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B-B̄ mixing and relations among quark masses, angles and phases
Volume 195, n u m b e r 4
PHYSICS LETTERS B
17 September 1987
B - B M I X I N G A N D R E L A T I O N S A M O N G Q U A R K M A S S E S , A N G L E S A N D P H A S E S "~ Haim H A R A R I and Yosef N I R Stanford Lmear Accelerator Center, Stanford Umverslty, Stanford, CA 94305, USA Received 15 June 1987
Using the recent data on B°-l]° mixing we estabhsh constraints on the allowed values of the standard model parameters mt, 013 and & The allowed range is inconsistent ruth the Stech scheme for the quark mass mamces. For almost all allowed values of the parameters, the Fritzsch form of the mass matrices ~s also excluded. Only in the unlikely event that the experimental and theoretIcal quantities xd, ms~rob, 023, BK and fB assume extreme values within their allowed ranges, a set of parameters exists which is consistent with all data and with the Fritzsch scheme. We then obtain the unique solution mt ~ 85 GeV, sin 013~ 0 0035, fi ~ 100 °, yleldmg F(b~u~92 )/F (b~c~9~) ~0 01, maximal B~-B° mixing, relatively small E'/E and BR(K-*nvg) ~ 1× 10 ~o
The quark sector of the three-generation standard model contains ten independent parameters: six quark masses, three mixing angles and one Kobayashi-Maskawa ( K M ) phase. Future theories may lead to a calculation of these parameters or at least to relations among them. Seven o f the ten parameters are reasonably well determined: the masses o f the first five quarks and the mixing angles 012 and 023. For the remaining three parameters mi, 013 and d, we have fairly weak bounds. Several theoretical and phenomenological models which go beyond the standard model have led to the derivation of relations among mixing angles, phases and quark mass ratios. The best known scheme is the one proposed by Fritzsch [ 1 ]. Another interesting scheme has been suggested by Stech [ 2 ] and variations on the two themes were proposed by Gronau, Johnson and Schechter [ 3 ] and by Shin [ 4 ]. The recent observation of Bd-Bd o -o mixing by the A R G U S collaboration [ 5 ] in DESY provides us with important new information on the three poorly determined parameters of the quark sector of the standard model. It also leads to severe constraints on the relations among masses, angles and phases. In this note we investigate the implications of the new Work supported by the Department of Energy, Contract DEAC03-76SF00515 I On leave from the Weizmann Institute of Science, 76100 Rehovot, Israel
586
data and show that the Stech scheme [ 2 ] (as well as its extension by Gronau, Johnson and Schechter [ 3 ]) is now clearly inconsistent with the data. We also show that the Fritzsch [ 1 ] matrix (and its extension suggested by Shin [ 4 ]) disagrees with the central values o f the various experimental parameters and with almost all other allowed values of the parameters. The Fritzsch scheme will survive only if all the following 0 - 0 mixing is at the lowest conditions are met: Bd-Bd level allowed by the error quoted by the A R G U S experiment; the B-meson decay constant is approximately 0.2 GeV (at the top of its "reasonable" range); the hadronic factor BK in the expression for the CP violating parameter e obeys BK ~ 1; the ratio ms/mb is around 0.022 (e.g. m s = 120 MeV, m u = 5 . 4 GeV); the value of the mixing angle 023 is at the top o f its allowed range. Only If all of these constraints are s~multaneously fulfilled will the Fritzsch matrix remain consistent with the data. In such a case, we have m i ~ 8 5 GeV; s13~0.0035; d ~ 1 0 0 ° and the phases o f the Fritzsch matrix must be close to the ones suggested by Shin. We also comment on several consequences o f these results for other experimental quantities. In our analysis we use a choice of mixing angles based on the combined work of Maiani [ 6], Wolfenstein [ 7 ], Chau and Keung [ 8 ] and others. We follow the notation and the precise form of this matrix as given by Harari and Leurer [9]. For the purpose of the
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
Volume 195, number 4
PHYSICS LETTERS B
analysis presented here it is sufficient to neglect corrections of order 0~ and to assume cos0~2= cos 023=c0s 013= 1. The resulting form of the mixing matrix is 1 V~
--s12 --$23s13 e~o S12S23 --S13 e~a
s~2
Sl3 e-~6)
1
$23 1
--$23
(1)
where s,j = sin 0u.We have the following direct information on the three angles: (i) s12=0.221 +0.002. (ii) ~23 ~ -u.,,-,-,_o-n nAa+ooo7009. This result is based on the following assumptions: % = (1.16+0.16) × 10 -12 s [10]; B R ( b ~ c ~ 9 0 = 0 . 1 2 1 + 0 . 0 0 8 [11]; m b = 5.0_+ 0.3 GeV; m~= 1.5 _+0.2 GeV. The values of rnb and rn~ appear in the phase space factors relating the measured b lifetime and the relevant matrix element. It is not entirely clear which values should be used and we have allowed for fairly generous errors. In some cases we will need to use the product %s223. Since the errors on s23 contain the errors on %, the product has smaller errors than s~3 alone. We obtain ZbS~3= (3.25 + 1.10) × 109 G e V - ~. (iii) The only direct limit on s~3 is obtained from the experimental bound [ 11 ]
17 September 1987
cases we can use the data in order to exclude certain ranges of the parameters. The results of the first part of our analysis are similar of those recently obtained by several groups of authors [ 13 ]. We therefore do not give here all the details of the analysis, but quote the assumptions and the main results. We use the following expression [ 14] for E: Ie ] =
CBKs~3q sin ~{ [ r/3f3(Yt) -- ~]1]yes12
+ q2Ytf2(yt)s~3(S12 -- q COS~) }, where
G 2~f2KMK M w2
C~- ~ = 4 × 104, 6zc x/2AMK
y,=m]/M2w
(t=c, t),
3 yt(l+yt) ( l + ~ l n ( y t ) )
f2(Yt) = 1 4 (1 - y t ) 2 f3(Yt) = I n ( y ~ )
43 1 -Yt y t (l+lY_~ytln(yt)).
In order to minimize the ambiguities which are due to the b-quark mass, we use the ratio q = s~3/s23 rather than using s~3. For the values of mb and m~ listed above, we obtain Sl3~<0.011 and
We use Gv= 1.166× 10 -5 GeV-2; f ~ = (0.16 GeV)2; MK= 0.498 GeV; M w = 8 2 GeV; AMK= 3.52 × 10 -15 GeV; r/l=0.7; r/2=0.6; r/3=0.4. The three parameters q, are QCD corrections. We also use the values of me and s23 listed above. The parameter BK is unknown and is usually believed to be somewhere in the range ~ ~
q-~S13/$23~<0.22.
ra =x~/( 2 + xaa) .
The direct lower experimental bound on m t is still 23 GeV. A recent combined analysis of all measurements of sin20w showed [ 12 ] that the standard model radiative corrections are consistent with the data only if mr ~<180 GeV. We therefore consider only the range 23 GeV~
The expression for xa is [ 13 ]
F ( b ~ u ~ 9 0 < 0.08 F(b~cl~9~)
Xa = %( G~/6zc 2) rlMB(BBJ~B)M 2
y ~ ( y t ) l V*~dV~bl2 • In addition to the previous parameters we now also use MB= 5.28 GeV; Vtb= 1; r/=0.85. Here r/is, again, a QCD correction. We also have ] Vtd]2 ~___$223($22+q2 --2s12q COS ~5) .
Note that the expression for xa contains the combination rbS~3 for which we use the value quoted earlier. We also have here two additional unknown parameters: The B decay constant fB and the had587
Volume 195, number 4
PHYSICS LE'FFERS B
ronic parameter B~ which is analogous to the usual BK mentioned above. Based on previous analysis [13] we assume BB~I and fB=0.15+0.05 GeV. What we really use is BBf~ = (0.15+0.05 GeV) :. We can write
Xd = C' ytf2(Yt)( ~2 +
q~
17 September 1987 mt:45GeV, BK:I
m r : 6 5 GeV, BK=I
0.4 ~0.2 n ET
- 2st2q cos 5 ) ,
where
C' = ( %s23)( G~/6~r2)tlM~(B~f~)M~w l Vtb j2
f ~
f t ~'.~................... ' ~~ ',~ " te,~¢*
| 0
I I00 B
I 50
I | 150 O
mr=85 GeV, BK=I
I 50
I I00 8
I I 150
mr= 180 GeV, BK=I
q 6. ~,,,.x_3 1+70
We now use the experimental values ~= 2.3 X 1O-3 and Xd=0.73+0.18 (corresponding to rd=0.21 --+0.08) in order to limit further the allowed range of the three poorly determined parameters rot, s l3 and 5. We do it in the following way: In principle, we express the known quantities d C and xdC' as functions of rot, $13, ~. We select a value of mt and plot the two allowed curves for d C and xdC' in the q-5 plane. If the two curves intersect below the upper limit q~<0.22, the intersection point is an allowed solution for (mr, s13, 5). In practice, Xd has a large experimental error and C' has both a theoretical ambiguity and an experimental error. We therefore obtain a wide band rather than a curve for X d l C t in the q-5 plane for each possible value of mt. The central curve and the two edges of the band correspond to xJC' = 0.14 +0.47 -o 09, as given by the most extreme values assumed above for Xd, ( B j ~ ) , (ZbS~3). Note that the edge of the band corresponds to the very unlikely situation that all these parameters simultaneously obtain their most extreme allowed values. For e/C, we produce a band whose width is determined by the allowed range of s23 and m~. It is narrower than the xJC' band. However, we must repeat the analysis for several different values of BK in the usually accepted range. The overlap regions of the two bands which also lie below q=0.22 provide us with a conservative evaluation of the allowed range of m , s~3, 5. Fig. 1 shows the relevant bands for BK= I and mr=45, 65, 85, 180 GeV. In each case the allowed range of q and 5 values is shown by the shaded area. There is no allowed region for any value of m t below 43 GeV. The central values of the two bands meet in the allowed region only for mt>/85 GeV. The lowest allowed value of q is obtained for the largest m~ and 588
I -
~, 0.2
\ ~
. ...........
.~[
ob 0
50
lib\
x
I00 B
\
"..
IIII
..................... L~.~. \
.S ......................
4....S
150 0
50
I00 B
150
Fig 1. Allowedrange ofq=sl3/S23 and 6 for mr=45, 65, 85, 180 GeV and BK= 1. Heavy and hght sohd lines gwe, respectively, the central value and the edges of the band allowed by c. Heavy and light dashed lines give the correspondingband for xd. The upper edgeof the xa band is often outside the graph The dotted line is the upper hmlt for q The shaded area is the final allowed region
is around 0.03 implying S13>/0.0015. At that same mt v a l u e , ~ can be anywhere between 50 ° and 175 °. On the other hand, for mt values around 50-60 GeV, s13 must be very close to 0.01 and 5 is between 100 ° and 165 ° . Values near the boundaries of the two bands are improbable in the sense that they require all parameters to assume their extreme values simultaneously. For lower values of BK, the E band corresponds to higher values of s13 and the allowed region is substantially smaller. A sample of such results for BK=0.4 is shown in fig. 2 for mr=65, 85 GeV. mr=65 GeV, BK=O.4
I'-'~
"D j
O
IOO 8
50
m t =85 GeV, BK=O.4
'I/F.
150 O
",\~
50
IOO 8
Jill
150
Ftg. 2. Allowed range of q and ~ for mt= 65, 85 GeV and BK = 0.4. The meaning of the lines and shaded area are as in fig. i.
Volume 195, number 4
PHYSICS LETTERSB
It is easy to see from the figures that, for low mt values, the new information obtained from Ba-Bd o -o mixing excludes a substantial part of the region which remained allowed by all previous experiments. For larger mt values, the new data do not provide us with much new information. Additional details of the above analysis will be presented elsewhere. The results discussed so far are consistent with those of previous authors [ 13 ]. Having estabhshed the allowed range for the parameters mt, s~3 and ~, we now proceed to consider the proposed relations among masses, angles and phases. We first note that all quark masses mentioned in our phenomenological analysis up to now are essentially the "physical" masses. In QCD language, the quark masses are "running" parameters, defined only for a specific energy scale. The "physical" mt values appearing in figs. 1 and 2 and in the above analysis should be interpreted as values of rot(mr). In fact, we should also include a first-order QCD correction, obtaining [ 15 ] m tphys ~---mt(mt)[ 1 + (4/3n)as(mt)] .
Ma=aS+A,
where S, A are, respectively, a symmetric and an antisymmemc matrix and o~ is a free parameter. Without loss of generality we can always choose a basis in which M u is a real diagonal matrix with matrix elements m~, - rn~, mr. In such a basis M d has the form
M d=
( a m ~ i A i B ) -iA - am~ iC -iB -iC amt
.
The Stech matrices have seven real parameters and they lead to three relations among quark masses, angles and phases. We use as input the seven bestdetermined parameters (masses of the first five quarks and Sl2, s23). The masses should all be chosen at the same energy scale and we select 1 GeV and use the values advocated by Gasser and Leutwyler [ 15 ]:
md/ms = 0.051 _+0.04, m,flm~ =0.0038_+ 0.0012,
In order to relate mtphys to, say, mt(1 GeV) we must use the usual equation for the running mass [ 15 ] m(u) =m(1
Mu=S,
17 September 1987
ms/mb =0.033 _+0.011 , rnc(1 GeV) = 1.35+0.05 G e V .
2 f l l ) , o l n ( L + l ) . 8 , 1 " ~ ( L ) -2r°za° L
-
,
where po= 11-~Nf,
?0=2,
All relations involve only mass ratios, but we need the value of mc in order to derive m t from the ratio mffmt. We use the S~e, s23 values quoted earlier. One of the predictions of the Stech scheme is $23 ~ x/ms/mb -- mclmt
flo = 1 0 2 - a}Nf, L=ln(ktZ/AZ),
~1 = ll'~-- l"~Nf , A=0.1 G e V ,
and n~ is the renormalization group invariant mass. A useful approximate relation for m t °hys in the interesting range between 40 GeV and 180 GeV is m~phYS,,~0.6mt(1 GeV). However, in the following discussion we use the full expression for the running mt. We consider two main schemes which were proposed for the structure of quark mass matrices. The first is the Stech scheme [ 2 ] according to which the mass matrices M u and M d are both hermitian and can be described by
•
I f we scan the entire range of all the relevant parameters, this gives rnt(1 GeV)~<72 GeV, leading to m t phys ~<46 GeV. Our earlier analysis of the e and Xd data showed that even when all uncertainties were pushed to their limits, we obtained mPhYs~>43 GeV. That means that the only remaining hope for the Stech scheme is to have mr~ 45 GeV. We may now use another prediction of the Stech scheme which restricts s13 and & We obtain q cos ~ ~ - (ms/ms)s12 • This correspond to a curve in the q-~ plane. The resulting curve for the case of mt ~ 45 GeV is shown in fig. 3a. While the Stech curve passes through the 589
Volume 195, number 4
PHYSICS LETTERS B
mt=45GeV,
I',
i--
B K
co
I
mt=85GeV,
I
IL
I-
"%.~.
03 0 . 2
17 September 1987
BK < 1
I
1
II
S12 ,~, I ~ - /
\
.................. 1....... ~ " ~ '
e -'~l~uu/mcc I , -- 1~2
.............. X ......................
s13 ~ I (rn~/rnb) m,,/-~a/mb 0 / 0
I 50
I' ~ I00
, 4~,| 150 0
I 50
s
I - - - --'t ---'! [00 t50
s
Fig. 3 (a) Allowed range (shaded) ofq and 5 for mr=45 GeV, BK< 1 and the curve (dot-dash) representing the Stech scheme. The two do not overlap Higher mt values are excluded by the Steeh scheme. Lower mt values are excluded by the data Solid, dashed and dotted hnes are defined as in fig 1. (b) Allowed range ofq and 8 for mr=85 GeV, BK~<1 The small shaded area is the overlap of the region allowed by the data and the region allowed by the Fntzsch form of the mass matrices Higher mt values are inconsistent with the Fritzsch assumptions Lower mt values show no overlap between the two allowed regions Solid, dashed and dotted hnes are defined as in fig 1 region allowed by all earlier data, it is clearly in contradiction with the tiny allowed region which remains 0 -0 mixing data. The Stech scheme after the new Bd-Bd is therefore now excluded by the data. Gronau, Johnson a n d Schechter [ 3] p r o p o s e d a scheme in which the Stech assumptions are supplem e n t e d by the assumptions o f the Frltzsch scheme. Having excluded the Stech scheme, we are automatically excluding any m o d e l which uses the Stech assumptions as part o f its input. That applies, o f course, to the m o d e l o f ref. [3]. We now consider the Fritzsch scheme [ 1 ]. Without loss o f generality we can choose a basis [ 16 ] in which the Fritsch matrices have the form
au o) MU=
0 bu
M d=
de - , o t
bu cu
,
a
0 )
0 bde -,02
bae '~: cd
.
Here we have eight real parameters and therefore two predictions for relations among masses, angles and phases. The Fritzsch form enables us to express the three angles and the K M - p h a s e in terms o f the six-quark masses a n d the two arbitrary phases ~l, ~2 which a p p e a r in the matrices. The relations are [ 16 ] 590
+e
'°'
(
-e
sin 5
) l ,
sin ~1
S12S23/S13--COS ~ ~COS ~1 -~/rnamc/msmu" The second o f these four equations gives the b o u n d mc mt ~ ( ~ _ s 2 3 1 2
•
Using the entire allowed range o f the parameters, this leads to mi(1 GeV)~<145 G e V which yields m phys <~88 GeV. This b o u n d for m t is valid only for the lowest allowed value o f rns/mb a n d the highest allowed value o f s23, n a m e l y mdmb=O.022; s23= 0.05. F o r other values, stronger b o u n d s for mt are obtained. We can now restrict our attention only to m, values below 88 GeV. F o r each such value o f mr, we have already extracted (fig. 1 ) the region in the q-~ plane which is allowed by all present data. N o w we take the same mt value a n d find, in the same q-O plane, the region allowed by the Fritzsch conditions. We again cover all allowed values o f the various parameters. We find that for m t phys values below 82 GeV there is no overlap between the allowed Fritzsch d o m a i n a n d the region o f the q - 8 plane allowed by the data. The only overlap is o b t a i n e d for m t ~ 85 GeV. The tiny overlap region is shown in fig. 3b. It is clear from the figure that all values o f all p a r a m e t e r s are essentially determined by that solution. We obtain mt ~ 85 GeV; q ~ 0 . 0 7 leading to s13~0.0035; 3 ~ 1 0 0 ° ; B K ~ I ; xa~0.55; B e f 2 ~ (0.2 GeV)2; $a3~'0.05 and mdmb ~ 0.22. At this p o i n t one m a y take two different points o f view: A pessimist would say that the p r o b a b i l i t y that all the experimental a n d theoretical p a r a m e t e r s will eventually settle on their m o s t extreme allowed values is very small and that it is therefore unlikely that the Fritzsch form o f the mass matrix will survive. A n o p t i m i s t would argue that the Fritzsch scheme now p r o v i d e s us with a complete d e t e r m i n a t i o n o f all the p a r a m e t e r s o f the three-generation s t a n d a r d m o d e l
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PHYSICS LETTERS B
and allows us to predict all the results o f all future experiments which depend on these parameters. We leave it for the reader to decide which point o f view is more likely to prevail. The (unlikely) unique solution which we have obtained here leads to some interesting consequences: (i) Shin [4] proposed that the two arbitrary phases o f the Fritzsch matrices, ~ and 42, might have the simple values ¢ ~ = 9 0 ° ; ~2=0. We find that, with some effort, our unique Fritzsch solution can be consistent with this hypothesis. If, in addition to pushing all the above parameters to their extreme values, we also take the highest allowed value ofs~2 (namely 0.223) and the lowest allowed values for mJms and mu/mc, we find that for r o t = 8 8 GeV, s~3=0.003, d = 93 ° we can have a solution which is consistent with the Fritzsch form, the Shin hypothesis and all the data. This situation is even more unlikely but cannot be completely ruled out. (ii) The low value s~3 ~ 0.0035 suggests that it will be difficult to observe noncharmed b-decays and that their actual rates are a full order o f magnitude below the present limits. It is therefore clear that any direct observation o f b--,u~9~ in the near future will eliminate our unlikely unique solution and will exclude the Fritzsch scheme (and the Shin hypothesis). (iii) The low value o f s~3 also leads to relatively small values for the parameter ( / e . It is well known that the theoretical determination of this quantity is subject to major uncertainties. Our unique solution gives ImC6 (nnlQ6lK ° e: ~ 0.005 (~-0-~-. 1 ) ( ' 1-~ ~-eV3 ) ) , where C6, Q6 are not well determined and are defined in ref. [ 17 ]. For any value o f the uncertain theoretical parameters, this corresponds to ( - v a l u e s which are approximately one third of the values obtained by the current upper limit on s~3. This is certainly consistent with the present data. (iv) Using our unique solution we can predict that -0 B~0 -B~ mixing is practically maximal. Assuming that MB, BB, f,, and Zb are essentially the same for B ° and B °, we find
xdxa~ I Es 12/I E~ 12 With our values, we must have x d ~ 0 . 5 5
and
17 September 1987
[Vts[ =0.05; [Vtdl =0.011. Consequently, we find xs ~ 11 and rs =X2s/(2 + x ~ ) = 0.98. Note, however, that if, instead o f choosing the Fritzsch solution, one uses the central values of xd, the parameter r~ is even closer to one. The UA1 data [ 18] on B-l] mixing is consistent with r~~ 1. (v) Our complete set o f parameters also predicts that, for three light neutrinos B R ( K ~ r t v g ) ~ 1)<10 -I°" We summarize: we have used the recent A R G U S data for Bd-B~ o -o mixing in order to restrict the allowed domain o f values for rot, s~3 and d. We then compared this domain with the Stech and Fritzsch forms of the quark mass matrices. The Stech form is excluded by the new data, even if all uncertainties are stretched to their limits. The Fritzsch form is almost excluded, with a tiny allowed region remaining for m r ~ 8 5 GeV, s~3~0.0035, d ~ 100 °. This last unique solution leads to many definite experimental predictions. It should not be too difficult to rule it out by further data. We thank Ikaros Bigi, Fred Gilman, Miriam Leurer and Stephen Sharpe for helpful discussions. One o f us (Y.N.) acknowledges the support of a Fulbright fellowship.
References [ 1] H. Fntzsch, Phys. Lett B 70 (1977) 436, B 73 (1987) 317 [2] B Stech, Phys. Lett B 130 (1983) 189. [3] M. Gronau, R. Johnson and J. Schechter, Phys Rev. Lett. 54 (1985) 2176. [4] M. Slam, Phys. Lett B 145 (1984) 285, H. Fntzsch, Phys. Lett B 166 (1986) 423. [5] ARGUS Collab., H Albrecht et al, Phys. Lett B 192 (1987) 245 [6] L. Malam, Phys Len. B 62 (1976) 183, Proe 1977 Lepton-photon Symp (Hamburg), p.867. [7] L. Wolfenstem, Phys. Rev. Lett. 51 (1984) 1945 [ 8 ] L L. Cbau and W Y. Keung,Phys Rev Lett. 53 (1984) 1802 [9] H. Haran and M Leurer, Phys. Lett. B 181 (1986) 123. [ 10] M G D Gilchraese, Cornell Umverslty preprmt CLNS86/754 (1986) [ 11 ] E.H Thorn&ke m. Proc. 1985 Intern Symp. on Lepton and photon lnteractxons at high energies, eds M Konuma and K Takahashl (Kyoto, 1985). [ 12] U. Arnold1et al., University ofPennsylvama preprmt UPR0331T (1987) [ 13] J Ellis, J S. Hagehn andS. Rudaz, Phys. Lett B 192 (1987) 201, 591
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I I. Blgl and A I. Sanda, SLAC preprlnt SLAC-PUB-4299 (1987), L L. Chau and W.Y. Keung, UC Davis preprmt UCD-8702 (1987); V. Barger et al, Plays Lett B 194 (1987) 312. [ 14] T Inaml and C.S Lira, Progr. Theor. Plays 65 (1981) 297; F.J Gilman and M.B Wise, Phys Rev. D 27 (1983) 1128.
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[ 15] J. Gasser and H Leutwyler, Phys. Rep. 87 (1982) 77. [ 1 6 ] M Leurer, Weizmann Institute Ph.D. thesis (1985), unpubhshed. [ 17 ] F.J. Gilman and J.S. Hagehn, Phys. Lett. B 133 (1983 ) 443 [ 18] UA 1 Collab., C. Albajar et al., Phys. Lett. B 185 (1987) 241.
PHYSICS LETTERS B
17 September 1987
B - B M I X I N G A N D R E L A T I O N S A M O N G Q U A R K M A S S E S , A N G L E S A N D P H A S E S "~ Haim H A R A R I and Yosef N I R Stanford Lmear Accelerator Center, Stanford Umverslty, Stanford, CA 94305, USA Received 15 June 1987
Using the recent data on B°-l]° mixing we estabhsh constraints on the allowed values of the standard model parameters mt, 013 and & The allowed range is inconsistent ruth the Stech scheme for the quark mass mamces. For almost all allowed values of the parameters, the Fritzsch form of the mass matrices ~s also excluded. Only in the unlikely event that the experimental and theoretIcal quantities xd, ms~rob, 023, BK and fB assume extreme values within their allowed ranges, a set of parameters exists which is consistent with all data and with the Fritzsch scheme. We then obtain the unique solution mt ~ 85 GeV, sin 013~ 0 0035, fi ~ 100 °, yleldmg F(b~u~92 )/F (b~c~9~) ~0 01, maximal B~-B° mixing, relatively small E'/E and BR(K-*nvg) ~ 1× 10 ~o
The quark sector of the three-generation standard model contains ten independent parameters: six quark masses, three mixing angles and one Kobayashi-Maskawa ( K M ) phase. Future theories may lead to a calculation of these parameters or at least to relations among them. Seven o f the ten parameters are reasonably well determined: the masses o f the first five quarks and the mixing angles 012 and 023. For the remaining three parameters mi, 013 and d, we have fairly weak bounds. Several theoretical and phenomenological models which go beyond the standard model have led to the derivation of relations among mixing angles, phases and quark mass ratios. The best known scheme is the one proposed by Fritzsch [ 1 ]. Another interesting scheme has been suggested by Stech [ 2 ] and variations on the two themes were proposed by Gronau, Johnson and Schechter [ 3 ] and by Shin [ 4 ]. The recent observation of Bd-Bd o -o mixing by the A R G U S collaboration [ 5 ] in DESY provides us with important new information on the three poorly determined parameters of the quark sector of the standard model. It also leads to severe constraints on the relations among masses, angles and phases. In this note we investigate the implications of the new Work supported by the Department of Energy, Contract DEAC03-76SF00515 I On leave from the Weizmann Institute of Science, 76100 Rehovot, Israel
586
data and show that the Stech scheme [ 2 ] (as well as its extension by Gronau, Johnson and Schechter [ 3 ]) is now clearly inconsistent with the data. We also show that the Fritzsch [ 1 ] matrix (and its extension suggested by Shin [ 4 ]) disagrees with the central values o f the various experimental parameters and with almost all other allowed values of the parameters. The Fritzsch scheme will survive only if all the following 0 - 0 mixing is at the lowest conditions are met: Bd-Bd level allowed by the error quoted by the A R G U S experiment; the B-meson decay constant is approximately 0.2 GeV (at the top of its "reasonable" range); the hadronic factor BK in the expression for the CP violating parameter e obeys BK ~ 1; the ratio ms/mb is around 0.022 (e.g. m s = 120 MeV, m u = 5 . 4 GeV); the value of the mixing angle 023 is at the top o f its allowed range. Only If all of these constraints are s~multaneously fulfilled will the Fritzsch matrix remain consistent with the data. In such a case, we have m i ~ 8 5 GeV; s13~0.0035; d ~ 1 0 0 ° and the phases o f the Fritzsch matrix must be close to the ones suggested by Shin. We also comment on several consequences o f these results for other experimental quantities. In our analysis we use a choice of mixing angles based on the combined work of Maiani [ 6], Wolfenstein [ 7 ], Chau and Keung [ 8 ] and others. We follow the notation and the precise form of this matrix as given by Harari and Leurer [9]. For the purpose of the
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
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analysis presented here it is sufficient to neglect corrections of order 0~ and to assume cos0~2= cos 023=c0s 013= 1. The resulting form of the mixing matrix is 1 V~
--s12 --$23s13 e~o S12S23 --S13 e~a
s~2
Sl3 e-~6)
1
$23 1
--$23
(1)
where s,j = sin 0u.We have the following direct information on the three angles: (i) s12=0.221 +0.002. (ii) ~23 ~ -u.,,-,-,_o-n nAa+ooo7009. This result is based on the following assumptions: % = (1.16+0.16) × 10 -12 s [10]; B R ( b ~ c ~ 9 0 = 0 . 1 2 1 + 0 . 0 0 8 [11]; m b = 5.0_+ 0.3 GeV; m~= 1.5 _+0.2 GeV. The values of rnb and rn~ appear in the phase space factors relating the measured b lifetime and the relevant matrix element. It is not entirely clear which values should be used and we have allowed for fairly generous errors. In some cases we will need to use the product %s223. Since the errors on s23 contain the errors on %, the product has smaller errors than s~3 alone. We obtain ZbS~3= (3.25 + 1.10) × 109 G e V - ~. (iii) The only direct limit on s~3 is obtained from the experimental bound [ 11 ]
17 September 1987
cases we can use the data in order to exclude certain ranges of the parameters. The results of the first part of our analysis are similar of those recently obtained by several groups of authors [ 13 ]. We therefore do not give here all the details of the analysis, but quote the assumptions and the main results. We use the following expression [ 14] for E: Ie ] =
CBKs~3q sin ~{ [ r/3f3(Yt) -- ~]1]yes12
+ q2Ytf2(yt)s~3(S12 -- q COS~) }, where
G 2~f2KMK M w2
C~- ~ = 4 × 104, 6zc x/2AMK
y,=m]/M2w
(t=c, t),
3 yt(l+yt) ( l + ~ l n ( y t ) )
f2(Yt) = 1 4 (1 - y t ) 2 f3(Yt) = I n ( y ~ )
43 1 -Yt y t (l+lY_~ytln(yt)).
In order to minimize the ambiguities which are due to the b-quark mass, we use the ratio q = s~3/s23 rather than using s~3. For the values of mb and m~ listed above, we obtain Sl3~<0.011 and
We use Gv= 1.166× 10 -5 GeV-2; f ~ = (0.16 GeV)2; MK= 0.498 GeV; M w = 8 2 GeV; AMK= 3.52 × 10 -15 GeV; r/l=0.7; r/2=0.6; r/3=0.4. The three parameters q, are QCD corrections. We also use the values of me and s23 listed above. The parameter BK is unknown and is usually believed to be somewhere in the range ~ ~
q-~S13/$23~<0.22.
ra =x~/( 2 + xaa) .
The direct lower experimental bound on m t is still 23 GeV. A recent combined analysis of all measurements of sin20w showed [ 12 ] that the standard model radiative corrections are consistent with the data only if mr ~<180 GeV. We therefore consider only the range 23 GeV~
The expression for xa is [ 13 ]
F ( b ~ u ~ 9 0 < 0.08 F(b~cl~9~)
Xa = %( G~/6zc 2) rlMB(BBJ~B)M 2
y ~ ( y t ) l V*~dV~bl2 • In addition to the previous parameters we now also use MB= 5.28 GeV; Vtb= 1; r/=0.85. Here r/is, again, a QCD correction. We also have ] Vtd]2 ~___$223($22+q2 --2s12q COS ~5) .
Note that the expression for xa contains the combination rbS~3 for which we use the value quoted earlier. We also have here two additional unknown parameters: The B decay constant fB and the had587
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PHYSICS LE'FFERS B
ronic parameter B~ which is analogous to the usual BK mentioned above. Based on previous analysis [13] we assume BB~I and fB=0.15+0.05 GeV. What we really use is BBf~ = (0.15+0.05 GeV) :. We can write
Xd = C' ytf2(Yt)( ~2 +
q~
17 September 1987 mt:45GeV, BK:I
m r : 6 5 GeV, BK=I
0.4 ~0.2 n ET
- 2st2q cos 5 ) ,
where
C' = ( %s23)( G~/6~r2)tlM~(B~f~)M~w l Vtb j2
f ~
f t ~'.~................... ' ~~ ',~ " te,~¢*
| 0
I I00 B
I 50
I | 150 O
mr=85 GeV, BK=I
I 50
I I00 8
I I 150
mr= 180 GeV, BK=I
q 6. ~,,,.x_3 1+70
We now use the experimental values ~= 2.3 X 1O-3 and Xd=0.73+0.18 (corresponding to rd=0.21 --+0.08) in order to limit further the allowed range of the three poorly determined parameters rot, s l3 and 5. We do it in the following way: In principle, we express the known quantities d C and xdC' as functions of rot, $13, ~. We select a value of mt and plot the two allowed curves for d C and xdC' in the q-5 plane. If the two curves intersect below the upper limit q~<0.22, the intersection point is an allowed solution for (mr, s13, 5). In practice, Xd has a large experimental error and C' has both a theoretical ambiguity and an experimental error. We therefore obtain a wide band rather than a curve for X d l C t in the q-5 plane for each possible value of mt. The central curve and the two edges of the band correspond to xJC' = 0.14 +0.47 -o 09, as given by the most extreme values assumed above for Xd, ( B j ~ ) , (ZbS~3). Note that the edge of the band corresponds to the very unlikely situation that all these parameters simultaneously obtain their most extreme allowed values. For e/C, we produce a band whose width is determined by the allowed range of s23 and m~. It is narrower than the xJC' band. However, we must repeat the analysis for several different values of BK in the usually accepted range. The overlap regions of the two bands which also lie below q=0.22 provide us with a conservative evaluation of the allowed range of m , s~3, 5. Fig. 1 shows the relevant bands for BK= I and mr=45, 65, 85, 180 GeV. In each case the allowed range of q and 5 values is shown by the shaded area. There is no allowed region for any value of m t below 43 GeV. The central values of the two bands meet in the allowed region only for mt>/85 GeV. The lowest allowed value of q is obtained for the largest m~ and 588
I -
~, 0.2
\ ~
. ...........
.~[
ob 0
50
lib\
x
I00 B
\
"..
IIII
..................... L~.~. \
.S ......................
4....S
150 0
50
I00 B
150
Fig 1. Allowedrange ofq=sl3/S23 and 6 for mr=45, 65, 85, 180 GeV and BK= 1. Heavy and hght sohd lines gwe, respectively, the central value and the edges of the band allowed by c. Heavy and light dashed lines give the correspondingband for xd. The upper edgeof the xa band is often outside the graph The dotted line is the upper hmlt for q The shaded area is the final allowed region
is around 0.03 implying S13>/0.0015. At that same mt v a l u e , ~ can be anywhere between 50 ° and 175 °. On the other hand, for mt values around 50-60 GeV, s13 must be very close to 0.01 and 5 is between 100 ° and 165 ° . Values near the boundaries of the two bands are improbable in the sense that they require all parameters to assume their extreme values simultaneously. For lower values of BK, the E band corresponds to higher values of s13 and the allowed region is substantially smaller. A sample of such results for BK=0.4 is shown in fig. 2 for mr=65, 85 GeV. mr=65 GeV, BK=O.4
I'-'~
"D j
O
IOO 8
50
m t =85 GeV, BK=O.4
'I/F.
150 O
",\~
50
IOO 8
Jill
150
Ftg. 2. Allowed range of q and ~ for mt= 65, 85 GeV and BK = 0.4. The meaning of the lines and shaded area are as in fig. i.
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PHYSICS LETTERSB
It is easy to see from the figures that, for low mt values, the new information obtained from Ba-Bd o -o mixing excludes a substantial part of the region which remained allowed by all previous experiments. For larger mt values, the new data do not provide us with much new information. Additional details of the above analysis will be presented elsewhere. The results discussed so far are consistent with those of previous authors [ 13 ]. Having estabhshed the allowed range for the parameters mt, s~3 and ~, we now proceed to consider the proposed relations among masses, angles and phases. We first note that all quark masses mentioned in our phenomenological analysis up to now are essentially the "physical" masses. In QCD language, the quark masses are "running" parameters, defined only for a specific energy scale. The "physical" mt values appearing in figs. 1 and 2 and in the above analysis should be interpreted as values of rot(mr). In fact, we should also include a first-order QCD correction, obtaining [ 15 ] m tphys ~---mt(mt)[ 1 + (4/3n)as(mt)] .
Ma=aS+A,
where S, A are, respectively, a symmetric and an antisymmemc matrix and o~ is a free parameter. Without loss of generality we can always choose a basis in which M u is a real diagonal matrix with matrix elements m~, - rn~, mr. In such a basis M d has the form
M d=
( a m ~ i A i B ) -iA - am~ iC -iB -iC amt
.
The Stech matrices have seven real parameters and they lead to three relations among quark masses, angles and phases. We use as input the seven bestdetermined parameters (masses of the first five quarks and Sl2, s23). The masses should all be chosen at the same energy scale and we select 1 GeV and use the values advocated by Gasser and Leutwyler [ 15 ]:
md/ms = 0.051 _+0.04, m,flm~ =0.0038_+ 0.0012,
In order to relate mtphys to, say, mt(1 GeV) we must use the usual equation for the running mass [ 15 ] m(u) =m(1
Mu=S,
17 September 1987
ms/mb =0.033 _+0.011 , rnc(1 GeV) = 1.35+0.05 G e V .
2 f l l ) , o l n ( L + l ) . 8 , 1 " ~ ( L ) -2r°za° L
-
,
where po= 11-~Nf,
?0=2,
All relations involve only mass ratios, but we need the value of mc in order to derive m t from the ratio mffmt. We use the S~e, s23 values quoted earlier. One of the predictions of the Stech scheme is $23 ~ x/ms/mb -- mclmt
flo = 1 0 2 - a}Nf, L=ln(ktZ/AZ),
~1 = ll'~-- l"~Nf , A=0.1 G e V ,
and n~ is the renormalization group invariant mass. A useful approximate relation for m t °hys in the interesting range between 40 GeV and 180 GeV is m~phYS,,~0.6mt(1 GeV). However, in the following discussion we use the full expression for the running mt. We consider two main schemes which were proposed for the structure of quark mass matrices. The first is the Stech scheme [ 2 ] according to which the mass matrices M u and M d are both hermitian and can be described by
•
I f we scan the entire range of all the relevant parameters, this gives rnt(1 GeV)~<72 GeV, leading to m t phys ~<46 GeV. Our earlier analysis of the e and Xd data showed that even when all uncertainties were pushed to their limits, we obtained mPhYs~>43 GeV. That means that the only remaining hope for the Stech scheme is to have mr~ 45 GeV. We may now use another prediction of the Stech scheme which restricts s13 and & We obtain q cos ~ ~ - (ms/ms)s12 • This correspond to a curve in the q-~ plane. The resulting curve for the case of mt ~ 45 GeV is shown in fig. 3a. While the Stech curve passes through the 589
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PHYSICS LETTERS B
mt=45GeV,
I',
i--
B K
co
I
mt=85GeV,
I
IL
I-
"%.~.
03 0 . 2
17 September 1987
BK < 1
I
1
II
S12 ,~, I ~ - /
\
.................. 1....... ~ " ~ '
e -'~l~uu/mcc I , -- 1~2
.............. X ......................
s13 ~ I (rn~/rnb) m,,/-~a/mb 0 / 0
I 50
I' ~ I00
, 4~,| 150 0
I 50
s
I - - - --'t ---'! [00 t50
s
Fig. 3 (a) Allowed range (shaded) ofq and 5 for mr=45 GeV, BK< 1 and the curve (dot-dash) representing the Stech scheme. The two do not overlap Higher mt values are excluded by the Steeh scheme. Lower mt values are excluded by the data Solid, dashed and dotted hnes are defined as in fig 1. (b) Allowed range ofq and 8 for mr=85 GeV, BK~<1 The small shaded area is the overlap of the region allowed by the data and the region allowed by the Fntzsch form of the mass matrices Higher mt values are inconsistent with the Fritzsch assumptions Lower mt values show no overlap between the two allowed regions Solid, dashed and dotted hnes are defined as in fig 1 region allowed by all earlier data, it is clearly in contradiction with the tiny allowed region which remains 0 -0 mixing data. The Stech scheme after the new Bd-Bd is therefore now excluded by the data. Gronau, Johnson a n d Schechter [ 3] p r o p o s e d a scheme in which the Stech assumptions are supplem e n t e d by the assumptions o f the Frltzsch scheme. Having excluded the Stech scheme, we are automatically excluding any m o d e l which uses the Stech assumptions as part o f its input. That applies, o f course, to the m o d e l o f ref. [3]. We now consider the Fritzsch scheme [ 1 ]. Without loss o f generality we can choose a basis [ 16 ] in which the Fritsch matrices have the form
au o) MU=
0 bu
M d=
de - , o t
bu cu
,
a
0 )
0 bde -,02
bae '~: cd
.
Here we have eight real parameters and therefore two predictions for relations among masses, angles and phases. The Fritzsch form enables us to express the three angles and the K M - p h a s e in terms o f the six-quark masses a n d the two arbitrary phases ~l, ~2 which a p p e a r in the matrices. The relations are [ 16 ] 590
+e
'°'
(
-e
sin 5
) l ,
sin ~1
S12S23/S13--COS ~ ~COS ~1 -~/rnamc/msmu" The second o f these four equations gives the b o u n d mc mt ~ ( ~ _ s 2 3 1 2
•
Using the entire allowed range o f the parameters, this leads to mi(1 GeV)~<145 G e V which yields m phys <~88 GeV. This b o u n d for m t is valid only for the lowest allowed value o f rns/mb a n d the highest allowed value o f s23, n a m e l y mdmb=O.022; s23= 0.05. F o r other values, stronger b o u n d s for mt are obtained. We can now restrict our attention only to m, values below 88 GeV. F o r each such value o f mr, we have already extracted (fig. 1 ) the region in the q-~ plane which is allowed by all present data. N o w we take the same mt value a n d find, in the same q-O plane, the region allowed by the Fritzsch conditions. We again cover all allowed values o f the various parameters. We find that for m t phys values below 82 GeV there is no overlap between the allowed Fritzsch d o m a i n a n d the region o f the q - 8 plane allowed by the data. The only overlap is o b t a i n e d for m t ~ 85 GeV. The tiny overlap region is shown in fig. 3b. It is clear from the figure that all values o f all p a r a m e t e r s are essentially determined by that solution. We obtain mt ~ 85 GeV; q ~ 0 . 0 7 leading to s13~0.0035; 3 ~ 1 0 0 ° ; B K ~ I ; xa~0.55; B e f 2 ~ (0.2 GeV)2; $a3~'0.05 and mdmb ~ 0.22. At this p o i n t one m a y take two different points o f view: A pessimist would say that the p r o b a b i l i t y that all the experimental a n d theoretical p a r a m e t e r s will eventually settle on their m o s t extreme allowed values is very small and that it is therefore unlikely that the Fritzsch form o f the mass matrix will survive. A n o p t i m i s t would argue that the Fritzsch scheme now p r o v i d e s us with a complete d e t e r m i n a t i o n o f all the p a r a m e t e r s o f the three-generation s t a n d a r d m o d e l
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and allows us to predict all the results o f all future experiments which depend on these parameters. We leave it for the reader to decide which point o f view is more likely to prevail. The (unlikely) unique solution which we have obtained here leads to some interesting consequences: (i) Shin [4] proposed that the two arbitrary phases o f the Fritzsch matrices, ~ and 42, might have the simple values ¢ ~ = 9 0 ° ; ~2=0. We find that, with some effort, our unique Fritzsch solution can be consistent with this hypothesis. If, in addition to pushing all the above parameters to their extreme values, we also take the highest allowed value ofs~2 (namely 0.223) and the lowest allowed values for mJms and mu/mc, we find that for r o t = 8 8 GeV, s~3=0.003, d = 93 ° we can have a solution which is consistent with the Fritzsch form, the Shin hypothesis and all the data. This situation is even more unlikely but cannot be completely ruled out. (ii) The low value s~3 ~ 0.0035 suggests that it will be difficult to observe noncharmed b-decays and that their actual rates are a full order o f magnitude below the present limits. It is therefore clear that any direct observation o f b--,u~9~ in the near future will eliminate our unlikely unique solution and will exclude the Fritzsch scheme (and the Shin hypothesis). (iii) The low value o f s~3 also leads to relatively small values for the parameter ( / e . It is well known that the theoretical determination of this quantity is subject to major uncertainties. Our unique solution gives ImC6 (nnlQ6lK ° e: ~ 0.005 (~-0-~-. 1 ) ( ' 1-~ ~-eV3 ) ) , where C6, Q6 are not well determined and are defined in ref. [ 17 ]. For any value o f the uncertain theoretical parameters, this corresponds to ( - v a l u e s which are approximately one third of the values obtained by the current upper limit on s~3. This is certainly consistent with the present data. (iv) Using our unique solution we can predict that -0 B~0 -B~ mixing is practically maximal. Assuming that MB, BB, f,, and Zb are essentially the same for B ° and B °, we find
xdxa~ I Es 12/I E~ 12 With our values, we must have x d ~ 0 . 5 5
and
17 September 1987
[Vts[ =0.05; [Vtdl =0.011. Consequently, we find xs ~ 11 and rs =X2s/(2 + x ~ ) = 0.98. Note, however, that if, instead o f choosing the Fritzsch solution, one uses the central values of xd, the parameter r~ is even closer to one. The UA1 data [ 18] on B-l] mixing is consistent with r~~ 1. (v) Our complete set o f parameters also predicts that, for three light neutrinos B R ( K ~ r t v g ) ~ 1)<10 -I°" We summarize: we have used the recent A R G U S data for Bd-B~ o -o mixing in order to restrict the allowed domain o f values for rot, s~3 and d. We then compared this domain with the Stech and Fritzsch forms of the quark mass matrices. The Stech form is excluded by the new data, even if all uncertainties are stretched to their limits. The Fritzsch form is almost excluded, with a tiny allowed region remaining for m r ~ 8 5 GeV, s~3~0.0035, d ~ 100 °. This last unique solution leads to many definite experimental predictions. It should not be too difficult to rule it out by further data. We thank Ikaros Bigi, Fred Gilman, Miriam Leurer and Stephen Sharpe for helpful discussions. One o f us (Y.N.) acknowledges the support of a Fulbright fellowship.
References [ 1] H. Fntzsch, Phys. Lett B 70 (1977) 436, B 73 (1987) 317 [2] B Stech, Phys. Lett B 130 (1983) 189. [3] M. Gronau, R. Johnson and J. Schechter, Phys Rev. Lett. 54 (1985) 2176. [4] M. Slam, Phys. Lett B 145 (1984) 285, H. Fntzsch, Phys. Lett B 166 (1986) 423. [5] ARGUS Collab., H Albrecht et al, Phys. Lett B 192 (1987) 245 [6] L. Malam, Phys Len. B 62 (1976) 183, Proe 1977 Lepton-photon Symp (Hamburg), p.867. [7] L. Wolfenstem, Phys. Rev. Lett. 51 (1984) 1945 [ 8 ] L L. Cbau and W Y. Keung,Phys Rev Lett. 53 (1984) 1802 [9] H. Haran and M Leurer, Phys. Lett. B 181 (1986) 123. [ 10] M G D Gilchraese, Cornell Umverslty preprmt CLNS86/754 (1986) [ 11 ] E.H Thorn&ke m. Proc. 1985 Intern Symp. on Lepton and photon lnteractxons at high energies, eds M Konuma and K Takahashl (Kyoto, 1985). [ 12] U. Arnold1et al., University ofPennsylvama preprmt UPR0331T (1987) [ 13] J Ellis, J S. Hagehn andS. Rudaz, Phys. Lett B 192 (1987) 201, 591
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I I. Blgl and A I. Sanda, SLAC preprlnt SLAC-PUB-4299 (1987), L L. Chau and W.Y. Keung, UC Davis preprmt UCD-8702 (1987); V. Barger et al, Plays Lett B 194 (1987) 312. [ 14] T Inaml and C.S Lira, Progr. Theor. Plays 65 (1981) 297; F.J Gilman and M.B Wise, Phys Rev. D 27 (1983) 1128.
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[ 15] J. Gasser and H Leutwyler, Phys. Rep. 87 (1982) 77. [ 1 6 ] M Leurer, Weizmann Institute Ph.D. thesis (1985), unpubhshed. [ 17 ] F.J. Gilman and J.S. Hagehn, Phys. Lett. B 133 (1983 ) 443 [ 18] UA 1 Collab., C. Albajar et al., Phys. Lett. B 185 (1987) 241.