OHSTPY-HEP-T-10-007 1 1 Gauge Coupling Unification in Heterotic String 0 2 Models with Gauge Mediated SUSY Breaking n a J 0 1 ] h p - p Archana Anandakrishnan and Stuart Raby e h [ 1 Department of Physics, The Ohio State University, v 6 191 W. Woodruff Ave, Columbus, OH 43210, USA 7 9 1 . 1 0 1 Abstract 1 : v We calculate the weak scale MSSM spectrum starting from a heterotic i X string theory compactified onan anisotropicorbifold. Supersymmetry break- r ing is mediated by vector-like exotics that arise naturally in heterotic string a theories. The messengers that mediate SUSY breaking come in incomplete GUT multiplets and give rise to non-universal gaugino masses at the GUT scale. Models with non-universal gaugino masses at the GUT scale have the attractive feature of allowing for precision gauge coupling unification at the GUT scale with negligible contributions from threshold corrections near the unification scale. The unique features of the MSSM spectrum are light gluinos and also large mass differences between the lightest and the next-to- lightest neutralinos and charginos which could lead to interesting signatures at the colliders. 1 1 Introduction Grand unification of the fundamental forces is a very appealing idea. It was noticed in as early as 1974, that when the couplings of the three fundamental interactions are run to high energies, they seem to meet at a point [1],[2]. Supersymmetry is required for precise unification [3], without which there is a discrepancy of about 12 σ [4]. With supersymmetry, assuming universal scalar and gaugino masses at the GUT scale, precision electroweak data requires [5], for the strong coupling constant to match experiments, that α 3 be about 3 − 4% smaller than α and α at the GUT scale. This conflict 1 2 between the coupling constants at the GUT scale can be eased by including ‘threshold corrections’ from extra states around the GUT scale. In grand unified theories, the Higgs fields have to respect the GUT sym- metryandthus existence of Higgsdoublets alsoimplies theexistence ofHiggs triplets. In order to avoid rapid proton decay the triplets necessarily have mass greater than the GUT scale which introduces some unpleasantness into these SUSY GUT theories. In addition, complicated symmetry breaking potentials are required to break the GUT symmetry. Theories with extra dimensions have gained popularity in this respect, since they can eliminate some of the problems with 4D SUSY GUTs. In theories with extra dimen- sions, connectionismadetothelow-energyworld, bycompactifyingtheextra dimensions. The choice of boundary conditions then can lead to natural and simple solutions to the problems hindering SUSY GUTs. The threshold cor- rections required to match precision electroweak data can come from massive states around the GUT scale and from Kaluza-Klein states living between the compactification scale of the extra dimensions, M and the cut-off scale, C M∗; in string theory M∗ is the string scale, MS. Recent searches for the MSSM from heterotic string theory have yielded interesting results. Orbifold compactifications of the E ×E heterotic string 8 8 theoryhave beenshowntoyieldrealisticmodelsthatincludethegaugegroup and the matter content of the MSSM [12, 13]. In addition, the models also have vector-like exotics with Standard Model charges that obtain mass in the supersymmetric limit. They may couple to the SUSY breaking field, and mediate supersymmetry breaking. The mechanism of supersymmetry break- ing plays a very important role in understanding the low-energy spectrum, and in this case the possibility of Gauge Mediated Supersymmetry breaking 2 (GMSB) [9]. In GMSB, the gauginos receive mass at one-loop as: α hFi i M ∼ (1) i 4πhM i φ where, φ is the messenger field with mass, M , and hFi is the SUSY breaking φ VEV. Thus, a heavier messenger (in this case, the exotics) corresponds to a lighter gaugino. It was shown earlier in [6] that light(of the order 109−1013 GeV) vector- like exotic states were required for gauge coupling unification, assuming the standard scenario with universal gaugino masses at the GUT scale and threshold corrections of about -3%. Solutions to gauge coupling unification were constrained by the bounds on proton decay. It was also assumed that all the vector-like exotics obtain mass at the same scale. The exotics come in incomplete GUT multiplets and hence, in general could obtain mass at dif- ferent scales. In this work, we generalize the solutions, allowing the exotics that carry SU(3) and SU(2) charges to obtain mass at different scales. We build a consistent MSSM spectrum at the weak scale with these exotic mes- sengers. We find that this generalization increases the number of solutions satisfying gauge coupling unification. In addition, the weak scale spectrum now allows unification with moderate or even zero threshold corrections at the GUT scale. The low energy spectrum in such a case has light gluinos which should be detected at the Tevatron and/or LHC. 2 Gauge Mediated Supersymmetry Breaking Gauge Mediated SUSY Breaking(GMSB) [18] models have chiral supermul- tiplets called messenger fields that mediate supersymmetry breaking. The messenger fields carry SU(3) × SU(2) × U(1) charges and hence couple to the matter fields of the MSSM through the usual SU(3) × SU(2) × U(1) gauge interactions. The messenger fields are very massive at some scale, denoted by M . Sources of flavor violation near the messenger scale are mess given by (4+d)-dimension operators that are suppressed by 1 . Hence, (Mmess)d the major source of flavor violation is due to Yukawa couplings, similar to the Standard Model(SM). This suppression of flavor changing neutral currents (FCNC) is the most attractive feature of GMSB. In string theories where the extra-dimensions are compactified on an orbifold, there exist extra vector-like non-standard model particles, usually 3 called “exotics”. These exotics need to be heavy in order for them to de- couple from the low-energy theory. The exotics carry charges under the SM gauge group, and hence are perfect candidates to mediate supersymmetry breaking via gauge mediation. We build a consistent MSSM spectrum at the weak scale with these exotic messengers. By consistent, we mean that if we start at the highest scale in the model and run the coupling constants and soft SUSY breaking parameters all the way down to the weak scale, integrat- ing out heavy states during this running, we must end up with the coupling constants that match the experimental values at the weak scale. Since this requires knowledge of the spectrum of exotics and Kaluza-Klein modes, as well as the MSSM spectrum, we perform our analysis in two steps: Step #1WeconcentrateonaclassofmodelsbasedonSU(6)gauge-Higgs unification in 5D [12, 13]. Starting from heterotic string theory compactified onananisotropic orbifold, T6/Z -IIandapplying the‘Phenomenological Pri- 6 ors’ - Inequivalent models with the SM gauge group, 3 SM families, Higges, and non-anomalous U(1) ⊂ SU(5); the authors end up with 15 models Y consisting of low energy spectrum that is similar to that of MSSM. In addi- tion, the spectrum consists of heavy vector-like exotics that decouple from the low-energy theory. Gauge coupling unification was studied [6] in 2 out of the above 15 models [13] - “Model 1A” and “Model 2”. The matter content of both these models are very similar and is summarized in Table 3. For the gauge couplings to unify in the heterotic orbifold theory, it was noted in ′ Ref.[6] that there had to be at least ~n = (n ,n ,(n ,n )) ‘light’ exotics at 3 2 1 1 some intermediate scale M , below the 4D unification(GUT) scale. This EX scale, M was determined by matching the Renormalization Group Equa- EX tions(RGE) from the two theories - heterotic orbifold model and the 4D MSSM at some low energy scale µ, where both the theories predict the same running for the couplings. In the 4D MSSM, the gauge couplings unify at the GUT scale with some threshold corrections from new physics near the GUT scale whereas, on the heterotic side, the gauge couplings unify at the string scale, M (See Fig.3). The analysis in Ref.[6] was done in the context of a S minimal scenario where all the light exotics obtained mass at the same scale and to accommodate precision electroweak data, a -3 % threshold correction was assumed at the GUT scale. The exotics come in incomplete SU(5) GUT multiplets and in general could obtain mass at different scales. We therefore relax the previous as- 4 sumption that the light exotics obtain mass at the same scale and allow those that carry SU(3) and SU(2) charges to obtain mass at different scales. This leads to non-universal gaugino masses at the GUT scale, as a conse- quence of which, the threshold corrections required to match precision data at this scale is no longer of order -3%. The GUT scale threshold correction is a priori a free parameter which depends on the spectrum of states near the GUT scale. However, it’s value needs to be fixed by evaluating the 2-loop RGE running from the string scale to the weak scale and including one loop threshold corrections at both the weak and the GUT scales, self-consistently. This is done in the next step. The new intermediate scales, M and M EX3 EX2 are determined self-consistently using the RGEs. The details of the calcula- tion of the light exotics mass spectrum is given in Appendix A. Step #2 Once theexotic masses are determined, the soft SUSY breaking terms are calculated at this scale, i.e. the messenger scale [9]. We then use SOFTSUSY [11] to run them down to the weak scale.1 SOFTSUSY uses the 2-loop RG running to determine the weak scale MSSM spectrum and the 1-loop weak scale threshold corrections. The MSSM parameters are then run back again to the GUT scale and the GUT scale threshold corrections are calculated, i.e. the (output) values fixed by SOFTSUSY (see Eq. (7)). We compare this value of the GUT threshold corrections with the (input) value determined independently by the exotic mass spectrum (see Eq. (11)). We vary the arbitrary parameters of the orbifold string theory and save only those cases where the input (determined by the exotic mass spectrum) and output (required by the low energy MSSM spectrum) threshold corrections match. We also only keep cases consistent with the bound on the proton lifetime and a lower bound on the Higgs mass. 1Note, SOFTSUSY runs from the presumed 4D GUT scale to the weak scale. In our case the soft masses are only determined at the messenger scale. The error made by matching the 5D theory to the 4D theory at the messenger scale is however,small. Using the fact that up to 1-loop, the ratio M /α = constant, we calculate the soft-masses at i i the GUT scale where they get small 2-loop corrections. As shown in Appendix. B, the corrections to the gaugino masses are found to be less than 1% and can be neglected. A detailed discussion of the soft SUSY breaking masses is in Section 2.1. 5 2.1 Soft Masses Theexotics oftheorbifoldtheorymediatesupersymmetry breaking by acting as messengers of gauge mediation. Due to the gauge interactions of the messengers, soft terms are generated at the messenger scale. We assume that the gravity-mediated SUSY breaking contributions to gaugino masses are much smaller than the gauge-mediated contribution. There can be anomaly mediatedcontributionstothesoftmasses, proportionaltothegravitinomass, m , or dilaton contributions proportional to F /M .2 We allow for a large 3/2 S Pl gravitino mass of the order of a few TeV. However, if the ratio, F ≫ m , MEX 3/2 we can ignore the gravitino contribution to the gaugino masses. At one loop, the gauginos masses are given by: α F α F M = bEX3 i +bEX2 i (2) i i 4πM i 4πM EX3 EX2 where F is the SUSY breaking VEV, which at this point is chosen to be arbitrary. The scalars obtain mass at two-loops, and the dominant contribution to their mass is from the gravitino. In addition, in string models, it is natural to have an anomalous U(1) gauge interaction. Such interactions can add X an additional Fayet-Iliopoulos D-term to the scalar potential. In such cases, the scalar masses can receive a contribution from the D-term that is of the same order as the gauge mediation contribution. This was discussed in [21] where the contribution to scalar masses was modeled by a term: δm2 = d QXM2 (3) φi i 2 with, d, anarbitraryparameterand, QX, theU(1) chargeofthefieldφ . For i X i the matter fields this charge is taken to be +1, and for the Higgs fields, it is set equal to -2; i.e. U(1) is the U(1) in SO(10) commuting with SU(5). M X 2 represents the wino mass calculated earlier in Eq. (2). With contributions from the gravitino, gauge mediation, and the D-term, the scalar masses are 2We assume the SUSY breaking dilaton VEV, F , is negligibly small. Kahler and S complex structure moduli (denoted generically by T) contribute to gaugino masses via one loop corrections to the gauge kinetic function with scale set by F /M . F is then T Pl assumed to be a linear combination of geometric moduli and chiral matter moduli. 6 given by: 2 2 α F α F m2 = m2 +2 bEX3 3 C (i)+2 bEX2 2 C (i) φi 3/2 (cid:18) 3 4πM (cid:19) 3 (cid:18) 2 4πM (cid:19) 2 EX3 EX2 2 α F F +2 3 bEX3 +bEX2 C (i)+dQXM2 (4) (cid:18)4π (cid:18) 1 M 1 M (cid:19)(cid:19) 1 i 2 EX3 EX2 where, C s represent the quadratic Casimir invariants [10]. The low energy i spectrum is now computed for different values of~n, ǫ , F, m , and d. Table 3 3/2 1 shows the GUT scale parameters for four sample points. Table 1: The GUT scale parameters for four different cases. g is discussed string in Appendix A.3.2. Dimensionful quantities in units of GeV unless specified. Observable Case 1 Case 2 Case 3 Case 4 ~n (4,2,(2,1)) (4,2,(2,1)) (4,2,(1,1)) (4,2,(2,0)) m 4 TeV 10 TeV 10 TeV 4 TeV 3/2 d 0 5 5 1 g 0.99412 0.99604 0.8233 0.8588 string M 6.04 ×1017 6.05 ×1017 7.39 ×1017 7.27 ×1017 S M 1.2 ×1016 1.2 ×1016 3.2 ×1016 2.8 ×1016 C M 5.03 ×1013 1.10 ×1014 1.07 ×1014 5.05 ×1013 EX3 M 1.69 ×1013 8.54 ×1013 5.35 ×1013 8.87 ×1012 EX2 M 2.5 ×1016 2.0 ×1016 3.25 ×1016 1.75 ×1016 GUT ǫ -2.5 % 0 % -2.5 % -0.5 % 3 F 1.0×1018GeV2 1.0×1018GeV2 1.0×1018GeV2 1.0×1018GeV2 M (M ) 257.296 155.269 120.882 260.894 3 GUT M (M ) 392.844 600.865 119.793 747.307 2 GUT M (M ) 124.900 128.947 39.666 260.894 1 GUT 3 Features of the Spectrum The MSSM spectrum is calculated using SOFTSUSY [11]. For the four cases shown in Table 1, the spectrum from SOFTSUSY is shown in Table. 2. Non-universal gaugino masses: The split exotics give rise to non-universal gaugino masses at the GUT scale, as is clear from Eq.(2). As a result of this 7 3 At MGUT At MZ 2.5 2 2 M 3/1.5 M 1 0.5 0 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 ε3 Figure 1: The scatter plot of consistent points for the case, ~n = (4,2,(2,1)). The gravitino mass, m was taken to be 4 TeV. The running of the couplings 3/2 depend only on the ratio M /M since the scalars are very heavy. We find an 3 2 anti-correlation between this ratio and the value of ǫ . Precision unification favors 3 M /M ∼ 0.3 at the GUT scale. 3 2 non-universality, the GUT scale threshold corrections required to match the precision electroweak data need not necessarily be of order -3 % . In fact, we notice that it is possible to obtain precision unification when M < M . 3 2 This requirement was observed in [7] in a variety of SUSY breaking scenarios including a Higgs-messenger mixing model where SUSY is broken via gauge mediation. The weak scale MSSM spectrum now has a light gluino. Case 2 in Table 2 illustrates this feature of the gluino being the second lightest sparticle after the neutralino. Figure 1 demonstrates the correlation between the GUT scale threshold corrections ǫ and the ratio of gaugino masses at 3 the weak scale and the GUT scale. Note, the scalar masses are heavy and degenerate. As a result, they do not introduce differential running of the coupling constants. SUSY breaking scale, gravitino mass, and D-Term: The low energy spec- trum depends on the following parameters that are chosen arbitrarily - the SUSY breaking scale F, the gravitino mass m and the d parameter in 3/2 the D-term. Although the doublet and the triplet exotics could couple to 8 Figure 2: The plot shows the consistent points with varying m and d. The 3/2 plot on the left is for m = 4 TeV and the one on the right is for m = 10 TeV. 3/2 3/2 different SUSY breaking fields, for simplicity we use a single SUSY break- ing VEV, F. The gravitino mass m , is the dominant contribution to the 3/2 scalar masses. In order to obtain consistent solutions, we find m ≥ 2 TeV, 3/2 otherwise the scalars become non-degenerate at the GUT scale and spoil uni- ficationthroughdifferential running. At thesametime, ifm >10TeV, the 3/2 assumption that gauge mediation is the dominant contribution to gaugino masses no longer holds and the gravitino corrections to the gaugino masses must be included. The D-term introduces a splitting between the sparticle masses and the Higgs masses, since they carry different charges under the U (1). For the two cases of m = 4 TeV and 10 TeV, the graph 2 shows X 3/2 the set of consistent points with varying d = 0, 5. MSSM Spectrum: The MSSM spectrum for the four particular cases (Ta- ble 1) is given in Table 2. The m contribution makes the scalars very 3/2 heavy, with the third family being slightly lighter. The gauginos receive the dominant contributions only from the gauge messengers and are light in comparison with the scalars. The LSP is the lightest neutralino, χ˜0, which is 1 predominantly “bino-like”. The gluino and chargino masses depend on the threshold corrections at the GUT scale, as is seen for the four cases given in Table 2. Most of the points that were found to be consistent with the 9 low energy data have small values of tanβ < 10. Finally, increasing the d parameter gives a handle on the possible values for tanβ. Collider prospects: We have found light gauginos which can be produced at the LHC or possibly the Tevatron. Since the gauginos are lighter than the scalars, they will decay only through off-shell squarks. Gluinos can decay via the process: g˜ → q q¯χ˜ . The produced χ˜ would then undergo cascade decay i i until the final product is the LSP and Standard Model leptons. This would give a striking signature of at least 4 jets + missing E [16]. The heavier T χ˜0 and χ˜± could decay into their lighter counterparts and leptons that could i i be the cleanest signature at the LHC. The unique feature of the spectrum is the mass difference between the heavier neutralinos and the LSP - about 150 GeV in one case and close to 500 GeV in the other. This could lead to very high energy leptons and a lot of missing energy making this a very favorable channel at the LHC. Once detected, this would give useful information about the GUT scale threshold corrections. 4 Summary Wehavecalculatedthespectrumofexoticsaswell theMSSMspectrumstart- ing from a heterotic string theory compactified on an anisotropic orbifold. Allowing the exotics, that come in incomplete GUT multiplets to obtain mass according to their quantum numbers allows for more possible solutions to gauge coupling unification. We find that we can build consistent MSSM spectra starting with such theories with the exotics acting as messengers of SUSY breaking through the gauge mediation mechanism. The gaugino masses in the low energy spectrum depend on the threshold corrections at theGUTscale. Theyarelighterthanthescalarsandarewithinthekinematic reach of the LHC. The unique features of the spectrum are light gluinos and also large mass differences between the lightest and the next-to-lightest neu- tralinos and charginos which could lead to interesting signatures at the LHC. Acknowledgments We would like to thank Ben Dundee and Konstantin Bobkov for useful dis- cussions. We also received partial support for this work from DOE grant DOE/ER/01545-891. 10