fig. 12. The superparticle spectrum against K5,Z for m3/2 = 1.0 TeV, θ = π/2, φ = π/4, 2α Re(T ) = 0.5 and K5,Z Z¯ = 1.
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
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The 5-brane effect can relax the electroweak symmetry breaking condition, but it makes the m2τ˜1 > 0 constraint severe. The value K5,Z Z¯ = O(1) is favorable. The value K5,Z is constrained more strongly than K5,Z Z¯ . In the case with the 5-brane the region with the neutralino LSP appears. The positive D-term contribution m2D2 can relax the constraint m2τ˜1 > 0 and lead to the region with the neutralino LSP. The negative D-term contribution m2D1 can relax the electroweak symmetry breaking conditions. In the suitable value of m2D1 we have the widest region with the neutralino LSP. For small tan β, in particular tan β < 50, the electroweak symmetry breaking is realized easily. However, for large α(T + T¯ ) and large θ , the regions with m2τ˜ < 0 and the region 1 with the stau LSP remain. Soft terms in M-theory with 5-branes include more free parameters. Thus, we will leave to future work systematic study of the whole parameter space on other phenomenological aspects, e.g., cosmological implications and the b → sγ decay [63–69], which leads to another constraint for large tan β and lighter superparticles in the universal case.
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Electroweak symmetry breaking and s-spectrum in M-theory Tatsuo Kobayashi a,b , Jisuke Kubo c , Hitoshi Shimabukuro c a Department of Physics, High Energy Physics Division, University of Helsinki, Helsinki, Finland b Helsinki Institute of Physics, P.O. Box 9, FIN-00014 Helsinki, Finland c Institute for Theoretical Physics, Kanazawa University, Kanazawa, 920-1192, Japan
Received 31 March 1999; revised 19 October 1999; accepted 29 February 2000
Abstract We study soft SUSY breaking parameters in M-theory with and without 5-brane moduli fields. We investigate successful electroweak symmetry breaking and the positivity of stau mass squared. A value of α(T + T¯ ) is phenomenologically constrained. Also we have phenomenological constraints on magnitudes of derivatives of the Kähler potential for the 5-brane moduli field. We study mass spectra in the allowed regions. 2000 Elsevier Science B.V. All rights reserved. PACS: 04.65.+e; 11.25.Mj; 12.60.Jv Keywords: SUSY GUT; Coupling reduction; Soft SUSY breaking parameters; Radiative electroweak breaking
1. Introduction Supersymmetry (SUSY) breaking is very important to connect superstring theory to low-energy physics. Induced soft SUSY breaking terms are significant to currently running and future experiment, and they also have cosmological implications. Actually, soft SUSY breaking terms have been derived, e.g., within the framework of weakly coupled heterotic string theory under the assumption that only F -terms of the dilaton S and moduli fields T i contribute to SUSY breaking [1–5]. Also various phenomenological aspects of these string-inspired soft terms have been studied. Recently strongly coupled superstring theories as well as strongly coupled SUSY gauge theories have been studied. Strongly coupled E8 × E8 heterotic string theory has been considered as M-theory compactified on S 1 /Z2 [6–8]. In particular, the 4-dimensional effective theory has been obtained and SUSY breaking has been studied [9–23]. Soft parameter formulae between the weakly and strongly coupled regions are connected by the parameter α(T + T¯ ). Actually, several phenomenological aspects of soft SUSY breaking 0550-3213/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 1 3 8 - 3
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parameters have been investigated, e.g., radiative electroweak symmetry breaking and s-spectrum [24–29], although the large tan β region has not been studied extensively. The standard embedding leads to the gauge group E6 × E80 , where the top, bottom and tau Yukawa couplings are unified, that is, we have a large value of tan β. In this sense, the large tan β scenario is one natural target to study. When E6 breaks field-theoretically into the standard model (SM) gauge group GSM = SU (3)C × SU (2)L × U (1)Y , integrating-out of heavy modes induces additional soft scalar terms, i.e., D-term contributions. On the other hand, non-standard embedding including Wilson line breaking is also possible [30–33] and that can lead to smaller gauge groups, e.g., SU (3) × SU (2) × U (1) or SU (5). Thus, in non-standard embedding a small value of tan β can also be realized. Furthermore, non-standard embedding, in general, has 5-branes, that is, in 4-dimensional effective field theory we have a new type of moduli fields Z n whose vacuum expectation values (VEVs) correspond to positions of 5-branes zn within [0, 1] along the orbifold S 1 /Z2 . It is possible that F -terms of the 5-brane moduli fields Z n also contribute to SUSY breaking. Recently 4-dimensional effective theory with Z n has been studied [34,35] and SUSY breaking terms have also investigated [36]. The purpose of this paper is to take into account several effects which are mentioned above and to investigate successful electroweak breaking conditions and the lightest superparticle (LSP) systematically. M-inspired soft SUSY breaking parameters have a certain difference from those derived from weakly coupled string models. Such difference depends on the parameter α(T + T¯ ) and the limit α(T + T¯ ) → 0 corresponds to the weakly coupled model without threshold corrections. First of all, the gauge kinetic function has a α(T + T¯ )-dependent term [9,11,14,15] and that implies we have non-vanishing gaugino masses even in the moduli-dominant SUSY breaking. Threshold corrections in the weakly coupled models have similar effects, that is, even for the moduli-dominant SUSY breaking non-vanishing gaugino masses can appear in the weakly coupled models with threshold corrections. However, in M-theory α(T + T¯ )-dependent terms appear in the soft scalar masses and the A-terms, too. Thus, since the parameter α(T + T¯ ) is important, we investigate α(T + T¯ ) dependence of phenomenological aspects, e.g., the mass spectrum. As we shall show, sfermion mass squared can be negative in certain parameter spaces and radiative electroweak symmetry breaking is not realized. Thus, we have phenomenological constraints on the value of α(T + T¯ ) and other free parameters. In particular, in the large tan β scenario, the stau (mass)2 becomes small and negative because of large and negative radiative corrections from the large Yukawa coupling. Thus the stau mass leads to a significant constraint for large tan β. We cover the whole realistic region of tan β including large tan β. We investigate α(T + T¯ )-dependence of electroweak symmetry breaking and s-spectrum. Furthermore, we study the effect due to F -terms of the moduli fields Z n . Their Kähler potential have significant effects on soft SUSY breaking terms. Even though we lack in knowledge about an explicit form of the Kähler potential of Z n , we can derive phenomenological constraints on magnitudes of its derivatives. This paper is organized as follows. In Section 2, after we describe the structure of the soft parameters in M-theory without 5-brane, we investigate the electroweak symmetry
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
5
breaking conditions and the positivity of stau mass squared under the assumption of top-bottom-tau Yukawa coupling unification, that is, the large tan β scenario. D-term contributions, which appear in E6 breaking into GSM , are also taken into account. We also consider the case with small tan β. In Section 3, we consider soft parameters in M-theory that the 5-brane moduli field also contribute to SUSY breaking. In this case, electroweak symmetry breaking and mass spectra are studied and allowed regions are compared with the case without 5-brane effects. Section 4 is devoted to conclusions.
2. SUSY breaking in M-theory without 5-branes In this section we study SUSY breaking in M-theory without 5-branes. For example, standard embedding corresponds to this class. We discuss the soft breaking parameters derived within this framework and study their phenomenological implications. 2.1. Soft parameters The 4-dimensional effective supergravity action of M-theory can be analyzed by expanding it in powers of the two dimensionless variables: ε1 = κ 2/3πρ/V 2/3 and ε2 = κ 2/3/(πρV 1/3 ), likewise the weakly coupled heterotic string theory expanded by the string coupling εs = e2φ /(2π)5 and the world sheet sigma model coupling εσ = 4πα 0 /V 1/3 , where κ 2 , ρ and V denote the 11-dimensional gravitational coupling, the 11-dimensional length and the Calabi–Yau volume respectively. Here we consider only the overall moduli field T . Thus, the 4-dimensional effective supergravity includes the dilaton S and the moduli field T in addition to chiral multiplets Φ p , gauge multiplets and the graviton multiplet. Scalar components of S and T can be identified as 1 (4πκ 2 )−2/3 V , 2π 61/3 (4πκ 2 )−1/3 πρV 1/3 . (1) Re(T ) = 8π The Kählar potential is computed by M-theory expansion for matter fields up to order of κ 4/3 [18], α 3 ¯ − 3 log(T + T¯ ) + + |Φ|2 . (2) K = − log(S + S) T + T¯ S + S¯ Re(S) =
Similarly the gauge kinetic functions of the observable and hidden sectors are calculated [9,11,14,15], fE6 = S + αT , where 1 (8π )α = 64π 3 α 0 2
fE8 = S − αT , Z
1 ω ∧ tr(F ∧ F ) − tr(R ∧ R) . 2
(3)
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In addition, the tree-level superpotential is obtained, W = Ypqr Φ p Φ q Φ r .
(4)
Here α(T + T¯ ) is of the order one at the “physical” point where gauge coupling unification can be realized [9]. Note that the superpotential W and gauge kinetic functions fE6 and fE8 are exact up to nonperturbative corrections, while there are small additional perturbative corrections to the Kählar potential which are of order 1/[4π 2Re(S)] or 1/[4π 2Re(T )]3 ∼ O(αGUT /π) in the strong coupling limit εs 1. Now we know the forms of the Kähler potential K, the gauge kinetic function fE6 and the tree-level superpotential W . Thus, we can derive the formula of soft SUSY breaking parameters following Refs. [3,37]. We assume that a nonperturbative superpotential of S and T is induced and F -terms of S and T contribute to SUSY breaking. The gravitino mass m3/2 is obtained 3m23/2 =
|F S |2 |F T |2 + − V0 , ¯ 2 (T + T¯ )2 3(S + S)
(5)
where V0 is the vacuum energy. Hence, following Ref. [4], we parameterize F -terms, √ ¯ sin θ e−iγS , (6) F S = 3 m3/2C(S + S) T −iγ T , (7) F = m3/2 C(T + T¯ ) cos θ e where C 2 = 1 + V0 /(3m3/2). Using these parameters, we can write the soft parameters, i.e., the gaugino mass M1/2 , the soft scalar mass m and the A-parameter as follows [19] √ 3 Cm3/2 M1/2 = ¯ (S + S) + α(T + T¯ ) ¯ ¯ sin θ e−iγS + α(T√+ T ) cos θ e−iγT , (8) × (S + S) 3 3C 2 m23/2 m2 = V0 + m23/2 − ¯ + α(T + T¯ ) 3(S + S) α(T + T¯ ) sin2 θ × α(T + T¯ ) 2 − ¯ + α(T + T¯ ) 3(S + S) ¯ 3(S + S) ¯ cos2 θ + (S + S) 2 − ¯ + α(T + T¯ ) 3(S + S) √ ¯ 2 3 α(T + T¯ )(S + S) sin θ cos θ cos(γS − γT ) , (9) − ¯ + α(T + T¯ ) 3(S + S) √ 3α(T + T¯ ) sin θ e−iγS A = 3 Cm3/2 −1 + ¯ + α(T + T¯ ) 3(S + S) ¯ √ 3(S + S) −iγT cos θ e + 3 −1 + . (10) ¯ + α(T + T¯ ) 3(S + S) In following analyses we concentrate to the case with the vanishing vacuum energy, V0 = 0 (C = 1) and no CP phase γS = γT = 0.
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Since the gauge kinetic function fE6 provides the gauge coupling constant, that is, 2 , the phenomenological requirement g −2 ' 2 leads to Re fE6 = 1/gE GUT 6 ¯ + α(T + T¯ ) ' 4. (S + S)
(11)
¯ > α(T + T¯ ). Therefore, we have Moreover, the constraint for Re fE8 > 0 leads to (S + S) 0 < α(T + T¯ ) . 2.
(12)
Note that there are three free parameters m3/2 , θ and α(T + T¯ ), and in particular, α(T + T¯ ) is the characteristic parameter of M-theory. The limit α(T + T¯ ) → 0 corresponds to weakly coupled theory. 2.2. Electroweak symmetry breaking and mass spectrum We now analyze phenomenology of M-theory by using Eqs. (8)–(10) as boundary conditions of renormalization group equations (RGEs) at the GUT scale. We investigate successful radiative electroweak symmetry breaking and positive masses squared for any sfermion. That gives constraints for the parameters m3/2 , α(T + T¯ ) and θ . If we assume a certain type of µ-term generation mechanism, we could fix magnitudes of the supersymmetric Higgs mixing mass µ and the corresponding SUSY breaking parameter B. However, we do not take such a procedure here. Because we would like to study generic case. We determine these magnitudes by using the following minimization conditions of the MSSM Higgs potential, m21 + m22 = −
2µB , sin β
m21 − m22 = − MZ2 + m21 + m22 cos 2β,
(13)
where m21,2 = m2H1 ,H2 + µ2 , and m2H1 ,H2 are soft scalar masses of Higgs fields. The Higgs fields for the down sector and the up sector are denoted by H1 and H2 . Using these equations, µ and B are written in terms of the soft scalar masses and tan β. Realization of successful electroweak symmetry breaking leads to the condition, m21 m22 < |µB|2 ,
(14)
and the bounded-from-below condition along the D-flat directions in the Higgs potential, that is, m21 + m22 > 2|µB|,
(15)
should be satisfied in order for stability of the potential along the D-flat direction. In the large tan β scenario, these conditions lead to m2H1 − m2H2 & MZ2 /2.
(16)
Also, we require that a physical mass squared should be positive for any sfermion. Since the sleptons have no large positive-definite radiative corrections from loops involving the gluino, the sleptons become lighter than squarks at low energy. In particular, in the case
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of large tan β, the stau mass squared has a sizable and negative radiative correction due to the large Yukawa coupling. The lighter stau mass becomes significant for large tan β. 1 Furthermore, what is the LSP is an important issue. One candidate is the lightest neutralino and another candidate is the lighter stau for large tan β from the above reason. 2 The stau mass matrix is 2 mτ˜L + (− 12 + sin2 θW )MZ2 cos 2β vYτ (Aτ cos β − µ sin β) , (17) m2τ˜ = m2τ˜R − MZ2 sin2 θW cos 2β vYτ (Aτ cos β − µ sin β) where v 2 ≡ hH1 i2 + hH2 i2 , and we have neglected the tau (mass)2 Mτ2 term here. The lighter stau mass squared is obtained as 1 1 2 m2τ˜L + m2τ˜R − MZ2 cos 2β mτ˜1 = 2 2 2 1 − 2 sin2 θW MZ2 cos 2β − m2τ˜L − m2τ˜R − 2 1/2 2 + 4vYτ (Aτ cos β − µ sin β) . (18) We take the following input parameters, Mτ = 1.777 GeV,
MZ = 91.188 GeV, Mt 8 −1 log (MZ ) = 127.9 + , αEM 9π MZ sin2 θW = 0.2319 − 3.03 × 10−5 ∆t − 8.4 × 10−8 ∆2t ,
(19)
where 1t = Mt [GeV] − 165.0. Here Mτ and Mt are physical tau lepton and top quark masses. We define the SUSY scale as the arithmetic average of the stop squared mass eigenvalues [44,45], 2 MSUSY
=
m2t˜ + m2t˜ 1
2
2
.
(20)
2.2.1. Standard embedding First of all, we consider the case of standard embedding, where the gauge group E6 × E80 is obtained and the top, bottom and tau Yukawa couplings are unified at the GUT scale. From the experimental value of Mτ , we obtain tan β = 53 and Mt = 175 GeV. We assume at the GUT scale E6 breaks into GSM field-theoretically and the MSSM matter content remains light. On top of that, we assume this symmetry breaking induces no extra contributions to soft parameters. Then we use Eqs. (8)–(10) as the initial conditions of RGEs at the GUT scale. The latter assumption will be relaxed later. Before studying the constraints, it is useful to show how the soft parameters, i.e., the gaugino mass M1/2 , A-parameter and the soft scalar mass m at the GUT scale, depend 1 For example, in Refs. [38,39] the constraint m2 > 0 has been considered in details for (finite) SU(5) GUTs τ˜1
and SO(10) GUTs. 2 In Refs. [40–43] cosmological implications of the stau LSP have been discussed. The stau LSP could not be a candidate for cold dark matter. In the case with the stau LSP we need another candidate for cold dark matter.
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Fig. 1. The ratios −A/M1/2 and m/M1/2 against 2α Re(T ). The three solid lines correspond to −A/M1/2 for θ = 0, π/5 and π/2. The two dotted ones correspond to m/M1/2 for θ = π/5 and π/2.
on the parameters, 2α Re(T ) and θ . Note that the A-parameter is always negative. Fig. 1 shows the ratios −A/M1/2 and m/M1/2 against 2α Re(T ) for various values of θ , where the three solid lines and the two dotted lines stand for −A/M1/2 and and m/M1/2 . The uppermost, middle and lowermost solid lines are for −A/M1/2 with θ = 0, π/5 and π/2, respectively. Similarly, the upper and lower dotted lines are for m/M1/2 with θ = π/5 and π/2, respectively. In the weak coupling limit we obtain the universal SSB parameters for for any value of θ 3 : A = −1, M1/2
m 1 =√ . M1/2 3
(21)
For any value of 2α Re(T ) in the region 0.4 < tan θ < 1.0, we find A/M1/2 ≈ −1. In the dilaton-dominant (moduli-dominant) case, however, −A/M1/2 becomes smaller (larger). √ The ratio m/M1/2 = 1/ 3 holds approximately for any value of 2α Re(T ) in the case with 2 decreases as 2α Re(T ) 0.01 . tan θ . 1.0. For most of the other values of tan θ , m2 /M1/2 increases. Also the soft scalar mass squared m2 itself decreases except 0.0 . tan θ . 0.5 as 2α Re(T ) increases. Furthermore, the soft scalar mass squared m2 can become negative. For example, for 2α Re(T ) = 0.5, 1.0 and 1.5 m2 is negative in the regions, −0.2 . tan θ . −1/70, −0.7 . tan θ . −1/30 and −6 . tan θ . −1/20, respectively. Thus, the value of 2α Re(T ) is constrained phenomenologically for fixed tan θ , in particular for negative tan θ . 3 These relations have an implication for finiteness and RG invariance [38,46–52].
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Moreover, negative radiative corrections due to Yukawa couplings make such constraint more severe for the third family of sfermions, stop, sbottom and stau. Now we consider the conditions for the electroweak symmetry breaking (14) and (15) and look for the parameter region in which these conditions are satisfied, where we investigate the cases for θ = π/5, π/4, π/2, and 3π/4. The region I with lines slanting from upper right downwards in Figs. 2(a)–(d) is the region in which the conditions (14) and (15) are NOT satisfied. As one can see from Figs. 2(a)–(d), there are two regions, leftand right-hand sides of the figures. In the region II (with lines slanting from upper left downwards) m2τ˜1 is negative, and so it is not desirable, because the electromagnetic U (1) will be broken in this region. The shaded region III denotes the parameter space in which we find m2τ˜ < m2 0 . Consequently, the parameter space in the region III which does not 1
χ˜ 1
overlap with I and II leads to an MSSM with a stau as the LSP. The constraints coming from a successful electroweak symmetry breaking (14) and (15) allow a wider space with increasing 2α Re(T ), and in particular, the larger θ is, the wider is the allowed space. But too large θ in the case of a large 2α Re(T ) is not desirable, because we would enter in the region II (mτ2˜1 < 0), as we can see from Figs. 2(c) and 2(d). This results from the fact that due to Eq. (9) we have m2 < 0 at the GUT scale. Furthermore, the negative radiative correction due to the Yukawa coupling contributes to widen the region with m2τ˜1 < 0. Thus, we have the phenomenological upper bound of 2α Re(T ) for fixed values of θ and m3/2 . In any parameter space that is not excluded by I and II, in the range of 2α Re(T ), M3/2 and θ we have considered, the LSP is a stau. That is a unique property of this model. 2.2.2. D-term contributions We have assumed that the gauge symmetry E6 breaks down to GSM field-theoretically and the symmetry breaking has no effect on soft parameters. However, integrating-out of heavy modes, in general, affects soft scalar mass terms. In particular, in the case that a gauge group symmetry breaks reducing its rank, the integrating-out of heavy modes induces the so-called D-term contribution to soft scalar mass terms [53–60]. Here we take into effects of such D-term contributions to soft scalar masses. In general, these D-term contributions are proportional to quantum numbers of broken diagonal generators. In the present case, we have two independent universal factors of D-term contributions, which we denote as m2D1 and m2D2 , because the rank of E6 is larger than GSM by two. We regard these as free parameters and then analyze their effects on the parameter space. The soft scalar masses at the GUT scale including D-term contributions are written as m2Q˜ = m2t˜ = m2τ˜R = m227 − m2D1 + m2D2 , m2b˜ = m2τ˜L = m227 + 3m2D1 + m2D2 , m2H1 = m227 − 2m2D1 − 2m2D2 , m2H2 = m227 + 2m2D1 − 2m2D2 , where m27 is the soft scalar mass m in Eq. (9).
(22)
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(a)
(b)
Fig. 2. The excluded regions by the electroweak breaking conditions (14) and (15) (region I), the constraint m2τ˜ > 0 (region II) and the region with m2τ˜ > m2 0 (region III). (a) and (b) correspond to 1
the cases with θ = π/5, π/4.
1
χ˜ 1
12
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
(c)
(d)
Fig. 2. (Continued.) The excluded regions by the electroweak breaking conditions (14) and (15) (region I), the constraint m2τ˜ > 0 (region II) and the region with m2τ˜ > m2 0 (region III). (c) and (d) 1
correspond to the cases with θ = π/2, 3π/4.
1
χ˜ 1
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
13
We now analyze D-term contribution effects on the parameter space at tan β = 53, which has been obtained under the assumption of the top-bottom-tau Yukawa coupling unification. The term m2D1 with a negative and sizable value is helpful to create a positive gap m2H 1 − m2H 2 , which is favorable to successful electroweak symmetry breaking. Also, a negative value of m2D1 increases m2τ˜R , but it reduces m2τ˜L . The (mass)2 m2τ˜R is dominant to m2τ˜1 around the universal case with m2τ˜R = m2τ˜L . Therefore negative m2D1 may increase m2τ˜ and seems to be favorable to the constraint of m2τ˜ . However, as m2D1 decreases, m2τ˜L is 1 1 decreased and below a critical value of m2D1 m2τ˜L becomes dominant to mτ2˜1 . A similar effect of the D-term contribution has been discussed within the framework of SO(10) GUTs [39]. On the other hand, a positive value of m2D2 increases both mτ2˜R and m2τ˜L . Thus, a positive value of m2D2 is favorable to increase m2τ˜1 . The effects of m2D2 to m2H 1 and m2H 2 are same. Hence, m2D2 has little effect on electroweak symmetry breaking. These results are shown in Fig. 3(a), (b) and Fig. 4(a), (b) for θ = π/2. Fig. 3 show the m2D2 effects, where we have assumed that m2D1 = 0 and have chosen m2D2 = 0.5 and 1.0 (TeV)2 for Figs. 3(a) and 3(b), respectively. A large value of m2D2 reduces the region excluded by the constraint m2τ˜1 > 0, while it has little effect on electroweak symmetry breaking. The region with the neutralino LSP appears and it becomes wider as m2D2 increases. For example, in the case of m2D2 = 1.0 (TeV)2 , the region with the neutralino LSP has the largest value m3/2 = 3.1 TeV when α(T + T¯ ) = 0.8. In this case the lightest neutral Higgs boson mass mh0 is 151 GeV. Here we use the two loop Higgs mass mh0 [44, 45,61,62]. Similarly, we obtain mh0 = 129 GeV when m3/2 = 1.0 TeV and α(T + T¯ ) = 0.8. This value is stable between m3/2 = 0.4 TeV and 1.0 TeV. Fig. 4(a), (b) show the m2D1 effects, where we have assumed that m2D2 = 0. They show that the region excluded by the electroweak symmetry breaking conditions becomes more narrow and the region with m2τ˜1 < 0 becomes wider as m2D1 decreases. In suitable values of m2D1 , we have the region with the neutralino LSP. The critical value is around m2D1 = −2.0 (TeV)2 , which is shown in Fig. 4(b), where the neutralino LSP region is widest. This should be compared with Fig. 4(a) for which we have used m2D1 = −0.05 (TeV)2 .
2.2.3. Small tan β scenario It is also possible that we obtain a smaller gauge group, e.g., GSM or SU (5) at the GUT scale by non-standard embedding including Wilson line breaking. In this case, the assumption of the top-bottom-tau Yukawa coupling unification can be relaxed and a smaller value of tan β can be realized. We assume that below the GUT scale the matter content is same as the MSSM. Here, we vary tan β holding Eqs. (8)–(10) as the initial conditions of RGEs at the GUT scale and fixing Mt = 175 GeV. Fig. 5(a), (b) show that the allowed parameter region for tan β and α(T + T¯ ). Fig. 5(a) shows that for a small value of | sin θ |; there is no excluded region at tan β < 50. However, for large values of | sin θ | and α(T + T¯ ), the excluded regions I and II appear even in the case of small tan β, which is shown in Fig. 5(b).
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T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
(a)
(b)
Fig. 3. Effects of the D-term contribution m2D2 on the excluded region for θ = π/2. (a) and (b) correspond to the cases with (m2D1 , m2D2 ) = (0, 0.5) and (0, 1.0) [(TeV)2 ], respectively.
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
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(a)
(b)
Fig. 4. Effects of the D-term contribution m2D1 on the excluded region for θ = π/2. (a) and (b) correspond to the cases with (m2D1 , m2D2 ) = (−0.05, 0) and (−2.0, 0) [(TeV)2 ], respectively.
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T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
(a)
(b)
Fig. 5. The excluded region in the tan β − 2α Re(T ) space. (a) corresponds to the case with θ = π/5 and m3/2 = 1.0 TeV for Mtop = 175.0 GeV. Similarly, (b) corresponds to the case with θ = π/2 and m3/2 = 2.0 TeV for Mtop = 175.0 GeV.
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
17
3. M-theory with 5-brane 3.1. Soft parameters In general, non-standard embedding includes 5-branes, that is, 4-dimensional effective theory includes 5-brane moduli fields Z n . It is possible that Z n also contribute to SUSY breaking. Thus, we consider such effects for generic case. If the 5-branes exist between the boundaries in M-theory, the Kähler potential and the gauge kinetic functions are modified. The relevant part of the moduli Kähler potential Kmod and the gauge kinetic functions of the observable and hidden sectors f (1) and f (2) with N 5-branes take the following form [34–36], ¯ − 3 ln(T + T¯ ) + K5 , Kmod = − ln(S + S) f (1) = S + εT β (0) +
N X
!
(23)
(1 − Z n )2 β (n) ,
n=1
f
(2)
= S + εT β
(N+1)
! N X n 2 (n) + (Z ) β ,
(24)
n=1
where K5 is the Kähler potential for the 5-brane position moduli Z n and the expansion parameter ε = (κ/4π)2/3 2π 2 ρ/V 2/3 . In addition, β (0) , β (N+1) are the instanton numbers on the observable sector boundary and the hidden sector boundary respectively, and β (n) (n = 1, . . . , N) is a magnetic charge on the each 5-brane. They should satisfy the cohomology constraint, N+1 X
(n)
βi
= 0,
(25)
n=0
which means physically that the net charges must be zero because the “flux” cannot get away anywhere in compact space. The part of the Kähler potential for the matter fields is obtained εζ 3 + |Φ|2 , (26) Kmat = ¯ (T + T¯ ) (S + S) where ζ = β (0) +
N X
(1 − zn )2 β (n) .
(27)
n=1
Following the generic formulae of Refs. [3,37] again, we can derive the soft parameters. We assume V0 = 0 again. The soft parameters are obtained, S 1 F + εζ F T + εT F n ζn , ¯ ¯ (S + S) + εζ (T + T ) S 2 9 1 2 2 − m = m3/2 − F ¯ 2 [3(S + S) ¯ + εζ (T + T¯ )]2 (S + S)
M1/2 =
(28)
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(εζ )2 1 − ¯ + εζ (T + T¯ )]2 (T + T¯ )2 [3(S + S) ¯ εζnm¯ (T + T ) ε2 ζn ζm¯ (T + T¯ )2 n ¯m ¯ − −F F ¯ + εζ (T + T¯ ) [3(S + S) ¯ + εζ (T + T¯ )]2 3(S + S) εζ εζm¯ (T + T¯ ) ¯ + 3F S F¯ m¯ + 3F S F¯ T 2 ¯ + εζ (T + T¯ )] ¯ + εζ (T + T¯ )]2 [3(S + S) [3(S + S) ε2 ζ ζm¯ (T + T¯ ) εζm¯ − − F T F¯ m¯ + (c.c.), ¯ + εζ (T + T¯ ) [3(S + S) ¯ + εζ (T + T¯ )]2 3(S + S) (29) 9 2 − A = FS ¯ + εζ (T + T¯ ) S + S¯ 3(S + S) εζ + FT ¯ + εζ(T + T¯ ) 3(S + S) 3εζn (T + T¯ ) n , (30) + F K5,n − ¯ + εζ (T + T¯ ) 3(S + S) 2 − F T
where K5,n = ∂K5 /∂Z n , ζn = ∂ζ /∂zn and ζnm = ∂ 2 ζ /(∂zn ∂zm ). We have normalized the soft scalar mass m as, m02 = m2
∂ 2 Kmat , |∂Φ|2
(31)
using the total Kähler metric, where m0 is the mass for the field Φ with non-canonical kinetic factor, ∂ 2 Kmat /|∂Φ|2 . That is different from Ref. [36], where the soft scalar mass is normalized as 3 . (32) m02 = m2 (T + T¯ )2 Here we ignore CP phases, i.e., F = F¯ . For simplicity, we assume that there is only one ¯ i.e., relevant 5-brane moduli Z and its Kähler potential is a function of only (Z + Z), ¯ K5 = K5 (Z + Z).
(33)
The gravitino mass is obtained, m23/2 =
2 |F S |2 |F T |2 1 + + K5,Z Z¯ F Z . ¯ 2 (T + T¯ )2 3 3(S + S)
Now we can parameterize each F -term as follows √ ¯ sin θ sin φ, F S = 3 m3/2 (S + S) F T = m3/2(T + T¯ ) cos θ sin φ, s 3 Z m3/2 cos φ. F = K5,Z Z¯
(34)
(35)
Still now we have several unknown parameters and most of them are due to detailed information of the 5-brane. Thus, we fix z and β (1) , e.g.,
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
1 z= , 2
4 β (1) = β (0). 3
19
(36)
Note that both the instanton number β (0) and the intersection number β (1) are integers. Eq. (36) leads to the following simple relations with Eq. (27) as 1 (37) ζ |z=1/2 = −ζ,z |z=1/2 = ζ,zz |z=1/2 = β (1) . 2 In this case we can rewrite the soft parameters, ( √ m3/2 ¯ sin θ sin φ 3(S + S) M1/2 = ¯ + εζ (T + T¯ ) (S + S) ) s ¯) 3 εζ (T + T cos φ , + εζ (T + T¯ ) cos θ sin φ − 2 K5,Z Z¯ ( ¯ 2 9(S + S) 2 2 sin2 θ sin2 φ m = m3/2 1 − 3 1 − ¯ + εζ (T + T¯ )]2 [3(S + S) " 2 # εζ (T + T¯ ) cos2 θ sin2 φ − 1− ¯ + εζ (T + T¯ ) 3(S + S) εζ (T + T¯ ) 3εζ (T + T¯ ) 2− cos2 φ − ¯ + εζ (T + T¯ )] ¯ + εζ (T + T¯ ) K5,Z Z¯ [3(S + S) 3(S + S) √ ¯ (T + T¯ ) 6 3(S + S)εζ sin θ cos θ sin2 φ + ¯ + εζ (T + T¯ )]2 [3(S + S) s √ ¯ (T + T¯ ) 3 6 3(S + S)εζ sin θ sin φ cos φ − 2 ¯ ¯ K [3(S + S) + εζ (T + T )] 5,Z Z¯ εζ (T + T¯ ) 2εζ (T + T¯ ) 1− + ¯ + εζ (T + T¯ ) ¯ + εζ (T + T¯ ) 3(S + S) 3(S + S) ) s 3 cos θ sin φ cos φ , × K5,Z Z¯ ( ¯ √ 9(S + S) sin θ sin φ 3 2− A = m3/2 ¯ + εζ (T + T¯ ) 3(S + S) 3εζ (T + T¯ ) cos θ sin φ ¯ + εζ (T + T¯ ) 3(S + S) ) s 3εζ (T + T¯ ) 3 cos φ , + K5,Z + ¯ + εζ (T + T¯ ) K5,Z Z¯ 3(S + S)
−
(38)
where we have neglected the phase of T . In addition, from the gauge kinetic functions (24) we obtain constraints for values of moduli S and T as, ¯ + εζ (T + T¯ ) = 2g −2 ' 4, (39) 2 Re f (1) = (S + S) GUT
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and ¯ − 3 εζ (T + T¯ ) > 0. (40) 2 Re f (2) = (S + S) 2 Here we use Eqs. (36), (37), (25), (39) and (40), so that we find the parameter region as 0 < εζ (T + T¯ ) < 8/5. That is almost comparable to the region 0 < α(T + T¯ ) < 2 in the case without 5-branes. 3.2. Electroweak symmetry breaking and mass spectrum Using Eq. (38), we analyze the 5-brane effects on electroweak symmetry breaking and the mass spectrum. Eq. (38) has total 6 free parameters, that is, m3/2 , εζ (T + T¯ ), θ , φ, and K5,Z Z¯ , K5,Z . Here we use the same notation for εζ (T + T¯ ) as the case without 5-brane, i.e., α(T + T¯ ) ≡ εζ (T + T¯ ). Note that K5,Z Z¯ is positive definite value due to Eq. (34), (33), but K5,Z is any real number. The parameter K5,Z appears only in the A-parameter and K5,Z -dependence of A is easy to understand, that is, A/ cos φ increases linearly as K5,Z increases. As we shall see, we can derive phenomenological constraints on the magnitudes of K5,Z Z¯ and K5,Z although we lack in their explicit forms. Now, let us concentrate ourselves to the region with |F S | |F T |, that is, sin θ ≈ 1. In this region, Eq. (38) is reduced to ( ) s ¯) m3/2 √ α(T + T 3 3 4 − α(T + T¯ ) sin φ − cos φ , M1/2 = 4 2 K5,Z Z¯ ( 9[4 − α(T + T¯ )]2 2 2 sin2 φ m = m3/2 1 − 3 1 − ¯ + α(T + T¯ )]2 [3(S + S) α(T + T¯ ) 3α(T + T¯ ) 2− cos2 φ − ¯ + α(T + T¯ )] ¯ + α(T + T¯ ) K5,Z Z¯ [3(S + S) 3(S + S) ) s √ 3 6 3[4 − α(T + T¯ )]α(T + T¯ ) sin φ cos φ , (41) − ¯ + α(T + T¯ )]2 K5,Z Z¯ [3(S + S) ( √ 9[4 − α(T + T¯ )] sin φ 3 2− A = m3/2 ¯ + α(T + T¯ ) 3(S + S) ) s 3α(T + T¯ ) 3 cos φ . (42) + K5,Z + ¯ + α(T + T¯ ) K5,Z Z¯ 3(S + S) For example, in the case with α(T + T¯ ) = 1 K5,Z Z¯ = 0.5 and 0.2 lead to m2 < 0 for −0.17 6 tan φ 6 1.9 and −1.1 6 tan φ 6 3.9, respectively. Thus, we have the constraint on the value of K5,Z Z¯ and the region with K5,Z Z¯ > O(1) is favorable. The parameter K5,Z in the A-term also contribute sfermion masses. Because the radiative corrections of the A-terms to the soft scalar masses squared is negative. Furthermore, after the electroweak symmetry breaking the A-terms contribute to the off-diagonal elements of sfermion mass matrices and that reduces the lighter eigenvalues of the masses. Hence, a large value of |K5,Z | makes the constraint m2τ˜ > 0 more severe. 1
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
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Fig. 6. The soft parameters M1/2 , |A| and m with fixed values of m3/2 = 2.0 TeV, θ = π/2, φ = π/4 and K5,Z Z¯ = K5,Z = 1 as functions of 2α Re(T ) when the 5-brane effects are taken into account.
To compare with the results in Section 2 without 5-brane, here we take tan β = 53. Actually, in the case with such large tan β we have stronger constraints. At first, we consider the case with K5,Z Z¯ = K5,Z = 1. Fig. 6 shows M1/2 , m and A for φ = π/4 as well as m3/2 = 2.0 TeV, θ = π/2, and K5,Z Z¯ = K5,Z = 1. Obviously, the ratios, A/M1/2 and m/M1/2 , are different from Eq. (21). The A-parameter is always positive when φ = π/4. For small α(T + T¯ ), the dominant terms are the first term of M1/2 , and the second term of m2 . The 5-brane SUSY breaking effect F Z within 0 < sin φ < 1 reduces M1/2 , but increases m2 . That is obvious from the comparison between Fig. 6 and Eq. (21). Therefore we obtain m2τ˜1 > m2 0 ∼ M12 at small α(T + T¯ ) and the lightest neutralino is almost Binoχ˜ 1
like. When α(T + T¯ ) become larger, the third and forth term in the right-hand side of (41) for m2 become more sizable and m2 itself is decreased. Thus the excluded region by the constraint m2τ˜1 > 0 is wider than the case without the 5-brane effect. The analyses are shown in Fig. 7(a), (b), where as before the region I is the excluded region by the electroweak symmetry breaking conditions (14) and (15), the region II stands for m2τ˜1 < 0, and the region III for m2τ˜1 < m2 0 . Here we have used θ = π/2, K5,Z Z¯ = K5,Z = 1, and χ˜ 1
φ = π/4 and φ = π/5 for Fig. 7(a) and Fig. 7(b). In the allowed region (white), the lightest neutralino is almost Bino-like. In the 5-brane dominant case, cos φ → 1, most of the region is excluded. Figs. 8(a) and 8(b) show the α(T + T¯ )-dependence and φ-dependence of representative superparticle masses, e.g., the lightest neutralino, chargino, stop and stau. The φdependence is slightly larger than α(T + T¯ )-dependence. When we vary φ from 0.25 to
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(a)
(b)
Fig. 7. 5-brane effects on the electroweak symmetry breaking, the constraint m2τ˜ > 0 and the LSP 1 for θ = π/2 and K5,Z Z¯ = K5,Z = 1. (a) and (b) correspond to φ = π/4 and φ = π/5.
T. Kobayashi et al. / Nuclear Physics B 580 (2000) 3–28
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(a)
(b)
Fig. 8. The superparticle spectra against 2α Re(T ). (a) shows 2α Re(T )-dependence of the lightest neutralino, chargino, stau and stop masses for m3/2 = 1.0 TeV, θ = π/2, π/4 and K5,Z Z¯ = K5,Z = 1. Similarly, (b) shows φ-dependence of the superparticle masses for m3/2 = 1.0 TeV, θ = π/2, 2α Re(T ) = 0.5 and K5,Z Z¯ = K5,Z = 1.
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0.5, the lightest Higgs mass mh0 change from 132 GeV to 138 GeV in the allowed region for m3/2 = 1 TeV and θ = π/2. The lightest Higgs mass is rather stable for change of α(T + T¯ ). Next, we calculate K5,Z Z¯ contribution at K5,Z = 1. Fig. 9 shows M1/2 , m and |A| for m3/2 = 2.0 TeV, θ = π/2, φ = π/4, 2α Re(T ) = 1.0 and K5,Z = 1. Obviously, the A/M1/2 and m/M1/2 , are different from Eq. (21), again. The A-parameter changes from positive to negative around K5,Z Z¯ = 2.5 for φ = π/4 and 2α Re(T ) = 1.0. The terms p of the 5-brane contribution of M1/2 and m2 are proportional to 3/K5,Z Z¯ or 3/K5,Z Z¯ . When K5,Z Z¯ increases, then these terms are suppressed and these soft parameters become close to those without 5-brane except for the existence of the “reducing” factor sin φ. Therefore the effects of sin φ is favorable for the m2τ˜1 > 0 condition and it leads to mτ2˜1 > m2 0 . Fig. 10 shows such K5,Z Z¯ -dependence of the regions excluded by the electroweak χ˜ 1
symmetry breaking conditions (region I) and the constraint m2τ˜1 > 0 (region II) as well as the stau LSP condition (region III). The larger K5,Z Z¯ is, the higher is mτ2˜ , but it is not 1 favorable for a successful electroweak symmetry breaking, as we can see from Fig. 10. Finally, we show K5,Z -dependence, which has the linear effect only on A but affects the mass spectrum through A as mentioned above. Fig. 11 show the K5,Z effect on the electroweak symmetry breaking and m2τ˜1 for m3/2 = 2.0 TeV, φ = π/4 and K5,Z Z¯ = 1. As K5,Z increases, the electroweak symmetry breaking conditions exclude only smaller values of α(T + T¯ ) (region I). Consequently, there exists a white region which has no overlap with I, II and III. The region I (unsuccessful electroweak symmetry breaking) disappears at K5,Z > 1.4, but at K5,Z > 3.2 all the region is excluded because of m2τ˜1 < 0. Therefore, K5,Z is constrained by 1 . K5,Z . 2. As a final investigation, we compute representative superparticle masses (i.e., mχ˜ 0 , mτ˜1 , 1 mχ˜1c and mt˜1 ) to see their K5,Z -dependence. This is shown in Fig. 12, and we observe that the masses except for mτ˜1 are rather stable against change of K5,Z .
4. Conclusions We have studied soft SUSY breaking parameters in M-theory with and without 5brane moduli effects. We investigate successful electroweak symmetry breaking and the constraint m2τ˜1 > 0. We have derived the phenomenological constraints on the values of α(T + T¯ ), K5,Z and K5,Z Z¯ , which are characteristic parameters in M-theory. In the allowed regions, we have shown mass spectra. We have considered the large tan β scenario intensively, because these constraints are severe for large tan β. In the case without 5-brane effects, the electroweak symmetry breaking conditions are relaxed in the region with large α(T + T¯ ). In such region, however, the constraint m2τ˜1 > 0 becomes strong in particular for large θ . Then we have the phenomenological upper bound of α(T + T¯ ). In any allowed region, the LSP is the stau.
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Fig. 9. The soft parameters M1/2 , |A| and m when we vary K5,Z Z¯ . The other parameters are fixed as m3/2 = 2.0 TeV, θ = π/2, φ = π/4, 2α Re(T ) = 1.0 and K5,Z = 1.
Fig. 10. K5,Z Z¯ -dependence of the electroweak symmetry breaking, the constraint m2τ˜ > 0 and the 1 LSP for m3/2 = 2.0 TeV, θ = π/2, φ = π/4 and K5,Z = 1.
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Fig. 11. K5,Z -dependence of the electroweak symmetry breaking, the constraint m2τ˜ > 0 and the 1 LSP for m3/2 = 2.0 TeV, θ = π/2, φ = π/4 and K5,Z Z¯ = 1. Fig. 11 shows the regions with −1.0 6 K5,Z 6 1.0 .
Fig. 12. The superparticle spectrum against K5,Z for m3/2 = 1.0 TeV, θ = π/2, φ = π/4, 2α Re(T ) = 0.5 and K5,Z Z¯ = 1.
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The 5-brane effect can relax the electroweak symmetry breaking condition, but it makes the m2τ˜1 > 0 constraint severe. The value K5,Z Z¯ = O(1) is favorable. The value K5,Z is constrained more strongly than K5,Z Z¯ . In the case with the 5-brane the region with the neutralino LSP appears. The positive D-term contribution m2D2 can relax the constraint m2τ˜1 > 0 and lead to the region with the neutralino LSP. The negative D-term contribution m2D1 can relax the electroweak symmetry breaking conditions. In the suitable value of m2D1 we have the widest region with the neutralino LSP. For small tan β, in particular tan β < 50, the electroweak symmetry breaking is realized easily. However, for large α(T + T¯ ) and large θ , the regions with m2τ˜ < 0 and the region 1 with the stau LSP remain. Soft terms in M-theory with 5-branes include more free parameters. Thus, we will leave to future work systematic study of the whole parameter space on other phenomenological aspects, e.g., cosmological implications and the b → sγ decay [63–69], which leads to another constraint for large tan β and lighter superparticles in the universal case.
Acknowledgement The authors would like to thank K. Enqvist and S.-J. Rey for useful discussions. This work was partially supported by the Academy of Finland (no. 44129). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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