- Email: [email protected]
Model independent determination of |Vub| in heavy meson effective theory
27 April 1995 PHYSICS
LETTERS
B
Physics LettersB 349 (1995) 541-547
ELSEVIER
Model independent determination of 1I& 1in heavy meson effective theory Noriaki Kitazawa
’
Department of Physics, Tokyo Metropolitan University, 1-l Minami Ohsawa, Hachioji-shi, Tokyo 192-03, Japan
Received 24 September 1994; revised manuscript received 29 December 1994 Editor: H. Georgi
Abstract
We propose a strategy of the model-independent determination of the Kobayashi-Maskawa matrix element l&l within the framework of the heavy meson effective theory. The effective theory is model-independent, since no assumption on the dynamics is introduced except for the symmetry of QCD. Although the effective theory can only be applied to the decay process with soft pions, high detection efficiency on the soft pions in the future B-factories makes this theory effective. The two exclusive processes, B -+ ?rl~ and B -+ n?rlv, are used to determine the value of l&l with l/Mb correction, under the condition that the values of the decay constants of B and B’ mesons are given.
The precise measurement of the KobayashiMaskawa matrix elements is one of the approaches to detect the new physics beyond the standard model at B-factories. The unitarity relation of the matrix in the B”-tio system ,
c
I$&
= 0,
(1)
1=u,c,t is described as a triangle in the complex plane. If the measured values of the sides and angles are not consistent with the triangle, the contribution from the new physics should exist. Various strategies to measure the sides and angles are proposed. Focusing on the sides, 1V&l in 1V$,Vc.J can be model-independently determined with enough accuracy by virtue of the heavy quark symmetry [ l31. But the determinations of the other two sides, lvtVtd( N l&d1 and IV$,&j N I&,[, are still suffered ’ E-mail: [email protected].
from the ambiguity due to the theoretical estimation of the hadron matrix elements. At present, IVubJ is extracted as a ratio I&,/b&J from the inclusive semi-leptonic B decay [ 41. The measured spectrum of the charged lepton momentum is understood as the combination of the two components of b + clv and b --) ulv, which are proportional to IvC# and IVub1*, respectively. The theoretical predictions of both spectra and decay rates of these two components are necessary to extract (Vub/Vcb(. There are some theoretical models [5-71 to calculate the hadron form factors. But they give different predictions, and we have almost no criteria to select one of them at present. The extracted values of 1Kb/Vcbl are model dependent, and the ambiguity is larger than the present experimental error. The other strategy which is free from the model dependence is expected. Heavy meson effective theory [ 81 is not the model of the hadrons, since no assumption on the dynamics is introduced except for the symmetry of QCD. Spin-
0370-2693/95/%09.50 @I 1995 Elsevier Science B.V. All rights reserved SSDlO370-2693 (95)00304-5
542
N. Kitazawa /Physics Letters B 349 (1995) 541-547
flavor symmetry of the heavy quarks and the chiral symmetry of the light quarks restrict the interactions between the heavy-light mesons (D, D*, B, and B*) and the light pseudo-scalar mesons ( ‘IZ, K, and q), and the effect of the QCD dynamics is represented by the coupling constants of the interactions. Once the coupling constants are fixed by using the experimental data, we can predict the form factors in modelindependent way. The form factors are effective only for the soft pions, since the chiral expansion is used in the construction of the effective theory. In this letter, we propose a model-independent strategy of extracting jVub] through the fixing of the parameters of the exclusive processes B + dv and B + mdv with soft pions, under the condition that the values of the decay constants of B and B* mesons are given by the lattice calculations. Both the above two processes are needed, if we consider the l/Mb correction. The importance of the correction in the heavy-to-light transition is suggested by the lattice calculations of the decay constants of heavy-light mesons. The Lagrangian of the heavy meson effective theory has been written down including the breaking effect of the spin-flavor symmetry [9,10]. Up to 0(p2) in the chiral expansion and 0( l/M;) in the l/Me expansion,
+ (Anti-particle),
(2)
with l/M = diag( l/MC l/Mb). There are five dimensionless parameters (K, K’, A, and At.2) and a parameter A with mass dimension. The heavy mesons are described by the field
fj)++
’
[lYSP"
7j-
”
Y&x
+
I
(3)
where v is the velocity of the heavy quark inside. The fields P, and PJ are the heavy pseudo-scalar and heavy vector fields defined as pc*, = ”
Do B-
(*I
Df
0,’
Et?
Do
(4)
s
which have the mass dimension 3/2. The lightpseudoscalar mesons (r, K, and 7s) are described by the following fields as the Nambu-Goldstone bosons, (5)
t=
ein/f=
and
fI=I’F$.
(6)
All the possible independent terms which are consistent with the symmetry (including C, P, and reparametrisation invariance) are enumerated in the Lagrangian. The transformation properties of the fields and the details of the construction of the Lagrangian are described in the paper of Ref. [9] 2 . The heavy-to-light weak current can be written down in the same level of the expansion as J:(O)
= F[tr{(@)jiyP(l
+ &tr{b?)jiy,(l
- y5)Hzj) - ys)[yp,iDPHzjl}]
+atktr{(@)iiy,(l a
- y~)Hzj}
+a2~tr{(tt)jiy,(l D
- ydypH$y,}
(7)
* In that paper, the light vector mesons (p, K*, and ws) are introduced using the method of the hidden local symmetry [ 111. But here we do not introduce them, since the method is a model,
N. Kitazawa / Physics Letters B 349 (1995) 541-547
with three parameters of the mass dimension 3/2 (F and at,z). The decay constants of the heavy-light mesons are parametrized as
fP
= J
fv=
-zMQ
+2Ly*)
F+&
&
F+
-/pi
$
+
B*‘
1
(8)
, I
MQ
{
543
-2a2)
>
0
(9)
3
where fp and fv are the decay constants of pseudoscalar and vector heavy mesons, respectively. The Lagrangian gives the mass formulae for the heavy mesons
Fig. 1. The diagrams for B” -t ?r+lP decay. Black circle denotes the strong vertex, and black square denotes the weak vertex.
are used. If the quark masses are set as M, = 1500 MeV and Mb = 5000 MeV [ 121 (WV- up)A3 N (695 MeV)3.
(16)
Then we obtain the magnitude of the universal 1/MQ correction to the squared mass difference as
1+2h+2~'$-+12~-$ MQ +WP-$-
A3 Q
3
Q
Q
(10)
3
I+2~+2~'$-4& MQ Q fWV$
A3 Q
>
-16~A~
Q
(11)
>
where mp and mv arethe masses of pseudo-scalar and vector heavy mesons, respectively. The l/ML correction to the un-squared masses mp and mv aregenerally attached as the A3/Mi terms with the coefficients wp and WVin the brackets. Using a relation m2,- m'p= -16~A~ + (WV- wp)A2n MQ
(12)
=
(rnt., -m$0)--(w--w~)~
c 21 (597 MeV)2.
(17)
We can see that the convergency of the l/M, expansion is not good, since the 1/MC and 1 /Mz corrections to the rniarethe same order of magnitude. The magnitude of the corrections relative to the leading are
(18) A3 A3 WpY$ N (WV- WP’~ 210.10. C C
I /I N
1.1 x 10-2, (wv-wp)-
(20) A3 M3b
N (396 MeV)2,
(13)
where the values of the meson masses [ 121 mDo 2! 1860 MeV,
MD’0N 2010 MeV,
(14)
mg0 215280 MeV,
mg.0 215320 MeV,
(15)
(19)
But the l/M; correction to rniis one order of magnitude smaller than the 1/Mb correction:
following from the above mass formulae, we can estimate the convergency of the ~/MQ expansion. The magnitude of the l/ML correction is given as
= (m&O -m2,0)- (m&O -miO>
A3
~22.7~ 10-3.
(21)
This indicates that the convergency of the l/Mb expansion is good. Therefore, we can safely apply this effective theory only to the B meson system. The decay amplitude of the process B” + 7.r+ZVis obtained by calculating the diagrams of Fig. 1. The form factors defined by (r(Pr)
IJib
(B’(pB))
= f+(q2)(PB+Pn)~++-(92)(pB-P~)~
(22)
544
N. Kitazawa / Physics Letters B 349 (1995) 541-547
are obtained as
f*(q2) = ;g [I - g x
A 1+ E)
+ ((Al -
{(
x
(4-m:
A21 -
‘F(mi-mi+q2)
-s2)
q2 - rn&
where o.p,
q2 = (pi
=
- p,)
A)
&}
]v
and
2
m~+m~-q2 2ms
(23)
’
(24)
The explicit dependence on the bottom quark mass Mb is replaced by the dependence on the B meson mass mg up to the order of l/M; by using the mass formula. The B* meson pole contribution emerges with the coupling constant A and the combination A1- AZ. The mass of the pion is remained non-zero in the phase space integration only. At the tree level, all the effect of the explicit chiral symmetry breaking due to the light quark masses, m,, and md, is reduced to the isospin breaking effect which is negligibly small. Then we can use the coupling constants of the chiral limit. There are no contribution from the higher order terms of chiral expansion at the tree level. Therefore, these form factors should be applied as far as the chiral expansion converges 3 . Following to the naive dimensional analysis [ 133, the chiral expansion converges if
&
(-1 4rfn
2<
dq2
where the masses of the charged lepton and neutrino are neglected. Once we fix the values of all the unknown parameters, A, ((Ai - AZ) - A)A/mB, fs, and fs*, the value ]t&,] can be obtained by fitting the spectrum of the above region. If we consider the very soft pions, namely
E,r
(-> 4rfTr
1,
(25)
{E; _
m2,}3/2 1f+(q2>)2,
(26)
with the above f+ ( q2) is effective in the region (mLi-mm,)2 >q2 >m~i-m~-2me.4~f,,
(27)
3 This corresponds to the fact that only the 0(p2) and S(p4) terms in the non-linear chiral Lagrangian can contribute to the r-r scattering amplitude at the tree level. 4 Reparametrisation invariance always allow us to take pt~ = mgu.
2<
0.1
(28)
or (1128 -
where E,, = v . ps denotes the energy of the pion in B meson rest frame4 and fT N 88 MeV denotes the decay constant of the pions in the chiral limit. The q2-spectrum
d!-=9
Fig. 2. The diagrams for B- --+ T+?T-/~ decay. Black circle denotes the strong vertex, and black square denotes the weak vertex.
F&j2
> q2
> rni + rn$ - 2rnB. d(4TfT)2/10,
(29)
the E,/2mB dependence in the form factors can be approximately neglected in the 10% level, and only a combination of the couplings, A E A + ( (AI - AZ) A)A/??ZB,emerges in the form factor. To fix the values of the coupling constants, we must consider the other independent processes of B meson decay. The decay amplitude of the process B- -+ IT+T-Z~ is obtained by calculating the diagrams of Fig. 2. The form factors which are defined as [ 141 (~+(P+)~-(P~-)IJ;~(O)IB-(PB)) =
w+k+, + w-k-,
+ rqP + hiew,qrpik$k?., (30)
where kk = p,,+ fp,-
and q = pl +pv, are obtained as
N. Kitazawa /Physics
“+
=
i(
l+PB’k+-PB’~-
((4
-
4m2,
A21
-
A)
>
&}
G;IV,bi2f;
drs-wave
1
2m;-3(pB.k+-pB.k_) rni - m$. -pB’k++pB*k-
545
AZ,emerges in h. Taking the S-wave configuration of pions in B meson rest frame, the contributions of the p meson resonance, which can couple with the pion pair, and the form factor h are dropped out. The decay spectrum of the S-wave pions is given as
---
A
Letters B 349 (1995) 541-547
(31)
dk2,dq2
= 12(4r)5mi 4
X
-5
E
-42)2
_dii,)3’2
K
x
X
X [I+~(~+((hl-A2)4+-)
{(
l+Ps*k+-Ps*k4m2,
A
x
(32)
h= X
1 rni
-
rn&
Y4:;21 r
-pBsk++pB.k_q2-m&
PB.k+-Ps*k-
2
[(*+p*)+2(A,-A)A
XA A
mB
--
A
.
(33)
mB
We do not write r. since it does not contribute if the masses of the lepton and neutrino are neglected ’ . The relations (w+--w-1
.
P,+ =o
= -1-
1
‘2f+((PB
2(m&
-mi)+mi+k$-$
The coupling constants are included only in the combination of 1s A + ( (At - AZ) - A) A/mB, since we are considering the very soft pions which satisfy . +
1
B
4m& (36)
1
2mi-(pB’k+-pi?‘k_) rni - rn& -pB*k++ps*k-
{
3-
2
< 0.1,
(37)
where u. k+/2 denotes the average of the pion energies in B meson rest frame. The ET-/2mB dependence in the form factors w+ and w_ are approximately neglected in the 10% level. The next order terms of the chiral expansion ( 0(p2)) can contribute to this process. To keep the good approximation within 10% error, we must set the condition of Fq. (37). The width of D*+ -+ Do& decay can be calculated as
-?‘T-)~)
Jzf?r
(34) (w++o-),
__&=o, VT
(35)
which follow form the chiral symmetry, are satisfied. The explicit dependence on the bottom quark mass Mb is replaced by the dependence on the B meson mass mB up to the order of l/M; by using the mass formula. A new combination of the parameters, At +
(38) where E,, =
rn& -m2,+m2, 2mD8
.
The experimental upper bound on this width [ 151 i-( D*+ 4 DOT+) < 131 keV
5 Two of the five diagrams with the derivative coupling of the pseudo-scalar B meson with W bosons only contribute to r.
gives the upper bound on the parameter
(39)
546
A + ((Al - AZ) - A) &
N. Kifazawa / Physics Letters B 349 (1995) 541-547
< 0.417.
(40)
Although the l/M, expansion may not converge, this result gives the order of magnitude of the couplings. We set > = 0.4 in the following estimates. We assume that the values of f~ and f~. are given. The value of f~ will be obtained within 10% accuracy by the lattice calculation in near future6 . Although the value of fs* is phenomenologically less important 7 , the lattice calculation of the value of f~ can be interesting with a view to check the convergency of the l/Mb expansion. We set f~ = fs* = 120 MeV in the following estimates. Once we obtain the values of the decay constants, and if we neglect the l/Mb correction, we can obtain the value of I& 1by using the q2-spectrum of B + rlv decay alone. (If the decay constant fB is not given, we can extract only the value of a combination of f~ 1&I. The process B --+ (mr)~+,~~~ Iv does not give any additional information.) The process B -+ mrZv is not needed in this case. The unknown parameters are only (&,I and A in the form factor f + ( q2). These parameters can be fixed simultaneously by fitting the q2spectrum. But the 1/Mb correction must be important in the process of heavy-to-light transition, as the recent lattice calculations of the decay constants of the heavy-light mesons strongly indicate it. In the heavy mass limit, the quantity fp & should be a constant which is independent of MQ, and the MQ dependence of the quantity is introduced as the ~/MQ correction (see EQ. (8)). But recent lattice calculations show that the quantity is strongly depend on MQ in the region of MQ 21 Mb. To handle the l/Mb correction in the determination of 1Vubj,we need more one process B -+ m~lv. The decay rate of the B” + &EF with very soft pion is obtained by integrating the spectrum of Eq. (26) over the phase space restricted by Eq. (28). The expected number of the events of such decay with lo8 B mesons is 6 If the lattice calculation can give the form factor f+ (q*) with the enough accuracy, it is straightforward to extract the value of IV&l form the B - lrlv decay. But since the calculation of the form factor is more difficult than that of the decay constant, it can be imagined that the accuracy is not enough compared with the experimental accuracy in the future B factories. ’ The weak decays of the B* meson are hard to observe, since the branching ratio is very small.
N( B0 + 7r+lD) IveryS&St @” N 1040-
lv,b12 (l/20)3’
(41)
The decay rate of the B- -+ ST+TT-/~ with very soft pions is obtained in the same way by integrating the spectrum of Eq. (36) over the phase space restricted by Eq. (37). The expected number of the events with lo8 B mesons is
N(B- + T+T-/~)Ivery softpion (42) These number can be detectable in the future Bfactories within the 10% accuracy. Therefore, it can be expected that the value of I&,[ is extracted by fitting both i and l&l with these numbers. The model-independent determination of l&j within 10% accuracy can be expected. We may obtain both the values of A and (( A1 AT) - A)Alrn~ by fitting the spectrum of Do + rr+ZD in the region of Eq. (27). The spectrum should be normalized by the number of Eq. (42) to factor out IVubI. The convergency of the l/Mb expansion can be estimated by comparing the values of A and ( ( hl - AZ) A)h/mB. Once these parameters are fixed, the value of [&,I can be obtained by fitting the un-normalized spectrum of B” --+ ?r+/5. The extracted value is also the model-independent. The accuracy of the extracted value depend on the experimental ability to detect the very soft pions, since the main theoretical ambiguity comes from the higher order contribution of the chiral expansion. If we want the theoretical ambiguity of less than 5%, the phase space should be restricted as (&/47rfr)* < 0.05 in the process Do --+ ~r+Zti, for example. The resultant number of the events will be very few and hard to detect. If the precise measurement of the pure leptonic decays of B mesons, B- + p3 or B- + ~ii, is easier, this strategy have no advantage. Another large theoretical ambiguity may come form the n--n= re-scattering effect in B- + &rr-lF decay. But we can carry out the improvement by using the experimental data of the 7r-r scattering phase shift from the process of K ---+mrlu. In this letter we proposed a model-independent strategy to extract the value of I&,[ from the experiments. Two kind of decay processes, a0 + <
N. Kitazawa/Physics Letters B 349 (1995) 541-547
and B- ---) rr+r-lfi, with very soft pions are used. The effect of the QCD dynamics is represented by only one parameter except for the decay constants, since the phase space of the decays is restricted so as to include only the very soft pions. The values of the decay constants, f~ and fs*, are expected to be obtained within the enough accuracy by the future lattice calculations. The theoretical ambiguity can be less than 10% depending on the experimental ability to detect the soft pions, if the decay constants of B and B* are known in good accuracy. I am grateful to Yasuhiro Okada, Minoru Tanaka, and Masaharu Tanabashi of KEK theory group and Takeshi Kurimoto of Osaka University for fruitful discussions and comments. I also would like to thank S.Uno of KEK for the helpful advice from the experimental point of view. References [l] N. Isgur and M.B. Wise, Phys. L&t. B 232 (1989) 113; B 237 (1990) 527.
547
I21 M.E. Luke, Phys. Len. B 252 (1990) 447; C.G. Boyd and D.E. Brahm, Phys. I&t. B 257 (1991) 393. [31 M. Neubert, Phys. L&t. B 264 (1991) 455. r41 CLBO Collaboration, Phys. Rev. L&t. 71 ( 1993) 4111. [51 N. Isgur, D. Scora, B. Grinstein and M. Wise, Phys. Rev. D 39 (1989) 799. [61 M. Wirbel, B. Stech and M. Bauer, 2. Phys. C 29 (1985) 637. [71 G. Altamlli, N. Cabibbo, Cl. Corbo, L. Maiani and G. Martinelli, Nucl. Phys. B 208 (1982) 365. 181 H. Georgi, Lectures delivered at TASI, published in Boulder TASI 91, 589 (HUTP-91-A039); G. Burdman and J.P. Donoghue, Phys. Lett. B 280 (1992) 287; M.B. Wise, Phys. Rev. D 45 (1992) R2188. [91 N. Kitazawa and T. Kurimoto, Phys. Lett. B 323 ( 1994) 65. [lOI C.G. Boyd and B. Grinstein, UCSD-PTH-93-46. [Ill M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. L&t. 54 (1985) 1215; for a review see, M. Bando, T. Kugo and K. Yamawaki, Phys. Rep. 164 (1988) 217. [I21 Particle Data Group, Phys. Rev. D 45 ( 1992). [I31 H. Georgi and L. Randall, Nucl. Phys. B 276 ( 1986) 241. [I41 A. Pais and S.B. Treiman, Phys. Rev. 168 (1968) 168; C.L.Y. Lee, M. Lu and M.B. Wise, Phys. Rev. D 46 ( 1992) 5040. [I51 ACCMOR Collaboration, Phys. Lett. B 278 ( 1992) 480.
LETTERS
B
Physics LettersB 349 (1995) 541-547
ELSEVIER
Model independent determination of 1I& 1in heavy meson effective theory Noriaki Kitazawa
’
Department of Physics, Tokyo Metropolitan University, 1-l Minami Ohsawa, Hachioji-shi, Tokyo 192-03, Japan
Received 24 September 1994; revised manuscript received 29 December 1994 Editor: H. Georgi
Abstract
We propose a strategy of the model-independent determination of the Kobayashi-Maskawa matrix element l&l within the framework of the heavy meson effective theory. The effective theory is model-independent, since no assumption on the dynamics is introduced except for the symmetry of QCD. Although the effective theory can only be applied to the decay process with soft pions, high detection efficiency on the soft pions in the future B-factories makes this theory effective. The two exclusive processes, B -+ ?rl~ and B -+ n?rlv, are used to determine the value of l&l with l/Mb correction, under the condition that the values of the decay constants of B and B’ mesons are given.
The precise measurement of the KobayashiMaskawa matrix elements is one of the approaches to detect the new physics beyond the standard model at B-factories. The unitarity relation of the matrix in the B”-tio system ,
c
I$&
= 0,
(1)
1=u,c,t is described as a triangle in the complex plane. If the measured values of the sides and angles are not consistent with the triangle, the contribution from the new physics should exist. Various strategies to measure the sides and angles are proposed. Focusing on the sides, 1V&l in 1V$,Vc.J can be model-independently determined with enough accuracy by virtue of the heavy quark symmetry [ l31. But the determinations of the other two sides, lvtVtd( N l&d1 and IV$,&j N I&,[, are still suffered ’ E-mail: [email protected].
from the ambiguity due to the theoretical estimation of the hadron matrix elements. At present, IVubJ is extracted as a ratio I&,/b&J from the inclusive semi-leptonic B decay [ 41. The measured spectrum of the charged lepton momentum is understood as the combination of the two components of b + clv and b --) ulv, which are proportional to IvC# and IVub1*, respectively. The theoretical predictions of both spectra and decay rates of these two components are necessary to extract (Vub/Vcb(. There are some theoretical models [5-71 to calculate the hadron form factors. But they give different predictions, and we have almost no criteria to select one of them at present. The extracted values of 1Kb/Vcbl are model dependent, and the ambiguity is larger than the present experimental error. The other strategy which is free from the model dependence is expected. Heavy meson effective theory [ 81 is not the model of the hadrons, since no assumption on the dynamics is introduced except for the symmetry of QCD. Spin-
0370-2693/95/%09.50 @I 1995 Elsevier Science B.V. All rights reserved SSDlO370-2693 (95)00304-5
542
N. Kitazawa /Physics Letters B 349 (1995) 541-547
flavor symmetry of the heavy quarks and the chiral symmetry of the light quarks restrict the interactions between the heavy-light mesons (D, D*, B, and B*) and the light pseudo-scalar mesons ( ‘IZ, K, and q), and the effect of the QCD dynamics is represented by the coupling constants of the interactions. Once the coupling constants are fixed by using the experimental data, we can predict the form factors in modelindependent way. The form factors are effective only for the soft pions, since the chiral expansion is used in the construction of the effective theory. In this letter, we propose a model-independent strategy of extracting jVub] through the fixing of the parameters of the exclusive processes B + dv and B + mdv with soft pions, under the condition that the values of the decay constants of B and B* mesons are given by the lattice calculations. Both the above two processes are needed, if we consider the l/Mb correction. The importance of the correction in the heavy-to-light transition is suggested by the lattice calculations of the decay constants of heavy-light mesons. The Lagrangian of the heavy meson effective theory has been written down including the breaking effect of the spin-flavor symmetry [9,10]. Up to 0(p2) in the chiral expansion and 0( l/M;) in the l/Me expansion,
+ (Anti-particle),
(2)
with l/M = diag( l/MC l/Mb). There are five dimensionless parameters (K, K’, A, and At.2) and a parameter A with mass dimension. The heavy mesons are described by the field
fj)++
’
[lYSP"
7j-
”
Y&x
+
I
(3)
where v is the velocity of the heavy quark inside. The fields P, and PJ are the heavy pseudo-scalar and heavy vector fields defined as pc*, = ”
Do B-
(*I
Df
0,’
Et?
Do
(4)
s
which have the mass dimension 3/2. The lightpseudoscalar mesons (r, K, and 7s) are described by the following fields as the Nambu-Goldstone bosons, (5)
t=
ein/f=
and
fI=I’F$.
(6)
All the possible independent terms which are consistent with the symmetry (including C, P, and reparametrisation invariance) are enumerated in the Lagrangian. The transformation properties of the fields and the details of the construction of the Lagrangian are described in the paper of Ref. [9] 2 . The heavy-to-light weak current can be written down in the same level of the expansion as J:(O)
= F[tr{(@)jiyP(l
+ &tr{b?)jiy,(l
- y5)Hzj) - ys)[yp,iDPHzjl}]
+atktr{(@)iiy,(l a
- y~)Hzj}
+a2~tr{(tt)jiy,(l D
- ydypH$y,}
(7)
* In that paper, the light vector mesons (p, K*, and ws) are introduced using the method of the hidden local symmetry [ 111. But here we do not introduce them, since the method is a model,
N. Kitazawa / Physics Letters B 349 (1995) 541-547
with three parameters of the mass dimension 3/2 (F and at,z). The decay constants of the heavy-light mesons are parametrized as
fP
= J
fv=
-zMQ
+2Ly*)
F+&
&
F+
-/pi
$
+
B*‘
1
(8)
, I
MQ
{
543
-2a2)
>
0
(9)
3
where fp and fv are the decay constants of pseudoscalar and vector heavy mesons, respectively. The Lagrangian gives the mass formulae for the heavy mesons
Fig. 1. The diagrams for B” -t ?r+lP decay. Black circle denotes the strong vertex, and black square denotes the weak vertex.
are used. If the quark masses are set as M, = 1500 MeV and Mb = 5000 MeV [ 121 (WV- up)A3 N (695 MeV)3.
(16)
Then we obtain the magnitude of the universal 1/MQ correction to the squared mass difference as
1+2h+2~'$-+12~-$ MQ +WP-$-
A3 Q
3
Q
Q
(10)
3
I+2~+2~'$-4& MQ Q fWV$
A3 Q
>
-16~A~
Q
(11)
>
where mp and mv arethe masses of pseudo-scalar and vector heavy mesons, respectively. The l/ML correction to the un-squared masses mp and mv aregenerally attached as the A3/Mi terms with the coefficients wp and WVin the brackets. Using a relation m2,- m'p= -16~A~ + (WV- wp)A2n MQ
(12)
=
(rnt., -m$0)--(w--w~)~
c 21 (597 MeV)2.
(17)
We can see that the convergency of the l/M, expansion is not good, since the 1/MC and 1 /Mz corrections to the rniarethe same order of magnitude. The magnitude of the corrections relative to the leading are
(18) A3 A3 WpY$ N (WV- WP’~ 210.10. C C
I /I N
1.1 x 10-2, (wv-wp)-
(20) A3 M3b
N (396 MeV)2,
(13)
where the values of the meson masses [ 121 mDo 2! 1860 MeV,
MD’0N 2010 MeV,
(14)
mg0 215280 MeV,
mg.0 215320 MeV,
(15)
(19)
But the l/M; correction to rniis one order of magnitude smaller than the 1/Mb correction:
following from the above mass formulae, we can estimate the convergency of the ~/MQ expansion. The magnitude of the l/ML correction is given as
= (m&O -m2,0)- (m&O -miO>
A3
~22.7~ 10-3.
(21)
This indicates that the convergency of the l/Mb expansion is good. Therefore, we can safely apply this effective theory only to the B meson system. The decay amplitude of the process B” + 7.r+ZVis obtained by calculating the diagrams of Fig. 1. The form factors defined by (r(Pr)
IJib
(B’(pB))
= f+(q2)(PB+Pn)~++-(92)(pB-P~)~
(22)
544
N. Kitazawa / Physics Letters B 349 (1995) 541-547
are obtained as
f*(q2) = ;g [I - g x
A 1+ E)
+ ((Al -
{(
x
(4-m:
A21 -
‘F(mi-mi+q2)
-s2)
q2 - rn&
where o.p,
q2 = (pi
=
- p,)
A)
&}
]v
and
2
m~+m~-q2 2ms
(23)
’
(24)
The explicit dependence on the bottom quark mass Mb is replaced by the dependence on the B meson mass mg up to the order of l/M; by using the mass formula. The B* meson pole contribution emerges with the coupling constant A and the combination A1- AZ. The mass of the pion is remained non-zero in the phase space integration only. At the tree level, all the effect of the explicit chiral symmetry breaking due to the light quark masses, m,, and md, is reduced to the isospin breaking effect which is negligibly small. Then we can use the coupling constants of the chiral limit. There are no contribution from the higher order terms of chiral expansion at the tree level. Therefore, these form factors should be applied as far as the chiral expansion converges 3 . Following to the naive dimensional analysis [ 133, the chiral expansion converges if
&
(-1 4rfn
2<
dq2
where the masses of the charged lepton and neutrino are neglected. Once we fix the values of all the unknown parameters, A, ((Ai - AZ) - A)A/mB, fs, and fs*, the value ]t&,] can be obtained by fitting the spectrum of the above region. If we consider the very soft pions, namely
E,r
(-> 4rfTr
1,
(25)
{E; _
m2,}3/2 1f+(q2>)2,
(26)
with the above f+ ( q2) is effective in the region (mLi-mm,)2 >q2 >m~i-m~-2me.4~f,,
(27)
3 This corresponds to the fact that only the 0(p2) and S(p4) terms in the non-linear chiral Lagrangian can contribute to the r-r scattering amplitude at the tree level. 4 Reparametrisation invariance always allow us to take pt~ = mgu.
2<
0.1
(28)
or (1128 -
where E,, = v . ps denotes the energy of the pion in B meson rest frame4 and fT N 88 MeV denotes the decay constant of the pions in the chiral limit. The q2-spectrum
d!-=9
Fig. 2. The diagrams for B- --+ T+?T-/~ decay. Black circle denotes the strong vertex, and black square denotes the weak vertex.
F&j2
> q2
> rni + rn$ - 2rnB. d(4TfT)2/10,
(29)
the E,/2mB dependence in the form factors can be approximately neglected in the 10% level, and only a combination of the couplings, A E A + ( (AI - AZ) A)A/??ZB,emerges in the form factor. To fix the values of the coupling constants, we must consider the other independent processes of B meson decay. The decay amplitude of the process B- -+ IT+T-Z~ is obtained by calculating the diagrams of Fig. 2. The form factors which are defined as [ 141 (~+(P+)~-(P~-)IJ;~(O)IB-(PB)) =
w+k+, + w-k-,
+ rqP + hiew,qrpik$k?., (30)
where kk = p,,+ fp,-
and q = pl +pv, are obtained as
N. Kitazawa /Physics
“+
=
i(
l+PB’k+-PB’~-
((4
-
4m2,
A21
-
A)
>
&}
G;IV,bi2f;
drs-wave
1
2m;-3(pB.k+-pB.k_) rni - m$. -pB’k++pB*k-
545
AZ,emerges in h. Taking the S-wave configuration of pions in B meson rest frame, the contributions of the p meson resonance, which can couple with the pion pair, and the form factor h are dropped out. The decay spectrum of the S-wave pions is given as
---
A
Letters B 349 (1995) 541-547
(31)
dk2,dq2
= 12(4r)5mi 4
X
-5
E
-42)2
_dii,)3’2
K
x
X
X [I+~(~+((hl-A2)4+-)
{(
l+Ps*k+-Ps*k4m2,
A
x
(32)
h= X
1 rni
-
rn&
Y4:;21 r
-pBsk++pB.k_q2-m&
PB.k+-Ps*k-
2
[(*+p*)+2(A,-A)A
XA A
mB
--
A
.
(33)
mB
We do not write r. since it does not contribute if the masses of the lepton and neutrino are neglected ’ . The relations (w+--w-1
.
P,+ =o
= -1-
1
‘2f+((PB
2(m&
-mi)+mi+k$-$
The coupling constants are included only in the combination of 1s A + ( (At - AZ) - A) A/mB, since we are considering the very soft pions which satisfy . +
1
B
4m& (36)
1
2mi-(pB’k+-pi?‘k_) rni - rn& -pB*k++ps*k-
{
3-
2
< 0.1,
(37)
where u. k+/2 denotes the average of the pion energies in B meson rest frame. The ET-/2mB dependence in the form factors w+ and w_ are approximately neglected in the 10% level. The next order terms of the chiral expansion ( 0(p2)) can contribute to this process. To keep the good approximation within 10% error, we must set the condition of Fq. (37). The width of D*+ -+ Do& decay can be calculated as
-?‘T-)~)
Jzf?r
(34) (w++o-),
__&=o, VT
(35)
which follow form the chiral symmetry, are satisfied. The explicit dependence on the bottom quark mass Mb is replaced by the dependence on the B meson mass mB up to the order of l/M; by using the mass formula. A new combination of the parameters, At +
(38) where E,, =
rn& -m2,+m2, 2mD8
.
The experimental upper bound on this width [ 151 i-( D*+ 4 DOT+) < 131 keV
5 Two of the five diagrams with the derivative coupling of the pseudo-scalar B meson with W bosons only contribute to r.
gives the upper bound on the parameter
(39)
546
A + ((Al - AZ) - A) &
N. Kifazawa / Physics Letters B 349 (1995) 541-547
< 0.417.
(40)
Although the l/M, expansion may not converge, this result gives the order of magnitude of the couplings. We set > = 0.4 in the following estimates. We assume that the values of f~ and f~. are given. The value of f~ will be obtained within 10% accuracy by the lattice calculation in near future6 . Although the value of fs* is phenomenologically less important 7 , the lattice calculation of the value of f~ can be interesting with a view to check the convergency of the l/Mb expansion. We set f~ = fs* = 120 MeV in the following estimates. Once we obtain the values of the decay constants, and if we neglect the l/Mb correction, we can obtain the value of I& 1by using the q2-spectrum of B + rlv decay alone. (If the decay constant fB is not given, we can extract only the value of a combination of f~ 1&I. The process B --+ (mr)~+,~~~ Iv does not give any additional information.) The process B -+ mrZv is not needed in this case. The unknown parameters are only (&,I and A in the form factor f + ( q2). These parameters can be fixed simultaneously by fitting the q2spectrum. But the 1/Mb correction must be important in the process of heavy-to-light transition, as the recent lattice calculations of the decay constants of the heavy-light mesons strongly indicate it. In the heavy mass limit, the quantity fp & should be a constant which is independent of MQ, and the MQ dependence of the quantity is introduced as the ~/MQ correction (see EQ. (8)). But recent lattice calculations show that the quantity is strongly depend on MQ in the region of MQ 21 Mb. To handle the l/Mb correction in the determination of 1Vubj,we need more one process B -+ m~lv. The decay rate of the B” + &EF with very soft pion is obtained by integrating the spectrum of Eq. (26) over the phase space restricted by Eq. (28). The expected number of the events of such decay with lo8 B mesons is 6 If the lattice calculation can give the form factor f+ (q*) with the enough accuracy, it is straightforward to extract the value of IV&l form the B - lrlv decay. But since the calculation of the form factor is more difficult than that of the decay constant, it can be imagined that the accuracy is not enough compared with the experimental accuracy in the future B factories. ’ The weak decays of the B* meson are hard to observe, since the branching ratio is very small.
N( B0 + 7r+lD) IveryS&St @” N 1040-
lv,b12 (l/20)3’
(41)
The decay rate of the B- -+ ST+TT-/~ with very soft pions is obtained in the same way by integrating the spectrum of Eq. (36) over the phase space restricted by Eq. (37). The expected number of the events with lo8 B mesons is
N(B- + T+T-/~)Ivery softpion (42) These number can be detectable in the future Bfactories within the 10% accuracy. Therefore, it can be expected that the value of I&,[ is extracted by fitting both i and l&l with these numbers. The model-independent determination of l&j within 10% accuracy can be expected. We may obtain both the values of A and (( A1 AT) - A)Alrn~ by fitting the spectrum of Do + rr+ZD in the region of Eq. (27). The spectrum should be normalized by the number of Eq. (42) to factor out IVubI. The convergency of the l/Mb expansion can be estimated by comparing the values of A and ( ( hl - AZ) A)h/mB. Once these parameters are fixed, the value of [&,I can be obtained by fitting the un-normalized spectrum of B” --+ ?r+/5. The extracted value is also the model-independent. The accuracy of the extracted value depend on the experimental ability to detect the very soft pions, since the main theoretical ambiguity comes from the higher order contribution of the chiral expansion. If we want the theoretical ambiguity of less than 5%, the phase space should be restricted as (&/47rfr)* < 0.05 in the process Do --+ ~r+Zti, for example. The resultant number of the events will be very few and hard to detect. If the precise measurement of the pure leptonic decays of B mesons, B- + p3 or B- + ~ii, is easier, this strategy have no advantage. Another large theoretical ambiguity may come form the n--n= re-scattering effect in B- + &rr-lF decay. But we can carry out the improvement by using the experimental data of the 7r-r scattering phase shift from the process of K ---+mrlu. In this letter we proposed a model-independent strategy to extract the value of I&,[ from the experiments. Two kind of decay processes, a0 + <
N. Kitazawa/Physics Letters B 349 (1995) 541-547
and B- ---) rr+r-lfi, with very soft pions are used. The effect of the QCD dynamics is represented by only one parameter except for the decay constants, since the phase space of the decays is restricted so as to include only the very soft pions. The values of the decay constants, f~ and fs*, are expected to be obtained within the enough accuracy by the future lattice calculations. The theoretical ambiguity can be less than 10% depending on the experimental ability to detect the soft pions, if the decay constants of B and B* are known in good accuracy. I am grateful to Yasuhiro Okada, Minoru Tanaka, and Masaharu Tanabashi of KEK theory group and Takeshi Kurimoto of Osaka University for fruitful discussions and comments. I also would like to thank S.Uno of KEK for the helpful advice from the experimental point of view. References [l] N. Isgur and M.B. Wise, Phys. L&t. B 232 (1989) 113; B 237 (1990) 527.
547
I21 M.E. Luke, Phys. Len. B 252 (1990) 447; C.G. Boyd and D.E. Brahm, Phys. I&t. B 257 (1991) 393. [31 M. Neubert, Phys. L&t. B 264 (1991) 455. r41 CLBO Collaboration, Phys. Rev. L&t. 71 ( 1993) 4111. [51 N. Isgur, D. Scora, B. Grinstein and M. Wise, Phys. Rev. D 39 (1989) 799. [61 M. Wirbel, B. Stech and M. Bauer, 2. Phys. C 29 (1985) 637. [71 G. Altamlli, N. Cabibbo, Cl. Corbo, L. Maiani and G. Martinelli, Nucl. Phys. B 208 (1982) 365. 181 H. Georgi, Lectures delivered at TASI, published in Boulder TASI 91, 589 (HUTP-91-A039); G. Burdman and J.P. Donoghue, Phys. Lett. B 280 (1992) 287; M.B. Wise, Phys. Rev. D 45 (1992) R2188. [91 N. Kitazawa and T. Kurimoto, Phys. Lett. B 323 ( 1994) 65. [lOI C.G. Boyd and B. Grinstein, UCSD-PTH-93-46. [Ill M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. L&t. 54 (1985) 1215; for a review see, M. Bando, T. Kugo and K. Yamawaki, Phys. Rep. 164 (1988) 217. [I21 Particle Data Group, Phys. Rev. D 45 ( 1992). [I31 H. Georgi and L. Randall, Nucl. Phys. B 276 ( 1986) 241. [I41 A. Pais and S.B. Treiman, Phys. Rev. 168 (1968) 168; C.L.Y. Lee, M. Lu and M.B. Wise, Phys. Rev. D 46 ( 1992) 5040. [I51 ACCMOR Collaboration, Phys. Lett. B 278 ( 1992) 480.