arX

iv:1

103.

4782

v1 [

hep-

ph]

24

Mar

201

1

Neutralino and Chargino Masses beyond MSSM

Katri Huitu, 1 ∗ P. N. Pandita, 2 † and Paavo Tiitola 1 ‡

1 Department of Physics, and Helsinki Institute of Physics,P. O. Box 64, FIN-00014 University of Helsinki, Finland and

2 Department of Physics, North Eastern Hill University, Shillong 793 022, India

We study the implications of dimension five operators involving Higgs chiral superfields for themasses of neutralinos and charginos in the minimal supersymmetric standard model (MSSM). Theseoperators can arise from additional interactions beyond those of MSSM involving new degrees offreedom at or above the TeV scale. We discuss the effect of these operators for different models ofsupersymmetry breaking gaugino masses. In addition to the masses of the neutralinos and charginos,we show that a sum rule involving the squares of the masses of these particles can be used to studythe presence of the dimension five operators in the context of MSSM.

PACS numbers: 11.30.Pb, 12.60.Jv, 14.80.Ly

I. INTRODUCTION

Supersymmetry (SUSY) is at present a leading candidate [1] for physics beyond the Standard Model (SM).In supersymmetric theories the Higgs sector, so essential for the internal consistency of the SM, is technicallynatural [2]. Supersymmetry is, however, not an exact symmetry in nature. The precise manner in whichSUSY is broken is not known at present. The necessary SUSY breaking can be introduced through softsupersymmetry breaking terms that do not reintroduce quadratic divergences in the Higgs mass, and therebydo not disturb the stability of the hierarchy between the weak scale and the large (grand unified or Planck)scale. Such terms can typically arise in supergravity theories, in which local supersymmetry is spontaneouslybroken in a hidden sector, and is then transmitted to the visible sector via gravitational interactions. Aparticularly attractive implementation of the idea of supersymmetry, with soft supersymmetry breaking termsgenerated by supergravity, is the Minimal Supersymmetric Standard Model (MSSM) obtained by introducingthe supersymmetric partners of the of the SM states, and introducing an additional Higgs doublet, with oppositehypercharge to that of the SM Higgs doublet, in order to cancel the gauge anomalies and generate masses forall the fermions of the Standard Model [3]. In order for broken supersymmetry to be effective in protectingthe weak scale against large radiative corrections, the supersymmetric partners of the SM particles should havemasses of the order of a few hundred GeV. Their discovery is one of the main goals of present and futureaccelerators.Because of underlying gauge invariance and supersymmetry, the Higgs sector of the MSSM is very tightly

constrained, and, therefore, is especially interesting. At the tree level, the mass of the lightest Higgs boson isbounded from above by the mass of the Z boson. The LEP experimental lower bound [4] on the Standard ModelHiggs boson, mhSM

>∼ 114 GeV, violates this tree level bound. However, there are large radiative correctionsto the tree level bound [5–7]. The dominant radiative corrections to the Higgs mass come from the top-stoploops, and in order for these to be significant one of the stop mass eigenstates should be heavy. However,for the radiative corrections to account for the current lower limit on the lightest Higgs boson mass, the topsquarks must be so massive, that it makes the theory appear as finely tuned. Alternatively, there must be largeleft-right mixing between scalar top quarks. While such large mixing is possible, it is rather difficult to obtainin specific models and can arise only from rather special points in the parameter space [8]. As pointed out byDine, Seiberg and Thomas [9], this suggests that there are likely to be additional degrees of freedom in thetheory beyond those of the MSSM.There are several candidates for such additional physics beyond the MSSM [10–13]. If this new physics lies

at an energy scale M , which is above the masses of the MSSM degrees of freedom, one can study the effects ofsuch additional degrees of freedom by using an effective Lagrangian from which the the physics at scale M hasbeen integrated out. The most general superpotential for the MSSM Higgs sector up to dimension five is [9]

W5 = µHuHd +λ

M(HuHd)

2, (I.1)

∗ Electronic address: katri.huitu@helsinki.fi† Electronic address: ppandita@nehu.ac.in‡ Electronic address: paavo.tiitola@helsinki.fi

2

where µ is the higgs(ino) mixing parameter in the superpotential of MSSM, M is an energy scale which is muchabove the typical masses of the MSSM fields, and λ is a dimensionless coupling. The dimension five operatorin (I.1) raises the lightest Higgs boson mass above the LEP limit without fine tuning, and, hence, without lossof naturalness [9].Another class of dimension five operators involve supersymmetry breaking. The effects of these dimension

five operators may be included by considering superpotential and Kahler potential operators with additionalpowers of spurion superfields with expectation values for auxiliary fields, especially the phenomenologicallyinteresting case of an F -term auxiliary expectation value. This class of operators may be represented by adimensionless chiral superfield spurion

Z = θ2mSUSY, (I.2)

where mSUSY is the supersymmetry breaking scale. The dimension five operator with a spurion field

∫d2θZ λ

M(HuHd)

2, (I.3)

gives a holomorphic effective dimension five correction to the renormalizable MSSM Higgs potential. IfmSUSY ≈|µ|, the correction to the lightest Higgs mass comes dominantly from the supersymmetric operator (I.1) ratherthan from (I.3). Therefore, the effects of dimension five operator (I.1) are of considerable importance.In this paper we study the effects of the dimension five operator (I.1) on the spectrum of neutralinos and

charginos. In Section II we write down the mass matrices for the neutralinos and charginos in the presenceof the dimension five operator (I.1). Here we also review the experimental constraints on the parameters ofthe neutralino and chargino mass matrices, and discuss different patterns for the soft supersymmetry breakinggaugino mass parameters that arise in different models of low energy supersymmetry. We also discuss theimplications of the dimension five operator (I.1) for different models of supersymmetry breaking. In Section IIIwe present our results for the spectrum of charginos and neutralinos, and the effect of dimension five operatoron this spectrum. Here we also discuss a sum rule involving the squared masses of neutralinos and charginoswhich can be used to study the effect of the dimension five operator. We conclude with a summary in SectionIV.

II. NEUTRALINO AND CHARGINO MASS MATRICES

A. Higgsino sector

The superpotential (I.1) leads, up to fimension five, to the following interaction Lagrangian involving only

the Higgsino (Hu, Hd) and the Higgs (Hu, Hd) fields [9]:

L = µ(HuHd)−ǫ1µ∗[2(HuHd)(HuHd) + 2(HuHd)(HuHd) + (HuHd)(HuHd) + (HuHd)(HuHd)

]+ H.c.,

(II.1)

where SU(2)L contraction between the fields in round parentheses is implied, and where

ǫ1 = λµ∗/M. (II.2)

For definiteness, we shall take µ to be real in this paper.The first and second terms in (II.1) with scalar Higgs expectation values correct the charged and neutral

higgsino Dirac masses. The third and fourth terms in (II.1) with scalar Higgs expectation values give rise toneutral higgsino Majorana masses which are absent in the tree-level neutralino mass matrix. Precision fits toboth masses and couplings of neutralinos and charginos would be sensitive to the dimension five Higgs-higgsinointeractions. It is important to note that the interactions (II.1) are all proportional to a single coupling, ǫ1,which is the same as the coupling affecting the Higgs mass [9].When the electroweak symmetry is broken, with the neutral components of the Higgs fields Hu and Hd

obtaining vacuum expectation values

vu = |〈H0u〉| = v sinβ,

vd = |〈H0d 〉| = v cosβ, (II.3)

3

the neutralino mass matrix in the bino-wino-higgsino basis

ψ0j = (−iλ′, − iλ3, ψ1

H1, ψ2

H2), j = 1, 2, 3, 4, (II.4)

following from (II.1) can be written as [14, 16]

M0 =

M1 0 −MZ cosβ sin θW MZ sinβ sin θW

0 M2 MZ cosβ cos θW −MZ sinβ cos θW

−MZ cosβ sin θW MZ cosβ cos θW2λM v2 sin2 β −µ+ 4λ

M v2 sinβ cosβ

MZ sinβ sin θW −MZ sinβ cos θW −µ+ 4λM v2 sinβ cosβ 2λ

M v2 cos2 β

, (II.5)

whereM2 andM1 are the SU(2)L and U(1)Y soft supersymmetry breaking gaugino masses,M2Z = 1

2(g2+g′2)v2,

M2W = 1

2g2v2, g and g′ are SU(2)L and U(1)Y gauge couplings, and v = (23/2GF )

−1/2 ≃ 174 GeV is the Higgs

vacuum expectation value. We shall denote the eigenstates of the neutralino mass matrix by χ01, χ

02, χ

03, χ

04 with

eigenvalues Mχ0

i=1,2,3,4, labeled in order of increasing mass.

In the wino-higgsino basis

ψ+j = (−iλ+, ψ1

H2), ψ−

j = (−iλ−, ψ2H1

), j = 1, 2, (II.6)

where λ± = (1/√2)(λ1 ∓ λ2), and the superscripts 1, 2 refer to SU(2)L indices, the chargino mass matrix at

dimension five can be written as

M± =

(M2

√2MW sinβ

√2MW cosβ µ− 2λ

M v2 sinβ cosβ

), (II.7)

We shall denote the eigenstates of the chargino mass matrix (II.7) as χ±1 and χ±

2 , with eigenvalues Mχ±

i=1,2,

respectively.Bounds on ǫ1 have been discussed in [14]. The dimension five operator (I.1) causes a shift in the mass of

the lightest Higgs boson, and if one assumes shift in mh0 to be at most 20% − 30%, then ǫ1 is limited tovalues smaller than 0.05 [14]. Larger shifts in the Higgs mass could in principle disrupt the vacuum stability bycreating a new global minimum for the potential. This issue was examined in [15], and a criterion was foundto exclude transitions to such a vacuum. Furthermore, ǫ1 is restricted by the scale of new physics appearingbeyond the MSSM. If the scale of new physics is taken to be M/λ > 5 TeV, then using (II.2) one arrives at alimit |ǫ| <∼ 0.04 for µ = 200 GeV, whereas M/λ > 2 TeV allows for |ǫ| <∼ 0.1. This limit is further increased atlarger values of µ. However, for large µ the neutralinos and charginos are mostly gauginos and the contributionfrom ǫ1 to their masses is much less significant. In the following we limit our discussion to |ǫ1| <∼ 0.1.We note that the dimension five operator (I.1) contributes to the lower right 2×2 submatrix of the neutralino

mass matrix (II.5). We have included the most significant MSSM one-loop radiative corrections to the neutralinoand chargino mass matrices in our analysis. Although these loop corrections are small (of the order of few GeV),the corrections in the (3, 3) and (4, 4) elements of the neutralino mass matrix can be important, since theseelements vanish in the absence of dimension five contribution. Therefore, it is important to include radiativecorrections in our analysis, since the effect of dimension five operator would be competing with the effects ofone-loop radiative corrections.

B. Experimental Constraints

Collider experiments have searched for the supersymmetric partners of the standard model particles. Nosupersymmetric partners of the SM particles have been found in these experiments. At present only lowerlimits on their masses have have been obtained. In particular, the search for the lightest chargino state at LEPhas yielded lower limits on its mass [17]. The limit depends on the spectrum of the model [19]. Assuming thatm0 is large, from the chargino pair production one obtains the lower bound

Mχ±

1

>∼ 103 GeV. (II.8)

For small m0, the bound is lowered, so that for mν < 200 GeV, but mν > mχ±

1

, the limit becomes [19]

Mχ±

1

>∼ 85 GeV. (II.9)

4

For the parameters of the chargino mass matrix (II.8) implies an approximative lower limit [20, 21]

M2, µ >∼ 100 GeV. (II.10)

The limits (II.10) on the parameters M2 and µ are found from scanning over the MSSM parameter space andare thus model independent.

C. Gaugino Mass Patterns

Having constrained the parametersM2 and µ, which enter the chargino as well as the neutralino mass matrix,we now turn to the theoretical models for the supersymmetry breaking gaugino mass parameters M1,M2, andM3. Theoretically, a simple set of patterns has emerged for these SUSY breaking parameters, which can bedescribed as follows.

1. Gravity mediated breaking

The first pattern, which has been the object of extensive studies, is the one which arises in the gravitymediated supersymmetry breaking models, usually referred to as the mSUGRA pattern. In the gravity mediatedminimal supersymmetric standard model, the soft gaugino masses Mi and the gauge couplings gi satisfy therenormalization group equations (RGEs) (|M3| ≡Mg, the tree level gluino mass)

16π2 dMi

dt= 2biMig

2i , bi =

(33

5, 1,−3

), (II.11)

16π2 dgidt

= big3i (II.12)

at the leading order, where i = 1, 2, 3 refer to the U(1)Y , SU(2)L and the SU(3) gauge groups, respectively.Furthermore, g1 = 5

3g′, g2 = g, and g3 is the SU(3)C gauge coupling. With the boundary conditions (αi =

g2i /4π, i = 1, 2, 3)

M1 = M2 =M3 = m1/2, (II.13)

α1 = α2 = α3 = αG (II.14)

at the GUT scale MG, the RGEs (II.11) and (II.12) imply that the soft supersymmetry breaking gauginomasses scale like gauge couplings:

M1(MZ)

α1(MZ)=

M2(MZ)

α2(MZ)=M3(MZ)

α3(MZ)=m1/2

αG. (II.15)

The relation (II.15) reduces the three gaugino mass parameters to one, which we take to be the gluino massMg. The other gaugino mass parameters are then determined through

M1(MZ) =5α

3α3 cos2 θWMg ≃ 0.14 Mg, (II.16)

M2(MZ) =α

α3 sin2 θWMg ≃ 0.28 Mg, (II.17)

where we have used the value of various couplings at the Z0 mass

α−1(MZ) = 127.9, sin2 θW = 0.23, α3(MZ) = 0.12. (II.18)

For the this leads to the ratio

M1 :M2 :M3 ≃ 1 : 2 : 6. (II.19)

This pattern is typical of any scheme obeying Eqs. (II.11) and (II.13), like e.g. gauge mediated models inaddition to the mSUGRA models [22]. Note that the gluino mass used above is the running mass evaluated atthe scale of the gluino mass.

5

This discussion of the gaugino mass parameters is valid at tree level. Including the radiative corrections [18],the ratio for these parameters in mSUGRA is modified to

M1 :M2 :M3 ≃ 1 : 1.9 : 6.2. (II.20)

Using the ratio (II.20) and the lower limit (II.10), we have the constraint

M1 >∼ 50 GeV, (II.21)

in the gravity mediated supersymmetry breaking models.We note that in the gravity mediated supersymmetry breaking models, the parameter µ is not constrained.

As such µ can be smaller or larger than M1,2. If µ≫M1,M2, then the lightest neutralino is mostly a gaugino,whereas in the opposite case µ≪M1,M2, it is dominantly a higgsino.

2. Anomaly mediated breaking

The second pattern of gaugino masses, which is distinct from the mSUGRA pattern and emerges undertheoretical assumptions that are appealing, arises in the anomaly mediated supersymmetry breaking mod-els (AMSB). Since the soft supersymmetry breaking parameters are determined by the breaking of the scaleinvariance, they can be written in terms of the beta functions and anomalous dimensions in the form of rela-tions which hold at all energies. In the minimal supersymmetric standard model (MSSM), the pure anomalymediated contributions to the soft supersymmetry breaking parameters Mλ (gaugino mass), m2

i (soft scalarmass squared), and Ay (the trilinear supersymmetry breaking coupling, where y refers to the Yukawa coupling)can be written as

Mλ =βggm3/2, (II.22)

m2i = −1

4

(∂γi∂g

βg +∂γi∂y

βy

)m2

3/2, (II.23)

Ay = −βyym3/2, (II.24)

where m3/2 is the gravitino mass, β’s are the relevant β functions, and γ’s are the anomalous dimensionsof the corresponding chiral superfields. An immediate consequence of these relations is that supersymmetrybreaking terms are completely insensitive to physics in the ultraviolet. The degrees of freedom that are excitedat a given energy determine the anomalous dimensions and beta functions, thus completely specifying the softsupersymmetry breaking parameters at that energy. We note that the gaugino masses are proportional totheir corresponding gauge group β functions with the lightest supersymmetric particle being mainly a wino.However, it turns out that the pure scalar mass-squared anomaly contribution for sleptons is negative [23].There are a number of proposals for resolving this problem of tachyonic slepton masses [24–28], but some of thesolutions may spoil the most attractive feature of the anomaly mediated models, i.e., the renormalization group(RG) invariance of the soft terms and the consequent ultraviolet insensitivity of the mass spectrum. A simplephenomenologically attractive way of parametrizing the nonanomaly mediated contributions to the sleptonmasses, so as to cure their tachyonic spectrum, is to add a common mass parameter m0 to all the squaredscalar masses [29], assuming that such an addition does not reintroduce the supersymmetric flavor problem.Such an addition of a nonanomaly mediated term destroys the attractive feature of the RG invariance of softmasses. However, the RG evolution of the resulting model, nevertheless, inherits some of the simplicity of thepure anomaly mediated relations.There are several alternative ways to generate these extra contributions to the soft squared masses in the

anomaly mediated supersymmetry breaking scheme. In particular there are models of supersymmetry breakingmediated through a small extra dimension, where SM matter multiplets and a supersymmetry breaking hiddensector are confined to opposite four-dimensional boundaries while gauge multiplets lie in the bulk. We notethat in this scenario the soft gaugino mass terms are due to the anomaly mediated supersymmetry breaking,and, therefore, are governed by (II.22). On the other hand, scalar masses get contributions from both anomalymediation and a tiny hard breaking of supersymmetry by operators on the hidden sector boundary. Theseoperators contribute to scalar masses at one loop and this contribution is dominant, thereby making all squaredscalar masses positive. The gaugino spectrum is unaltered, and the model resembles an anomaly mediatedsupersymmetry breaking model with nonuniversal scalar masses [30].

6

Using Eq. (II.22), we then have the following pattern for the gaugino masses at tree level:

M1 :M2 :M3 ≃ 3.3 : 1 : 9, (II.25)

which, after radiative corrections [18] (assuming m3/2 = 40 TeV) are included, becomes

M1 :M2 :M3 ≃ 2.8 : 1 : 7.1, (II.26)

in the minimal supersymmetric standard model with anomaly mediated supersymmetry breaking. Schemes inwhich this pattern is realized require a strict separation of hidden sector that breaks SUSY from the visiblesector of the MSSM. This implies a strong sequestering, and requires that all supersymmetry breaking fieldsare sequestered from the visible sector. Nevertheless, it may be achieved in certain class of theories with extradimensions or a conformal field theory sector.Using (II.10) and the anomaly pattern of the gaugino masses (II.26), we have

M1 >∼ 280 GeV. (II.27)

This is to be contrasted with the corresponding result (II.21) for the gravity mediated supersymmetry breaking.We further note that in the anomaly mediated supersymemtry breaking mechanism, the higgs(ino) parameter µcannot be smaller thanM1 due to the constraints following from electroweak symmetry breaking condition [29].This implies that the dominant component of the lightest neutralino will be a gaugino. Thus, the effect of thedimension five operator on the lightest neutralino mass will be negligible, since it affects the higgsino componentonly.

3. Mirage mediated supersymmetry breaking

A third simple gaugino mass pattern arises from the mirage (or mixed modulus) mediation supersymmetrybreaking, which is a hybrid between anomaly mediated supersymmetry breaking and mSUGRA pattern, andhas low energy values for masses quite distinct from either of the two. Mirage mediation is naturally realizedin KKLT-type moduli stabilization [31] and its generalizations, a well known example being KKLT modulistabilization in type IIB string theory [32]. Phenomenology and cosmology of mirage mediation have beenstudied in [33–41]. Signatures of the scenario at LHC and the spectrum of neutralino mass in particular havebeen studied in [42, 43]. The boundary conditions for the soft supersymmetry breaking terms that produce themirage mediation scheme can be written as [44]

Ma = M0

[1 +

ln(MPl/m3/2)

16π2bag

2aα],

Aijk = M0

[(ai + aj + ak)−

ln(MPl/m3/2)

16π2(γi + γj + γk)α

],

m2i = M2

0

[ci −

ln(MPl/m3/2)

16π2θiα−

(ln(MPl/m3/2)

16π2

)2

γiα2], (II.28)

where M0 ∼ 1 TeV is a mass parameter characterizing the moduli mediation, MPl is the reduced Planckmass, ga are the gauge couplings and ba the corresponding one-loop beta function coefficients, γi are theanomalous dimensions, and α = m3/2/[M0 ln(MPl/m3/2)] = O(1) is a parameter representing the ratio ofanomaly mediation to moduli mediation, and

θi = 4∑

a

g2aCa2 (φi)−

∑

jk

|yijk|2(ai + aj + ak), (II.29)

γi = 8π2 dγid logµ

, (II.30)

where Ca2 (φi) is the quadratic Casimir operator for the gauge group with gauge coupling ga, and has a value

(N2 − 1)/(2N) for the SU(N) gauge group.Thus the generic mirage mediation is parametrized by

M0, α, ai, ci = 1− ni, tanβ. (II.31)

7

The parameter values ci = ai = 1 and α = 1 correspond to the minimal KKLT compactification of type IIBtheory with modular weight ni = 0, but other parameter values are also possible for different scenarios, forexample choice of α = 2 with aHU

= cHU= 0 and aU3

+ aQ3= cU3

+ cQ3= 0 can possibly alleviate the fine

tuning problem for the electroweak symmetry breaking [45]. In our studies we have used the values ci = ai = 1.At low energies, the gaugino masses in mirage mediation can be written as

Ma(µ)

g2a(µ)=

(1 +

ln(MPl/m3/2)

16π2g2GUT baα

)M0

g2GUT

. (II.32)

This leads to a unification of the soft gaugino masses at the mirage messenger scale [46]

Mmir =MGUT

(m3/2

MPl

)α/2

, (II.33)

which is lower than GUT scale for positive values of α. For g2GUT ≃ 1/2 the resulting low energy values yieldthe mirage mass pattern

M1 :M2 :M3 ≃ (1 + 0.66α) : (2 + 0.2α) : (6 − 1.8α). (II.34)

When the radiative corrections are included [18] for the supersymmetry breaking gaugino masses, we obtain

M1 :M2 :M3 ≃ 1 : 1.5 : 2.1 for α = 1, (II.35)

M1 :M2 :M3 ≃ 1 : 1.2 : 0.92 for α = 2. (II.36)

where we have used the value M0 = 1 TeV. Thus, for the mirage mediation, we find

M1 >∼ 67 GeV for α = 1, (II.37)

M1 >∼ 83 GeV for α = 2. (II.38)

III. NUMERICAL RESULTS AND A SUM RULE FOR NEUTRALINO AND CHARGINO

MASSES

For large values of µ, the lightest neutralino and chargino are almost pure gauginos. In this case, thecorrections to the lightest neutralino and chargino masses from BMSSM operators are small, since they affectthe higgsino sector. If, on the other hand, the µ parameter is small compared to the gaugino mass parameters,i.e. if the lightest neutralino and chargino are dominantly higgsinos, the BMSSM corrections to their massescan be significant. We have, therefore, calculated the effects of the dimension five operator (I.1) on the massesof the lightest neutralino and chargino when the higgsino component in these is the dominant component. Wewill also show that in general that a sum rule involving all the neutralino and chargino masses can be used tostudy the effects of the dimension five operator.We have calculated the masses when the higgsino component in the neutralino is large. We will show that

the case when the lightest neutralino and chargino are dominantly gauginos may be possible to study by usingthe sum rule containing all neutralinos and charginos.Since radiative corrections will be competing with the corrections coming from the dimension five operators,

it is important to compare the ǫ1 corrections with one-loop radiative corrections. In Fig. 1 we have plotted thelightest neutralino mass in the mSUGRA pattern of gaugino masses with µ = 200 GeV, tanβ = 10, and m0 = 1TeV. We have plotted the lightest neutralino mass at tree level, with radiative corrections, with correctionscoming only from ǫ1, and with both the radiative and ǫ1 corrections. The radiative corrections are calculatedusing small µ approximation [47, 48], that is, corrections to the gaugino mass parameters are neglected andonly the corrections to the lower right 2×2 submatrix of of the neutralino mass matrix and to the (2, 2) elementof the chargino mass matrix are included. Only the contributions from quark-squark loops are included andsquark masses are taken to be 1 TeV. It is seen that radiative corrections are small (at a few GeV level),while for ǫ1 = 0.1 the corrections can be O(10 GeV). The kink in the BMSSM corrections shows that at thecorresponding value of the parameterM1, the lightest neutralino changes from an eigenstate containing a smallgaugino component to another mass eigenstate, which is almost a pure higgsino.In Fig. 2 we show the lightest neutralino and chargino masses for several values of ǫ1, ǫ1 = 0, ±0.05, ±0.1. We

have plotted these masses for the mSUGRA model. The mass of the lightest neutralino in mirage mediationmodels with α = 1, 2 is almost identical since the only difference in the masses comes from the gaugino

8

mχ01

(GeV)

M1(GeV)

200

200

160

170

180

190

400 600 800 1000

FIG. 1: The lightest neutralino mass in mSUGRA at tree level, and with one-loop radiative corrections as a function ofthe gaugino mass parameter M1. The blue solid line corresponds to the tree level mass with ǫ1 = 0. The other curvesare in order of increasing dash length: tree level mass with ǫ1 = 0.1 (violet); one-loop mass with ǫ1 = 0.1 (ochre); andone-loop mass with ǫ1 = 0 (green). Here µ = 200 GeV and tan β = 10.

mχ01(GeV)

M1(GeV)

160

170

180

190

200

200 400 600 800 1000

mχ±

1

(GeV)

M1(GeV)

195

200

200

205

210

400 600 800 1000

FIG. 2: The lightest neutralino and chargino masses in mSUGRA for several values for the parameter ǫ1 = λ

Mµ. The

blue solid line corresponds to ǫ1 = 0, and the thick dashed lines in order of increasing dash length represent ǫ1 = 0.05(violet), ǫ1 = 0.1 (ochre). The thin dashed lines denote the lightest neutralino mass for ǫ1 = −0.05 (green), ǫ1 = −0.1(blue), again in the order of increasing dash length. Here µ = 200 GeV, tan β = 10, and one-loop radiative correctionsare included.

nonuniversality in the radiative corrections. This is because for µ < M1,M2 the higgsino sector stronglydominates the lightest masses. Thus, the plot for mSUGRA is a representative for the mirage mediationmodels as well. It is seen that the effect of BMSSM operators in the case of mSUGRA pattern of gauginomasses is a few GeV, depending on the parameters. The effect for small M1 values is opposite for neutralinoand chargino masses, while for large values of M1, the neutralino mass is always less than its value when theǫ1 correction is absent. For chargino mass the correction is positive for negative ǫ1 and is negative for positiveǫ1. Thus, the effect of dimension five operator is enhanced for negative ǫ1 in the difference of chargino andneutralino masses, as seen in Fig. 3. We have not shown the results for the AMSB case, since in the AMSB µcannot be smaller than M1 due to the electroweak symmetry breaking condition [29], and thus in this case thedimension five contribution is negligible.If the µ parameter is large compared to the soft gaugino masses, it is obvious that the effect of the dimension

five operator on the lightest neutralino mass is too small to be observed, since in this case it is dominantly agaugino. The two heavy neutralinos are mostly higgsinos, but the contribution of the dimension five operatorto the mass for a heavy particle from the BMSSM operators is relatively small. Thus, one needs other ways toprobe such a case. One possiblity that we have explored is to study the difference between the sum of squaresof masses of all charginos and the sum of squares of masses of all neutralinos. Such a sum rule depends on theµ parameter only through BMSSM operators, and the dependence on gaugino masses enters only through thedifference of the squares of the these masses.

9

mχ±1

−mχ01(GeV)

M1(GeV)

5

10

15

20

25

30

200 400 600 800 1000

FIG. 3: The difference between the lightest neutralino and the lightest chargino mass in mSUGRA plotted for severalvalues of the parameter ǫ1 = λ

Mµ. The blue solid line corresponds to ǫ1 = 0, and the thick dashed lines in order of

increasing dash length represent ǫ1 = 0.05 (violet), ǫ1 = 0.1 (ochre). The thin dashed lines correspond to ǫ1 = −0.05(green), ǫ1 = −0.1 (blue), again in the order of increasing dash length. Here µ = 200 GeV, tan β = 10, and one-loopradiative corrections are included.

From the trace of the squares of the neutralino and chargino mass matrices, one obtains the sum rule

∑(ǫ1) ≡ 2

2∑

i=1

M2

χ±

i

−4∑

i=1

M2

χ0

i=[M2

2 −M21

]+ 4M2

W − 2M2Z + 4ǫ1v

2 sin 2β. (III.1)

at leading order in ǫ1. Using relations (II.14), (II.22), (II.32), and (II.33 ) the gaugino mass parameters M1

and M2 can be expressed in terms of observable quantities. For mSUGRA, AMSB and mirage mediation thesum rule can then be written as

∑

mSUGRA

(ǫ1) =M2

g

α23

(α22 − α2

1) + 4M2W − 2M2

Z + 4ǫ1v2 sin 2β, (III.2)

∑

AMSB

(ǫ1) =M2

g

9

[α22

α23

− (33

5)2α21

α23

]+ 4M2

W − 2M2Z + 4ǫ1v

2 sin 2β, (III.3)

∑

mirage

(ǫ1) =M2

g

α23

[1− 3

ln(MGUT /Mmir)

16π2

]−2[α22

(1 +

ln(MGUT /Mmir)

16π2

)2

−α21

(1 +

33

5

ln(MGUT /Mmir)

16π2

)2]+ 4M2

W − 2M2Z + 4ǫ1v

2 sin 2β. (III.4)

The sum rule (III.1) is independent of µ except through dependence on the parameter ǫ1. We note that whilefor large µ the neutralino and the chargino masses are not significantly affected by dimension five contribution,the sum rule can, nevertheless, be used to distinguish the effect of such a contribution. The fraction of thecontribution to the sum rule (III.1) arising from ǫ1 is plotted in Fig. 4. As seen in the Fig.4, the effect of thedimension five operator can be at the level of a few percent, and would require a precision measurement ofneutralino and chargino masses.In Fig. 5 we show the fraction of the contribution to the sum rule as a function of the ratio of the mass

parameters M2 and M1. Interestingly, at the point M2/M1 = 1 the contributions of the gaugino mass param-eters cancel in the sum rule, thus making the sum completely independent of the gaugino masses, with thecontribution of the ǫ1 becoming the dominant one. This point corresponds to mirage mediation with α = 2.17.More generally, mirage mediation with α close to 2 produces much smaller gradient for

∑along the gaugino

masses than the other models, and can have significant contributions from the dimension five operator evenwith heavy gauginos and small ǫ1.

IV. SUMMARY

We have studied the contribution of the dimension five BMSSM operators involving chiral Higgs superfieldson the neutralino and chargino masses. The contribution can be substantial when the higgs(ino) parameter

10

ǫ1

0.01

0.01

0.02

0.02

0.03

0.03

0.050.05 0.100.10

-

-

-

--

∑(ǫ1)−

∑(0)∑

(ǫ1)

mSUGRA

AMSB

Mirage, α = 1

Mirage, α = 2

FIG. 4: The contribution arising from ǫ1 to the total sum of (III.1) in mSUGRA, AMSB, and mirage mediation models.Here tanβ = 10, Mg = 2 TeV, and M0 = 1 TeV.

µ is small compared to the soft supersymmtry breaking gaugino mass parameters, as we have illustrated inthis paper. If the µ parameter is large, its effect is negligible on the mass of the lightest neutralino, which aredominantly a gaugino. Thus, the sensitivity to the BMSSM in different supersymmetry breaking mechanismsis very different in different supersymmetry breaking models, since in the mSUGRA and mirage meditaionmodels the µ parameter can be smalllight, while in the anomaly mediation models it is always larger than thegaugino mass parameters. The effect of the dimension five operators on the masses of the heavier neutralinosis relatively small as compared to the lightest neutralino mass, and thus more difficult to isolate.However, as shown here, even in the case of large values of µ, the difference of the squares of chargino masses

and neutralino masses can be a good measure of the BMSSM effects. This is due to the fact that this sumrule depends on the µ parameter only through the BMSSM operators, and that the dependence on the gauginomass parameters enters through their squares, thereby reducing the dependence on these parameters. Indeed,the contribution of the gaugino mass parameters to the sum rule vanishes when these soft parameters are equal.

Acknowledgments KH and PT gratefully acknowledge support from the Academy of Finland (Project No.137960). PNP would like to thank Department of Physics, University of Helsinki, and Helsinki Institute ofPhysics for hospitality while part of this work was done. The work of PNP is supported by the J. C. BoseNational Fellowship, and by the Council of Scientific and Industrial Research, India.

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