Aspects of Gauge-Mediated Supersymmetry Breaking

Transcription

1 Racah Institute of Physics, The Hebrew University, Jerusalem Aspects of Gauge-Mediated Supersymmetry Breaking M.Sc Thesis by Latif Eliaz This work was done under the supervision of Prof. Amit Giveon August 2012

2 Acknowledgements I would like to thank my advisor, Prof. Amit Giveon for his patient and clear guidance. Special thanks are due to Dr. Zohar Komargodski for useful discussions and counseling throughout the process of this work. I am thankful for and would like to acknowledge many members of the high-energy physics group at the Racah Institute. I am deeply grateful to Dr. Roberto Auzzi who kindly helped me in every aspect of the work. I am very grateful to Dr. S. Bjarke Gudnason especially for his patient help with the details of my work. I would like to thank my office roommate Ofek Birnholtz for his presence and support, influencing my work directly and indirectly. I would also like to thank Tomer Shacham, and Shahar Hadar. I am very grateful to my roommates, friends and family who supported me over the course of this work: Elad Shtilerman, Lior Chervinsky and Maayan Kreitzman; my parents and sisters who encouraged me throughout this work; I am deeply grateful to Matan Field for special support and inspiration. Lastly and most importantly I am grateful to my partner for life Yael, being with me and supporting me over the last year of this work. Your presence with me was of great help in making this work come true. Latif Eliaz i

3 Abstract In this work we study the parameter space of a model of gauge-mediated supersymmetry (SUSY) breaking with two pairs of weakly-coupled messengers. This model generalizes minimal gauge mediation, including the possibility of generic renormalizable interactions among the messengers. We rely on leading-loop formulae for the soft masses, which are exact to all orders in the supersymmetry-breaking order parameter. We find that the scalar masses squared are positive in all of the parameter space where the messengers are non-tachyonic. The ratio between the sfermion and the gaugino masses is maximized in the subspace where the model reduces to minimal gauge mediation. The parameter space of the model includes also a subspace where the gaugino masses vanish to leading order in SUSY breaking. This subset of parameters corresponds to models of direct gauge mediation which may be embedded in supersymmetric QCD; we find that the exact calculation yields sizable contributions to the gaugino masses. Finally, we compute the pole masses of the superpartners. In order to achieve a 126 GeV Higgs, we find that the squark masses need to be above 10 TeV. However the gaugino masses can be rendered lighter than 1 TeV, a region which may be accessible to LHC searches. This mild split-susy spectrum is achieved also in the subspace of parameters which can be realized in SQCD. ii

4 Contents 1. Introduction Background N =1 Supersymmetry SUSY Breaking The Minimal-SSM Gauge-Mediated SUSY Breaking Renormalization Group Flow RG Flow Using Soft-SUSY SQCD Embedding Outline of Results General Setup Introducing the Model MGM Case Two Messengers (N = 2) Parameter Space MGM Case for N = Positivity of m 2 f N = 2 Numeric Scanning of the Parameter Space Analyzing the Limit 0 Using STrM Effective Theory for Expansion of m 2 f for x The Ratio m2 f M 2 g 7. On the Subspace of Vanishing Gaugino Mass to Leading Order in SUSY Breaking RG Evolution to the Weak Scale Summary References iii

5 1. Introduction Supersymmetry (SUSY) is considered a promising explanation for the high energy physics beyond the currently accepted model, called the Standard Model (SM) of particle physics (for a review see e.g. [1]). The SM 1 provides a successful description of the particles and interactions in the electromagnetic, strong and weak sectors. Nonetheless, this model leaves behind a few questions which are not resolved in a satisfactory manner. This motivates a search for alternative models, which are in agreement with the SM in the familiar regimes. Some of the main questions of interest are: 1. How to unite the SM with gravity, in order to provide a full description of the four known interactions of nature? Attempts to build such a field theoretical model fail, as the produced theories are non-renormalizable, and hence do not allow one to handle computations. On the other hand, another attractive approach for dealing with this question is String Theory. 2. How to resolve the Hierarchy problem? In the (successful) renormalization of the SM, the Higgs mass squared receives corrections of order of the cut-off squared (Λ 2 ), for each particle which is coupled to the Higgs. For example, a coupling L = λ f H ff in the Lagrangian gives rise to the contribution m 2 H = λ2 f Λ2 /8π 2 to the Higgs mass at one loop. The cut-off scale is expected to be of the order of the Planck mass, in the absence of a new theory completing the SM in some intermediate scale. On the other hand, the ATLAS and CMS collaborations have presented clear evidence for the production of a natural particle with mass near 126 GeV, which is compatible with a SM Higgs boson [2,3]. The identification of this new particle with a SM Higgs is also supported by electroweak precision measurements. Hence, a fine-tuned adjustment of corresponding counter terms is required to set the physical Higgs mass at the right scale. The second problem might be softened by requiring an extra symmetry: SUSY, and requiring it to be spontaneously broken near the weak scale. In addition, spontaneously broken SUSY is inspired by providing a natural gauge coupling unification. Moreover, SUSY is supported by its requirement to allow consistent (super-)string Theory which contains fermionic excitations. 1 augmented by neutrino masses. 1

6 In a supersymmetric theory, the Poincare group (or the conformal group in the massless case) is extended with additional, fermionic, symmetry generators (denoted by Q α ), which do not commute with the Poincare group 2. These operators transform a bosonic state into a fermionic one, and vice versa. It turns out that the freedom in building such an extension, keeping the required properties of the SM (and especially allowing for Chiral fermions), is very restricted. After choosing the number of SUSY generator-pairs (N =1,2,4), and the central charge, the SUSY algebra is uniquely defined. Only N =1 allows for chiral matter, and thus it will be the case considered here. In the representation of this group on the field space there is a correspondence between the fermionic degrees of freedom and the bosonic ones. Also, the generated SUSY theories have specific relations between the different parameters of the theory. Specifically, each bosonic particle corresponds to a fermionic particle (the superpartner) with the same mass, and it carries the same properties under the gauge group (and vice versa). No particles in the SM can be SUSY partners one of the other. Hence, in order to turn the SM into a SUSY model it is required to add to its content the superpartners of the known particles (named sfermion for a fermion scalar superpartner, and gaugino for a gauge boson fermionic superpartner). In such an extension of the SM the Higgs mass is protected from corrections, since the fermionic and bosonic loop contributions cancel each other (since a fermionic loop contributes with a relative minus sign). If SUSY exists in nature it needs to be spontaneously broken, to comply with the known particles and dynamics. Yet, SUSY breaking cannot be too strong in order to preserve the Higgs mass protection. Hence, the models considered obey a spontaneous and soft SUSY breaking. In such models SUSY is broken by the ground state and, moreover, in the effective theory the leading order dimensionless coupling constant preserves the special relations of the SUSY theory, hence preventing the Higgs mass from acquiring possible quadratic divergent corrections. Softly-broken SUSY extensions of the SM (SSM) allow to obtain a particles spectrum that is not in contradiction with the experimental results so far, and partly solves the hierarchy problem. The solution is not complete, it leaves open issues which are attended differently in future models. These issues include 2 The Coleman-Mandula theorem clarifies that it is impossible to extend the Poincare group with bosonic operators, which are not commuting with Poincare, without breaking the continuity of the scattering matrix. 2

7 CP and flavor problems, and also topics realized in the required fine tuning of the Higgs mass: the little hierarchy problem, and the µ problem (see section 2.3 for further details). Recent LHC data [4] is constraining the allowed region of parameters of many SUSY models as a natural explanation to the hierarchy problem (see e.g. [5], and references therein). SUSY breaking within the Minimal-SSM (MSSM) is impossible, since it predicts light particles which are unacceptable experimentally. Hence, it is commonly assumed that SUSY is broken by particles which are unknown and contained in a hidden sector, which is (indirectly) connected to the visible sector. There are a few possible mechanisms for mediating SUSY breaking from the hidden sector to the MSSM, among them are gauge mediation, gravity mediation, anomaly mediation etc. A combination of the different mediation mechanisms is also a possibility. Gauge-mediated SUSY breaking (GMSB) states that the SM gauge interactions are responsible for communicating soft-susy breaking to the visible sector (for reviews on GMSB see, e.g. [6] for earlier work, and [7] for more recent work). Models of this class carry the nice property of transmitting SUSY breaking via flavor blind interactions, hence avoiding undesirable flavor-changing neutral currents, that might be encountered in a general SSM. The simplest model of this kind is known as Minimal Gauge Mediation (MGM), and was studied extensively, for example in [8], where a full derivation of the gaugino and the sfermion masses (the soft masses 3 ) is given, by an explicit loop calculation. In this model a pair of messenger superfields is introduced. The messengers are charged under the SM gauge groups and are coupled to a spurion field, which artificially introduces SUSY breaking to the model. Therein, SUSY breaking is communicated to the MSSM through the messenger, and then by the gauge interactions. This work studies a generalization of MGM, containing two pairs of messengers (N = 2). The model is introduced in sections 2.4 and 4.1. Models of this kind were studied extensively, e.g. in [9], where the soft masses are calculated to the leading order in the SUSY-breaking order parameter (at the messenger scale). Moreover, exact formulae (true to all orders in SUSY breaking) for the soft masses in this model were recently given [10,11]. The current work is motivated by several recent results [11], in which this generalized messenger model was studied from a more general perspective, using the 3 The masses of the gauginos and the sfermions are acquired by soft-susy breaking, hence named soft masses. 3

8 framework of General Gauge Mediation (GGM) [12]. Among these results there was a bound on the possible ratio between the masses of the sfermions (m f ) and the gauginos (M g ): m 2 f M 2 g Y f 2 N. (1.1) This inequality is formulated for a simplified case, which consists of only a U(1) gauge group, where Y f is the sfermion charge with respect to this group. This relation is correct at the leading order in SUSY breaking. It applies a limitation on the possible spectra of the model. The main motivation for this work is to generalize this limitation based on the exact soft-mass formulae in the MSSM mentioned above, and study its consequences. This work is organized as follows. In 2 we review some of the theoretical background of SUSY breaking and its mediation to the MSSM through the messengers via the gauge interactions. In 3-8 we present the results of this work. These sections include a study of the sfermion mass squared positivity, which is necessary for the viability of the model; a study of the limitation on the soft-mass parameter space covered by this model; a study of the subspace of vanishing gaugino mass to leading order in SUSY breaking; and a study of the RG evolution of the mass spectrum to the electroweak scale. Finally, a summary of the main results and conclusions is given in 9. 4

9 2. Background The following sections shortly review the theoretical foundation of this work 4. We begin with N =1 supersymmetry, and continue through SUSY breaking, the MSSM, and gauge mediation, to the underlying model of this work. Then we review the renormalization group flow, being used in this work to compute the low energy effective masses of the different particles. For completeness we review the topic of SQCD embedding, which is relevant for some aspects of this work N =1 Supersymmetry The minimal supersymmetric extension of the Poincare group contains an extra pair of fermionic symmetry generators Q α and Q β, which satisfy the following SUSY algebra: {Q α, Q β} = 2P α β = 2σ µ α β P µ {Q α, Q β } = { Q α, Q β} = 0 [P α β, Q γ ] = [P α β, Q γ ] = 0 (2.1) Here σ µ α β are the Pauli-matrices, and α, β = 1, 2 are spinor indices. These generators are represented in the superspace fields manifold by: Q α = θ α i θ βσ µ α β µ Q α =. (2.2) θ α iθβ σ µ β α µ Here θ α and θ β are complex anti-commuting spinor coordinates, which compose superspace together with the usual space-time coordinates x µ. In order to construct a general supersymmetric theory we introduce the chiral supermultiplet Φ (also: the chiral superfield), and the gauge supermultiplet V (the vector superfield), satisfying D α Φ = 0 and V = V, where D α = θ α + iθβ σ µ β α µ (2.3) is the chiral covariant derivative 5, satisfying { D α, Q β } = { D α, Q β} = 0. 4 The references for studying these topics include Martin s review [1], SUSY course notes in the Hebrew University by S. Elitzur, Argyres notes [13], Weinberg s book [14], and more. 5 D α and D α (defined similarly) form a second representation of the SUSY algebra. 5

10 The chiral supermultiplet can be compactly written in terms of the variables y µ = x µ + i θσ µ θ, in the general form Φ(y µ, θ, θ) = φ(y) + 2θψ(y) + θθf (y). (2.4) In this expression φ and F are complex scalar fields and ψ is a weyl-fermion. The chiral superfield expressed in terms of the superspace coordinates (x, θ, θ) is Φ(x µ, θ, θ) = φ(x) + i θ σ µ θ µ φ(x) θθ θ θ µ µ φ(x) + 2θψ(x) i 2 θθ θ σ µ µ ψ(x) + θθf (x). (2.5) By means of the chiral superfield it is possible to write renormalizable supersymmetric theories of the following general form: ( S = d 4 xd 2 θd 2 θ K[Φi, Φ i ] + ) d 4 xd 2 θ W [Φ i ] + c.c. K[Φ, Φ] = 1 4 Φ i Φ i (2.6) W [Φ] = f i Φ i M ij Φ i Φ j yijk Φ i Φ j Φ k. The action is composed of two contributions known as the Kähler-potential K, and the superpotential W. Note that the composition of superfields in W [Φ i ] is a chiral superfield itself. Taking the integral d 2 θ then produces the F-term of this chiral superfield. Similarly the Kählerpotential contribution is a D-term of a produced vector superfield (introduced shortly). Hence, the Kähler-potential, and the superpotential are also called the D-term and the F-term contributions to the action, respectively. By integrating over the fermionic coordinates one obtains the space-time Lagrangian. From the D-term one obtains S D = d 4 x [ φi µ µ φ i + i2 ( µψ i σ µ ψi ) 14 ] F i F i, (2.7) the action of n pairs of massless free scalars particles φ i, free weyl fermions ψ i, ψ i, and non-dynamical auxiliary fields F i. The F-term contribution is: [ 1 S F = d 4 W x F i 1 2 ] W ψ i ψ j + c.c. = 2 Φ i 4 Φ i Φ j [ d 4 1 x 2 (f i + M ij φ j yijk φ j φ k )F i 1 ] 4 (M ij + y ijk φ k )ψ i ψ j + c.c.. 6 (2.8)

11 Since F i are auxiliary, and contribute quadratically they can be replaced by their classical equations of motion (EOM) solutions: F i = 2 W Φ i F i = 2 W Φ i (2.9) producing the full equivalent action: S = [ d 4 x φi µ µ φ i + i 2 ( µψ i σ µ ψi ) 1 ] 4 (M ij+y ijk φ k )ψ i ψ j f i +M ij φ j +y ijk φ j φ k 2 +c.c.. This action contains mass terms for the scalars and fermions, Yukawa coupling y ijk, and quartic and quadratic coupling for the scalars, with coupling constants depending on M ij and y ijk. The fermions and scalars carry the same mass squared Mij 2 in the case of f i = 0 (only), which is the case of unbroken SUSY 6. The vector (generally matrix) superfield, introduced to allow gauge interactions, is written in the following general form V (x, θ, θ) = a + θξ + θξ + θθb + θ θb + θ σ µ θa µ + θ θθ(λ i 2 σµ µ ξ ) +θθ θ(λ i 2 σµ µ ξ) + θθ θ θ( 1 2 D µ µ a). (2.10) Supergauge transformations are of the form V e iω e V e iω, where Ω is a (possibly matrix) chiral superfield gauge parameter. In the Abelian case the transformation law reduces to V V + i(ω Ω). This produces the familiar gauge transformation for the gauge-boson A µ. The supergauge transformation contains extra parameters with respect to the regular gauge transformation, which are conventionally used for the Wess-Zumino (WZ) gauge fixing, producing the following form of the vector superfield: V W Z = θ σ µ θa µ + θ θθλ + θθ θλ θθ θ θd (2.11) The remaining components are the gauge boson A µ, the weyl fermion gaugino λ, and an auxiliary gauge field D. After choosing the WZ supergauge we are left with an ordinary 6 A term f i 0 is contributing to spontaneously-susy breaking as will be discussed below. Also this term is allowed only for a gauge singlet Φ i, which is not required to be present in the Minimal-SSM. 7

12 gauge freedom. Fixing the supergauge in this way breaks the invariance under SUSY transformations, but re-fixing the gauge again after any SUSY transformation is possible. This means that SUSY is unbroken, but it is not manifest in WZ gauge. The following assumes the WZ gauge. To construct a supersymmetric gauge invariant action one defines the following gauge transformation of the chiral superfield Φ a (e 2ig a Ω a T a )Φ a (e iω a )Φ a, Φa Φ a (e 2ig a Ω a T a ) Φ a (e iω a ). In this expression the gauge coupling g a, and the gauge group(s) generators T a are absorbed into Ω a. Simultaneously V a is redefined similarly to absorb 2g a T a. The general renormalizable supersymmetric gauge invariant Kähler-potential is now written in the following way: K = 1 Φ a e V a Φa. (2.12) 4 a The sum is over different gauge groups and representations. The supergauge superpotential can be written in the form: where W α a W = W chiral iτ a 16π T r(w α a W αa ) (2.13) = 1 4 D 2 (e V a D α e V a ) is the field strength chiral superfield, containing the ordinary field strength Fµν a as a component, and τ a = Θ a 2π are the holomorphic coupling constants. For an Abelian gauge group an extra gauge invariant term (Fayet- Iliopoulos term) can be included in the Kähler-potential + 4πi g 2 a δk = 2κ a V a. (2.14) Similar to the non-gauged case, by integrating over the fermionic variables one obtains the equivalent space-time action: S = S chiral [ µ D µ ] + [ d 4 x 1 4 F a µν F µνa + i λ a σ µ D µ λ a D ad a 2g a ( φ a T a φ a )λ a 2g a λa ( ψ a T a φ a ) (κ a g a ( φ a T a φ a ))D a + g2 aθ a 64π 2 ɛµνρσ F µνa F ρσa ] (2.15) The first term is the chiral action of (2.7) and (2.8), with substitution of the ordinary derivatives with gauge-covariant derivatives. The Θ term is a total derivative, but may 8

13 have a physical effect for a non-abelian gauge group, related to the instantons number. Again one can use the equations of motion of the auxiliary fields D,F to obtain an equivalent on-shell action. The resulting scalar potential is of the form V (φ, φ) = F i F i D ad a = f i + M ij φ j + y ijk φ j φ k (κ a g a ( φ a T a φ a )) 2. (2.16) It is clearly non-negative, hence assuring non-negative vacuum energy for the supersymmetric theory. Note that with (unbroken) SUSY the number of degrees of freedom (DOF) remains equal for the fermionic and the bosonic sectors, both off-shell and on-shell, due to the presence of the auxiliary fields. A chiral superfield carries 4 bosonic and 4 fermionic real off-shell DOF: 2 bosonic DOF from φ, another 2 bosonic DOF from F (complex scalars), and 4 fermionic DOF come from the off-shell weyl-spinor ψ (having 2-complex component). The on-shell ψ has 2 helicity states, and the auxiliary field vanishes so 2 fermionic and 2 bosonic real DOF remain. A similar argument holds for the vector superfield having 4 bosonic and fermionic real off-shell DOF, and 2 real DOF of each kind on-shell (in the WZ gauge, off-shell A µ remains with 3 DOF and one DOF is removed by the remaining gauge freedom). SUSY implies specific restrictions on the low energy effective action, stated by nonrenormalization theorems. These properties were first found by Grisaru et al. [15] at 1979 using the supergraph technique developed at that time, and revisited by N. Seiberg [16] at 1993 using holomorphicity and symmetry arguments. These arguments can be applied in different circumstances to guarantee a significant similarity between the Wilsonian effective action and the original UV action, stating that the superpotential is almost nonrenormalized perturbatively and also restricted non-perturbatively. For a gauge invariant and renormalizable theory like (2.12) and (2.13) 7 the general perturbative statement (to all orders in perturbation theory) is that the effective action under a cutoff λ has the following structure: S λ = d 4 xd 2 θd 2 θdλ [Φ, Φ, V, D... ] + [ d 2 θ W chiral (Φ) + 1 ] 2gλ 2 T r(w α W α ) + c.c. (2.17) Here W chiral is exactly the same chiral superpotential of the source UV theory, and g λ is the one loop effective gauge coupling. On the other hand the Kähler term D λ can 7 More precisely the theorem is stated in this way only with g a and T a taken out of V. 9

14 be a general Lorentz and gauge invariant function, depending on the cutoff λ, on the chiral, anti-chiral and vector superfields, and possibly on terms involving superderivatives or spacetime derivatives of the superfields 8. The one loop correction to the gauge coupling g λ may induce a run-away potential in the effective action, stabilized according to the matter content of the theory. A possible Θ term in the original action is ignored, having no effect in perturbation theory. This theorem holds as long as the integrating out is done supersymmetrically and gauge invariant (that is to say the effective action scale λ is taken above SUSY-breaking scale, if it exist). A version of this theorem also holds for nonrenormalizable theories [17]. Another similar result is the non-renormalization of Fayet- Iliopoulos terms. Non-perturbatively the requirement for a holomorphic superpotential remains, also implying restrictions on the superpotential in the effective action SUSY Breaking As stated in the introduction, we are interested in models with softly broken SUSY. As explained, in the SM the Higgs H carries corrections to its mass squared term from each particle interacting with it. These corrections include divergences which are quadratic in the cut-off Λ UV. For example, a coupling of the form L = λ f H ff to a Dirac fermion f may allow the one loop contribution (figure 1.a): m 2 H = λ f 2 8π 2 Λ2 UV +.. These contributions give rise to the hierarchy and fine tuning problems 9. SUSY neutralizes these divergence by forcing the existence of the (superpartner s) coupling L = λ s H 2 s 2 with a complex scalar field s, and requiring λ s = λ f 2, producing the one loop contribution (figure 1.b): m 2 H = λ s 8π 2 [Λ2 UV 2m 2 s ln(λ UV /m s ) +..], 8 Note that derivatives terms that might be included in the superpotential are equivalently included instead in the Kähler potential. 9 By choosing a different regularization scheme the problem is expressed differently, but a significant fine tuning of the high-energy theory parameters shows-up anyway. Considering dimensional regularization instead of an high energy cut-off, still any heavy particle (Q) which possibly exist in the high energy theory, and which share some of the SM gauge interactions, induce a contribution to the Higgs mass squared proportional to m 2 Q, resulting fine tuning. 10

15 Fig. 1: One-loop contributions to the Higgs mass, from the interaction with a Dirac fermion (a), and a complex scalar (b). Figure taken form [1]. exactly canceling the fermionic quadratic divergent contribution. In order to preserve this property, and to avoid quadratic divergences, the special relation between the coupling constants λ s = λ f 2, needs to hold under SUSY breaking. This is done by considering spontaneous softly broken SUSY, permitting only SUSY-breaking terms of positive mass dimension. These terms do not alter the relation between λ f and λ s which are dimensionless. The possible soft-susy breaking terms, that might be included in the effective action, take the general form: L soft = ( 1 2 M aλ a λ a aijk φ i φ j φ k bij φ i φ j + t i φ i + c.c.) (m 2 ) i jφ j φ i M a Diracλ a ψ a + c.c. (2.18) L maybe soft = 1 2 cjk i φ i φ j φ k + c.c. These terms are assumed to emerge by some spontaneous SUSY breaking dynamics. As partly explained in the following this dynamics might take different forms, and impose different constraints on the possible SUSY-breaking terms. Some possible terms, like fermion mass term m ij ψ i ψ j, are excluded from (2.18), to avoid redundancy (considering the terms already included in the superpotential). The c jk i term possibly enables quadratic divergences, when some chiral supermultiplet singlets under all gauge symmetries exist. These divergences might exist despite this term is of positive mass dimension. In practice the possibility of the c jk i term is always neglected, although such chiral singlet does not exist in the MSSM, since it is hard to construct spontaneous SUSY breaking models with this term effective. In L soft the coupling constants are of positive mass dimension, and it has been proven that they do not lead to quadratic divergences to all orders in perturbation theory. The M a Dirac term is possible only if the theory contains a gauge singlet chiral superfield, or a chiral superfield in the adjoint representation of a simple factor of the 11

16 gauge group. theory. Also other terms in (2.18) might be restricted by the symmetries of the We turn now to the possible SUSY breaking mechanism. In general when SUSY is broken, the vacuum state 0 is not invariant under SUSY, so Q α 0 = 0 or Q β 0 = 0. The SUSY algebra (2.1) implies that H = P 0 = 1 4 (Q 1 Q 1 + Q 1 Q 1 + Q 2 Q2 + Q 2 Q 2 ), hence SUSY is broken if and only if 0 H 0 > 0 (using the fact that 0 H 0 = 1 4 ( Qα Q α 0 2 ) 0, assuming the Hilbert space has positive norm). Moreover if space-time dependent effects can be neglected and neglecting fermion condensates 10 it is reasonable to assume that 0 H 0 = 0 V 0, where V is the scalar potential of (2.16). Therefore in this case SUSY is broken if and only if F i or D a acquires a non-zero VEV; These possibilities are known respectively as F-term SUSY breaking, and D-term SUSY breaking. Another possibility is having a metastable vacuum [20], for which SUSY is necessarily broken. Anyway, if 0 V 0 > 0 the current vacuum (whether it is stable or not) breaks SUSY. The basic example for D-term tree-level SUSY breaking is known as Fayet-Iliopoulos mechanism. It is occurring if a U(1) gauge group exist, with the introduction of an FI term (2.14). For simplicity, consider the case of a (single) D-component field and a superpotential with no f, and y terms (i.e. taking f i = 0, and y ijk = 0 in (2.6)). Then, the scalar potential (2.16) is V = 1 2 (κ gd i q i φ i 2 ) 2 + i M ij φ j 2, (2.19) where q i are the charges of the fields φ i under the U(1) gauge group. Assuming that all the eigenvalues of M ij (corresponding to the scalar fields charged under the U(1) gauge group) are nonzero, a nonzero κ would imply that this contribution is non-vanishing, and hence SUSY is broken. The U(1) gauge group may or may not be broken according to the parameters. The basic example for F-term tree-level SUSY breaking is known as O Raifeartaigh models. Using the EOM for the F i auxiliary fields (2.9), tree level F-term SUSY breaking is realized only if there is no solution to the set of equations W Φ i = 0, (2.20) 10 There are known examples of SUSY breaking due to fermion condensates, see [18,19]. 12

17 for the (constant) vacuum state. O Raifeartaigh idea is to pick a superpotential with no such solution. A simple example for such a tree-level model is given by W = kφ 1 + mφ 2 Φ 3 + y 2 Φ 1Φ 2 3, (2.21) where the parameters k, m and y are chosen to be real and nonzero. The linear kφ 1 term is possible only when Φ 1 is a gauge singlet. Such a term is required to allow tree-level F-term SUSY breaking in renormalizable theories, since otherwise taking all φ i = 0 will always be a solution of (2.20). In the given model the (F-term) scalar potential is V = k y 2 φ mφ mφ 2 + yφ 1 φ 3 2, which clearly cannot vanish. Without governing symmetries the condition (2.20) contains N complex analytic equations with N complex variables, hence generally admit a solution, assuming the coefficients are choose freely 11. Constraining the superpotential with a (standard) global internal symmetry does not change the possibility for a solution in the general case: The number of equations stays the same as the number of variables, either if the vacuum spontaneously breaks this symmetry or not. In a supersymmetric theory there is another possible global internal symmetry, called R-symmetry. This symmetry allows different components of the chiral superfields to transform differently. In theories with N =1 SUSY the algebra allows only a single Hermitian U(1) generator of this kind, satisfying the commutation relation 12 : [R, Q α ] = Q α. (2.22) Imposing such an R-symmetry, and requiring it to be spontaneously broken, is the only possibility to generically obtain F-term SUSY breaking at tree-level 13 (this is the Nelson- Seiberg theorem, [21]). [In this case, we can assume Φ 1 0 whose R-charge is r 1 0. Then the superpotential can be written as W = (Φ 1 ) 2/r 1 f(u 2,..., U N ), where U k = Φ k /(Φ 1 ) r k/r 1. Now, 11 Considering quantum corrections, usually the parameters of the theory may change, therefore generally cannot be fixed to specific values to allow SUSY breaking. 12 In the conventional normalization. 13 Note that the model (2.21) obeys such a (broken) R-symmetry, with charges R(Φ 1 ) = R(Φ 2 ) = 2, R(Φ 3 ) = 0. 13

18 SUSY is unbroken if there is a solution to the N equations n f = 0 (n = 2..N) and f = 0, over the space of the N 1 variables U k. Since there are more equations than variables, generically there is no solution, and SUSY is broken]. This argument is refereed to as the rank condition mechanism of SUSY breaking. However, a spontaneously broken R-symmetry implies the existence of a Goldstone boson, which is not observed. Hence, a different path is needed to allow for natural SUSY breaking in a SSM. Moreover, the non-renormalization theorem result also that if 0 V 0 = 0 at tree level, the stable vacuum cannot break SUSY to any order in perturbation theory. Hence, loops corrections do not improve the possibility for natural F-term SUSY breaking [17]. Non-perturbativly an analog of the non-renormalization theorem exists, but allows extra contributions in the effective action that might break SUSY [16]. Now, a passageway for natural F-term SUSY breaking is possible, due to the nonperturbative non-renormalization properties of the theory. These restrictions allow effective non-generic models in which SUSY is broken by F-term, without spontaneously broken R-symmetry, e.g. a variation of the 2-3 model given by [21]. Another possible solution for natural F-term SUSY breaking is to consider metastable SUSY-breaking vacua. Another outcome of the non-renormalization properties, reveals that it is impossible to obtain F-term SUSY breaking perturbatively (to any order of perturbation theory) if it does not exist at tree level. Moreover, generally in the absence of an FI term, if SUSY is unbroken by a tree level F-term, then SUSY is unbroken perturbatively. The proof makes use of gauge-invariance to conclude that if the F-term contribution to SUSY breaking vanishes (2.20), it is also possible to find a solution for vanishing D-term [17]. Hence, the D-term can contribute to SUSY breaking perturbatively only when an FI term exists at tree level. Recent studies by [22] on dynamical D-term SUSY breaking show that an effective FI term is impossible in a theory with supergravity. However, this conclusion does not apply to models with a field-dependent FI term, emerged as the effective theory of some string models [23]. To conclude, following these arguments the possibilities for SUSY breaking include: 1. Tree-level SUSY breaking from D or F-terms. D-term SUSY breaking is possible only when an FI term exists, allowed only in the existence of an Abelian gauge symmetry. Then, a nonzero D-term VEV is acquired in some cases, e.g. if all the particles gauged under this symmetry are massive. Allowing F-term SUSY breaking in renormalizable theory, requires a linear term in the superpotential, and hence the existence of a 14

19 gauge singlet chiral superfield. Natural F-term tree-level SUSY breaking is possible if a spontaneously broken R-symmetry exists, or in other cases by considering the effective action of a theory with non-perturbative SUSY breaking, or with metastable vacua. 2. Non-perturbative SUSY breaking. Dynamical SUSY breaking (DSB) by F-term was demonstrated in the literature, e.g. the 2-3 model, metastable vacua and fermion condensates. Dynamical D-term SUSY breaking carries some complications, as mentioned. As a result of spontaneous SUSY breaking the effective theory carries some special properties. First there is the appearance of a massless fermion, the goldstino, which is analog to the goldston boson appearing if spontaneous breaking of a global symmetry occurs. This result holds in general, also non-perturbatively. The goldstino is the fermion component of the supermultiplet whose auxiliary field obtains a VEV. For example, in the FI scenario the goldstino is the gaugino of the U(1) symmetry, and in the example of O Raifeartaigh model given (2.21) the goldstino is the massless fermion ψ 1. In supergravity models the goldstino is eaten by the gravitino, making the gravitino massive. Second, a theory with spontaneous SUSY breaking keeps, at tree level, some properties of the original theory, known as the sum rule, or the supertrace formula: mass 2 2 mass mass 2 = 0. (2.23) spin 0 spin 1/2 spin 1 This result holds separately for particles with different conserved quantities (charge, color, baryon and lepton numbers). As an implication of this sum rule, tree-level SUSY breaking is not allowed within the MSSM The Minimal-SSM In order to extend the SM into a supersymmetric model it is necessary to introduce a chiral superfield for each of the fermions in the SM, and a vector superfield for each of the gauge particles 14. A list of the above chiral superfields is represented in the following table. The different family and color indices are not expressed in this notation. Following 14 For a full review of the topics highlighted in this section see e.g. [1]. 15

20 a standard convention, all superfields are defined in terms of left-handed Weyl spinors, hence u R, d R and e R appear conjugated in the table. The tilde denotes the superpartners of the SM particles (the sparticles). Names Symbol spin 1 2 spin 0 SU(3), SU(2), U(1) Y quarks, squarks Q (u L d L ) (ũ L dl ) 3,2, 1 6 ū u R ũ R d d R d R 3, 1, 2 3 3, 1, 1 3 leptons, sleptons L (ν e L ) ( ν ẽ L ) 1, 2, 1 2 ē e R ẽ R 1, 1, 1 The Higgs is represented in the MSSM by introducing two (SU(2) doublets) chiral superfields H u and H d, differentiated by their U(1) charges, as represented in the following table. Names Letter spin 0 spin 1 2 SU(3), SU(2), U(1) Y Higgs, higgsinos H u (H u + Hu) 0 ( H u + H u) 0 1,2, 1 2 H d (Hd 0 H d ) ( H d 0 H d ) 1,2, 1 2 The two Higgs doublets are required to allow Yukawa couplings of the Higgs to all of the quarks and leptons. Explicitly, the holomorphicity of the superpotential does not allow a term like the ordinary SM term dqh. Hence, an extra Higgs superfield is required to allow a gauge-invariant mass for the down quark. Moreover, the extra Higgs field is demanded to cancel a U(1) anomaly appearing with the addition of a single higgsino field 15. The MSSM vector (gauge) superfields are represented in the following table. The superpatners of the gauge fields named in general gaugino, and particularly the gluino, the winos and the bino. Names spin 1 spin 1 2 SU(3), SU(2), U(1) Y gloun, gluino g g 8,1,0 W bosons, winos W ± W 0 ± 0 W W 1,3,0 B boson, bino B 0 B0 1,1,0 15 One could imagine taking one of the L superfields to be the second Higgs, since they carry the same quantum numbers. This is impossible because there is strong experimental evidence that the lepton-number symmetry is not spontaneously broken. 16

21 The superpotential of the MSSM is constructed in the form of (2.6), by W MSSM = ūy u QH u dy d QH d ēy e LH d + µh u H d, (2.24) where y u, y d and y e are dimensionless Yukawa coupling constants, which are 3 3 matrices in the family space. They are related to the known SM-particles masses and mixing angles, after first considering the Higgs fields acquiring a VEV. Since the third family of the quarks and leptons is significantly heavier than the first and second families, usually an approximation is taken by neglecting the rest of the matrix elements, taking: y u y t, y d y b, y e y τ. (2.25) Writing the SU(2) multiplets in components, and considering this approximation, the MSSM superpotential is: W MSSM y t ( tth 0 u tbh + u ) y b ( bth d bbh 0 d) y τ ( τν τ H d ττh0 d) + µ(h + u H d H0 uh 0 d). (2.26) As opposed to the (non-supersymmetric) SM, in an SSM gauge invariance allows extra (renormalizable) terms in the superpotential (2.24) that violate the total baryon (B) and lepton (L) numbers. Including such terms in the theory contradicts the stability of the proton. However, imposing B and L symmetries in the MSSM by hand is not considered a viable option, since they are theoretically known to be violated by non-perturbative electroweak effects. Instead, the MSSM is usually assumed to obey another global and discrete 16 symmetry known as R-parity, defined by the conservation of: P R = ( 1) 2s ( 1) 3(B L), (2.27) where s denotes the spin of the particle considered. Imposing this symmetry allows only terms with P R = 1 in the action, hence permits only the terms of (2.24) in the (renormalizable) superpotential. Thus, the total baryon and lepton numbers are effectively conserved, but can be violated by (small) non-renormalizable terms in the action. As alternative to (2.27) the symmetry can be imposed without the ( 1) 2s factor, resulting in the same 16 A discrete symmetry is preferred, because of motivation from string theory & gravity to allow only local continuous (exact) symmetries. 17

22 constraint on the renormalizable superpotential. This form is known as matter parity. R-parity is preferred since it predicts a possible candidate for the unexplained dark matter in cosmology. The underlying reason is that all the SM sparticles carry odd R-parity P R = 1, while the SM particles have even R-parity P R = 1. Hence, R-parity implies that the lightest sparticle (the LSP) is stable. If the LSP is electrically natural it can be a candidate for dark matter. Exact R-parity conservation is motivated theoretically by examples of emergent R-parity from gauge symmetry breaking and from string theory. The µ term in (2.24) is dimensionful coupling, contributing to the Higgs and higgsino masses. Electroweak symmetry breaking cannot be understood from the superpotential W MSSM, and is introduced via SUSY-breaking terms. Regardless of the exact SUSY-breaking dynamics at high energy, the MSSM assumes spontaneous soft-susy breaking, and introduces an effective theory, in the form of (2.18). Imposing gauge invariance and matter parity, the most general possible effective MSSM soft-susy breaking Lagrangian is: L soft = 1 2 (M 3 g g + M 2 W W + M1 B B + c.c.) ( ūa u QHu dad QHd ēa e LHD + c.c.) Q m 2 Q Q L m 2 L L ūm ū ū 2 dm 2 d d ēm ēs ē 2 m 2 H u H uh u m 2 H d H dh d (bh u H d + c.c.). (2.28) Here, the terms in the first line correspond to the M a term of (2.18), and are mass terms for the gauginos. The second line corresponds to the a ijk of (2.18), where each of a u, a d, a e is a 3 3 complex matrix matching one of the Yukawa terms in the superpotential (2.24). The terms in the third line correspond to the (m 2 ) i j of (2.18), and are (3 3 matrices of) mass terms for the squarks and sleptons. The last line contains contributions to the Higgs potential, with two more terms corresponding to the (m 2 ) i j term, and the last term is the only b ij term of (2.18) occurring in the MSSM. Gauge (and family) indices are suppressed in the above Lagrangian. The SUSY-breaking sector introduces many new arbitrary parameters to the MSSM. However, the form of (2.28) is severely restricted by experimental constraints on CPviolating, and flavour mixing processes. In fact, all non-diagonal elements of the matrices a u, a d, a e and m 2 Q, m2 L, m2 ū, m 2 d, mē 2 are constrained to be extremely small 17. Moreover, 17 These restrictions depend on the MSSM spectrum, and are slightly less significant if the superparticles are relatively heavy, at the TeV scale. 18

23 to avoid CP-violation, the new parameters should not introduce new complex phases in the MSSM. Hence, often the hypothesis of SUSY-breaking universality is assumed, i.e. the new mass parameters are diagonal in family space, the a ijk terms are proportional to the Yukawa couplings, and new complex phases are absent: m 2 Q = m 2 Q1, m 2 L = m 2 L1, mū 2 = m 2 ū1, m 2 d = m 2 d1, mē 2 = m 2 ē1 a u = A u0 y u, a d = A d0 y d, a e = A e0 y e arg(m i ) = arg(a u0 ) = arg(a d0 ) = arg(a e0 ) = 0 or π. (2.29) Alternatively, different explanations of the flavour mixing in (2.28) exist, among them there are the hypotheses: 1. Irrelevancy. Assuming very heavy (much larger than 1 TeV) sparticles, and hence weaker restrictions on (2.28). 2. Alignment. Assuming that the mass matrices of the sparticles are aligned with the Yukawa matrices, such that large flavour mixing is avoided. One of the hypotheses listed above, or another variation, is generally assumed to emerge from properties of the dynamics that breaks SUSY. However, evolving the renormalization group (RG) equations (elaborated in section 2.6) typically implies that the theory at the weak scale does not always satisfy exactly the universality or alignment relations (or other such relations) that are imposed at the original (high) scale. Electroweak symmetry-breaking (EWSB) is obtained by considering the Higgs scalar potential, induced from (2.24) and (2.28). By gauge fixing, the (classical) potential at the minimum is expressed by specifying H u + = H d = 0, and takes the following form V = ( µ 2 +m 2 H u ) Hu 0 2 +( µ 2 +m 2 H d ) Hd 0 2 (bhuh 0 d 0 +c.c.)+ 1 8 (g2 +g 2 )( Hu 0 2 Hd 0 2 ) 2. Thereafter by field redefinition one can conclude that at the minimum b, Hu, 0 Hd 0 0. Now, EWSB is possible if there exist a stable vacuum with nonzero Higgs VEV, requiring: 2b < 2 µ 2 + m 2 H u + m 2 H d b 2 > ( µ 2 + m 2 H u )( µ 2 + m 2 H d ). (2.30) These constraints involve the parameters of the effective renormalized Lagrangian, after RG evolution, and are corrected by loop contributions. The loop corrections to the effective tree-level parameters are minimized when the RG scale is chosen at the average scale of the 19

24 top squarks masses. Note that the SUSY-breaking terms b and m 2 H u m 2 H d are necessary to allow EWSB. Both the VEV Hu 0 = v u and Hd 0 = v d are nonzero (in order to be able to write fermionic mass terms for both top and bottom quarks). The Higgs VEV pair satisfies vu 2 + vd 2 = v2 (174 GeV) 2, and their ratio is written as tan β = v u /v d. Using this notation, the relation satisfied by the VEV is written as: m 2 H u + µ 2 b cot β (m 2 Z/2) cos(2β) = 0 m 2 H d + µ 2 b tan β + (m 2 Z/2) cos(2β) = 0 (2.31) The value of tan β is not determined by experiments, but can be used together with m 2 H u, m 2 H d and (2.31) to eliminate the dependence on b and µ, but not the phase of µ. However µ with complex phase possibly causes CP problems, and hence a real µ is almost always assumed, leaving only the sign undetermined. Theoretically, tan β is constrained by the approximate bounds 1.2 tan β 65 to prevent non-perturbative Yukawa couplings (however, the upper bound is not absolute). After EWSB the MSSM acquires different mass eigenstates. From the Higgs doublets, which has 8 real DOF, three are eaten by the gauge bosons W ± and Z 0, and the remaining five are arranged into three natural scalars h 0, H 0, A 0 and two charged scalars H ±. The scalar h 0 is defined to be the lightest of the two natural CP-even states (h 0, H 0 ), and commonly (but not always) behaves in a very similar way to a SM Higgs boson. Another outcome of EWSB is that the higgsinos mix with the electroweak gauginos. The resulting mass eigenstates of the MSSM are denoted as in the following table. Names Higgs bosons neutralinos charginos Gauge eigenstates H 0 u, H 0 d, H+ u, H d B 0, W 0, H 0 u, H 0 d W ±, H + u, H d Mass eigenstates h 0, H 0, A 0, H ± χ 0 1, χ 0 2, χ 0 3, χ 0 4 χ ± 1, χ± 2 Moreover, as a result of the Yukawa mixing terms (possibly) present in W MSSM and more terms of L soft, the squarks and sleptons may be mixed as well. Considering the assumptions (2.25) and (2.29) 18, the complementary mass eigenstate composition of the MSSM is 18 and neglecting small mixing due to the RG evolution 20

25 denoted as in the following table. Names squarks sleptons gluino goldstino Gauge eigenstates ũ L, ũ R, d L, d R ẽ L, ẽ R, ν e g G1/2 s L, s R, c L, c R µ L, µ R, ν µ t L, t R, b L, b R τ L, τ R, ν τ Mass eigenstates (same) (same) (same) (gravitino (same) (same) G3/2 ) t 1, t 2, b 1, b 2 τ 1, τ 2, ν τ As stated in the introduction, in a SSM, SUSY cannot be broken by renormalizable interactions at tree level within the MSSM. This is forbidden since the sum-rule (2.23) predicts the existence of light sparticles, which should have been found 19. Moreover, in this case the non-renormalization theorem also forbids (renormalizable) loop contributions to SUSY breaking, from within the MSSM content (if SUSY is not already broken). Hence SUSY breaking is required to emerge from a hidden sector separated from the MSSM visible sector. There are various models extending the MSSM which include SUSY breaking, or specify the interactions between the hidden and the visible sectors. Commonly, the interactions are assumed to be indirectly, through loop corrections or non-renormalizable terms 20. In order to naturally maintain a relation like (2.29), SUSY-breaking mediation by flavor-blind interactions is considered. Gravity mediation assumes that SUSY breaking is related to the dynamics at the Planck scale and is mediated to the visible sector by gravity. The constraint of (2.23) is evaded since gravity contributions are non-renormalizable. On the other hand, gauge mediation assumes that the known gauge interactions are responsible for transmitting SUSY breaking to the visible sector. In this case SUSY breaking appears in the effective MSSM action as non-renormalizable loop corrections, and evades (2.23). In these models, SUSY is assumed to be broken at some intermediate scale, between the weak scale and the Planck scale. Fixing a tree-level SUSY-breaking scale 19 Also, none of the superfields of the MSSM can acquire a nonzero F or D-term VEV. A gauge singlet chiral superfield is missing to acquire an F-term VEV. Also, a D-term VEV for the U(1) Y gauge superfield results unacceptable spectrum. 20 However, other possibilities exist, e.g. [24]. 21

26 by hand might cause fine tuning, if there is no underlying reason to choose it at a low scale relative to the Planck scale. Thus, DSB is favored, where the corresponding scale is possibly fixed by the high-energy dynamics (like the QCD scale). Some possible models of DSB were mentioned in section 2.2. Conveniently, SUSY breaking can be included in the gauge-mediation action by assuming the existence of a spurion superfield, acquiring an F-term VEV, a D-term VEV or both. The spurion can effectively represent DSB, or alternatively stand for some tree-level model of the hidden sector (i.e. O Raifeartaigh or FI model). The current work focuses on models of gauge-mediated SUSY breaking (GMSB), which will be described extensively in the following section. One general distinct property between GMSB and gravity mediation, is the gravitino mass. Assuming F-term SUSY breaking, it is given in both models by a dimensional-analysis estimate: m 3/2 F /M P. (2.32) A similar estimation shows that in gravity mediation this quantity is of the same order of the soft masses m soft, while gauge mediation predicts a suppression m 3/2 (M/M P ) m soft, where M is a typical mass scale of the hidden sector, usually assumed to be much smaller than M P. Hence, in GMSB the LSP is almost certainly a very light gravitino, while in gravity mediation the gravitino is not so light and not necessarily the LSP. As mentioned in the introduction, the MSSM Higgs coupling terms give rise to a possible fine tuning, and hierarchy problem. First, the µ term appearing in (2.24) is dimensionful, and hence is naturally expected to be of the order of the Planck scale (this is the µ problem ). Secondly, the terms b, m 2 H u, m 2 H d appearing in (2.28) are of the soft- SUSY breaking scale m 2 soft. However, all these terms contribute to the Higgs potential, and are related to known quantities of the weak scale, as expressed for example in relation (2.31). Hence the magnitude of µ 2, b, m 2 H u, m 2 H d ought to be not too far above the weak scale. Moreover, if these parameters are at a scale much larger than m 2 Z, as in many SSM models, a significant cancellation is needed. Without an underlying reason it is amazing to accept such cancellation between µ and the m soft parameters, which seem to be unrelated. This is the little hierarchy problem. Thus, it is commonly believed that the MSSM has to be extended with a high-energy mechanism that will relate the effective value of µ to the effective SUSY-breaking scale. Also obtaining the correct value of b µ 2 in GMSB is generically problematic (the µ/bµ problem ). For some proposals, see e.g. [25,26,27]. 22