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Upper bound on hot dark matter density from SO(10) Yukawa unification
8 September 1994 PHYSICS LETTERS B
Physics Letters B 335 (1994) 345-354
I'll ^~t!\.'II;R
Upper bound on hot dark matter density from unification *
SO(10) Yukawa
Andrea Brignole a'l , Hitoshi M u r a y a m a ~,2, Riccardo Rattazzi
b
a Theoretical Physics Group. Lawrence Berkeley l,aboratory. University of California, Berkeley. CA 94720, USA h Department of Physics and Astronomy, Rutgers University, Piscatawa); NJ 08855. USA Received 12 July 1994 Editor: M. Dine
Abstract
We study low-energy consequences of supersymmetric SO(10) models with Yukawa unification ht = hN and hb = hT. We find that it is difficult to reproduce the observed mb/m~ ratio when the third-generation right-handed neutrino is at an intermediate scale, especially for small tan/3. We obtain a conservative lower bound on the mass of the right-handed neutrino M,,~ > 6 × 10 ~3 GeV lbr tan/3 < 10. This bound translates into an upper bound on the r-neutrino mass, and therelore on its contribution to the hot dark matter density of the present universe, lq,,h 2 < 0.004. Our analysis is based on the full two-loop renormalization group equations with one-loop threshold effects. However, we also point out that physics above the GUTscale could modify the Yukawa unification condition hh = hT for tan/3 < 10. This might affect the prediction of mh/m~ and the constraint on MN.
I. G r a n d unification [ l ] is o n e o f the leading guiding principles to build models o f fundamental forces in naturc. T h e simplest m o d e l s also lead to a partial unification o f Yukawa c o u p l i n g s [ 2 ]. S u p c r s y m m e t r y ( S U S Y ) , on the other hand, appears as a necessary principle to m a k e grand unified m o d e l s theoretically viable 13]. It is then remarkable that the hypothesis o f s u p e r s y m m e t r y and o f a "grand desert" between the Fermi and G U T scales is p h e n o m e n o l o g i c a l l y suc* This work was supported 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-AC03-76SF(K)098, and by NSF grants PHY-91-21039 and PHY-89-04035. i Supported by a fellowship of lstituto Nazionale di Fisica Nuclc~e, ,~zione di Padova, Padua, Italy. 2 On leave of abscnce front Department of Physics, Tohoku University, Scndai, 980 Japan. Elsevier Science B.V. SSDI 0 3 7 0 - 2 6 9 3 ( 9 4 ) 0 0 9 3 9 - 2
cessful for both gauge (see, e.g. [4] and references therein) and Yukawa (hb = h~,) [5,6] unification. On the other hand, astrophysics and c o s m o l o g y suggest that there may be an intermediate scale in the "desert". O n e indication c o m e s from the observed deficit in the solar neutrino flux, which favors neutrino oscillations ~ la M i k h e e v - S m i r n o v - W o l f e n s t e i n ( M S W ) [ 7 ] . If one assumes v,,-v~ oscillation and SO( 10)-like mass matrices, then the data suggest the existence o f right-handed neutrinos N with an intermediate mass MN ~ 101°-1013 G e V (see, e.g. 18] ). A n o t h e r indication c o m e s from the c o s m i c density fluctuations at C O B E [9] and galaxy [101 scales. These are in moderate contradiction with each other when a scale-invariant spectrum and dark matter o f cold type only are assumed. C o e x i s t e n c e o f cold and hot dark matter can solve this p r o b l e m [ 11 ]. If
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A. Brignole et al. / Physics Letters B 335 (1994) 345-354
hot dark matter (HDM) is given by ~,r, then, again, MN "~ 1012 GeV is favored. Moreover, right-handed sneutrinos N can play interesting cosmological roles in baryogenesis and inflation [ 121. Therefore, it is important to study the phenomenology of SUSY GUTs in which right-handed neutrinos are the only thresholds in the grand desert. Obviously, gauge unification is not significantly affected since right-handed neutrinos are gauge singlets. This letter studies the unification of the Yukawa couplings of the third family in SUSY GUTs with righthanded neutrinos at an intermediate scale. We consider SO(IO) models with Yukawa unification ht = hN and hb = h,-. We find that an intermediate scale right-handed neutrino has a critical influence on the predicted value ofmh/m~. Consistency with the experimental value o f m h / m T implies a lower bound on MN. This constraint translates into an upper bound on the v~ contribution ~L, to the HDM density in the present universe. Cosmologically interesting values of IL, are strongly disfavored for small values of tan/3.3 2. We first describe the theoretical framework of our study. We assume a simple SO(IO) GUT scenario where only small irreducible representations (irreps) like 10, 16, 16, 45, 54 and possibly singlets are present, 4 and matter fields come only in 16's. As for the Yukawa interactions, the large value of the top Yukawa coupling suggests that it originates from a renormalizable coupling. With our assumptions this can only be hc,16310H 163. (Even a 120 would not couple because of antisymmetry.) In general, the two light doublets H~.d of the minimal supersymmetric stanThere is a claim 1131 that b - r Yukawa unification combined with the value of mt recently suggested by CDF excludes the region 1.5 < tan/3 <_ 30. This claim, however, depends crucially on a's(mz) -~ 0.125 and mt ~< 174 GeV, for which the experimental support is not yet established. Thus. we take tan/3 as a free parameter in this letter. 4 Notice that large irreps drive the theory out of the penurbative regime; for example, if a 126 is present, the SO(10) gauge coupling constant blows up below < 8MG even with the minimum particle content 45 + 54 + 126 + 126 + 3 × 16. In superstring models, the above mentioned small irreps appear already at Kac-Moody level k = 2. whereas 126 can appear only for 5 < k < 7 1141. Phenomenologically, the presence of a 126 may also give an unacceptable relation h~ = 3ha 115 I. Finally, it is noteworthy that the absence of 126 and R-odd gauge singlets forbids MN at the renormalizablc level, thus leading to MN << Me,.
dard model (MSSM) contain only fractions of those in 1011, due to mixing with doublets in other irreps. In our case these can be 10 u' 's, 16n's and 16n's, the latter ones being necessarily present in order to reduce the rank of the gauge group. The mixing strengths are in general different for H~ and Hd because S U ( 2 ) R is broken at Me,. Then the GUT boundary condition on the Yukawa couplings is ht = hN = s,h~, ht, = hr = sdha, where S,.d are the Higgs mixing angles, and in general s~ ~ Sd. Here the meaning of hi, hN, hb, and h~, is the obvious one. The particular case su = Sd has been studied before [ 16,15]. Actually the boundary condition hb = hT
ht = hN,
( 1)
is just a result of the underlying Pati-Salam SU(4) symmetry, so that one may imagine scenarios, maybe in strings, where Eq. ( 1 ) holds even without a unified simple gauge group. There are also more exotic scenarios with mixings between 16 and 10 matter fields [17], while Eq. (1) keeps holding. Moreover, there is a class of models where hb and hr originate solely from higher-dimension operators, while predicting the same relation [ 18]. We thus conclude that there is a wide class of models ( S O ( I O ) and beyond) which lead to the boundary condition Eq. ( I ). 3. In order to illustrate the low-energy consequences of Yukawa unification when MN < MG, we first consider a simplified picture based on one-loop renormalization group equations (RGE), without threshold corrections. This also allows for a clearer comparison with the pure MSSM case (MN "~ M 6 ) . The RGE for the individual gauge and Yukawa couplings can be derived from the general formulae in Ref. [ 19], both at one-loop order (needed here) and at two-loop order (to be used below), and we do not write them down explicitly. Here we rather focus on the evolution of the ratio R(/z) = hb(Iz)/h~(Iz) from the GUT-scale value R ( M G ) = 1 to the low-energy value R ( m z ) . The latter should be compared with the experimental value of mb/mr scaled up to mz, which we denote by R e x P ( m z ) . The RGE for R reads
R[ 77
+
-
-
-16 2
- tTg3-
42]
~gl) ,
(2)
347
A. Brignoleet al. /PhysicsLettersB 335 (1994)345-354 where t - log/z and h 2 has to be omitted f o r / z < M N. The solution of Eq. (2) can be formally written as R ( m z ) = R ee -v, where Rg :-- (a3(mz)/Cr3(MG))8/9 x
(a'l ( m z ) / a l (MG)) 10/99
where hh and hr are negligible. Then we can write Y = Yi + Y2, where YI and 112 are the first and second terms in Eq. (3). We denote by ho(t) the solution of the RGE for hi (t) in the pure MSSM case with boundary condition ho(M(;) ---* co. Its explicit expression is [20] h~( t) = 47r2(-F' ( t) /3F( t) ), where &;
and
F(t) =- f d; tN
it; t
' "2 [.f h : ( t ) d t + / ( h ~ ( t ) - h ~ ( t ) ) d t Y = 16~ lZ
× (a2(M(;)/ce2(ff)) 3 (oq(M(;)/ot, (ff))13/99.
IN
I(;
IZ
with ta -- log MA. Notice that Y should be considered as a functionofhh(M¢;), ht(M6) and MN. For MN = M(;, the second term in Y drops and the pure MSSM case is recovered. Since R~ is larger than Re~p (mz) by 30-80%, the Yukawa contributions are essential in reducing the value of R(mz). In fact, the three terms in Y are all positive. In the pure MSSM case, one could formally make Y arbitrarily large (i.e. R(mz) arbitrarily small) by taking h,(Mc,) large enough [201. In practice, h,(MG) ~-- (.9(1) is required to reproduce the correct Rexp(mz). In contrast, for MN < Ma a full cancellation of Yukawa contributions occurs at p, M¢;, due to the boundary conditions ( 1 ). Thus Y cannot be larger than afinite maximum value Yma~(MN), obtained in the limit hr(MG) ---* :)c. Since Ymax(MN) decreases as Mx decreases, it may be impossible to get R(mz) as low as Rexp(mz) when MN is lighter than a certain critical value. This is why a lower bound on MN appears. Actually, Ym~,,~(MN) crucially depends on tan/3. For fixed MN, Ymax(MN) is smaller when tan/3 is small, because h5 and hr are negligible and the last term in Y is missing. In fact this is the situation where stronger constraints on MN will be found. On the contrary, when tan/3 is very large, the last term in Y is significant (notice also the factor 3). Then, even if MN is light, Y can be made large enough as to reduce the value of R appropriately. The above discussion is supported not only by a numerical analysis, but also by the following analytical argument. We show that Y cannot be arbitrarily large even with ht(M~D ---* oc, in contrast to the pure MSSM case. We consider the most interesting case
A study on the RGE of hi(t) for t < MN and t > M s , with a given boundary condition ht(MG), leads easily to the upper bound hi(t) < ho(t), valid for any t and independent of ht(MG). This gives an upper bound on YI,
F(mz) Yl < Y~*( M~/) - ~ log F( MN~
(4)
where we have replaced for convenience the arguments ta with MA. A simple bound on Y2 can bc derived, e.g. by studying the RGE for the ratio/~(t) h t ( t ) / h N ( t ) , which reads l d]~ 1 [3(h~ - h 2) -- (16g2 /~ dt - 1697"2 .3- 3 + ~ g ~ ) ] "
(5)
If one formally integrates the above equation term by term between tu and tG, the right-hand side contains 112 and easily integrable gauge terms. The left-hand side gives just - log/~(tN), which is negative (one can easily prove that /~(t) > 1 from Eq. ( 5 ) ) . Theretore one obtains an absolute bound o~3(MN)
Y2 < Yf(M,v) =" ~ l o g cr3(Mo-------~
2 l o cet g (MN) - 297 cr;(Mc;) (6)
Notice that the bounds (4) and (6) are independent of the initial values ht (Me;) = hN(Mc;), so they hold even in the formal limit ht(MG) ---* oG. In conclusion, for a given MN < MG we explicitly get an absolute upper bound Ymax(MN) on Y which is a monotonically increasing function of MN 5 z Although Ymax(M.,v) is always monotonically increasing, the same applies to the analytic bound Ymax(MN) for the region of interest MN ~ 10~1' GeV. We add that stronger analytic bounds than that described here can be found.
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A. Brignole et al. / Physics Letters B 335 (1994) 345-354
Y _< Ymax(MN) < Ymax(MN) = YI*(MN) + YZ* ( M s )
.
(7) This in turn gives an absolute lower bound on R(mz)
R(mz) >
R
"X~ (F(mz) e-r:~, (M~') = &, g \F(MN) )
\ 3-U oi j
/
•
l/ l2
the same scale. 6 On the other hand, we take ots(mz) as an independent parameter, and allow a3 v~ a l = a2 at MG. This difference could be accounted for by GUT threshold effects. The gauge coupling constants in the MSSM at mz are o,11(mz)=59.1+1(12
(8)
27r
msusY
.-3-log - ?nZ
+ 1 log
mA'] , ?nZ/I
a~-I ( m z ) = 29.4 + ~ ( 4 1 o g - - s m s u s~-v I log __),mA mz mz 4
msusY mz
a31 (mz) = or.?I (mz) + ~ log - This bound can be compared with e e x P ( m z ) , s o that a lower bound on M s can be interred.
4. We have shown an analytic bound on R(mz) in which hN(M6) was allowed to go to infinity. In practice hN will not be taken larger than O ( 1 ). Then the effect o f MN '~ 1013 GeV can be roughly estimated by treating it as a threshold correction to R at Me; [ 15]. This gives R(M¢~) ~- 1 + (h2N/16zr z) log(MaiM~v), which amounts to an O ( 1 0 % ) increase in R(Mc,) and hence in R(mz). A 10% is critical when comparing to R~Xp(mz), since already in the pure MSSM the predicted R(mz) lies in the upper part of the allowed range. Nonetheless it is clear that all other effects of the same order may have a huge impact on the bound on M s , because R(mz) depends roughly logarithmically on M/v. Then to make our analysis more accurate, we have to discuss SUSY and G U T thresholds and use two-loop RGE. In fact the effects of these, and especially the first two, can possibly pile up to O ( 1 0 % ) . We also attempt an estimate of the possible effects from physics above M6, even though our understanding of these can be only qualitative. Finally, by varying the QCD coupling a s ( m z ) within its experimental range we get an O ( 1 0 - 2 0 % ) effect, which we treat with the proper attention. Our reference value of as(mz) is the one from the Z ° hadronic width as(mz) = 0.124 + 0.007 [21], and we allow a 2o" range. To obtain quantitative constraints on M s , we perform a numerical analysis based on two-loop RGE with one-loop threshold corrections. In order to be able to discuss the dependence on a.~(mz) [6,15], we take the following procedure. We define MG as the scale where a l and az meet, and take ht = h s , hb = hr at
Here the logarithms take care o f the matching between the SM and the MSSM; we take the additional Higgs doublet at ma and the rest o f the SUSY particles at msusv. (The non-logarithmic parts and the translation from MS to DR are numerically small and have been neglected.) We also match the couplings in the M S S M + N to those in the M S S M a t / x = Ms, 7 h b ( M s ) [MSSM = hb(Ms)]MSSM~ N, h r ( M N ) [MSSM = h e ( M N ) [MSSM-N( I -- e ) ,
h t (M/v) [MSSM = h t ( MN ) [MSSM ~-N( l -- e ) , hN(MN)IMSSM = hN(MN)!MSSM~-N(I -- 2 e ) ,
(9)
where e = h~(Mu)/327r 2. h/v(#) in the M S S M is defined in such a way that the coefficient o f the dimension-five operator ( L H , ) 2 is h2u(i.t)/MN. For any fixed values o f (hi, hb) at M6, the actual calculation is done as follows. We use the two-loop RGE of the M S S M and the M S S M + N . By using an iterative procedure, we find the values of M c , O ' I ( M G ) and a 3 ( M ¢ ' ; ) which reproduce the values of a l ( m z ) , a 2 ( m z ) and a3(mz) defined above. At the same time we obtain the values o f the Yukawa couplings at mz. The resulting b-r Yukawa ratio R(mZ)IMSSM is then matched to the ratio o f the corresponding masses in the broken electroweak theory R ( m z ) = rnb(mz)/mr(mz),
R(mz) = ( l + k h - k r + f R )
R(rnz)IMSSM,
(10)
6 Numerically, MG ~ 2 x 1016 GeV. GUT threshold corrections to Yukawa couplings will be discussed later. 7 We define the right-handed neutrino mass MN by MN = M s ( M s ) , where MS(p,) is the DR running mass.
A. Brignole et al. / Physics Letters B 335 (1994) 345-354
where kb, k~, are the threshold corrections due to the SUSY particles, while fR is that from the additional Higgs doublet in the MSSM. The exact expressions are found in Ref. [ 15]. In the following we will only focus on the logarithmic terms in kb.T. We will discuss the potentially large non-logarithmic ones (k~, in Ref. [ 151 ) at the end of the letter. Note that also R(mZ)IMSSM depends implicitly on mSUSY and mA due to the threshold dependence of the MSSM gauge coupling constants a71(mz) (see above). For instance, in the small tan 13 region, the dependence of R on the SUSY particle masses turns out to be approximately 8 -~SR - - ~ - - 1 ( - 0 . 1 logmSUSY + 0 . 1 log ma "]
R
27r
mz
mz J '
(11)
which essentially vanishes for msusY = ma. This agrees with the results in Ref. [6]. In the final step, the predicted R(mz) is compared to ReXp(mz). Notice that also R~Xp(mz) has an important dependence on ce~(mz), since it is obtained by scaling the b-z mass ratio up to mz. GUT-scale threshold corrections could also affect the boundary condition in Eq. (1). Though we neglect such corrections in the following numerical analysis, we will discuss their possible effects later. We first illustrate the dependence of R ( m z ) on ht(Mc;) for the region tan13 ~ 10, where both hb, h~. get rcnormalized homogeneously and the result depends very weakly on tan 13. We show curves for different values of MN in Fig. 1, taking msus¥ = ma = mz, hi, = h~, = 0 . 0 1 at Me;, and crs(mz) =0.11 (Fig. la) or 0.12 (Fig. lb). For comparison, we also show the values of R e x P ( m z ) corresponding to the MS mass mt,(mb) = 3.9, 4.15, and 4.4 GeV. This is the range which is obtained from QCD sum rules [ 1 5 ] . 9 First of all, it is clear from the figures that it becomes harder 8 Even if the SUSY spectrum is nondegenerate, the dependence of R on the SUSY panicle masses can again be summarized by an effective msusY. It turns out that the dependence of R on colorless panicle masses is very weak, and msusY is approximately (m,?n~) 1/2. After considering generic splinings in the SUSY spectrum. we found that the value of R varied only between two extreme cases, ms,us;y = I TeV, ma = mz (smallest R), and msusY = mz, ma = 1 TeV (largest R). We stress that the approximate formula Eq. ( 1 I) cannot be applied for ma ~ mz. '~ The uncertainty on rob(mr,) is dominated by the yet-unknown (_9(a~) corrections to QCD sum rules [151.
L[(a) 2 4
i
349
"
a~(mz)-O i I ....
~"!" ..... '' I''"~ v('b) a , ( r n z ) = 0 . 1 ~
--
M:~= i013 'MN= .... MN=IO Is
:~.~ ~ ~ ~ .
I
--
~,
1014 "~
-Ms=10~°
t
2
4
"MN=IO'2t
MN=MG
_
I
1 (3 ~
-..........
(J
:-. . . . 0
J__ l
2
ht(Mc)
I .... 3 4 0
1
3
ht(MG)
Fig. 1. An illustration of the effect of the right-handed neutrino on R ( mz ) = mb ( mz )/mr( mZ ). The curves show the dependence of R ( m z ) on ht(MG) = hN(M(;), for the values of M,v = MG, 1015 GeV, l 0 In GeV. 1013 GeV, 1012 GeV. and 10 m GeV from below. The experimentally allowed values of R ( m z ) are also shown. These correspond to mb(mh) = 3.9, 4.15 and 4.4 GeV. Other input narameter~, are taken as m s u s y = mA = mz, ht~(MG) = hr(MG) = 0.01. and cts(mz) = 0.11 (a) or 0.12 ( b ) .
to reconcile R ( m z ) with ReXp(mz) for larger a,.(mz ) and lower MN. For a~.(mz) = 0.12, one needs very large ht(MG) > 3 even for MN = M~. 1o Moreover, even if increasing ht(M6) makes R(mz) dccrease, the suppression effect becomes weaker as we lower Ms. Note also that the value of R(mz) depends only weakly on ht(Mc;) for ht(MG) ~ 2, especially for MN < 1014 GeV. Such a "fixed point" behaviour and the dependence on MN were expected on the basis of the analytic discussion above (see, e.g. Eqs. (7), ( 8 ) ) . As we lower MN, we can check whether the curves reach the region of R~XP(mz) and then infer a lower bound on MN. In the following, when scanning the space of (hi(M(;), hb(M(;)), we will only take for definiteness h t , b ( M G ) '< 2. This reference value is motivated both by the above observation and by perturbativity reasons (see also point (b) below). Next we show lower bounds on MN as functions of tan 13 in Fig. 2. I1 In order to be conservative we take the maximum value ofmh(mh) = 4.4 GeV. For most of l ° F o r ot~(mz) ~ 0.12 and ht(MG) < 2, the predicted value of R ( m z ) is larger than the experimental upper bound. Consistency could be possibly restored by allowing 0 ( 5 % ) GUT- or SUSYscale threshold corrections, see section 5. 11 We define tan.8 using m r ( m z ) = h r ( m z ) t ; c o s . 8 , where hr ( m z ) is the MSSM one. Other definitions of tan 8 are related to this one via one-loop corrections.
A. Brignole et al. / Pl~vsics Letters B 335 (I 994) 345-354
350 •k | c .
h2uv2 sin2/3
/ ....................
~
"-
my,
\
101,1
•
...
\,,
-~
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:,
. . . . . . . . . . .
t~l° i
(x s(r[:.,:) - 0 . 1 1 "/
10 t~ ' . . . . . . . . .
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'~
,~o(,,,~):0:2:
,
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~(rnz) ~
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:'o
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-\,
-
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t~ra fi Fig. 2. Lower bounds on the right-handed neutrino mass MN from Yukawa unification, Curves for three values of as = 0 . I t 0 ( s o l i d ) , 0.117 (dots) and 0.125 (dash) are shown, both for the SUSY panicle spectra ( a ) msusY = I TeV, ma = m z (lower) and (b) m s u s ¥ = mA = mz ( u p p e r ) ; see text. The maximal possible hot dark matter density l~vh 2 from z-neutrino is also shown for comparison in dot-dashed lines.
the tan/3 values, we obtain stringent lower bounds on MN. We show the cases of two representative SUSY particle spectra (a) msusY = 1 TeV (conservative) and (b) msusv = mz (more stringent), while always keeping ma = mz (conservative). Curves are shown for three values of ee.,.(mz) = 0.110 (solid), 0.117 (dotted), and 0.125 (dashed), where the lowcr ones correspond to case (a) and the upper ones to (b). The values of MN above the curves are consistent with Yukawa unification. The constraint becomes stronger for larger values of e,. (rex), smaller values of msus¥, and larger values of mA. The curves do not extend to the region tan/3 > 58 because of the constraint hi, = h~- < 2 at MG. When the curves reach MG, it means that the lower values of tan/3 are not consistent with Yukawa unification even when M,v = M~; (with h,(MG) < 2). Possible effects from the nonlogarithmic SUSY threshold corrections and physics above the GUT-scale will be discussed in points (a) and (b) below. The lower bound on MN has a very interesting implication for the value of ~L,, the ~,, contribution to the present HDM density. The mass of vT is related to MN and hE via the seesaw formula, 12
-
MN
(12)
'
where t, = 174 GeV and hjv in the effective theory below ME was defined after Eq. (9). Moreover, the cosmic energy density of a neutrino o f mass m,, < I MeV is simply proportional to m,, [22],
~ , h 2 ~_
m,,
(13)
91.5 eV Here, h is the normalized Hubble constant, h = H0/( 100 k m s -j Mpc -j ). Note that hN(mz) cannot be larger than an infrared fixed point value, which is hN(mz) < 0.8 for the cosmologically interesting region MN < 1013 GeV. Using that value in Eqs. ( 12)(13), we obtain the maximum possible value of ~ h z for a given MN. By requiring that t,~, gives a certain contribution to L~, one can put an upper bound on MN. We indicated such bounds for ~L,h 2 = 0.01, 0.1, and 1 in Fig. 2 with dotted lines. We find that the cosmologically interesting range ~1,,h2 > 0.1 survives only for large tan 13, especially for larger values of e~.(mz). For c¢,(mz) >~ 0.125, we lind solutions only for very large tan/3, and they completely disappear for o¢~(mz) > 0.13. In other words, this analysis puts an upper bound on the cosmic HDM density as a function of tan/3. For instance, we get ~ v h 2 < 0.004,
(14)
for tan/3 < I0 even with the most conservative parameters a , ( m z ) = 0. II and case (a). 5. Wc now discuss the GUT-scale threshold corrections o n ht = hN, hb = hr. J3 They mainly come from the mass splittings within the heavy gauge multiplet as far as there are no other sizable Yukawa couplings of 163 to other GUT-scale fields [ 15]. For instance, mass splittings of 69(10) modify the boundary condition by less than 5%. A 5% correction on h,/hN leads to less than 1% change in the prediction of R ( m z ) so that it will not change the constraint at I~ As e x p l a i n e d above, o u r procedure i m p l i c i t l y requires a G U T scale threshold c o r r e c t i o n on a3 ( MG ) to r e p r o d u c e a', ( m z ) ; h o w -
12 Here wc neglect SUSY- and weak-scale threshold corrections on m,,,, which are at most (.9(5%). Neglecting them is also justified because they affect the constraint o n M N only linearly. In contrast. the analogous corrections to R ( m z ) discussed above are larger and affect the constraint o n M N exponentially.
ever, the required c o r r e c t i o n a m o u n t s to at most 5 % i f we vary
a ~ ( m z ) = 0 . 1 1 - 0 . 1 4 and mst;sY = m z - I rections on ot3 and
TeV
T h o u g h the cor-
hh/hr are not directly correlated, this value
gives us a rough idea of the magnitude of the GUT-scale threshold corrections.
A. Brignole et al. / Physics letters B 335 (1994) 345-354
all. However a 5% threshold correction on hh/h~ results in a 5% correction in R ( m z ) , leading to a two order of magnitude change on MN (see Fig. I) and one might imagine a situation where the combined effects of" oe~.(mz) = 0.1 1, SUSY and GUT thresholds pile up to allow MN = 10 ~2 GeV, even at small tan/3. Situations like this are however rather extreme. When a~(mz) ~ 0.12 (close to the centre of its range) and SUSY particles are light (as they should be), then GUT threshold corrections must amount to 15% to make M,v = 1012 GeV possible, which requires (,9(1000) mass splittings in the gauge multiplet. Relaxation of the maximum allowed ht(MG) up to 3.3 16] could also weaken the constraint on l],,h 2, but no more than by a factor of 7. Thus far we neglected family mixings, and one might ask whether, or how, these could affect our bound. The following two requirements are needed in order to evade our bounds: i) The interaction eigenstate N3 C 163 with Yukawa coupling hN = ht has an C)( 1 ) overlap with a mass eigenstate of mass ~ Mo.. Then the effective hN appearing in the RGE below M c is smaller than ht, and the effect of ht in Eq. (2) is no longer "exactly" compensated, ii) There still is a left-handed neutrino with mass m,, ~ C9(eV). We illustrate the consequences of these two requirements in the case of two families, which may represent the second and third ones. The superpotential reads as W = hieJL,.ejH d + JliJNLiNjH u q- ½ ~ l i j N i N j with i , j =
2, 3, where L, e and N are, respectively, lepton doublet, right-handed charged lepton, and right-handed neutrino superfields. The smallness of CKM angles and SO(IO) relations among Yukawa matrices suggest that both J~j and/z~ are hierarchical in the same basis. By going to the basis where hN is diagonal, we parametrize
0 (M22 ~ 1 = Iv M23
h2
=
0
I
hN,
M23"~ M33// '
(15)
and we expect • ~ m,./m, ,~ 10 -2. Requirement i) implies that at least one of the M 0 should be O(M~;), but not M22 alone. However, requirement ii) leads to
2 + 2e'M23 ~ 2 + M~2)I/2MG, }Det.,QI < '5 ( e4 M33
(16)
351
where ,5 ~ 10 -3 and we used hA, ,--, I. The above inequality implies a hierarchy in the eigenvalues of ,,Q. If all M 0 are O(Mc,), we need a fine-tuning of O('5) to obtain the small determinant in Eq. (16). 14 We can avoid such a fine-tuning only if h4 is hierarchical in the basis where h is diagonal with M22 < 'se2M~7, M23 ~ '51/2eMG a n d M33 "-' MG. T h i s c o u l d result
from an abelian horizontal symmetry, but the expansion parameter '51/2e would be rather small. Moreover, the HDM candidate would be predominantly v u, and the MSW oscillation should occur between v,, and Yr. This is a new possibility which may be worth 0t" further study. Nonetheless both the latter possibility and the fine-tuned case mentioned above show rather extreme fcatures, thus enhancing the importance of the cases to which our analysis correctly applies. These cases include the reasonable situation in which there is no hierarchy among right handed neutrino masses. 6. Finally wc point out two generic uncertainties in b - r Yukawa unification which might affect the predictivity on R. These exist even in the pure MSSM case. J5 Of course, such uncertainties may also affect the bound on MN. (a) SUSY-scale non-logarithmic threshold corrections to R for large tan/3 [ 15,231. Typically the largest corrections appear in the b mass via two diagrams, one involving gluino propagation and the other involving higgsinos. The resulting correction to mb can be written as
(2,.,__ p~m~ m,~e-----g " = tan/3 \ 3rr m~ + ~16rr
m2 } '
(17)
where m~ and At are, respectively, the gluino mass and the stop trilinear coupling, while mt and m2 represent effective SUSY masses out of the loop integrals. In particular mt roughly corresponds to the maximum between the sbottom and gluino masses, while m2 represents that between stop and higgsino (more exact expressions are found in [ 15] ). When the SUSY parameters are all of the same order of magnitude, the typical size of the expression inside brackets in Eq. (17) ¢4The tuning implies that the leading contribution to ,'~! is rotated by O ( I ) , with respect to the leading one in il,,.,v. It may be interesting to ask whether this can result from a flavour symmetry. 15These uncertainty actually arise due to our lack of understanding why m I, ~ rot.
352
A. Brignole et al. / Physics Letters B 335 (1994) 345-354
is ,~ i%. Then for tanfl = (9(1) we can safely neglect them. On the other hand, for tan fl ,'~ 10 their effect could be a 10% reduction of R (notice that the sign of Eq. (17) is not fixed). As already stated, a 10% reduction in R is critical for the bounds on MN, so that D.,, ~ 0.1 could be allowed for tanfl > 10. Notice however that t~mh depends rather strongly on the features of the SUSY spectrum and becomes negligible when either /z or the gluino mass (and At) are somewhat smaller than the squark masses. However after an experimental determination of the SUSY mass parameters, we will be able to make conclusive statements on the tan fl > 10 region. It remains however true that in a relevant region of parameter space (with small 6mh), the bounds shown in the figure are valid. 16 (b) Corrections from physics above the GUTscale. In our analysis we allowed a relatively large h t ( M G ) <" 2, which implies that the Yukawa Landau pole, and presumably some new physics, can be as close as 8Me,.17 In general, the presence of new physics at a scale A > Me; induces nonrenormalizable operators in the effective theory at the GUT-scale. Such operators will in general affect Eq. ( 1 ) with contributions scaling as powers of M c , / A = ~7, and one could ~xpect 71 ,~ 0.1 (e.g. from the above observation). Indeed, the order of magnitude of this ratio naturally suggests that flavour mixings and mass ratios are generated by the same type of effects. From this viewpoint, it is legitimate to make the order of magnitude interpretation r/ Vet, ~- 0.05. Flavour scenarios of this type have been depicted in Refs. [27,18]. We will therefore use the indicative value above to estimate the effects of nonrenormalizable operators, which we classify below. A first class of operators is obtained by inserting S O ( 1 0 ) breaking Higgs fields in the renormalizable ~6 It must be added that the approximate symmetries which render large tan/3 natural suppress t~mb [151. On the other hand, these approximate symmetries require m~,us¥ ~ tan/3m 2 because of the LEP constraint/1,, ml/2 ~ mz. Therefore the case tzmJm2usv I and the case/zm,~/mstls¥ ~ I / t a n / 3 require a comparable finetuning in the Higgs sector 124,251. However, constraints from b ~ sy again seem to favor the small 6rob case 125,261. 17 If one requires perlurbativity up to M p l a ~ k / V ~ , one needs ht (MG) < 1.5. On the other hand, notice that the value ht ( MG ) = 3.3 allowed in previous studies 16] implies that the Landau pole is closer than 2Mc,.
term 16310H163. Our assumption that there be small irreps only (45's, 54's, 16's, 16's and 10's ) constrains the lowest possible correction to be O('q2) ,.~ 1%. These might come from 16310H (45) (45) 163/A 2 and 16310H (16a) (1-'6a) 163/A 2. 18 Such effects can be safely neglected. A second class of operators involves the other Higgs fields 10h, 16H, and 16H, rather than the 10H appearing in the renormalizable term. The only possible limension five operators are 16310~..54)163, 16316h(45)163, 16316H(16H)I63, 16316H(16H)163. None of them affects h b = hr, while the last one may induce a correction O('r/) ~ 10% to ht = hN. Such a correction does not affect our analysis, as we already stated (see Section 5). On the other hand, dimension six operators can affect hh = hr. Actually, the relevance of this class of operators strongly depends on the magnitude of hh, h~. themselves, i.e. on tanfl. When hh,T > r/ (or tanfl > r l m t / m b ,~ I0), dimension six terms can give corrections which are O(rl~-/ht,,r) <~ r I ,-~ 5-10%. We recall that, lbr h~,.r << 1, Hd sits mainly in other Higgs fields than 10n. In fact, for hh,¢ ~ r/, the above dimension five operators become a natural source for these Yukawa couplings. When hh.~. < r/, i.e. tanfl < 10, the dimension five operators themselves have to be suppressed, since they would typically yield O ( r / ) Yukawa couplings. This could result from a flavor symmetry. In this situation the dimension six operators become relevant and can finally even dominate h/,.~. Since these terms can give both hb = h,. and 3hb = -h~. (via "composite" 126 combinations), or a mixture of both, the only way not to lose predictivity completely is to suppose that only the first type of terms exists. With this assumption the corrections to hb = hr are again expected to be at most O ( r / ) , and could be actually O(r/") (with n > 1 ) in more specific flavour models. In short, ht, = h,- is an automatic consequence of gauge symmetry and field content alone for hl,.~. ~ "r/. For smaller hb.,, the relation ht, = h~. should probably result from tlavor symmetries. We are aware that the last class of corrections is less under our control than all the other ones discussed so far. In fact, an opposite viewpoint would 18 Note that (54) preserves 0 ( 6 ) x 0 ( 4 ) (Pati-Salam group), and cannot modify Eq. ( 1). Also, 16310H(45)163 respects Eq. ( 1 ) due to the symmetry between two 163.
A. Brignole et al. /Physics Letters B 335 (1994) 345-354
emerge if neutrino-oscillation experiments like CHORUS or N O M A D should find m,,, > eV. This discovery could indeed be used to probe the existence of higher dimensional effects. For instance, should the observations suggest M,v = 1012 GeV, then wc would need R(Mc;) = 1.05, 1.13, 1.23, for ec,(mz) =0.110, 0.117, 0.125, respectively. It is manifest that for the larger values of ce, we need the second class of effects (and larger than expected), while for the smaller a , GUT thresholds alone could account for the deviation of R(Mc,) from unity. These remarks could be a useful guide in building realistic GUT models. For instance we stress that when only first class effects arc present, like in Ref. [27], then the condition in Eq. ( 1 ) is robust and so are our bounds on MN. 7. In summary, we have studied the impact on the low energy value of m~,/mr of an intermediate mass right-handed neutrino (N) in a class of S O ( I 0 ) models. The analysis generally results in a lower bound on M,v, and thus in an upper bound on the r neutrino mass and on its contribution ~),,, to the HDM density in the present universe. In order to do so we have performed a two-loop study of the RGE and discussed threshold effects at the GUT and SUSY scales. Depending on the values of ot.~ and tan/3 the bound varies from a very strong one to a weaker one. i) When tan/3 > 10, there are uncertainties in the bound on M,v from the yet-unknown non-logarithmic SUSY threshold corrections. When these effects are maximal, then M N is allowed to span all the phenomenologically interesting region (down to MN ~" 1012 GeV). On the other hand, when these effects are small, then only for very large tan/3 >~ 50 can we reach the cosmologically interesting region MN < 1013 GeV. We stress that effects at the SUSY-scale threshold will be known once SUSY particles are discovered and their parameters measured, so that such effects will be no longer ambiguities. Instead, we will be able to evaluate the corrections, which might even make the constraint stronger. Also, physics beyond M~ does not affect the boundary condition since h~,.¢ are relatively large. ii) For small tan fl, we get rather stringent bounds on iL,. For o~,(mz) = 0.11,0.117,0.125 we respectively get IL, h 2 < 4 x 10-3,3 × 10-5,6 × 10 - 6 for tan/3 _< 10. SUSY thresholds do not modify thcse bounds. If wc allow for "maximal" GUT threshold ef-
353
fects of O ( 5 % ) , these bounds can go up by about two orders of magnitude. We point out that there might be additional corrections to ht,/hr = 1 at Mcl from physics beyond Me. Assuming that the new physics is responsible for the tlavor structure, we estimate the size of the corrections to be O(V~.h) ~ 5%. Even taking all these possible ambiguities into account, the cosmologically interesting region [L, ~ 0.1 is allowed typically only for a.~ ( m z ) < 0.12. Improvement in the experimental knowledge of as [ 28 ] will allow tighter bounds, especially if the value will converge towards the present central one ~ 0.12. This work was supported by the United States Department of Energy under contract DE-AC0376SF00098 (A.B. and H.M.), and by NSF grants PHY-91-21039 (R.R.), PHY-89-04035 (H.M. and R.R.). The work of A.B. is also sponsored by an INFN fellowship. H.M. and R.R. would like to acknowledge the hospitality and support of the Institute for Theoretical Physics at Santa Barbara, where this work was started.
Note added. After completing the main part of the work described in this letter and presenting it at a conference [ 29 ], we received a preprint by Vissani and Smirnov discussing a similar topic [30]. Their basic conclusion is that MN < 1013 GeV disfavors Yukawa unification, or vice versa, which is the same as ours. However when deriving quantitative lower bounds on M,v, their analysis is based on one-loop RGE with tree-level matching, while ours is based on two-loop RGE with one-loop matching. We consider this to be necessary due to the high sensitivity of the bounds on MN to 69(5-10%) effects which might arise from SUSY-scale and GUT-scale threshold corrections. In addition, they do not discuss the possible relevance of points (a) and (b) we discussed above. However, in the cases where all such effects are negligible, our result is consistent with theirs.
References I11 H. Georgi and S.L. Glashow, Phys. Rcv. Lctt. 32 (1974) 438. 121 M. Chanowitz, J. Ellis and M. Gaillard, Nucl. Phys. 128 (1977) 506:
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A. But'as, J. Ellis, M. Gaillard and D.V. Nanopoulos, Nucl. Phys. 135 (1978) 66; J. Harvey, P. Ramond and D.B. Reiss, Nucl. Phys. B 199 (1982) 223. S. Dimopoulos and H. Georgi, Nucl. Phys. B 193 (1981) 150; N. Sakai, Z. Phys. C 11 (1981) 153. P. Langacker, UPR-O553-T, hep-ph/9305255, invited talk at 30-years of Neutral Currents. From Weak Neutral Currents to the W/Z and beyond, Santa Monica, Feb 3-5, 1993. H. Arason et al.. Phys. Rev. Lett. 67 (1991) 2933; Phys. Rcv. D 47 (1992) 232; M. Carena, S. Pokorski, C.EM. Wagncr, Nucl. Phys. B 406 (1993) 59. V. Bargcr, M.S. Berger and P. ()hmann, Phys. Rev. D 47 (1993) 1093: P. Langacker and N. Polonsky, Phys. Rev. D 49 (1994) 1454. L. Wolfenstein, Phys. Rev. D 17 (1978) 2369; P. Mikheyev and A. Smimov, Nuovo Cimento C 9 (1986) 17; H. Bethe, Phys. Rev. Lctt. 56 (1986) 1305. M. Fukugita and T. Yanagida, in: Physics and Astrophysics of Neutrinos, eds. M. Fukugita and A. Suzuki (SpringerVerlag, Tokyo, Japan, 1994) p. I. G.E Smoot et al., Astrophys. J. Lett. 396 (1992) LI. S.J. Maddox, G. Efstathiou and W.J. Sutherland, Mon. Not. R. Astron. Soc. 246 (1990) 433; M. Davis, G. Efstathiou, C.S. Frenk and S.D.M. White, Nature 356 (1992) 489. A. Klypin, J. Holzman, J. Primack and E. Reg(is, Astrophys. J. 416 (1993) I. H. Murayama and T. Yanagida, Phys. Lctt. B 322 (1994) 349; FI. Murayama, H. Suzuki, T. Yanagida and J. Yokoyama, Phys. Rev. Lctt. 70 (1993) 1912. V. Barger, M.S. Berger and R Ohmann, talk presented at IFT Workshop on Yukawa Couplings and the Origin of Mass, Gainesville, Florida, Feb 11-13, 1994, MAD-PIt-831, hepph/9405245;
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C. Kolda, L. Roszkowski, J.D. Wells and G.L. Ka TH-94-03, hep-ph/9404253; N. Polonsky, talk presented at SUSY '94, May 14-i Michigan, UPR-0595T, hep-ph/9406225. A. Font, L.E. lb,'lfiez and E Quevedo, Nucl. Phy: (1990) 389. L.J. Hall, R. Rattazzi and U. Sarid, LBL-339~ ph/9306309. B. Ananthanarayan, G. Lazarides and Q. Shaft, PP D 44 (1991) 1613. J. Hisano, H. Murayama and T. Yanagida, Phys. R, (1994) 4966. D.B. Kaplan and M. Schmaltz, Phys. Rev. D 49 3741. J.E. Bj~rkman and D.R.T. Jones, Nucl. Phys. B 25 c 533. L.E. Ib,'ifiez and C. L6pez, Nucl. Phys. B 233 (19[ R. Miquel, talk presented at 22nd INS Symp. Phy~ High Energy Colliders, 8-10 Mar 1994. Tokyo, PPE/94-70 (1994). E. Kolb and M. Turner, The Early Universe (, Wesley, 1990). R. Hempfling, DF.kSY-93-092 (1993). A. Nelson and L. Randall, Phys. Lett. B 316 ( 199 R. Rattazzi, U. Sarid and L.J. Hall, SU-ITP-94-15, 37, hep-ph/9405313 (1994). R. Rattazzi and U. Sarid, in preparation. G. Anderson et al., Phys. Rev. D 49 (1994) 3660. K. Fujii, T. Matsui and Y. Sumino. KEK-PREPR 125, Oct (1993). H. Mumyama, talk presented at SUSY '94, May I Michigan. F. Vissani and A. Yu. Smirnov, SISSA-63-94-1 ph/9405399, May 30, 1994.
Physics Letters B 335 (1994) 345-354
I'll ^~t!\.'II;R
Upper bound on hot dark matter density from unification *
SO(10) Yukawa
Andrea Brignole a'l , Hitoshi M u r a y a m a ~,2, Riccardo Rattazzi
b
a Theoretical Physics Group. Lawrence Berkeley l,aboratory. University of California, Berkeley. CA 94720, USA h Department of Physics and Astronomy, Rutgers University, Piscatawa); NJ 08855. USA Received 12 July 1994 Editor: M. Dine
Abstract
We study low-energy consequences of supersymmetric SO(10) models with Yukawa unification ht = hN and hb = hT. We find that it is difficult to reproduce the observed mb/m~ ratio when the third-generation right-handed neutrino is at an intermediate scale, especially for small tan/3. We obtain a conservative lower bound on the mass of the right-handed neutrino M,,~ > 6 × 10 ~3 GeV lbr tan/3 < 10. This bound translates into an upper bound on the r-neutrino mass, and therelore on its contribution to the hot dark matter density of the present universe, lq,,h 2 < 0.004. Our analysis is based on the full two-loop renormalization group equations with one-loop threshold effects. However, we also point out that physics above the GUTscale could modify the Yukawa unification condition hh = hT for tan/3 < 10. This might affect the prediction of mh/m~ and the constraint on MN.
I. G r a n d unification [ l ] is o n e o f the leading guiding principles to build models o f fundamental forces in naturc. T h e simplest m o d e l s also lead to a partial unification o f Yukawa c o u p l i n g s [ 2 ]. S u p c r s y m m e t r y ( S U S Y ) , on the other hand, appears as a necessary principle to m a k e grand unified m o d e l s theoretically viable 13]. It is then remarkable that the hypothesis o f s u p e r s y m m e t r y and o f a "grand desert" between the Fermi and G U T scales is p h e n o m e n o l o g i c a l l y suc* This work was supported 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-AC03-76SF(K)098, and by NSF grants PHY-91-21039 and PHY-89-04035. i Supported by a fellowship of lstituto Nazionale di Fisica Nuclc~e, ,~zione di Padova, Padua, Italy. 2 On leave of abscnce front Department of Physics, Tohoku University, Scndai, 980 Japan. Elsevier Science B.V. SSDI 0 3 7 0 - 2 6 9 3 ( 9 4 ) 0 0 9 3 9 - 2
cessful for both gauge (see, e.g. [4] and references therein) and Yukawa (hb = h~,) [5,6] unification. On the other hand, astrophysics and c o s m o l o g y suggest that there may be an intermediate scale in the "desert". O n e indication c o m e s from the observed deficit in the solar neutrino flux, which favors neutrino oscillations ~ la M i k h e e v - S m i r n o v - W o l f e n s t e i n ( M S W ) [ 7 ] . If one assumes v,,-v~ oscillation and SO( 10)-like mass matrices, then the data suggest the existence o f right-handed neutrinos N with an intermediate mass MN ~ 101°-1013 G e V (see, e.g. 18] ). A n o t h e r indication c o m e s from the c o s m i c density fluctuations at C O B E [9] and galaxy [101 scales. These are in moderate contradiction with each other when a scale-invariant spectrum and dark matter o f cold type only are assumed. C o e x i s t e n c e o f cold and hot dark matter can solve this p r o b l e m [ 11 ]. If
.346
A. Brignole et al. / Physics Letters B 335 (1994) 345-354
hot dark matter (HDM) is given by ~,r, then, again, MN "~ 1012 GeV is favored. Moreover, right-handed sneutrinos N can play interesting cosmological roles in baryogenesis and inflation [ 121. Therefore, it is important to study the phenomenology of SUSY GUTs in which right-handed neutrinos are the only thresholds in the grand desert. Obviously, gauge unification is not significantly affected since right-handed neutrinos are gauge singlets. This letter studies the unification of the Yukawa couplings of the third family in SUSY GUTs with righthanded neutrinos at an intermediate scale. We consider SO(IO) models with Yukawa unification ht = hN and hb = h,-. We find that an intermediate scale right-handed neutrino has a critical influence on the predicted value ofmh/m~. Consistency with the experimental value o f m h / m T implies a lower bound on MN. This constraint translates into an upper bound on the v~ contribution ~L, to the HDM density in the present universe. Cosmologically interesting values of IL, are strongly disfavored for small values of tan/3.3 2. We first describe the theoretical framework of our study. We assume a simple SO(IO) GUT scenario where only small irreducible representations (irreps) like 10, 16, 16, 45, 54 and possibly singlets are present, 4 and matter fields come only in 16's. As for the Yukawa interactions, the large value of the top Yukawa coupling suggests that it originates from a renormalizable coupling. With our assumptions this can only be hc,16310H 163. (Even a 120 would not couple because of antisymmetry.) In general, the two light doublets H~.d of the minimal supersymmetric stanThere is a claim 1131 that b - r Yukawa unification combined with the value of mt recently suggested by CDF excludes the region 1.5 < tan/3 <_ 30. This claim, however, depends crucially on a's(mz) -~ 0.125 and mt ~< 174 GeV, for which the experimental support is not yet established. Thus. we take tan/3 as a free parameter in this letter. 4 Notice that large irreps drive the theory out of the penurbative regime; for example, if a 126 is present, the SO(10) gauge coupling constant blows up below < 8MG even with the minimum particle content 45 + 54 + 126 + 126 + 3 × 16. In superstring models, the above mentioned small irreps appear already at Kac-Moody level k = 2. whereas 126 can appear only for 5 < k < 7 1141. Phenomenologically, the presence of a 126 may also give an unacceptable relation h~ = 3ha 115 I. Finally, it is noteworthy that the absence of 126 and R-odd gauge singlets forbids MN at the renormalizablc level, thus leading to MN << Me,.
dard model (MSSM) contain only fractions of those in 1011, due to mixing with doublets in other irreps. In our case these can be 10 u' 's, 16n's and 16n's, the latter ones being necessarily present in order to reduce the rank of the gauge group. The mixing strengths are in general different for H~ and Hd because S U ( 2 ) R is broken at Me,. Then the GUT boundary condition on the Yukawa couplings is ht = hN = s,h~, ht, = hr = sdha, where S,.d are the Higgs mixing angles, and in general s~ ~ Sd. Here the meaning of hi, hN, hb, and h~, is the obvious one. The particular case su = Sd has been studied before [ 16,15]. Actually the boundary condition hb = hT
ht = hN,
( 1)
is just a result of the underlying Pati-Salam SU(4) symmetry, so that one may imagine scenarios, maybe in strings, where Eq. ( 1 ) holds even without a unified simple gauge group. There are also more exotic scenarios with mixings between 16 and 10 matter fields [17], while Eq. (1) keeps holding. Moreover, there is a class of models where hb and hr originate solely from higher-dimension operators, while predicting the same relation [ 18]. We thus conclude that there is a wide class of models ( S O ( I O ) and beyond) which lead to the boundary condition Eq. ( I ). 3. In order to illustrate the low-energy consequences of Yukawa unification when MN < MG, we first consider a simplified picture based on one-loop renormalization group equations (RGE), without threshold corrections. This also allows for a clearer comparison with the pure MSSM case (MN "~ M 6 ) . The RGE for the individual gauge and Yukawa couplings can be derived from the general formulae in Ref. [ 19], both at one-loop order (needed here) and at two-loop order (to be used below), and we do not write them down explicitly. Here we rather focus on the evolution of the ratio R(/z) = hb(Iz)/h~(Iz) from the GUT-scale value R ( M G ) = 1 to the low-energy value R ( m z ) . The latter should be compared with the experimental value of mb/mr scaled up to mz, which we denote by R e x P ( m z ) . The RGE for R reads
R[ 77
+
-
-
-16 2
- tTg3-
42]
~gl) ,
(2)
347
A. Brignoleet al. /PhysicsLettersB 335 (1994)345-354 where t - log/z and h 2 has to be omitted f o r / z < M N. The solution of Eq. (2) can be formally written as R ( m z ) = R ee -v, where Rg :-- (a3(mz)/Cr3(MG))8/9 x
(a'l ( m z ) / a l (MG)) 10/99
where hh and hr are negligible. Then we can write Y = Yi + Y2, where YI and 112 are the first and second terms in Eq. (3). We denote by ho(t) the solution of the RGE for hi (t) in the pure MSSM case with boundary condition ho(M(;) ---* co. Its explicit expression is [20] h~( t) = 47r2(-F' ( t) /3F( t) ), where &;
and
F(t) =- f d; tN
it; t
' "2 [.f h : ( t ) d t + / ( h ~ ( t ) - h ~ ( t ) ) d t Y = 16~ lZ
× (a2(M(;)/ce2(ff)) 3 (oq(M(;)/ot, (ff))13/99.
IN
I(;
IZ
with ta -- log MA. Notice that Y should be considered as a functionofhh(M¢;), ht(M6) and MN. For MN = M(;, the second term in Y drops and the pure MSSM case is recovered. Since R~ is larger than Re~p (mz) by 30-80%, the Yukawa contributions are essential in reducing the value of R(mz). In fact, the three terms in Y are all positive. In the pure MSSM case, one could formally make Y arbitrarily large (i.e. R(mz) arbitrarily small) by taking h,(Mc,) large enough [201. In practice, h,(MG) ~-- (.9(1) is required to reproduce the correct Rexp(mz). In contrast, for MN < Ma a full cancellation of Yukawa contributions occurs at p, M¢;, due to the boundary conditions ( 1 ). Thus Y cannot be larger than afinite maximum value Yma~(MN), obtained in the limit hr(MG) ---* :)c. Since Ymax(MN) decreases as Mx decreases, it may be impossible to get R(mz) as low as Rexp(mz) when MN is lighter than a certain critical value. This is why a lower bound on MN appears. Actually, Ym~,,~(MN) crucially depends on tan/3. For fixed MN, Ymax(MN) is smaller when tan/3 is small, because h5 and hr are negligible and the last term in Y is missing. In fact this is the situation where stronger constraints on MN will be found. On the contrary, when tan/3 is very large, the last term in Y is significant (notice also the factor 3). Then, even if MN is light, Y can be made large enough as to reduce the value of R appropriately. The above discussion is supported not only by a numerical analysis, but also by the following analytical argument. We show that Y cannot be arbitrarily large even with ht(M~D ---* oc, in contrast to the pure MSSM case. We consider the most interesting case
A study on the RGE of hi(t) for t < MN and t > M s , with a given boundary condition ht(MG), leads easily to the upper bound hi(t) < ho(t), valid for any t and independent of ht(MG). This gives an upper bound on YI,
F(mz) Yl < Y~*( M~/) - ~ log F( MN~
(4)
where we have replaced for convenience the arguments ta with MA. A simple bound on Y2 can bc derived, e.g. by studying the RGE for the ratio/~(t) h t ( t ) / h N ( t ) , which reads l d]~ 1 [3(h~ - h 2) -- (16g2 /~ dt - 1697"2 .3- 3 + ~ g ~ ) ] "
(5)
If one formally integrates the above equation term by term between tu and tG, the right-hand side contains 112 and easily integrable gauge terms. The left-hand side gives just - log/~(tN), which is negative (one can easily prove that /~(t) > 1 from Eq. ( 5 ) ) . Theretore one obtains an absolute bound o~3(MN)
Y2 < Yf(M,v) =" ~ l o g cr3(Mo-------~
2 l o cet g (MN) - 297 cr;(Mc;) (6)
Notice that the bounds (4) and (6) are independent of the initial values ht (Me;) = hN(Mc;), so they hold even in the formal limit ht(MG) ---* oG. In conclusion, for a given MN < MG we explicitly get an absolute upper bound Ymax(MN) on Y which is a monotonically increasing function of MN 5 z Although Ymax(M.,v) is always monotonically increasing, the same applies to the analytic bound Ymax(MN) for the region of interest MN ~ 10~1' GeV. We add that stronger analytic bounds than that described here can be found.
348
A. Brignole et al. / Physics Letters B 335 (1994) 345-354
Y _< Ymax(MN) < Ymax(MN) = YI*(MN) + YZ* ( M s )
.
(7) This in turn gives an absolute lower bound on R(mz)
R(mz) >
R
"X~ (F(mz) e-r:~, (M~') = &, g \F(MN) )
\ 3-U oi j
/
•
l/ l2
the same scale. 6 On the other hand, we take ots(mz) as an independent parameter, and allow a3 v~ a l = a2 at MG. This difference could be accounted for by GUT threshold effects. The gauge coupling constants in the MSSM at mz are o,11(mz)=59.1+1(12
(8)
27r
msusY
.-3-log - ?nZ
+ 1 log
mA'] , ?nZ/I
a~-I ( m z ) = 29.4 + ~ ( 4 1 o g - - s m s u s~-v I log __),mA mz mz 4
msusY mz
a31 (mz) = or.?I (mz) + ~ log - This bound can be compared with e e x P ( m z ) , s o that a lower bound on M s can be interred.
4. We have shown an analytic bound on R(mz) in which hN(M6) was allowed to go to infinity. In practice hN will not be taken larger than O ( 1 ). Then the effect o f MN '~ 1013 GeV can be roughly estimated by treating it as a threshold correction to R at Me; [ 15]. This gives R(M¢~) ~- 1 + (h2N/16zr z) log(MaiM~v), which amounts to an O ( 1 0 % ) increase in R(Mc,) and hence in R(mz). A 10% is critical when comparing to R~Xp(mz), since already in the pure MSSM the predicted R(mz) lies in the upper part of the allowed range. Nonetheless it is clear that all other effects of the same order may have a huge impact on the bound on M s , because R(mz) depends roughly logarithmically on M/v. Then to make our analysis more accurate, we have to discuss SUSY and G U T thresholds and use two-loop RGE. In fact the effects of these, and especially the first two, can possibly pile up to O ( 1 0 % ) . We also attempt an estimate of the possible effects from physics above M6, even though our understanding of these can be only qualitative. Finally, by varying the QCD coupling a s ( m z ) within its experimental range we get an O ( 1 0 - 2 0 % ) effect, which we treat with the proper attention. Our reference value of as(mz) is the one from the Z ° hadronic width as(mz) = 0.124 + 0.007 [21], and we allow a 2o" range. To obtain quantitative constraints on M s , we perform a numerical analysis based on two-loop RGE with one-loop threshold corrections. In order to be able to discuss the dependence on a.~(mz) [6,15], we take the following procedure. We define MG as the scale where a l and az meet, and take ht = h s , hb = hr at
Here the logarithms take care o f the matching between the SM and the MSSM; we take the additional Higgs doublet at ma and the rest o f the SUSY particles at msusv. (The non-logarithmic parts and the translation from MS to DR are numerically small and have been neglected.) We also match the couplings in the M S S M + N to those in the M S S M a t / x = Ms, 7 h b ( M s ) [MSSM = hb(Ms)]MSSM~ N, h r ( M N ) [MSSM = h e ( M N ) [MSSM-N( I -- e ) ,
h t (M/v) [MSSM = h t ( MN ) [MSSM ~-N( l -- e ) , hN(MN)IMSSM = hN(MN)!MSSM~-N(I -- 2 e ) ,
(9)
where e = h~(Mu)/327r 2. h/v(#) in the M S S M is defined in such a way that the coefficient o f the dimension-five operator ( L H , ) 2 is h2u(i.t)/MN. For any fixed values o f (hi, hb) at M6, the actual calculation is done as follows. We use the two-loop RGE of the M S S M and the M S S M + N . By using an iterative procedure, we find the values of M c , O ' I ( M G ) and a 3 ( M ¢ ' ; ) which reproduce the values of a l ( m z ) , a 2 ( m z ) and a3(mz) defined above. At the same time we obtain the values o f the Yukawa couplings at mz. The resulting b-r Yukawa ratio R(mZ)IMSSM is then matched to the ratio o f the corresponding masses in the broken electroweak theory R ( m z ) = rnb(mz)/mr(mz),
R(mz) = ( l + k h - k r + f R )
R(rnz)IMSSM,
(10)
6 Numerically, MG ~ 2 x 1016 GeV. GUT threshold corrections to Yukawa couplings will be discussed later. 7 We define the right-handed neutrino mass MN by MN = M s ( M s ) , where MS(p,) is the DR running mass.
A. Brignole et al. / Physics Letters B 335 (1994) 345-354
where kb, k~, are the threshold corrections due to the SUSY particles, while fR is that from the additional Higgs doublet in the MSSM. The exact expressions are found in Ref. [ 15]. In the following we will only focus on the logarithmic terms in kb.T. We will discuss the potentially large non-logarithmic ones (k~, in Ref. [ 151 ) at the end of the letter. Note that also R(mZ)IMSSM depends implicitly on mSUSY and mA due to the threshold dependence of the MSSM gauge coupling constants a71(mz) (see above). For instance, in the small tan 13 region, the dependence of R on the SUSY particle masses turns out to be approximately 8 -~SR - - ~ - - 1 ( - 0 . 1 logmSUSY + 0 . 1 log ma "]
R
27r
mz
mz J '
(11)
which essentially vanishes for msusY = ma. This agrees with the results in Ref. [6]. In the final step, the predicted R(mz) is compared to ReXp(mz). Notice that also R~Xp(mz) has an important dependence on ce~(mz), since it is obtained by scaling the b-z mass ratio up to mz. GUT-scale threshold corrections could also affect the boundary condition in Eq. (1). Though we neglect such corrections in the following numerical analysis, we will discuss their possible effects later. We first illustrate the dependence of R ( m z ) on ht(Mc;) for the region tan13 ~ 10, where both hb, h~. get rcnormalized homogeneously and the result depends very weakly on tan 13. We show curves for different values of MN in Fig. 1, taking msus¥ = ma = mz, hi, = h~, = 0 . 0 1 at Me;, and crs(mz) =0.11 (Fig. la) or 0.12 (Fig. lb). For comparison, we also show the values of R e x P ( m z ) corresponding to the MS mass mt,(mb) = 3.9, 4.15, and 4.4 GeV. This is the range which is obtained from QCD sum rules [ 1 5 ] . 9 First of all, it is clear from the figures that it becomes harder 8 Even if the SUSY spectrum is nondegenerate, the dependence of R on the SUSY panicle masses can again be summarized by an effective msusY. It turns out that the dependence of R on colorless panicle masses is very weak, and msusY is approximately (m,?n~) 1/2. After considering generic splinings in the SUSY spectrum. we found that the value of R varied only between two extreme cases, ms,us;y = I TeV, ma = mz (smallest R), and msusY = mz, ma = 1 TeV (largest R). We stress that the approximate formula Eq. ( 1 I) cannot be applied for ma ~ mz. '~ The uncertainty on rob(mr,) is dominated by the yet-unknown (_9(a~) corrections to QCD sum rules [151.
L[(a) 2 4
i
349
"
a~(mz)-O i I ....
~"!" ..... '' I''"~ v('b) a , ( r n z ) = 0 . 1 ~
--
M:~= i013 'MN= .... MN=IO Is
:~.~ ~ ~ ~ .
I
--
~,
1014 "~
-Ms=10~°
t
2
4
"MN=IO'2t
MN=MG
_
I
1 (3 ~
-..........
(J
:-. . . . 0
J__ l
2
ht(Mc)
I .... 3 4 0
1
3
ht(MG)
Fig. 1. An illustration of the effect of the right-handed neutrino on R ( mz ) = mb ( mz )/mr( mZ ). The curves show the dependence of R ( m z ) on ht(MG) = hN(M(;), for the values of M,v = MG, 1015 GeV, l 0 In GeV. 1013 GeV, 1012 GeV. and 10 m GeV from below. The experimentally allowed values of R ( m z ) are also shown. These correspond to mb(mh) = 3.9, 4.15 and 4.4 GeV. Other input narameter~, are taken as m s u s y = mA = mz, ht~(MG) = hr(MG) = 0.01. and cts(mz) = 0.11 (a) or 0.12 ( b ) .
to reconcile R ( m z ) with ReXp(mz) for larger a,.(mz ) and lower MN. For a~.(mz) = 0.12, one needs very large ht(MG) > 3 even for MN = M~. 1o Moreover, even if increasing ht(M6) makes R(mz) dccrease, the suppression effect becomes weaker as we lower Ms. Note also that the value of R(mz) depends only weakly on ht(Mc;) for ht(MG) ~ 2, especially for MN < 1014 GeV. Such a "fixed point" behaviour and the dependence on MN were expected on the basis of the analytic discussion above (see, e.g. Eqs. (7), ( 8 ) ) . As we lower MN, we can check whether the curves reach the region of R~XP(mz) and then infer a lower bound on MN. In the following, when scanning the space of (hi(M(;), hb(M(;)), we will only take for definiteness h t , b ( M G ) '< 2. This reference value is motivated both by the above observation and by perturbativity reasons (see also point (b) below). Next we show lower bounds on MN as functions of tan 13 in Fig. 2. I1 In order to be conservative we take the maximum value ofmh(mh) = 4.4 GeV. For most of l ° F o r ot~(mz) ~ 0.12 and ht(MG) < 2, the predicted value of R ( m z ) is larger than the experimental upper bound. Consistency could be possibly restored by allowing 0 ( 5 % ) GUT- or SUSYscale threshold corrections, see section 5. 11 We define tan.8 using m r ( m z ) = h r ( m z ) t ; c o s . 8 , where hr ( m z ) is the MSSM one. Other definitions of tan 8 are related to this one via one-loop corrections.
A. Brignole et al. / Pl~vsics Letters B 335 (I 994) 345-354
350 •k | c .
h2uv2 sin2/3
/ ....................
~
"-
my,
\
101,1
•
...
\,,
-~
~
:,
. . . . . . . . . . .
t~l° i
(x s(r[:.,:) - 0 . 1 1 "/
10 t~ ' . . . . . . . . .
~
'~
,~o(,,,~):0:2:
,
lo
[]~h z,3~L:, -0.0 l
~(rnz) ~
0110
!S,,~
:'o
.t. . . .
3o
-\,
-
......
'\ \,
:,:
\\ '\ I " !
::
~ _ x _ . _ I~
,~o
I
:,o
_'
t~ra fi Fig. 2. Lower bounds on the right-handed neutrino mass MN from Yukawa unification, Curves for three values of as = 0 . I t 0 ( s o l i d ) , 0.117 (dots) and 0.125 (dash) are shown, both for the SUSY panicle spectra ( a ) msusY = I TeV, ma = m z (lower) and (b) m s u s ¥ = mA = mz ( u p p e r ) ; see text. The maximal possible hot dark matter density l~vh 2 from z-neutrino is also shown for comparison in dot-dashed lines.
the tan/3 values, we obtain stringent lower bounds on MN. We show the cases of two representative SUSY particle spectra (a) msusY = 1 TeV (conservative) and (b) msusv = mz (more stringent), while always keeping ma = mz (conservative). Curves are shown for three values of ee.,.(mz) = 0.110 (solid), 0.117 (dotted), and 0.125 (dashed), where the lowcr ones correspond to case (a) and the upper ones to (b). The values of MN above the curves are consistent with Yukawa unification. The constraint becomes stronger for larger values of e,. (rex), smaller values of msus¥, and larger values of mA. The curves do not extend to the region tan/3 > 58 because of the constraint hi, = h~- < 2 at MG. When the curves reach MG, it means that the lower values of tan/3 are not consistent with Yukawa unification even when M,v = M~; (with h,(MG) < 2). Possible effects from the nonlogarithmic SUSY threshold corrections and physics above the GUT-scale will be discussed in points (a) and (b) below. The lower bound on MN has a very interesting implication for the value of ~L,, the ~,, contribution to the present HDM density. The mass of vT is related to MN and hE via the seesaw formula, 12
-
MN
(12)
'
where t, = 174 GeV and hjv in the effective theory below ME was defined after Eq. (9). Moreover, the cosmic energy density of a neutrino o f mass m,, < I MeV is simply proportional to m,, [22],
~ , h 2 ~_
m,,
(13)
91.5 eV Here, h is the normalized Hubble constant, h = H0/( 100 k m s -j Mpc -j ). Note that hN(mz) cannot be larger than an infrared fixed point value, which is hN(mz) < 0.8 for the cosmologically interesting region MN < 1013 GeV. Using that value in Eqs. ( 12)(13), we obtain the maximum possible value of ~ h z for a given MN. By requiring that t,~, gives a certain contribution to L~, one can put an upper bound on MN. We indicated such bounds for ~L,h 2 = 0.01, 0.1, and 1 in Fig. 2 with dotted lines. We find that the cosmologically interesting range ~1,,h2 > 0.1 survives only for large tan 13, especially for larger values of e~.(mz). For c¢,(mz) >~ 0.125, we lind solutions only for very large tan/3, and they completely disappear for o¢~(mz) > 0.13. In other words, this analysis puts an upper bound on the cosmic HDM density as a function of tan/3. For instance, we get ~ v h 2 < 0.004,
(14)
for tan/3 < I0 even with the most conservative parameters a , ( m z ) = 0. II and case (a). 5. Wc now discuss the GUT-scale threshold corrections o n ht = hN, hb = hr. J3 They mainly come from the mass splittings within the heavy gauge multiplet as far as there are no other sizable Yukawa couplings of 163 to other GUT-scale fields [ 15]. For instance, mass splittings of 69(10) modify the boundary condition by less than 5%. A 5% correction on h,/hN leads to less than 1% change in the prediction of R ( m z ) so that it will not change the constraint at I~ As e x p l a i n e d above, o u r procedure i m p l i c i t l y requires a G U T scale threshold c o r r e c t i o n on a3 ( MG ) to r e p r o d u c e a', ( m z ) ; h o w -
12 Here wc neglect SUSY- and weak-scale threshold corrections on m,,,, which are at most (.9(5%). Neglecting them is also justified because they affect the constraint o n M N only linearly. In contrast. the analogous corrections to R ( m z ) discussed above are larger and affect the constraint o n M N exponentially.
ever, the required c o r r e c t i o n a m o u n t s to at most 5 % i f we vary
a ~ ( m z ) = 0 . 1 1 - 0 . 1 4 and mst;sY = m z - I rections on ot3 and
TeV
T h o u g h the cor-
hh/hr are not directly correlated, this value
gives us a rough idea of the magnitude of the GUT-scale threshold corrections.
A. Brignole et al. / Physics letters B 335 (1994) 345-354
all. However a 5% threshold correction on hh/h~ results in a 5% correction in R ( m z ) , leading to a two order of magnitude change on MN (see Fig. I) and one might imagine a situation where the combined effects of" oe~.(mz) = 0.1 1, SUSY and GUT thresholds pile up to allow MN = 10 ~2 GeV, even at small tan/3. Situations like this are however rather extreme. When a~(mz) ~ 0.12 (close to the centre of its range) and SUSY particles are light (as they should be), then GUT threshold corrections must amount to 15% to make M,v = 1012 GeV possible, which requires (,9(1000) mass splittings in the gauge multiplet. Relaxation of the maximum allowed ht(MG) up to 3.3 16] could also weaken the constraint on l],,h 2, but no more than by a factor of 7. Thus far we neglected family mixings, and one might ask whether, or how, these could affect our bound. The following two requirements are needed in order to evade our bounds: i) The interaction eigenstate N3 C 163 with Yukawa coupling hN = ht has an C)( 1 ) overlap with a mass eigenstate of mass ~ Mo.. Then the effective hN appearing in the RGE below M c is smaller than ht, and the effect of ht in Eq. (2) is no longer "exactly" compensated, ii) There still is a left-handed neutrino with mass m,, ~ C9(eV). We illustrate the consequences of these two requirements in the case of two families, which may represent the second and third ones. The superpotential reads as W = hieJL,.ejH d + JliJNLiNjH u q- ½ ~ l i j N i N j with i , j =
2, 3, where L, e and N are, respectively, lepton doublet, right-handed charged lepton, and right-handed neutrino superfields. The smallness of CKM angles and SO(IO) relations among Yukawa matrices suggest that both J~j and/z~ are hierarchical in the same basis. By going to the basis where hN is diagonal, we parametrize
0 (M22 ~ 1 = Iv M23
h2
=
0
I
hN,
M23"~ M33// '
(15)
and we expect • ~ m,./m, ,~ 10 -2. Requirement i) implies that at least one of the M 0 should be O(M~;), but not M22 alone. However, requirement ii) leads to
2 + 2e'M23 ~ 2 + M~2)I/2MG, }Det.,QI < '5 ( e4 M33
(16)
351
where ,5 ~ 10 -3 and we used hA, ,--, I. The above inequality implies a hierarchy in the eigenvalues of ,,Q. If all M 0 are O(Mc,), we need a fine-tuning of O('5) to obtain the small determinant in Eq. (16). 14 We can avoid such a fine-tuning only if h4 is hierarchical in the basis where h is diagonal with M22 < 'se2M~7, M23 ~ '51/2eMG a n d M33 "-' MG. T h i s c o u l d result
from an abelian horizontal symmetry, but the expansion parameter '51/2e would be rather small. Moreover, the HDM candidate would be predominantly v u, and the MSW oscillation should occur between v,, and Yr. This is a new possibility which may be worth 0t" further study. Nonetheless both the latter possibility and the fine-tuned case mentioned above show rather extreme fcatures, thus enhancing the importance of the cases to which our analysis correctly applies. These cases include the reasonable situation in which there is no hierarchy among right handed neutrino masses. 6. Finally wc point out two generic uncertainties in b - r Yukawa unification which might affect the predictivity on R. These exist even in the pure MSSM case. J5 Of course, such uncertainties may also affect the bound on MN. (a) SUSY-scale non-logarithmic threshold corrections to R for large tan/3 [ 15,231. Typically the largest corrections appear in the b mass via two diagrams, one involving gluino propagation and the other involving higgsinos. The resulting correction to mb can be written as
(2,.,__ p~m~ m,~e-----g " = tan/3 \ 3rr m~ + ~16rr
m2 } '
(17)
where m~ and At are, respectively, the gluino mass and the stop trilinear coupling, while mt and m2 represent effective SUSY masses out of the loop integrals. In particular mt roughly corresponds to the maximum between the sbottom and gluino masses, while m2 represents that between stop and higgsino (more exact expressions are found in [ 15] ). When the SUSY parameters are all of the same order of magnitude, the typical size of the expression inside brackets in Eq. (17) ¢4The tuning implies that the leading contribution to ,'~! is rotated by O ( I ) , with respect to the leading one in il,,.,v. It may be interesting to ask whether this can result from a flavour symmetry. 15These uncertainty actually arise due to our lack of understanding why m I, ~ rot.
352
A. Brignole et al. / Physics Letters B 335 (1994) 345-354
is ,~ i%. Then for tanfl = (9(1) we can safely neglect them. On the other hand, for tan fl ,'~ 10 their effect could be a 10% reduction of R (notice that the sign of Eq. (17) is not fixed). As already stated, a 10% reduction in R is critical for the bounds on MN, so that D.,, ~ 0.1 could be allowed for tanfl > 10. Notice however that t~mh depends rather strongly on the features of the SUSY spectrum and becomes negligible when either /z or the gluino mass (and At) are somewhat smaller than the squark masses. However after an experimental determination of the SUSY mass parameters, we will be able to make conclusive statements on the tan fl > 10 region. It remains however true that in a relevant region of parameter space (with small 6mh), the bounds shown in the figure are valid. 16 (b) Corrections from physics above the GUTscale. In our analysis we allowed a relatively large h t ( M G ) <" 2, which implies that the Yukawa Landau pole, and presumably some new physics, can be as close as 8Me,.17 In general, the presence of new physics at a scale A > Me; induces nonrenormalizable operators in the effective theory at the GUT-scale. Such operators will in general affect Eq. ( 1 ) with contributions scaling as powers of M c , / A = ~7, and one could ~xpect 71 ,~ 0.1 (e.g. from the above observation). Indeed, the order of magnitude of this ratio naturally suggests that flavour mixings and mass ratios are generated by the same type of effects. From this viewpoint, it is legitimate to make the order of magnitude interpretation r/ Vet, ~- 0.05. Flavour scenarios of this type have been depicted in Refs. [27,18]. We will therefore use the indicative value above to estimate the effects of nonrenormalizable operators, which we classify below. A first class of operators is obtained by inserting S O ( 1 0 ) breaking Higgs fields in the renormalizable ~6 It must be added that the approximate symmetries which render large tan/3 natural suppress t~mb [151. On the other hand, these approximate symmetries require m~,us¥ ~ tan/3m 2 because of the LEP constraint/1,, ml/2 ~ mz. Therefore the case tzmJm2usv I and the case/zm,~/mstls¥ ~ I / t a n / 3 require a comparable finetuning in the Higgs sector 124,251. However, constraints from b ~ sy again seem to favor the small 6rob case 125,261. 17 If one requires perlurbativity up to M p l a ~ k / V ~ , one needs ht (MG) < 1.5. On the other hand, notice that the value ht ( MG ) = 3.3 allowed in previous studies 16] implies that the Landau pole is closer than 2Mc,.
term 16310H163. Our assumption that there be small irreps only (45's, 54's, 16's, 16's and 10's ) constrains the lowest possible correction to be O('q2) ,.~ 1%. These might come from 16310H (45) (45) 163/A 2 and 16310H (16a) (1-'6a) 163/A 2. 18 Such effects can be safely neglected. A second class of operators involves the other Higgs fields 10h, 16H, and 16H, rather than the 10H appearing in the renormalizable term. The only possible limension five operators are 16310~..54)163, 16316h(45)163, 16316H(16H)I63, 16316H(16H)163. None of them affects h b = hr, while the last one may induce a correction O('r/) ~ 10% to ht = hN. Such a correction does not affect our analysis, as we already stated (see Section 5). On the other hand, dimension six operators can affect hh = hr. Actually, the relevance of this class of operators strongly depends on the magnitude of hh, h~. themselves, i.e. on tanfl. When hh,T > r/ (or tanfl > r l m t / m b ,~ I0), dimension six terms can give corrections which are O(rl~-/ht,,r) <~ r I ,-~ 5-10%. We recall that, lbr h~,.r << 1, Hd sits mainly in other Higgs fields than 10n. In fact, for hh,¢ ~ r/, the above dimension five operators become a natural source for these Yukawa couplings. When hh.~. < r/, i.e. tanfl < 10, the dimension five operators themselves have to be suppressed, since they would typically yield O ( r / ) Yukawa couplings. This could result from a flavor symmetry. In this situation the dimension six operators become relevant and can finally even dominate h/,.~. Since these terms can give both hb = h,. and 3hb = -h~. (via "composite" 126 combinations), or a mixture of both, the only way not to lose predictivity completely is to suppose that only the first type of terms exists. With this assumption the corrections to hb = hr are again expected to be at most O ( r / ) , and could be actually O(r/") (with n > 1 ) in more specific flavour models. In short, ht, = h,- is an automatic consequence of gauge symmetry and field content alone for hl,.~. ~ "r/. For smaller hb.,, the relation ht, = h~. should probably result from tlavor symmetries. We are aware that the last class of corrections is less under our control than all the other ones discussed so far. In fact, an opposite viewpoint would 18 Note that (54) preserves 0 ( 6 ) x 0 ( 4 ) (Pati-Salam group), and cannot modify Eq. ( 1). Also, 16310H(45)163 respects Eq. ( 1 ) due to the symmetry between two 163.
A. Brignole et al. /Physics Letters B 335 (1994) 345-354
emerge if neutrino-oscillation experiments like CHORUS or N O M A D should find m,,, > eV. This discovery could indeed be used to probe the existence of higher dimensional effects. For instance, should the observations suggest M,v = 1012 GeV, then wc would need R(Mc;) = 1.05, 1.13, 1.23, for ec,(mz) =0.110, 0.117, 0.125, respectively. It is manifest that for the larger values of ce, we need the second class of effects (and larger than expected), while for the smaller a , GUT thresholds alone could account for the deviation of R(Mc,) from unity. These remarks could be a useful guide in building realistic GUT models. For instance we stress that when only first class effects arc present, like in Ref. [27], then the condition in Eq. ( 1 ) is robust and so are our bounds on MN. 7. In summary, we have studied the impact on the low energy value of m~,/mr of an intermediate mass right-handed neutrino (N) in a class of S O ( I 0 ) models. The analysis generally results in a lower bound on M,v, and thus in an upper bound on the r neutrino mass and on its contribution ~),,, to the HDM density in the present universe. In order to do so we have performed a two-loop study of the RGE and discussed threshold effects at the GUT and SUSY scales. Depending on the values of ot.~ and tan/3 the bound varies from a very strong one to a weaker one. i) When tan/3 > 10, there are uncertainties in the bound on M,v from the yet-unknown non-logarithmic SUSY threshold corrections. When these effects are maximal, then M N is allowed to span all the phenomenologically interesting region (down to MN ~" 1012 GeV). On the other hand, when these effects are small, then only for very large tan/3 >~ 50 can we reach the cosmologically interesting region MN < 1013 GeV. We stress that effects at the SUSY-scale threshold will be known once SUSY particles are discovered and their parameters measured, so that such effects will be no longer ambiguities. Instead, we will be able to evaluate the corrections, which might even make the constraint stronger. Also, physics beyond M~ does not affect the boundary condition since h~,.¢ are relatively large. ii) For small tan fl, we get rather stringent bounds on iL,. For o~,(mz) = 0.11,0.117,0.125 we respectively get IL, h 2 < 4 x 10-3,3 × 10-5,6 × 10 - 6 for tan/3 _< 10. SUSY thresholds do not modify thcse bounds. If wc allow for "maximal" GUT threshold ef-
353
fects of O ( 5 % ) , these bounds can go up by about two orders of magnitude. We point out that there might be additional corrections to ht,/hr = 1 at Mcl from physics beyond Me. Assuming that the new physics is responsible for the tlavor structure, we estimate the size of the corrections to be O(V~.h) ~ 5%. Even taking all these possible ambiguities into account, the cosmologically interesting region [L, ~ 0.1 is allowed typically only for a.~ ( m z ) < 0.12. Improvement in the experimental knowledge of as [ 28 ] will allow tighter bounds, especially if the value will converge towards the present central one ~ 0.12. This work was supported by the United States Department of Energy under contract DE-AC0376SF00098 (A.B. and H.M.), and by NSF grants PHY-91-21039 (R.R.), PHY-89-04035 (H.M. and R.R.). The work of A.B. is also sponsored by an INFN fellowship. H.M. and R.R. would like to acknowledge the hospitality and support of the Institute for Theoretical Physics at Santa Barbara, where this work was started.
Note added. After completing the main part of the work described in this letter and presenting it at a conference [ 29 ], we received a preprint by Vissani and Smirnov discussing a similar topic [30]. Their basic conclusion is that MN < 1013 GeV disfavors Yukawa unification, or vice versa, which is the same as ours. However when deriving quantitative lower bounds on M,v, their analysis is based on one-loop RGE with tree-level matching, while ours is based on two-loop RGE with one-loop matching. We consider this to be necessary due to the high sensitivity of the bounds on MN to 69(5-10%) effects which might arise from SUSY-scale and GUT-scale threshold corrections. In addition, they do not discuss the possible relevance of points (a) and (b) we discussed above. However, in the cases where all such effects are negligible, our result is consistent with theirs.
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