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The rare decay B→Kτ+τ− in heavy meson chiral perturbation theory
PhysicsLettersB 317 (1993) 179-182 North-Holland
PHYSICS LETTERS B
The rare decay B Kz + z in heavy meson chiral perturbation theory Dongsheng Du, C h u n Liu and D a x i n Z h a n g Institute of High EnergyPhysics,Academia Sinica, P.O.Box 918(4),Belting100039, China Received 12 August 1993 Editor: H. Georgi
The rare decayB~Kz+z- is calculatedby applyingthe heavy mesonchiral perturbation theory which is a reliablemethod in this case. The typicalorder of maguitude of the branchingratio is 10-7. The inclusivedecayrate b--,sz+z- is also given.
Rare B meson decays induced by penguin diagrams are very important for testing the standard model and discovering new physics. A recent experiment has measured the rare process B~K*y [ 1 ]. Hopefully, in the near future, the decay channels B - , K ( K * ) I + I - (l=e,/z, z) will be measured also, at least at B-factories. Therefore, in addition to calculating b--.sl+l- [2,3 ], it is important to calculate the corresponding exclusive decays. However, almost all the calculations of these exclusive decay rates are model dependent with some uncertainties [ 3 ]. Here we calculate the channel B--,Kz +T- in the standard model by using the heavy meson chiral perturbation theory [ 4 ] which is a reliable method in this case. This mode has a branching ratio of the order of 10-7 and may likely be measured at B-factories. The heavy meson chiral perturbation theory is essentially a model-independent method in calculating the lowmomentum properties of mesons containing a single heavy quark. As a low energy effective field theory of QCD, by combining the chiral symmetry and the heavy quark symmetry, it can be used systematically to study the low energy strong interactions among the heavy mesons and the pseudo Goldstone bosons (g, K, ~/) [4]. Calculations in this theory are reliable when the energies of the Goldstone bosons are small compared to the typical scale of the chiral symmetry breaking A c s s - 4~f=/x/~--- 1.2 GeV [ 5 ]. There are also arguments in the literature that Acss may be even as large as 1.5 GeV [ 6 ]. In the decay B--,Kz + z - , the maximum energy of the K meson in the B meson rest frame is (m~ + m ~ - 4m ~ ) / 2roB" 1.5 GeV. IfAcss is at 1.5 GeV, then the heavy meson chiral perturbation theory is entirely reliable in the calculation. Indeed, even if the chiral symmetry breaking scale is AcsB-~ 1.2 GeV, most of the phase space lies in the region where the kaon has energies small enough for the reliable application of chiral perturbation theory. Therefore, the calculation of the rare decay B-~Kz +~- using heavy meson perturbation theory is a reasonable method. The hadronic matrix elements involved in the decays B--, KI +l - are <•(PK) lY?"( 1 - ?5)b[/~(PB) > =f+ (Pz +Px)~+f- (PB --PK)",
=is[ (PB + Px)~(Pn - P K ) ~ - (PB --PK)I'(PB + Px)P] --SeUVXP(Pn+ Px)a(Ps --PK)p .
(l )
TO the leading order in the heavy meson chiral perturbation theory, the form factors f÷ and f_ are [ 4 ] 0370-2693/93/$ 06.00 © 1993ElsevierSciencePublishers B.V. All rights reserved.
179
Volume 317, number 1,2
PHYSICSLETTERSB
v'~x+J}'
4 November 1993
(2)
and Jr+ - f - =
gdrama
fx(v.px+A) ,
(3)
where fn and fx are the decay constants of the B and K mesons respectively, v=pn/mn is the four-velocity of the B meson, 3 = m } , - ms and g is the coupling constant describing the interactions of heavy mesons with pseudo Goldstone bosons. The form factor s is related to f± by the heavy quark symmetry [ 7 ]
s=f+ -f2ms
(4)
In the standard model, the effective Hamiltonian relevant for the b ~ sl +1- decay is [2 ]
~ f r = - -~2 V,,V~ ~ cj(/~) ~-(/~), j = l , 2 , 7 , 8 , 9 ,
(5)
where the Wilson coefficients cj(#) and the operators ~.(~t) can be found in ref. [2]. The energy scale# should be at/~--mb in evaluating B meson decays, cj(mb) are obtained from cj(mw), which are functions of the top quark mass rnt, through the analysis of the renormalization group. While the x lepton mass is not negligible, we obtain the differential decay rates from the effective Hamiltonian
dl" ~" =
G2vIVt°Vgl2mSn°t2 1 -7 3 X297rn
[(1-r)2-25(l+r)+~2] 1/2
--
×[[ [cseff(mb)f+ + 2mbc7(mb)sl2+ lcg(mb)f+12][ (1--r)2--2$( l +r)+'2](l + 2~) + 12 [eg(mb)[2t[ (1 + r - - ½~)f% + (1 --r)f+f_ + ½~f2 ] 1 '
(6)
where
cater(rob) =cs(mb) + [3Cl (rob) d-c2(mb)
]g(~2) ,
(7)
where g(~2 ) can be found in ref. [ 2 ], r = m2/m 2, t= m 2]m2 and ~= m2/m 2 with m,, being the invariant mass of the tau lepton pair. Taking t=0, eq. (3) will reproduce the result given in ref. [2] for the B--,Kl+l - decay rate with the lepton mass being neglected. Using form factors in eqs. (5), (6) and (7), we can calculate the decay rate from eq. (3). In the numerical evaluation we use the universal chiral coupling constant g = 0.7, the decay constant fn = 0.2 GeV, the Bs meson mass ran, = 5.4 GeV, the CKM matrix element I Vu[ = 0.042 and the lifetime of the B meson zB= 1.3 ps. The differential branching ratio of B--,KT + z - as function of the invariant mass of the z pair m~ is plotted in fig. 1 using the values mt = 150 GeV and AQcD= 200 MeV. The numerical results of the decay branching ratios with various values of AQcD and mt are listed in table 1, where R is the ratio of the exclusive to inclusive decay rates. The branching ratios are typically of the order of 10 -7. From the table we see that the uncertainties of AQcD have no significant influence on the results. For the sake of comparison, we have also calculated the inclusive decay rate for b--,sz+x - from the Hamiltonian ( 1 ) without neglecting the lepton mass and the strange quark mass, 180
Volume 317, number 1,2
PHYSICS LETTERS B
4 November 1993
0.24
b.
0.2 o
U
o
o
0.16 lC
E o.12
"0
"~ 0.01~
o.o~ o
~4
16
19
20
Fig. l. The differential b ranching ratio of B--~K~ + ~ - decays ( mt ffi 150 GcV, A ~
2,~
J
ffi 200 M e V ).
Table 1 Branching ratios of B--,Kz+z- and the ratio R as functions of AQcvand mr A~-ua(MeV)
m,(OeV)
Br(B~Kr+z-)
100
120 150 180
1.0 1.6 2.3
41 45 49
200
120 150 180
1.1 1.7 2.5
48 52 57
300
120 150 180
1.0 1.6 2.5
49 53 58
dFdx- G~lVtbV~12m~'×°~2[( 277t 1-?)3s
( X l 0 -~)
R(%)
[(l+x/~)2-x][(1-v/z)2-x]l '/2
x / \ x I c71211 + I cs,ffl212 + 12
Re(c7c~off)Ia + ½l c91214
,
(8)
where
I]=2(1-z)2-x(l+x+z),
I2=(1-z)2+x(l+z)-2x 2, I 3 = l - x - z ,
I.=3[(l-z)2-x(x-4y)-4y(l+z)]-(1-?)[(l+v/z)2-x][(1-X/~)2-x],
(9)
181
Volume 317, number 1,2
PHYSICS LETTERS B
4 November 1993
with x = m 2 / m 2, y = m 2 / m 2 and z = m 2 / m ~,. Taking y = z = 0, eq. (6) is consistent with the result given in ref. [2] for the b - - , s l + l - decay with the lepton mass being neglected. The b o t t o m quark mass mb and the strange quark mass ms are taken as 4.8 GeV and 0.15 GeV respectively in the numerical evaluations. It is found that the ratio of the exclusive to inclusive decay rates is about 50%. This means that B ~ K T + ~ - is the dominant mode o f b--,s~ + T - . It should be remarked that we have not considered the long distance contribution which in this case can come from B--,K~(2S). Comparison with B ~ K I + I - ( l = e , / ~ ) in which the most important long distance contribution lies in B - - , K J / ~ , the long distance contributions in B - - , K T + T - are much less important. We have calculated the rare decay B - , K r + ~ - by applying the heavy meson perturbation theory which is a reliable method in this case. The typical order of magnitude o f the branching ratio is 10 -7. This process is very difficult to measure experimentally, except in the B-factories. Considering the detection efficiency, about 109 events o f B mesons are required.
References [ 1] CLEO Collab, Report No. CLNS 93/1212, CLEO 93-06. [ 2 ] B. Gdnstein, M.J. Savageand M.B. Wise, Nuel. Phys. B 319 (1989) 271. [3] Fora review, see R. Grigjanis, P.J. O'Donnell, M. Suthedand and H. Navelet, Phys. Rep. 228 (1993) 93. [4] M.B. Wise, Phys. Rev. D 45 (1992) R2188; G. Burdman and J.F. Donoghue, Phys. Lett. B 280 (1992) 287; T.M. Yan, H.Y. Cheng, C.Y. Cheung, G.L. Lin, Y.C. Lin and I-LL.Yu, Phys. Rev. D 46 (1992) 1148. [ 5 ] H. Georgi, Weak interactions and modern particle theory (Benjamin/Cummings, Menlo Park, CA, 1984). [6] J. Gasser and H. Leutwyler,Ann. Phys. (NY) 158 (1984) 142;Nuel. Phys. B 250 (1985) 465; J.L. Goity, Phys. Rev. D 46 (1992) 3929. [7] N. Isgur and M.B. Wise, Phys. Rev. D 42 (1990) 2388.
182
PHYSICS LETTERS B
The rare decay B Kz + z in heavy meson chiral perturbation theory Dongsheng Du, C h u n Liu and D a x i n Z h a n g Institute of High EnergyPhysics,Academia Sinica, P.O.Box 918(4),Belting100039, China Received 12 August 1993 Editor: H. Georgi
The rare decayB~Kz+z- is calculatedby applyingthe heavy mesonchiral perturbation theory which is a reliablemethod in this case. The typicalorder of maguitude of the branchingratio is 10-7. The inclusivedecayrate b--,sz+z- is also given.
Rare B meson decays induced by penguin diagrams are very important for testing the standard model and discovering new physics. A recent experiment has measured the rare process B~K*y [ 1 ]. Hopefully, in the near future, the decay channels B - , K ( K * ) I + I - (l=e,/z, z) will be measured also, at least at B-factories. Therefore, in addition to calculating b--.sl+l- [2,3 ], it is important to calculate the corresponding exclusive decays. However, almost all the calculations of these exclusive decay rates are model dependent with some uncertainties [ 3 ]. Here we calculate the channel B--,Kz +T- in the standard model by using the heavy meson chiral perturbation theory [ 4 ] which is a reliable method in this case. This mode has a branching ratio of the order of 10-7 and may likely be measured at B-factories. The heavy meson chiral perturbation theory is essentially a model-independent method in calculating the lowmomentum properties of mesons containing a single heavy quark. As a low energy effective field theory of QCD, by combining the chiral symmetry and the heavy quark symmetry, it can be used systematically to study the low energy strong interactions among the heavy mesons and the pseudo Goldstone bosons (g, K, ~/) [4]. Calculations in this theory are reliable when the energies of the Goldstone bosons are small compared to the typical scale of the chiral symmetry breaking A c s s - 4~f=/x/~--- 1.2 GeV [ 5 ]. There are also arguments in the literature that Acss may be even as large as 1.5 GeV [ 6 ]. In the decay B--,Kz + z - , the maximum energy of the K meson in the B meson rest frame is (m~ + m ~ - 4m ~ ) / 2roB" 1.5 GeV. IfAcss is at 1.5 GeV, then the heavy meson chiral perturbation theory is entirely reliable in the calculation. Indeed, even if the chiral symmetry breaking scale is AcsB-~ 1.2 GeV, most of the phase space lies in the region where the kaon has energies small enough for the reliable application of chiral perturbation theory. Therefore, the calculation of the rare decay B-~Kz +~- using heavy meson perturbation theory is a reasonable method. The hadronic matrix elements involved in the decays B--, KI +l - are <•(PK) lY?"( 1 - ?5)b[/~(PB) > =f+ (Pz +Px)~+f- (PB --PK)",
=is[ (PB + Px)~(Pn - P K ) ~ - (PB --PK)I'(PB + Px)P] --SeUVXP(Pn+ Px)a(Ps --PK)p .
(l )
TO the leading order in the heavy meson chiral perturbation theory, the form factors f÷ and f_ are [ 4 ] 0370-2693/93/$ 06.00 © 1993ElsevierSciencePublishers B.V. All rights reserved.
179
Volume 317, number 1,2
PHYSICSLETTERSB
v'~x+J}'
4 November 1993
(2)
and Jr+ - f - =
gdrama
fx(v.px+A) ,
(3)
where fn and fx are the decay constants of the B and K mesons respectively, v=pn/mn is the four-velocity of the B meson, 3 = m } , - ms and g is the coupling constant describing the interactions of heavy mesons with pseudo Goldstone bosons. The form factor s is related to f± by the heavy quark symmetry [ 7 ]
s=f+ -f2ms
(4)
In the standard model, the effective Hamiltonian relevant for the b ~ sl +1- decay is [2 ]
~ f r = - -~2 V,,V~ ~ cj(/~) ~-(/~), j = l , 2 , 7 , 8 , 9 ,
(5)
where the Wilson coefficients cj(#) and the operators ~.(~t) can be found in ref. [2]. The energy scale# should be at/~--mb in evaluating B meson decays, cj(mb) are obtained from cj(mw), which are functions of the top quark mass rnt, through the analysis of the renormalization group. While the x lepton mass is not negligible, we obtain the differential decay rates from the effective Hamiltonian
dl" ~" =
G2vIVt°Vgl2mSn°t2 1 -7 3 X297rn
[(1-r)2-25(l+r)+~2] 1/2
--
×[[ [cseff(mb)f+ + 2mbc7(mb)sl2+ lcg(mb)f+12][ (1--r)2--2$( l +r)+'2](l + 2~) + 12 [eg(mb)[2t[ (1 + r - - ½~)f% + (1 --r)f+f_ + ½~f2 ] 1 '
(6)
where
cater(rob) =cs(mb) + [3Cl (rob) d-c2(mb)
]g(~2) ,
(7)
where g(~2 ) can be found in ref. [ 2 ], r = m2/m 2, t= m 2]m2 and ~= m2/m 2 with m,, being the invariant mass of the tau lepton pair. Taking t=0, eq. (3) will reproduce the result given in ref. [2] for the B--,Kl+l - decay rate with the lepton mass being neglected. Using form factors in eqs. (5), (6) and (7), we can calculate the decay rate from eq. (3). In the numerical evaluation we use the universal chiral coupling constant g = 0.7, the decay constant fn = 0.2 GeV, the Bs meson mass ran, = 5.4 GeV, the CKM matrix element I Vu[ = 0.042 and the lifetime of the B meson zB= 1.3 ps. The differential branching ratio of B--,KT + z - as function of the invariant mass of the z pair m~ is plotted in fig. 1 using the values mt = 150 GeV and AQcD= 200 MeV. The numerical results of the decay branching ratios with various values of AQcD and mt are listed in table 1, where R is the ratio of the exclusive to inclusive decay rates. The branching ratios are typically of the order of 10 -7. From the table we see that the uncertainties of AQcD have no significant influence on the results. For the sake of comparison, we have also calculated the inclusive decay rate for b--,sz+x - from the Hamiltonian ( 1 ) without neglecting the lepton mass and the strange quark mass, 180
Volume 317, number 1,2
PHYSICS LETTERS B
4 November 1993
0.24
b.
0.2 o
U
o
o
0.16 lC
E o.12
"0
"~ 0.01~
o.o~ o
~4
16
19
20
Fig. l. The differential b ranching ratio of B--~K~ + ~ - decays ( mt ffi 150 GcV, A ~
2,~
J
ffi 200 M e V ).
Table 1 Branching ratios of B--,Kz+z- and the ratio R as functions of AQcvand mr A~-ua(MeV)
m,(OeV)
Br(B~Kr+z-)
100
120 150 180
1.0 1.6 2.3
41 45 49
200
120 150 180
1.1 1.7 2.5
48 52 57
300
120 150 180
1.0 1.6 2.5
49 53 58
dFdx- G~lVtbV~12m~'×°~2[( 277t 1-?)3s
( X l 0 -~)
R(%)
[(l+x/~)2-x][(1-v/z)2-x]l '/2
x / \ x I c71211 + I cs,ffl212 + 12
Re(c7c~off)Ia + ½l c91214
,
(8)
where
I]=2(1-z)2-x(l+x+z),
I2=(1-z)2+x(l+z)-2x 2, I 3 = l - x - z ,
I.=3[(l-z)2-x(x-4y)-4y(l+z)]-(1-?)[(l+v/z)2-x][(1-X/~)2-x],
(9)
181
Volume 317, number 1,2
PHYSICS LETTERS B
4 November 1993
with x = m 2 / m 2, y = m 2 / m 2 and z = m 2 / m ~,. Taking y = z = 0, eq. (6) is consistent with the result given in ref. [2] for the b - - , s l + l - decay with the lepton mass being neglected. The b o t t o m quark mass mb and the strange quark mass ms are taken as 4.8 GeV and 0.15 GeV respectively in the numerical evaluations. It is found that the ratio of the exclusive to inclusive decay rates is about 50%. This means that B ~ K T + ~ - is the dominant mode o f b--,s~ + T - . It should be remarked that we have not considered the long distance contribution which in this case can come from B--,K~(2S). Comparison with B ~ K I + I - ( l = e , / ~ ) in which the most important long distance contribution lies in B - - , K J / ~ , the long distance contributions in B - - , K T + T - are much less important. We have calculated the rare decay B - , K r + ~ - by applying the heavy meson perturbation theory which is a reliable method in this case. The typical order of magnitude o f the branching ratio is 10 -7. This process is very difficult to measure experimentally, except in the B-factories. Considering the detection efficiency, about 109 events o f B mesons are required.
References [ 1] CLEO Collab, Report No. CLNS 93/1212, CLEO 93-06. [ 2 ] B. Gdnstein, M.J. Savageand M.B. Wise, Nuel. Phys. B 319 (1989) 271. [3] Fora review, see R. Grigjanis, P.J. O'Donnell, M. Suthedand and H. Navelet, Phys. Rep. 228 (1993) 93. [4] M.B. Wise, Phys. Rev. D 45 (1992) R2188; G. Burdman and J.F. Donoghue, Phys. Lett. B 280 (1992) 287; T.M. Yan, H.Y. Cheng, C.Y. Cheung, G.L. Lin, Y.C. Lin and I-LL.Yu, Phys. Rev. D 46 (1992) 1148. [ 5 ] H. Georgi, Weak interactions and modern particle theory (Benjamin/Cummings, Menlo Park, CA, 1984). [6] J. Gasser and H. Leutwyler,Ann. Phys. (NY) 158 (1984) 142;Nuel. Phys. B 250 (1985) 465; J.L. Goity, Phys. Rev. D 46 (1992) 3929. [7] N. Isgur and M.B. Wise, Phys. Rev. D 42 (1990) 2388.
182