Improving the fine tuning in models of low energy gauge mediated supersymmetry breaking

E! ELSEVIER

Nuclear Physics B 507 (1997) 3-34

Improving the fine ttming in models of low energy gauge mediated supersymmetry breaking* K. Agashe l, M.

Graesser z

Theoretical Physics Group, Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720, USA

Received 3 April 1997; accepted 20 August 1997

Abstract

The fine tuning in models of low energy gauge mediated supersymmetry breaking required to obtain the correct Z mass is quantified. To alleviate the fine tuning problem, a model with sprit (5 + 5) messenger fields is presented. This model has additional triplets in the low energy theory which get a mass of 0(500) GeV from a coupling to a singlet. The improvement in fine tuning is quantified and the spectrum in this model is discussed. The same model with the above singlet coupled to the Higgs doublets to generate the/z term is also discussed. A Grand Unified version of the model is constructed and a known doublet-triplet spfitting mechanism is used to split the messenger (5 + 5)'s. A complete model is presented and some phenomenological constraints are discussed. (~) 1997 Elsevier Science B.V. PACS: l l.30.Pb; 12.60.Jv; 14.80.Ly; 12.10.Dm Keywords: Supersymmetryphenomenology;Gauge mediated supersymmetrybreaking; Grand unified model building

1. I n t r o d u c t i o n

One of the outstanding problems of particle physics is the origin of electroweak symmetry breaking ( E W S B ) . In the Standard Model ( S M ) , this is achieved by one Higgs doublet which acquires a vacuum expectation value (vev) due to a negative mass This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundationunder grant PHY-90-21139. 1 E-mail: [email protected] 2 E-mall: [email protected] 0550-3213/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PII S0550-3213 (97)00569-5

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

squared which is put in by hand. The SM has the well-known gauge hierarchy problem (see, for example, Ref. [ 1] ). It is known that supersymmetry (SUSY) (for reviews of supersymmetry and supersymmetry phenomenology, see, for example, Ref. [2] ) stabilizes the hierarchy between the weak scale and some other high scale without any fine tuning if the masses of the superpartners are less than a few TeV [3,4]. The Minimal Supersymmetric Standard Model (MSSM) is considered as a low energy effective theory in which the soft SUSY breaking terms at the correct scale are put in by hand. This raises the question: what is the origin of these soft mass terms, i.e., how is SUSY broken? If SUSY is broken spontaneously at tree level in the MSSM, then there is a colored scalar lighter than the up or down quarks [5]. So, the superpartners have to acquire mass through radiative corrections. Thus, we need a "hidden" sector where SUSY is broken spontaneously at tree level and then communicated to the MSSM by some "messengers". There are two problems here: how is SUSY broken in the hidden sector at the right scale and what are the messengers? There are models in which a dynamical superpotential is generated by non-perturbative effects which breaks SUSY [6]. The SUSY breaking scale is related to the Planck scale by dimensional transmutation. Two possibilities have been discussed in the literature for the messengers. One is gravity which couples to both the sectors [7]. In a supergravity theory, there are non-renormalizable couplings between the two sectors which generate soft SUSY breaking operators in the MSSM once SUSY is broken in the "hidden" sector. In the absence of a flavor symmetry, this theory has to be fine tuned to give almost degenerate squarks and sleptons of the first two generations which is required by Flavor Changing Neutral Current (FCNC) phenomenology [5,8]. The other messengers are the SM gauge interactions [9]. In these models, the scalars of the first two generations are naturally degenerate since they have the same gauge quantum numbers. This is an attractive feature of these models, since the FCNC constraints are naturally avoided and no fine tuning between the masses of the first two generation scalars is required. If this lack of fine tuning is a compelling argument in favor of these models, then it is important to investigate whether other sectors of these models are fine tuned. In fact, we will argue (and this is also discussed in Refs. [ 10-12] ) that the minimal model (to be defined in Section 2) of low energy gauge mediated SUSY breaking requires a minimum 7% fine tuning to generate a correct vacuum ( Z mass) if no superpartners are discovered at LEP2. Further, if a gauge-singlet and extra vector-like quintets are introduced to generate the "/z" and "B/x" terms, then the minimal model of low energy gauge mediated SUSY breaking requires a few percent fine tuning to correctly break the electroweak symmetry. These fine tunings make it difficult to understand, within the context of these models, how SUSY is to offer some understanding of the origin of electroweak symmetry breaking and the scale of the Z and W gauge boson masses. Our paper is organized as follows. In Section 2, we briefly review both the "messenger sector" in low energy gauge mediated SUSY breaking models that communicates SUSY breaking to the Standard Model and the pattern of the sfermion and gangino masses that follow. Section 3 quantifies the fine tuning in the minimal model using

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

the Barbieri-Giudice criterion [3]. Section 4 describes a toy model with split (5 + 5) messenger representations that improves the fine tuning. To maintain gauge coupling unification, additional triplets are added to the low energy theory. They acquire a mass of 0 ( 5 0 0 ) GeV by a coupling to a singlet. The fine tuning in this model is improved to ,-~ 40%. The sparticle phenomenology of these models is also discussed. In Section 5, we discuss a version of the toy model where the abovementioned singlet generates the/z and/~2 terms. This is identical to the Next-to-Minimal Supersymmetric Standard Model (NMSSM) [ 13] with a particular pattern for the soft SUSY breaking operators that follows from gauge mediated SUSY breaking and our solution to the fine tuning problem. We show that this model is tuned to ~ 20%, even if LEP does not discover SUSY/light Higgs. We also show that the NMSSM with one complete messenger (5 + 5) is fine tuned to N 2%. We discuss, in Section 6, how it is possible to make our toy model compatible with a Grand Unified Theory (GUT) [ 14] based upon the gauge group SU(5) x SU(5). The doublet-triplet splitting mechanism of Barbieri, Dvali and Strumia [ 15] is used to split both the messenger representations and the Higgs multiplets. In Section 7, we present a model in which all operators consistent with symmetries are present and demonstrate that the low energy theory is the model of Section 5. In this model R-parity ( R p ) is the unbroken subgroup of a Z4 global discrete symmetry that is required to solve the doublet-triplet splitting problem. Our model has some metastable particles which might cause a cosmological problem. In Appendix A we give the expressions for the Barbieri-Giudice parameters (for the fine tuning) for the MSSM and the NMSSM.

2. Messenger sector In the models of low energy gauge mediated SUSY breaking [ 10,16] (henceforth called LEGM models), SUSY breaking occurs dynamically in a "hidden" sector of the theory at a scale Adyn that is generated through dimensional transmutation. SUSY breaking is communicated to the Standard Model fields in two stages. First, a nonanomalous U(1) global symmetry of the hidden sector is weakly gauged. This U( 1)x gauge interaction communicates SUSY breaking from the original SUSY breaking sector to a messenger sector at a scale Amess ~ olxAdyn/477"as follows. The particle content in the messenger sector consists of fields ~b+, q~_ charged under this U( 1 )x, a gauge singlet field S, and vector-like fields that carry Standard Model quantum numbers (henceforth called messenger quarks and leptons). In the minimal LEGM model, there is one set of vector-like fields, c7, l, and q, [ that together form a ( 5 + 5 ) of SU(5). This is a sufficient condition to maintain unification of the SM gauge couplings. The superpotential in the minimal model is Wmess = A~bq~+~ ) - S "}- 1,~.sS3 --]- AqSqq ~-

The scalar potential is

AISIL

(1)

K. Agashe, M.

Graesser/NuclearPhysicsB 507 (1997)3-34

[Nil2 + m~_l~b+[2 + m2__l~b_[2.

V=~

(2)

i

In the m o d e l s o f [ 10,16], the ~b+, q~_ fields communicate (at two loops) with the hidden sector fields through the U ( 1 ) gauge interactions. Then, S U S Y breaking in the original sector generates a negative value ~ - (axAdyn) 2 / ( 4 ¢ r ) 2 for the mass parameters me+, m 2_ o f t h e q~+ and ~b_ fields. This drives vevs o f O (Amess) for the scalar components o f both ~b+ and ~b_, and also for the scalar and F - c o m p o n e n t o f S if the couplings As, gx and A,~ satisfy the inequalities derived in Refs. [ 11,17]. 3 Generating a vev for both the scalar and F - c o m p o n e n t o f S is crucial, since this generates a non-supersymmetric spectrum for the vector-like fields q and l. The spectrum of each vector-like messenger field consists o f two complex scalars with masses M 2 + B and two Weyl fermions with mass M where M = AS, B = AFs and A is the coupling of the vector-like fields to S. Since we do not want the SM to be broken at this stage, M 2 - B ~> 0. In the second stage, the messenger fields are integrated out. As these messenger fields have SM gauge interactions, SM gauginos acquire masses at one loop and the sfermions and Higgs acquire soft scalar masses at two loops [9]. The gaugino masses at the scale at which the messenger fields are integrated out, Amess ~ M are [ 16] M G = CeG(Amess) ASUSY~

4 zr

m

N~(m)f,

~

.

(3)

The sum in Eq. ( 3 ) is over messenger fields ( m ) with normalization Tr(TaT b) = N~(rn)6 "b where the T's are the generators o f the gauge group G in the representation R, f l ( x ) = l + O ( x ) , and AsusY --= B/M = Fs/S = xAmess with x = B/M2. 4 Henceforth, we will set AsusY ~ Amess- The exact one-loop calculation [ 18] o f the gaugino mass shows that f l ( x ) ~< 1.3 for x ~< 1. The soft scalar masses at Amess are [16]

(aG(Amess)'~2 : ( FS ) mi2=2A2usyZNG(m) CGR(si) ~ ~ ) J2 ~ ,

(4)

lr~,a

where Cg(si) is the Casimir o f the representation o f the scalar i in the gauge group G and f 2 ( x ) = 1 + O(x). The exact two-loop calculation [18] which determines f2 shows that for x ~< 0.8 (0.9), f2 differs from one by less than 1% ( 5 % ) . Henceforth we shall use f l ( x ) = 1 and f2(x) = 1. In the minimal L E G M model M ~ (Amess) -

O~G(Amess) Amess, 4¢r

(5)

3This point in field space is a local minimum. There is a deeper minimum where SM is broken [11,17]. To avoid this problem, we can, for example, add another singlet to the messenger sector [ I 1]. This does not change our conclusions about the fine tuning. 4 If all the dimensionless couplings in the superpotential m'e of O( 1), then x cannot be much smaller than 1.

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

m 2 (Amess)

(

- 2Amess C3 \

47r

J

÷

C2 ~

4¢r

+ 53-k,

7

4~-

J J '

(6) where Q = T3L ÷ Y and al is the SU(5) normalized hypercharge coupling. Further, C3 = 4 and C2 = 3 for colored triplets and electroweak doublets respectively. The spectrum in the models is determined by only a few unknown parameters. As Eqs. (3) and (4) indicate, the SUSY breaking mass parameters for the Higgs, sfermions and gauginos are mo, m ~ : rn"E, r n m , m ~ : m~R,m'ff ~ a3 : Ol2 : Oq.

(7)

The scale of Amess i s chosen to be ,-~ 100 TeV, so that the lightest of these particles escapes detection. It follows that the intrinsic scale of supersymmetry breaking, adyn, is ,-~ 10000 TeV. The goldstino decay of the lightest standard model superpartner then occurs outside the detector [ 19]. The phenomenology of the minimal LEGM model is discussed in detail in for example Ref. [ 19] and references therein.

3. Fine tuning in the minimal LEGM A desirable feature of gauge mediated SUSY breaking is the natural suppression of FCNC processes since the scalars with the same gauge quantum numbers are degenerate [9]. But, the minimal LEGM model introduces a fine tuning in the Higgs sector unless the messenger scale is low. This has been previously discussed in Refs. [ 10,11 ] and quantified more recently in Ref. [ 12]. We outline the discussion in order to introduce some notation. The superpotential for the MSSM is W = I x H , Hd + WYukawa.

(8)

The scalar potential is W -- /d,2 [nu 12 ÷/xalHd 2 I2 - ( t z 2 H . H d ÷ h.c.) ÷ D-terms + ~-loop,

(9)

where Vl-loop is the one-loop effective potential. The vev of H . (Ha), denoted by v. (vd), is responsible for giving mass to the up- (down-)type quarks, /x~ = m 2 + / z 2, / z22 = m~,, ÷ / x 2 and /z32,5 rn~,,, m L are the SUSY breaking mass terms for the Higgs fields. 6 Extremizing this potential determines, with tan fl -- v , / v d , 1 m z 2 = /22 _ / 2 2 tan 2 fl

tan 2 f l - 1

'

5/.t2 is often written as B/.t. 6 The scale dependence of the parameters appearing in the potential is implicit.

(10)

8

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34 tz 2

sin2/3 = 2 ~2-~3 - 2 ' /z 1 ± / z 2

(11)

where #~ = ~2 + 2avl_,oop/aV2.For large tan/3, m2z/2 ~ -(m~,, + ix2). This indicates that if Im2,1 is large relative to m 2, the /~2 term must cancel this large number to reproduce the correct value for rn2z. This introduces a fine tuning in the Higgs potential, that is naively of the order mZ/21m2 I. We shall show that this occurs in the minimal LEGM model• In the minimal LEGM model, a specification of the messenger particle content and the messenger scale Amess fixes the sfermion and gaugino spectrum at that scale. For example, the soft scalar masses for the Higgs fields are ~ a2(Amess) Amess/47r. Renormalization group (RG) evolution from Amess to the electroweak scale reduces m~,, due to the large top quark Yukawa coupling, At, and the squark soft masses• The one-loop Renormalization Group Equation (RGE) for m 2 is (neglecting gaugino and the trilinear scalar term (H.Q~t c) contributions)

dt

8~-2

mn,,(t) +m~c(t) +too(t)

(12)

which gives m~,, ( t ~ In ( ~ ) ) ~m2(0)

- 3"~t2 (m~,(O)+m~c(O)+mQ(O))ln(Amess~ \ mT/"

(13)

2 has been neglected. On the right-hand side of Eq. (13) the RG scaling of mLQ and m~c Since the logarithm It[ ~ ln(Amess/mT) is small, it is naively expected that m 2Hu will not be driven negative enough and will not trigger electroweak symmetry breaking. However, since the squarks are ~ 500 GeV (1 TeV) for a messenger scale Amess = 50 TeV ( 100 TeV), the radiative corrections from virtual top squarks are large since the squarks are heavy. A numerical solution of the one-loop RGE (including gaugino and the trilinear scalar term (HuO~t c) contributions) determines - m 2 , = (275 GeV) 2 ((550 GeV) 2) for Amess = 50 TeV (100 TeV) and setting ,~t = 1. Therefore, m2z/Z]m~,,[ ~., 0•06 (0.01), an indication of the fine tuning required. To reduce the fine tuning in the Higgs sector, it is necessary to reduce Im21; ideally so that m~o ~ -0.5m2z. The large value of Im2,1 at the weak scale is a consequence of the large hierarchy in the soft scalar masses at the messenger scale: rn2e-R < m 2 u << mL Q,gtc

2 2 at the messenger Models of Sections 4, 5 and 7 attempt to reduce the ratio m~/mrl,,

scale and hence improve the fine tuning in the Higgs sector• The fine tuning may be quantified by applying one of the criteria of Refs. [3,4]• The value O* of a physical observable O will depend on the fundamental parameters (Ai) of the theory• The fundamental parameters of the theory are to be distinguished from the free parameters of the theory which parameterize the solutions to O(Ai) = O*. If the value O* is unusually sensitive to the underlying parameters (Ai) of the theory then

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

a small change in ,~i produces a large change in the value of O. The Barbieri-Giudice function

c( O,,~i)- O'a*0~00 o=o*

(14)

quantifies this sensitivity [ 3 ]. This particular value of O is fine tuned if the sensitivity to ,~i is larger at O = O* than at other values of O [4]. If there are values of O for which the sensitivity to Ai is small, then it is probably sufficient to use c(O, Ai) as the measure of fine tuning. To determine c(m 2, Ai), we performed the following. The sparticle spectrum in the minimal LEGM model is determined by the four parameters Amess, /-62, ~, and tan ft. 7 The scale Amess fixes the boundary condition for the soft scalar masses, and an implicit dependence on tan fl from At, ab and ~ arises in RG scaling 8 from/ZRa = Amess to the ~2 ). The extremization conditions weak scale, that is chosen to be t z ~ = m~ + ~1 ( m~2t + rrttc of the scalar potential (Eqs. (10) and (11)) together with mz and mt leave two free parameters that we choose to be Amess and tan/3 (see Appendix A for the expressions for these functions). A numerical analysis yields the value of c(m2z, tz2) that is displayed in Fig. 1 in the (tan/3, Amess) plane. We note that c(m2z, Ix2) is large throughout most of the parameter space, except for the region where tan/3 ~> 5 and the messenger scale is low. A strong constraint on a lower limit for Amess is from the right-handed selectron mass. Contours rn~R = 75 GeV ( ~ the LEP limit from the run at v/s ~ 170 GeV [20]) and 85 GeV ( ~ the ultimate LEP2 limit [21]) are also plotted. The (approximate) limit on the neutralino masses from the LEP run at ~ ~ 170 GeV, mxo + mxo = 160 GeV and the ultimate LEP2 limit, mxo + mxo ~ 180 GeV are also shown in Figs. la,c for sgn(/z) = - 1 and Figs. lb,d for sgn(/x) = +1. The constraints from the present and the ultimate LEP2 limits on the chargino mass are weaker than or comparable to those from the selectron and the neutralino masses and are therefore not shown. If mz were much larger, then c ~ 1. For example, with mz = 275 GeV (550 GeV) and Amess = 50 (100) TeV, c(m2;/x 2) varies between 1 and 5 for 1.4 < tan/3 < 2, and is ~ 1 for tan/3 > 2. This suggests that the interpretation that a large value for c(m2;/x 2) implies that mz is fine tuned is probably correct. From Fig. 1 we conclude that in the minimal LEGM model a fine tuning of approximately 7% in the Higgs potential is required to produce the correct value for mz. Further, for this fine tuning the parameters of the model are restricted to the region tan/3 ~> 5 and Amess ' ~ 45 TeV, corresponding to m~R ~ 85 GeV. We have also checked that adding more complete (5 + 5)'s does not reduce the fine tuning.

7 We allow for an arbitrary/,2 at Amess. 8 The RG scaling of At was neglected.

10

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

(a)

A (-rev) ~ 9o

A

allowed

60 / ' ~

/

80 70

(b)

40

/" 25

6o

25

60

20 5O 4O

20

15

'i

5O

10 5

i0

15

20

(c)

A

25

70

i

~

I10 0

5

10

15

20

(-regv~

allowed 80

40

40

/

7O

60

601

50

501

40

401 15

20

25

t a n I]

30

tanl~

allowed

i0

25

(d)

A

fre~ 8(

40

30

tanl3

15

30

0

t

25

1II

10

f i0

15

20

25

30

tanl3

Fig. 1. Contours of c(mZz;/~2)= ( i0,15, 20, 25,40, 60) for an MSSM with a messengerparticle content of one (5 + 5). In (a) and (c) sgn(#) = -1 and in (b) and (d) sgn(/z) = +1. The constraintsconsideredare: (I) m~R = 75 GeV, (II) m~0 -I- m2o = 160 GeV, (III) m~R = 85 GeV, and (IV) m2o + m?o = 180 GeV. A central value of mtop = 175 GeV is assumed.

4. A toy model to reduce fine tuning

4.1. Model In this section the particle content and couplings in the messenger sector that are 2 2 at the scale Amess. sufficient to reduce is discussed. The aim is to reduce rn.~/mi4,,

Im~,,I

The idea is to increase the number of messenger leptons ( S U ( 2 ) doublets) relative 2 - 2 and to the number of messenger quarks ( S U ( 3 ) triplets). This reduces both m -~/mH,

mQ/m~

at the scale Amess (see Eq. ( 4 ) ) . This leads to a smaller value of

tm~,,I in

the R G scaling (see Eq. ( 1 3 ) ) and the scale Amess can be lowered since m~R is larger. For example, with three doublets and one triplet at a scale Amess = 30 TeV, so that me, ~ 85 GeV, we find ImP. (mff) l ~ (100 GeV) 2 for At = 1. This may be achieved by

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

11

the following superpotential in the messenger sector: 1 3 W = ,~qlSqlCll -+- .~llSIlll q- Ai2Sl212 + ,~1~Sl313 -1- gAsS

q-A~bSq~-q~+ -k- 1ANN3 -+- aqzNq2Ut2 -q- Aq3Nq3q3,

(15)

where N is a gauge singlet. The two pairs of triplets q2, ~72 and q3, c/3 are required at low energies to maintain gauge coupling unification. In this model the additional leptons le,/2 and 13, [3 couple to the singlet S, whereas the additional quarks couple to a different singlet N that does not couple to the messenger fields ~b+, ~b_. This can be enforced by discrete symmetries (we discuss such a model in Section 7). Further, we assume the discrete charges also forbid any couplings between N and S at the renormalizable level (this is true of the model in Section 7) so that SUSY breaking is communicated first to S and to N only at a higher loop level. 4.2. Mass spectrum

Before quantifying the fine tuning in this model, the mass spectrum of the additional states is briefly discussed. While these fields form complete representations of S U ( 5 ) , they are not degenerate in mass. The vev and F-component of the singlet S gives a mass Amess to the messenger lepton multiplets if the F-term splitting between the scalars is neglected. As the squarks in qiq-Cli (i = 2, 3) do not couple to S, they acquire a soft scalar mass from the same two-loop diagrams that are responsible for the masses of the MSSM squarks, yielding m~ ~ ce3 (Amess) AsusY/( v'~rr). The fermions in q + ~ also acquire mass at this scale since, if either ~q2 or ~q3 ,--.a0 ( 1 ), a negative value for m 2 (the soft scalar mass squared of N) is generated from the /lqNqgl coupling at one loop and thus a vev for N N m~ is generated. The result is m l / m q ~ v~'n'/oz3 (Amess)(Amess/AsusY) ~ 85. The mass splitting in the extra fields introduces a threshold correction to sin 20w if it is assumed that the gauge couplings unify at some high scale M6UT ~ 1016 GeV. We estimate that the splitting shifts the prediction for sin 20w by an amount ~ - 7 x l O - 4 1 n ( m t / m q ) n , where n is the number of split (5 + ~).9 In this case n = 2 and m l / m q ~ 85, so ~sin20w ~ - 6 × 10 -3. If oz3(Mz) and ceem(Mz) are used as input, then using the two-loop RG equations sin 2 0w(MS) = 0.233 4 - 0 ( 10 -3) is predicted in a minimal S U S Y - G U T (see for example Ref. [22] ). The error is a combination of weak scale SUSY and G U T threshold corrections [22]. The central value of the theoretical prediction is a few percent higher than the measured value o f sin 2 0w(MS) = 0.231 _4_ 0.0003 [23]. The split extra fields shift the prediction of sin 20w to ~ 0.227 4- O( 10 -3) which is a few percent lower than the experimental value. In Sections 6, 7 we show that this spectrum is derivable from an SU(5) × SU(5) GUT in which the GUT threshold corrections to sin20w could be ,.o O ( 1 0 -3) - O ( 1 0 -2) [24]. It is possible that the combination of these GUT threshold corrections and the split extra field threshold corrections make the prediction of sin 20w more consistent with the observed value. 9 The complete (5 + 5), i.e., ll, fl and ql, g/l, that couples to S is also split because .,ll v~ ,~q at the messenger scale due to RG scaling from MGUTto Amess.This splitting is small and neglected.

12

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34 A

(a)

A

(TeV) 70

(b)

(TeV) 70

10~ " 60

allowed 7

allowed

10~"

~___--~

60

"

75 50

50

40

40

3

II

_

30

30

1

f 5

i0

15

(c)

A

20

25

30

tanl3

5

I0

15

20

25

(d)

A

30

tanl3

(TeV)0

(TeV7)0

10 ~

10~

allowed

allowed

60

60 50

5O

40

40

75

3

1]I 30~

30

0

i0

15

20

25

30

\

0

-

-

IV

..-" . . . . . . . . . . . . . . . . . . . . . . . '

" 5

tanl~

~ i0

15

~ 20

25

30

tanl3

Fig. 2. Contours of c(m2;I.L2) = (1,2, 3,5,7, 10) for a MSSM with a messenger particle content of three (l + 1)'s and one (q + 4). In (a) and (e) sgn(/z) = -1 and in (b) and (d) sgn(/z) = +1. The constraints considered are: (1) tneR = 75 GeV, (II) m20 + m ~ = 160 GeV, (III) mgR = 85 GeV, and (IV) m~l1 + mko = 180 GeV. A central value of mtop = 175 GeV is assumed.

4.3. Fine tuning To quantify the fine tuning in these class of models the analysis of Section 3 is applied. In our R G analysis the R G scaling of It, the effect of the extra vector-like triplets on the R G scaling of the gauge couplings, and weak scale SUSY threshold corrections were neglected. We have checked a postiori that this approximation is consistent. As in Section 3, the two free parameters are chosen to be Amess and tan/3. Contours of constant c(m2z,tZ 2) are presented in Fig. 2. We show contours of mxo + mxo = 160 GeV, and m~R = 75 GeV in Fig. 2a for sgn(/z) = - 1 and in Fig. 2b for sgn(/z) = +1. These are roughly the present limits from LEP (including the run at v/S ~ 170 GeV [20] ). The (approximate) ultimate LEP2 reaches [21] mxo + mxo = 180 GeV, and m~R = 85 GeV are shown in Fig. 2c for sgn(/x) = - 1 and in Fig. 2d for s g n ( # ) = +1. Since / z 2 ( ~

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

13

(100 GeV) 2) is much smaller in these models than in the minimal LEGM model, the neutralinos (X ° and X2°) are lighter so that the neutralino masses provide a stronger constraint on Amess than does the slepton mass limit. The chargino constraints are comparable to the neutralino constraints and are thus not shown. It is clear that there are areas of parameter space in which the fine tuning is improved to ~ 40% (see Fig. 2). While this model improves the fine tuning required of the/x parameter, it would be unsatisfactory if further fine tunings were required in other sectors of the model, for example, the sensitivity of m2z t o / z 2, Amess and /It and the sensitivity of mt t o / z 2,/x 2, amess and At. We have checked that all these are less than or comparable to c(m2;/z2). We now discuss the other fine tunings in detail. For large tan/3, the sensitivity of m2z to/z~, c ,tm 2z ,./ ' 32,) o( 1/tan 2/3, and is therefore smaller than c( m2z ; tz2). Our numerical analysis shows that c( m2z ; Ix~ ) < c( m2z; tz 2) for all tan/3. In the one-loop approximation m~,, and m 2 at the weak scale are proportional to 2 Amess since all the soft masses scale with Amess and there is only a weak logarithmic dependence on Amess through the gauge couplings. We have checked numerically that 2 2 (Om2/OA2ess) ,-.o 1. Then, c(mz, 2. Amess) 2 (Amess/mH,,) c(m2; m~,,) + c(m2z; m~,,). We find that c(m2z; Amess)2 ~ c(m2z;/x 2) + 1 over most of the parameter space. In the one-loop approximation, m~,,(t) is m 2 ( t ) ~ m 2 , , ( 0 ) + (mQ ( 0 ) + m ~ ; ( 0 ) + m 2 , ( 0 ) ) ( e -3a2'/s~)t- 1 ) . Then, using t ~ ln(Amess/m~3) ~ ln(v/-6cr/ce3) -.~ 4.5 and /It ~ 1, Appendix A)

4 Om2(t) m2 03 c(m2; ~It) ~ m2 O/I} ~ 50(600 GeV) 2 .

c(m2;/I,)

(16) is (see

(17)

This result measures the sensitivity of m 2 to the value of /It at the electroweak scale. While this sensitivity is large, it does not reflect the fact that ,~t(Mel) is the fundamental parameter of the theory, rather than At(Mweak). We find by both numerical and analytic computations that, for this model with three (5 + 5)'s in addition to the MSSM particle content, ~/It(Mweak) ~ 0.1 × ~$At(Mp1), and therefore

c(m2;/It(Mel) ) ,-~ 5

m2 Q3 (600 GeV) 2"

(18)

For a scale of Amess = 50 TeV (m~" ~ 600 GeV), c ( m 2 ; a t ( M e l ) ) is comparable to c(m2;/x 2) which is ,-~ 4 to 5. At a lower messenger scale, Amess ~ 35 TeV, corresponding to squark masses of ,,~ 450 GeV, the sensitivity of m 2 to ht(Mp1) is ~ 2.8. This is comparable to c(m2z;/z 2) evaluated at the same scale. We now discuss the sensitivity of mt to the fundamental parameters. Since m 2 = ½v2 sin 2/3/It2, we get

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

14

Table 1 Soft scalar masses in GeV for messenger particle content of three (1 + 1)'s and one q + ~ and a scale Amess = 50 TeV in N Q1,2

mvtc 1,2

mdc ,

titsLi,Ha

?neC

mN Q3

mpc3

687

616

612

319

125

656

546

c(mt;

Ai)

l c t m 2"

-= ~Atai -~- ~

t

c°s3fl 0 tanfiA-

Z ' A i ) "~- ~

OA i

"

(19)

Numerically we find that the last term in c(mt; Ai) is small compared to c(m2z; Ai) and thus over most of parameter space c(mt; Ai) ~ ~1cgm ~ 2 z ,. Ai). As before, the sensitivity of mt to the value of At at the GUT/Planck scale is much smaller than the sensitivity to the value o f At at the weak scale.

4.4. Sparticle spectrum The sparticle spectrum is now briefly discussed to highlight deviations from the mass relations predicted in the minimal L E G M model. For example, with three doublets and one triplet at a scale of A = 50 TeV, the soft scalar masses (in GeV) at a renormalization 2 3 + m 2 ) ,~ (630 GeV) 2, for At = 1, are shown in Table 1. scale /Z2RG= m 2 + 2! (m~" Two observations that are generic to this type of model are: (i) By construction, the spread in the soft scalar masses is less than in the minimal L E G M model. (ii) The gaugino masses do not satisfy the one-loop SUSY-GUT relation Mi/oli = constant. In this case, for example, M3/ce3 : M2/ce2 ~ 1 : 3 and M3/ce 3 : M1/Cel ~ 5 : 11 to one loop. We have also found that for tan fl > 3, the Next Lightest Supersymmetric Particle (NLSP) is one o f the neutralinos, whereas for tan fl < 3, the NLSP is the fighthanded stau. Further, for these small values of tan fl, the three right-handed sleptons are degenerate within ~ 200 MeV.

5. N M S S M In Section 3, the/x term and the SUSY breaking mass/~23 were put in by hand. There it was found that these parameters had to be fine tuned in order to correctly reproduce the observed Z mass. The extent to which this is a "problem" may only be evaluated within a specific model that generates both the/x and #23 terms. For this reason, in this section a possible way to generate both t h e / z term and /z~ term in a manner that requires a minimal modification to the model of either Section 2 or Section 4 is discussed. The easiest way to generate these mass terms is to introduce a singlet N and add the interaction NHuHd to the superpotential (the NMSSM) [ 13]. The vev of the scalar component of N generates /z and the vev of the F-component of N generates /x~.

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

15

We note that for the "toy model" solution to the fine tuning problem (Section 4), the introduction of the singlet occurs at no additional cost. Recall that in that model it was necessary to introduce a singlet N, distinct from S, such that the vev of N gives mass to the extra light vector-like triplets, qi, qi ( i = 2,3) (see Eq. (15)). Further, discrete symmetries (see Section 7) are imposed to isolate N from SUSY breaking in the messenger sector. This last requirement is necessary to solve the fine tuning problem: if both the scalar and F-component of N acquired a vev at the same scale as S, then the extra triplets that couple to N would also act as messenger fields. In this case the messenger fields would form complete (5 + 5)'s and the fine tuning problem would be reintroduced. With N isolated from the messenger sector at tree level, a vev for N at the electroweak scale is naturally generated, as discussed in Section 4. We also comment on the necessity and origin of these extra triplets. Recall that in the toy model of Section 4 these triplets were required to maintain the SUSY-GUT prediction for sin 20w. Further, we shall also see that they are required in order to generate a large enough - m 2 (the soft scalar mass squared of the singlet N). Finally, in the GUT model of Section 7, the lightness of these triplets (as compared to the missing doublets) is the consequence of a doublet-triplet splitting mechanism. The superpotential in the electroweak symmetry breaking sector is AN N 3 + AqNq(l- AHNHuHd, W = -~-

(20)

which is similar to an NMSSM except for the coupling of N to the triplets. The superpotential in the messenger sector is given by Eq. (15). The scalar potential is l0 IF,.I2 + m2NIN[2 + m2,lnul 2 + m~,lHdl 2 + D-terms

V-- ~ i

- ( A H N H u H d + h.c.) + Vl-loop.

(21)

The extremization conditions for the vevs of the real components of N, Hu and lid, denoted by VU, vu and va respectively (with u =

ua

2 ~ 250 GeV), are

VN(I~t2N-~~2 V2 2 -}-A2NV2N--AHANUuUd)-- ~/~AHUuvd 1

2

2mz =

(22)

~2 - - / z~22 t a n 2 / 3 /xl

t~nf-~7 i

'

(23)

/.z2

sin2fl = 2

3

(24)

with tz2 = ~ 1-2 AHU2N,

(25)

AH

I0 In models of gauge mediated SUSY breaking, = 0 at tree level and a non-zero value of AH is generated at one loop. The trilinear scalar term is generated at two loops and is neglected.

ANN3

K. Agashe, M. Graesser/NuclearPhysics B 507 (1997) 3-34

16

2 = --~AHUuUd 1 2 1 2N -}- AH __~ 13N, 1.63 -'1-~AHANU

(26)

m i - m 2i + 2 ggl/~_loop ~ ,

(27)

~

2

_

i=(u,d,N).

We now comment on the expected size of the Yukawa couplings Aq, AN and An. We must use the RGE's to evolve these couplings from their values at MGIyr or Mp1 to the weak scale. The quarks and the Higgs doublets receive wavefunction renormalization from SU(3) and SU(2) gauge interactions respectively, whereas the singlet N does not receive any wavefunction renormalization from gauge interactions at one loop. So, the couplings at the weak scale are in the order Aq -,, O( 1 ) > An > ,iN if they all are O( 1 ) at the GUT/Planck scale. We remark that without the Nqg7 coupling, it is difficult to drive a vev for N as we now show below. The one-loop RGE for m 2 is

dm2u 6AZm2 r t~ 222 dt ~ Sqr2 N' " + - ~ 2 (m~"(t)-f-m2Hd(t)-t-m2N(t)) 3A2q

2

+-ff-~2 (m~(t) + m 2 ( t ) ) .

(28)

Since N is a gauge singlet, m2N = 0 at Amess. Further, if kq = 0, an estimate for m 2 at the weak scale is then 2A2H

m2 ~ --ff-~2 (rnk,(O) + m2,,(O))ln

(Amess "]

\ mHj I

,

(29)

i.e., An drives m 2 negative. The extremization condition for VN, Eq. (22), and using Eqs. (24) and (26) (neglecting An) shows that m 2 + ~ *tn 2 v 22

A2( ~

2 2 (m2H"(0)+m2H"(O))ln(Amess'~ ) 8rr

(30)

\ mild ./

has to be negative for N to acquire a vev. This implies that m2H,, and m2,~ at Amess have to be greater than ~ (350 GeV) 2 which implies that a fine tuning of a few percent is required in the electroweak symmetry breaking sector. With Act ~ O( 1 ), however, there is an additional negative contribution to m 2 given approximately by

--3A2 ( m 2 ( O ) + m 2 ( O ) ) l n ( A m e s s ) 87r2 \ mo / "

(31)

This contribution dominates the one in Eq. (29) since Aq ~ AH and the squarks q, have soft masses larger than the Higgs. Thus, with Aq 4= 0, m 2 q- A2V2/2 is naturally negative. Fixing mz and m , we have the following parameters: Amess, Aq, AH, AN, tan fl, and vrc. Three of the parameters are fixed by the three extremization conditions, leaving three free parameters that for convienence are chosen to be Amess, tan fl >~ 0, and An. The signs of the vevs are fixed to be positive by requiring a stable vacuum and no

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

17

spontaneous C P violation. The three extremization equations determine the following relations: AN = AHU'----~N N 1~2 Jr- gall sin2flv 2 --

AHUN

,

(32)

/Z,

UN~'X/g~H -2

I

(33) 2

2

1.2

2

'TlN = ANAH~ sin 2/3v 2 - ANUN -- ~AHU -~-

1 02 ~--v~AH sin 2/3~--N,

(34)

where 1~2 =

1 2 /'7/2,, t a n 2 / 3 - ~ . --2mz + 1 -- tan2/3

2/* 2 = sin 2 / 3 ( 2 # 2 ÷ r~2, + r~2,).

'

(35) (36)

The superpotential term NHuHa couples the RGE's for m2,, m2, and m 2. Thus the values o f these masses at the electroweak scale are, in general, complicated functions o f the Yukawa parameters At, AH, AN and Aq. In our case, two of these Yukawa parameters (Aq and AN) are determined by the extremization equations and a closed form expression for the derived quantities cannot be found. To simplify the analysis, we neglect the dependence of m 2 and m 2H,, on AH induced in RG scaling from Amess to the weak scale. Then rn2H,, and m 2 depend only on Amens and tan/3 and thus closed form solutions for AN, vN and ~2N can be obtained using the above equations. Once r~2N at the weak scale is obtained, the value of Aq is obtained by using an approximate analytic solution. An exact numerical solution of the RGE's then shows that the above approximation is consistent. 5.1. Fine tuning and phenomenology The fine tuning functions we consider below are c(O;AH), c(O;/~N), c(O;At),

c(O; Aq) and c(O; Amess) where O is either m~ or mr. The expressions for the fine tuning functions and other details are given in Appendix A. In our RG analysis the approximations discussed in Subsection 4.3 and above were used and found to be consistent. Fine tuning contours of c ( m 2 ; AH) are displayed in Figs. 3a,b for An = 0.1 and Figs. 3c,d for AH = 0.5. We have found by numerical computations that the other fine tuning functions are either smaller or comparable to c(m~; AH). 11 We now discuss the existing phenomenological constraints on our model and also the ultimate constraints if LEP2 does not discover SUSY/light Higgs (h). These are shown in Figs. 3a,c and Figs. 3b,d respectively. We consider the processes e+e - --+ Zh, 0 0 and e+e - ---+ ene~ e+e - ~ (h + pseudoscalar), e+e - ~ X + X - , e+e - --+ X1X2, I I In computing these functions the weak scale value of the couplings AN and An has been used. But since AN and A,u do not have a fixed point behavior, we have found that AH(MGuT)/AH(mZ) aa,q(mz)/ 0AH(MGuT) ~ 1 so that, for example, c(m2z;An(MGuT)) ~ c(m2z;aH(mz)).

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

18 A

5O

A (TeV) 5=

(a)

(TeV) 55

allowed 1

I

/1

5

(b) allowed

.,'

IVi

5C

/

45 4O

35

,,\

V

30

......

25 0

2

4

A fray)

451

(c) .

8

i0

.

8

i0

tanl~

IV/

allowed

! 2

~

"'\

/-

30

,/ 4

1

tanl3 Fig. (l + (d), (II)

_

6

45f

"'\

3

,_

(d)

(XeV)| 5oi

,

10 / ' ~ ~ - ~

2

2

A55 f

/ / /

0

tanl3

\\

35

..~..?, _

i

6

allowed 20/ ~ ~ ~

40

,,

30~

if,

2

~" 3

i

4

tan

of

3. Contours c(m2Z;tH) for the NMSSM of Section 5 and a messenger particle content of three D's and one ( q + ,'7). In (a) and (b), c(mez;AH) = (4,5,6,10,15) and AH = 0.1. In (c) and c(m2z;An) = (3,4,5, 10, 15,20) and AH = 0.5. The constraints considered are: (I) mh + ma = mz, m~n = 75 GeV, (II1) m~0 + m ~ 2 = 160 GeV, (IV) mh = 92 GeV, (V) me R = 85 GeV, and (VI)

m~,0 + m2o = 180 GeV. For AH = 0.5, the limit mh ~ 70 GeV constrains tanfl ~ 5 (independent of Amess) and is thus not shown. A central value of mtop = 175 GeV is assumed. o b s e r v a b l e at LEP. S i n c e this m o d e l also has a light pseudoscalar, w e also c o n s i d e r upsilon decays 7' --+ ( 3 / + p s e u d o s c a l a r ) . W e find that the m o d e l is p h e n o m e n o l o g i c a l l y v i a b l e and requires a N 2 0 % tuning even i f no n e w particles are discovered at L E P 2 . W e begin w i t h the constraints on the scalar and pseudoscalar spectra o f this model. T h e r e are three neutral scalars, two neutral pseudoscalars and o n e c o m p l e x charged scalar. W e first c o n s i d e r the mass s p e c t r u m o f the pseudoscalars. A t the b o u n d a r y scale Amess, S U S Y is softly b r o k e n in the visible sector o n l y by the soft scalar masses and the g a u g i n o masses. Further, the superpotential o f Eq. ( 2 0 ) has an R-symmetry. Therefore, at the tree level, i.e. with A n = 0, the scalar potential o f the v i s i b l e sector ( E q . ( 2 1 ) )

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

19

has a global symmetry. This symmetry is spontaneously broken by the vevs of N R, H R, and H R (the superscript R denotes the real component of fields), so that one physical pseudoscalar is massless at tree level. It is

a=

1 (VN NI + vsin2/3cos/3HIu + v sin2/3sin/3HId), V/v~ + U2 sin 2 2/3

(37)

where the superscripts I denote the imaginary components of the fields. The second pseudoscalar,

A~

2NI+

HI

(38)

UCOS/3' acquires a mass m2A = ½AH,~Nv2(tan/3 + cot/3) + ~.HaN v2 sin2/3

(39)

through the ]FNI 2 term in the scalar potential. The pseudoscalar a acquires a mass once an An-term is generated, at one loop, through interactions with the gauginos. Including only the wino contribution in the one-loop RGE, A n is given by

An ~

6°d2(Amess).Me~Hln(~)

4¢r

20An

28() G e V

GeV,

(40)

where M2 is the wino mass at the weak scale. Neglecting the mass mixing between the two pseudoscalars, the mass of the pseudo-Nambu-Goldstone boson is computed to be m a2

9 2 -'~AVNVuVa/(V N + V2 sin 2 2t9)

~" (40)2

~

280 G e ~

\ s i n 2 2 / 3 + (vN/250 G e V ) 2 J (GeV)2" (41)

If the mass of a is less than 712 GeV, it could be detected in the decay T -+ a + y [23]. Comparing the ratio of decay width for T --+ a + y to T - + / z - + / z + [23,25], the limit sineetan e

< 0.43

\~

(42)

~/(VN/250 GeV) 2 + sin 2 2/3 is found. Further constraints on the spectra are obtained from collider searches. The nondetection of Z --+ scalar + a at LEP implies that the combined mass of the lightest Higgs scalar and a must exceed ,-~ 92 GeV. Also, the process e+e - --+ Zh may be observable at LEP2. For AH = 0.1, the constraint mh + ma ~ 92 GeV is stronger than mh ~ 70 GeV which is the limit from LEP at V~ ~ 170 GeV [20]. The contour

20

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

of mh q- ma = 92 GeV is shown in Fig. 3a. In Fig. 3b, we show the contour of mh = 92 GeV (~-, the ultimate LEP2 reach [26] ). For An = 0.5, we find that the constraint m h ~ 70 GeV is stronger than mh q- ma >~ 92 GeV and restricts tan/3 < 5 independent of A m e s s . The contour mh = 92 GeV is shown in Fig. 3d. We note that the allowed parameter space is not significantly constrained. We find that these limits make the constraint of Eq. (42) redundant. The left-right mixing between the two top squarks was neglected in computing the top squark radiative corrections to the Higgs masses. The pseudo-Nambu-Goldstone boson a might be produced along with the lightest scalar h at LEP. The (tree-level) cross section in units of R = 87/s nb is

° - ( e + e - --+ ha) ~ O ' 1 5 ( s -

m2z) 2A2v

(

l'mh'S

'

(43)

where g A / cos Ow is the Z ( aOh-hOa) coupling, and v( x, y, z ) = V/ ( X - y - z )2 _ 4yz. If h = c N N r~ + c . H ~ + caH~, then A

sin 213

cos/3c, - sin ~3ca

(44)

~/(VN/250 GeV) 2 + sin 2 2/3 We have numerically checked the parameter space allowed by mh > 70 GeV and A~/ ~< 0.5 and have found the production cross section for ha to be less than both the current limit set by DELPHI [27] and a (possible) exclusion limit of 30 fb [26] at v/~ ~ 192 GeV. The production cross section for hA is larger than for ha and A is therefore in principle easier to detect. However, for the parameter space allowed by mh ~ 70 GeV, numerical calculations show that ma ~> 125 GeV, so that this channel is not kinematically accessible. The charged Higgs mass is m 2 ± = m 2 + m2, + m2,, + 2/.,2,

(45)

which is greater than about 200 GeV in this model since m ~ > (200 GeV) 2 for Amess >~, 35 TeV and as /x2 ,.o - m ~ , . The neutralinos and charginos may be observable at LEP2 at V~ ~ 192 GeV if mx+ ~< 95 GeV and mxol + mxo ~< 180 GeV. These two constraints are comparable, and thus only one of these is displayed in Figs. 3b,d, for ,~r~ = 0.1 and An = 0.5 respectively. Also, contours of mxo +mxo = 160 GeV (N the LEP kinematic limit at x/s ~ 170 GeV) are shown in Figs. 3a,c. Contours of 85 GeV (,-, the ultimate LEP2 limit) and 75 GeV (N the LEP limit from x/s ~ 170 GeV) for the right-handed selectron mass further constrain the parameter space. The results presented in all the figures are for a central value of mt = 175 GeV. We have varied the top quark mass by 10 GeV about the central value of mt = 175 GeV and have found that both the fine tuning measures and the LEP2 constraints (the Higgs mass and the neutralino masses) vary by ~ 30%, but the qualitative features are unchanged.

21

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

A (TeV)

A

(a) I

allowed ./

200 ~

80

(b)

(TeV) /

/

i

80

IV

allowed/' /./:"

200~

\:

70

70

60

60[

50

50[

'\

100

/:

40 . . . . .

1

2

.

3

4

5

tanl3

1

2

3

4

5

tanlg

Fig. 4. Contours of c(mZ;tH) = (50,80, 100, 150,200) for the NMSSM of Section 5 with AH = 0.1 and a messenger particle content of one (5 + 3). The constraints considered are: (I) mh + ma = mz, (II) me, = 75 GeV, (III) m~ + m20 = 160 GeV, (IV) mh = 92 GeV, (V) m~R = 85 GeV, and (VI) m2o + mX,o = 180 GeV. A central value of mtop= 175 GeV is assumed. We see from Fig. 3 that there is parameter space allowed by the present limits in which the tuning is ~ 30%. Even if no new particles are discovered at LEP2, the tuning required for some region is ~ 20%. It is also interesting to compare the fine tuning measures with those found in the minimal L E G M model (one messenger (5 ÷ 5 ) ) with an extra singlet N to generate the /x and /x 32 terms. 12 In Fig. 4 the fine tuning contours for c(m2z; AH) are presented for ~t4 = 0.1. Contours of me, = 75 GeV and mxo1 + mxo = 160 GeV are also shown in Fig. 4a. For .L~ = 0.1, the constraint mh + ma >~ 92 GeV is stronger than the limit mh > 70 GeV and is shown in Fig. 4a. In Fig. 4b, we show the (approximate) ultimate LEP2 limits, i.e. mh = 92 GeV, mxo + rex0 = 180 GeV and me, = 85 GeV. Of these constraints, the bound on the lightest Higgs mass (either mh ÷ ms >~ 92 GeV or mh >~ 92 GeV) provides a strong lower limit on the messenger scale. We see that in the parameter space allowed by present limits the fine tuning is ~< 2% and if LEP2 does not discover new particles, the fine tuning will be < 1%. The coupling Au is constrained to be not significantly larger than 0.1 if the constraint mh + ma > 92 GeV (or mh >~ 92 GeV) is imposed and if the fine tuning is required to be no worse than 1%.

6. M o d e l s d e r i v e d f r o m a G U T

In this section, we discuss how the toy model of Section 4 could be derived from a G U T model. 12 We a s s u m e that the m o d e l contains s o m e m e c h a n i s m to generate -m~v N ( 1 0 0 G e V ) 2 _ ( 2 0 0 G e V ) 2 ; for

example, the singlet is coupled to an extra (5 + 5).

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

22

In the toy model of Section 4, the singlets N and S do not separately couple to complete SU(5) representations (see Eq. (15)). If the extra fields introduced to solve the fine tuning problem were originally part of (5 + 5) multiplets, then the missing triplets (missing doublets) necessarily couple to the singlet S ( N ) . T h e triplets must be heavy in order to suppress their contribution to the soft SUSY breaking mass parameters. If we assume the only other mass scale is MGUT, they must acquire a mass at MGUT. This is just the usual problem of splitting a (5 + 3) [ !4]. For example, if the-s~p~rpotential in the messenger sector con~airis four (5 + 5)'s, W = ,~1 $511511 ~-/~2S512512 ~-/~3S513513 -~ ~4S5q5q,

(46)

then the SU(3) triplets in the (5t + 5t)'s and the SU(2) doublet in (5q + 5q) must be heavy at MGUT so that in the low energy theory there are three doublets and one triplet coupling to S. This problem can be solved using the method of Barbieri, Dvali and Strumia [15] that solves the usual Higgs doublet-triplet splitting problem. The mechanism in this model is attractive since it is possible to make either the doublets or triplets of a quintet heavy at the GUT scale. We next describe their model. The gauge group is SU(5) x SU(5) ~, with the particle content ~S(24, 1), ~P(1,24), ~0(5, 5) and ¢ ( 5 , 5) and the superpotential can be written as

-I-1Az Tr 2;3 + 1Az, Tr2; '3.

(47)

A supersymmetric minimum of the scalar potential satisfies the F-flatness conditions 0 = F~r, =

M~,~Z

+ aZ}~Z, + a ' -/(~

'~

'

fit

(48) With the ansatz 13 X = vz diag(2, 2, 2, - 3 , - 3 ) ,

N~ = vz, diag(2, 2, 2, - 3 , - 3 ) ,

(49)

the F¢~ = 0 condition is diag[ M3, M3, M3, M2, M2] • diag[v3, o3, u3, u2, u2] = 0,

(50)

where M3 = M~p + 2Avz + 2A~vz, and M 2 = M e - 3 A v s - 3 A ~ v z , and the second matrix is the vev of @. To satisfy this condition, there is a discrete choice for the pattern of vev of ~: (i) v3 4= 0 and M3 = 0 or (ii) v2 ~ 0 and M2 = 0. Substituting either (i) or 13The two possible solutions to the F-flatness conditions are 2: = vzdiag(2,2,2,-3,-3) and £ = v,2diag( 1, 1, 1, 1,-4).

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

23

(ii) in the Fz and Fz, conditions then determines v3 (or v2). With two sets of fields, ~ l , ~1 with v3 ~ 0 and q~2, qSz with v2 v~ 0, we have the following pattern of symmetry breaking: SU(5) × SU(5)

I Uv,Uvt

"~+" (SU(3) × SU(2) × U ( 1 ) ) × (SU(3) × SU(2) × U ( 1 ) ) '

U3 ,t12

SM (the diagonal subgroup).

(51)

If the scales of the two stages of symmetry breaking are about equal, i.e. vz, vz, v3, v2 ~ MGtjT, then the SM gauge couplings unify at the scale MGUT.14 The particular structure of the vevs of ~bl and 452 can be used to split representations as follows. Consider the Higgs doublet-triplet splitting problem. With the particle content 5h (5, 1 ), 5h (5, 1 ) and X ( 1 , 5 ) , 3f( l, 5) and the superpotential ,=.d~,~aa' + 5hX,~'~bb~ -a ~' , W = 5 haA

(52)

the SU(3) triplets in 5h, 5h and X, J( acquire a mass of order M6uw whereas the doublets in 5h, 5h and X, Jf are massless. We want only one pair of doublets in the low energy theory (in addition to the usual matter fields). The doublets in X, )? can be made heavy by a bare mass term MotjTXX. Then the doublets in 5h, 5h are the standard Higgs doublets. But if all terms consistent with symmetries are allowed in the superpotential, then allowing MGUTq~I~I, MGUTX)(, 5hX~l and 5hX~l implies that a bare mass term for 5h5h is allowed. Of course, we can by hand put in a / z t e r m ],Z5h5 h of the order of the weak scale as in Section 4. However, it is theoretically more desirable to relate all electroweak mass scales to the original SUSY breaking scale. So, we would like to relate the/z term to the SUSY breaking scale. We showed in Section 5 that the NMSSM is phenomenologically viable and "un-fine tuned" in these models. The vev structure of q~2, ~2 can be used to make the doublets in a 5 + 5 heavy. Again, we get two pairs of light triplets and one of these pairs can be given a mass at the GUT scale. We can use this mechanism of making either doublets or triplets in a (5 + 5) heavy to show how the model of Section 4 is derivable from a GUT. The model with three messenger doublets and one triplet is obtained from a GUT with the following superpotential: W = $5~ + S5t5t + SXl21 + 5t21~1 + 51Xl~bl -]-5qXq~2 "~ 5qXq~P2 -~- MGUTXhXh -}- 5h2h~f) 1 + 5 h X h ~ l ']- ]..1.,5h5h

(53)

-q-N 3 .--}-N5q5q -Jr"N X q X q .

Here, some of the "extra" triplets and doublets resulting from splitting (5 + 5 ) ' s are massless at the GUT scale. For example, the "extra" light doublets are used as the additional messenger leptons. After inserting the vevs and integrating out the heavy states, this corresponds to the superpotential in Eq. (15) with the transcription 14See Refs. [ 15,24] for models which give this structure of vevs for the ~ fields without using the adjoints. \ ,\

24

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34 5, 5 ---~q], c~l + 11,/], 51, 5l ---+12, [2, Xl , Xl --+13, [3, 5q, 5q ----+q2, c12, Xq, ~'q ~ q3, t73.

(54)

We conclude this section with a remark about light singlets in SUSY-GUT's with low energy gauge mediated SUSY breaking. 15 In a SUSY-GUT with a singlet N coupled to the Higgs multiplets, there is a potential problem of destabilising the mweak/MGtrr hierarchy, if the singlet is light and if the Higgs triplets have a SUSY invariant mass of O(MGuT) [28]. In the L E G M models, a B-type mass for the Higgs triplets and doublets is generated at one loop with gauginos and Higgsinos in the loop, and with SUSY breaking coming from the gaugino mass. Since SUSY breaking (the gaugino mass and the soft scalar masses) becomes soft above the messenger scale, Amess 100 TeV, the B-type mass term generated for the Higgs triplets is suppressed, i.e., it is O((a/4qr)M2AZmess/MGuT). Similarly the soft mass squared for the Higgs triplets 2 2 2 ). Since the triplets couple to the singlet N, the soft scalar are O(mweakAmess/MGtrr mass and B-term generates at one loop a linear term for the scalar and F-component of N respectively. These tadpoles are harmless since the SUSY breaking masses for the triplets are so small. This is to be contrasted with supergravity theories, where the B-term ~ O(mwe~MGuT) and the soft mass ~ O(mweak) for the triplet Higgs generate a mass for the Higgs doublet that is at least ,-~ O(x/mweakMGuT/4~r).

7. One complete model The model is based on the gauge group Gqoc = SU(5) × S U ( 5 ) ' and the global symmetry group Gglo = Z3 × Z~ × Z4. The global symmetry acts universally on the three generations o f the SM. The particle content and their Gloe × Gglo quantum numbers are given in Table 2. The most general renormalizable superpotential that is consistent with these symmetries is

w=wl +w2+w3+Wn+Ws+W6+WT,

(55)

where

1 ~2 + ½A~, Tr X~3 WI= ½M~Tr£2+ ½,~Tr£3 + ~M~,Tr2 q-~02 (M@2 q- Aq52~ -Jr"A~zX)@2+q~I(M,t,] t t + Aa,~X+ A.blX ' , )qOl -

(56)

W2 = Ml fflX,

(57)

W 3 = .~15h~)lX h Jr ~15h~)lX h -q- ~25l~91Xl Jr- ~251q~1Xl,

(58)

15The authors thank H. Murayama for bringing this to their attention.

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

25

Table 2 SU(5) x SU(5) ~ × Z3 × Z~ x Z4 quantum numbers for the fields of the model discussed in Section 7. The generators of Z3 x Z~ x Z4 are (a, b, c). The three SM generations are labeled by the index i. Gloc

Z3

Z 3t

Z4

51 101 5h 5h

(5, 1) ( 10,1) (5, 1 ) (5, 1 )

1 a a a2

b 1 1 b2

c c c2 c2

.~ ~' ~2 • 2 ~bt q51

(24,1 ) ( 1,24 ) (3, 5) (5,3) (3, 5) (5, 5)

1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 c2 C2

51 31

(5, 1 ) (3, 1 ) (1,5) (1,3) (5, 1 ) (3, 1)

a2 1 1 a 1 a2

1 1 1 1 b2 b

c2 c2 1 1 1 1

(1,5) ( I, 3) (1,5) (1,5) ( l, 5) (1,3)

a 1 a a2 a2 a

b2 b b 1 l b2

1 1 1 1 1 1

S N

a 1

1 b

l 1

Nt ~b+ cb_

a a a

b2 1 1

1 1 1

X! Rt 5q 5q

Xq .~q Xh Xh X R

W4 = ,~35qCP2Xq + ~35q~2-,Yq,

(59)

W5 = ~6S5151 + t~7S5q5q q- /~8S~ff~hXl -4- ~ 9 S X X h + 11~$83,

(60)

1 ~ lV r~rt3, W6 = --)tH5h5hN + ½)tN N3 + ~ q N X f ( + A I o N t X X q + A l l N t f ~ q X + "~,tN, (61)

W7 = Ai~3ilOi3h +

A~10il0j5h.

T h e o r i g i n o f e a c h o f t h e Wi's a p p e a r i n g In computing

(62) in the superpotential

t h e F = 0 e q u a t i o n s at t h e G U T

come from fields appearing a c q u i r e v e v s at t h e G U T diag[2,2,2,-3,-3],

is e a s y to u n d e r s t a n d .

scale, t h e o n l y n o n - t r i v i a l c o n t r i b u t i o n s

i n W1, s i n c e all o t h e r Wis are b i l i n e a r i n f i e l d s t h a t d o n o t scale. T h e f u n c t i o n o f W1 is to g e n e r a t e t h e v e v s Z , 2 t

~ " = q~2 ~ d i a g [ 0 , 0 , 0 ,

1, 1] a n d ~1r = ~bl ~ d i a g [ 1, 1, 1 , 0 , 0 ] .

T h e s e v e v s a r e n e c e s s a r y t o b r e a k Gloc --+ S U ( 3 ) c x S U ( 2 )

x U(1)y

(this was explained

26

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

in Section 6). The role of W3 and W4 is to generate the necessary splitting within the many (5 + 5 ) ' s of Glo e that is necessary to solve the usual doublet-triplet splitting problem, as well as to solve the fine tuning problem that is discussed in Sections 3, 4 and 5. The messenger sector is given by Ws. It will shortly be demonstrated that at low energies this sector contains three vector-like doublets and one vector-like triplet. The couplings in W6 and W7 at low energies contain the electroweak symmetry breaking sector of the NMSSM, the Yukawa couplings of the SM fields, and the two light vectorlike triplets necessary to maintain the few percent prediction for sin 20w as well as to generate a vev for N. We now show that the low energy theory of this model is the model that is discussed in Section 5. Inserting the vevs for qh and ~1 into W3, the following mass matrix for the colored triplet chiral multiplets is obtained:

(5h,2h,51,Xt)

l0 iv 0 0 0/(51/ ~lv~,

0

0

0

0

Xh

0

0

0

~2VcI)l

0

5l

0

0

A2V,~,

0

M1

(63)

and all other masses are zero. There are a total of four vector-like colored triplet fields that are massive at MGUT- These are the triplet components of (5h,Xh), (5h,Xh), (5l, Xl) and (XI,TH), where Tu is that linear combination of triplets in 5l and X that marries the triplet component of Xl. The orthogonal combination to Tu, TL, is massless at this scale. The massless triplets at MGUT are (5q,5q), (Xq, ffq) and (X, TL), for a total of three vector-like triplets. By inspection, the only light triplets that couple to S at a renormalizable level are 5q and 5q, which was desirable in order to solve the fine tuning problem. Further, since X contains a component of TL, the couplings of the other light triplets to the singlets N and N ~ are Weft =/~loNIXXq q- ~ l l Nt XqTL -}- ,~qNTL X,

(64)

where ,~q = ~q COS Olt, ~11 = All COS Off and d is the mixing angle between the triplets in 51 and X, i.e., TL = cos d X - sin d 5 t . The AqNTLX coupling is also desirable to generate acceptable/~ and/z~ terms (see Section 5). In Sections 4, 5 it was also demonstrated that with a total of three messenger doublets the fine tuning required in electroweak symmetry breaking could be alleviated. By inserting the vev for q~2 into W4, the doublet mass matrix is given as

(X/, 5q, Xq)

0

0

0

7~3va,2

1~3vq)2

5q

0

Xq

(65)

and all other masses are zero. At MGUT the heavy doublets are (XI, X), (5q, ffq) and (5q, Xq), leaving the four vector-like doublets in (5h, 5h)' (51, 5t), (X, Xl) and (Xh, Xh)

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

27

massless at this scale. Of these four pairs, (5h, 5h) are the usual Higgs doublets and the other three pairs couple to S. The (renormalizable) superpotential at scales below Moot is then W = AqNq2q2 + ½ANN 3 + AloNt q3q2 + AllNt q2q3 - AHNHuHd + ½AN,N t3 q-h6S[lll q- ~-7S#IQ1 -~"/~8S72/2 "-~A9S13/3 q- ½AsS 3 + W7,

(66)

where the fields have been relabeled to make, in an obvious notation, their SU(3) × SU(2) × U(1) quantum numbers apparent. We conclude this section with comments about both the choice of Z4 as a discrete symmetry and about non-renormalizable operators in our model. The usual R-parity violating operators 10SM5SM5SM are not allowed by the discrete symmetries, even at the non-renormalizable level. In fact, R-parity is a good symmetry of the effective theory below MGuT. By inspection, the fields that acquire vevs at Moor are either invariant under Z4 or have a Z4 charge of 2 (for example, q~l ), so that a Z2 symmetry is left unbroken. In fact, the vevs of the other fields S, N, N ~ and the Higgs doublets do not break this Z2 either. By inspecting the Z4 charges of the SM fields, we see that the unbroken Z2 is none other than the usual R-parity. So at MGtrr, the discrete symmetry Z4 is broken to Rp. We also note that the Z4 symmetry is sufficient to maintain, to all orders in lIMP1 operators, the vev structure of ~0i and 42, i.e., to forbid unwanted couplings between 41 and ~b2 that might destabilize the vev structure [24]. This pattern of vevs was essential to solve the doublet-triplet splitting problem. It is interesting that both R-parity and requiring a viable solution to the doublet-triplet splitting problem can be accommodated by the same Z4 symmetry. The non-SM matter fields (i.e., the messenger 5's and X's and the light triplets) have the opposite charge to the SM matter fields under the unbroken Z2. Thus, there is no mass mixing between the SM and the non-SM matter fields. Dangerous proton decay operators are forbidden in this model by the discrete symmetries. Some higher dimension operators that lead to proton decay are allowed, but are sufficiently suppressed. We discuss these below. Renormalizable operators such as 10sM10SM5q and 10SM5SM5q are forbidden by the Z3 symmetries. This is necessary to avoid a large proton decay rate. A dimension-6 proton decay operator is obtained by integrating out the colored: triplet scalar components of 5q or 5q. Since the colored scalars in 5q and 5q have a mass ~ 0 ( 5 0 TeV), the presence of these operators would have led to an unacceptably large proton decay rate. The operators 10SMl 0SM10SM5SM/Mpl and 10SM10SM10SM5SM(~/M2pl) n/Mpl ' which give dimension-5 proton decay operators, are also forbidden by the two Z3 symmetries. The allowed non-renormalizable operators that generate dimension-5 proton decay operators are sufficiently suppressed. The operator 10SM10SM10SM5sMN~/(MpI) 2, for example, is allowed by the discrete symmetries, but the proton decay rate is safe since UN, ,-,o 1 TeV. The operators 10i5./~1 ()~ or f(q)/Mpl could, in principle, also lead to a large proton decay rate. Setting ~1 to its vev, the superpotential couplings, for example,

28

1'2.Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

& i ( U ~ D } 2 ( 3 ) + Q i L j X ( 3 ) ) are generated with A~i suppressed only by v~t,~/Mpt. In this model the colored triplet (scalar) components of X and Xq have a mass rn0 500 GeV, giving a potentially large proton decay rate. But, in this model these operators are forbidden by the discrete symmetries. The operator lOi5j~lffS/M~l is allowed giving a four SM fermion proton decay operator with coefficient ~ (v4hvs/Mp1)2 2/l,ng12 1 0 - 3 4 GeV -2. This is smaller than the coefficient generated by exchange of the heavy gauge bosons of m a s s M G U T , which is ,.o gGuT/MGuT 2 2 ,-o 1/2" 10 .32 GeV -2 and so this operator leads to proton decay at a tolerable rate. With our set of discrete symmetries, some of the messenger states and the light color triplets are stable at the renormalizable level. Non-renormalizable operators lead to decay lifetime for some of these particles of more than about 100 seconds. This is a problem from the viewpoint of cosmology, since these particles decay after Big-Bang Nucleosynthesis (BBN). With a non-universal choice of discrete symmetries, it might be possible to make these particles decay before BBN through either small renormalizable couplings to the third generation (so that the constraints from proton decay and FCNC are avoided) or non-renormalizable operators. This is, however, beyond the scope of this paper.

8. Conclusions We have quantified the fine tuning required in models of low energy gauge-mediated SUSY breaking to obtain the correct Z mass. We showed that the minimal model requires a fine tuning of order ~ 7% if LEP2 does not discover a right-handed slepton. We discussed how models with more messenger doublets than triplets can improve the fine tuning. In particular, a model with a messenger field particle content of three (l + 7)'s and only one ( q + g/) was tuned to ~ 40%. We found that it was necessary to introduce an extra singlet to give mass to some color triplets (close to the weak scale) which are required to maintain gauge coupling unification. We also discussed how the vev and F-component of this singlet could be used to generate the/z and B/z terms. We found that for some region of the parameter space this model requires N 25% tuning and have shown that limits from LEP do not constrain the parameter space. This is in contrast to an NMSSM with extra vector-like quintets and with one (5 + 5) messenger fields, for which we found that a fine tuning of a few percent is required and that limits from LEP do significantly constrain the parameter space. We further discussed how the model with split messenger field representations could be the low energy theory of an SU(5) x SU(5) GUT. A mechanism similar to the one used to solve the usual Higgs doublet-triplet splitting problem was used to split the messenger field representations. All operators consistent with gauge and discrete symmetries were allowed. In this model R-parity is the unbroken subgroup of one of the discrete symmetry groups. Non-renormalizable operators involving non-SM fields lead to proton decay, but at a safe level.

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

29

Acknowledgements The authors would like to acknowledge N. Arkani-Hamed, C.D. Carone, H. Murayama, I. Hinchliffe and M. Suzuki for many useful discussions. This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S, Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-90-21139. M.G. would also like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for their support.

Appendix A In this section the Barbieri-Giudice parameters for both the MSSM and NMSSM in a gauge mediated SUSY breaking scenario are presented. In an MSSM with gauge mediated SUSY breaking, the fundamental parameters of the theory (in the visible sector) are: Amess; At; /z; and/z 32" Once electroweak symmetry " breaking occurs, the extremization conditions determine both m2z and tan/3 as a function of these parameters. To measure the sensitivity of m 2 to one of the fundamental parameters Ai, we compute the variation in m2z induced by a small change in one of the Ai. The quantity 6m2z =-- c ( m2z ; ,~i ) ~ ~_.~i m2z Ai '

(A.1)

where

Ai ~gm~ c(m2; a,) = ,--~ Oa, '

(A.2)

measures this sensitivity [3]. In the case of gauge mediated SUSY breaking models, there are four functions c(m2z; hi) to be computed. They are c ( m 2 ; / X 2) = 2/z2 m2

×(,

tan 2 fl + 1 4 tan 2/3(/212 --/22) ) + (t-~n2/~--- i) 2 (/22 - / 2 2 ) ( t a n 2 / 3 7 i) --m--~z(tan2B-- 1)) ' (A.3)

tan2,8+1 c(m2z;/z32) = 4 tang/:¢ (t~n2 ~ _- B 3 4 /22 - / 2 ~ - tan2/3 m2 ,

/22-/22 m2z for large tan/3,

(A.4)

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

30

c(m2; A,) = 2 a2 0m2z Om~,, em ,, a a 2,

c9m2"

= - -4

2 tan 2/3 ( ~ 2 -- /d, ~22 tan 2/3 + 1 __ 9/2'1 m~A'tan2/3 1 oat2 ,1+-/2-~+/2---~ (ta-a-~--i-)aJ

40m~,,

for large tan/3.

~ m2z 0A~ '

(A.5)

This measures the sensitivity of m2z to the electroweak scale value of At, A t ( M w e a k ) . The Yukawa coupling A t ( M w e a k ) is not, however, a fundamental parameter of the theory. The fundamental parameter is the value of the coupling at the cutoff A° = McuT or Mpl of the theory. We really should be computing the sensitivity of m2z to this value of At. The measure of sensitivity is then correctly given by At(A°)

c(m2; At(A°) ) - At (Mweak)

c(m2; A t ( M w e a k ) )

OAt(Mweak)

3At(A °)

(A.6)

We remark that for the model discussed in the text with three l + [ and one q + messenger fields, the numerical value of (At(A°)/At(Mweak))OAt(Mweak)/OAt(A °) is typically ~ 0.1 because a,(Mweak) is attracted to its infra-red fixed point. This results in a smaller value for c(m2z; At) than is obtained in the absence of these considerations. With the assumption that m 2 and m2,~ scale with Amess 2 , w e get c(m~; Amess) 2 2 ,) + c(m2z;mH,,) 2 = c(m~; mH, /X2 tan 2/3 + 1 = l + 2 r n ~ -- (tan 2 / 3 _ 1 ) 2 4 tan2/3(m2H,, + m H 2 , , ) ( l-Z2 l - - t '~22 ) / m/ 2 Z

x (/22 _/22) (tan 2/3 + 1) - m2z(tan 2/3 - 1)"

(A.7)

The Barbieri-Giudice functions for mt are similarly computed. They are 1

2

c(mt;/x~) = ~c(mz; c( mt; ~ 2)

2

1

].g3) -'}- 1 -- tan 2/3'

7c(mz;/xl 2 2) +

~ 2/2~ +/22 tan2/3 _ 1'

At 1 Om~,, c(mt;at) =1 + ½c(m2;at) + tan2/3 - 1 /22 +/222 OAt ' c(rrtt;

2

1

2.

2

/212 q - / 2 2 __ 2/,£2

Amess) = 2c(mz' Amess) - (1 - tan 2/3) (/22 +/212)"

(A.8) (A.9)

(A.10) (a.11)

Since rnz and mt are measured, two of the four fundamental parameters may be eliminated. This leaves two free parameters, which for convenience are chosen to he Amess and tan/3. In an NMSSM with gauge mediated SUSY breaking, the scalar potential for N, H~ and Ha at the weak scale is specified by the following six parameters: Ai = mN,mH,2 2 m 2HJ,

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

31

the NHuHd coupling AH, the scalar NHuHd coupling AH, and the N 3 coupling, AN. In minimal gauge mediated SUSY breaking, the trilinear soft SUSY breaking term NHuHd is zero at tree level and is generated at one loop by wino and bino exchange. In this case, A H ( 2 i ) = 2HA(2i). Since the trilinear scalar term N 3 is generated at two loops, it is small and is neglected. The extremization conditions which determine mz = g2zv2/4 (v = ~ / v 2 + v 2 ) , tan/3 = V,/Vd and VN as a function of these parameters are given in Section 5. Eq. (22) can be written, using/z = 2HVN/X/~ as m2u + 2 ~--~H/z2-- AHANIu2sin2fl+ 2I 22v2 1 AHV22Hsin2fl=O. H -- 7-41z

(A.12)

Eq. (23) is 1 2 2 ~gz v + / z 2 --

2

tan 2/3

mH,, l ~-~a~-~ /3 +

m2

Hd 1

1 = 0. -tan2/3

(A.13)

Substituting v~ from Eq. (22) in Eq. (26) and then using this expression for /z23 in Eq. (24) gives

(m2H'+m2Ha+2tz2) sin2/3+A'H(m2N+'

~AHO''22,) + AH (

AN

2,An 41V222H 2/3~iZAN sin ]

= 0.

(A.14)

The quantity c = (2i/m 2) (,)rn2z/3&) measures the sensitivity of mz to these parameters. This can be computed by differentiating Eqs. (A.12), (A.13) and (A.14) with respect to these parameters to obtain, after some algebra, the following set of linear equations:

(A

+ AAH)X i =

B ~+ BiA,,,

(A.15)

where /Z 2 - - / Z 2

2 tan/3

1

v2

( 1 - tan 2/3) 2

1

23 1 -- tan 2/3 '2 2 0 (1 + tan2/3) 2

1

A=

23 ( 2H

- - AN sin 2/3) ~z-~u

2~

sin 2fly 2

1 -- tan 2/3

g2Z (]/,2 + /.~2) AN

/J'l2 -~" ]Z22

( 1 + t a n 2/3) 2

v2

'

(A.16)

AH AA. = --

tz

0 X

__ A~/sin2/3 2g~za2 A21 v2 sin 2/3

0

v2

A~ sin 2/3 v2 16A2 b7 ( A~ sin 213 v2

0

~'~

tanZ fl--I ( 1+tan2 fl) 2 4a~ tan2/3--1 A ~ v2

) ,

(A.17)

32

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3 - 3 4

10m 2

I

v20Ai

1 0tl,2

X ~II,'~N

(A.18)

V2 0 A i

'

0 tariff

am~ ##2

X rll2

(A.19)

(i = u , d , N ) ,

V20 tan fl with

2

2

1~i = m N, mrt,, , m~,,,

AM, AN, and 0

Bm~ + Bm~ All =

2 A2v a~

(A.20)

v2

a . 2(#12 + #2) tan2 fl

)

1 7~a~-fl +

An' =

0

_

u2

,

(A.21)

,

(A.22)

sin 213

2(#12 + #~)

/ 1 ) tan2 fl

//12~I

B'U~l'l + BA n' =

-- 1

0 __

V2

sin 2fl

2(#12 + t.2)

Ba"=

/

° /

A3 3 A~ sin2fl ,~H m2N - "~'--~N+ 4 AN A2 V2 1 (l m~v 3©2 A 2H'~

,

(A.23)

(o)

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

33

~2 sin 2/3

an

AH

~

BAH = "

0 ) { 0) 1 a~ sin 2/3

_

2

AH

BAH

-

-

~

(A.24)

aT7 7.-7

B]tN ..~

-

,

a u v 2 sin 2/3

(

ax

-

1 a4H

A~ m 2

(A.25)

2 x 1 ~(m2N + ~1v . 2 ~all) + .2) aN

A~sin2/3 4h3N a 2 U22_____ sin fl

.

(A.26)

In deriving these equations An (Ai) = h n A ( a i) was assumed and OA/OaH was neglected. Inverting this set of equations gives the c functions. We note that these expressions for the various c functions are valid for any NMSSM in which the N 3 scalar term is negligible and the NHuHa scalar term is proportional to AH. In general, these six parameters might, in turn, depend on some fundamental parameters, ~i. Then, the sensitivity to these fundamental parameters is ~i ~ ogaj Om2z ~ ~i Oa.] ~ = Ai Om2 m--~ c9~, -- m--Tz~ • aAi c~ami = ~ . ~J. c(m2;A]) " cgAi

(A.27)

For example, in the NMSSM of Section 5, the fundamental parameters are Amess, all, /IN, /it and Aq (At-1 is a function of ,in and Amess)- Fixing mz and mt leaves three flee parameters, which we choose to be Amess, AH and tan/3. As explained in that section, the effect o f / i n in the RG scaling of rn2H,, and m2H,~was neglected, whereas the sensitivity of m2u to AH could be non-negligible. Thus, we have e(m~; An) = c(rn2z; An) + c(m2z; m 2) AH 8m2N

(A.28)

m~ OAH" We find, in our model, that c(m2z; m2N) is smaller than c(m2z; aH) by a factor of ,-~ 2. Also, using approximate analytic and also numerical solutions to the RG equation for m 2, we find that (au/m~)(am2x/aan) is ~< 0.1. Consequently, in the analysis of Section 5 the additional contribution to ~(m2; an) due to the dependence of m 2 on An was neglected. A similar conclusion is true for aN. Also,

aq 0m2N

c(m2; "~q) = c(m2z; m2) m 2

aa/

(A.29)

34

K. Agashe, M. Graesser/Nuclear Physics B 507 (1997) 3-34

W e find t h a t ()tq/m 2) (OmzN/OAq) is ~ 1 so t h a t 0 ( m ~ ; Aq) is s m a l l e r t h a n 0(m~z; AH) b y a f a c t o r o f 2.

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