Radiative corrections to Higgs masses in Z' models

We calculate radiative corrections to the masses of the Higgs bosons in a minimal supersymmetrical model that contains an additional non-anomalous U(1)' gauge symmetry. With some fine-tuning of the U(1)' charges of the Higgs fields, it is possible to suppress the Z-Z' mixing. We use this fact, along with the lower bound on the lightest Higgs mass after LEP II era, as a criterion to restrict the set of parameters in our analysis. We calculate the mass of the lightest Higgs and its mixing with the other Higgs bosons in a large region of the parameter space.


Tree-level effective potential
We first describe the structure of the effective potential at the tree level. The gauge group is the same as that of the SM, but with an additional U (1) factor, i.e. G = SU (3) c ×SU (2) L ×U (1) Y × U (1) Y , with coupling constants g 3 , g 2 , g 1 , and g 1 , respectively. The Higgs sector contains two Higgs doublets H 1 and H 2 , and one singlet S. The matter multiplets are given by left-handed chiral superfields. For specific charge assignments of the chiral superfields with respect to the gauge group, we refer the readers to [10]. For our purposes, we only need to know the charges of the Higgs fields under the extra U (1) gauge symmetry. We denote the Higgs charges by Q 1 , Q 2 , and Q s . The superpotential is given by The most important feature of the above superpotential that distinguishes it from either MSSM or NMSSM, is the absence of a cubic term inŜ and a term proportional toĤ 1 ·Ĥ 2 , usually called the µ term 1 . The gauge invariance of the superpotential under the U (1) forbids the appearance of such terms. Although a µ term is absent from the superpotential, an effective µ parameter is generated by the vacuum expectation value (VEV) of the scalar field S. We use the hatted fields to denote chiral Higgs superfields, and the unhatted fields to denote scalar Higgs fields. We parametrize the explicit soft breaking of SUSY by whereλ i are the gaugino fields, and the Hermitian conjugate terms are assumed to keep the potential real. The tree-level Higgs potential follows from the L SB , F and D terms: where Q i are the U (1) charges of H i and Q s + Q 1 + Q 2 = 0 due to the gauge invariance of the superpotential under the extra gauge symmetry. Above we notice that g 1 always appears in combination with Q i . Hence, it is convenient to absorb it in the definition of Q i and define new charges Q i = g 1 Q i . Therefore, g 1 will not explicitly appear in our formulae unless stated otherwise.
We assume that all the coupling constants in the above potential are real. At the tree level, the potential cannot violate CP symmetry either explicitly or spontaneously. A possible phase could come from Ah s ; however, such a phase could be absorbed into the global phases of the Higgs fields. At the one-loop level, CP symmetry can be explicitly broken due to the complex phases in the scalar quark sector. However, we set all the CP-violating phases equal to zero and consider only the CP-conserving scenario.

49.4
The Higgs sector of the theory contains ten real degrees of freedom. Each Higgs doublet contains four real fields and the Higgs singlet contains two real fields. After the electroweak symmetry breaking, four of the ten fields become the longitudinal components of the four vector bosons in our model. The remaining six fields result in three scalars, one pseudoscalar and one charged Higgs. We decompose the Higgs fields as where the neutral components will be further separated into scalar and pseudoscalar bosons below.
The vacuum state of the theory is defined by the Higgs VEVs: The effective µ parameter is generated by the VEV of S, and is defined by For this to be a physical minimum, the potential must be negative when evaluated at the point (v 1 , v 2 , x), and the masses of the Higgs bosons must be positive. Even when these conditions are satisfied, the above point is not guaranteed to be the absolute minimum. Whether it is still acceptable depends on the location and depth of the other minima and the width between them. At the minimum point, the potential has vanishing first derivatives with respect to the three CP-even scalars, i.e. all tadpoles vanish. This enables one to trade the soft mass-squared parameters for their VEVs.
The tree-level masses of the Higgs bosons are obtained by diagonalizing their corresponding field-dependent mass-squared matrices. To this end, we need to substitute φ 0 i = (v i +φ i +iϕ i )/ √ 2 and S = (x + φ 3 + iϕ 3 )/ √ 2 into the potential. Here φ i and ϕ i stand for CP-even and CP-odd directions, respectively. Using the basic definitions we form the Higgs mass-squared matrix. Evaluation of equation (4) at the tree level is straightforward. Defining m 2 3 = Aµ s , we obtain For the pseudoscalar mass-squared matrix, we get The eigenvalues of the scalar mass-squared matrix correspond to the masses of the Higgs bosons. Although these eigenvalues can be obtained analytically, they are often too complicated to be useful. However, from the structure of the above matrix we can obtain useful information about its smallest eigenvalue, which corresponds to the lightest Higgs mass. Namely, for any symmetrical n × n matrix, its smallest eigenvalue is less than the smaller eigenvalue of its left upper 2 × 2 sub-matrix. With this observation, we get where Q H = Q 1 cos 2 β + Q 2 sin 2 β. The first two terms are familiar from NMSSM [1], while the third term is unique to the model under consideration. Notice that this term allows the lightest Higgs mass to be larger than that predicted by either MSSM or NMSSM. After appropriate rotations, the pseudoscalar mass-squared matrix gives one non-zero eigenvalue corresponding to the physical pseudoscalar mass. The other two eigenvalues which are zero, are the Goldstone degrees of freedom. After electroweak symmetry breaking, these become the longitudinal components of Z and Z . The Z-Z mass-squared matrix is given by where The eigenvalues of the above matrix, together with the Z-Z mixing angle, are given by The mixing angle α Z−Z has to be smaller than a few times 10 −3 , so that M Z 1 would correspond to the observed Z boson mass. For completeness, we also give the expressions for the pseudoscalar mass and the charged Higgs mass From the above, it is clear that the pseudoscalar mass is never negative, while the charged Higgs mass can be lower than the W boson mass, and can even run to negative values for some choices of the parameters.
In the next section, we include the main one-loop contributions to the tree-level effective potential. In general, the tree-level potential is written in terms of the running coupling constants and masses, which are defined at some renormalization point Q. However, the tree-level effective potential written in terms of running parameters is too sensitive to the choice of Q, and one cannot make reliable calculations. The situation is considerably improved when one includes the oneloop contributions to the effective potential [11]. We take into account the one-loop top/stop and sbottom effects, which are the main corrections. 49.6

One-loop effective potential
As explained at the end of the previous section, the most important one-loop contribution to the tree-level effective potential comes from the top and scalar top quarks. However, the contribution of the bottom scalar quarks can also be sizable when tan β ∼ 40 or larger. We take both contributions into account. For the rest of this section, we will state our results in full generality, making no assumptions about the numerical values of our parameters.
The stop and sbottom mass-squared matrices are given by where k t Using the stop and sbottom masses from above, we can express the one-loop correction to the effective potential by the Coleman-Weinberg formula [12] where Q is the renormalization scale in the MS scheme and k = 3/(32π 2 ). We sum over q = (t, b) and j = (1, 2). Here, The one-loop scalar and pseudoscalar mass-squared matrices are given by where the second term in the brackets is due to the fact that the position of the minimum has shifted because of the one-loop effects. By substituting equation (12) into (13) and substituting the resulting expression into equation (14), we get  (11). The functions F and G that appear in equation (15), are the usual loop amplitudes which also appear in the MSSM effective potential. These are given by where one particularly notices that F depends explicitly on the renormalization scale. By combining equations (6), (10), and (14), we can diagonalize the total pseudoscalar mass-squared matrix to obtain the mass of the pseudoscalar We can check the validity of our expressions for M 2 P and M ij by comparing them to the well known MSSM results. Our model reduces to MSSM if we fix µ s = h s x/ √ 2 ≡ µ and set h s = g 1 = Q i = 0. In this limiting case, we identically recover the usual MSSM results computed in [7]. We can also obtain a useful upper bound for the lightest Higgs mass at the one-loop level.
Although we did not take into account the two-loop effects, we do not expect these corrections to be very large. Indeed, as was shown in [13] for MSSM, the two-loop effects are less than a few gigaelectronvolts. In our future work, we will include the two-loop effects in the model under consideration. But given the fact that not even the one-loop effects have been worked out in this model, it is important to have these corrections before higher order effects can be taken into account.

Numerical examples
We numerically diagonalize the total one-loop scalar mass-squared matrix, given by equation (15), to obtain the mass of the lightest Higgs scalar. In order to get concrete results, we must fix some of the parameters in our model, which include , h s , tan β, and Q as free parameters against which the lightest Higgs mass will be plotted. Furthermore, because we are considering the electroweak symmetry breaking driven by a large VEV of the singlet field S, much larger than v, we fix x = 1.5 TeV.
The values of g 1 , Q 1 and Q 2 are not directly constrained by the experimental data. However, in most GUT-motivated models with an extra U (1) factor, g 1 = ( 5λ g /3)g 1 tan ϑ w , where g 1 is the hypercharge gauge coupling, and λ g is of the order of one with the exact value depending on how the GUT gauge group is broken down to the SM gauge group [14]. For our calculations, we take g 1 = 0.3. To fix the charges Q i , we take the Z − Z mixing angle α Z−Z to be smaller than a few times 10 −3 . Without excessive fine-tuning of the theory, this requires h s ∼ g 1 Q S (see [10] for details). This together with 0.35 ≤ h s ≤ 0.9 imply Q s ∼ O(1). Furthermore, the gauge invariance of the superpotential under the U (1) gives Q 1 + Q 2 = −Q s . Because Q 1 Q 2 > 0 as is argued in [10], we conclude that |Q i | ∼ 1. A natural choice would be to take Q 1 = Q 2 = −1.
However, according to equation (9), this specific choice restricts the value of tan β severely. In fact, α Z−Z ≤ 3 × 10 −3 implies 1 ≤ tan β ≤ 1.3. It turns out that tan β is highly sensitive to the ratio of the charges. Fortunately, the masses of the Higgs bosons are not very sensitive to the charges at all. This is because in the mass-squared matrix, each charge Q i is accompanied by a factor of g 1 . At the one-loop level, it is the top quark Yukawa coupling that gives the largest contribution to the Higgs masses. Therefore, we are free to choose Q i without effecting our results significantly. We have also verified this numerically. Because tan β is the only quantity that is really sensitive to the ratio of the charges, we choose charges that allow tan β to vary over a wide range of values. If we take Q 1 = 3Q 2 = −3/2, then we can have 1 tan β 60 without violating the bound on α Z−Z . If we choose other values for the charges that are still of the order of one, the masses of the Higgs bosons would not change by a significant amount.
We find that the Higgs masses are sensitive to the value of h s . For this reason, we calculate the Higgs masses for h s = 0.4, 0.6, and 0.8. Finally, we fix the scale at which we carry out the numerical calculations. We take the scale, denoted by Q not to be confused with the charges Q i , to be of the order of the electroweak scale. This is indeed necessary for the consistency of our analysis. We have also verified that as Q varies between 100 and 2000 GeV, the Higgs masses change by less than 2-3 GeV almost independently of the other parameters, see figure 2(c). It is clear that the change in the lightest Higgs mass is because of different choices of h s , and not because of Q. That the Higgs masses are relatively stable against the choice of the scale, is a restatement of the stability of the effective potential against the scale when one-loop effects are taken into account, as we mentioned above. In the actual calculations we fix Q = 300 GeV.
In summary, we are studying the lightest Higgs mass, which is phenomenologically the most interesting one, as we vary m U 2000 GeV. This is demonstrated in figure 2(c), for different values of h s . We have also plotted the lightest Higgs mass as a function of µ s , figures 2(a), (b). Notice the extreme fine-tuning of the charges Q 1 and Q 2 in order to have tan β large with no restriction on x, and respecting the experimental bound on the Z − Z mixing angle. With figure 2(b), we arrive at the same conclusion as above. Namely, for large tan β, h s ≤ 0.8. Larger values of tan β, however, are not generally favoured by the above model.
We have verified numerically that for nominal values of tan β and h s , say, tan β ∼ 4-15 and h s ∼ 0.4-0.6, the lightest Higgs mass is about 130 GeV, which is very close to the MSSM prediction. This can be seen from our graphs by examining the dashed black and red curves. Based on the MSSM analysis [13], we do not expect the two-loop corrections to add more than a few gigaelectronvolts to this value. That there is an agreement between the above model and MSSM is no surprise, because the above model is just an extension of MSSM. This is in fact a check on the validity of our results.
The above conclusions might be slightly effected when CP-violating effects are taken into account. However, the inclusion of CP-violating phases coming from h s A s and h t A t mixes the scalar and pseudoscalar mass-squared matrices. The resulting mass-squared matrix is a 4 × 4 matrix giving four Higgs bosons with no definite CP properties. This requires a different analysis than what we have done in this paper. We consider CP violation in a separate work.
Finally, for phenomenological reasons, it is desirable to know the Higgs mixing angles. In figure 2(f), we have plotted the three Higgs mixing angles against tan β. The derivation of the Higgs mixing angles is shown in the appendix, which is already well known. From the graph, one can see that for smaller values of tan β, the lightest Higgs state is dominated by a mixture of H 0 1 and H 0 2 , while its mixing with the singlet stays less than about 10%. However, when tan β is larger than about 10, the lightest state is dominated by H 0 2 , and the total mixing with H 0 1 and S stays less than 10%. This result is not surprising because larger tan β implies a larger VEV for H 0 2 .

Conclusions
In this work, we studied the one-loop effects on the lightest Higgs mass in a minimal supersymmetrical model augmented by an Abelian U (1) gauge symmetry. We calculated the top and stop/sbottom one-loop effects in the framework of the effective potential approach. The most important issue concerning the minimal models extended by a U (1) factor, is the mixing of the SM Z boson with the Z boson associated with the extra gauge symmetry. In order for such models to be phenomenologically viable, the Z-Z mixing angle has to be very small, less than a few times 10 −3 . We used the smallness of the Z-Z mixing angle, together with the lower bound set on the lightest Higgs mass, 114 GeV, by LEP II data [9], to constrain our parameter space. We showed numerically that the one-loop effects due to the top and stop/sbottom quarks are non-negligible. The radiative corrections to the Higgs boson masses and mixing angles are crucial for interpreting and predicting the Higgs production and decay rates in upcoming colliders. In linear colliders, for example NLC, the main production mechanisms are the Bjorken process and pair-production process, each of which requires a precise knowledge of Higgs boson masses and their couplings to the gauge bosons. (For the analysis of these processes at the tree level, see e.g. [15].) The model at hand predicts a larger upper bound on the Higgs boson masses than the MSSM. Therefore, even if the MSSM bounds are violated in the near-future colliders, the model at hand, which generates the µ parameter dynamically, will accommodate larger Higgs masses.