Raising the Higgs mass with Yukawa couplings for isotripletsin vector-like extensions of minimal supersymmetry

# Raising the Higgs mass with Yukawa couplings for isotriplets in vector-like extensions of minimal supersymmetry

Stephen P. Martin Department of Physics, Northern Illinois University, DeKalb IL 60115, and
Fermi National Accelerator Laboratory, P.O. Box 500, Batavia IL 60510.
###### Abstract

Extra vector-like matter with both electroweak-singlet masses and large Yukawa couplings can significantly raise the lightest Higgs boson mass in supersymmetry through radiative corrections. I consider models of this type that involve a large Yukawa coupling between weak isotriplet and isodoublet chiral supermultiplets. The particle content can be completed to provide perturbative gauge coupling unification, in several different ways. The impact on precision electroweak observables is shown to be acceptably small, even if the new particles are as light as the current experimental bounds of order 100 GeV. I study the corrections to the lightest Higgs boson mass, and discuss the general features of the collider signatures for the new fermions in these models.

## I Introduction

Supersymmetry as an extension of the Standard Model addresses the hierarchy problem associated with the small ratio of the electroweak breaking scale to the Planck scale or other very high energy scales. However, the lack of a signal for the lightest neutral scalar boson, , at the CERN LEP2 collider imposes some tension on the minimal supersymmetric standard model (MSSM) parameter space, motivating an examination of further extensions that can increase the theoretical prediction of the mass of .

In minimal supersymmetry, the biggest radiative corrections to come from one-loop diagrams with top quarks and squarks, and are proportional to the fourth power of the top Yukawa coupling. This suggests that one could improve the situation by introducing new supermultiplets with large Yukawa couplings that would raise the mass still further. This has been considered for the case of a fourth chiral family Fok:2008yg (); Litsey:2009rp (). However, in supersymmetry, the Yukawa couplings of a fourth chiral family would have to be so large (in order to evade discovery by LEP2 and the Tevatron) that perturbation theory would break down not far above the electroweak scale Fok:2008yg (). This would mean that the apparent success of gauge coupling unification in the MSSM is merely an illusion. Even accepting this, the corrections to precision electroweak physics would be too large in most of the parameter space, unless there are rather specific splittings of fermion masses Kribs:2007nz ().

Instead, one can consider models with extra matter in chiral supermultiplets comprised of vector-like representations of the Standard Model gauge group , i.e., those that allow tree-level superpotential mass terms before spontaneous electroweak symmetry breaking. These bare mass terms are responsible for most of the vector-like fermion masses. However, if the extra vector-like matter includes appropriate representations differing by unit of weak isospin, then they can also have Yukawa couplings to the MSSM Higgs supermultiplets. If large enough, these new Yukawa couplings can yield a significant enhancement of through one-loop effects, helping to explain why was not kinematically accessible to LEP2.

Earlier model-building work Moroi:1991mg (); Babu:2004xg (); Babu:2008ge (); Martin:2009bg (); Graham:2009gy () along these lines has considered extra vector-like matter transforming in gauge representations of the types already present in the MSSM, and their conjugates. Under , these candidate extra superfields transform like:

 Q=(3,2,1/6),¯¯¯¯Q=(¯¯¯3,2,−1/6),U=(3,1,2/3),¯¯¯¯U=(¯3,1,−2/3), D=(3,1,−1/3),¯¯¯¯¯D=(¯¯¯3,1,1/3),L=(1,2,−1/2),¯¯¯¯L=(1,2,1/2), E=(1,1,−1),¯¯¯¯E=(1,1,1),N,¯¯¯¯¯N=(1,1,0). (1.1)

(Each bar appearing here is part of the name of the field, and does not denote any kind of conjugation.) Requiring that these new particles are not much heavier than 1 TeV, and that the gauge couplings still unify perturbatively, there are three types of models, with new (non-MSSM) chiral supermultiplets:

 LNDn models: n×(L,¯¯¯¯L,N,¯¯¯¯¯N,D,¯¯¯¯¯D)for(n=1,2,3), (1.2) QUE model: Q,¯¯¯¯Q,U,¯¯¯¯U,E,¯¯¯¯E, (1.3) QDEE model: Q,¯¯¯¯Q,D,¯¯¯¯¯D,E,¯¯¯¯E,E,¯¯¯¯E. (1.4)

In each case, the number of singlets or is actually arbitrary, since they do not directly affect the running of the gauge couplings, but including the in the (LND) models allows new Yukawa couplings. There is also a possible model with new supermultiplets:

 QUDLE model:Q,¯¯¯¯Q,U,¯¯¯¯U,D,¯¯¯¯¯D,L,¯¯¯¯L,E,¯¯¯¯E. (1.5)

However, to avoid the gauge couplings become non-perturbative in the ultraviolet before they have a chance to unify,To correctly implement this perturbativity requirement, it is mandatory to use 2-loop (or higher) beta functions. The numerical results in this paper always use 2-loop beta functions for all parameters. These can be obtained straightforwardly from the general results listed in betas:1 (); betas:2 (), and so are not listed explicitly here. the average of the new particle masses in the QUDLE model would have to be at least about 2.5 TeV. This does not rule out the QUDLE model, but it goes strongly against the motivation of avoiding fine tuning. (If the large masses of the new fermions are due mostly to supersymmetric mass terms, then one cannot have a large enough hierarchy between scalar and fermion masses to increase appreciably, unless the soft supersymmetry breaking scalar masses are much larger still.) Up to the inclusion of singlets, the LND model content corresponds to a of , the QUE model to a of , and the QUDLE model to a of , although one need not subscribe to a belief in those groups as grand unified gauge symmetries.

In ref. Martin:2009bg (), I showed that the LND, QUE and QDEE models are compatible with precision electroweak constraints, even if the new Yukawa couplings are as large as their quasi-fixed-point values and the new quarks and leptons are approximately as light as their present direct search limits from Tevatron and LEP2.

However, the new vector-like matter may include other representations not listed in eq. (1.1). Let us denote possible triplet and octet chiral supermultiplets by:

 T=(1,3,0),O=(8,1,0). (1.6)

These are real representations of the gauge group, and so can have Majorana-type superpotential mass terms by themselves. If we denote by the number of pairs, and similarly for , , , and , and denote by and the number of and supermultiplets respectively, then the one-loop beta functions for the gauge couplings (with a GUT normalization ) are:

 Qdg1dQ=βg1 = g3116π2(33+nQ+8nU+2nD+3nL+6nE)/5, (1.7) Qdg2dQ=βg2 = g3216π2(1+3nQ+nL+2nT), (1.8) Qdg3dQ=βg3 = g3316π2(−3+2nQ+nU+nD+3nO), (1.9)

where is the renormalization scale. Perturbative unification requires that the one-loop contributions to the beta functions from the new fields are equal and not too large, so that

 (nQ+8nU+2nD+3nL+6nE)/5=3nQ+nL+2nT=2nQ+nU+nD+3nO≡N, (1.10)

where is 1, 2, or 3. (The details and precise quality of the unification depend also on 2-loop effects, including the effects of new Yukawa couplings. However, these effects do not make a dramatic difference, provided that .) This allows us to recognize some model possibilities different from those in eqs. (1.2)-(1.4). Consider models with extra chiral supermultiplets beyond the MSSM:

 TUD model: T,U,¯¯¯¯U,D,¯¯¯¯¯D, (1.11) TEDD model: T,E,¯¯¯¯E,D,¯¯¯¯¯D,D,¯¯¯¯¯D, (1.12) OLLLE model: O,L,¯¯¯¯L,L,¯¯¯¯L,L,¯¯¯¯L,E,¯¯¯¯E, (1.13) OTLEE model: O,T,L,¯¯¯¯L,E,¯¯¯¯E,E,¯¯¯¯E, (1.14) TLUDD model: T,L,¯¯¯¯L,U,¯¯¯¯U,D,¯¯¯¯¯D,D,¯¯¯¯¯D, (1.15) TLEDDD model: T,L,¯¯¯¯L,E,¯¯¯¯E,D,¯¯¯¯¯D,D,¯¯¯¯¯D,D,¯¯¯¯¯D. (1.16)

The first two have , and the last four have . As before, these models can be augmented by any number of gauge singlet supermultiplets,For example, the OTLEE model augmented by five singlets would correspond to an adjoint representation of the GUT group . which do not affect the gauge coupling running.

There are other possible representations of that one could try to include. However, if one requires no unconfined fractional electric charges, then all such vector-like combinations, which include for example or or or or or or , would contribute too much to and can not be consistent with perturbative gauge coupling unification, unless the average of the new particle masses is at least in the multi-TeV range. So with these requirements, and are the only new possibilities beyond eq. (1.1). Restricting the new supermultiplets to those in eqs. (1.1) and (1.6) assures that small mixings with the MSSM quark and lepton or gaugino and higgsino fields can eliminate stable exotic particles, which otherwise could be disastrous relics of the early universe. For some other recent discussions of vector-like supermultiplets in supersymmetry, see Liu:2009cc ()-Li:2010hi ().

In this paper, I will reserve the capital letters as above for new extra supermultiplets, and use lowercase letters for the usual chiral MSSM quark and lepton supermultiplets:

 qi=(3,2,1/6),¯¯¯ui=(¯¯¯3,1,−2/3),¯¯¯di=(¯¯¯3,1,1/3),ℓi=(1,2,−1/2), ¯¯¯ei=(1,1,1),Hu=(1,2,1/2),Hd=(1,2,−1/2), (1.17)

with denoting the three families. The MSSM part of the superpotential, in the approximation that only third-family Yukawa couplings are included, is:

 W=μHuHd+yt¯¯¯u3q3Hu−yb¯¯¯d3q3Hd−yτ¯¯¯e3ℓ3Hd. (1.18)

[Products of weak isospin doublet fields implicitly have their indices contracted with an antisymmetric tensor , with the first component of every doublet having and the second component having . So, for example , with the minus signs working out to give positive masses after the neutral components of the Higgs fields get vacuum expectation values (VEVs).]

Because of their vector-like representations, any Yukawa coupling-induced mixing between the new fields and their MSSM counterparts will not be governed by a GIM mechanism, and so must be highly suppressed. Therefore, to first approximation one can consider only Yukawa couplings that connect pairs of new fields. This can be enforced by an (approximate) symmetry, for example a under which the new superfields are odd and the MSSM quark and lepton superfields are even, or vice versa. The TUD and TEDD models do not have any allowed Yukawa couplings between pairs of new fields, and the OLLLE model allows only Yukawa couplings of the form and (and and if singlets are present), which are qualitatively similar to the ones in the LND model already studied in refs. Babu:2008ge (); Martin:2009bg (), with fixed points that are not large enough to raise the mass by a very significant amount.

In contrast, the OTLEE, TLUDD, and TLEDDD models all allow§§§The OTLEE and TLEDDD models can also have Yukawa couplings , (and and if singlets are present), but I will assume these are absent or negligible for simplicity. If present, they would reduce the quasi-fixed point values of . the qualitatively new possibility of (doublet)-(triplet)-(doublet) superpotential Yukawa couplings and involving the MSSM Higgs fields and the weak isotriplet field and the new vector-like isodoublet fields and . Including also the relevant gauge-singlet mass terms, the superpotential is:

 W=kHuTL+k′HdT¯¯¯¯L+12MTT2+ML¯¯¯¯LL. (1.19)

In this paper, I will examine the features of models that include this structure. In particular, when is large, it can induce a significant positive correction to . The infrared quasi-fixed point for is not too small to do so, in part because of the larger Casimir invariant for the triplet compared to a doublet (2 compared to ). In the following, I will use the OTLEE model as an example, but many of the results apply also to the TLUDD and TLEDDD models with only small numerical changes. The unification of the gauge couplings in the OTLEE model is shown in Figure 1, with for simplicity.

Although the gauge coupling would not run according to the one-loop renormalization group (RG) equations, two-loop effects are seen to cause it to get stronger in the ultraviolet, but not enough to become non-perturbative before unification takes place. The runnings in the TLUDD and TLEDDD models are only slightly different; all three of these models have from eq. (1.10).

## Ii The new particles and their masses

In this section, I consider the fermion and scalar content of the , , and supermultiplets. After the mixing implied by the Yukawa couplings and in eq. (1.19), the fermions will consist of three neutral Majorana fermion mass eigenstates, and two charged Dirac mass eigenstates denoted here by for , and for , respectively. To find the mass eigenstates and their mixing angles, the superpotential eq. (1.19) can be written explicitly in terms of the different electric charge components of the gauge eigenstate fields as:

 W = MT(T+T−+12T0T0)+ML(L−¯¯¯¯L+−L0¯¯¯¯L0) (2.1) +k(T0L−H+u+T0L0H0u+√2T+L−H0u−√2T−L0H+u) +k′(T0¯¯¯¯L+H−d+T0¯¯¯¯L0H0d+√2T+¯¯¯¯L0H−d−√2T−¯¯¯¯L+H0d).

Therefore, the mass matrices after electroweak symmetry breaking are, in two-component fermion notation DHM ():

 L = (2.2) M0 = (2.3)

where and are the VEVs of the Higgs fields and , with . The real positive fermion mass eigenvalues and unitary mixing matrices , , and are defined by

 N∗M0N† = diag(mψ01,mψ02,mψ03), (2.4) U∗M±V† = diag(mψ+1,mψ+2), (2.5)

with and and .

The scalar components of the , , supermultiplets mix to form four complex charged scalars for , and six real neutral scalars for . The general form of the soft supersymmetry-breaking Lagrangian is:

 −Lsoft = ak(T0L−H+u+T0L0H0u+√2T+L−H0u−√2T−L0H+u) (2.6) +ak′(T0¯¯¯¯L+H−d+T0¯¯¯¯L0H0d+√2T+¯¯¯¯L0H−d−√2T−¯¯¯¯L+H0d) +bT(T+T−+12T0T0)+bL(L−¯¯¯¯L+−L0¯¯¯¯L0)+c.c. +m2T(|T0|2+|T+|2+|T−|2)+m2L(|L0|2+|L−|2)+m2¯¯¯L(|¯¯¯¯L+|2+|¯¯¯¯L0|2).

It follows that the gauge-eigenstate squared-mass matrix for the neutral scalars is

 (CD†DC), (2.7)

in blocks, where

 C = M†0M0+diag(m2T,m2L+Δ12,0,m2¯¯¯L+Δ−12,0) (2.8)

with electroweak -term contributions defined by , and

 D=⎛⎜⎝bTakvu−kμ∗vdak′vd−k′μ∗vuakvu−kμ∗vd0−bLak′vd−k′μ∗vu−bL0⎞⎟⎠. (2.9)

For the charged scalars the gauge-eigenstate squared-mass matrix is:

 (EG†GF), (2.10)

where the blocks are

 E = M†±M±+diag(m2T+Δ1,1,m2¯¯¯L+Δ12,1), (2.11) F = M±M†±+diag(m2T+Δ−1,−1,m2L+Δ−12,−1), (2.12) G = (bT√2(−ak′vd+k′μ∗vu)√2(akvu−kμ∗vd)bL). (2.13)

The tree-level scalar squared masses and are the eigenvalues of eqs. (2.7) and (2.10). I will assume that, as usual in phenomenologically viable supersymmetric models, the soft terms , , and are large enough to make the scalar mass eigenstates and much heavier than their fermion counterparts and .

An important feature of these models is that infrared quasi-fixed points PRH () govern the new Yukawa couplings. This can be seen qualitatively from the one-loop parts of the RG equations:

 QdkdQ=βk = k16π2(8k2+2k′2+3y2t−7g22−35g21), (2.14) Qdk′dQ=βk′ = k′16π2(8k′2+2k2+3y2b+y2τ−7g22−35g21). (2.15)

The infrared quasi-fixed points occur when the positive contributions from Yukawa couplings nearly cancel the negative contributions from gauge couplings. In the following, we will be most interested in the case that is a large as possible, because when this leads to the largest possible contribution to the mass of ; this is obtained when . The two-loop RG running of for various different input values at the unification scale is shown in Figure 2.

More generally, the contour of quasi-fixed points in the plane is shown in Figure 3, obtained by requiring the perturbativity conditionThis criterion is somewhat arbitrary, but the fixed point values are not very sensitive to it. at the unification scale.

Although there is coupling between and in their RG equations, the quasi-fixed point value of does not vary much as long as is not too large. In the following, I will use as the fixed point value, motivated by the fact that a wide range of input values at the unification scale will end up close to this fixed point.

The phenomenology of these models will depend strongly on the fermion masses. These masses are shown in Figure 4 for and and varying superpotential mass parameters and , for three different fixed ratios , 1, and 2.

One-loop radiative corrections to the masses are potentially important, and so are included using the results of Appendix A. In all cases, the lightest of the new fermions turns out to be the neutral .

When , the lightest fermions and form a very nearly degenerate triplet, but the presence of the Yukawa coupling and one-loop radiative corrections ensures a non-zero splitting. When , the lightest fermions , , are mostly a Dirac pair of doublets, with a much larger mass splitting than the light triplet case. When , there is significant mixing between the doublets and the triplet, although the splitting between and can be seen to remain fairly small. The mass splitting between the lowest-lying states

 Δm≡mψ±1−mψ01 (2.16)

plays an important role in collider signals, and so is shown in Figure 5 for cases with the lightest fermions mostly doublets , mixed , and mostly triplet ( and ). The one-loop radiative corrections always increase . The mass splitting is smallest in the extreme limit of pure winos () where it asymptotically approaches GeV Cheng:1998hc (); Feng:1999fu (); Gherghetta:1999sw (). However, in most cases the mass splitting is considerably larger because of the Yukawa coupling, and it is always larger than the charged pion mass.

The RG running of the soft supersymmetry breaking terms has several interesting features that are comparable to those found in the LND, QUE and QDEE models studied in Martin:2009bg (). [To be concrete, here I use , , and at the unification scale. It cannot be under-emphasized that working to only one-loop order would yield very misleading results, because of the large values of the gauge couplings and non-trivial running of at high scales.] First, if one assumes that the gaugino masses are unified with a value at the same scale as the gauge couplings, then one finds that the RG running leads to quite different ratios than in the MSSM,

 (M1,M2,M3) = (0.13,0.23,0.47)m1/2,(OTLEE model) (2.17) = (0.13,0.24,0.62)m1/2,(TLUDD, TLEDDD models) (2.18) = (0.41,0.77,2.28)m1/2,(MSSM), (2.19)

evaluated at TeV. In particular, the ratios and are both much smaller in the extended models than in the MSSM. The extended models therefore predict a more compressed spectrum of superpartners than is found in the MSSM with unified gaugino masses.

If one takes the soft scalar squared masses and (scalar) terms to vanish at the unification scale, corresponding to the “no-scale” or “gaugino-mediated” boundary conditions and , then one finds for the ordinary first- and second-family squark and slepton mass parameters at TeV:

 (m˜q1,m˜u1,m˜d1,m˜ℓ1,m˜e1) (2.20) = (1.15,1.08,1.08,0.50,0.30)m1/2(OTLEE model% ), = (1.29,1.23,1.22,0.51,0.30)m1/2(TLUDD, % TLEDDD models), (2.21) = (2.08,2.01,2.00,0.67,0.37)m1/2(MSSM). (2.22)

Again, one sees a compression of the mass spectrum for the extended models compared to the MSSM. The soft masses for the new scalars in the OTLEE, TLUDD, and TLEDDD models are, respectively:

 (m˜T,m˜L,m˜¯¯¯L,m˜E,m˜¯¯¯¯E,m˜O) = (0.73,0.29,0.51,0.29,0.30,1.51)m1/2, (2.23) (m˜T,m˜L,m˜¯¯¯L,m˜U,m˜¯¯¯¯U,m˜D,m˜¯¯¯¯D) = (0.74,0.33,0.51,1.23,1.23,1.22,1.22)m1/2, (2.24) (m˜T,m˜L,m˜¯¯¯L,m˜E,m˜¯¯¯¯E,m˜D,m˜¯¯¯¯D) = (0.75,0.33,0.51,0.29,0.30,1.23,1.23)m1/2. (2.25)

Comparing eqs. (2.20)-(2.25) to eqs. (2.17)-(2.19) shows that, unlike the MSSM, the extended models considered here permit gaugino mass domination for the soft supersymmetry breaking terms at the unification scale while still having a bino-like neutralino as the LSP. (This feature was also observed in the QUE and QDEE models in ref. Martin:2009bg ().)

Another important consideration is the running of the (scalar) coupling . The coupling will play an important role in the corrections to to be discussed below. It turns out that when is near its quasi-fixed point trajectory, then the quantity

 Ak≡ak/k (2.26)

itself has a strongly attractive quasi-fixed point near small multiples of , as shown in Figure 6 for the OTLEE model.

I have checked that the TLUDD and TLEDDD models give very similar results, and that this behavior is not very sensitive to the assumption of gaugino mass unification, if is replaced by the value of at the unification scale.

## Iii Corrections to the lightest Higgs scalar boson mass

In this section, I consider the contribution of the new doublet and triplet supermultiplets , , and to the lightest Higgs scalar boson mass. The effective potential approximation provides a simple way to estimate this contribution, and is equivalent to neglecting non-zero external momentum effects in self-energy diagrams. (This is an accurate approximation since .) The one-loop contribution to the effective potential due to the particles in the , , and supermultiplets is:

 ΔV=6∑i=1F(m2ϕ0i)−23∑i=1F(m2ψ0i)+24∑i=1F(m2ϕ±i)−42∑i=1F(m2ψ±i). (3.1)

Here , , , and are the VEV-dependent tree-level squared-mass eigenvalues from eqs. (2.3)-(2.5), (2.7), and (2.10), and with the renormalization scale. I will assume the decoupling approximation that the neutral Higgs mixing angle (in the standard convention, described e.g. in primer ()) is , which is valid if . Then the correction to is

 (3.2)

In the OTLEE, TLUDD, and TLEDDD models, the other new fields do not make a significant radiative contribution to the Higgs mass, as they do not have Yukawa couplings to and .

Before obtaining numerical results in a realistic model, it is useful to first consider a relatively simple analytical result for the case that the superpotential mass parameters are equal () and the non-holomorphic soft supersymmetry-breaking squared masses are also equal (), and neglecting the holomorphic terms and . Then, writing

 x = M2S/M2F,M2S≡M2F+m2=%averagescalarmass (3.3) Xk = Ak−μcotβ, (3.4)

and, expanding in , I find

 Δm2h0 = v24π2{k4sin4β[f(x)+X2kM2S(5−2x)−5X4k12M4S] (3.5) +34(g2+g′2)k2sin2βcos(2β)[ln(M2S/Q2)+X2k/2M2S]}

where

 f(x)=5ln(x)−92+112x−1x2. (3.6)

Note that is approximately the ratio of the mean squared masses of the scalars to the fermions and is therefore assumed greater than 1, while the mixing between the new triplet and the new doublet scalars is parameterized by . Similar to the models discussed in Moroi:1991mg (); Babu:2004xg (); Babu:2008ge (); Martin:2009bg (); Graham:2009gy (), the contribution to does not decouple with the overall new particle mass scale, provided that there is a hierarchy maintained between the scalars and the fermions. The electroweak -term contribution involving is quite small, provided one chooses a RG scale , and is neglected below. The maximum possible contribution to occurs when , leading to a “maximal mixing” result given by , where

 fmax(x)=f(x)+35(5−2/x)2. (3.7)

In Figure 7,

I show an estimate of these corrections to , using (83 GeV), corresponding to the quasi-fixed point with reasonably large , and assuming that the predicted mass before the correction is 110 GeV, so that 110 GeV.

As found in the previous section, the quasi-fixed point behavior of the running of the scalar trilinear coupling implies that the mixing parameter is probably actually much smaller than in the “maximal mixing” case. Also, the soft supersymmetry breaking squared masses , , and need not be degenerate. A perhaps better-motivated scenario is therefore the gaugino mass dominated case shown in Figure 8, where I take and [see eq. (2.23) and Figure 6], with adjusted so that the lightest new charged fermion mass is , 125, 150, 200, 250, and 400 GeV.

These results were computed using the complete expressions in eqs. (2.3), (2.7)-(2.13) and (3.1), (3.2), not from the simplified expansion in . The correction to turns out to be not dramatically sensitive to and (taken to be 0 here), or (set to here) or (set to 10 here) provided it is not too small. For a given value of , the upper bound on corrections to is nearly saturated when .

Figure 8 shows that the corrections to are moderate, but can easily exceed 5 GeV for an average new scalar mass less than 1 TeV, provided that at least one new charged fermion is lighter than about 200 GeV. However, it should be kept in mind that the actual corrections can be larger or smaller than indicated in Figure 8, depending on the details of the new particle spectrum. If the fixed point behavior for noted above is evaded somehow, then the corrections to could be substantially larger. The contribution to also monotonically increases with the scalar masses (for fixed fermion masses), and so in principle could be much larger, subject to considerations of fine-tuning that intuitively should get worse with larger supersymmetry breaking. Due to the impossibility of defining an objective measure of fine tuning, I will not attempt to quantify the merits of this trade-off, but simply note that that even a contribution of a few GeV to is quite significant in the context of the supersymmetric little hierarchy problem. Smaller fermion masses may be considered preferred in the sense that this maximizes .

## Iv Precision electroweak effects

The Yukawa couplings and break the custodial symmetry of the Higgs sector, and therefore contribute to virtual corrections to , , and photon self-energies, of the type that are constrained by precision electroweak observables. Similarly to the cases analyzed in Martin:2009bg (), these corrections are actually benign, at least if one uses , , and -peak observables as in the LEP Electroweak Working Group analyses LEPEWWG (); LEPEWWG2 (). (A different set of observables are used in RPP (), leading to a worse fit.) This is because the corrections decouple with larger vector-like masses and , even if the Yukawa couplings are large and soft supersymmetry breaking effects including , and produce a large scalar-fermion hierarchy. In particular, they decouple even when the corrections to do not.

The most important new physics contributions to the precision electroweak observables can be summarized in terms of the Peskin-Takeuchi and parameters Peskin:1991sw () (similar parameterizations of oblique electroweak observables were discussed in Golden:1990ig ()). In this paper, I will use the updated experimental values

 s2eff = 0.23153±0.00016ref.~{}\@@cite[cite]{\@@bibref{Auth% ors Phrase1YearPhrase2}{LEPEWWG}{\@@citephrase{(}}{\@@citephrase{)}}} (4.1) MW = 80.399±0.025