arXiv:hep-ph/9910439v1 22 Oct 1999
Inclusive 0 production in B decaysand the
Enhancem entdue to charged technipions
G ongru Lua;b ,Zhenjun X iaoa;b;c,H ongkaiG uob,Linxia Lub
a. C C A ST (W orld Laboratory) P.O .B ox 8730,B eijing 100080,P.R .C hina
b. D epartm ent ofPhysics,H enan N orm alU niversity,X inxiang,453002 P.R .C hina.
c. D epartm ent ofPhysics,Peking U niversity,B eijing,100871 P.R .C hina.
A ugust 31,2016
A bstract
T he new contributions to the charm less B decay B ! X s 0 from the unitcharged technipions P and P 8 are estim ated. T he technipions can provide a
large enhancem ent to the inclusive branching ratio: B r(B ! X s 0) 7 10 4 for
m p1 = 100G eV and m p8 = 250 350G eV w hen the e ectofQ C D gluon anom aly is
also taken into account. T he new physics e ect is essentialto interpret the C LEO
data.
PA C S num bers: 12.60.N z,12.15.Ji,13.20.Jf
E-m ailaddress:dphnu@ public.zz.ha.cn
1
R ecently C LEO hasreported [1]a very large branching ratio forthe inclusive production of 0:
B r(B !
0
X s)= (6:2
1:6
1:3)
10 4; for 2:0
E
0
2:7G eV
(1)
and a corresponding large exclusive branching ratio [2]
B r(B
!
0
2:5
K )= (7:1+ 2:
1
0:9) 10
5
(2)
w here the acceptance cut was used to reduce the background from events w ith charm ed
m esons. By using the Standard M odel(SM ) factorization one nds [3]B r(B ! 0X s)
(0:5 2:5) 10 4 including the experim entalcut,w ith the largest yields corresponding
to a fairly lim ited region ofparam eter space,w hich is m uch sm aller than the observed
inclusive rate in eq.(1).
U p to now a num berofinterpretationshave been proposed [4,5,6,7,8,3]to account
fortheobserved largebranching ratio ofB ! 0X s and/ortheexclusivebranching fraction
B r(B ! 0K ).T hese include: (a)conventionalb ! sqq w ith constructive interference
between the uu,dd and ss com ponentsofthe 0[4],(b)b! ccs decay enhanced cccontent
ofthe 0 [5,6],(c)b! sg ! sg 0 from Q C D gluon anom aly [7]orfrom both Q C D gluon
anom aly w ith running s and the new physics e ects [8,3],(d) non-spectator e ects[9].
From the above works,the follow ing m ajor points about the inclusive and exclusive
branching ratios B r(B ! 0X s) and B r(B ! 0K ) can be reached;
1. T he SM factorization can,in principle, account for the exclusive 0 yield w ithout
the need ofnew physics[3,9].A lthough a SM "cocktail" solution forlarge inclusive
rateB r(B ! 0X s)involving contributionsfrom severalm echanism sisstillpossible,
butthe intervention ofnew physicsin the form ofenhanced chrom o-m agnetic dipole
operatorsprovides a sim ple and elegantsolution to the puzzle in question[8,3]. O n
theotherhand,theshort-distanceb! 0sg subprocessm ostpossibly doesnota ect
the exclusive B ! 0K branching ratios[3].
2. T he observed inclusive branching fraction is larger than w hat is expected from
scenario (a). Furtherm ore,the data show thatthe invariant m ass spectrum M (X s)
ofthe particles recoiling against the 0 peaks above 2G eV [1].
3. T he large inclusive rate m ay be connected to the standard m odelQ C D penguins
via the gluon anom aly,w hich leads to the subprocess b ! sg ! sg 0. Taking a
constantgg 0 vertex form factorH (0;0;m 20)[7],the observed large branching ratio
in eq.(1)can be achieved. Butasargued by H ou and T seng [8],ifone considers the
2
running of s im plicit in H (0;0;m 20),the result presented in ref.[7]w illbe reduced
by roughly a factor of 3. In other words, the new physics e ect is essential to
interpret the observed large inclusive rate[8].
4. A s pointed by K agan and Petrov [3], the m 20=(q2 m 20) dependence of the gg 0
coupling should be considered. Including this dependence nom inally reduce the
form er result [7]to 1:6 10 5 including the cut, w hich is signi cantly sm aller
than the observed inclusive rate.T hisfactw illstrengthen the need fornew physics.
5. It is possible to enhance the chrom o-dipole bsg coupling by new physics at the
TeV scale w ithout jeopardizing the electrodipole bs coupling[10,11]. O ne explicit
exam plein thefram ework oftheM inim alSupersym m etric Standard M odel(M SSM )
has been studied in ref.[11].
In this letter we w illshow that the unit-charged technipions P and P 8 appeared in
alm ost allnonm inim altechnicolor m odels [12,13]can provide the required enhancem ent
to account for the observed large rate B r(B ! 0X s)[1].
In the fram ework ofthe SM ,the loop induced e ective bsg coupling was calculated
long tim e ago[14],
SM
h
GF
= gs p 2 VisVibsT a F 1i(q2
4 2
q 6q)
iF 2i
i
q (m sL + m bR ) b
(3)
w here the gs is the strong coupling constant,V is the C K M m atrix,T a = a=2 and a is
the G ell-M ann m atrix,q = pb ps and the charge radius form factors F 1i and the dipole
m om ent F 2i ( i= u;c;t) are
F 1i =
F 2i =
i
xi h
2yi xi 2
yi + 13yi2 6yi3 +
(4yi + 5yi3
12
3
6
i 3x2y3
xi h
i i
yi + 3yi2 + 6yi3 +
ln[xi]
4
2
3yi4) ln[xi];
(4)
(5)
w here xi = m 2i=M W2 and yi = 1=(xi 1) for i= u;c;t.
In the fram ework ofTechnicolor theory,the new e ective bsg coupling can be derived
by replacing the internalW -linesin the one-loop diagram sthatinduce bsg coupling in the
SM w ith the charged technipion lines. U sing the gauge and e ective Yukawa couplingsas
given in refs.[12],one nds the new e ective bsg coupling induced by P and P 8 ,
N ew
h
GF
(i;N ew ) 2
(q
= gs p 2 VisVibsT a F 1
4 2
(i;N ew )
q 6q) iF 2
3
i
q (m sL + m bR ) b (6)
w ith
D 0( i)
8D 0( i)
p
p
+
3 2G F F 2 3 2G F F 2
#
"
8D ( i)+ E ( i)
D ( i)
p
p
+
3 2G F F 2
3 2G F F 2
F 1N ew ( i; i) =
F 2N ew ( i; i) =
(7)
(8)
and
5 + 19
20 2 4 2 2 3
+
ln[ ]
24(1
)3
4(1
)4
32 23
7 29 + 16 2
ln[ ]
D 0( ) =
72(1
)3
12(1
)4
12 15
52 9
18 2
E( ) =
+
ln[ ]
8(1
)3
4(1
)4
(9)
D( ) =
(10)
(11)
w here = m 2p1=m 2t and = m 2P 8=m 2t,and m p1 and m p8 denote the m asses ofthe colorsingletand color-octettechnipion P and P 8 respectively. T he technipion decay constant
F = 123G eV in the O ne-G eneration Technicolor M odel(O G T M ) [12]. T he G F is the
Ferm icoupling constant G F = 1:16639 10 5(G eV ) 2.
C om paring the e ective bsg coupling N ew in eq.(6) w ith the S M in eq.(3),one can
(i;N ew )
(i;N ew )
see thatthe form factorsF 1
and F 2
arethe counterpartsofthe F 1i and F 2i in the
(i;N ew )
(i;N ew )
SM .T he new form factors F 1
and F 2
describe the contributions to the decay
b ! sg from the charged technipions P and P 8 .
In the num ericalcalculation we use the branching ratio form ula forB ! 0+ X s w ith
gluon anom aly as given in ref.[8],
"
#2
d2B r(b! 0sg)
gs(m b)
= 0:2
dxdy
4 2
"
a2gm 2b
c1
c2
j F 1j2c0 + R e( F 1F 2 ) + jF 2j2 2
4
y
y
#
(12)
2 2
w here 0:2 com es from (Vcb
G F m 2b)=(192 3) ’ 0:2 B via the standard trick ofrelating to
B s:l (see ref.[7]). T he factors c0;c1 and c2 in eq.(12) are
h
c0 =
c1 =
c2 =
2x2y + (1
(1
y)(y
y)(y
x0)2;
(1
y)(y
h
2x2y2
x0)(2x + y
x0)(2xy
i
x0) =2;
i
y + x0) =2
(13)
w here x = m 2=m 2b w ith m isthe physicalrecoilm assagainstthe 0 m ason,and y = q2=m 2b
w ith q = pb ps and x0 = m 20=m 2b. T he term F 1 was de ned as F 1 = F 1(xt)
q
F 1(xc).T he factorag = N f s( )=( f 0)isthe e ective anom aly coupling H (q2;k2;m 20)
4
as de ned in ref.[7] and f 0 = 131M eV . For the running of s, we use the two-loop
approxim ation as given for instance in ref.[15].
T he Fig.1 show s the m ass dependence ofform factors in the SM and in the O G T M .
T he dot-dashed line corresponds to the F 1 = 5:25 in the SM ,w hile the long dashed
line show s the F 1N ew = F 1N ew ( t; t) F 1N ew ( c; c)
4:6 in the O G T M , assum ing
m p1 = 100G eV and m p8 = 250 600G eV . T he short-dashed line is the F 2 = 0:2 in
the SM for m t = 180G eV and m W = 80:2G eV ,w hile the solid curve is the F 2N ew in the
O G T M ,assum ing m p1 = 100G eV and m p8 = 250 600G eV . It is easy to see that the
size ofF 2N ew can be m uch larger than the F 2 in the SM for light color-octet technipion.
Furtherm ore,the P 8 dom inates the totalcontribution to the F 1N ew and F 2N ew .
Because we do not know the "correct" form ofgg 0 vertex form factor H (q2;k2;m 20),
we consider the follow ing two di erent cases respectively.
C ase-1: W econsiderthee ectdueto therunning of s [8]aswellasthenew contribution
from the charged technipions.
A ftertheinclusion oftherunning of s one ndsB r(B ! 0X s) 3:4 10 4 including
the cut,as show n in Fig.2 (the dot-dashed line). T he horizontalband in Fig.2 represents
the C LEO data in eq.(1). T he long dashed curve corresponds to the total inclusive
branching ratio B r(B ! 0X s) w hen the new physics e ects are also included. N um erically,B r(B ! 0X s)= (48:9 5:7) 10 4 form p1 = 100G eV and m p8 = (250 600)G eV .
T he theoreticalprediction isnow wellconsistentw ith the C LEO data form p8 350G eV .
T he color-octettechnipion P 8 dom inates the totalnew contribution: the increase due to
the P is only about 10% at the levelofthe corresponding branching ratio.
C ase-2: W e consider the e ect ofthe m 20=(q2 m 20) suppression and the new physics
contribution from P and P 8 .
W hen the new suppression factor m 20=(q2 m 20) is taken into account one nds
B r(B ! 0X s) = 2:3 10 5 including the cut as show n by the short dashed line in
Fig.2, w hich is m uch sm aller than the C LEO m easurem ent. W hen the new contributions from the charged technipions are included,the inclusive branching ratio B r(B !
0
X s) can be enhanced greatly as illustrated by the solid curve in Fig.2. N um erically,
B r(B ! 0X s) = (15:2 0:7) 10 4 for m p1 = 100G eV and m p8 = (250 600)G eV .
T he theoretical prediction is now consistent w ith the C LEO data for m p8
280G eV .
T he new physics e ectisessentialto interpretthe C LEO data forthe C ase-2. A gain,the
color-octet technipion P 8 dom inates the totalcontribution as that in C ase-1.
In this letter we show a realexam ple that the observed large ratio B r(B ! 0X s)
can be explained by the new physics contributions from the unit-charged technipions P
and P 8 . Because the m ajor properties ofthe technipions in di erent technicolor m odels
5
are generally very sim ilar,the analyticaland num ericalresults obtained in this letter are
representative and can be extended to other new technicolor m odels easily.
In thisletter,we rstly evaluate the new one-loop penguin diagram sw ith the internal
P and P 8 lines and obtained the new form factors F 1N ew ( i; i) and F 2N ew ( i; i) w hich
describe the new physics contributions to the decay in question. T he size ofF 2N ew can be
rather large for relatively light charged technipions. Secondly,we com bine the new form
factorsF iN ew (i= 1,2)w ith theircounterpart F 1 and F 2 in the SM properly and use them
in the num ericalcalculation. W e nally calculate the inclusive branching ratios for both
C ase-1 and C ase-2. A s illustrated in Fig.2,the unit-charged technipions can provide a
large enhancem ent to account for the large rate B r(B ! 0X s) observed by C LEO [1].
T he authorswould like to thank D ongsheng D u,K uangta C hao and Yadong Yang for
helpfuldiscussions. T his work is supported by the N ationalN aturalScience Foundation
ofC hina under G rant N o.19575015 and by the funds from H enan Scienti c C om m ittee.
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F igure C aptions
F ig.1: T he dot-dashed line show s F 1 = 5:25 in the SM ,the long dashed line corresponds to the F 1N ew ,the short dashed line is the F 2S M = 0:2 and the solid curve
is the F 2N ew induced by unit-charged technipions.
F ig.2: T he horizontal solid band represents the C LEO data B r(B ! 0X s) = (6:3
2:1) 10 4. T he dot-dashed (short dashed) line is the SM prediction in C ase-1
(C ase-2), w hile the long-dashed and solid line show the totalinclusive branching
ratio B r(B ! 0X s)w hen the new physics e ects are included in C ase-1 and C ase2,respectively.
7