Yukawa unification

Physics Letters B 274 ( 1992) 387-392 North-Holland

PHYSICS LETTERS B

Yukawa unification S.

Kelley, Jorge L. Lopez

i and D.V.

Nanopoulos

Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA and Astroparticle Physics Group. Houston Advanced Research Center (HARC), The Woodlands, TX 77831, USA Received 30 October 1991

We present a complete one-loop calculation of the top- and bottom-quark and the tau lepton masses in two classes of unified supersymmetricmodels with minimal (SSM) and non-minimal (EVSSM) matter content. In the EVSSM heavy extra vector-like fields are added with masses chosen to hit the string unification scale of O( l0 ~s GeV), while keeping the successful value of sin20w~ 0.233. We examine both SU (5)-type (2b = 2-~)and SO ( 10)-type (2t= 2b= 2~) Yukawacouplingrelations for both models. Our results show that in both models the SU (5)-type Yukawa relation predicts values of mb in terms of tan fl, mr, and ct3, which include the range mb= 4.2-5.0 GeV. The SO ( 10)-type Yukawa relation predicts mt and tan fl in terms of a3 and mb, with mt ~ 90160 GeV ( SSM ) and mt ,~ 140-170 GeV ( EVSSM) for a3 = 0.113 + 0.004 and mb= 4.2-5.0 GeV. We also show that most of the SU (5)-type and some of the SO ( 10)-type allowed values of mt and tan fl in the SSM satisfy the constraints from radiative electroweak symmetry breaking.

The standard model (SM) becomes a highly predictive theory only after a rather large set of parameters is specified. In particular, all fermion Yukawa couplings are unknown. This weakness of the SM may just provide the "smoking gun" of a more complete theory which predicts the values of these couplings, such as a so-called theory of everything ( T O E ) . The only chance for this class of theories to make a real prediction in the fermion mass spectrum is in the mass of the top quark, since we know that there are only three generations [ 1,2 ] a n d five quarks have already been observed. In this paper we leave open the actual nature of the TOE, although we do assume the experimentally viable beyond-the-SM properties of low-energy supersymmetry and gauge coupling unification at very high energies. We examine two classes of supersymmetric unified models which differ in their matter content and therefore in their unification masses. We consider models (i) with m i n i m a l matter content, i.e., the m i n i m a l supersymmetric SM with universal soft supersymmetry breaking terms, which unifies at M u = O ( 1016 GeV) [3,4]; we call this the standard supersymmetric model (SSM), and (ii) with extra Supported by an ICSC-WorldLaboratory Scholarship. Elsevier SciencePublishers B.V.

heavy vector-like fields with masses chosen to delay unification until a mass scale M y = O ( 1018 G e V ) ; we refer to it as a the extra vector SSM (EVSSM). The latter are string-inspired models since string gauge coupling unification is predicted to occur at such high scales [5] (the actual value depends on the particular string model). Also, the added representations occur naturally in many string models [ 6 ]. Here we add one extra vector-like quark doublet Q and one vector-like conjugate down-type quark D e. This model is representative of a large class of extra vector models, distinguished by requiring Q and at least one other type of extra vector-like representation, giving both the correct string unification scale and the correct sin20w [7] ~l For the above two classes of models we examine relationships among the third-generation Yukawa couplings of two types ~2: (a) 2b=2~, as obtained in models with S U ( 5 ) gauge symmetry [ 11 ], and (b) 2t = 2 b = 2~, which occurs in SO ( 10 ) [ 12 ] and E 6 [ 13 ] unification schemes. Let us point out that almost all #l For an alternative viewpoint on this matter see ref. [ 8 ]. ~2 For recent studies in actual SU(5) models (i.e., SSM) see refs. [4,9 ]; for an SO ( 10) SSM study see ref. [ 10]. We have found it difficult to compare our results with those in refs. [ 10,9] due to the lack of details in those papers (see the appendix). 387

Volume 274, number 3,4

PHYSICS LETTERSB

of the known string models have been constructed using level-one Kac-Moody algebras, and thus grand unified groups such as the one naively needed to obtain the above Yukawa relations are not allowed [ 14 ], and it is therefore rather surprising that relations of this kind arise at all. However, realistic string-derived models which contain relations of both kinds do exist [ 6,1 5 ]. Beyond all the model-dependent details, what remains valid is that definite predictions for the fermion Yukawa couplings can be made and thus tested in a large class of string models. In this paper we focus for concreteness on the simple SU( 5 )and SO ( 10)-like relations to illustrate their predictive power. We perform all our calculations using the full oneloop renormalization group equations (RGEs) between and mz, dropping the top-quark Yukawa coupling at m,. Below mz, we compute corrections to the masses in the form 01(Q') ]y where we include two-loop 013 terms in the calculation of the large value of 013 in this region. Comparison with complete two-loop calculations [4] shows little difference. There appears to be disagreement in the literature as to what value of mb to be used to compare with the output of the RGE calculations. Invoking various theoretical frameworks, values between 4.2 and 5.0 GeV have been recently considered [4,10,9]. To avoid-settling such a detailed question prematurely, in what follows we just take mb=4.2--5.0 GeV as a representative interval for our quantitative comparisons, although intuitively we would be more inclined to take rnb~ 5 GeV. A unified model with a particular field content has for present purposes five relevant parameters at the unification scale: the unification scale Mu, the unified gauge coupling 01(Mu ), and the third-generation Yukawa couplings 2t(Mv), 2b(Mu), and 2 d M u ) . Extrapolation of the low-energy gauge couplings aem and a3 gives and 01(Mtj) and predicts sin20w. Since the error in OZemis negligible, both Mu and (Mu), and sin20w are determined by a3 in the SSM. In the EVSSM, the scale of the extra vector-like Q and D c fields is determined by the unification scale Mu and the low-energy values ofsin20w, 013, and t~em as follows [ 7 ] :

My

m(Q)=m(Q')[a(Q)/

Mu

16 January 1992

ln(m~)= 3

ln(Mv)+n(sin2Ow+\ mz / \ 01em

, 3Ol3

,) 20~em

(la)

ln(Mv)=71n(_~z)_n(Sin20w \mD¢/

-~

\ ~

In(m~) ,

7

1 )

3013 + 2~mem

(lb)

where the (small) effect of the top-quark threshold (for m , > m z ) has been included as well. Taking M u = l018 GeV, sin20w=0.2331 _+0.0013 [4], 01era -~ 127.9+0.2, 013=0.113+0.004 [16], and ~3 mr: 9 0 - 1 9 0 GeV, we obtain m q = 2 . 6 × 1 0 ~ 2 × ( 2 . 9 -+~) GeV and m D c = 3 . 0 X 1 0 5 × ( 2 9 -+~) GeV. The allowed latitude in the extra vector-like masses clearly indicates that no fine-tuning is involved. Once the gauge couplings are determined, the Yukawa couplings at the unification scale are related to their values at low energies, which can be parameterized by rn, rob, ms, and tan ft. Since the error in the tau mass is negligible (and we fix sin20w=0.233 and --1 = 127.9 ), our models have only four parameters: aem o~3, mr, mb, and tan ft. The SU (5)-type Yukawa relation gives one constraint, allowing one of these four parameters to be predicted from the other three. The SO ( 10 )-type Yukawa relation gives two constraints, allowing two of these four parameters to be predicted from the other two. Figs. 1 and 2 show contours of constant mb in the (mr, tan fl) plane in the SSM and the EVSSM respectively, resulting from the SU (5)-type Yukawa relation for the extreme values of o/3 ---- 0.109 (figs. 1a and 2a) and a 3-- 0. l 17 (figs. lb and 2b). The corresponding dotted lines impose the SO (10)-type Yukawa relation. Requiring mb= 4.2--5.0 GeV gives little constraint in the SU(5)-type SSM allowing the whole region between the lower bound m, > 89 GeV and the mb=4.2 GeV contour on fig. lb for some value of a3. However, mb=4.2-5.0 GeV in the EVSSM model restricts the (mr, tan fl) plane to the narrow band between the mb= 5.0 GeV contour in fig. 2a and the rob= 4.2 GeV contour in fig. 2b, giving ~3 Note that some recent analyses of this nature [ 10,9] assume

significantly lower central values of a3.

388

Volume 274, number 3,4

PHYSICS LETTERS B

EVSSM SU(5) unification, aa=0 109

SSM SU(5) unification, aa=0,109 . . . .

50

16 January 1992

5o ~ - , ~ _ ~ , ,

,

~

......

4O

4O

tan fl

tan fl

30

3O

20

10

100

120

140

160

180

100

200

120

140

160

IBO

200

m t (OeV)

m t (GeV) SSM SU(5) unification, a3=0.117

EVSSM SU(5) unification, aa=O.117 50

50

,b1

4O

tan fl

~

tan

30

3O

20

20

10

100

120

140

160

180

200

m t (GeV)

Fig. 1. Contours of constant mb in the (mr, tan fl) plane for the SSM with SU (5)-type Yukawa relations with (a) cq = 0.109 and (b) a3=0.117. The dotted line represents the constraint imposed by the SO( 10)-type Yukawa relations.

a tight constraint between m~ and tan ft. The SO ( 10 )type Yukawa relations further restrict the allowed regions in the SSM and EVSSM to be between the dotted lines in the a3=0.109 and a3=0.117 figures. These allowed bands (due to the a3 spread) are very narrow, and in the EVSSM practically predict tan fl from mt or viceversa. Figs. 3 and 4 show the values of the unified SO ( 10)type Yukawa couplings at the unification scale for the SSM and EVSSM respectively. Note that in the stringinspired EVSSM these values are closer to the magnitude expected in typical string models [i.e., O ( g ) ].

100

120

140 m t

160

1B0

200

(GeV)

Fig. 2. Same as fig. 1 for EVSSM. All the above results have been obtained by assuming that the supersymmetric particles all decouple at the scale mz. We now show how decoupling at scales between mz and 1 TeV affect our previous results. We express these threshold corrections in terms of the variable ri defined as rn, (SUSY thresholds) rt = rn~(SUSY thresholds = mz) '

(2)

for i = t , b, where SUSY thresholds stand for the masses above mz of the decoupled supersymmetric particles. This means that for values of SUSY thresholds above mz, ri gives the factor needed to convert the values ofmb and/or mr, shown in figs. 1 and 2 for all SUSY thresholds at mz, to the actual values of 389

Volume 274, number 3,4

PHYSICS LETTERS B

16 January 1992 4/7

SSM SO(IO) unified Yukawa, a 3 = 0 1 1 3 ± 0 0 0 4 = (OL3 ( m z _ ) ' ]

r~ 2 o ~ ( m , ) / 08

j/

4.~

/~

4.3

l/

4.4

./

j--" i

06

/

04

---- / - -

02

~ ~---~

oo

0.105

,

.....

__-/~/

I

511

~ ....... ~

I

0115

~3

4.7

_~__

,

012

Fig. 3. The value of the unified Yukawa coupling corresponding to the SSM predictions for mt and tan fl in fig. 1 (dotted line).

EVSSM SO(IO) unified Yukawa, c~3=0 113±0 004 10 i

I

]

/

08

~

06

~

43 44

46

Z_ Z?IZ/IZ2 ~illk~Z~ 4547 __ - - -- _- ~ _ - - ~

54

~T~ZTZZ-----~ --

_ --

\o~(m~)/

\a3(Mb)/

t~33 (M~u))

/

. . . .

8/9 "~

(OL3(m2)

× { o~ ( mz)'~ -~/~

50 /

4/5 (oz3 (ml)'~

49

5o

(3)

'

where m l = m i n ( m 4 , rn~) and m 2 - - m a x ( m 4 , m~). Note that the value o f Mm and therefore o~3( M u ) , depends on the SUSY thresholds and differs between the n u m e r a t o r and d e n o m i n a t o r o f eq. ( 3 ) , i.e., o6 ( M ~ ) ( o ~ 3 ( M u ) ) has been calculated with given SUSY thresholds above ( a t ) mz. The numerical resuits are plotted in fig. 4. As an example, the contour o f rob=4.9 GeV in fig. 1 for m ~ = m ~ = m z becomes a contour o f ~ 4 . 8 OeV for m ~ m ~ 2 0 0 OeV, and so on. In figs. 1 and 2, the m, scale in reading the d o t t e d line [ SO ( 10)-type Yukawa relation ] gets shifted by the corresponding reduction factor. A similar calculation for the EVSSM [ 17 ] gives very different values o f ri than those for the SSM, and not only the squark and gluino thresholds are significant. However, for SUSY thresholds less than 1 TeV these contributions never give more than a few percent shift in the quark masses. F o r these models to be acceptable, it must also be possible to generate dynamically (i.e., radiatively) the a p p r o p r i a t e electroweak s y m m e t r y breaking VEVs corresponding to the value o f tan fl and mz. F o r the

02 K

0o ...... 0105

[. . . . .

011

C(3

]

0115

~

r

~

,

I O0

0 IE

Fig. 4. Same as fig. 3 for EVSSM.

, ] . . . .

, , ] ....

rn~= 100 CeV

r i

200

098

SUSY thresholds. F o r the SU ( 5 ) - t y p e Yukawa relations only rb applies (i.e., q - 1 ) , whereas for the SO ( 10 )-type relations both factors must be used. The function r, is o b t a i n e d by studying the effect o f the decoupling o f heavy particles on the beta function and the a n o m a l o u s d i m e n s i o n s o f the respective quarks. Because the strong coupling is responsible for most o f the r e n o r m a l i z a t i o n o f the quark masses, calculating only this c o n t r i b u t i o n to r, is an excellent approximation. In the SSM only m~ and m~ give significant contributions to r~ and the result m a y be expressed as 390

500

096 ,

l. . . . .

200

I

400

....

I

....

600

l~_, 800

,

j

1000

m~ (CeV) Fig. 5. The value of the correction factor ri to be applied to the mb masses shown in figs. I and 2 due to the decoupling of heavy squark and gluino fields with the given masses. This factor should also be applied to the top-quark mass scale used to read the dotted line in these figures.

Volume 274, number 3,4

PHYSICS LETTERS B

SSM, the analysis of ref. [ 18 ] gives acceptable eleco troweak breaking in a t o o = A = 0 supersymmetry breaking scenario overlapping a substantial region in the ( m , tan fl) plane where mb has its preferred value of ~ 5 GeV. However, this supersymmetry breaking scenario does not give acceptable electroweak breaking overlapping the region required by the SO ( 10)type Yukawa relations. We have checked scenarios with non-zero values o f mo and Ab and found that electroweak breaking is possible at tree-level for the restricted range in the (mr, tan fl) plane required by the SO(10)-type relation. However, since the required values of ml/2 are rather large ( > 500 GeV), a definitive answer requires a study o f the one-loop Higgs potential. A similar analysis for the EVSSM model is currently under investigation [ 17 ]. In sum, we have investigated the predictions of Yukawa coupling relations of SU (5) and SO (10) types in supersymmetric unified models with different motivations. The predictive power of these models is least for the SSM with SU ( 5 )-type Yukawa relation and most for the EVSSM with SO( 10)-type relations. In fact, in the latter a relation between m t and tanfl parameterized by mb, can be obtained largely independently of the value of c~3 (within its allowed range). We conclude that the physics of the string-inspired EVSSM deserves detailed analysis, which is currently underway. This work has been supported in part by DOE grant DE-FG05-91-ER-40633.

Appendix To facilitate comparison with other results [ 10,9] and to help understand the importance of different effects in our calculation, we present some of the details of the analysis for the SSM, neglecting Yukawa and light threshold effects. Neglecting Yukawa couplings, the one-loop RGEs may be solved analytically obtaining for the SSM

mb(mb)=m,:(m.~)f~(°~3(mz)-) 8/9 \Ot3 (Mu),]

X (o/y(mz)_)

\o(v(Mu)J where

--7/198

(Oly(mz)~ 3/22 \ay(Mu)/

'

(A.1)

16 January 1992

fb.~_( O/3(rob) ~

12/23 ( o t = ( m b ) ~- 1/40

\Ot3(mz)J

/

,

\aem(mz),/

x \--9/40/

,

(A.2a)

. \--9/38

f _laemtm.~)~

(Otemtmb)~

:~-- kO~em(mb),]

kO/em(mz)}

(A.2b)

account for the effects below mz. For the central values a 3 = 0 . 1 1 3 and aem-I= 127.9, our numerical resuits are fb(one-loop) = 1.299, fb(two-loop) = 1.326, f~=l.015, and mb=5.272 GeV. For fb(two-loop), Ot3(mb) has been calculated from ot3(mz) using twoloop RGEs neglecting O (O£emO/2) terms. All other results in the present paper include Yukawa effects at one-loop, where the top Yukawa coupling is dropped at the scale mt along with the appropriate change in the gauge beta functions, and the other Yukawa couplings are dropped at rnz. The analogous results for the EVSSM will be presented elsewhere [ 17 ].

References

[ 1] A.J. Buras, J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B 135 (1978) 66; D.V. Nanopoulos and D.A. Ross, Nucl. Phys. B 157 ( 1979) 273; Phys. Lett. B 108 (1982) 351;B 118 (1982) 99. [2] L3 Collab., B. Adeva et al., Phys. Len. B 231 (1989) 509; ALEPH Collab., D. Decamp, et al., Phys. Lett. B 231 (1989) 519; OPAL Collab., M. Akrawayet al., Phys. Lett. B 231 (1989) 530; DELPHI Collab., P. Aarnio et al., Phys. Lett, B 231 (1989) 539. [3] J. Ellis, S. Kelley and D.V. Nanopoulos, Phys. Lett. B 249 (1990) 441;B 260 (1991) 131; U. Amaldi, W. de Boer and H. Fiirstenau, Phys. Lett. B 260 (1991) 447; P. Langacker and M.-X. Luo, Phys. Rev. D 44 ( 1991 ) 817. [4] J. Ellis, S. Kelley and D.V. Nanopoulos, CERN preprint CERN-TH.6140/91. [ 5 ] V. Kaplunovsky, Nucl. Phys. B 307 ( 1988) 145; I. Antoniadis, J. Ellis, R. Lacazeand D.V. Nanopoulos, Phys. Len. B 268 (1991) 188; S. Kalara, J.L. Lopez and D.V. Nanopoulos, Phys. Len. B 269 ( 1991 ) 84. [6]See, e.g., I. Antoniadis, J. Ellis, J. Hagelin and D.V. Nanopoulos, Phys. Lett. B 231 (1989) 65. [ 7 ] I. Antoniadis, J. Ellis, S. Kelley and D.V. Nanopoulos, Phys. Len. B 272 (1991) 31. [8] L. Ibfifiez, D. Liist and G. Ross, Phys. Len. B 272 (1991) 251.

391

Volume 274, number 3,4

PHYSICS LETTERS B

[ 9 ] H. Arason et al., University of Florida preprint UFIFT-HEP91-16; A. Giveon, L. Hall and U. Sarid, University of California, Berkeley, preprint LBL-31084. [ 10] B. Ananthanarayan, G. Lazarides and Q. Shaft, Phys. Rev. D 44 (1991) 1613. [ 11 ] H. Georgi and S. Glashow, Phys. Rev. Lett. 32 (1974) 438. [ 12 ] H. Georgi and D.V. Nanopoulos, Nucl. Phys. B 155 ( 1979 ) 52. [ 13] R. Barbieri and D.V. Nanopoulos, Phys. Lett. B 91 (1980) 369.

392

16 January 1992

[ 14] J. Ellis, J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 245 (1990) 375; A. Font, L. Ib~ifiez and F. Quevedo, Nucl. Phys. B 345 (1990) 389. [ 15] J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 251 (1990) 73; Phys. Lett. B 268 ( 1991 ) 359. [16] J. Ellis, D.V. Nanopoulos and D. Ross, Phys. Lett. B 267 (1991) 132. [ 17 ] S. Kelley, J.L. Lopez and D.V. Nanopoulos, in preparation. [18] S. Kelley, J.L. Lopez, D.V. Nanopoulos, H. Pois and K. Yuan, Phys. Lett. B 273 ( 1991 ) 423.