A little more Gauge Mediation and the light Higgs mass

We consider minimal models of gauge mediated supersymmetry breaking with an extra $U(1)$ factor in addition to the Standard Model gauge group. A $U(1)$ charged, Standard Model singlet is assumed to be present which allows for an additional NMSSM like coupling, $\lambda H_u H_d S$. The U(1) is assumed to be flavour universal. Anomaly cancellation in the MSSM sector requires additional coloured degrees of freedom. The $S$ field can get a large vacuum expectation value along with consistent electroweak symmetry breaking. It is shown that the lightest CP even Higgs boson can attain mass of the order of 125 GeV.

However, phenomenologically 1 the minimal versions of gauge mediation are severely constrained due to the discovery of a Higgs particle with a mass around 125 GeV. In MSSM, for the lightest CP even Higgs to be around 125 GeV would require, stop mixing parameter X t to be large, While this holds true as long as stops are light ∼ 1 TeV, for very heavy stops 4 TeV, the mixing parameter can be smaller. This would however push stops out of the reach of the LHC. In spite of theoretically appealing features, unfortunately, in minimal gauge mediation, the only way to fit a light Higgs mass ∼ 125 GeV is by making stops very heavy. The typical trilinear couplings in these models are very small at the mediation scale ∼ 0.
Renormalisation group (RG) effects do generate them at the weak scale, however their magnitude is not large enough unless one makes gluinos ultra heavy ∼ several TeV [15]. It should be noted that the constraints from 125 GeV Higgs boson are stronger even if one moves away from minimal mediation models to general gauge mediation models as long as A t remains zero at the messenger scale [16].
Several possible solutions have been explored in the literature [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. One of the directions which is popular with many authors is to introduce direct Yukawa couplings between messenger fields and the MSSM fields in addition to gauge interactions [35,36]. In some cases, these interactions could also violate flavour [37]. In most of the models it is possible to generate large enough A t at the weak scale to fit the 125 GeV light CP even Higgs boson mass. In a recent survey [19,38] it has been pointed out that a particular class of messenger-matter interactions, messenger-stop mixings, has the least fine tuning of all the possible models which fit the light Higgs mass. Another direction which has been considered is to add additional vector like quarks close to the weak scale which couple to the Higgs superfields. These lead to additional corrections to the light Higgs boson thus lifting its mass without the need of increasing the stop masses (see for example, [30][31][32]).
In the following we would like to take an alternate route. We would like to keep the minimal mediation structure in tact, thus would not like to introduce direct couplings between matter and 1 For an early phenomenology of these models, please see, [12][13][14].
messenger fields. Adding an additional singlet field, like in NMSSM could help to raise the light Higgs mass. There are however, problems with electroweak symmetry breaking while incorporating NMSSM in minimal gauge mediation. These are well documented in literature [39,40]. There are ways out, either by adding additional matter fields or dynamics through which NMSSM can be made viable with minimal gauge mediation [41][42][43][44][45][46][47][48][49][50]. Post 125 GeV Higgs boson, a model within this class has been explored in [26].
In the present work, we will consider an additional U(1) gauge group under which the 'singlet' of the NMSSM is charged. This U(1) factor also participates in gauge mediation. Anomaly cancellation requires additional vector like matter to be present. Such vector like matter is typically introduced to generate correct electroweak symmetry breaking while incorporating NMSSM in minimal mediation models [40]. In the present case, it is motivated from anomaly cancellation requirements. It should be noted that this kind of model has been considered earlier by the authors of Ref. [41]. Ours is a more explicit realisation of it in the sense that we have taken care of U (1) charges and anomaly cancellation conditions. Furthermore, we have performed a more detailed analysis of the Higgs masses in the light of 125 GeV Higgs discovery.
We found that it is possible to find an appropriate set of rational U(1) charges which satisfy the anomaly cancellation conditions as well as allow the correct set of terms in the superpotential.
Electroweak symmetry breaking is possible as the U(1) charged singlet can achieve a reasonable vacuum expectation value (vev). Two factors contribute to the raise in the lightest CP even Higgs mass: the effective µ term is sufficiently large ∼ 0.5 − 1 TeV and secondly the RG generated A t term is large compared to minimal gauge mediation. The later is because at the 1-loop level, the SU (3) beta function, b 3 is zero in this model and the 2-loop b 3 is not sufficiently large. Together they result in sufficient X t to ensure large mixing in the stop mass matrix. It is possible to find reasonable parameter space which gives correct lightest CP even Higgs mass and satisfy direct constraints from LHC as well as constraints from Z − Z mixing.
The rest of the paper is organised as follows: In the next section particle spectrum and the model are presented. The details of supersymmetric spectrum and various constraints on the parameter space are discussed in section 3. Numerical results are presented in section 4. We close with an outlook in section 5.

II. MODEL AND THE PARTICLE SPECTRUM
The basic premise of the model is that the singlet of the NMSSM should no longer be a singlet, but instead, it is charged under an extension of the Standard Model gauge group such that it receives non-zero supersymmetry breaking contributions at the mediation scale. As it will be detailed in the next section this would help in attaining a large enough vacuum expectation value for the field 'S'. In this present work, we try to do this by considering the simplest extension in terms of a U (1 where the first three represent the usual Standard Model gauge group and the additional gauge group is represented by a subscript A. U (1) A is a chiral gauge group and hence introduces an extra set of anomalies which need to be canceled for a consistent quantum field theory. This imposes a set of conditions on the U (1) A charges; they are listed in Appendix A. We insist that the anomalies cancel independently for the NMSSM sector and the Messenger sector. It is easily verified that the MSSM particle spectrum along with the new field S is not sufficient to cancel all the anomalies.
In particular, from (U (1) A − [SU (3) c ] 2 ) anomaly condition we get where A 1 (exotics) is the contribution of the new exotic fields which need to be added and s is the U (1) A charge of the field S. The U (1) A charge s cannot be zero as per our requirements.
Furthermore, to generate the effective µ term (λSH u H d ) in the super-potential, the charge s should be equal to where h 1 and h 2 are the U (1) A charges of H 1 and H 2 respectively. We thus need coloured exotics to anomaly. The number of the exotics is fixed by other anomaly conditions as well as by the U (1) A gauge invariance of the super-potential. It turns out that one possible minimal set of exotic fields would be three families of SU (2) L singlet coloured exotics. We introduce a pair of colour fundamental and anti-fundamentals D i andD i , which are SU (2) singlets, for each of the three families. In addition to the QCD interactions they are allowed to couple with the field where through an abuse of notation, we have expanded the spurion as < X >= X + θ 2 F X and defined Λ = F X /X. C i (f ) are quadratic Casimirs for the fields f under the four gauge groups.
The index i here runs over all the four gauge groups of Eq.(1). We denote the gauge coupling corresponding to U (1) A as g 4 and we can see, the soft mass of S has the following non-zero value at the X scale : where we used the standard notation ofα i = α i /(4π) and α i = g 2 i /(4π). Similarly, we christen M 4 to be the neutral gaugino corresponding to U (1) A group. It's mass is given by The presence of additional U (1) A also introduces additional splittings between the mass squared terms at the mediation scale X. For example, the slepton doublets and the Higgs which are degenerate at the high scale in Minimal case, get split as: However, as we will see later the freedom of these splits is limited as the choice of U (1) A is quite restricted due to phenomenological constraints and anomaly cancellation conditions. Finally, just as in the minimal messenger model, the trilinear A -terms and bilinear B terms remain zero at the mediation scale X.

III. WEAK SCALE SPECTRUM
The soft terms at the weak scale can be evaluated by using the relevant Renormalisation Group (RG) equations with the above boundary conditions, Eq. (7). One interesting aspect about the one loop beta functions for the gauge couplings is that the beta function of SU (3), b 3 = 0. This is due to the presence of the additional colour triplets D,D in three generations 3 . As the α s does not run at the 1-loop level, most coloured particles receive larger corrections in RGE running, compared to the Minimal messenger model. This has consequences for the running of y t and subsequently to all the parameters which depend on y t or A t . We have used 1-loop RGE for the soft terms and added 2-loop RGE's for the gauge couplings and Yukawa couplings in this analysis. The relevant RGE for this model are given in Appendix C.
Before proceeding further, a comment about kinetic mixing is in order. The U(1) gauge fields can mix through the kinetic terms of the type χ dθ W A W Y . The current bounds on χ limit it to 10 −3 [54]. We expect that the implications on the phenomenology to be discussed in our paper will be minimally affected due to the presence of the kinetic mixing. For this reason, we will neglect all its effects in the subsequent discussion.
At the weak scale, M SU SY ∼ 1 TeV, we impose electroweak symmetry breaking conditions along with the U (1) A breaking. The neutral Higgs scalar potential is given by where G 2 = g 2 1 + g 2 2 . The minimisation conditions are modified compared to the standard NMSSM case due to the presence of terms proportional to g 4 . Subsequently, we can see from Eq. (18), that which is the typical vev one expects in extra U(1) models [41,55]. At the high scale, X, m 2 S which is positive and proportional toα 2 4 Λ 2 can be driven negative at the electroweak scale by the Yukawa couplings of the exotics k 1 , k 2 , k 3 .
This should be contrasted with the vev in minimal gauge mediation, without the U (1) factor.
See for example,Refs.[ [39,56] ]. From the minimization conditions of NMSSM, we get which is too small to get µ ef f ( λvs √ 2 ) of the order of electroweak symmetry breaking. To achieve a significant value either λ has to be very large (> 1) or κ has to be too small. In both the cases, achieving electroweak symmetry breaking is highly constrained [57]. We now turn our attention to the Higgs sector. The CP-even tree-level Higgs mass squared matrix, , and the elements of the matrix are given as: Given that the physical Higgs spectrum should be non-tachyonic, we can derive constraints on the parameter space of the model. Firstly the sign of the determinant of the matrix, in the limit v s >> v 1,2 is crucially dependent on the sign of the A λ . This is obvious, by considering the full determinant of the 3 × 3 mass matrix, which is given by  Table I and tanβ is chosen to be 10.
breaking is not possible for the shaded region (Det < 0) in the parameter space. From the figure, it is seen that for g 4 0.1, large values of λ 0.6 are disfavoured as they do not allow electroweak symmetry breaking.
The question then arises, whether A λ > 0 ?. Typically the A terms are negative due to the RG running from the high scale. However, in this case, A λ turns out to be O(10) and positive at the weak scale. This positive A λ ensures us a safe electroweak vacuum. This is shown in the left panel of Figure 2 , where we have plotted A λ with respect to running scale. As we see from the figure 2, A λ initially turns negative and then increases turning positive at the weak scale. This happens because of the complicated coupling between A t and A λ RGE. The RGE of these parameters are presented in the Appendix C along with the other parameters. In the below, we reproduce them: Compared to the minimal gauge mediated models, the running effects on the parameter A t are very large as α 3 barely runs in this models. As mentioned above, b 3 = 0 at 1-loop and is very small, at the 2-loop. For this reason, after the SUSY threshold M S ∼ 1TeV, α s barely runs all the way to the mediation scale. Due to this Y t and A t receive comparatively large corrections due to the relatively large α s . Additional corrections from g 4 , k i and A k i also contribute in the running of the A λ . This feeds into A λ , making it positive at the weak scale. In the right panel of the Fig   [2], we show the running of the A t for the same parameters In the NMSSM, it is well known that the tree level contribution can be appreciably enhanced from the MSSM tree level values only for large values of λ 0.7. The above bound is thus saturated only for special values of the parameters. For most of the parameter space, however the actual eigenvalue is far below the above bound. As in MSSM, one loop corrections would play a major role.
The total number of parameters are Λ, M X , g 4 , λ and the U (1) charges. Before proceeding to present the numerical results, we discuss the possible constraints on the various parameters. The first constraint we discuss is from the neutral gauge boson mixing. The neutral gauge bosons Z and Z mix with their mass matrix given by where The mixing of the matrix is given by The current limits on M Z require it to be greater than 1 TeV [58]. For g 4 ∼ g 1 , these limits already push v s to be much larger than 1 TeV. Θ ZZ is constrained by electroweak precision data, it should be less than O(10 −3 ) [54]. As v s is already very heavy with M Z of a mass of TeV order, the constraint on mixing angle is avoided easily.
A second constraint comes from the mass spectrum of the scalar super-partners. The D-terms due to the new U (1) A group play an important role in determining the sfermion mass spectrum due to the large vev of the S field. The strongest effects are felt in the stau mass matrix which is given as: where The chargino mass matrix remains unaltered compared to the MSSM whereas the neutralino mass matrix is now expanded to include the neutral gauging of U (1) A as well as the fermionic partner of the S field. Note that the fermonic partner of the S is not exactly the singlino as it carries a U (1) A charge unlike the NMSSM case. To summarise the constraints, we have :    (3). This process is repeated several times to obtain a parameter space which satisfy electroweak symmetry breaking conditions. Subsequent to this, we impose phenomenological constraints from direct SUSY searches at LHC [60, 61] as well as the flavour constraints from b → s + γ and b → s + µ + µ − .
In the numerical analysis, we fix the U (1) A charges to be as given in Table I. It should be noted that these are not the only solutions available from anomaly cancellation conditions. A list of five solutions is presented in Appendix A. Of the remaining parameters, we have fixed tan β = 10 and varied the remaining parameters within a range presented in Table (IV).
Instead of presenting the results in terms of regions of allowed parameter space, we present the correlations of the parameters with the lightest CP even Higgs boson mass. In Fig. (3 In Fig. (4), we present the correlation between m h and λ in the left panel and m h and g 4 in the right panel. We find a surprising relation between λ and m h . The Higgs mass seems to be lower for higher values of λ. This is contrary to expectations based on NMSSM. This is because for higher values of λ achieving electroweak symmetry breaking becomes harder. Similarly, larger values of λ typically mean lighter values of v s . Similarly, larger values of g 4 are not preferred by the data as they can lead to Landau poles. This can be seen from the right panel of Fig.(4). Thus, the regular NMSSM like enhancement of the tree level Higgs mass is not possible in this model. lightest neutralino is still the LSP and could be the dark matter candidate. A study of collider signatures and dark matter issues could be interesting and will be pursued in a future work.
Finally, we have not concentrated on the issue of fine tuning in this model. Though we have not explicitly measured it, it is expected that it could be large as long as M X and Λ are close as we have chosen. A reasonable separation between the scales can perhaps reduce the fine tuning (see for example, discussion in [63]).

Acknowledgments
We acknowledge discussions and important inputs from E. J. Chun. We also acknowledge discussions with P. Bandopadhyaya. We also thank L. Calibbi to bringing to our notice a reference. SKV is supported by DST Ramanujan Grant SR/S2/2008/RJN-25 of Govt of India. VSM is supported by CSIR fellowship 09/079(2377)/2010-EMR-1.

Appendix A: Anomaly Conditions
In the following we present the anomaly cancellation conditions and U(1) charges which are solutions to them. More elaborate analysis of anomaly cancellations pertinent to U(1) extensions of MSSM has been presented in [64]. To begin with, the U (1) A gauge invariance of the superpotential Eq.(5) leads to the below equations which should be satisfied by the U (1) A charges.
In addition, the following five anomaly cancellation conditions should also be satisfied.
In the following, we analyse each of these conditions and the corresponding solutions for U (1) A charges.
a. Anomaly The constraint here is given as This anomaly condition puts constraints on the hypercharges of the exotic fields. The anomaly condition is give by By taking the combination of Eqs. (A11) + (A9) -8 (A2) + 2 (A1) -6 (A3) + 6 (A4), we get which has several solutions. In the present work, we choose The final two anomalies do not have simple algebraic solutions. These are given as A 4 : We looked for integer solutions for the U (1) A charges. We could not find any as long as the charges are restricted to lie below 10. We then resorted to rational charges. There are several solutions which have been found. In Table III, we present five sample solutions which satisfy the anomaly conditions as well as the superpotential requirements. In addition to this set of charges, one can also find sets where all the z i andz i are equal. It should also be noted that each of the set of the charges has a completely different phenomenology. This is because the charges decide the U (1) A one loop beta function, b 4 , which could vary drastically. This in turn modifies the values of λ and κ i allowed and their respective ranges.
Appendix B: One loop corrections to the CP even Higgs mass matrix In the following we present the one loop corrections to the CP even Higgs mass matrix. There are two main contributions, one from the stop-top sector and the second one from from the vector like exotic quarks.To derive the one loop corrections, we use the well known effective potential  methods. The one loop effective potential is given by [65] where m 2 are the eigenvalues of the field dependent sfermion mass matrix.m f is the corresponding fermion mass.
The corrections to the CP even mass matrices can be written as By denoting mass matrix can be written as To include corrections to the Higgs mass matrix from the stop-top loop and all the three exotic quarks, we need to calculate B3 in each case separately and add them. We have presented below corrections from the stop-top loop and one exotic quark.

Top-Stop correction
Dominant one loop correction to the Higgs mass matrix comes from the top and stop loop. The stop mass squared matrix is given as