- Email: [email protected]
Effects of extended Higgs sector on loop-induced B decays
Nuclear Physics B326 (1989) 54-72 North-Holland, Amsterdam
EFFECTS OF E X T E N D E D HIGGS S E C T O R ON LOOP-INDUCED B DECAYS Wex-Shu H O U
Max-PlamA-lnsntut fur Physlk und Astrophystk, Postfach 40121, 8000 Mumch 40, FRG R S WILLEY
Department of Physzcs and Astronomy, Unwerstty of Pittsburgh, PA, USA Received 18 August 1988 (Revised 30 March 1989)
Extended Hlggs sectors generally revolve charged Hxggs bosons (H +) Effects of these charged Hlggs bosons on vanous loop-induced processes (b --* s'y, b ~ sg* and b ---, sY +Y ) in the B-meson system are discussed, w~thm the framework of two I-hggs doublet models (2HDM's) In these models there are just two addmonal parameters associated with the effect of the H + - m H and the ratio of the two vacuum expectatmn values The H + contribution to the dipole moment form factor turns out to be very slgmficant, furthermore, it leads to rather different effects for the two types of 2HDM's Bd mixing constrains the above parameters Even with these constraints, strong enhancements of decay rates are possible In the case of b ---, sT, strong suppressions are also possible in one of the 2HDM's Constraints on the parameters of an extended Haggs sector coming from continued improvement of experimental hmats on these processes are as important as (and complementary to) constraints from Bd mixing We also briefly consider the prospects of dlstmgmshlng between the effects of charged Higgs bosons and the effects of a fourth fermlon generation on these processes
1. Introduction The theoretical pred~cuon of, and experimental search for, rare flavor changing neutral current decays is of great interest In the standard model (SM) these do not occur in lowest order Their calculation, at the one-loop order, is thus a test of the renormallzable structure of the SM Furthermore, these one-loop calculations are sensmve to the presence of new physics, beyond the SM This paper [1] is devoted to a detailed study of the effects of an extended Hlggs sector on the rare flavor changing neutral current decays b ~ s g + f -, b ~ sy, b ~ sg* Aside from simple r e p e t m o n of the fermlon generations of the SM, almost all proposals to go beyond the SM mvolve enlargement of the Hlggs sector (or its equivalent) The heavy b quark decay processes offer a number of theoretical advantages The "spectator approximation" is expected to be reliable, particularly for inclusive semlleptonlc 0550-3213/89/$03 50 ©Elsevier Science Publishers B V (North-Holland Physics Pubhshmg Division)
W 7 Hou, R S Wilier / Loop-induced B deca~s
55
decays, and because the b quark has large K o b a y a s h l - M a s k a w a (KM) couphng to the t quark, the one-loop integrals should be short-distance dominated In order to maintain the empirically successful result of the minimal model, M w2 / M z c2o s 2 0 w = 1, we restrict the generahzatlon to the inclusion of additional SUw(2 ) (complex) doublets of Hlggs bosons [2] To avoid introduction of a large n u m b e r of new undetermined parameters we will mostly deal wnh the mlmmal extension from one Hlggs doublet to two Hlggs doublets With more than one Hlggs doublet one has to take care to avoid tree level flavor changing neutral current decays mediated by the additional neutral Hlggs bosons Th~s may be accomplished [3] without "fine tuning" of parameters if all fermlons of a given charge are coupled to only one neutral Hlggs boson m the starting lagranglan In the context of a two doublet Hlggs sector, this can be implemented m one of two ways (1) charge 2 / 3 quarks obtain mass from the vacuum expectation value (v e v ) of one HIggs doublet and charge - 1 / 3 quarks obtain mass from the v e v of the other Hlggs doublet (model I) (it) Both types of quarks obtain mass from the v e v of one and the same Hlggs doublet (model II) It is of interest to note that the structure of model I occurs as a natural feature in theories with the Peccel-Qulnn type of symmetry, or in theories with supersymmetry
2
Formulas for b ~ s f + f -, b --, sT, b - , sg* in the standard model
We first review the one-loop calculations of these processes in the standard model In figs 1, 2, and 3 we give the Feynman diagrams which contribute to b--* s d + f in a renormahzable gauge (we calculate in the F e y n m a n - ' t Hooft gauge) The diagrams for b --4 s~, and b ~ sg* are a subset of these Corresponding
Fig I Box, Z exchange, and y-exchange diagrams for b ~ sfl + E
Fig 2 Box diagrams Solid lines at the bottom are the quark hnes sohd lanes at the top are the lepton lines (as in fig 1) The wavy lines are W +- the dashed hnes are charged scalars (unphyslcal and physical)
W S Hou, R S Wtllev / Loop-mdu~ed B de¢ays
56
_LA_A/ "t-
+
Ca)
(b)
(c)
& +,A + (9)
÷
",
+
(d)
t
Z~'
(e)
t
#/ ", +
,
•
(f)
+ A-.,
(h)
(t)
(1)
Fig 3 b s Z or bs7 vertex diagrams Note that m (g) to (j) the two-point subgraph is a t r a n s m o n graph (b --* s), not a self-energy graph The sohd lines are quark hnes The wavy lines an the loop are W -+ The external wavy hne is Z ° or 7 The dashed lanes are charged scalars (unphysmal and physical)
to fig 1, we write d d ( b --+ s d + d - ) =,-/~/box q-./~/Z q-,-/~ Y
(2 1)
F o r the box contribution we have
g4 ;//box = E V,25~-~2 [ g ( p ' ) 7 . ( 1 - 7 , ) b ( p ) ] [ / ( k ) 7 " ( 1
- )'5) [ ' ( k ' ) ]
1 ~SGbo×(X,)
I
(2 2) We use M = M w and the sum over t is a sum over generations of virtual intermediate charge 2 / 3 quarks, x, = m2,/M 2, Ot ~- Vts~Vlb,
Eui i
(2 3)
= 0
T h e second equation of (2 3) follows from unltarlty of K M elements V,j and embodies the G I M mechanism So terms i n G b o x which do not depend on x, may be dropped Then G b°x(X)
- 1--X
F o r Z exchange we have ( M 2 =
+ (1 - -- x ) 2 In
(2 4)
M2/c 2)
•//gz=~.[v, 2@6~2[g(P')7.(X-75)b(P) ] 1
x[d(k)v"(1-4s 2 - 75)d'(k')]~sGz(x,), 5 G
(x) =
x
+ 7 (1 - x )
x + 3x2/2 -
(1 - x )
(2 5)
/ 1
2 ln/-)
\x
(2 6)
T h e term which grows as x, for large x, is the dominant term if m t is substantially larger than M [4]
57
W S Hou, R S Wtllev / Loop-lndu6ed B decays
For photon exchange (e 2= g2s2) the calculations are more comphcated because electromagnetm gauge lnvarlance requires one to extract two powers of external momenta before neglecting external momenta relative to large internal masses (mr, M) in the remaining factors g2e2 [ [ -4F 1 "//[v --= ~-~v'2~-~zs(P')L(q23"-q"q)(l- Ys)[ --M-5-)
+'°~xqX(mb(1+ Y')
t
+ms(1-Ys))
M2jjb(P) ×~-(/(k)r"/'(k')), (q=p-p'=k + k')
(27)
The %q term multiplying F 1 gives zero contracted with the lepton current (or external photon) and we neglect m s (multiplying Fz) relanve to m b The coefflcmnts of F 1 and F 2 are chosen such that these are the functions originally calculated by Inaml and Lun [5] For the process b ~ sy ( q 2 = 0), there is no l/q 2 or lepton current and only the F 2 term contributes The function Ft(x ) has a logarithmic singularity at vamshlng x, which we extract explicitly for the u and c quarks (x u << x c << 1) [
~
1
v'Fl(x')= -~Q vuL(x°'q2)+vcL(xc'q2)+ ~'V'l- x, lnx, '
] "l- E u t F I ( X t )
,
1
(
2
L(x,,q2) =6 foldZz(1- z)ln M2 m2
~ i z ( 1 - z)
(28)
)
(29)
Then
=xlQ [
1
1 --+ 12 1 - x
13
1 - - + 12 ( l - x ) 2
1
2
1
2 -( l -- x )+ 3
1
1
+
7
1 - - + 6 (l-x) z
6 1-x
1 - - + 6 (l-x) 3
2
1
12 ( l - x )
5
5
3 (l-x)
13
lln(l/] 7 1 2 ( 1 2 X ) 4 J 1 7 ] ] "~- 12 1 - x 1
1
3
6 (l-x)
1 2
1
1
2 (l-x)
3
1 1 )(1t)
2 (1 ""-X) 4 In
(210) 1
1
3
1 - - + 4 (l-x) 2
4 1-x 1
1
21-x
+
9
1
4 (l-x)
3
1
2 (l-x)
2 (1
3
2
1 - - + 2 (l-x) 3
3
3x _- ~ - ~ l n (1)] ,x2
2 (1 - - x ) 41n
x
( I1
(2 11)
58
W S Hou, R S Wlllev / Loop-reduced B decays
For the processes b ~ sg+d - and b ~ s3,, the charge Q is 2 / 3 For b---, sg* [6] drop the terms independent of Q and replace Q by the appropriate color charge generator Xa/2 We have taken advantage of eq (2 3) to drop all constant (independent of x,) terms so G box, G z, F1, F2 are all of order x (or x In x) for small x Only G z has a term which grows linearly in x for large x Combining terms according to eq (2 1) we have the lnvariant matrix element for
b --+ sg+:
g2( 2 I ~/~=
P )T T
(P))~
(kj'}t/zT
( k ) ) [ ~ ( G b o x q- GZ),]
1-'[5 (p))(d(k)T~,:'(k'))[-S2w (Gz + /71) ,]
+(g(p '") 7 ~ ' ~ b +((£(p,)
q"
l+Ts..].
t%~-qgrn b - - - ~ b ( p ) )( d( k )7"d~( k') )[ s2wF2]
,}
(2 12) To compute the decay rate, one has to square the matrix element (2 12) and do the spin sums and the integrals over phase space The q2 dependence in eq (2 9) complicates the phase space integrals Because vu << vc the i = u contribution is negligible relative to ~= c For the c quark contribution, we approximate by neglecting the q2 dependence in eq (2 9),
L(xc,q 2) ~-ln(m2c/M 2)
(2 9')
Although this ts a crude approximation to the function, we have checked that tt does not lead to a bad approximation to the integral over all phase space* With this approximation and the neglect of the final state masses m S, m: relative to m b, the phase space mtegrals can be done analytically The result is [4]
192~r3\4~rs 2 ]
8 a +
4ac+4c 2
m2 t
(2 13)
where
a=~_.v,[Gbox+(1-4s2)Gz-4s2Fl],, I
b = E o t [ G b o x -~ G z ] , , I
c =
~'~v,[4s2F2],
(2 14)
1
* By keeping the q2 dependence m eq (2 9) and doing the complete phase space integral over (the square of) this term alone, one finds an error of not more than 20% in rate
59
W S Hou, R S Wtllev / Loop-mduced B decavs
Expressmn (2 13) Is the rate for the "on-shell" quark process b ~ sg+g - We make the usual ansatz for the semi-inclusive branching ratio F(b ---, st~+g - ) BR(B ~ K ~ + f - X ) = F(b ~ cgv) + F(b ~ ug'v) BR(B ~ {~X), with
/'(b ~ c t ; ) + F(b ~ ug'9)
2,[
GFmb
2
c
192~r3 I Vcb] f
+ I rubl2],
(2 15)
(2 16)
SO
BR(B ~ K g + ~ - X ) (
= ~
[a2+b2-4ac+4c2(ln(m~/4m~)-4/3)] BR(B ~ ~ X ) 8 i-Vcb~12f---((m2/m2~+ iVub--]~
c~2 ) 1
(2 17) The function f(mc/mb) 2 2 is the standard three-pamcle phase space factor for b -~ cgv which has a value close to 0 5 for reasonable values of m c, m b We now dtscuss briefly the values of the physical parameters we use in the calculation and the sensitivity of the results to the uncertamUes in their values First note that the strong rn~ dependence of the absolute rate (2 13) has been ehmlnated by forming the ratio in eq (2 15) The result (2 17) depends on the quark masses mc,m b through the phase space factor in the denominator and through the logarithm m the numerator and the logarithm (2 9') We take rnc= 1 5 GeV and m b = 4 9 GeV Then f(mJmb) 2 2 = 0 5 1 The result (2 17) depends on the KM elements and there are large percent uncertalnt~es in some of these, however the particular combination winch enters into (2 17) is not so sensmve to these uncertainties Using three generation unltanty, one has [Vu] << Iv~], Ivt], so to a first approximation we have v. = 0 and vt = - v c, (and also I gub[ 2-- 0 m the denominator) In tins approximation, the entire dependence on the KM elements enters in the factored form Ivcl2/I Vcbl2 = 1
(assuming three generation unltarlty)
(2 18')
In our actual calculations we used the parametnzatlon of Wolfensteln [7] w~th the values X = 0 2 2 , A = 1 0 5 , p = - l , 6 = 0 , giving Vu = - 0 00246, ~cb = 0 0508,
~ = 0 04951, gub = - 0 0 1 1 2
v t = - 0 04705, (2 18)
60
W S Hou, R S Wtllev / Loop-mduced B de¢avs
As pointed out after eq (2 11) all of the functions appearing in eq (2 12) are p o w e r suppressed for small x, except for the logarithm extracted from F 1 in eq (2 8) Ttus m e a n s that eqs (2 12) and (2 14) are completely dominated by the t = t terms with the simple exception of the c-quark logarithm term (2 9') The W - b o s o n m a s s M a p p e a r s only in the ratio x t = mZ/M 2 and in the logarithm (2 9') We take M -- 82 G e V Then the c-quark logarithm (2 9') is m fact the largest single term in eq (2 12) for m t ~ M (For large m t the term linear in x in G z (2 6) dominates ) Since this large logarithm occurs in F I which is multlphed by S2w in eq (2 14), we see from eq (2 13) that the rate is also relatively insensitive to the value of s 2 , which we take to be 0 2 2 Finally, we take the P D G value for the inclusive s e m l l e p t o m c branching ratio, BR(B ---, g~X) = 0 12 As a measure of the sensitivity to the values of the K M elements, we c o m p a r e the results of a calculation in the approximation (2 18') with the calculation using the values (2 18) We give the results for the case m t = M (x t = 1) and E= e We have f r o m (2 4), (2 6), (2 10), and (2 11) Gbox (1) _~_15,
G z ( 1 ) _-3a,
ff~(1) -
2t6ss,
F2(1 ) = ~
(2 19)
T h e n putting the approximation (2 18') in (2 17) gwes BR = 3 02 × 10 6 while the values (2 18) lead to BR = 2 50 × 10 - 6 F o r the process b ---, s~, (B ~ K~,X) we have
3c~ BR(B
KyX)
I~v, F2( x,)l 2
2 2 2~r(lgcbl2f(mc/mb)+lgub[Z)BR(B
g'~X)
(2 20)
Again we evaluate this for m t = M, first in the a p p r o x i m a t e (2 18'), second with the values (2 18) T h e results are respectively BR -- 3 62 × 10 .5 and BR = 2 78 × 10 .5 F o r the process b ~ sg*(B ~ K without charm) we refer to the detailed discussion of ref [61 W e now turn to the question of Q C D corrections We have used the standard inclusive semlleptonlc rate to normalize the rate for the rare flavor changing neutral current decays (2 15), (2 16) The leading Q C D correction to this rate has been calculated [8]
/'(b --, c~) With
2 2 ~ I(mc//mb)
_ 1 9f- ~F3 b
c i gcb ] 2 1
3~r
I(m2c/m~)
(2 21)
2 41 the bracket in (2 21) has the numerical value 0 88 In our numerical work, we multiphed the complete (2 16) (including the Vu~) by this factor
W S Hou, R S WtlleI / l oop-mdu(ed B de{ms
61
Then the formula for the BR's (2 17) and (2 20), and the numerical values quoted are each enhanced by the factor (0 88) i This factor is included in all of our subsequent calculations It has been pointed out [9] that since F2(x ) has no unsuppressed logarithm, QCD corrections to b ~ sy might be quite important A renormahzatlon group improved leading logarithm calculation of this has been given [10] by Grlnsteln, Springer and Wise (GSW) For C t ~ ( m b ) = 0 235 and Q ( M ) = 0 13 their result is numerically 'Y2v,F2 (x,) ---}'0 662(0 577 + F2(xt)) vt
(2 22)
l
For x t = 1, F2(1 ) = 0 208 is corrected to 0 520 and the BR to b --0 s7 is multiphed by a factor 6 25 BR(B ~ y K X )
--
2 0 × 10 4
(SM, QCD corrected)
(2 23)
mt=M
N o complete calculation has been done for the QCD corrections for the process b ~ sS+g If we include the GSW correction to F 2 we have, first the BR 2 50 × 1 0 - 6 replaced by 2 84 × 10 6 for the QCD correction to the denominator, and second, 2 84 × 1 0 - 6 replaced by 4 15 × 1 0 - 6 for the F 2 QCD correction Since Fl(Xc) has a large logarithm (2 9% a large QCD logarithm multlphed by a,/~r would not be such an important correction to ~2v,Fi(x,) With the two corrections stated above we have BR(B ~ K e + e X)
=
4 1 × 10
6
(SM, partial QCD corrected)
(2 24)
m t = M
The reader who has followed the discussion can make his/her own assessment of the accuracy of these results Our feehng is that (2 23), (2 24) probably represent the SM values for these BR (for m t = M ) u p t o a factor of 2 An experimental result differing by one order of magnitude from either of these results (or the correspondlng result for the appropriate value of mr) would strongly indicate physics beyond the standard model
3 Non-minimal Higgs sector
In order to maintain the empirically successful relation of the minimal SM, M E = M 2 / c 2 ) , we only consider extension from one to N SUw(2 ) complex doublets of Hlggs fields The extended Hlggs potential introduces a large number of parameters (Hlggs masses and quartlc couplings) Then there may be N different
62
w S Hou, R S Wdley / Loop-reduced B decays
vacuum expectation values (of the isoscalar neutral members of the N doublets) These may be varied by varying the parameters in the Higgs potential The first step is to form linear combinations of these N original doublets such that only one of the new doublets has a nonvanlshing v e v (v 2 = E t v 2) The doublet thus singled out plays the role of the single doublet of the mimmal model The isovector neutral and charged members of this doublet do not appear in the quadratic Hlggs potential They are the would-be Goldstone bosons whmh provide the longatudmal modes of the Z and W + All of thmr couplings to the standard gauge fields and fermlon fields are the same as m the minimal standard model Then one must dlagonahze the quadratm terms in the Hlggs potential, i e dlagonahze the Hlggs mass matrices (analogous to the K M dlagonahzatlon of the fermaon mass matrix for each fermlon charge) For the charged Hlggs fields we write
el+ = ~2,jHJ+,
I, J = 1,2,
, N,
(3 1)
where the 0 / , ~Y_ = (q~1+), are the charged members of the original doublets ( ~ t ) in the lagrangian and the H J+ for J >1 2 are the physical charged Hlggs (mass elgenstates) We have conventionally chosen Ht+ to be the unphyslcal bosons of the mlmmal sector This fixes fa n = v l / v
(3 2)
In order that the KM dlagonahzatlon of the fermion mass matrices should also diagonahze the tree level neutral current couplings we must couple the v e v of only one of the tb I to each of the Q = 2/3, - 1/3, - 1 fermlon fields [3] Let ~lu, ~xd be the doublets with v e v coupled to the Q = 2/3, - 1 / 3 quarks, respectively (Iu and I d m a y be the same or they may be different ) Then the lagranglan coupling the quarks to the charged Hlggs fields Is
g LqH~-- 2v~-
rnu,(~)~,(l_ys)V, sds~,u,H1++hc M
2v'2
M
f i ' ( 1 + 75)V'sds~2'JH1+ + h c
(33)
N o t e that (3 2) lmphes that (3 3) for the H i is the same as for the minimal SM The large number of new parameters m (3 3) for general N does not provide much more freedom for the processes we consider in this paper than is provided by
w s Hou, R S Wdlev / Loop-mdu~edB de~a}s
63
the s i m p l e e x t e n s m n s to N = 2 F o r N = 2 [11]
~= (cosB -slnB] sin fl cos/3 = v i / v ,
cos fl ]
(3 4)
sm 13 = v2/v
a n d the f e r m l o n couphngs to the extra, physical, H ± are
~L~ H+= ~
~.(1 -- ~,)~jd~
~'~.(1 + Vs)V.jd~ . + + h c . (3 5)
w h e r e ~ a n d ~' are ratios of the two v e v v 1 a n d vz T h e r e are a n u m b e r of possible c o n v e n t i o n s b u t all lead to one of two possible m o d e l s Model 1
I u ~ 1d ,
then ~' = - 1 / ~ ,
Model 2
I u = Ia,
then ~' =
(3 6)
Since the r a t i o U1/U2 IS an u n d e t e r m i n e d p a r a m e t e r of the theory, the Y u k a w a c o u p h n g s of the H + to the fermlons can be rather e n h a n c e d or s u p p r e s s e d relative to those of the u n p h y s l c a l charged scalars of the m i n i m a l S M The Z H+H and ~, H + H (derivative) c o u p h n g s are the same for all H ~ b e c a u s e they are the S U ( 2 ) × U(1) gauge c o u p h n g s The Z W+_H~ a n d 6 W ± H z_ c o u p l i n g s are generated b y a v e v a n d hence are s t a n d a r d for the u n p h y s l c a l Ha+ and zero for all the physical H s , J > 1 N o w referring to fig 2, we see that t~Lq H gives neghglble c o n t n b u t m n s to Gbo x b e c a u s e it will be c o m b i n e d with L r H CCm J M Referring to fig 3 we see that t~Lq H will l e a d to c o n t r i b u t i o n s from & a g r a m s (b), (f), (h), (j) (There is no c o n t r i b u t i o n from (d), (e) because of the v a m s h l n g of the Z W + H v couplings for the p h y s i c a l H + ) W e note two p r o p e r t i e s of this subset of d i a g r a m s (x) W h e n this s u b s e t is s u m m e d ultraviolet divergences which are n o t killed b y G I M cancel one a n o t h e r (u) T h e sum of this subset of d m g r a m s satisfies the t r a n s v e r s a h t y c o n d i t i o n for the p h o t o n vertex F o r the b -~ sZ vertex the result is G z --, G z + 3RG z w~th*
8.Gz(X, x,) = ~2x( 1 x, = m~,/M 2,
1
x'
1)
1 - x ' + (1 - - - - ~ - l n ~ 7
x; = m~,/m~,
* Throughout this paper m H as the mass of these physical charged Hlggs
'
(3 7)
(3 8)
W S Hou, R S Wtllev / Loop-mduced B decays
64
For N H > 2 , (3 7) becomes a sum of terms with ~2 and x ; i = m z / m 2 and •=2, ,N H For the b ---. s7 vertex, dmgrams (h), (j) have no q dependence They cancel the contnbuuons from &agrams (b) and (f) evaluated at q = 0 and leave the transverse form (expanded m powers of q/m)
3F.
Lelg2-'p')[(
2~I
q u - q.4)(1 - Ts)(-3F1/M 2)
-= 64~r2St
+ t%.qV(mb(1 + 75) + ms(1
- "~5))(~F2/m2)]
b(p),
q=p-p'
(3 9)
with contnbunons only from diagrams (b), (f) The results are
= 2x,/Q [
7
1
5
1 m + 12 (1 - x
36 1 - x'
1
11
_1 )(1)]
2 (1-x') 3
1
5
+
6 ( 1 - x ' ) 4 In 1
1
1x,3
1
- -- x') 3 + 6 ( 1 7 x ' ) 41n 77 6 (1
12 (1 - x') 2
36 1 - x '
6 (1 - x') 3
, ]l 2
(1 1 +
1
'
(3 10) 1
a.F~=Ux'{Q
1
1
1 m + 4 (1-xq 2
12 1 - x ' 1
1
!
3
+
Q
-
-
-
1
-
x'
11 + 2 1-x'
0I-
I
1
(1
-- X ' ) 2
1
1
2
(1--x')' 1
4 (1-x') 2
6 1-x'
+ ~'x'
1
1
1 x' {1)] 2 (1 - x') a ln~ 77
1
2 (1-x') 3 + 2 (1-
1
x') 4
(1)]
(1 _7 x,) 3 In ~-;
x,
( 1 - x ' ) 2 + ~( 1 -lx ' )n
(1)) 77
(311)
Note that only ~HF2 depends on both ~, ~' and hence distinguishes between models 1 and 2 (3 6) For quantmes m whose calculation one can neglect external
W S Hou, R S Wllley / Loop-induced B de~avs
65
( Q - - 1 / 3 quark) masses compared to internal (t quark and W) masses from the beginning, ~' does not enter The parameter ~' enters 6HF2 because EM gauge Invarlance requires an expansion in powers of q / M which leads to the explicit q m b / M 2 term in (3 9) In particular, consideration of A m K [12], E~: [13], and B d [14] mixing can put bounds on (~, mn+ ) but does not distinguish between models 1 and 2
4. Discussion of results W e have chosen to present a n u m b e r of representative results in a series of graphs (figs 4 - 1 5 ) F o r each of the three processes b --* sg'+8 , b ~ s7, b --* sg*, we plot the B R as a function of m n / m t for r o t = 80 GeV and four values of ~ (0, 2, 4, 8), for m o d e l 1 and for model 2* For fixed ~ and m H / m t there is not great sensitivity to m t F o r m H / r n , >~ 2 or 3 the BR's increase by a factor of 2 or 3 as m t is increased f r o m 40 to 120 GeV For small m H / m t there is greater enhancement from the H + and less sensitivity to m t Note that for ~ = 0, the results for model 2 are just the SM results ( C o m p a r e the m t = 80 GeV results with (2 23) and (2 24) ) F o r model 1, ~ ' = - 1 and 8Hi~]2 IS nonzero even for ~ = 0 Also, for m H / m t >> 1, the BR's for either model 1, 2 approach the SM results One can see from the graphs that for mH -- mt there can be very large enhancements (e g three orders of magnitude for ~ = 8 and m H - m t In this situation, the rates are proportional to (4) A feature of the graphs is the dramatic suppression of the b ~ s¥ BR (relative to the SM result) in model 2 for some range of values of ((, m H / m t ) This happens because the 8HF2 term dominates F 2 for large ~ and for model 2, with ~ ' = ~2, it is always of opposite sign to the SM F 2 As mentioned above, ~ is an undetermined parameter of the extended Hlggs sector, but it 1s constrained by empirical data, particularly the recent observation [15] of B a mixing We have gone through the exercise [14] of computing the correction to the SM prediction of this mixing in the presence of a second Hlggs doublet T h e n for fixed values of mr, rntj we determine the m a x i m u m value of consistent with the experimental 90% C L upper limit x d -%<1 02 from the A R G U S result r d = 0 21 + 0 08 The result obtained depends on the choices of various parameters in the calculation For tins calculation, we have taken f B v ~ - = 100 MeV and Via = 0 01 These values are toward the low end of the allowable range and lead to larger values of ~m~, The results of this calculation are summarized in table 1 * In our calculations based on the ansatz (2 15) we use calculated values for the quark semlleptonlc widths, but the experimental value for the B-meson semaleptomc BR This procedure is correct as long as the result is a BR much less than one For the strong proces~ b---. sg*, with large enhancements for large ~ this condition is not satisfied In the calculations presented m figs 8, 9 and 15 we have enforced the u m t a n t y constraint by setting B = " ( 2 15)"/(1 + "(2 15)") where "(2 15)" is the value computed from (2 15)
66
W S Hou, R S Wtlley / Loop-mduced B decays
16 z I
I
I
I
I
I
i
I
I
I
I
vs mH/mt
B R ( b - > s.O#)
~
i
i
i
~ x
Model I
i
,
i
i
J
i
I
i
t
l
I
i
i
i
i
B R ( b - > s~Q) vs mH/m t
-2
I0
mt = 8 0 GeV 10-3
i
mt = 8 0 GeV \
Model 2
\ I0 !
15
5
I0
t5
20
t5
20
mH/m t Fig 4 -6
10
I
L
I
I
[
I
I
I
[
1 2 3 4 5
I
t0
mH/m t Fig 5
tO-'
to-3
10- 4
I
i
L
I
I
i
1 2 3 4 5
I
I
I
I0
166
5
tO
mH/m t
mH/m t
Fig 6
Fag 7
Figs 4 - 9 BR for four values of the parameter 4(0,2,4,8) F o r small values of mH/mt, the mcreasmg values of the BR correspond to increasing values of ~ (m H is the mass of the physical charged scalar )
W S Hou, R S Wdley / Loop-mdueed B de~.ays f
I
I
I
I
BR(b->s
I
I
I
I
F
67
I
w / o c ) vs m H / m t
1 .
mt = 80 GeV
_
del
1
-I
10
-2
10
I
I
I
I
2
3 4 5
~
I
I
I
I
I
1
I0
m H / mt Fig 8 ,
,
,
i
i
i
,
,
,
I
BR(b->s
T
,
,
i
I
,
,
,
r
mH/rn t
w/o c)vs
mt = 8 0 G e V -t
10
-2
10
I
I
5
I
I
I
I
I
I
J
I0
I
I
I
=
I
I
15
I
20
mH/m t Fig 9 Figs 8 and 9 See caption on facing page 2 -- l O m H / m , (for m t ~ 40 R o u g h interpolating formulas for these results are ~m~ GeV), ~max----45mH/mt 2 (for r n t ~ 80 GeV), and 3rnH/m t (for m t 120 GeV) One m a y c o m p a r e these results with ref [14] which found corresponding interpolating formula ranging from 2 m n / m t to 1 2 m H / m t (for m r = 4 5 GeV) d e p e n d i n g on choice of parameters We present another set of graphs of the BR for the three processes as functions of m H / m t, for three values of mt, with ~ = ~max determined above for each m n, m t, for model 1 and for model 2, Note, however, one subtlety Because of the destructive interference in F 2 for model 2, we have the result B(~m~ ) = Bmax when ~ = ~m~, leads to enhancement, but Br.~, = BsM when = ~ma~ leads to suppression (This is the case for m t = 120 GeV See fig 13 )
W S Hou, R S Wtllev / Loop-reduced B deems
68
-3
10
-3
i
i
~
i
I
i
BR(b->stl~)
i
i
I0
i
i
i
~
i
1
i
BR(b->sQI~)
mH/m t
vs
Model
i
io-4
10
i
i
1
mH/m t
vs
Model
1
I
2
-____ BO
lO-
120 80 40
I~ 6
i 1
i I , 2 3 4 5
i
i
i
l
I
1(~6
L
10
I
I
I
I
I
I
I
I
I
mH/m t
I
10
1 2 3 4 5
mH/m t
Fig l 0
Fig 11
-1
10
-1
10
BR(b->sT')
vs Model
mH/m
t
1
i
1
i
i
i
i
i
1
r
i
vs mH/m t
BR(b->sT') Model
2
40
10-2
-2
i
10
8O
IG 3
-3
10
8O 10-4
I
I
I
I
I
1
2
3
4
5
I
mH/m t Fig 12
I
I
i(54
I
10
I
I
I
I
I
t
2
3
4
5
t
i
I
I
I
10
mH/m t Fig 13
Figs 10-15 BR computed for maximum values of ( consastent with observed hmlt on Bd (see text and table 1) The three curves are for m t = 40, 80, and 120 GeV, as marked (m H Is the mass of the physical charged scalar)
W S Hou, R S Wllle~ / Loop-mdu~ed B de~avs
t
Model I
2[ ~
_
tff'
'
I
i
I
I
I
_
,20
I
I
I
1 2 5 4 5 mH/m t
J
10
Fig 14
10
BR(b->s
w/o c) vs mH/m t Model 2
40
8O
-I
10
f 120
-2
10
I
J
I
I
I
I
I
I
J
1 2345
I
10 mH/m t Fig 15
Figs 14 and 15 See caption on facing page
69
W S Hou, R S Wdley / Loop-reduced B decays
70
TABLF. 1 ~,],.~ vs m H / m t for several m t values (M = Mw)
mH/mt
~max (mr = M/2)
~ma~ 2 (rnt = M )
2 (grit = V/~M ) ~max
05 10 20 30 40
100 137 219 304 401
41 58 96 136 177
25 36 60 85 114
50
488
220
140
10 0
96 9
44 2
29 8
We discuss again briefly the situation with Q C D corrections The processes we consider are driven by one-loop integrals dominated by m o m e n t a of order M w, hence plausibly short distance on the Q C D scale Thus one m a y expect Q C D corrections to be perturbatlve, of order (e~s/~r)ln(x,/xs) Because F2(x,) has no I n ( l / x , ) (except multiphed by x,), the Q C D correction to b---, s~, in the standard model is relatively large [9,10] but m the nonmlnimal model with large enhancements, 8 H F 2 dominates over such Q C D corrections (But the values of ~, m~, m t for which there is large suppression of b ~ s~ in model 2 are sensitive to Q C D corrections ) In the results presented in the figures we have included the k n o w n (incomplete) Q C D corrections already discussed in sect 2, eqs (2 21) and (2 22) As mentioned above, the SM results for the BR's can be seen from the (straight line) curves for ~ = 0 in the model 2 graphs Then the m a x i m u m enhancement of these BR's c o m i n g from the presence of a second Hlggs doublet, and consistent with the observed limit on B d mixing, may be determined by c o m p a n n g these values with the second set of graphs giving the BR's for ~ = ~max The first striking observation f r o m tbas c o m p a r i s o n is that the largest possible relative enhancements occur for the smallest value of rn t considered This Is a reflection of the constraint imposed b y B a mixing The mlmmal SM can produce the observed value with a "large" value of m t Conversely, large m t does not leave much r o o m for enhancement by n o n - S M physics, but small m t requires it T h e experimental study of rare B decays is still at an early stage, but there are already some results For b - - , s f + g - C L E O [16] has an Inclusive bound, BR(B -~ f + f X) < 1 2 × 10 -3, still an order of magnitude larger than the largest multl-Hlggs enhanced BR consistent with the observed B d mixing F o r b ~ s~, A R G U S [17] has reported a limit BR(B ~ K*~,) < 4 × 10 4 Taking a conservative estimate for the exclusive decay / ' ( B - ~ K*~,)/F(b ~ s~,)> 6%, we arrive at BR(b --, s~,) < 6 × 10 -3 This is already exceeded by the enhanced B R in model 2 for " l i g h t " m t (40 GeV) and m H , / m t ~ 3 Conversely the present limit on B ~ K * y already excludes a region of the parameter space for the two doublet Hlggs sector which IS not excluded by the b o u n d on B d mixing F o r b ~ sg*, which dominates
W S Hou, R S Wtlley / Loop-Induced B deca~,s
71
the total flavor changing neutral current b ~ s transition rate, there is still no definite experimental mformat~on However, one can obtain some indications from the reclusive K yield reported by CLEO [18] and the reclusive charmed particle production m B decays reported by A R G U S and CLEO [17,19] charmless B ~ K translnons at the 10-20% level are not yet excluded Weak as these limits are, they already place constraints on enhancement by additional charged Hlggs which are competlnve wtth corresponding limits obtained from B d mixing Since the Bd mixing is already a measured number [15], one does not expect drastically improved bounds on the extended Higgs sector from subsequent more accurate measurements But for the rare B decays considered above, as the experimental upper limits continue to improve*, stronger constraints on the parameters of an extended Hlggs sector will result Of course, an extended Hlggs sector is only one possible source of enhancement of loop-reduced flavor changing neutral current B decays Another possibility ~s extension of the fermlon sector, 1 e the existence of a fourth generation of fermlons In a previous paper [4] we have &scussed possible enhancement of b ~ s f + { decays in the presence of a fourth generation Here we point out that a fourth generation with a very heavy t' quark can be most effective in enhancing the b--* sE+g - decay because of the linear x, term m the Z-exchange function G z (2 6) A fourth generanon does not lead to strong enhancement of b --* sy or b --* sg because these are determined by Fz(x ) which actually decreases for x >> 1 (GIM kills the asymptoncally constant term) The sltuatton is reversed for enhancement by enhanced couplings of the charged Hxggs bosons of an extended Higgs sector Since B d m~xmg also has a hnear x, dependence, the observed hmlt strongly constrains the possible muln-H~ggs enhancement of b ~ sE+E -, ~f there are only three generations** On the other hand, 6 n F 2 which dormnates the enhancement of b ~ sT, b --* sg depends on m t only through the ratio m t / m n with m n (mass of physical charged Higgs) unknown So for smaller values of m t, for which the B d mixing limit allows large values of (~2, m r ~ r a n ) ' one can have large enhancements (by scahng down m n with m t One can not do this with 8riG z which depends also on m t / M ) In summary a greatly enhanced b ~ s f + g - rate would most likely indicate the existence of a fourth generation, while greatly enhanced rates for b ~ sT a n d / o r b ~ sg* (signature a greatly enhanced subprocess b --* sg with g "on-shell", see ref [6] for discussion) would more likely be attributed to enhancements from an extended Htggs sector
* New hmlts on loop induced flavor changing B decays have been reported by the A R G U S and CLEO collaborations at the XXIV International Conference on High Energy Physlcs, Munich, August 4 - 1 0 1988 they strengthen our statement made here See the talks by A Jawahery and A Golutvln given at the conference * * The observed limit on Ba mixing does not strongly limit the fourth generation enhancement of b--,s•+g because it bounds VttbVtdXt while what enters into b - - * s f + f is I/tbVt~x t If n o n m a x l m a l B, mixing is subsequently observed thas situation will be changed [20]
W S Hou, R S Wdley / Loop-reduced B decays
72
Either the presence of a fourth generation or an extended Hlggs sector can lead to suppression of the b ~ sy rate due to Interference between different contributions to the single form factor F 2 which determines this rate Since neither, either, or both of these extensions of the minimal SM may be realized in nature, additional experimental information (beyond the BR's, or hmlts, for these loop induced B decays) will be required to sort out the various possibilities with confidence In the short term, discovery of the t quark (and its mass) and neutrino counting at L E P / S L C , combined with new data on the B decays dxscussed here, will substantially improve the constraints on these possibilities Ultimately, if there ~s a fourth generation, or an extended Hlggs sector, experlmentahsts must discover the additional quark(s) a n d / o r charged Hlggs bosons We thank T Ferguson and A Soni for discussions
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
A brief report of some of the results of this paper has appeared in Phys Lett B202 (1988) 591 D A Ross and M Veltman, Nucl Phys B45 (1975) 135 S Glashow and S Welnberg, Phys Rev D15 (1977) 1958 W S Hou, R S Wdley and A Sore, Phys Rev Lett 58 (1987) 1608, 60 (1988) 2337(E) T Inaml and C S Llm, Prog Theor Phys 65 (1981) 297, 1772(E) W S Hou Nucl Phys B308 (1988)561 L Wolfenstein, Phys Rev Lett 51 (1983)1945 N Cablbbo and L Maaanl, Phys Lett B79 (1978) 109 J L Cortes, X Y Pham and A Tounsl, Phys Rev D25 (1982) 188 M A Shlfman, A I Vamshte, n and V I Zakharov, Phys Rev D18 (1978) 2583 S Bertohm, F Borumati and A Mas/ero, Phys Rev Lett 59 (1987) 180, N G Deshpande P Lo, J Trampetic, G Ellam and P Singer, Phys Rev Lett 59 (1987)183 B Grlnstem, R Springer, and M Wise, Phys Lett B202 (1988) 138 H E Haber, G L Kane and T Sterling, Nucl Phys B161 (1979) 493 This has also been apphed to the b ---,sT process by B Grlnsteln and M Wise, Phys Lett B201 (1988) 274 L F Abbott, P Slklvle and M Wise, Phys Rev D21 (1980)1391 G G Athanasm and F J Gdman, Phys Lett B153 (1985) 274 G G Athanaslu, P J Franzlm and F J Gilman, Phys Rev D32 (1985) 3010 ARGUS Collaboration H Albrecht et al, Phys Len B192 (1987) 245 CLEO Collaboration, A Bean et al, Phys Rev D35 (1987) 3533 W Schmldt-Parzefall, Nucl Phys B (Proc Suppl) 3 (1988) 257 CLEO Collaboration, M S Alam et al, Phys Rev Lett 58 (I987) 1814 CLEO Collaboration, D Bertoletto et al, Phys Rev D35 (1987) 19 W S Hou and A Sonl Phys Lett BI96 (1987)92
EFFECTS OF E X T E N D E D HIGGS S E C T O R ON LOOP-INDUCED B DECAYS Wex-Shu H O U
Max-PlamA-lnsntut fur Physlk und Astrophystk, Postfach 40121, 8000 Mumch 40, FRG R S WILLEY
Department of Physzcs and Astronomy, Unwerstty of Pittsburgh, PA, USA Received 18 August 1988 (Revised 30 March 1989)
Extended Hlggs sectors generally revolve charged Hxggs bosons (H +) Effects of these charged Hlggs bosons on vanous loop-induced processes (b --* s'y, b ~ sg* and b ---, sY +Y ) in the B-meson system are discussed, w~thm the framework of two I-hggs doublet models (2HDM's) In these models there are just two addmonal parameters associated with the effect of the H + - m H and the ratio of the two vacuum expectatmn values The H + contribution to the dipole moment form factor turns out to be very slgmficant, furthermore, it leads to rather different effects for the two types of 2HDM's Bd mixing constrains the above parameters Even with these constraints, strong enhancements of decay rates are possible In the case of b ---, sT, strong suppressions are also possible in one of the 2HDM's Constraints on the parameters of an extended Haggs sector coming from continued improvement of experimental hmats on these processes are as important as (and complementary to) constraints from Bd mixing We also briefly consider the prospects of dlstmgmshlng between the effects of charged Higgs bosons and the effects of a fourth fermlon generation on these processes
1. Introduction The theoretical pred~cuon of, and experimental search for, rare flavor changing neutral current decays is of great interest In the standard model (SM) these do not occur in lowest order Their calculation, at the one-loop order, is thus a test of the renormallzable structure of the SM Furthermore, these one-loop calculations are sensmve to the presence of new physics, beyond the SM This paper [1] is devoted to a detailed study of the effects of an extended Hlggs sector on the rare flavor changing neutral current decays b ~ s g + f -, b ~ sy, b ~ sg* Aside from simple r e p e t m o n of the fermlon generations of the SM, almost all proposals to go beyond the SM mvolve enlargement of the Hlggs sector (or its equivalent) The heavy b quark decay processes offer a number of theoretical advantages The "spectator approximation" is expected to be reliable, particularly for inclusive semlleptonlc 0550-3213/89/$03 50 ©Elsevier Science Publishers B V (North-Holland Physics Pubhshmg Division)
W 7 Hou, R S Wilier / Loop-induced B deca~s
55
decays, and because the b quark has large K o b a y a s h l - M a s k a w a (KM) couphng to the t quark, the one-loop integrals should be short-distance dominated In order to maintain the empirically successful result of the minimal model, M w2 / M z c2o s 2 0 w = 1, we restrict the generahzatlon to the inclusion of additional SUw(2 ) (complex) doublets of Hlggs bosons [2] To avoid introduction of a large n u m b e r of new undetermined parameters we will mostly deal wnh the mlmmal extension from one Hlggs doublet to two Hlggs doublets With more than one Hlggs doublet one has to take care to avoid tree level flavor changing neutral current decays mediated by the additional neutral Hlggs bosons Th~s may be accomplished [3] without "fine tuning" of parameters if all fermlons of a given charge are coupled to only one neutral Hlggs boson m the starting lagranglan In the context of a two doublet Hlggs sector, this can be implemented m one of two ways (1) charge 2 / 3 quarks obtain mass from the vacuum expectation value (v e v ) of one HIggs doublet and charge - 1 / 3 quarks obtain mass from the v e v of the other Hlggs doublet (model I) (it) Both types of quarks obtain mass from the v e v of one and the same Hlggs doublet (model II) It is of interest to note that the structure of model I occurs as a natural feature in theories with the Peccel-Qulnn type of symmetry, or in theories with supersymmetry
2
Formulas for b ~ s f + f -, b --, sT, b - , sg* in the standard model
We first review the one-loop calculations of these processes in the standard model In figs 1, 2, and 3 we give the Feynman diagrams which contribute to b--* s d + f in a renormahzable gauge (we calculate in the F e y n m a n - ' t Hooft gauge) The diagrams for b --4 s~, and b ~ sg* are a subset of these Corresponding
Fig I Box, Z exchange, and y-exchange diagrams for b ~ sfl + E
Fig 2 Box diagrams Solid lines at the bottom are the quark hnes sohd lanes at the top are the lepton lines (as in fig 1) The wavy lines are W +- the dashed hnes are charged scalars (unphyslcal and physical)
W S Hou, R S Wtllev / Loop-mdu~ed B de¢ays
56
_LA_A/ "t-
+
Ca)
(b)
(c)
& +,A + (9)
÷
",
+
(d)
t
Z~'
(e)
t
#/ ", +
,
•
(f)
+ A-.,
(h)
(t)
(1)
Fig 3 b s Z or bs7 vertex diagrams Note that m (g) to (j) the two-point subgraph is a t r a n s m o n graph (b --* s), not a self-energy graph The sohd lines are quark hnes The wavy lines an the loop are W -+ The external wavy hne is Z ° or 7 The dashed lanes are charged scalars (unphysmal and physical)
to fig 1, we write d d ( b --+ s d + d - ) =,-/~/box q-./~/Z q-,-/~ Y
(2 1)
F o r the box contribution we have
g4 ;//box = E V,25~-~2 [ g ( p ' ) 7 . ( 1 - 7 , ) b ( p ) ] [ / ( k ) 7 " ( 1
- )'5) [ ' ( k ' ) ]
1 ~SGbo×(X,)
I
(2 2) We use M = M w and the sum over t is a sum over generations of virtual intermediate charge 2 / 3 quarks, x, = m2,/M 2, Ot ~- Vts~Vlb,
Eui i
(2 3)
= 0
T h e second equation of (2 3) follows from unltarlty of K M elements V,j and embodies the G I M mechanism So terms i n G b o x which do not depend on x, may be dropped Then G b°x(X)
- 1--X
F o r Z exchange we have ( M 2 =
+ (1 - -- x ) 2 In
(2 4)
M2/c 2)
•//gz=~.[v, 2@6~2[g(P')7.(X-75)b(P) ] 1
x[d(k)v"(1-4s 2 - 75)d'(k')]~sGz(x,), 5 G
(x) =
x
+ 7 (1 - x )
x + 3x2/2 -
(1 - x )
(2 5)
/ 1
2 ln/-)
\x
(2 6)
T h e term which grows as x, for large x, is the dominant term if m t is substantially larger than M [4]
57
W S Hou, R S Wtllev / Loop-lndu6ed B decays
For photon exchange (e 2= g2s2) the calculations are more comphcated because electromagnetm gauge lnvarlance requires one to extract two powers of external momenta before neglecting external momenta relative to large internal masses (mr, M) in the remaining factors g2e2 [ [ -4F 1 "//[v --= ~-~v'2~-~zs(P')L(q23"-q"q)(l- Ys)[ --M-5-)
+'°~xqX(mb(1+ Y')
t
+ms(1-Ys))
M2jjb(P) ×~-(/(k)r"/'(k')), (q=p-p'=k + k')
(27)
The %q term multiplying F 1 gives zero contracted with the lepton current (or external photon) and we neglect m s (multiplying Fz) relanve to m b The coefflcmnts of F 1 and F 2 are chosen such that these are the functions originally calculated by Inaml and Lun [5] For the process b ~ sy ( q 2 = 0), there is no l/q 2 or lepton current and only the F 2 term contributes The function Ft(x ) has a logarithmic singularity at vamshlng x, which we extract explicitly for the u and c quarks (x u << x c << 1) [
~
1
v'Fl(x')= -~Q vuL(x°'q2)+vcL(xc'q2)+ ~'V'l- x, lnx, '
] "l- E u t F I ( X t )
,
1
(
2
L(x,,q2) =6 foldZz(1- z)ln M2 m2
~ i z ( 1 - z)
(28)
)
(29)
Then
=xlQ [
1
1 --+ 12 1 - x
13
1 - - + 12 ( l - x ) 2
1
2
1
2 -( l -- x )+ 3
1
1
+
7
1 - - + 6 (l-x) z
6 1-x
1 - - + 6 (l-x) 3
2
1
12 ( l - x )
5
5
3 (l-x)
13
lln(l/] 7 1 2 ( 1 2 X ) 4 J 1 7 ] ] "~- 12 1 - x 1
1
3
6 (l-x)
1 2
1
1
2 (l-x)
3
1 1 )(1t)
2 (1 ""-X) 4 In
(210) 1
1
3
1 - - + 4 (l-x) 2
4 1-x 1
1
21-x
+
9
1
4 (l-x)
3
1
2 (l-x)
2 (1
3
2
1 - - + 2 (l-x) 3
3
3x _- ~ - ~ l n (1)] ,x2
2 (1 - - x ) 41n
x
( I1
(2 11)
58
W S Hou, R S Wlllev / Loop-reduced B decays
For the processes b ~ sg+d - and b ~ s3,, the charge Q is 2 / 3 For b---, sg* [6] drop the terms independent of Q and replace Q by the appropriate color charge generator Xa/2 We have taken advantage of eq (2 3) to drop all constant (independent of x,) terms so G box, G z, F1, F2 are all of order x (or x In x) for small x Only G z has a term which grows linearly in x for large x Combining terms according to eq (2 1) we have the lnvariant matrix element for
b --+ sg+:
g2( 2 I ~/~=
P )T T
(P))~
(kj'}t/zT
( k ) ) [ ~ ( G b o x q- GZ),]
1-'[5 (p))(d(k)T~,:'(k'))[-S2w (Gz + /71) ,]
+(g(p '") 7 ~ ' ~ b +((£(p,)
q"
l+Ts..].
t%~-qgrn b - - - ~ b ( p ) )( d( k )7"d~( k') )[ s2wF2]
,}
(2 12) To compute the decay rate, one has to square the matrix element (2 12) and do the spin sums and the integrals over phase space The q2 dependence in eq (2 9) complicates the phase space integrals Because vu << vc the i = u contribution is negligible relative to ~= c For the c quark contribution, we approximate by neglecting the q2 dependence in eq (2 9),
L(xc,q 2) ~-ln(m2c/M 2)
(2 9')
Although this ts a crude approximation to the function, we have checked that tt does not lead to a bad approximation to the integral over all phase space* With this approximation and the neglect of the final state masses m S, m: relative to m b, the phase space mtegrals can be done analytically The result is [4]
192~r3\4~rs 2 ]
8 a +
4ac+4c 2
m2 t
(2 13)
where
a=~_.v,[Gbox+(1-4s2)Gz-4s2Fl],, I
b = E o t [ G b o x -~ G z ] , , I
c =
~'~v,[4s2F2],
(2 14)
1
* By keeping the q2 dependence m eq (2 9) and doing the complete phase space integral over (the square of) this term alone, one finds an error of not more than 20% in rate
59
W S Hou, R S Wtllev / Loop-mduced B decavs
Expressmn (2 13) Is the rate for the "on-shell" quark process b ~ sg+g - We make the usual ansatz for the semi-inclusive branching ratio F(b ---, st~+g - ) BR(B ~ K ~ + f - X ) = F(b ~ cgv) + F(b ~ ug'v) BR(B ~ {~X), with
/'(b ~ c t ; ) + F(b ~ ug'9)
2,[
GFmb
2
c
192~r3 I Vcb] f
+ I rubl2],
(2 15)
(2 16)
SO
BR(B ~ K g + ~ - X ) (
= ~
[a2+b2-4ac+4c2(ln(m~/4m~)-4/3)] BR(B ~ ~ X ) 8 i-Vcb~12f---((m2/m2~+ iVub--]~
c~2 ) 1
(2 17) The function f(mc/mb) 2 2 is the standard three-pamcle phase space factor for b -~ cgv which has a value close to 0 5 for reasonable values of m c, m b We now dtscuss briefly the values of the physical parameters we use in the calculation and the sensitivity of the results to the uncertamUes in their values First note that the strong rn~ dependence of the absolute rate (2 13) has been ehmlnated by forming the ratio in eq (2 15) The result (2 17) depends on the quark masses mc,m b through the phase space factor in the denominator and through the logarithm m the numerator and the logarithm (2 9') We take rnc= 1 5 GeV and m b = 4 9 GeV Then f(mJmb) 2 2 = 0 5 1 The result (2 17) depends on the KM elements and there are large percent uncertalnt~es in some of these, however the particular combination winch enters into (2 17) is not so sensmve to these uncertainties Using three generation unltanty, one has [Vu] << Iv~], Ivt], so to a first approximation we have v. = 0 and vt = - v c, (and also I gub[ 2-- 0 m the denominator) In tins approximation, the entire dependence on the KM elements enters in the factored form Ivcl2/I Vcbl2 = 1
(assuming three generation unltarlty)
(2 18')
In our actual calculations we used the parametnzatlon of Wolfensteln [7] w~th the values X = 0 2 2 , A = 1 0 5 , p = - l , 6 = 0 , giving Vu = - 0 00246, ~cb = 0 0508,
~ = 0 04951, gub = - 0 0 1 1 2
v t = - 0 04705, (2 18)
60
W S Hou, R S Wtllev / Loop-mduced B de¢avs
As pointed out after eq (2 11) all of the functions appearing in eq (2 12) are p o w e r suppressed for small x, except for the logarithm extracted from F 1 in eq (2 8) Ttus m e a n s that eqs (2 12) and (2 14) are completely dominated by the t = t terms with the simple exception of the c-quark logarithm term (2 9') The W - b o s o n m a s s M a p p e a r s only in the ratio x t = mZ/M 2 and in the logarithm (2 9') We take M -- 82 G e V Then the c-quark logarithm (2 9') is m fact the largest single term in eq (2 12) for m t ~ M (For large m t the term linear in x in G z (2 6) dominates ) Since this large logarithm occurs in F I which is multlphed by S2w in eq (2 14), we see from eq (2 13) that the rate is also relatively insensitive to the value of s 2 , which we take to be 0 2 2 Finally, we take the P D G value for the inclusive s e m l l e p t o m c branching ratio, BR(B ---, g~X) = 0 12 As a measure of the sensitivity to the values of the K M elements, we c o m p a r e the results of a calculation in the approximation (2 18') with the calculation using the values (2 18) We give the results for the case m t = M (x t = 1) and E= e We have f r o m (2 4), (2 6), (2 10), and (2 11) Gbox (1) _~_15,
G z ( 1 ) _-3a,
ff~(1) -
2t6ss,
F2(1 ) = ~
(2 19)
T h e n putting the approximation (2 18') in (2 17) gwes BR = 3 02 × 10 6 while the values (2 18) lead to BR = 2 50 × 10 - 6 F o r the process b ---, s~, (B ~ K~,X) we have
3c~ BR(B
KyX)
I~v, F2( x,)l 2
2 2 2~r(lgcbl2f(mc/mb)+lgub[Z)BR(B
g'~X)
(2 20)
Again we evaluate this for m t = M, first in the a p p r o x i m a t e (2 18'), second with the values (2 18) T h e results are respectively BR -- 3 62 × 10 .5 and BR = 2 78 × 10 .5 F o r the process b ~ sg*(B ~ K without charm) we refer to the detailed discussion of ref [61 W e now turn to the question of Q C D corrections We have used the standard inclusive semlleptonlc rate to normalize the rate for the rare flavor changing neutral current decays (2 15), (2 16) The leading Q C D correction to this rate has been calculated [8]
/'(b --, c~) With
2 2 ~ I(mc//mb)
_ 1 9f- ~F3 b
c i gcb ] 2 1
3~r
I(m2c/m~)
(2 21)
2 41 the bracket in (2 21) has the numerical value 0 88 In our numerical work, we multiphed the complete (2 16) (including the Vu~) by this factor
W S Hou, R S WtlleI / l oop-mdu(ed B de{ms
61
Then the formula for the BR's (2 17) and (2 20), and the numerical values quoted are each enhanced by the factor (0 88) i This factor is included in all of our subsequent calculations It has been pointed out [9] that since F2(x ) has no unsuppressed logarithm, QCD corrections to b ~ sy might be quite important A renormahzatlon group improved leading logarithm calculation of this has been given [10] by Grlnsteln, Springer and Wise (GSW) For C t ~ ( m b ) = 0 235 and Q ( M ) = 0 13 their result is numerically 'Y2v,F2 (x,) ---}'0 662(0 577 + F2(xt)) vt
(2 22)
l
For x t = 1, F2(1 ) = 0 208 is corrected to 0 520 and the BR to b --0 s7 is multiphed by a factor 6 25 BR(B ~ y K X )
--
2 0 × 10 4
(SM, QCD corrected)
(2 23)
mt=M
N o complete calculation has been done for the QCD corrections for the process b ~ sS+g If we include the GSW correction to F 2 we have, first the BR 2 50 × 1 0 - 6 replaced by 2 84 × 10 6 for the QCD correction to the denominator, and second, 2 84 × 1 0 - 6 replaced by 4 15 × 1 0 - 6 for the F 2 QCD correction Since Fl(Xc) has a large logarithm (2 9% a large QCD logarithm multlphed by a,/~r would not be such an important correction to ~2v,Fi(x,) With the two corrections stated above we have BR(B ~ K e + e X)
=
4 1 × 10
6
(SM, partial QCD corrected)
(2 24)
m t = M
The reader who has followed the discussion can make his/her own assessment of the accuracy of these results Our feehng is that (2 23), (2 24) probably represent the SM values for these BR (for m t = M ) u p t o a factor of 2 An experimental result differing by one order of magnitude from either of these results (or the correspondlng result for the appropriate value of mr) would strongly indicate physics beyond the standard model
3 Non-minimal Higgs sector
In order to maintain the empirically successful relation of the minimal SM, M E = M 2 / c 2 ) , we only consider extension from one to N SUw(2 ) complex doublets of Hlggs fields The extended Hlggs potential introduces a large number of parameters (Hlggs masses and quartlc couplings) Then there may be N different
62
w S Hou, R S Wdley / Loop-reduced B decays
vacuum expectation values (of the isoscalar neutral members of the N doublets) These may be varied by varying the parameters in the Higgs potential The first step is to form linear combinations of these N original doublets such that only one of the new doublets has a nonvanlshing v e v (v 2 = E t v 2) The doublet thus singled out plays the role of the single doublet of the mimmal model The isovector neutral and charged members of this doublet do not appear in the quadratic Hlggs potential They are the would-be Goldstone bosons whmh provide the longatudmal modes of the Z and W + All of thmr couplings to the standard gauge fields and fermlon fields are the same as m the minimal standard model Then one must dlagonahze the quadratm terms in the Hlggs potential, i e dlagonahze the Hlggs mass matrices (analogous to the K M dlagonahzatlon of the fermaon mass matrix for each fermlon charge) For the charged Hlggs fields we write
el+ = ~2,jHJ+,
I, J = 1,2,
, N,
(3 1)
where the 0 / , ~Y_ = (q~1+), are the charged members of the original doublets ( ~ t ) in the lagrangian and the H J+ for J >1 2 are the physical charged Hlggs (mass elgenstates) We have conventionally chosen Ht+ to be the unphyslcal bosons of the mlmmal sector This fixes fa n = v l / v
(3 2)
In order that the KM dlagonahzatlon of the fermion mass matrices should also diagonahze the tree level neutral current couplings we must couple the v e v of only one of the tb I to each of the Q = 2/3, - 1/3, - 1 fermlon fields [3] Let ~lu, ~xd be the doublets with v e v coupled to the Q = 2/3, - 1 / 3 quarks, respectively (Iu and I d m a y be the same or they may be different ) Then the lagranglan coupling the quarks to the charged Hlggs fields Is
g LqH~-- 2v~-
rnu,(~)~,(l_ys)V, sds~,u,H1++hc M
2v'2
M
f i ' ( 1 + 75)V'sds~2'JH1+ + h c
(33)
N o t e that (3 2) lmphes that (3 3) for the H i is the same as for the minimal SM The large number of new parameters m (3 3) for general N does not provide much more freedom for the processes we consider in this paper than is provided by
w s Hou, R S Wdlev / Loop-mdu~edB de~a}s
63
the s i m p l e e x t e n s m n s to N = 2 F o r N = 2 [11]
~= (cosB -slnB] sin fl cos/3 = v i / v ,
cos fl ]
(3 4)
sm 13 = v2/v
a n d the f e r m l o n couphngs to the extra, physical, H ± are
~L~ H+= ~
~.(1 -- ~,)~jd~
~'~.(1 + Vs)V.jd~ . + + h c . (3 5)
w h e r e ~ a n d ~' are ratios of the two v e v v 1 a n d vz T h e r e are a n u m b e r of possible c o n v e n t i o n s b u t all lead to one of two possible m o d e l s Model 1
I u ~ 1d ,
then ~' = - 1 / ~ ,
Model 2
I u = Ia,
then ~' =
(3 6)
Since the r a t i o U1/U2 IS an u n d e t e r m i n e d p a r a m e t e r of the theory, the Y u k a w a c o u p h n g s of the H + to the fermlons can be rather e n h a n c e d or s u p p r e s s e d relative to those of the u n p h y s l c a l charged scalars of the m i n i m a l S M The Z H+H and ~, H + H (derivative) c o u p h n g s are the same for all H ~ b e c a u s e they are the S U ( 2 ) × U(1) gauge c o u p h n g s The Z W+_H~ a n d 6 W ± H z_ c o u p l i n g s are generated b y a v e v a n d hence are s t a n d a r d for the u n p h y s l c a l Ha+ and zero for all the physical H s , J > 1 N o w referring to fig 2, we see that t~Lq H gives neghglble c o n t n b u t m n s to Gbo x b e c a u s e it will be c o m b i n e d with L r H CCm J M Referring to fig 3 we see that t~Lq H will l e a d to c o n t r i b u t i o n s from & a g r a m s (b), (f), (h), (j) (There is no c o n t r i b u t i o n from (d), (e) because of the v a m s h l n g of the Z W + H v couplings for the p h y s i c a l H + ) W e note two p r o p e r t i e s of this subset of d i a g r a m s (x) W h e n this s u b s e t is s u m m e d ultraviolet divergences which are n o t killed b y G I M cancel one a n o t h e r (u) T h e sum of this subset of d m g r a m s satisfies the t r a n s v e r s a h t y c o n d i t i o n for the p h o t o n vertex F o r the b -~ sZ vertex the result is G z --, G z + 3RG z w~th*
8.Gz(X, x,) = ~2x( 1 x, = m~,/M 2,
1
x'
1)
1 - x ' + (1 - - - - ~ - l n ~ 7
x; = m~,/m~,
* Throughout this paper m H as the mass of these physical charged Hlggs
'
(3 7)
(3 8)
W S Hou, R S Wtllev / Loop-mduced B decays
64
For N H > 2 , (3 7) becomes a sum of terms with ~2 and x ; i = m z / m 2 and •=2, ,N H For the b ---. s7 vertex, dmgrams (h), (j) have no q dependence They cancel the contnbuuons from &agrams (b) and (f) evaluated at q = 0 and leave the transverse form (expanded m powers of q/m)
3F.
Lelg2-'p')[(
2~I
q u - q.4)(1 - Ts)(-3F1/M 2)
-= 64~r2St
+ t%.qV(mb(1 + 75) + ms(1
- "~5))(~F2/m2)]
b(p),
q=p-p'
(3 9)
with contnbunons only from diagrams (b), (f) The results are
= 2x,/Q [
7
1
5
1 m + 12 (1 - x
36 1 - x'
1
11
_1 )(1)]
2 (1-x') 3
1
5
+
6 ( 1 - x ' ) 4 In 1
1
1x,3
1
- -- x') 3 + 6 ( 1 7 x ' ) 41n 77 6 (1
12 (1 - x') 2
36 1 - x '
6 (1 - x') 3
, ]l 2
(1 1 +
1
'
(3 10) 1
a.F~=Ux'{Q
1
1
1 m + 4 (1-xq 2
12 1 - x ' 1
1
!
3
+
Q
-
-
-
1
-
x'
11 + 2 1-x'
0I-
I
1
(1
-- X ' ) 2
1
1
2
(1--x')' 1
4 (1-x') 2
6 1-x'
+ ~'x'
1
1
1 x' {1)] 2 (1 - x') a ln~ 77
1
2 (1-x') 3 + 2 (1-
1
x') 4
(1)]
(1 _7 x,) 3 In ~-;
x,
( 1 - x ' ) 2 + ~( 1 -lx ' )n
(1)) 77
(311)
Note that only ~HF2 depends on both ~, ~' and hence distinguishes between models 1 and 2 (3 6) For quantmes m whose calculation one can neglect external
W S Hou, R S Wllley / Loop-induced B de~avs
65
( Q - - 1 / 3 quark) masses compared to internal (t quark and W) masses from the beginning, ~' does not enter The parameter ~' enters 6HF2 because EM gauge Invarlance requires an expansion in powers of q / M which leads to the explicit q m b / M 2 term in (3 9) In particular, consideration of A m K [12], E~: [13], and B d [14] mixing can put bounds on (~, mn+ ) but does not distinguish between models 1 and 2
4. Discussion of results W e have chosen to present a n u m b e r of representative results in a series of graphs (figs 4 - 1 5 ) F o r each of the three processes b --* sg'+8 , b ~ s7, b --* sg*, we plot the B R as a function of m n / m t for r o t = 80 GeV and four values of ~ (0, 2, 4, 8), for m o d e l 1 and for model 2* For fixed ~ and m H / m t there is not great sensitivity to m t F o r m H / r n , >~ 2 or 3 the BR's increase by a factor of 2 or 3 as m t is increased f r o m 40 to 120 GeV For small m H / m t there is greater enhancement from the H + and less sensitivity to m t Note that for ~ = 0, the results for model 2 are just the SM results ( C o m p a r e the m t = 80 GeV results with (2 23) and (2 24) ) F o r model 1, ~ ' = - 1 and 8Hi~]2 IS nonzero even for ~ = 0 Also, for m H / m t >> 1, the BR's for either model 1, 2 approach the SM results One can see from the graphs that for mH -- mt there can be very large enhancements (e g three orders of magnitude for ~ = 8 and m H - m t In this situation, the rates are proportional to (4) A feature of the graphs is the dramatic suppression of the b ~ s¥ BR (relative to the SM result) in model 2 for some range of values of ((, m H / m t ) This happens because the 8HF2 term dominates F 2 for large ~ and for model 2, with ~ ' = ~2, it is always of opposite sign to the SM F 2 As mentioned above, ~ is an undetermined parameter of the extended Hlggs sector, but it 1s constrained by empirical data, particularly the recent observation [15] of B a mixing We have gone through the exercise [14] of computing the correction to the SM prediction of this mixing in the presence of a second Hlggs doublet T h e n for fixed values of mr, rntj we determine the m a x i m u m value of consistent with the experimental 90% C L upper limit x d -%<1 02 from the A R G U S result r d = 0 21 + 0 08 The result obtained depends on the choices of various parameters in the calculation For tins calculation, we have taken f B v ~ - = 100 MeV and Via = 0 01 These values are toward the low end of the allowable range and lead to larger values of ~m~, The results of this calculation are summarized in table 1 * In our calculations based on the ansatz (2 15) we use calculated values for the quark semlleptonlc widths, but the experimental value for the B-meson semaleptomc BR This procedure is correct as long as the result is a BR much less than one For the strong proces~ b---. sg*, with large enhancements for large ~ this condition is not satisfied In the calculations presented m figs 8, 9 and 15 we have enforced the u m t a n t y constraint by setting B = " ( 2 15)"/(1 + "(2 15)") where "(2 15)" is the value computed from (2 15)
66
W S Hou, R S Wtlley / Loop-mduced B decays
16 z I
I
I
I
I
I
i
I
I
I
I
vs mH/mt
B R ( b - > s.O#)
~
i
i
i
~ x
Model I
i
,
i
i
J
i
I
i
t
l
I
i
i
i
i
B R ( b - > s~Q) vs mH/m t
-2
I0
mt = 8 0 GeV 10-3
i
mt = 8 0 GeV \
Model 2
\ I0 !
15
5
I0
t5
20
t5
20
mH/m t Fig 4 -6
10
I
L
I
I
[
I
I
I
[
1 2 3 4 5
I
t0
mH/m t Fig 5
tO-'
to-3
10- 4
I
i
L
I
I
i
1 2 3 4 5
I
I
I
I0
166
5
tO
mH/m t
mH/m t
Fig 6
Fag 7
Figs 4 - 9 BR for four values of the parameter 4(0,2,4,8) F o r small values of mH/mt, the mcreasmg values of the BR correspond to increasing values of ~ (m H is the mass of the physical charged scalar )
W S Hou, R S Wdley / Loop-mdueed B de~.ays f
I
I
I
I
BR(b->s
I
I
I
I
F
67
I
w / o c ) vs m H / m t
1 .
mt = 80 GeV
_
del
1
-I
10
-2
10
I
I
I
I
2
3 4 5
~
I
I
I
I
I
1
I0
m H / mt Fig 8 ,
,
,
i
i
i
,
,
,
I
BR(b->s
T
,
,
i
I
,
,
,
r
mH/rn t
w/o c)vs
mt = 8 0 G e V -t
10
-2
10
I
I
5
I
I
I
I
I
I
J
I0
I
I
I
=
I
I
15
I
20
mH/m t Fig 9 Figs 8 and 9 See caption on facing page 2 -- l O m H / m , (for m t ~ 40 R o u g h interpolating formulas for these results are ~m~ GeV), ~max----45mH/mt 2 (for r n t ~ 80 GeV), and 3rnH/m t (for m t 120 GeV) One m a y c o m p a r e these results with ref [14] which found corresponding interpolating formula ranging from 2 m n / m t to 1 2 m H / m t (for m r = 4 5 GeV) d e p e n d i n g on choice of parameters We present another set of graphs of the BR for the three processes as functions of m H / m t, for three values of mt, with ~ = ~max determined above for each m n, m t, for model 1 and for model 2, Note, however, one subtlety Because of the destructive interference in F 2 for model 2, we have the result B(~m~ ) = Bmax when ~ = ~m~, leads to enhancement, but Br.~, = BsM when = ~ma~ leads to suppression (This is the case for m t = 120 GeV See fig 13 )
W S Hou, R S Wtllev / Loop-reduced B deems
68
-3
10
-3
i
i
~
i
I
i
BR(b->stl~)
i
i
I0
i
i
i
~
i
1
i
BR(b->sQI~)
mH/m t
vs
Model
i
io-4
10
i
i
1
mH/m t
vs
Model
1
I
2
-____ BO
lO-
120 80 40
I~ 6
i 1
i I , 2 3 4 5
i
i
i
l
I
1(~6
L
10
I
I
I
I
I
I
I
I
I
mH/m t
I
10
1 2 3 4 5
mH/m t
Fig l 0
Fig 11
-1
10
-1
10
BR(b->sT')
vs Model
mH/m
t
1
i
1
i
i
i
i
i
1
r
i
vs mH/m t
BR(b->sT') Model
2
40
10-2
-2
i
10
8O
IG 3
-3
10
8O 10-4
I
I
I
I
I
1
2
3
4
5
I
mH/m t Fig 12
I
I
i(54
I
10
I
I
I
I
I
t
2
3
4
5
t
i
I
I
I
10
mH/m t Fig 13
Figs 10-15 BR computed for maximum values of ( consastent with observed hmlt on Bd (see text and table 1) The three curves are for m t = 40, 80, and 120 GeV, as marked (m H Is the mass of the physical charged scalar)
W S Hou, R S Wllle~ / Loop-mdu~ed B de~avs
t
Model I
2[ ~
_
tff'
'
I
i
I
I
I
_
,20
I
I
I
1 2 5 4 5 mH/m t
J
10
Fig 14
10
BR(b->s
w/o c) vs mH/m t Model 2
40
8O
-I
10
f 120
-2
10
I
J
I
I
I
I
I
I
J
1 2345
I
10 mH/m t Fig 15
Figs 14 and 15 See caption on facing page
69
W S Hou, R S Wdley / Loop-reduced B decays
70
TABLF. 1 ~,],.~ vs m H / m t for several m t values (M = Mw)
mH/mt
~max (mr = M/2)
~ma~ 2 (rnt = M )
2 (grit = V/~M ) ~max
05 10 20 30 40
100 137 219 304 401
41 58 96 136 177
25 36 60 85 114
50
488
220
140
10 0
96 9
44 2
29 8
We discuss again briefly the situation with Q C D corrections The processes we consider are driven by one-loop integrals dominated by m o m e n t a of order M w, hence plausibly short distance on the Q C D scale Thus one m a y expect Q C D corrections to be perturbatlve, of order (e~s/~r)ln(x,/xs) Because F2(x,) has no I n ( l / x , ) (except multiphed by x,), the Q C D correction to b---, s~, in the standard model is relatively large [9,10] but m the nonmlnimal model with large enhancements, 8 H F 2 dominates over such Q C D corrections (But the values of ~, m~, m t for which there is large suppression of b ~ s~ in model 2 are sensitive to Q C D corrections ) In the results presented in the figures we have included the k n o w n (incomplete) Q C D corrections already discussed in sect 2, eqs (2 21) and (2 22) As mentioned above, the SM results for the BR's can be seen from the (straight line) curves for ~ = 0 in the model 2 graphs Then the m a x i m u m enhancement of these BR's c o m i n g from the presence of a second Hlggs doublet, and consistent with the observed limit on B d mixing, may be determined by c o m p a n n g these values with the second set of graphs giving the BR's for ~ = ~max The first striking observation f r o m tbas c o m p a r i s o n is that the largest possible relative enhancements occur for the smallest value of rn t considered This Is a reflection of the constraint imposed b y B a mixing The mlmmal SM can produce the observed value with a "large" value of m t Conversely, large m t does not leave much r o o m for enhancement by n o n - S M physics, but small m t requires it T h e experimental study of rare B decays is still at an early stage, but there are already some results For b - - , s f + g - C L E O [16] has an Inclusive bound, BR(B -~ f + f X) < 1 2 × 10 -3, still an order of magnitude larger than the largest multl-Hlggs enhanced BR consistent with the observed B d mixing F o r b ~ s~, A R G U S [17] has reported a limit BR(B ~ K*~,) < 4 × 10 4 Taking a conservative estimate for the exclusive decay / ' ( B - ~ K*~,)/F(b ~ s~,)> 6%, we arrive at BR(b --, s~,) < 6 × 10 -3 This is already exceeded by the enhanced B R in model 2 for " l i g h t " m t (40 GeV) and m H , / m t ~ 3 Conversely the present limit on B ~ K * y already excludes a region of the parameter space for the two doublet Hlggs sector which IS not excluded by the b o u n d on B d mixing F o r b ~ sg*, which dominates
W S Hou, R S Wtlley / Loop-Induced B deca~,s
71
the total flavor changing neutral current b ~ s transition rate, there is still no definite experimental mformat~on However, one can obtain some indications from the reclusive K yield reported by CLEO [18] and the reclusive charmed particle production m B decays reported by A R G U S and CLEO [17,19] charmless B ~ K translnons at the 10-20% level are not yet excluded Weak as these limits are, they already place constraints on enhancement by additional charged Hlggs which are competlnve wtth corresponding limits obtained from B d mixing Since the Bd mixing is already a measured number [15], one does not expect drastically improved bounds on the extended Higgs sector from subsequent more accurate measurements But for the rare B decays considered above, as the experimental upper limits continue to improve*, stronger constraints on the parameters of an extended Hlggs sector will result Of course, an extended Hlggs sector is only one possible source of enhancement of loop-reduced flavor changing neutral current B decays Another possibility ~s extension of the fermlon sector, 1 e the existence of a fourth generation of fermlons In a previous paper [4] we have &scussed possible enhancement of b ~ s f + { decays in the presence of a fourth generation Here we point out that a fourth generation with a very heavy t' quark can be most effective in enhancing the b--* sE+g - decay because of the linear x, term m the Z-exchange function G z (2 6) A fourth generanon does not lead to strong enhancement of b --* sy or b --* sg because these are determined by Fz(x ) which actually decreases for x >> 1 (GIM kills the asymptoncally constant term) The sltuatton is reversed for enhancement by enhanced couplings of the charged Hxggs bosons of an extended Higgs sector Since B d m~xmg also has a hnear x, dependence, the observed hmlt strongly constrains the possible muln-H~ggs enhancement of b ~ sE+E -, ~f there are only three generations** On the other hand, 6 n F 2 which dormnates the enhancement of b ~ sT, b --* sg depends on m t only through the ratio m t / m n with m n (mass of physical charged Higgs) unknown So for smaller values of m t, for which the B d mixing limit allows large values of (~2, m r ~ r a n ) ' one can have large enhancements (by scahng down m n with m t One can not do this with 8riG z which depends also on m t / M ) In summary a greatly enhanced b ~ s f + g - rate would most likely indicate the existence of a fourth generation, while greatly enhanced rates for b ~ sT a n d / o r b ~ sg* (signature a greatly enhanced subprocess b --* sg with g "on-shell", see ref [6] for discussion) would more likely be attributed to enhancements from an extended Htggs sector
* New hmlts on loop induced flavor changing B decays have been reported by the A R G U S and CLEO collaborations at the XXIV International Conference on High Energy Physlcs, Munich, August 4 - 1 0 1988 they strengthen our statement made here See the talks by A Jawahery and A Golutvln given at the conference * * The observed limit on Ba mixing does not strongly limit the fourth generation enhancement of b--,s•+g because it bounds VttbVtdXt while what enters into b - - * s f + f is I/tbVt~x t If n o n m a x l m a l B, mixing is subsequently observed thas situation will be changed [20]
W S Hou, R S Wdley / Loop-reduced B decays
72
Either the presence of a fourth generation or an extended Hlggs sector can lead to suppression of the b ~ sy rate due to Interference between different contributions to the single form factor F 2 which determines this rate Since neither, either, or both of these extensions of the minimal SM may be realized in nature, additional experimental information (beyond the BR's, or hmlts, for these loop induced B decays) will be required to sort out the various possibilities with confidence In the short term, discovery of the t quark (and its mass) and neutrino counting at L E P / S L C , combined with new data on the B decays dxscussed here, will substantially improve the constraints on these possibilities Ultimately, if there ~s a fourth generation, or an extended Hlggs sector, experlmentahsts must discover the additional quark(s) a n d / o r charged Hlggs bosons We thank T Ferguson and A Soni for discussions
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
A brief report of some of the results of this paper has appeared in Phys Lett B202 (1988) 591 D A Ross and M Veltman, Nucl Phys B45 (1975) 135 S Glashow and S Welnberg, Phys Rev D15 (1977) 1958 W S Hou, R S Wdley and A Sore, Phys Rev Lett 58 (1987) 1608, 60 (1988) 2337(E) T Inaml and C S Llm, Prog Theor Phys 65 (1981) 297, 1772(E) W S Hou Nucl Phys B308 (1988)561 L Wolfenstein, Phys Rev Lett 51 (1983)1945 N Cablbbo and L Maaanl, Phys Lett B79 (1978) 109 J L Cortes, X Y Pham and A Tounsl, Phys Rev D25 (1982) 188 M A Shlfman, A I Vamshte, n and V I Zakharov, Phys Rev D18 (1978) 2583 S Bertohm, F Borumati and A Mas/ero, Phys Rev Lett 59 (1987) 180, N G Deshpande P Lo, J Trampetic, G Ellam and P Singer, Phys Rev Lett 59 (1987)183 B Grlnstem, R Springer, and M Wise, Phys Lett B202 (1988) 138 H E Haber, G L Kane and T Sterling, Nucl Phys B161 (1979) 493 This has also been apphed to the b ---,sT process by B Grlnsteln and M Wise, Phys Lett B201 (1988) 274 L F Abbott, P Slklvle and M Wise, Phys Rev D21 (1980)1391 G G Athanasm and F J Gdman, Phys Lett B153 (1985) 274 G G Athanaslu, P J Franzlm and F J Gilman, Phys Rev D32 (1985) 3010 ARGUS Collaboration H Albrecht et al, Phys Len B192 (1987) 245 CLEO Collaboration, A Bean et al, Phys Rev D35 (1987) 3533 W Schmldt-Parzefall, Nucl Phys B (Proc Suppl) 3 (1988) 257 CLEO Collaboration, M S Alam et al, Phys Rev Lett 58 (I987) 1814 CLEO Collaboration, D Bertoletto et al, Phys Rev D35 (1987) 19 W S Hou and A Sonl Phys Lett BI96 (1987)92