The Dilaton-Dominated Supersymmetry Breaking Scenario in the Context Of

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〉 PostScript processed by the SLAC/DESY Libraries on 30 Jan 1995. HEP-PH-9501393 e.mail address:[email protected] 1 h string uary 1995 Jan SUSX-TH-95/11 y of the solution of scenario es provided byCERNDocumentServer mo del al Scienc non-minimal brought toyouby 1 yp e soft sup ersymmetry breaking d Kingdom CT al sup ersymmetric standard mo del are the CORE tal results. The viabilit al and Physic breaking w top quark mass. As a consequence suc of ABSTRA University of Sussex, ery lo G. V. KRANIOTIS t exp erimen tro duction of a gauge singlet eld is also brie y discussed. dilaton-dominate d text ol of Mathematic Brighton BN1 9QH, Unite ys nd a v y recen a Scho text of the next to minim con e alw sup ersymmetric The the term through the in estigated. W v sup ersymmetry acua are excluded b in the v in The phenomenological consequences of theterms dilaton-t in the con

The Standard Mo del remains unscathed in its confrontation with exp eriment.

Furthermore, the compatibility of the mo del with the recent CDF observation [1], is

very encouraging and enhances our b elief that we are on the right path towards the

realization of the uni cation program.

However, in order to talk ab out the uni cation of all the forces wehaveto

see the standard mo del as the e ective limit of a more fundamental theory.Today

the only theoretical framework which is a strong candidate for the uni cation of

all the interactions, including gravity, is heterotic string theory. Sup ersymmetry

which is necessary in taming the radiative corrections to the Higgs b oson mass, the

only still unseen particle of the standard mo del, is naturally emb edded in string

theory. However, sup ersymmetry has to b e softly-broken at an energy of order of

the electroweak scale in order to solve the hierarchy problem mentioned ab ove. In

the Minimal Sup ersymmetric extension of the Standard Mo del (MSSM) the terms

resp onsible for the breaking are customarily parametrized in terms of four universal

parameters, M , A, m , B , the universal gaugino mass, the trilinear scalar terms

1=2

asso ciated with the trilinear couplings in the sup erp otential, the scalar masses, and

the B term asso ciated with the higgs-doublet mixing term in the sup erp otential. In

string theory these parameters are in principle, predicted but a de nite answer is at

present lacking due to the fact that the sup ersymmetry breaking mechanism in the

theory is not well understo o d.

Nevertheless progress has taken place at the theoretical level, and therefore

we b elieve that it is an appropriate time to use all the present existing knowledge

to study string inspired scenarios and confront them with current exp erimental data.

In this way phenomenological criteria may help us select the correct string vacuum

state among the plethora of equivalent string vacua that app ear at the level of string

p ertubation theory, the ma jority of them leading to unacceptable phenomenology.

The pro cess of sup ersymmetry breaking has to have a non-p ertubative origin

since it is well known that SUSY is preserved order by order in p ertubation theory.

However, very little is known ab out non-p ertubative e ects in string theory, partic-

ularly in the four-dimensional case. This has led the authors of [3], to parametrize

the e ect of SUSY-breaking by the VEVs of the F -terms of the dilaton (S ) and the

mo duli (T ) chiral sup er elds generically present in large classes of four-dimensional

sup ersymmetric heterotic strings . In a way, if sup ersymmetry breaking is triggered

by these elds (i.e., hF i6=0 or hF i6= 0), this would b e a rather generic prediction

S T

of string theory.

There are various p ossible scenarios for sup ersymmetry breaking which are

obtained in this mo del indep endentway.To discriminate among these we consider a

simpli ed expression for the scalar masses

2 2 2

m = m (1 + n cos ) (1)

i 3=2

In string p ertubation theory, b oth the mo duli and the dilaton are exact at directions of the

e ective p otential, leaving their VEVs undetermined. In ref. [3] this p ertubative degeneracy is

assumed to b e completely lifted by the non-p ertubative dynamics and VEVs for mo duli and dilaton

to b e induced. 1

with tan =hF i/hF i [3]. Here m is the gravitino mass and the n are the mo dular

S T i

3=2

weights of the resp ective matter eld. There are twoways in which one can obtain

universal scalar masses, as strongly desired phenomenologically to avoid large avor-

changing-neutral currents (FCNCs) [4]: (i) setting = =2, that is hF ihF i;or

S T

(ii) in a mo del where all n are the same, as o ccurs for Z Z orbifolds [3] and free

i 2 2

fermionic constructions [5]. In the rst scenario sup ersymmetry breaking is triggered

by the dilaton F - term and if the vanishing of the cosmological constant is imp osed

as a constraint, this results in a universal scenario for the soft parameters involved

[3]. In particular wehave:

A = M ; m = M (2)

1=2 0 1=2

As one can see the four-dimensional soft-parameter space is then e ectively

two-dimensional. In the strict-dilaton case the B term is no longer an indep endent

parameter but is given by

M =2m (3) B =

1=2

The ab ove universal scenario leads to a natural supression of FCNC. The

dilaton dominated scenario has b een studied in the context of the minimal sup ersym-

metric mo del [8] and in the context of the ipp ed SU(5) [9]. However, in b oth cases

the necessary Higgs mixing term is provided by a bilinear mass term. The relevant

term in the sup erp otential is of the form

W = H :H : (4)

1 2

The asso ciated soft-breaking term in the scalar p otential will have the form

BH :H (5)

1 2

It is well known that, in order to get appropriate SU (2) U(1) breaking, the

parameter has to b e of the same order of magnitude as the SUSY-breaking soft terms.

This is in general unexp ected since the -term is a sup ersymmetric term whereas the

other soft terms are originated after sup ersymmetry breaking. This is called \the

problem", the reason why should b e of the order of the soft terms.

Avery attractive solution to the problem [10 ] is to add an extra singlet su-

p er eld N which couples with the two Higgs doublets sup er elds. The resulting

sup erp otential will contain b esides the usual standard mo del terms the following

term.

W = N H :H + kN (6)

N 1 2

If, the gauge singlet acquires a vacuum exp ectation value hN i then the role of is

played by hN i and the role of B is played by A . The resulting mo del is the so called

nonminimal sup ersymmetric standard mo del. We note that such a sup erp otential

which contains only trilinear couplings emerges naturally in sup erstrings. 2

It is also very interesting to p oint out that in general the soft terms computed

in a general class of string mo dels are complex. The resulting phases, are quite

constrained by limits on the electric dip ole moment of the neutron (EDMN), since

they give large one-lo op contributions to this CP-violating quantity. In the case of

the MSSM, there are three classes of phases asso ciated with the parameters A, M ,

1=2

B and which are candidates for CP violation. These are given by

= ar g (A = );

A ij k ij k

= ar g (B=);

= ar g (M ) (7)

C a

where M are the masses of the gaugino elds asso ciated with the three gauge group

factors of the MSSM and are the trilinear Yukawa couplings of the chiral su-

ij k

p er elds of the theory and A the usual trilinear soft terms. However, it has b een

ij k

shown [6]that after a rede nition there are only two CP violating phases

= ar g (AM ); = ar g (BM ) (8)

A B

1=2 1=2

These phases are constrained as we mentioned by the electric dip ole moment of the

neutron. One has in fact

; 10 (9)

A B

for sparticle masses around few hundreds GeV. The EDMN receives imp ortant contri-

butions from b oth and phases. In the dilaton-dominated scenario in the MSSM

A B

it has b een shown that the phases for M and A coincide and ! 0 [3]. However,

a A

is in general large and can b e suciently supressed only if further assumptions

ab out the origin of SUSY-breaking are made [3, 7 ].

In the case of the nonminimal mo del, the dilaton-dominated scenario leads to

a natural supression of the EDMN. Indeed, if the problem is solved by the addition

of a singlet N , the role of B is played by a trilinear coupling A whose phase is aligned

with that of the gauginos and naturally = =0 . Thus the nonminimal mo del

A A

with sup ersymmetry breaking by the VEV of the dilaton eld is a well motivated

scenario that deserves further study. However, as we shall see, phenomenological

considerations exclude the mo del.

Avery attractive feature of sp ontaneously broken e ective sup ergravities is the

radiatively induced breaking of the electroweak gauge symmetry [13].For the study

of electroweak symmetry breaking we use the one-lo op e ective p otential

V = V +V (10)

0 1

where the radiative corrections to the scalar p otential

" !#

1 M 3

V = M ln (11) Str

2 2

64 Q 2 3

dep end on the Higgs elds through the tree-level squared-mass matrix M . The

sup ertrace in equation (11) is given by

2 2J 2

Strf (M )= (1) (2J +1)f(m ) (12)

where m denotes the eld-dep endent mass eigenvalue of the ith particle of spin J .

In the ab ove expression for V , all parameters of the theory are running parameters,

functions of the renormalization p oint Q. The use of the one-lo op e ective p otential

guarantees the scale-indep endence of the solutions [12]. The tree level p otential V

contains the standard F - and D - terms, and the following soft-sup ersymmety breaking

terms:

c 3

(M + M + M + h A Q:H U + A H :H N + kA N )+h:c:

1 1 1 2 2 2 3 3 3 t t 2 1 2 k

2 2 2 2 2 2 2 2 2 c 2

+m jH j + m jH j + m jN j + m jQj + m jU j + ::: (13)

1 2

1 2 N Q U

where , , and denote the gauginos of the U (1) , SU(2) and SU(3) gauge

1 2 3 Y

groups, resp ectively.

For the calculation of radiative corrections in (11)we takeinto account the

corrections due to top, and stop lo ops. This approximation is correct as long as

tan

t b t b

and tan is the ratio of the VEVs of the two Higgs doublets

tan = h H i=h H i (14)

2 1

It has b een shown that sup ersymmetry prevents the sp ontaneous breaking of CP [11],

so that the vevs of the elds H , H and N are of the form

1 2

! !

v 0

hH i = ; hH i = ; hN i = x (15)

1 2

v 0

with v ,v ,x real.

1 2

The equations for extrema of the full scalar p otential in the directions (15) in

eld space read

1 1

2 2 2 2 2 2 2 2

(g +g )(v v )] + v x(kx + A )+ @V =@ v =0; (16) v [m + (v + x )+

2 1 1 1

1 2 1 2 1 2

4 2

1 1

2 2 2 2 2 2 2 2

(g +g )(v v )] + v x(kx + A )+ @V =@ v =0; (17) v [m + (v + x )+

1 1 2 2

1 2 2 1 2 1

4 2

2 2 2 2 2 2

x[m + (v + v )+2k x +2k v v + kA x]+ A v v + @ V =@ x =0 (18)

1 2 k 1 2 1

N 1 2

Wenow describ e our numerical pro cedure. At the string uni cation scale the

following relations hold

g = g = g g

1 2 3 U 4

M = M = M M

1 2 3

1=2

h = h ; = ;k = k (19)

t U U

2 2 2

m m = M ; i =1;2;N;Q;U

i 0 1=2

A = A = A A = M

t k 1=2

We scan the parameter space of the mo del as follows:First wecho ose a set of

t 16

initial values for the parameters,h , , k , M at the scale of O(10 )GeV. Then

U U 1=2

by using the well known set of RGE at the one-lo op order weevolve the relevant

parameters down to the electroweak scale of O(100)GeV and insert their numerical

values into the minimization equations (16) (18). The latter are highly non-linear

algebric equations with indep endentvariables the three vacuum exp ectation values. In

order to solve them we use numerical routines from the NAG library which iteratively

converge to a solution. We start with an initial guess for the three vacuum exp ectation

values and the routines employed quickly converge to the solution for v ;v ;x.We

1 2

seek solutions where three nonzero vacuum exp ectation values develop. Our solutions

are nontrivially constrained, by the physical requirement that the correct mass for the

Z b oson must b e repro duced. Furthermore, in order to guarantee a global minimum,

we demand that all the physical Higgs b osons have p ositive mass squared and that the

vacuum exp ectation value of the Higgs p otential is negative and therefore energetically

preferable over the symmetric minimum. An essential check of the correctness of the

minimization of the one-lo op e ective p otential is the app earance of the charged and

neutral massless Goldstone b osons, which indicate that the charged and neutral Higgs

eld dep endence has b een included prop erly in the M matrices app earing in (10). The

running top quark mass obtained in our pro cedure is related to the exp erimentally

observable p ole mass by [14 ]

3 2

4 (m ) (m )

s t s t

pole

5 4

m (20) = m (m ) 1+ +K

t t t

where

1 K =16:11 1:04 (21)

i=1

where M , i=1,...,5, represent the masses of the ve lighter quarks. We solve (20) by

using the nonlinear equation solution routines describ ed ab ove for the minimization

of the e ective p otential.

The exp erimental constraints that we imp ose are summarized in table 1. We

pole

also require that m > 0toavoid a L6=0 vacuum [10] and that m > 131GeV [2].

~ 5

As it turned out the mo del failed to satisfy all the constraints imp osed from

radiative electroweak breaking and the exp eriment. The results of our research are

summarized in table 2. In particular we list values of the relevant parameters which

0 2

repro duce the correct Z b oson mass, m > 0 and gluino masses ab ove 120GeV .

pole

The single entry for m in table 2 indicates the maximum value of the physical top

quark mass obtained in our study.For our notation in table 2 see Ellis et al. in ref.

[10].

As one can see from the solutions of the renormalization group the values of

the trilinear Yukawa couplings of the mo del that give correct electroweak breaking

are very constrained. In particular the top Yukawa coupling is always small implying

avery low top quark mass in con ict with published exp erimental results [2]. Higher

values of the top Yukawa coupling require very small gluino masses in order that the

constraint of correct electroweak breaking is satis ed. Also higher values for h tend

to generate VEVs for the sneutrino eld. We do not list in table 2 lower values than

0.16 for the h coupling.The lightest Higgs mass eigenstate, m , is always b elow

U 1

the threshold of 60 GeV. Also, from the study of the vacuum exp ectation value of

the gauge singlet Higgs eld we conclude that the generated term is only a few

GeV, implying therefore a lightchargino mass,m , sometimes b elow the current ex-

p erimental limit. At this p ointwemust mention that it is p ossible to generate a

di erent set of VEVs, which give m 130GeV ,by starting with higher values for the

top and Yukawas. The feature of the latter set is the large vacuum exp ectation

value of the gauge singlet of O(1)TeV. However, this set of VEVs lead to an unstable

vacuum since hV (1 l oop)i> 0 and the Higgs mass squared eigenvalues are not

H iggs

all p ositive. This is in agreement with the work in ref.[15] where an upp er b ound for

the Yukawa has b een obtained, < 0:55. In our case the values of which

U U U

generates the large VEV for the gauge singlet are such that 0:65 for h 0:35.

All the ab ove facts make the nonminimal mo del with dilaton-dominated soft-terms

excluded by exp eriment. It is worthwhile to mention that the no-scale scenario with

the sup ersymmetry breaking driven by the F -terms of the mo duli elds is also not

a viable scenario in the context of sup ersymmetric mo dels with a gauge singlet [15].

Thus the sup ersymmetric mo dels with a gauge singlet Higgs eld seem dicult to

reconcile with universal no-scale scenarios. We remind the reader that no-scale su-

p ergravity is the infrared limit of sup erstring theory [16 ]. Alternative mechanisms

for the generation of the term exist in the context of string theory [3]. From the

phenomenological p oint of view string inspired univeral no-scale sup ergravity mo dels

with two Higgs doublets are preferable to mo dels with the extra gauge singlet Higgs

eld. However, the latter with more general soft-sup ersymmetry breaking terms can

lead to acceptable phenomenology though is dicult to have de nite predictions for

the particle sp ectrum due to the high-dimensionality of the parameter space [15]. On

the other hand, if the problem of the term in string theory is solved by the intro duc-

tion of a gauge singlet sup er eld this might lead to a departure from the universality

The dilaton-dominated scenario is a no-scale scenario driven by the F -terms of the dilaton

sup er eld. 6

of the soft SUSY breaking terms. .

Acknowledgments

I wish to thank my sup ervisor Professor D. Bailin for useful suggestions and for

reading the manuscript.

Non universal scenarios for the soft-terms emerge naturally on orbifold constructions [3]. 7

References

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Particle Exp erimental Limit (GeV)

gluino 120

squark,slepton 45

chargino 45

neutralino 20

light higgs 60

Table 1: Exp erimental constraints

Parameter jk j =0:01 jk j =0:1 jk j=1:0

U U U

h 0.16-0.182 0.16-0.184 0.16-0.189

0.16-0.35 0.19-0.35 0.34-0.57

h 0.47-0.52 0.47-0.53 0.46-0.54

0.2-0.403 0.24-0.405 0.34-0.5

jk j 0.01 0.082-0.093 0.43-0.49

m (GeV) 71-91 71 -90 70-91

pole

m 97 96 96

tan 1.42-7.6 1.41-4.51 1.4-3.1

r 0.09-0.52 0.21-0.52 0.34-0.5

m (GeV) 120-192 121-200 120-218

m (GeV) 2.3-37 3.12-39 9.2-51

m (GeV) 40-68 41-66 41-69

m (GeV) 12-67 28-68 43-71

m 23-35 23-38 9-39 (GeV)

m (GeV) 77-89 78-89 82-98

m (GeV) 6.6-11 29-36 62-103

m (GeV) 62-108 74-104 104-114

m (GeV) 10-39 21-39 12-44

m (GeV) 49-92 59-88 89-97

m (GeV) 95-108 95-108 108-134

Table 2: Typical parameter values that emerge from the renormalization group anal-

ysis which are consistent with correct electroweak breaking 9