Bled Workshops in Physics Vol. 16, No. 2 A Proceedings to the 1 8th Workshop What Comes Beyond ... (p. 1) Bled, Slovenia, July 11-19, 2015 1 Aspects of String Phenomenology in Particle Physics and Cosmology I. Antoniadis * LPTHE, UMR CNRS 7589 Sorbonne Universites, UPMC Paris 6, 75005 Paris, France and Albert Einstein Center, Institute for Theoretical Physics, Bern University, Sidlerstrasse 5, 3012 Bern, Switzerland Abstract. We describe the phenomenology of a model of supersymmetry breaking in the presence of a tiny (tunable) positive cosmological constant. It utilises a single chiral multiplet with a gauged shift symmetry, that can be identified with the string dilaton (or an appropriate compactification modulus). The model is coupled to the MSSM, leading to calculable soft supersymmetry breaking masses and a distinct low energy phenomenology that allows to differentiate it from other models of supersymmetry breaking and mediation mechanisms. Povzetek. Avtor obravnava lastnosti modela za zlom supersimetrije, ko majhno pozitivno kozmolosko konstanto prilagaja fenomenoloskim lastnostim. Obravnava primer kiralnega multipleta, ko postane umeritvena simetrija dilatacijska simetrija strune (uporabiti pa je mogoce tudi kak drug model kompaktifikacije). Model poveže s standardnim modelom z minimalno supersimetrijo, kar omogoči izračun mas pri mehki zlomitvi supersimetrije. Model uspesno opise fenomenoloske lastnosti polj, kar ga loci od ostalih modelov za zlomitev supersimetrije. 1.1 Introduction If String Theory is a fundamental theory of Nature and not just a tool for studying systems with strongly coupled dynamics, it should be able to describe at the same time particle physics and cosmology, which are phenomena that involve very different scales from the microscopic four-dimensional (4d) quantum gravity length of 10-33 cm to large macroscopic distances of the size of the observable Universe -1028 cm spanned a region of about 60 orders of magnitude. In particular, besides the 4d Planck mass, there are three very different scales with very different physics corresponding to the electroweak, dark energy and inflation. These scales might be related via the scale of the underlying fundamental theory, such as string theory, or they might be independent in the sense that their origin could be based on different and independent dynamics. An example of the former constrained and more predictive possibility is provided by TeV strings with a fundamental scale at low energies due for instance to large extra dimensions transverse to a * E-mail: ignatios.antoniadis@polytechnique.edu 2 I. Antoniadis four-dimensional braneworld forming our Universe [1]. In this case, the 4d Planck mass is emergent from the fundamental string scale and inflation should also happen around the same scale [2]. Here, we will adopt the second more conservative approach, assuming that all three scales have an independent dynamical origin. Moreover, we will assume the presence of low energy supersymmetry that allows for an elegant solution of the mass hierarchy problem, a unification of fundamental forces as indicated by low energy data and a natural dark matter candidate due to an unbroken R-parity. The assumption of independent scales implies that supersymmetry breaking should be realized in a metastable de Sitter vacuum with an infinitesimally small (tunable) cosmological constant independent of the supersymmetry breaking scale that should be in the TeV region. In a recent work [3], we studied a simple N = 1 supergravity model having this property and motivated by string theory. Besides the gravity multiplet, the minimal field content consists of a chiral multiplet with a shift symmetry promoted to a gauged R-symmetry using a vector multiplet. In the string theory context, the chiral multiplet can be identified with the string dilaton (or an appropriate compactification modulus) and the shift symmetry associated to the gauge invariance of a two-index antisymmetric tensor that can be dualized to a (pseudo)scalar. The shift symmetry fixes the form of the superpotential and the gauging allows for the presence of a Fayet-Iliopoulos (FI) term, leading to a supergravity action with two independent parameters that can be tuned so that the scalar potential possesses a metastable de Sitter minimum with a tiny vacuum energy (essentially the relative strength between the F- and D-term contributions). A third parameter fixes the Vacuum Expectation Value (VEV) of the string dilaton at the desired (phenomenologically) weak coupling regime. An important consistency constraint of our model is anomaly cancellation which has been studied in [5] and implies the existence of additional charged fields under the gauged R-symmetry. In a more recent work [6], we analyzed a small variation of this model which is manifestly anomaly free without additional charged fields and allows to couple in a straight forward way a visible sector containing the minimal supersymmetric extension of the Standard Model (MSSM) and studied the mediation of super-symmetry breaking and its phenomenological consequences. It turns out that an additional 'hidden sector' field z is needed to be added for the matter soft scalar masses to be non-tachyonic; although this field participates in the supersymmetry breaking and is similar to the so-called Polonyi field, it does not modify the main properties of the metastable de Sitter (dS) vacuum. All soft scalar masses, as well as trilinear A-terms, are generated at the tree level and are universal under the assumption that matter kinetic terms are independent of the 'Polonyi' field, since matter fields are neutral under the shift symmetry and supersymmetry breaking is driven by a combination of the U(1) D-term and the dilaton and z-field F-term. Alternatively, a way to avoid the tachyonic scalar masses without adding the extra field z is to modify the matter kinetic terms by a dilaton dependent factor. A main difference of the second analysis from the first work is that we use a field representation in which the gauged shift symmetry corresponds to an ordinary U(1) and not an R-symmetry. The two representations differ by a Kahler 1 Aspects of String Phenomenology in Particle Physics and Cosmology 3 transformation that leaves the classical supergravity action invariant. However, at the quantum level, there is a Green-Schwarz term generated that amounts an extra dilaton dependent contribution to the gauge kinetic terms needed to cancel the anomalies of the R-symmetry. This creates an apparent puzzle with the gaugino masses that vanish in the first representation but not in the latter. The resolution to the puzzle is based to the so called anomaly mediation contributions [7,8] that explain precisely the above apparent discrepancy. It turns out that gaugino masses are generated at the quantum level and are thus suppressed compared to the scalar masses (and A-terms). 1.2 Conventions Throughout this paper we use the conventions of [9]. A supergravity theory is specified (up to Chern-Simons terms) by a Kahler potential K, a superpotential W, and the gauge kinetic functions fAB (z). The chiral multiplets za,xa are enumerated by the index a and the indices A, B indicate the different gauge groups. Classically, a supergravity theory is invariant under Kahler tranformations, viz. K(z,z) —» K(z,z) + J(z) + J(z), W(z) —> e-l<2j(z)W(z), (1.1) where k is the inverse of the reduced Planck mass, mp = k-1 = 2.4 x 1015 TeV. The gauge transformations of chiral multiplet scalars are given by holomorphic Killing vectors, i.e. Sza = eAkA(z), where eA is the gauge parameter of the gauge group A. The Kahler potential and superpotential need not be invariant under this gauge transformation, but can change by a Kahler transformation SK = eA [ta(z)+ ta(z)] , (1.2) provided that the gauge transformation of the superpotential satisfies SW = -eA k2ta (z) W. One then has from SW = WaSza WakA = -k2taw, (1.3) where Wa = 9aW and a labels the chiral multiplets. The supergravity theory can then be described by a gauge invariant function g = k2 K + log(K6WW). (1.4) The scalar potential is given by V = vf +vd VF = eK2'c (-3k2ww + VaWgaPVpW) vd = 2 (Ref)-1 AB PaPb, (1.5) where W appears with its Kahler covariant derivative VaW = 9aW(z) + k2(9«K)W (z). (1.6) 4 I. Antoniadis The moment maps PA are given by Pa = i(k£3aK - ta). (1.7) In this paper we will be concerned with theories having a gauged R-symmetry, for which rA (z) is given by an imaginary constant rA (z) = iK-2£,. In this case, k-2£ is a Fayet-Iliopoulos [10] constant parameter. 1.3 The model The starting point is a chiral multiplet S and a vector multiplet associated with a shift symmetry of the scalar component s of the chiral multiplet S 6s = -ic9, (1.8) and a string-inspired Kahler potential of the form —p log(s + s). The most general superpotential is either a constant W = K-3a or an exponential superpotential W = K-3aebs (where a and b are constants). A constant superpotential is (obviously) invariant under the shift symmetry, while an exponential superpotential transforms as W —» We-lbc0, as in eq. (1.3). In this case the shift symmetry becomes a gauged R-symmetry and the scalar potential contains a Fayet-Iliopoulos term. Note however that by performing a Kahler transformation (1.1) with J = K-2bs, the model can be recast into a constant superpotential at the cost of introducing a linear term in the Kahler potential 6K = b(s + s). Even though in this representation, the shift symmetry is not an R-symmetry, we will still refer to it as U(1 )R. The most general gauge kinetic function has a constant term and a term linear in s, f(s) = 6 + (3s. To summarise,1 K(s, s) = —p log(s + s) + b(s + s), W (s) = a, f(s)= 6 + (s, (1.9) where we have set the mass units k = 1. The constants a and b together with the constant c in eq. (1.8) can be tuned to allow for an infinitesimally small cos-mological constant and a TeV gravitino mass. For b > 0, there always exists a supersymmetric AdS (anti-de Sitter) vacuum at (s + s) = b/p, while for b = 0 (and p < 3) there is an AdS vacuum with broken supersymmetry. We therefore focus on b < 0. In the context of string theory, S can be identified with a compact-ification modulus or the universal dilaton and (for negative b) the exponential superpotential may be generated by non-perturbative effects. 1 In superfields the shift symmetry (1.8) is given by 6S = —icA, where A is the superfield generalization of the gauge parameter. The gauge invariant Kahler potential is then given by K(S, S) = —pK-2 log(S + S + cVR) + K-2b(S + S + cVR), where VR is the gauge superfield of the shift symmetry. 1 Aspects of String Phenomenology in Particle Physics and Cosmology 5 The scalar potential is given by: V = Vf + VD Vf = a2eblp-2 j 1 (pi - b)2 - 3l2J I = 1/(s + s) Vd = c2 p+25l(Pl - b)2 (L10) In the case where S is the string dilaton, VD can be identified as the contribution of a magnetized D-brane, while VF for b = 0 and p = 2 coincides with the tree-level dilaton potential obtained by considering string theory away its critical dimension [11]. For p > 3 the scalar potential V is positive and monotonically decreasing, while for p < 3, its F-term part VF is unbounded from below when s + s —} 0. On the other hand, the D-term part of the scalar potential VD is positive and diverges when s + s —» 0 and for various values for the parameters an (infinitesimally small) positive (local) minimum of the potential can be found. If we restrict ourselves to integer p, tunability of the vacuum energy restricts p = 2 or p = 1 when f(s) = s, or p = 1 when the gauge kinetic function is constant. For p = 2 and f(s) = s, the minimization of V yields: b/l = a « -0.183268 , p = 2 (1.11) a2 A —2 = A2(a) + B2(a)-A2 « -50.6602 + O(A), (1.12) bc2 b3c2 where A is the value of V at the minimum (i.e. the cosmological constant), — is the negative root of the polynomial -x5 + 7x4 - 10x3 - 22x2 + 40x + 8 compatible with (1.12) for A = 0 and A2(a), B2(a) are given by A2(a)= 2e-a -4 + 4 ^- f ; B2M = (1.13) 3 - 4 2 - 2 2 - 4 -2 It follows that by carefully tuning a and c, A can be made positive and arbitrarily small independently of the supersymmetry breaking scale. A plot of the scalar potential for certain values of the parameters is shown in figure 1.1. At the minimum of the scalar potential, for nonzero a and b < 0, supersym-metry is broken by expectation values of both an F and D-term. Indeed the F-term and D-term contributions to the scalar potential are 2 vfU_f = 2aW(1 - >°, b3C2 ( 2 — VdIs+5_a = - 1 - - >0. (1.14) S+S_ b - V — The gravitino mass term is given by n2b2 (ma/2)2 = eG = —r ea. (1.15) Due to the Stueckelberg coupling, the imaginary part of s (the axion) gets eaten by the gauge field, which acquires a mass. On the other hand, the Goldstino, which is 6 I. Antoniadis V Fig. 1.1. A plot of the scalar potential for p = 2, b = —1, 5 = 0, | = 1 and a given by equation (1.12) for c = 1 (black curve) and c = 0.7 (red curve). a linear combination of the fermion of the chiral multiplet x and the gaugino A gets eaten by the gravitino. As a result, the physical spectrum of the theory consists (besides the graviton) of a massive scalar, namely the dilaton, a Majorana fermion, a massive gauge field and a massive gravitino. All the masses are of the same order of magnitude as the gravitino mass, proportional to the same constant a (or c related by eq. (1.12) where b is fixed by eq. (1.11)), which is a free parameter of the model. Thus, they vanish in the same way in the supersymmetric limit a —» 0. The local dS minimum is metastable since it can tunnel to the supersymmetric ground state at infinity in the s-field space (zero coupling). It turns out however that it is extremely long lived for realistic perturbative values of the gauge coupling 1 ~ 0.02 and TeV gravitino mass and, thus, practically stable; its decay rate is [5]: 1.4 Coupling a visible sector The guideline to construct a realistic model keeping the properties of the toy model described above is to assume that matter fields are invariant under the shift symmetry (1.8) and do not participate in the supersymmetry breaking. In the simplest case of a canonical Kahler potential, MSSM-like fields ^ can then be added as: where WMssm(^) is the usual MSSM superpotential. The squared soft scalar masses of such a model can be shown to be positive and close to the square of r - e-B with B« 10300 . (1.16) k = — k 2 log(s + s) + K 2b(s + (( W = K-3a + Wmssm, (1.17) 1 Aspects of String Phenomenology in Particle Physics and Cosmology 7 SL = (1.18) the gravitino mass (TeV2). On the other hand, for a gauge kinetic function with a linear term in s, (3 = 0 in eq. (1.9), the Lagrangian is not invariant under the shift symmetry ,Pc T and its variation should be canceled. As explained in Ref. [5], in the 'frame' with an exponential superpotential the R-charges of the fermions in the model can give an anomalous contribution to the Lagrangian. In this case the 'Green-Schwarz' term ImsFF can cancel quantum anomalies. However as shown in [5], with the minimal MSSM spectrum, the presence of this term requires the existence of additional fields in the theory charged under the shift symmetry. Instead, to avoid the discussion of anomalies, we focus on models with a constant gauge kinetic function. In this case the only (integer) possibility2 is p = 1. The scalar potential is given by (1.10) with ( = 0, 6 = p = 1. The minimization yields to equations similar to (1.11), (1.12) and (1.13) with a different value of a and functions Ai and Bi given by: b(s + s) = a « -0.233153 bc2 A = A1 (a) + B1 (a)^- « -0.359291 + O(A) (1.19) a2 a2b Ai(a)= 2eaa3 - (a ~ ^ , Bi(a)= (a - 1)2 ' 11 J (a - 1)2 ' where a is the negative root of —3+(a-1 )2(2-a2/2) = 0 close to -0.23, compatible with the second constraint for A = 0. However, this model suffers from tachyonic soft masses when it is coupled to the MSSM, as in (1.17). To circumvent this problem, one can add an extra hidden sector field which contributes to (F-term) supersymmetry breaking. Alternatively, the problem of tachyonic soft masses can also be solved if one allows for a non-canonical Kahler potential in the visible sector, which gives an additional contribution to the masses through the D-term. Let us discuss first the addition of an extra hidden sector field z (similar to the so-called Polonyi field [12]). The Kahler potential, superpotential and gauge kinetic function are given by K = -K-2 log(s + s) + K-2b(s + s) + zz + pp , W = K-3a(1 + YKz)+ Wmssm(p) , f(s) = 1 , fA = 1/gA, (1.20) where A labels the Standard Model gauge group factors and y is an additional constant parameter. The existence of a tunable dS vacuum with supersymmetry 2 If f(s) is constant, the leading contribution to Vd when s + s —> 0 is proportional to 1/(s + s)2, while the leading contribution to VF is proportional to 1/(s + s)p. It follows that p < 2; if p >2, the potential is unbounded from below, while if p = 2, the potential is either positive and monotonically decreasing or unbounded from below when s + s —> 0 depending on the values of the parameters. 8 I. Antoniadis breaking and non-tachyonic scalar masses implies that y must be in a narrow region: 0.5