Strictly Anomaly Mediated Supersymmetry Breaking

The SM  Higgs-like particle of mass $125\hbox{GeV}$ recently discovered at the LHC [1, 2] strongly constrains future model building, while recent negative results from both the Tevatron and LHC in searches for sparticles place increasing pressure on models with low energy supersymmetry. Here we explore a specific supersymmetric model in which the low energy spectrum is that of the Minimal Supersymmetric Standard Model (MSSM), but the gauge symmetry is augmented by an extra gauged U(1) symmetry, $\textrm{U(1)}^{\prime}$, spontaneously broken at high energies in a manner which affects both physics at the supersymmetry breaking scale and physics at high scales characterising inflation and cosmic strings.

The broad features of the model are independent of the source of supersymmetry breaking, but if we assume that this source is in fact anomaly mediation (AMSB) [3]-[5], then there arises an interesting interplay between the low energy physics (and in particular the Higgs $\mu$-term) and the high energy physics involving strings and inflation. Moreover the breaking of $\textrm{U(1)}^{\prime}$ solves the tachyonic slepton problem characteristic of AMSB [5, 6].

We first presented this specific model in [7], in a form where we also introduced a Fayet-Iliopoulos (FI) term for the $\textrm{U(1)}^{\prime}$. Here we concentrate on the minimal formulation when there is no such term¹¹1Aside from the fact that this makes the model more appealing by removing an independent mass scale, we also thereby avoid confrontation with the conclusion of Komargodski and Seiberg [8] that a global theory with a FI term cannot be consistently embedded in supergravity. The model implements a form of AMSB which we refer to as strictly anomaly-mediated supersymmetry-breaking (sAMSB), by which we mean that there are no other sources of supersymmetry breaking beside the F-term of the conformal compensator field. As a consequence the soft parameters have an elegant renormalisation group (RG) invariant form. It therefore differs from so-called minimal AMSB(mAMSB), which posits an extra source of supersymmetry-breaking, instead of extra fields, in order to solve the tachyonic slepton problem. Our model is not quite a complete sAMSB implementation, in that it requires an extension to determine the soft parameter associated with the Higgs $\mu$-term.

We begin by describing the symmetries and field content of the model and explaining in detail how the spontaneous breakdown of the $\textrm{U(1)}^{\prime}$ symmetry at a large scale $M$ not only solves the AMSB tachyonic slepton problem, but also generates a Higgs $\mu$ term and the see-saw mechanism for neutrino masses. This outcome is achieved by the introduction of three new chiral superfields; $S$, which is a gauge singlet and a pair of $\textrm{SU(3)}\otimes\textrm{SU(2)}\otimes\textrm{U(1)}_{Y}$ singlet fields $\Phi,\overline{\Phi}$ which are oppositely charged under $\textrm{U(1)}^{\prime}$. We then exhibit characteristic sparticle spectra for the model; the calculations involved to obtain these are essentially as described in Refs. [9, 10], but allowing for a larger gravitino mass. We also discuss the fine-tuning issue raised by this, and compare the results of our model with results from the most popular (but, we will argue, less elegant) version of AMSB, generally called mAMSB. We will see that sAMSB generally keeps sleptons lighter than in mAMSB, which means that the contribution to the muon anomalous magnetic moment $a_{\mu}\equiv(g_{\mu}-2)/2$ is typically higher for a given Higgs mass.

The theory incorporates a natural mechanism for supersymmetric F-term inflation, with the scalar component of $S$ as the inflaton. Previously, we concentrated on a region of parameter space such that inflation ended with a transition to a state with only the $\textrm{U(1)}^{\prime}$ broken. There is, however, an interesting alternative that inflation ends with the development of vevs for the Higgs multiplets, $h_{1,2}$, breaking the electroweak symmetry. A combination of the Higgs fields $h_{1}\cdot h_{2}$ and the scalar components of the singlet fields $\phi\overline{\phi}$ is a flat direction, lifted by soft supersymmetry-breaking terms, and the normal low energy electroweak vacuum is achieved after a later period of thermal inflation. Approximately 17 e-foldings of thermal inflation reduce the number of e-foldings of high scale inflation, and therefore reduce the spectral index of scalar Cosmic Microwave Background (CMB) fluctuations to about 0.975. This is within about $1\sigma$ of the WMAP7 value.

The reheat temperature after this period of thermal inflation is around $10^{9}$ GeV, which means that there is no gravitino problem: gravitinos are very massive, more than 40 TeV, and so decay early enough not to be in conflict with nucleosynthesis. Indeed, the gravitino problem can turn into the gravitino solution for the typical AMSB  feature of too low a dark matter density generated at freeze-out: the lightest supersymmetric particle (LSP) is mostly wino and has a relatively high annihilation cross-section. In our model a critical density of LSPs can be generated by gravitino decays, if the gravitino (and hence the LSP) is heavy enough.

The model also has the possibility of baryogenesis via leptogenesis following thermal inflation, with CP violation supplied by the neutrino sector. The field giving mass to right-handed neutrinos has an inflation-scale ($10^{16}$ GeV) vacuum expectation value (vev), but if the lightest right-handed neutrino is sufficiently light to be generated at reheating after thermal inflation, a lepton asymmetry can be generated by its out-of-equilibrium decay.

There is a broken U(1) symmetry in the model, and cosmic strings with a $10^{16}$ GeV mass scale are formed, although not until the end of thermal inflation. There is a large Higgs condensate in the core of the string which spreads the string out to a width of order the supersymmetry-breaking scale, and reduces its mass per unit length by well over an order of magnitude. The strings satisfy CMB constraints on the mass per unit length from combined WMAP7 and small-scale observations. Their decays are constrained by pulsar timing observations in the case of gravitational waves and the diffuse $\gamma$-ray background in the case of particle production: the latter means that less than about 0.1% of the energy of the strings should end up as particles, and the former puts constraints on the average size of the loops at formation.

The model has the same field content as the $F_{D}$ hybrid inflation model [11, 12], but different charge assignments and couplings. $F_{D}$ hybrid inflation also has a singlet which is a natural inflaton candidate, but differs in other ways: for example, right-handed neutrinos have electroweak-scale masses, and the gravitino problem is countered by entropy generation.

The model also has the same field content as the B$-$L model of Refs. [13], although as explained in section 3, the U(1)^′ symmetry cannot be U(1)_B-L in AMSB. It is also closely related to the model of Ref. [14], in which the fields $\Phi$, $\overline{\Phi}$ are SU(2)_R triplets. This also has a flat direction involving the Higgs, although the authors did not pursue its consequences.

To summarise our results: at $\tan\beta=10$, sAMSB  can accommodate a Higgs mass above 120 GeV for gravitino masses over 80 TeV, while accounting for the discrepancy in $a_{\mu}$ between the Standard Model (SM) theory and experiment to within $2\sigma$ would have favoured 80 TeV or lower. Larger values of $\tan\beta$ allow a more massive Higgs: for $\tan\beta=16$ we find a Higgs mass of 125 GeV for a gravitino mass of 140 TeV.

sAMSB also allows for an observationally consistent dark matter density, if the gravitino mass is over about 100 TeV, with the dark matter deriving from the decay of gravitinos produced from reheating after thermal inflation. The spectral index of scalar cosmological perturbations is within 1$\sigma$ of the WMAP7 value, and the observational bounds on cosmic strings can be satisfied if the strings decay into gravitational radiation. The model has also has a natural mechanism for baryogenesis via leptogenesis through the decays of right-handed neutrinos.

We will assume that supersymmetry breaking arises via the renormalisation group invariant form characteristic of Anomaly Mediation, so that the soft parameters for the gaugino mass $M$, the $\phi^{3}$ interaction $h$ and the $\phi^{*}\phi$ and $\phi^{2}$ mass terms $m^{2}$ and $m_{3}^{2}$ in the MSSM take the generic RG invariant form

Here $\mu$ is the renormalisation scale, and $m_{\frac{3}{2}}$ is the gravitino mass; $\beta_{g_{i}}$ are the gauge $\beta$-functions and $\gamma$ is the chiral supermultiplet anomalous dimension matrix. $Y_{U,D,E,N}$ are the $3\times 3$ Yukawa matrices, and $\mu_{h}$ is the superpotential Higgs $\mu$-term. We will see that in our low energy theory, Eq. (3) is replaced, in fact, by

where the $Y_{i}$ are charges corresponding to a U(1) symmetry. This $kY$ term corresponds in form to the contribution of an FI D-term, and can be employed to obviate the tachyonic scalar problem characteristic of AMSB. How such a term can be generated (with AMSB) was first discussed in Ref. [5], and first applied to the MSSM in Ref. [6]. The basic idea was pursed in a number of papers [15]-[19]. For example, Ref. [15]  demonstrated explicitly the UV insensitivity of the result, and Ref. [16] emphasised that the tachyonic problem could be solved using a single U(1) rather than a linear combination of two, the approach followed in Ref. [6]. An extension of the MSSM such that the spontaneous breaking of a gauged $\textrm{U(1)}^{\prime}$ with an FI term gave rise to the $kY$ term was written down in Ref. [9]. In [7] we developed an improved version of this model, retaining the possibility of a primordial FI term for $\textrm{U(1)}^{\prime}$; here we will dispense with the FI term, and emphasise that we can nevertheless generate the $kY$ term naturally with $k$ of $O(m_{\frac{3}{2}}^{2})$, by breaking a $\textrm{U(1)}^{\prime}$ symmetry at a large scale, without introducing an explicit FI term.

At first sight Eq. (5) resembles the formula for the scalar masses employed in the so-called mAMSB model, where the $kY_{i}$ term is replaced by a universal scalar mass contribution $m_{0}^{2}$. The differences are as follows:

• •

  The mAMSB involves the introduction of an additional source of supersymmetry breaking independent of the gravitino mass, while, as we shall see, Eq. (5) does not.

• •

  The parameter $k$ in Eq. (5) turns out to be more constrained than $m_{0}^{2}$. This is associated with the fact that inevitably all the $Y_{i}$ cannot have the same sign.

• •

  The elegant RG invariance of Eq. (1)-Eq. (4) is preserved by Eq. (5).

It is these observations that prompts us to refer to our model as sAMSB. Note that, of course, we cannot “promote” the mAMSB into the sAMSB by the addition of additional heavier fields which cancel the associated $\textrm{U(1)}^{\prime}$ anomaly; with an unbroken $\textrm{U(1)}^{\prime}$, any massive chiral multiplets will obviously make no contribution to this anomaly.

Eq. (4) is the most general form for $m_{3}^{2}$ that is consistent with RG invariance, as first remarked explicitly in Ref. [9]; the parameter $\kappa$ is an arbitrary constant. For discussion of possible origins of $m_{3}^{2}$ from the underlying superconformal calculus formulation of supergravity see Refs. [5, 18, 19]. We will simply assume that the model can be generalised to produce such a term; the procedure which has, in fact, been generally followed. The presence of $\kappa$ means that in sparticle spectrum calculations one is free to calculate $m_{3}^{2}$ (and the value of the Higgs $\mu$-term, $\mu_{h}$) by minimising the Higgs potential at the electroweak scale in the usual way. (For $\kappa=1$, which is the value suggested by a straightforward use of the conformal compensator field [5], one might have hoped to use the minimisation conditions to determine $\tan\beta$, but it turns out this leads to a very small value of $\tan\beta$ incompatible with gauge unification, because of the correspondingly large top Yukawa coupling [17]). We will see, however, that in our model the result for $\mu_{h}$ has implications for other parameters in the underlying theory which are constrained by cosmological considerations.

The MSSM (including right-handed neutrinos) admits two independent generation-blind anomaly-free U(1) symmetries. The possible charge assignments are shown in Table 1.

The SM gauged $\textrm{U(1)}_{Y}$ is $q_{L}=-1,q_{E}=2$; this U(1) is of course anomaly free even in the absence of $N$. $\textrm{U(1)}_{B-L}$ is $q_{E}=-q_{L}=1$; in the absence of $N$ this would have $\textrm{U(1)}^{3}$ and U(1)-gravitational anomalies, but no mixed anomalies with the SM gauge group.

Our model will have, in addition, a pair of MSSM singlet fields $\Phi,\overline{\Phi}$ with $\textrm{U(1)}^{\prime}$ charges $q_{\Phi,\overline{\Phi}}=\pm(4q_{L}+2q_{E})$ and a gauge singlet $S$. In order to solve the tachyon slepton problem we will need that, for our new gauge symmetry $\textrm{U(1)}^{\prime}$, the charges $q_{L},q_{E}$ have the same sign at low energies. As explained in Ref. [10], however, it is in fact more appropriate to input parameters at high energies, when in fact although necessarily $q_{E}>0$, the range of acceptable values of $q_{L}$ includes negative ones; not negative enough, however, to allow $\textrm{U(1)}^{\prime}$ to be $\textrm{U(1)}_{B-L}$.

Thus sAMSB has three input parameters $m_{\frac{3}{2}}$, $kq_{L}$, $kq_{E}$, associated with the supersymmetry breaking sector, while mAMSB only has two: $m_{\frac{3}{2}}$, $m_{0}$. However, it turns out that because the allowed $(q_{L},q_{E})$ region is so restricted, sAMSB is the more predictive of the two. We will see this explicitly in section 6.

The complete superpotential for our model is:

where $W_{A}$ is the MSSM superpotential, omitting the Higgs $\mu$-term, and augmented by Yukawa couplings for the right-handed neutrinos, $N$:

and

where $M,\lambda_{1},\lambda_{3}$ are real and positive and $\lambda_{2}$ is a symmetric $3\times 3$ matrix. The sign of the $\lambda_{3}$ term above is chosen because with our conventions, in the electroweak vacuum where

we have $H_{1}H_{2}\to-\frac{1}{2}v_{1}v_{2}$.

The $\textrm{U(1)}^{\prime}$ symmetry forbids the renormalisable $B$ and $L$ violating superpotential interaction terms of the form $QLD$, $UDD$, $LLE$, $H_{1}H_{2}N$, $NS^{2}$, $N^{2}S$ and $N^{3}$, as well as the mass terms $NS$, $N^{2}$ and $LH_{2}$ and the linear term $N$. Moreover $W_{B}$ contains the only cubic term involving $\Phi,\overline{\Phi}$ that is allowed. Our superpotential Eq. (6) is completely natural, in the sense that it is invariant under a global $R$-symmetry, with superfield charges

which forbids the remaining gauge invariant renormalisable terms ($S^{2}$, $S^{3}$, $\Phi\overline{\Phi}$ and $H_{1}H_{2}$). This $R$-symmetry also forbids the quartic superpotential terms $QQQL$ and $UUDE$, which are allowed by the $\textrm{U(1)}^{\prime}$ symmetry, and give rise to dimension 5 operators capable of causing proton decay [20]-[22]. It is easy to see, in fact, that the charges in Eq. (10) disallow B-violating operators in the superpotential of arbitrary dimension. Of course this $R$-symmetry is broken by the soft supersymmetry breaking.

In this section we discuss the spontaneous breaking of the $\textrm{U(1)}^{\prime}$ symmetry and its consequences. We shall assume $M$ is much larger than the scale of supersymmetry-breaking. (Such a large tadpole term has been disfavoured in the past; moreover it has been argued that it would generally be expected to lead to a large vev $\mathopen{\langle}s\mathclose{\rangle}$, but as we shall see this does not happen in our model.) It is then clear from the form of the superpotential $W_{B}$ as given in Eq. (8) that for an extremum that is supersymmetric (when we neglect supersymmetry-breaking) we will require non-zero vevs for $\phi,\overline{\phi}$ and/or $h_{1,2}$ (in order to obtain $F_{S}=0$). The existence of competing vacua of this nature was noted in by Dvali et al in Ref. [14]; their model differs from ours in choice of gauge group (they have $\textrm{SU(3)}\otimes\textrm{SU(2)}_{L}\otimes\textrm{SU(2)}_{R}\otimes\textrm{U(1)}_{B-L}$) and supersymmetry-breaking mechanism.

Let us consider these two possibilities in turn.

In this section we calculate sparticle spectra for the sAMSB model, and compare the results with typical mAMSB spectra. We shall be interested in seeking regions of parameter space with a “high” Higgs mass - that is, close to about 125 GeV as suggested by recent LHC data [1, 2] - and a supersymmetric contribution to the muon anomalous magnetic moment $\delta a_{\mu}$ compatible with the experimental deviation from the Standard Model prediction, $\delta a_{\mu}^{\textrm{exp}}=29.5(8.8)\times 10^{-10}$ [25]. We will also wish to remain consistent with the negative results of recent LHC supersymmetry searches, see for example Refs. [26, 27].

We use the methodology of Ref. [10], which, as explained in Section 2, can also be applied to mAMSB  by replacing the characteristic $(L,e)$ FI-type terms of sAMSB  by a universal mass term $m_{0}^{2}$.

We begin by choosing input values for $m_{\frac{3}{2}}$, $\tan\beta$, $L$, $e$ and $\hbox{sign}\,\mu_{h}$ at the gauge unification scale $M_{X}$. Then we calculate the appropriate dimensionless coupling input values at the scale $M_{Z}$ by an iterative procedure involving the sparticle spectrum, and the loop corrections to $\alpha_{1\cdots 3}$, $m_{t}$, $m_{b}$ and $m_{\tau}$, as described in Ref. [28]. We define gauge unification by the meeting point of $\alpha_{1}$ and $\alpha_{2}$; this scale, of around $10^{16}\hbox{GeV}$, we assume to be equal or close to the scale of $\textrm{U(1)}^{\prime}$ breaking. For the top quark pole mass we use $m_{t}=172.9\hbox{GeV}$. All calculations are done in the approximation that we retain only third generation Yukawa couplings, $\lambda_{t,b,\tau}$; thus the squarks and sleptons of the second generation are degenerate with the corresponding ones of the first generation.

We then determine a given sparticle pole mass by running the dimensionless couplings up to a certain scale chosen (by iteration) to be equal to the pole mass itself, and then implementing full one-loop corrections from Ref. [28], and two-loop corrections to the top quark mass [29]. We use two-loop anomalous dimensions and $\beta$-functions throughout.