Static quantities of the W boson in the MSSM

18 August 1994 PHYSICS LETTERS B

ELSEVIER

Physics Letters B 334 (1994) 378-390

Static quantities of the W boson in the MSSM A.B. Lahanas a,1, V.C. Spanos b,2 a

Nuclear and Particle Phystcs Section, Physics Department, Untversity of Athens, lhssta, GR - 15771 Athens, Greece b NRCPS 'Democrttos', Instttute of Nuclear Phystcs, Aghta Paraskevi, GR - 15310 Athens, Greece

Received 20 May 1994 Editor' R Gatto

Abstract

We systematlcally analyze the anomalous dipole Akv and quadrupole AQv moments of the W gauge bosons m the context of the minimal supersymmetric standard model as functions of the soft SUSY breaking parameters A~ m~ M1/2 and the top quark mass. The severe constraints imposed by the radiative breaking mechanism of the electroweak symmetry SU(2) × U(1) are duly taken into account. The supersymmetric values of Akv and AQv can be largely different, in some cases, from the standard model predictions but of the same order of magnitude for values of A@m~ MI/2 ~
One of the most crucial tests of the standard theory of electroweak interactions will be the study of the three gauge boson couplings to be probed in forthcoming experiments in the near or remote future. Although there is little doubt that the nonabelian structure of the standard model is the right framework for describing the electroweak phenomena at low energies, in the vicinity of the electroweak scale, nevertheless we are still lacking a direct experimental verification of it. Such a study will take place at LEP2 as well as in next round experiments at HERA, NLC, LHC, with high accuracy putting bounds on the dipole Akv and quadrupole AQ~ form factors which are directly related to the anomalous magnetic m o m e n t / z w and the electric quadrupole moment Q w o f the W - b o s o n [ 1 ]. The standard model predicts Akv = AQv = 0 at the tree level but higher order corrections modify these values by finite amounts that can be tested in the laboratory with high accuracy of the order o f 1 0 - 2 - 1 0 -3 [2]. Such measurements can be o f vital importance not only for the self consistency of the standard model but also for probing possible structure beyond that of the standard theory signalling the presence of new physics. The last years there has been a revived interest towards supersymmetric extensions of the standard model. The supersymmetric structure seems to be a necessary ingredient in the efforts towards embedding the standard model in larger schemes unifying all existing forces o f nature and is also suggested by precision data on the gauge couplings E-mail address: alahanas@atlas uoa.ariadne-t gr 2 E-mail address: [email protected]. 0370-2693/94/$07.00 © 1994 Elsevier Science B V All rights reserved SSD103 70-2693 ( 94 ) 00741 -o

A.B. Lahanas, V.C.Spanos/ PhystcsLettersB 334 (1994)378-390

379

al, a2, ct3 which merge at a unification scale M G u T ~-- 1016 GeV, provided the SUSY breaking scale Ms lies in the TeV range [3,4]. Such relatively low values for the supersymmetry breaking scale - T e V may have important consequences for phenomenology. The TeV scale may be the onset of new physics and detection of supersymmetric particles with masses ~
(

1)

where Sasosv is its supersymmetric part derived from a superpotential 7/" bearing the form [ 13]

~g/'= (huO_'fflJ2fJC+hoa't-IJff)C+h~L'IYI~E c+/x/-I~/-IJ)E,j,

~12 = + 1

(2)

and ./asoft is its supersymmetry breaking part given by -.O~soft = ~] m21~, 12+ (huAvQH2UC+hDADQHIDC+heALLHIE¢+h.c.) t

1 + (lzBHIH2 +h.c.) + ~ y '

Ma~a)~ a .

(3)

a

In Eq. (3) the sum extends over all scalar fields involved and we have suppressed all familly indices. In our analysis we assume universal boundary conditions for the soft masses at a unification scale MGtrr = 1016 GeV. The evolution of all couplings as well as all soft masses and the mixing parameters/z, B from Mctrr down to energies E in the vicinity of the electroweak scale is given by their renormalization group equations (RGE) [6,14]. These are known up to two loop order [ 15]. As arbitrary parameters of the model we take the running top quark 3Even in Ref [ 12] the SUSY breakingeffects are actually ignored since only the supersymmetriclimit of the MSSM is considered in the discussion of the physicalresults.

380

A B. Lahanas, V C Spanos/ PhysicsLettersB 334 (1994) 378-390

mass m,(Mz) 4 at the Z-boson mass the angle/3 defined by tan/3=v2(Mz)/vl(Mz) and the soft SUSY breaking parameters Ao, mo, M1/z at M6trr. Vl.z(Mz) are the vacuum expectation values of the Higgses H1,2 and the values of the trilinear scalar couplings Ao, common scalar mass mo and common gaugino mass M1/z are meant at the unification scale Mctrr. In this approach, which has been adopted by other authors too [ 17,18], B,/z are not free parameters; in fact their values at Mz are determined through the minimizing equations of the scalar potential, M2z rh 2 -rh22 tan2/3 -7" = tan2/3 - 1 ,

sin 2/3=

2B/z n~2 +n~22 •

(4, 5)

In Eqs. (4) and (5) the masses appearing are defined by -2_

ml,2

2 --ml,2

OAV

-~-

- -

0u2,2 '

2 _

2

ml,2 - - m i l l , 2

..F].L2

(6)

where AV accounts for the one loop corrections to the effective potential which should be included in the minimizing equations. If they are not the results are known to depend drastically on the choice of the scale becoming ambiguous and untrustworthy [ 19-17]. Eqs. (4) and (5) cannot be analytically solved to obtain the values B(Mz), t~(Mz) and our numerical routines have to run several times to reach convergence. In solving the RGE's we have also to properly take into account the appearance of the thresholds of the various particles opened as we approach low energies. We have duly taken care of these effects along the lines of Ref. [20]. We have found that their presence little upsets the picture as long as one loop corrections to the Akv and AQ~ are concerned. The supersymmetric limit of the model is realized when all soft SUSY breaking terms vanish, Ao =mo = M~/2 = 0, the v.e.v's Vl and v2 become equal and the mixing parameter/.~ vanishes. It is in this limit that particles and their superpartners get a common mass. In that limit the quadrupole moment (AQv) susv vanishes but the same does not happen for the dipole moment (Akv)susv. For the latter the difference Akv-(L~kv)susv is a measure of the importance of SUSY breaking effects on the dipole moment. After this introductory remarks concerning the MSSM and an outline of the numerical procedure we shall follow we embark on discussing separately the contributions of the various particles involved to the quantities of interest. The most general W + W - y vertex consistent with current conservation can be written as F ~ B = - ie (f[ 2g=~A~ + 4 ( g ~ Q~ - g ~ Q~) ] + 2 A k ~ ( g ~ Q ~ - g ~ Q ~ )

+4~

Au(Q~Q/3-½Q2g,~))+ ....

(7)

where the W's are on their mass shell and the ellipsis denote C, P violating terms. The labelling of the momenta and the assignment of Lorentz indices is as shown in Fig. 1. To lowest order f = 1, Akv= AQv=0. The values of the form factors Ak,, AQ~,at zero momentum transfer are related to the actual magnetic dipole moment/Zw and electric quadrupole moment Qw by [ 1,2,9,10] e gw = ~ ( I + K r + A ~ , ) ,

Qw =

e M E (K~'--)t:')'

where Akv= Kv+ A v - 1 and AQ:,= - 2A~. The standard model one loop predictions for the dipole and quadrupole moments have been calculated and can be traced in the literature [ 9,10]. The contributions of gauge bosons and matter fermions of the MS SM are identical to those of the SM. These are displayed in Table 1 in units of g2/167r z. For the contributions of the fermions we agree with the findings of other authors as far as the isospin 7"3= - 1/2 fermions are concerned but we disagree in 4 The physmaltop quark massM, is definedby Mt= rnt(M,)/ [ 1+ 4aa(Mt)/3~'] when the one loopcorrectionsare takeninto account [ 16].

381

A.B Lahanas, V C Spanos / Physics Letters B 334 (1994) 378-390

-~(2Q)

7, 2Q

~

i

-- Q

w:

/

Q

T+

~..~--A w;

-~(2Q)

T_

B~+(p)

WZ- (P')

T_

W+(P)

(~)

Wa'- (P')

(b)

F~g I The W_ W+ 3, vertex Lorentz indices and momentum assignments are as shown m the figure Fig. 2. Triangle graph (a) and its crossed (b) contributing to the dipole and quadmpole moments. Q, T+, T_ denote electric charge and tsospm raising and lowering operators respectively.

the sign of the T3 = + 1/2 chiral fermion contributions. This means for instance that " u p " and "down" quarks' contributions to the dipole/quadrupole moments have the same sign despite the fact that they carry opposite electric charges. This is due to the fact that the triangle graph with the up quark coupled to the photon and the corresponding one with the down quark playing that role are crossed; the dipole and quadrupole terms however change sign under crossing, resulting to a same sign contribution, unlike the anomaly which preserves its sign yielding = Trace(Q). Since this is a rather delicate point which, we think, has been overlooked in previous works we shall discuss it in more detail. The fermionic Lagrangian relevant for the calculation of Ak v and AQ~ is -~=~g

( W+ ~LT~T+ ~L +h.c.) +eA~,(~LQ3,~L) .

(8)

In this ~L is a column involving all left handed fermions, T± = T1 + iT2 are weak isospin raising/lowering matrices and Q is the diagonal electric charge matrix. The graphs we have to calculate are shown in Fig. 2, where p, p' are the momenta carried in by the external W's. The Feynman integral of the graph shown in Fig. 2b has a momentum dependence which follows from that of Fig. 2a under the interchange c~~ fl and p ~ p ' or equivalently Q~ ~ Q., A . ~ - A.. One is easily convinced of that by explicitly writing down the expressions of the two graphs shown in Fig. 2. Therefore one has Graph(2a) =Tr(T+ T_ Q ) V ~ ( Q , A),

Graph(2b) =Tr(T_ T+ Q)V~,~(Q, - A ) ,

where the tensor structure of V . ~ ( Q , A) including the axial anomaly term is

V ~ ( Q , A) = aoe~f~aAa +/31g,~A~ +/32(g~Q~-g~,Q~) +/33A~Q,~Q~ + ....

(9)

The first term is the anomaly and/31,2.3 are form factors where the contributions to the charge renormalization, dipole and quadrupole moments are read from. The anomaly term does not flip its sign under ct ~/3, Q . ¢ Q., A . ~ - A . but the remaining terms do. As a result we get for the sum of the two graphs Trace(Q{T_, T+ } ) e ~ a A a + Trace(Q[T+, T_ ])(/31g~A~ +...) ,

(10)

382

A B Lahanas, V.C. Spanos IPhysicsLetters B 334 (1994) 378-390

Table 1 Radiative corrections to the dipole Akv and quadrupole moments AQv m units of g 2/( 161r2)

Akv= ~ sin20w, AQv= i9 sin20w

gauge bosuns

1

f

20

Z:

Akv= "~ -~+½

dt

t4+lOt3-36t2+32t-16 t2+R(l_t )

o

AQv=(.~R+ 1)i dt t2+R(1 t3(l-t)- t ) o

(R : (Mz/Mw)

leptons, quarks:

2)

Ak,= C---~Qf,i d t t 4 + ( r / - r / ' - l ) t 3 + ( 2 r f ' - r / ) t 2

2

o

']

t2+(rf'--r/--l)t+r/ --[f~f

(~,)L SU(2) doublets

AQ,=

2c_~ Q:,i tit"t2 + (rf,t3(1--t) 3 -- rf-- 1)t+ rf -- [ f ~ f o

']

(r/..,=-(mH,/Mw)z) sleptons, squarks:

Akv= -CgQf,

r v ] vi'~2 .... , , - j , , ~,J~ 1

i dt t2(t-1)(2t-l+Ry:-R~) t2+ro___O__lXt+o_ -[f'~f'] t, alf~ alfl I J'fj 0

(~,)L SU(2) doublets

AQv=-2CIQf'3 ~ tvlvl"2i . . . . l,Xjl, t,J-- 1

dt

tz+,,,/;t3(l ,a-. -t)-R:

l)t+Rl

0

-- [ f ~ f '1

R~j , =-( m~j , I Mw) 2; my.jr are sfermion masses In the SUSY limit K.f,p ~'- become unit matrices and R]~,~= r:, R]~2 = rf,

higgses.

(a) broken SUSY

(H+,Ho, ho, A)

A"

Ak7 =D2(RA, R+ ), AQ~ = Q(RA, R+ )

ho" Akv = sin20 Dl(Rh) + cos20 D2(Rh, R+ ) AQv=sin20 Q(Rh, 1) +c0s20 Q( Rh, R+ ) Ho:

As in ho with Rh--'R~ and sin20~ c0s20

Ro =-(mJMw) 2, a=ho, Ho, A, H ~=

(b)exactSUSY

A, ho: Akv=-~, +~t, AQv=~, ~ 1

Ho.

Ak~=~+½

f

t4--2t3

dtt=+R(l_t),AQv=~

o

(R= (Mz/Mw)2)

1

I

o

t3(l-t)

dtt2+R(l_t )

383

A B Lahanas, V C Spanos/Phystcs Letters B 334 (1994) 378-390

Table 1 (Continued) (c) standard model, Higgs contribution

Akr= Dl ( 8 ), AQr=Q ( 8, 1) (8= ( mr~gsJ Mw) z)

Dt(r)=-~

i . 2t4+(-2-r)t3+(4+r)t clt t2--~r(l-_- ~

2

0

D2(r, R) ~ ½i dt 2t4+ ( - 3 - r + R ) t 3 + (1 + r - R ) t 2 t2+ ( - 1 - - r + R ) t + r 0 1

Q(r,R)='~

f

t3(l - t) dtt2+(-i--"~-R)t+ r

0

neutrahnos (Z~,, ct= 1, . , 4), charginos (C,, t = 1, 2)

(a) broken SUSY

Akr= - ~_, F,~ ) dt t4+ (R,,-R, - 1)/3+ (2R, - R ~ ) t 2 t 2 + ( R , - R ~ - 1)t+Ro t,o, 0 4t 2 _ 2t + ~"slgn(m,m,~)Go, Rf~R~ d t t 2 + ( R _ R , _ l ) t + R i,a

~

o

4 ).., Fo, ) t3(1-t) A Q = - -~ dt t2 + (R, - R , , - 1)t+R,~ t,~

0

(R~,,,=- (m~.,/Mw) 2) (b) exact SUSY

Y/

A k r = - ~3 sin2Ow, A Q r = - ~9 sm20w

1

Akr=16_ 8_8_ f dt 3R 0

A Q r = - "~ +

t4+3ta-15t2+12t--4 t2 + R( 1 - t)

t2+R(l_t) 0

(R = (Mz/Mw) 2)

The symbols are explained in the main text

where for the sake of the argument we have suppressed the fermion mass dependence entering into fll.2.3 by assuming that all fermions have the same mass. In the standard model { T_, T+ } = constant × 1 and the anomaly is proportional to Trace(Q) which vanishes, a well known result. The second term of Eq. (10) however yields a contribution proportional to Trace(QT3), when all fermions have the same mass. From this it becomes obvious that " u p " and " d o w n " quarks's contributions to the dipole quadrupole moments have the same sign since they carry opposite isospin and electric charges. Since the fermion masses are different the graphs of Fig. 2 yield, ignoring the anomaly term,

~, (QyT})V~a~(Q,A, m}, m},),

(11)

f

where f, f ' are left handed fermions belonging to doublets of the weak SU(2). The exphcit expressions for Ak v, AQv of each fermion doublet (yY,)L are given in Table 1, where they are expressed in terms of the dimensionless

384

A.B Lahanas, V.C.Spanos/Phystcs LettersB 334 (1994)378-390

ratios rs~f, = (mcs,/Mw) 2. The factor Cg appearing in these formulae is the color factor, one for the leptons and three for the quarks. The discussion of the sleptons and squarks is complicated by the fact that the superpartnersj~L, f ~ of the known fernaionsfL, f ~ are not mass eigenstates. This is the case for the third family fermions and especially for the stops tL, t~ due to the heaviness of the top quark which results to large t'L, t-~ mixings. The first two families have small mass and such mixings are not large. In the stop sector the relevant mass matrix squared has the following form:

12=(m~3+m2t+M2(cos2fl)(½-2 mt(A+ ~ cot/3)

sin20w)

mt(a+/x cot/3) ) m ~ +m~ +g2z(COS 2/3)(~ sin20w) '

(12)

which is diagonalized by a matrix K;, i.e. K;..~ ~K ;r = diagonal. Since the top mass is large, K;deviates substantially from unity. In the same way the sbottom and stau mass matrices get diagonalized by K b, K e. However in that case the correspondingj~L,j7 ~. mixings are not substantial. The sfermion contributions to Ak~, AQv are presented in Table 1. They are expressed in terms of the diagonalizing matrices discussed previously and the dimensionless ratios Ri, = (m~/Mw) 2, where i = 1, 2 runs over the mass eigenstates. In the supersymmetric limit the matrices K : i ' corresponding to the sfermion doublet (~,)L are unit matrices and myi.2 (m: 1.2) become equal to the fermion mass my(my, ). In our numerical analysis we have taken only the top quark mass to be nonvanishing and therefore the aforementioned discussion regards only the stops. The inclusion of the bottom and tau masses little affects the quantities Ak~, AQ~, owing to the fact that the induced mixings in the corresponding sbottom and stau sectors are small. In Table 1 we also give the dipole moment in the supersymmetric limit discussed earlier. Notice that in that limit the fermion/sfermion contributions to the quadrupole moment of each family separately cancel against each other. We turn next to discuss the Higgs sector. In the MSSM we have two doublets of Higgs scalars H~,/-/2 giving masses to " d o w n " (/'3 = - 1/2) and " u p " (/'3 = 1/2) fermions when they develop nonvanishing v.e.v's as a result of electroweak symmetry breaking. The relative strength of their v.e.v's is parametrized by the angle fl defined as tan fl=v2/vl. These are running parameters depending on the scale, as is the running Z-boson mass m 2z= (g 2 + g ,2) ( v ~ + v 22) / 2. The physical Z-boson mass is defined as the pole of the Z-boson propagator occurring therefore at the point Mz for which mz(Mz) = Mz. Experimentally we know that Mz = 9 1.2 GeV. As a result of the electroweak symmetry breaking we have five physical Higgses A, ho, Ho, H+ with the following masses: A (neutral): m~ = m~ + m~, Ho, ho(neutral): m H,h 2 = ½[(m~ +M~) 2-1-~/(m~ +M2z) 2 - 4M2zrn~ cos22fl] H± (charged): m2H± = ma2 +M~v. The lightest of the neutral Higgses ho has always a tree level mass smaller than Mz. However it is known that radiative corrections due to the top/stops can be large, due to the heaviness of the top quark [ 2 1] and shift its upper limit to values that can exceed Mz. In that case ho can escape detection at LEP2. In the supersymmetric limit ho and the pseudoscalar A become massless, H± has a mass Mr,, and Ho a mass Mz. The Higgs contributions to the dipole and quadrupole moments are shown in Table 1 as functions of the dimensionless parameters R,~ = (m~,/Mw) 2, ot = ho, Ho, A, H± and a mixing angle 0; for sin20 = 1, ho becomes the standard model Higgs scalar 5. In the standard model only one neutral Higgs survives the spontaneous electroweak symmetry breaking having mass mH~gg~whose contributions to Ak~, AQ~, are expressed in terms of the ratio 8= (mmggJMw)2; for comparison these are also given in Table 1 along with the Higgs contributions in the supersymmetric limit. 5 sin20 = (me + MZzsin2 2 / 3 - m h~ ) / ( m ~ - m h 2 ). NO dependence of the Hlggs contributions on sin 0 appears in the results cited in Ref. [ 10], however we agree with the expressions given by the same authors m Ref [ 13] for the limiting cases discussed in that paper

A.B. Lahanas, V.C SpanosI Phystcs Letters B 334 (1994) 378-390

385

The neutralino and chargino sectors have the most complicated structure and will be discussed in more detail. Their weak and electromagnetic currents are given by J~+ = ~] 2~aT~(PRC~ + P L C ~ ) C , ,

Je~m= ]~ C,T~'C,,

~¢,1

(13)

I

where the sums are over neutralino, a = 1, 2, 3, 4, and/or chargino, i = 1, 2, indices and PR.L are the right and left handed projection operators ( 1 + 3'5)/2. Z,~, a = 1, 2, 3, 4 are four component Majorana spinors eigenstates of the symmetric neutralino mass matrix given by

Ma 0 girl/hi2

.,g,,N=

0 Mz -gvl/~12

g'v,/~12 -gv'/gr2 0

t-g'o21

-g'v2/V~' gv217r2 -I.~ '

(14)

o

-

while the charginos C,, i = 1, 2 are Dirac fermions, mixtures of winos ~'+ and Higgsinos/4+ and they are eigenstates of the chargino mass matrix,

.# _( M2

-gvz]

c-~-gv~

(15)

ix 1"

The right and left handed couplings C R appearing in (13) are given by 1 =

- -

O3,mU,2--

,

Co, = +

1

04,~V~2 - 02,,V,* •

(16)

The real orthogonal matrix O diagonalizes ¢gcN, and the unitary matrices U, V diagonalize --~'o i.e. O T •~ N O = diagonal and U..Cfc Vr -- diagonal. The neutralino and chargino contributions to the dipole and quadrupole moments are displayed in Table 1. For AQv we are in complete agreement with the findings of Ref. [ 12] but for Ak~,we disagree in the first term of the equation for Akv appearing in Table 1. However our result in the limit of vanishing right handed coupling, C R = 0, receives a form which up to group factors is exactly the same as that of a massive fermion family as it should. This consists a check of its correctness. The sign(m,m~,) in the second term of Akv takes care of the fact that we have not committed ourselves to a particular sign convention for the chargino and neutralino mass eigenvalues m, and m,~. The prefactors F,~, and G,~, appearing in the expressions for Akv and AQ~,are defined as F~ = i c ~ 1 2 + 1

L z, C,~l

G a , = ,¢cLc~* -~v,~ +h.c.)

(17)

In the supersyrnm._etric limit two of the neutralinos states, namely the photino ~/= (cos 0w)/~ + (sin 0w)if'3 and the axino t i = (1/X/2) (,~1 +,~2 ), have vanishing mass 6. The first combines with the photon and the second with the scalars ho, A to form supersymmetric multiplets of zero mass. The other two linear combinations ti x = ( 1/ V/2)(-X1 +,~2), orthogonal to ti, and the " Z i n o " Z = - ( s i n 0w)/~+ (cos 0w)ff'3, which is orthogonal to the photino state, have both mass Mz. These two form a new "Zino", ~', which is a Dirac fermion describing thus four physical states of mass Mz. This combines with the heavy neutral Ho and the Z-boson to forming a multiplet having mass Mz. Also in that limit we have two charginos, Dirac fermions of mass Mw, which along with the charged Higgses H± and the W-bosons belong the supermultiplets of mass Mw. This is how supersymmetry manifests itself in the SUSY limit discussed previously. In that limit the contributions of fermions and bosons of each multiplet to AQ:, cancel each other as can be seen from Table 1. Such a cancellation however does not hold for the dipole moment as is well known [ 11 ]. 6

X1.2 are phaserotated Higgslnofields,~1.2= t/~t.2.

A B. Lahanas, V.C. Spanos / Phystcs Letters B 334 (1994) 378-390

386

Table 2 The dipole and quadrupole moments, in units of g2/(167r2), for the input values mt(Mz), tan fl(Mz), Ao, mo and M~/2 shown below the table

Ak~, ~>0

q,l W,T,Z ho, H±.o, A

#,i total SUSY hmit standard model

AQ~, ~<0

~>0

- 1 973 1.179 0 945[0 945] (0 946) -0.004[-0.024](0.009) -0013[-0032](-0035) 0.01310015](0697) -0003[-0005](0.026) 0 15910 143](0859) 0 13510 115](0 143)

~<0

1 922 0.235 0 029[0.028] (0 028) 0.009[0 019] (0.027) 0.00910.019] (0 025) - 0 0 1 6 [ - 0 . 0 1 7 ] ( - 0 . 5 9 2 ) - 0 0 1 4 [ - 0 0 1 4 ] ( - 0 170) 2.17812.1881 ( 1 621 ) 2 18012 190] (2.041)

AksUsv = 1 237

AQrSUSV=0

AkTM =0.188, - 0 106, - 0 . 4 4 9 AQ TM =2 186, 2.174, 2.161

For comparison we display the standard model predictions for Hlggs masses 50, 100 and 300 GeV, the corresponding moments in the supersymmetnc limit are also shown m,= 160, tan 13= 2, Ao, mo, M~/2 300, 300, 300 [0, 0, 300] (300, 300, 80)

We pass now to discuss our physical results. As we have already mentioned in the beginning tan fl(Mz) and mt(Mz) are considered as free parameters; given these the top Yukawa coupling at Mz is known and so is its value at the unification scale McuT----1016 GeV. At all scales we have imposed the perturbative requirement h2/ (4~-) ~/15); accepting nonvanishing bottom and Yukawa couplings produces only minor changes to the one loop dipole and quadrupole moments. Then starting with Ao, mo, M~/2 at MtuT we run the RGE's of all parameters involved except those of B and/z whose values B(Mz), tx(Mz) are determined through the minimization conditions (4), (5). Their values at any other scale are found by running their corresponding RGE's using their near decoupling from the rest of the renormalization group equations. For their determination we properly take into account the radiative corrections to the effective potential as we have already discussed. For a given pair of tan/3(Mz), mt(Mz) the parameter space Ao, mo, M~/2can be divided into three main regions, (i) Ao, mo, M~/2 comparable. This includes also the ddaton dominated SUSY breaking mechanism with vanishing cosmological constant [ 24 ]. (ii) M1/2 >> Ao, mo. In this case the gaugino mass is the dominant source of SUSY breaking. This includes also the no-scale models (see Lahanas et al. [ 14] ) which in the physically interesting cases favor values Ao = mo = 0. (iii) Ao, mo >> M1/2.This includes the light gluino case. We have explored the entire parameter space for values of the top mass ranging from 130 to 180 GeV and in Table 2 we present sample results covering the previously discussed cases. In every case appearing in that table the dominant SUSY breaking terms have been taken equal to 300 GeV. One observes that the MSSM values for the quantities of interest, and especially for Aky, in some particular cases can be largely different from those of the SM but of the same order of magnitude indicating that possible supersymmetric structure is hard to be detected at LEP2. This behaviour characterizes the most of the parameter space provided that the typical SUSY breaking scale is m the range O ( 100 GeV-1 TeV). The distinction between SM and MSSM predictions can be therefore made detectable once the accuracy reaches the level of O ( 1 0 - 2 ' - 3 ) . In general the contributions of each sector separately to dipole and quadrupole moments, in units of g2/16~.2, are as follows:

387

A.B. Lahanas, V C. Spanos / Phys,cs Letters B 334 (1994) 378-390

20 f ............................................ ..----<-°2 ...... ::::::::::::::::::: ............

/../,) 0

1.0 ~

2.5 2.0 ..................................... (a) 1.5 1.0

0.5

0.5

00

(,,) (P)

0.0

-0.5 400

800

800

morea O

.(.P2......

1000

-0.5

I

i

r

400

600

800

1000

l~to= A 0

(b) Fig. 3. (a) Dipole (solid line) and quadrupole (dashed lme) moments,in units of g2/(161r2), as functions of too,Ao for M1/2= 80 GeV The cases shown are for m, = 140 GeV ((~) and m t 160 GeV (/3). For conveniencewe have taken mo= Ao. The sign of the parameter p. is positive. The honzontal lines are the standard model predictions for mr~sss= 100 GeV and m,= 160 GeV (b) Same as in (a) for negative sign of the parameter/z =

The matter fermions contributions to Akv are of order O ( 1 ) and negative while those to AQ~ are of the same order of magnitude but positive. The gauge bosons contributions to Ak~ is of the same order as that of the fermions but opposite in sign. However those to the quadrupole moment AQv are almost an order of magnitude smaller. The supersymmetric Higgses yield Akv~ O ( 1 ) , A Q ~ O( 10 -2) which little deviate from the SM predictions provided the standard model Higgs boson has a mass in the range ---50-100 GeV. Actually in the MSSM with radiative electroweak breaking one of the Higgses, namely ho, turns out to have a mass in the aforementioned range; this is predominantly the standard model Higgs. The rest have large masses, of the order of the supersymmetry breaking scale, giving therefore negligible contributions. The squark and slepton sector is the one yielding the smallest contributions. These are of order - O( 10 -2) or less having therefore negligible effect on both Akv and AQv. It remains to discuss the effects of the charginos and neutralinos to the dipole and quadrupole moments. In general this sector gives -- O( 10 -2) to both Ak:, and A Q r However in some cases and for positive values of the parameter /~, Akv can be substantially larger, -- O ( 1 ) . This occurs only when M~/2 << mo, Ao, case (iii), and the solution for as given from Eqs. (4) and (5) happens to allow for light chargino and neutralino states with masses me, m~ such that m e + m z = M w . In such cases the values of Ak,, AQv are enhanced and a structure is observed 7 (see Fig. 3). However even for such relatively large contributions of this sector we can not have values approaching the sensitivity limits of LEP2. Only in a very limited region of the parameter space and when accidentally m c + m2 turns out to be almost equal to W-boson mass, the chargino and neutralino contributions can be very large saturating the sensitivity limits of LEP2. We disregard such large contributions since they are probably outside the validity of the perturbation expansion. Even if it were not for that reason these cases are unnatural since they occupy a very small portion of the available parameter space which is further reduced if the lower experimental bounds imposed on the chargino mass are strictly observed. The results presented in Table 2 are representative of the cases discussed previously. To simplify the discussion the dominant SUSY breaking parameters have been assumed equal. Notice the large chargino/neutralino contri7 In those cases the integrations over the Feynman parameter are of the form f~f(t) dt/[ ( t - a)2 + 6.2 ] with 0 •

(X <(

1 and ~ small

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25

2.5

2.0

2.0

I

d

I

15

15

#)o

10

1.0

05

0.5

0.0

0.0 ,

-0.5 2

I

I

4

6

i

1

8

-0.5 10

t ,

4

4

,

i

6

tanfl

tan#

(a)

(b)

,

QI

8

10

Fig. 4 Dipole (solid line) and quadrupole (dashed hne) moments, in units of g2/(16¢r2), as funclaons of tan fl for p,> 0 (a) and/x < 0 (b) In both cases Ao =mo = 300 GeV, M,/2 = 80 GeV and mt= 160 GeV.

bution to Ak~, and AQ v in the case where M,/2 = 80 GeV, and mo, Ao = 300 GeV. Smaller values for MI/2 c a n lead to even larger contributions resulting to moments Ak,, AQ~, which may approach the sensitivity limits of LEP2. However the lower experimental bound put on the chargino mass ( >/45 GeV) constraints the situation a great deal not allowing for arbitrarily low Mº/2 values. In the same table the supersymmetric values Ak susY, AOSUSY~rare also shown. For the latter we know that AQ~,usY = 0. Its vanishing merely serves as a check of the correctness of the calculations. ~Ak y suse is nonvanishing however receiving the value 1.273(g2/167r 2) for m,= 160 GeV. For a comparison of our results we also display the standard model predictions for Higgs masses mmggs= 50, 100 and 300 GeV. In order to examine the behaviour of the dipole and quadrupole moments with varying the supersymmetry breaking scale we plot in Fig. 3 Akv, AQv as functions ofmo, Ao for the physically interesting case M1/E << mo, Ao. To compare with the previously discussed cases we have taken M1/z = 80 GeV and mo =Ao ranging from 200 GeV to 1 TeV for both/z > 0 and/~ < 0. Also in order to see how sensitive are our results to the top quark mass we plot Ak v, AQ v for mt= 140 GeV and m,= 160 GeV. The two cases yield almost identical results showing in a clear manner that are insensitive to the choice of the top quark mass. They differ appreciably when we are close to values of too, Ao for which a dip in Ak v, AQ~, is developed, occurring when m, = 160 GeV a n d / z > 0 , due to the large neutralino and chargino contributions discussed previously. For comparison in the same figure we have drawn the standard model predictions for a Higgs mass equal to 100 GeV and m, = 160 GeV. The dependence of the quantities of interest on the value of the parameter tan/3 is rather smooth. In Fig. 4 we display Akw AQ v as functions of tan/3 for values ranging from 2 to 10. No strong dependence on tan/3 is observed either although for/z > 0 the dipole moment is appreciably larger for small values of tan/3. The cases shown correspond to m o = A o = 3 0 0 GeV, Ml/2 = 80 GeV and mt = 160 GeV. However these are representative of the more general situation. Concerning our numerical results a last comment is in order. Scanning the parameter space we have intentionally ignored cases giving large contributions due to the presence of Landau singularities of our one loop expressions. These are encountered when the masses involved take values for which the integrands f ( t) develop double poles within the integration region [0,1] of the Feynman parameter t. We are aware of the fact that near a Landau singularity the dipole and quadrupole moments can become sizeable but these cases should be considered with a "grain of salt" not corresponding to an actual physical situation. Our ignorance how to treat these singularities forces us to keep a rather conservative view leaving aside values of the physical masses for which we are in the

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vicinity of a Landau singularity. These result to large and untrustworthy values for the dipole and quadrupole moments which cannot be handled perturbatively. The supersymmetric dipole and quadrupole moments though different, in general, from the corresponding standard model quantities are of the same order of magnitude in almost the entire parameter space mo, Ao and M1/2. The MSSM values of these quantities are not sensitive to either top mass or tan fl as long as the former lies in the physical region 140 < m, < 180 GeV. For M~/2 small, as compared to mo, Ao, the chargino and neutralino sector may give substantial contributions resulting to Ak,, AQ:, approaching the sensitivity limits of LEP2. However these cases should be considered with some caution for they may not be allowed within the perturbative regime; in addition these cases occupy a tiny region in the parameter space, being therefore unnatural, and hence to be disregarded on these grounds. Our conclusion is that deviations from the standard model predictions for the dipole and quadrupole moments due to supersymmetry are hard to be observed at LEP2. If such deviations are observed most likely these are not due to an underlying supersymmetric structure. The work of A.B.L. was partly supported by the EEC Science Program SCI-CT92-0792. The work of V.C.S. is supported by the Human Capital and Mobility programme CHRX-CT93-0319.

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