fourth family quark masses, b0b0 mixing, and rare decay modes, in a canonical structure of the 4×4...

Volume 201, number 4 PHYSICS LETTERS B 18 February 1988

F O U R T H FAMILY QUARK MASSES, B°-B ° M I X I N G , A N D R A R E DECAY M O D E S ,

I N A C A N O N I C A L S T R U C T U R E O F T H E 4 × 4 K M M A T R I X

Michae l S H I N

Department of Phystcs, Umverstty of Cahfornta at Irvme, Irvtne, CA 92717, USA

Received 25 November 1987

The tmphcatlons of a canonical structure of the 4X4 KM matrix proposed by Shin, Chlvukula and Flynn are r___eexammed in the Ba-Bd mixing and light of the recently reported ARGUS result on o o UAI's lower bound on mt The reported B°-B ° mixing is shown

to be consistent with mr= 50-90 GeV, provided BK<0,3, even if I Vubl/I Vcbl ts as small as 0.05-0 07. For BK>/0.3, mb iS predicted to be less than 55 GeV if mt >~ 60 GeV, implying the very posslbdlty of discovering the b'- or t-quark below 60 GeV m the near future. The Bs°-Bs° mixing parameter rs Is predicted to be maximal ffthe reported ARGUS result on rd is vahd. The branching ratios of some flavor-changing rare decay modes, which will be measured in the next generation of experiments, are predicted to be BR(K+-,n+vg) = (0 41-18.7) × 10-t0, BR(b~svg) = (0.27-6.2) × 10 -5, BR(b--,s~+J~ - ) = (2 2-9.9) × 10 -6, and BR(b --* s'/) = (1.0-7.0) × 10 -5. Among these BR(K +--,n +vg) and BR(b~svg) are shown to be the best candidates for distmgmshmg the four-family KM model from that of the three-famdy KM model.

1. Introduction

In recent years, there has b e e n a growing in teres t [ 1 ] in the four th fami ly o f quarks and leptons. Moreover , the recent ly repor ted A R G U S [ 2 ] result on u n u s u a l l y large Bo °-BoO mix ing m ay be an ind i rec t ev idence for the exis tence o f the four th - fami ly quarks . In this article, we revisi t a canon ica l s t ructure of the 4 × 4 K M mat r ix p roposed by Shin, C h i v u k u l a a n d F l y n n ( S C F ) [ 3 ], in which the ent i re K M angles are intrinsica_llly related to the qua rk mass ratios, a n d r e ex ami n e its imp l i ca t ions in the light o f the A R G U S result on o o Ba-Bd mix ing and U A I ' s recent ly repor ted [4] lower b o u n d on the t -qua rk mass mi>~44 G e V ~ . A par t icu la r emphas i s will be g iven to the a l lowed mass ranges o f the four th fami ly quarks a n d B ° - B ° mixings , while the analyses on flavor- chang ing rare b r a n c h i n g rat ios such as B R ( K ÷-- ,n ÷vg) , B R ( b - . s v g ) , B R ( b ~ s ~ + ~ - ), and B R ( b ~ s 7 ) are also presented .

2. SCF hypothesis on quark mass matrices and 4X4 KM matrix

The SCF hypothes is for four - fami ly qua rk mass mat r ices is a genera l iza t ion o f the F r i t z s c h - S h i n [ 5,6 ] structure ,2 for the th ree- fami ly case. It is based u p o n ( i ) sym m et r i c neares t -ne ighbor in te rac t ions in fami ly space and ( i i ) phases o f mul t ip les o f zc/2 genera ted by s p o n t aneous s y m m e t r y b r e a k d o w n of C P - i n v a n a n c e in a CP- i n v a r i a n t lagrangian. In its s imples t vers ion, the qua rk mass mat r ices take the fo rm

~ With somewhat stronger assumptions, the lower bound becomes 51 GeV See ref. [4]. ~2 For an incomplete list of various discussions on this structure In recent years, see refs. [7-12] For other hypotheses on quark mass

matrices, which are ruled out by UAI's recent data [4] on mr, see ref. [ 13]

0370-2693/88 /$ 03.50 © Elsevier Science Publ i shers B.V. ( N o r t h - H o l l a n d Physics Pub l i sh ing D i v i s i o n )

559


r° i °il Ii °° 1 +ia 0 b 0 b' Md= b 0 , M . = b' 0 0 , (1)

L°0 o c where a, b .... , c', d' are all real and positive, and are related to the quark mass eigenvalues ma, m ...... rot, rot, ( a - m x / - ~ d m~, b ~ mx/-~mb, etc.). A detailed discussion on this and the rest of this section can be found in ref. [3]. The 4 × 4 KM matrix Vis then given by [3]

V= (R~) T diag.(i, 1, 1, 1 )R~) , (2)

where Rg (R}) is the orthogonal matrix which diagonahzes the real symmetric phase-removed version of Md (Mu), and depends on the quark mass ranos only. Using the explicit expressions of R~ and R~ given in the appendix of ref. [ 3 ], one finds the following expressions for the elements of the KM matrix V:

Vua~-i (1- i )x / /~- l i (1- i~lx /~l )x /~- l ) , V u s ~ i ~ l - ~ l ,

Vub ~" (N~l ( ~ 2 - dx2(l--y3) [ 1N/~X3 3I-dX3Y3/(1-x3)]}+iy2y.j~y2)/x/1-y~,

Vub. = (V /~ {~/X2(1 --Y3)[ x/X3/(1 --X3) -- xfY3(1 --X3)] --Y3 Y.~X/~2Y3 } + i y 2 y ~ ~ ) / X / 1 --Y~,

V<b " ({--,~-~2 + x/X2 (1 --Y3)[X/1 --X3 +x/XsY31(1 --X3)] } +iy2 X I ~ Y 2 ) / l ~ - - y ~ ,

Vcb, "" ({x/X2(I --Y3)Ix/y3(1 --X3) -- x/X3/(1 --XD] +Y3 Y~2Y3} + +iy2y~ ~ ) / X / 1 --Y~,

Vta"(v l -~{x l i~2- - (X/ -~{x i l~2- -x /yz( l - -x3)X/y2(1- -X3) [~ +x/x3Y31(1--ys) ] } + l X 2 ~ ) l x / 1 - - X 2 ,

Vts '~({--~22+dy2(1--X3)[ l x f i - ~ - y 3 + d x 3 y 3 / ( 1 - - y 3 ) ] } + i x 2 ~ ) l lxf~--X~,

Vtb -- [(1 + ~ 3 x/~3 ) +ix2Y2x/X~X2y~Y2/( 1 --X3)(l --Y3)]/~/(1 +X3)( 1 +Y3),

Vtb,- [(x/r~3 -- ~ 3 ) +ix2y2y~lx~x2y~y2Y3/(1 --x3)(l --Y3)]/~/(1 +x3)(1 +Y3),

V,,d ~ ( - x/Y-~, {X/Y~(1- X3)[ ~/X3(1--Y3)- ~ 3 1 + X3 X~X3} + iX~X~,,i~X~ )ilx/'-f'L-~3--X~, Vt's "" ({dY2( 1 -x3 ) [ x/x3 (1 --Y3) - dY3/( 1 -Y3 )] "t-x3 ~ } - t - i x 2 x 2 d y i x i x z x 3 ) / ~ ,

vt,b = [(x/~3 - v/~3 ) +ix2y2x2~lXlX2X3y~yz/(1 -x3)(1 -y3)] /~/ (1 +x3)(1 +y3) ,

vt,l,, ,.. [(1 + x/~3 x/r~3 ) +ixzy2x~y~x/x~x2x3yiy2y3/(1 -x3)(1 -y3)] /~/ (1 +x3)(1 +y3) , (3)

where

x~=-mu/rnc, x 2 ~ m J m t , x3~mt /mi ,, y l=md/ms, y2=mJmb, y3=mb/mt, , (3')

are the ratios of the running quark masses.

F 3. Analysis on the allowed (mb,mt,), B°-B ° mixings, CP, and rare decay modes.

Given the running quark mass ratios, the entire 4 × 4 KM matrix is determined according to the previous section. In this section, we discuss the implications of these KM angles on the allowed mass ranges o fb ' and t' quarks and various KM phenomenologies including B °-B ° mixings and CP-violations, arid branching ratios of flavor-changing rare decay modes such as BR(K +--+n ÷ vg), BR(b-+s vg), BR(b--+s~+£ -) , and BR(b--+sT), which

560


will be measured m the next generation o f experiments. The following three choices o f the running quark mass ratios for the light quarks:

Case A: mb/m~=30.3, m~/m, =265 ,

Case B: mb/m~ =40.4, rnJmu = 389,

Case C: mb/m~ =25.1, m J m , = 197,

m~lma = 19.6+ 1.6,

mJmd = 19.6+ 1.6,

mJma = 19.6+ 1.6, (4)

and rnc (1 GeV) = 1.35 GeV, mb (1 GeV) = 5.3 GeV have been used as in ref. [ 3 ]. The constituent mass o f the t- quark m, has been chosen from the set { 50, 60, 70, 80, 90 GeV) in the light o f the recently reported lower bound m~ >/44 GeV of UA 1 [ 4 ]. A brief description o f the procedure used to determine the allowed region of (rob,rot,) is as follows:

(i) Given the constituent mass mt, mb,, and rot,, we determined the running quark mass ratios mt/mc, mt,/mt, and mb,/mb, using the one-loop renormalization group equation o f QCD as in ref [ 3 ], and computed the entire KM matrix with these and the light quark mass ratios o f (4).

(ii) The computed KM matrix is compared with the present experimental constraints; (a) I Vcbl = 0 . 0 3 4 - 0.058 ~3, I V, sl =0.221 +0.002 [ 15], and I Vubl/I Vcbl ~<0.22 [ 14]. (b) The box-diagram calculation o f the CP- violating parameter Re e in K ° - K ° is performed ~4 and we have demanded that the "bag factor" BK required to produce the observed strength o f Re ¢ is not too small (i.e. not too much CP-violation generated by the fourth family quarks). Thus the condit ion I BKI I> 0.1 ~5 has been applied as in ref. [3 ].

(iii) The constraint f rom Ap, I m t , -m b , I < 180 GeV [ 18] has been applied. For the constituent mass o f the t-quark m~ chosen from the set {50, 60, 70, 80, 90 GeV}, the result o f the

computat ion shows that case A of (4) has allowed regions for mt = 50 and 60 GeV, and case B has allowed regions for mt = 60, 70, 80, 90 GeV, while case C has no allowed regions for m , = 50-90 GeV. The results on the allowed regions are shown in figs. 1-6. The most stringent constraints turn out to come from I Vcb l, BK and Ap, while the constraints f rom I V~s I and [ V,b[/I Vcb[ are trivially satisfied.

Having determined the allowed region of (mb,, rot,) for each set o f input parameters we have computed the following quantities at each point on the (mb, , m t, )-plane using the corresponding KM matrix.

(1) Ba°-Bd° mixing ~6.. The amount o f mixing is given by the well-known formula

N +÷ + N - - x 2 rd=---N+- +N -+ -2+x~d ' xd=(AM/F')Bd " (5)

Since the standard box-diagram calculation Of Xd is proportional to the parameter BBf 2, whose precise value is not yet known although it is expected to be in the range (0.15 + 0.05 GeV)2 [ 19 ], we have computed the "calibrated" mixing parameter

x~ - (0.15 GeV/x/~afn)2Xd (6)

to represent the intrinsic amount o f mixing due to KM angles. The present experimental data on ro by A R G U S (rd=0.21 +0 .08 [2]) and CLEO(rd<0 .24 [21]) imply

rd = 0 . 1 8 + 0 . 0 6 , (7)

which, in turn, implies xd=0 .66 _+ 0.14. The corresponding value ofxfi is then

~3 Due to the uncertainties present in the phase space factor in the calculation of the semlleptomc branching ratio of the b-quark, we take this number from various estimates given in ref [ 14].

~4 Since the naive box-diagram calculation of Re e is valid in a phase convention in which the relative phase between d- and s-quarks vanishes, we have rotated the phases of the first and the second columns of Vso that Vu, and Vua are relatively real, as m ref. [3].

~5 Although BK lS not [ 16] precisely known at this time, It is most likely that BK is O(1 ) instead ofO(10-2), as is suggested by the naive vacuum insertion (BK = 1.0) and the current algebra calculation [ 17 ]. Thus, it is very hkely that BK is at least 0.1.

~6 For a recent review on this, see ref. [ 19 ]. For the recent analyses on the three-family KM model, see refs. [ 20,11 ].

561


, + xa=0.37_0 .07 for x//~-~fB =0.2 GeV,

xd=0.66_0.14 forv/ffa-afB=0.15 GeV

, -~- Xd= 1.5_0.3 for X/~-afB = 0.1 GeV (8)

So, the minimum value ofx~, consistent with the present B°-B ° mixing data, ts xd~- 0.3. The contours ofxfi are shown in fig__s. 1-6.

(2) B °-B ° mixing: We have computed the "calibrated" mixing parameter

x'~ =- (0.15 GeV/x/-~afB)2 x~ (9)

and will present the result in terms ofx'Jx~. (3) CP-violations in B ° and B °: The total charge asymmetry of all primary leptons coming from the decays

ofB ° mesons,

( N + - N - ) (10) T-ff=

and similarly for B ° mesons, are computed using the box diagrams and the KM matrix. (4) The flavor-changing neutral-current processes induced by the one-loop electroweak penguin diagrams,

BR(K +--,n +vg), BR(b--,svg), BR(b--,s£+£ - ), and BR(b- , s? ) are computed ~7, following ref. [22]. Branching ratios of these rare decay modes will be measured in the next generation of experiments and test the standard model at one-loop level, and may serve as indirect evidence for the existence of the fourth family of quarks.

The results of the calculation of the various quantities discussed above are shown in table 1. To compare with the three-family case, we have also calculated the corresponding quantities in the Fritzsch-Shin structure of the three-family quark mass matrices ~8 for the same input parameters and have presented the results in table 2.

4. Discussion

As one can see in figs. 1-6, there are substantially large regions with the B°-B ° mixing parameter x~ >/0.3 (required by the lower bound on ra given by ARGUS with v/if-a f8 ~<0.2 GeV). For mr=50 GeV (fig. 1 ), the region is rather small, while for mt>~60 GeV (figs. 2-6) , Xd-..- 0.3 In most regions and can be as large as 0.65 as rnt increases upto 90 GeV. Thus, if the lower bound on ro given by ARGUS (ro >/0.13) is indeed valid, it is most likely that mt >/60 GeV. A general feature present in figs. 1-6 is that the required bag factor Br. has to be rather small (<0 .3 ) in the regions with xd~-0.3. This is because the mixing angles ~9 of the fourth family with the lighter families, required to produce the enhanced B°-B ° mixing, also enhances the box-diagram contribution to the CP-impurity parameter Re e in K°-K °. We believe that this is a general feature of any realistic model of the 4 × 4 KM matrix which enhances B °-B ° mixing considerably through the fourth family of quarks, and is not particularly true for the SCF model only. However, in the absence [ 16 ] of a true theoretical knowledge on BK, BK smaller than 0.3 can not be regarded as unrealistic. On the other hand, ifBK>~0.3, figs. 2, 4, 5 indicate an upper bound on mb, of 55 GeV. Thus, if rnt>~ 60 GeV and BK>~ 0.3, the b'-quark should be seen below 55 GeV and should be discovered ~o in the near future. In that case, the ARGUS lower bound on rd can not be explained

,7 In the calculation of B R ( K +--,x +vg) and B R ( b ~ s v ~ ) , the fourth-family charged-lepton mass of 60 GeV has been used. The depen- dence of the result on this mass is very m i d .

,s For a recent discussion on this, see refs. [ 11,12 ] The results obtained m refs. [ 11,12 ] indicate that the phase structure should be very close to the one proposed m ref [ 6 ], i f the general structure o f ref [ 5 ] is to be consistent with the A R G U S [ 2 ] result on B°-B°mlxmg.

,9 For a fixed value of mr , the mixing angles I Vt d I, I Vt s I, I Vt b I . . . . increase as mb increases, as one can see m eq. (3) . ~t°The b ' -quark may show up at any t ime m the present e+e - (TRISTAN, SLC) and p~}(Sp~S, TEVATRON) colhders, even before the

discovery of the t-quark Thus, we encourage our experimental colleagues to look for this quark seriously m the present colhders.

562

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Volume 201, number 4

4 2 0 --

PHYSICS LETTERS B 18 February 1988

3 8 0

3 4 0

3 0 0

2 6 0 (.9

* ' 2 2 0

180

140

I 0 0

20 60 I00 140 leo 2Zo 260

m b, (GeV)

Fig 1. Allowed region of (mb , mt ), and contours ofx~, BK, and I V¢b[, for mb/m~=30.3, mJmu=265, mdmd=19 6+ 1.6, and

m~ = 50 GeV

4 2 0

380

3 4 0

300

~9 260

%. E 22(]

18¢

14(:

I0¢

zo eo ~oo ~4o =so zzo mb' (GeV)

Fig 2 Sameasf ig 1, with mr=60 GeV

260

360 ~ ~ l ~ ,

320

280

240 Q (.9

E ~ 2 0 0

1 6 0

120

80 20

.,..~9,~'-~ -" x~,. 9 a -

. .d- ' / ~ x;~°3z-

I I / . . i . . . . . . 5v;,i.o~ I ' ' ' ' ~ ' B K = O I

I I I I I I 60 I00 140 180 220 260 :300

m b, ( G e V )

Fig. 3 Allowed region of (rob, m r ) , and contours ofx~, BK, and I Vcbl, for mb/ms=40 4, mc/m,=389, mdmd= 19 6+ 1.6, and mt= 60 GeV

4 2 0

380 ~

340

300

260

E 220

180

1 4 0

IOC

~ , B K = 0 2 ,p~ .J I%bl" o 043

J / ' " / J r x'.-o 4 ,~ - " / Y ~ . o ) .

i j / . / i '

• i i "~

" " 7~,1 f l , i " l l " ...t_..1- . . . . , , .o ,

-

I I I I I I I 20 60 I00 140 180 220 1~60

m b, ( GeV )

Fig. 4 Same as fig. 3, with mr=70 GeV

564


4 4 0

400

360

,.-. 320

> ~ 280

E~240

200

160

120

I I I I I I

Ap

_

,..X~lv, d =o osz

O I

Bx" 0 2 , , ( / / ' i f ' /

:],71b>-_lv,,i .o.o, l'Lmx,.o,

I I I I J i ZO 60 IO0 140 leO 220 260

m b, ( G e V )

Fig. 5 Same as fig 3, with mr=80 GeV

A > ( P

E

260

240

220

200

180

160

140

I I 1 I I

Iv=d = o osr

• ~ = 0 65

B. .o .2 ,~ / ,/'~-x:=o8

_ ~ l / / , / 'Nv , d, oo55 tJ) • I

= , ~ . Iv==p o osz ~"~'X~ = 0 45

I I I I 1 20 50 40 50 60

m b, (GeV)

Fig. 6. Same as fig. 3, wzth mr=90 GeV.

70

by the SCF scheme and is to be explained by some physics beyond the KM picture, or the ARGUS result itself may not be valid ~11. Repeatedly speaking, if B~:~> 0.3 (as is suggested by the naive vacuum insertion (BK= l ) or the current algebra calculation [ 17 ]), either the t-quark or the b ' -quark should be seen below 60 GeV. This is the most interesting prediction that we find in this article. Turning to the discussion on B°-B ° mixing, table 1 shows that x'~/x)- 15-19 for all regions in figs. 1-6. This can be seen as follows: The dominant contributions to B°-B ° mixings in the box-diagram calculation come from those involving the t- and t '-quarks, while the relevant KM angles in eq. (3) satisfy the relation, Vtd ~ -- ~ Vt~ and Vt,d -~ - ~ Vr~. Thus, we expect x'~/x~- ( ~ ) 2 _ 20. This implies that the B°-~°~ ° mixing parameter rs should be (nearly) maximal if the reported ARGUS result on rd is indeed valid. As for the CP-violating parameters (F)ad and (~)as , table 1 gives the value (0.93-8.6) × 10- 5 and - (0.95-5.7) × 10- 5, respectively. These are not much different from the three- family case shown in table 2. For the rare decay mode branching ratios, table 1 gives BR(K+-~x+I ,P)= (0.41-18.7) X 10 -~°, B R ( b ~ s v g ) = (0.27-6.2) × 10 -5, BR(b~s~+£ - ) = (2.2-9.9) X 10 -6 , and BR(b--,ST) = (1.0-7.0) × 10 -5. Comparing with the corresponding quantities for the three-family case of table 2, we see that the best candidate modes which will distinguish the four-family case from the three-family case are K ÷ -*n +v9 and b~svg, whose branching ratios can be enhanced by an order of magnitude by the fourth family.

5. Conclusion

In this article we have examined the implications of the SCF hypothesis on the 4 × 4 KM matrix in the light of the recently reported o o Bd-Bd mixing and UAI ' s lower bound. The reported o o Bd-Bd mixing is shown to be conszstent with m, = 50-90 GeV, provided BK < 0.3, even if I V,b [ / I Vc~ [ is as small as 0.05-0.07. For BK >/0.3, rob, is predicted to be less than 55 GeV if mt >/60 GeV, implying the very possibility of discovering ~ o b ' of the t-quark below 60 GeV in the near future. The o o Bs-Bs mixing parameter r~ is predicted to be maximal if the

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r epo r t ed A R G U S resul t on rd is va l id . T h e b r a n c h i n g ra t ios o f s o m e rare decay modes , which will be m e a s u r e d

in the nex t gene ra t i on o f expe r imen t s , are p r e d i c t e d to be B R ( K + ~ T t + v g ) = ( 0 . 4 1 - 1 8 . 7 ) × 1 0 -~°,

BR(b- - , svg) --- ( 0 . 2 7 - 6 . 2 ) × 10 -s , B R ( b ~ s ~ + £ - ) = (2 .2 -9 .9 ) × 10 -6, and B R ( b ~ s T ) = (1 .0-7 .0) × 10 -5. A m o n g

these B R ( K ÷ - , ~ +vg) and B R ( b ~ s v ~ ) are shown to be the best cand ida te s for d i s t ingu ish ing the four - fami ly

K M m o d e l f r o m tha t o f the th ree - fami ly K M mode l .

Acknowledgement

I t hank M y r o n B a n d e r for m o s t va luab le d i scuss ions t h r o u g h o u t the course o f this work. I also thank G o r d o n

Shaw and D e n n i s S i l v e r m a n fo r l i s ten ing to me. H a i m H a r a r i and ¥ o s e f N i r are acknowledged for send ing m e

the i r works [ 11 ] as soon as they were comple t ed . Sekhar C h i v u k u l a and J o n F lynn are acknowledged for the

co l l abora t ion on ref. [ 3 ], u p o n which this w o r k is based. Th i s research is suppor t ed by the N S F u n d e r grant No .

N S F - P H Y - 8 6 0 5 5 5 2 .

~' ~Most of the upper half values of the ARGUS [ 2 ] result are already m conflict with the CLEO [ 21 ] upper bound ofrd < 0 24. This may be due to the underestimate of the background szgnals by the ARGUS group

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fourth family quark masses, b0b0 mixing, and rare decay modes, in a canonical structure of the 4×4...

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