Wave functions and decay constants of $B$ and $D$ mesons in the relativistic potential model

0.2 MB·English

by  Mao-Zhi Yang

Wave functions and decay constants of B and D mesons
in the relativistic potential model
∗
Mao-Zhi Yang
School of Physics, Nankai University, Tianjin 300071, P.R. China
(Dated: January 20, 2013)
With the decay constants of D and Ds mesons measured in experiment recently, we revisit the
study of the bound states of quark and antiquark in B and D mesons in the relativistic potential
model. The relativistic bound state wave equation is solved numerically. The masses, decay con-
stants and wave functions of B and D mesons are obtained. Both the masses and decay constants
obtainedherecanbeconsistentwiththeexperimentaldata. Thewavefunctionscanbeusedinthe
studyof B and D meson decays.
2
PACSnumbers: 12.39.Pn,14.40.Lb,14.40.Nd
1
0
2
I Introduction confining term. Such a potential will be consistent with

n
perturbativeQCD atshortdistance, itcanalsogenerate
a
J Thewavefunctionoftheboundstateofquarkandan- quark confinement at long distance [11, 18].
0 tiquark is determined by the strong interaction between The parameters in the effective potential can be con-
3 quark and antiquarks. The study of the wave functions strainedbycomparingtheeigenvaluesoftheboundstate
ofheavy-flavoredmesonslikeB andD areimportantnot wave equation with the masses of the relevant bound
h] only for studying the property of strong interaction be- states measured in experiment. Recently the decay con-
p tween heavy and light quarks, but also for investigating stantsofDandDsmesonshavebeenmeasuredbyCLEO
- the mechanism of heavy meson decays. The wave func- [19–21], Belle [22] and BABAR [23] Collaborations. The
p
e tiondeterminesthemomentumdistributionsofthequark measured values of D and Ds mesons’ decay constants
h and antiquark in mesons, which is an important quan- fD andfDs cangivefurtherinformationabouttheinter-
[ tity forcalculating the amplitude of heavymesondecays actions within the heavy-light quark-antiquark system.

[1,2]. Thelight-conemomentumdistributionamplitudes In this paper, with the recently measured decay con-
2
v forB mesonappearinthe amplitude ofB decays,which stants fD and fDs available, we revisit the study of the
9 are defined through the hadron-to-vacuum matrix ele- boundstatesofB, Bs,D andDs mesonsintherelativis-
1 mentofnon-localoperatorsofquarkandantiquarksepa- tic potential model. We solve the relativistic version of
8 ratedalongthelight-cone 0q¯β(z)bα(0)B¯(p) . Thelight- Schr¨odinger equation for the bound state wave function
3 h | | i
cone distribution amplitudes of B meson have been ex- ofheavy-lightquark-antiquarkmesonsystem. Thedecay
.
04 tensively studied in the recentseveralyears. Some prop- constantsfD andfDs canbeusedasafurtherconstriant
ertiesofthelight-conedistributionamplitudeshavebeen on the parameters in the potential model. The obtained
1
1 obtained. Based on these achievements, several models masses and decay constants of B and D mesons can be
: satisfying these constraints have been proposed in the wellconsistentwith the values measuredin experiments.
v
literature [3–9]. Methods to obtain the light-cone distri- Then the wave functions obtained here can be more re-
i
X butionamplitudeexactlyfromthefirstprincipleofQCD liable then ever. It can be useful for studying B and D
r are still under investigation. decays, where the momentum-distribution of the quarks
a
Alternatively, directly studying the wave functions of is needed.
the heavymesonsby solvingthe bound state waveequa- Althoughthe boundstates ofheavymesonshavebeen
tionisaneffectivewaytoobtaintheknowledgeaboutthe studied with the relativistic potential model in the lit-
boundstateofquarkandantiquark[10–15]. Foraheavy- erature several years before, these works need to be im-
light system, in the heavy quark limit, the heavy quark proved with the recent experimental data of the decay
canbeviewedasastaticcolorsourceintherestframeof constantsofD andDs mesonsavailable. Thedecaycon-
the heavy meson. The light antiquark is bound around stantsandboundstatemassesarecalculatedinRefs.[13–
the heavy quark by an effective potential. The heavy 15], where Richardson potential [24] was taken, here the
quark spin decouples from the interactions as m potential we considered is different from theirs. In ad-
Q
→ ∞
(m is the mass of the heavy quark) [16, 17]. The inter- dition, with the experimental values of the decay con-
Q
actions relevant to quark spin can be treated as pertur- stants fD and fDs available recently, the parameters in
bative correction. thepotentialcanbeconstrainedmorestringently. There-
Inspired by asymptotic freedom at short distance in fore our prediction on the decay constants for B and D
QCD and quark confinement at long distance, the effec- mesons are quite different from previous predictions in
tivepotentialbetweenquarkandantiquarkinmesoncan the relativistic potential model.
betakenasacombinationofaCoulombtermandalinear The paper is organized as follows. In section II, we
2
solve the relativistic wave equation for the heavy-light condition
quark-antiquarksystem. SectionIII givesthe decaycon-
stant in terms of the wave function. In section IV the dΩY (rˆ)Y (rˆ)=δ δ . (5)
l1n1 l2n2 l1l2 n1n2
QCD-inspired potential is presented. Section V is de- Z
voted to the numerical result and discussion. Section VI
Using the spherical harmonics decomposition of the ex-
is a brief summary. ponential in Eq.(4), and factorize the wave function into
the product of two parts: radial and angular wave func-
tions
II The relativistic wave equation for heavy-light
system and the solvement
ψ(~r)=Φ (r)Y (rˆ), (6)
l ln
TheB andDmesonsareassumedtobeapproximately then the wave equation of Eq.(3) can be transferred to
described in terms of heavy-light valence-quark configu- be
rations in the rest-frame of the mesons. The effective
2 k2
potential is one-gluon exchange dominant at short dis- V(r)Φ (r)+ dk dr′r′2( k2+m2
tances and a linear confinement at long distances [11]. l π¯h ¯h2 1
Z Z ′ q
The equation describing the bound state wave functions + k2+m2 )j (kr)j (kr )Φ (r′)=EΦ (r). (7)
is a Schr¨odinger-typewave equationwith relativistic dy- 2 l ¯h l ¯h l l
q
namics [11–15]
For convenience later, let us define a new reduced radial
wave function u (r) by
¯h2 2+m2+ ¯h2 2+m2+V(r) ψ(~r) l
− ∇1 1 − ∇2 2
(cid:20)q q (cid:21) ul(r)
=Eψ(~r), (1) Φl(r)= . (8)
r
where ~r = ~x1 ~x2 is the displacement of the light an- With this definition, and for the case l =0 which we are
−
tiquark from the heavy quark, and ~x1 and ~x2 are the interested in this work, Eq.(7) becomes
coordinates of the light and heavy quarks respectively.
∞ ∞
eTvhaentopteorathtoersco∇o1rdainndat∇es2oafr~exthaengdra~xdi.enmtopiesrtahtoermsraesls- V(r)u0(r)+ π2¯h dk dr′( k2+m21
of the light antiquark, and m12 the m2 ass1of the heavy + k2+m2 )sinZ(0kr)siZn0(kr′)uq(r′)=Eu (r), (9)
quark. V(r) is the effective potential of strong interac- 2 ¯h ¯h 0 0
tion between heavy and light quark-antiquark. In the q
rest frame of the bound state system, the eigenvalues in where we have used the explicit expression of the spher-
the waveequation will be the masses of the series bound ical Bessel function for l =0
states.
sinx
The wave function can be expressed in terms of spec- j0(x)= . (10)
x
trum integration
Eq.(9) is for taking c = 1, if recover the speed of light
ψ(~r) = d3r′δ3(~r ~r′)ψ(~r′) appearing in the formulas, Eq.(9) should be
−
Z ∞ ∞
= Z d3r′Z (2dπ3¯hk)3ei~k·(~r−~r′)/h¯ψ(~r′). (2) V(r)u0(r)+ π2¯hcZ0 dkZ0 ′dr′(qk2+m21
kr kr
SubstituteEq.(2)intoEq.(1),thewaveequationbecomes + k2+m22 )sin(¯hc)sin(¯hc)u0(r′)=Eu0(r). (11)
q
d3k d3r′( k2+m2+ k2+m2 ) In principle the integration over momentum k in the
(2π¯h)3 1 2 aboveequationcanbeperformedbecausethe wavefunc-
Z q q ′
ei~k·(~r−~r′)/h¯ψ(~r′)=(E V(r))ψ(~r). (3) tionu0(r )doesnotdependonthemomentum. However
′
× − theintegrationoverk willgiveasingulartermforr r
→
The exponential ei~k·~r/h¯ can be decomposed in spherical in the above equation [14]. In this work, we will take a
new step to continue to solve this equation, this method
harmonics
can circumvent the appearance of the singular integral
ei~k·~r/h¯ =4π ilj (kr)Y∗(kˆ)Y (rˆ), (4) equation.
l ¯h ln ln Foraboundstateoftwoparticles,whentheseparation
ln
X between them is large enough, the wave function will
where j(kr) is the spherical Bessel function, Y (rˆ) is effectivelyvanish. Weassumesuchalargeenoughtypical
l h¯ ln
thesphericalharmonics,whichsatisfiesthenormalization value for the separation between the heavy quark and
3
the light antiquarkis L, then the quark-antiquarkin the Theaboveequationisjustthe eigenstateequationinthe
bound state can be approximately treated as if they are matrix form. It is not difficult to solve it numerically.
restricted in a limited space 0 < r < L. In the limited The eigenvalues are the masses of the series of bound
space,theFourierexpansionofthereducedwavefunction states of the heavy-light quark-antiquark system. Once
u (r) is the eigen equation is solved, the eigenvectors composed
0
of c can be substituted into Eq.(16) to get the reduced
∞ n
nπ wave function u (r).
u (r)= c sin r , (12) 0
0 n L Togetthe wavefunctioninmomentumspace,one can
nX=1 (cid:16) (cid:17) use the Fourier transform of the wave function ψ(~r)
where the expansion coefficients c are
n
1
Ψ(~k)= d3re−i~k·~r/h¯cψ(~r). (18)
2 L nπ (2π¯hc)3/2
c = sin r u (r)dr. (13) Z
n 0
L L
Z0 (cid:16) (cid:17) Separate the variable-dependence of the momentum-
space wave function as
In the limited space, the momentum k should be dis-
cretized, the integration over k should be replaced by a
Ψ(~k)=Ψ (k)Y (θ,φ). (19)
summation,thefollowingsubstitutionshouldbemadein l lm
the wave equation (11)
AsinEq.(8), wedefinethe reducedwavefunctioninmo-
mentum space
k nπ dk π
, . (14)
¯hc → L ¯hc → L ϕ (k)
Z Ψ (k)= l . (20)
l
k
With the above replacement, and the integration over
′ ′
the distance r being limited within 0 < r < L, Eq.(11) ThenusingEqs.(4),(6),(8),(19)and(20),onecanderive
becomes from Eq.(18)
2 L nπ¯hc 2 2 ∞ kr kr
V(r)u0(r)+ n LZ0 dr′(s(cid:18) L (cid:19) +m21 ϕl(k)=(−i)lrπ¯hcZ0 dr¯hc jl(¯hc)ul(r). (21)
X
2 For the case l =0, we get
nπ¯hc nπ nπ
+ +m2 sin( r)sin( r′)u (r′)
s(cid:18) L (cid:19) 2  L L 0 2 ∞ kr
ϕ (k)= drsin( )u (r), (22)
0 0
=Eu0(r).  (15) rπ¯hcZ0 ¯hc
whichgivesthemomentumdistributionofthequarkand
The above equation can go back to Eq.(11) as L .
→ ∞ antiquark in the rest frame of the heavy meson.
NumericallyifthevalueofListakentobelargeenough,
thesolutionofthisequationonlyslightlydepends onthe
value of L. For the parameters we take in section V,
III The pseudoscalar bound state of heavy-light
we find that the solution of the wave equation will be system and the decay constant
stationary when L>5fm.
TruncatetheseriesoftheFourierexpansionofthewave
The pseudoscalar meson composed of a heavy quark
function u (r) as
0 andalightantiquarkQq¯(Qcanbeborcquark,q stands
foru,dorsquark)canbewritteninthemesonrestframe
N
nπ
as follows
u (r)= c sin r , (16)
0 n
L
nX=1 (cid:16) (cid:17) P(p~=0) = 1 d3kΨ (k) 1 [bi+(~k, )di+( ~k, )
where N is a large integer. Substitute this truncated | i √3 0 √2 Q ↑ q − ↓
i Z
X
expansion into the wave equation (15) and simplify it, bi+(~k, )di+( ~k, )]0 , (23)
one can finally get the equation about c − Q ↓ q − ↑ | i
n
where i is the color index. The factor 1/√3 is the nor-
nπ¯hc 2 nπ¯hc 2 malizationfactorforcolorindices,and1/√2thenormal-
+m2+ +m2 c
s L 1 s L 2  n ization factor for spin indices.
(cid:18) (cid:19) (cid:18) (cid:19)
The normalization of the meson state is
 N 2 L nπ mπ 
+mX=1LZ0 drV(r)sin(cid:16) L r(cid:17)sin(cid:16) L r(cid:17)cm hP(p~1)|P(p~2)i=(2π)32Eδ3(p~1−p~2), (24)
=Ec . (17) where E is the energy of the meson.
n
4
Substituting Eq.(23) into Eq.(24), we can finally get The running coupling constant α (Q2) in momentum
s
the normalization condition of the wave function in mo- space with N quark flavorsat large values of Q2, calcu-
f
mentum space lated in lowest-order QCD, is
12π
α (Q2)= . (29)
d3k|Ψ0(k)|2 =(2π)32E. (25) s (33−2Nf)ln(Q2/Λ2)
Z
This behavior of the strong coupling can be parameter-
The decayconstantofapseudoscalaris definedbythe ized in a simpler form which can be conveniently trans-
hadron-to-vacuum matrix element of the axial current formed into the r-space [11]
0q¯γ γ QP(p) =if p . (26) α (Q2)= α e−Q2/4γi2, (30)
µ 5 P µ s i
h | | i
i
X
Substituting the meson state of Eq.(23) into the above
where α are free parameters chosen to fit the behav-
i
equation in the rest frame, and contracting the quark ior of α (Q2) given by perturbative QCD (Eq.(29)). As
s
(antiquark) creation operators in the meson state with Q2 Λ2,thecouplingα (Q2)diverges,whichisbelieved
s
quark (antiquark ) annihilation operators in the quark tob→easignalofconfinement. However,asQ2 Λ2,per-
field of the axial current, we can get the expression of turbative QCD can not apply, the behavior of→α (Q2) at
s
the pseudoscalar decay constant smallQ2 giveninEq.(29)cannotbe the exactprediction
of QCD. One can make other choice for the behavior of
∞
3 1 1
f = dk~k 2Ψ (k) the strong coupling at small momentum transfer. As in
P r22π2mP Z0 | | 0 Ref. [11], we assume that the coupling αs saturate as a
(Eq +mq)(EQ+mQ) ~k 2 critical value αcsritical, where αcsritical = iαi. In prac-
−| | , (27)
tice,onlyseveralα areneededtobenon-zero,whichcan
× EqEQ(Eq +mq)(EQ+mQ) fitthebehaviorofαi (Q2)wellatperturbPativeregion,de-
s
p viation only occurs at small Q2.
where E and E are the energy of the heavy and light
Q q
The transformation of α (Q2) by using Eq. (30) in-
quarks. To be consistent with the wave equation, here s
stead of Eq. (29) is [11]
bothheavyandlightquarksaretakentobeon-shell. We
apsrsouxmimeatthelaytdtehsecrdibeecdayins toefrtmhseohfeoanv-yshmellesvoanlenccaenqbuearakps-, αs(r)= αi 2 γire−x2dx. (31)
√π
although the sum of the four-momenta of the valence Xi Z0
quarksare not equalto that of the mesonbecause of the Fig.1 is the behavior of α (r) with the parameters α =
s 1
existence of the color field within the hadron which can 0.15, α = 0.15, α = 0.20, and γ = 1/2, γ = √10/2,
2 3 1 2
carry both energy and momentum. γ = √1000/2, which is relevant to the critical value
2
We would like to mention that the leptonic decay of αcritical =0.5.
s
pseudoscalar mesons was considered several decades ago
0.8
inadifferentmethodby assumingthe couplingofmeson
)
with quark-antiquark pair [25]. (r 0.7
(cid:11)s 0.6
0.5
0.4
IV The QCD-inspired potential
0.3
0.2
The potential of strong interaction between the heavy
0.1
quark and light antiquark is taken as a combination of
0.5 1 1.5 2
a Coulomb term and a linear confining term inspired by
QCD [11, 18] r (fm)
4α (r) FIG. 1: The behavior of αs(r), with the parameters α1 =
V(r)= s +br+c. (28) 0.15, α2 = 0.15, α3 = 0.20, and γ1 = 1/2, γ2 = √10/2,
−3 r γ2=√1000/2.
The first term is the Coulomb term, which is consistent
with one-gluon-exchange contribution for short distance
calculated in perturbative QCD. The second term is the
linear-confinementterm, whichgeneratesconfinement in V Numerical result and discussion
longdistance. Thethirdtermisaphenomenologicalcon-
stant,whichisneededtoreproducethecorrectmassesfor The parameters are selected by comparing the pre-
heavy-light meson system. dicted heavy meson mass with the experimental data.
5
The recently measured values of the decay constants of At present the branching ratio of Br(B+ τ+ν ) has
τ
→
D and D mesons can give a further constraint on the been measured in experiment. The results still suffer
s
parameters. The parameters which we finally obtain are from large uncertainties. The measured value of Belle
collaboration is Br(B+ τ+ν ) = (1.79+0.56+0.46)
b=0.10GeV2, c= 0.19GeV2, 10−4 [28], while the value→s of BAτBAR colla−b0o.4r9a−ti0o.5n1ar×e
−
mb =4.98GeV, mc =1.54GeV, (B1r.8(B+0+.9 →0τ.4+ντ)0.=2)(0.910±−40.6[3±0].0.1T)h×e1c0o−m4b[i2n9e]darned-
ms =0.30GeV, mu =md =0.08GeV, sult−o0f.8B±ABAR± collab×oration is Br(B+ τ+ν ) =
αcsritical =0.5, (32) (1.2 0.4 0.3 0.2) 10−4 [31]. Consider→ing theτlarge
± ± ± ×
uncertainties of the experimental results, our predictied
and L=10 fm, N =100.
branching ratio of the decay mode B+ τ+ν is cosis-
The masses and decay constants of the B, Bs, D and → τ
tent with the experimental data.
D mesons calculated with the above parameters will
s
A super B factory will come into operation with the
be given in the following. The masses are given in Ta-
designed peak luminosity in excess of 1036 cm−2S−1 at
ble I. Here we do not consider the contribution of spin-
the Υ(4s) resonance in the next half decades [32]. The
dependent interactions in our calculation, it may give
integrated luminosity of 75 ab−1 would be collected in
errors about 100 200MeV for the masses. Varying the
∼ five years of data taking. Then the branching ratios of
parameters may also give errorsto the numerical values.
B+ τ+ν and µ+ν can be measured at the SuperB
We estimate the combination of both the errors can be → τ µ
factory with precisions of up to 4% and 5% respectively
about 7% for B and B mesons, and 10% for D and D
s s
[32]. Taking the value of the CKM matrix element V
mesons. | ub|
as input, the decay constant of f can be obtained at
B
superB.
TABLE I: Masses of pseudoscalar heavy mesons calculated The decay constants are also compared with previous
by solving the wave equation, and the comparison with ex-
theoretical results in Table III. Our predictions for the
perimental data. The data is quoted from the Particle Data
decay constants are quite different from previous results
Group [26].
calculatedintherelativisticpotentialmodel. Forf and
mB mBs mD mDs f ,ourresultsarelargerthanthatinRef[14],whiDleour
thiswork Ds
5.25 0.37 5.34 0.37 1.86 0.19 1.96 0.20 results for f and f are smaller than theirs. [45]
(GeV) ± ± ± ± B Bs
Exp. 5279.17 5366.3 1869.6 1968.47
(MeV) 0.29 0.6 0.16 0.33
± ± ± ± TABLE II: Decay constants of pseudoscalar heavy mesons
calculatedinthiswork,andthecomparisonwithexperimental
data. Allvalues are in unitsof MeV.
The decay constants obtained are
fB fBs fD fDs
fB =198±14MeV, fBs =237±17MeV, thiswork 198±14 237±17 208±21 256±26
f =208 21MeV, f =256 26MeV. (33) Exp. [19, 27] 205.8 8.5 2.5 254.6 5.9
D ± Ds ± − − ± ± ±
The comparison of the decay constants obtained in this
work with experimental data are given in Table II. Both Thereducedwavefunctionsincoordinateandmomen-
the masses and decay constants obtained in this work tum spaces (Eqs. (8) and (20) are depicted in Figs. 2
canbe well consistentwith experiment. For f andf , and 3. The wave function squared u (r)2 is the possi-
B Bs | l |
there are still no precise measured values in experiments bility density distributed along the quark-antiquark dis-
yet. Our prediction can be tested in experiment in the tance r. The curvesin Fig. 2 show that the most proba-
future. ble distribution of the quarks occurs at the distance 0.4
The leptonic decay rates of B meson relevant to the fm between the quark and antiquark in both B and D
decay constant f obtained in this work are mesons. The possibility density vanishes as the distance
B
larger than 2 fm. The mean square root of the distance
Br(B+ e+νe)=(1.11 0.26) 10−11, (34) is about 0.5 0.7 fm.
Br(B+ →µ+ν )=(4.7 ±1.1) ×10−7, (35) Thenumer∼icalsolutionofthewavefunctioninmomen-
µ
→ ± ×
Br(B+ τ+ν )=(1.1 0.2) 10−4, (36) tum space is given in Fig.3, which shows that the peak
τ
→ ± × ofthemomentum-distributionofthequarksintheheavy
where the errors are mainly caused by the uncertainties meson is at about 0.4 GeV. The reduced wave function
of the decay constant f and the CKM matrix element can be fitted with the analytical form as suggested in
B
V . The value of V is quoted from PDG [26] Ref.[15]
ub ub
| |
V =(3.93 0.36) 10−3. ϕ (k)=4π m α3ke−αk, (37)
ub 0 H
| | ± ×
p
6
25
TABLEIII:Thecomparisonofthedecayconstantscalculated
in this work with other theoretical results. All values are in ) 20
r
unitsof MeV. (0
u 15
fB fBs fD fDs ForB andBs
this work 198 14 237 17 208 21 256 26 10
Ref.[14]a 230±35 245±37 182±27 199±30
Ref.[33]b 203±23 236±30 205±20 235±24 5
± ± ± ±
Ref.[34]b 206 20 195 20
± − ± − 0 1 2 3 4 5
200 221 r (fm)
Ref.[35]c − − 230 270 12 (a)
Ref.[36]d 177 21 205 22 10
RReeff..[[3378]]ee 1961−9329 2161−9532 23023±825 24824±127 ur()0 8
Ref.[39]f 18±9 21±8 23±4 26±8 6 ForD andDs
Ref.[40]g 210±11.4 4
5.7 − − −
± 2
201 3 249 3
Ref.[41]h ± ±
− − 17 16
Ref.[42]h 195 11 243 11 207±11 249±11 0 1 (b) 2 3 4 r (fm5)
± ± ± ±
Ref.[43]h 207 4 241 3
− − ± ±
Ref.[44]h 248.0 2.5
− − − ± FIG.2: Thereducedwavefunctionu0(r)incoordinatespace.
(a) is for B and Bs mesons. The solid curve is for the wave
a Relativistic potential model. function of B meson, the dashed one is for Bs. (b) is for D
b QCD sum rule. andDs mesons. ThesolidcurveisforD,andthedashedone
c Light front quark model. for Ds.
d Finite energy sum rules.
e Quark model based on Bethe-Salpeter equation.
f Relativistic constituent Quark Model.
g Derived from result of Lattice QCD.
wave functions obtained here can be useful for studying
h Lattice QCD
heavy meson decays.
where m is the heavy meson mass. The factor
H Acknowledgments
4π√m α3 isthenormalizationfactorduetothenormal-
H
ization condition in Eq.(25). Note that the wave func-
This work is supported in part by the National Nat-
tion for B and/or D meson is Ψ (k) = ϕ (k)/k. Our
0 0
numerical solution gives α = 3.0 GeV−1, 2.6 GeV−1, ural Science Foundation of China under contracts Nos.
3.4GeV−1,and3.2GeV−1 forB, B , D andD mesons, 10575108, 10975077, 10735080, and by the Fundamen-
s s
tal Research Funds for the Central Universities No.
respectively.
65030021.
With the constraint of the measured values of f and
D
f considered, the wave functions obtained here can
Ds
be more reliable than before, which should be useful in
studying the decays of the B and D mesons. The appli-
cationofthewavefunctionsinstudyingtheheavymeson ∗ Electronic address: [email protected]
decays deserves a separate work. [1] M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda,
Phys. Rev. Lett. 83 (1999) 1914, hep-ph/9905312; M.
Beneke,G.Buchalla, M.Neubert,C.T.Sachrajda,Nucl.
Phys.B591(2000) 313, hep-ph/0006124;M.Beneke,G.
VI Summary
Buchalla,M.Neubert,C.T.Sachrajda,Nucl.Phys.B606
(2001) 245, hep-ph/0104110.
The wave functions and decay constants of B and D [2] H.N.Li,Phys.Rev.D52, 3958 (1995);H.N. Li,H.L.Yu,
mesons are revisited in the relativistic potential model. Phys. Rev. Lett. 74, 4388 (1995); C.H. Chang, H.N. Li,
The parameters in the potential model are further con- Phys.Rev.D 55,5577 (1997); T.W. Yeh,H.N.Li,Phys.
Rev.D56,1615(1997);Y.Y.Keum,H.-n.Li,A.I.Sanda,
strained with the experimental values of f and f
D Ds Phys.Lett.B504,6(2001);C.D.Lu¨,K.Ukai,M.Z.Yang,
available. The masses and decay constants of the heavy
Phys. Rev.D 63, 074009 (2001).
mesons are obtained, which can be well consistent with
[3] A. G. Grozin and M. Neubert, Phys. Rev. D 55, 272
the current experimental data. The wave functions both (1997); M. Beneke, T. Feldmann, Nucl. Phys. B592, 3
in coordinate and momentum spaces are obtained. The (2001); B. O. Lange and M. Neubert, Phys. Rev. Lett.
7
20
[17] Forareviewsee: M.Neubert,Int.J.Mod.Phys.A11,4173
(1996).
)
k 15
( [18] E. Eichten, K. Gottdried, T. Kinoshita, K.D. Lane, and
0
ϕ
T.-M.Yan,Phys.Rev.D17,3090(1978)[Erratum: Phys.
10 ForB andBs Rev. D21, 313 (1980); Phys. Rev.D21, 203 (1980).
[19] B.I.Eisensteinet al.(CLEOCollaboration), Phys.Rev.
5 D78, 052003 (2008).
[20] K. M. Ecklund et al. (CLEO Collaboration), Phys. Rev.
Lett. 100, 161801 (2008).
0 1 2 3 4 5 6
k (GeV) [21] J.P.Alexanderetal.(CLEOCollaboration), Phys.Rev.
(a)
14 D79,052001(2009);P.U.E.Onyisietal.(CLEOCollab-
) 12 oration), Phys. Rev. D79, 052002 (2009); P. Naik et al.
k
(010 (CLEO Collaboration), Phys.Rev. D80, 112004 (2009).
ϕ [22] K.Abeetal.(Bellecollaboration), Phys.Rev.Lett.100,
8 ForD andDs 241801 (2008).
6 [23] P. del Amo Sanchez et al. (BABAR Collaboration),
4 arXiv:1003.3063; Phys. Rev.D82, 091103 (2010).
2 [24] J. L. Richardson, Phys.Lett. B82, 272 (1979).
[25] S.S. Gershtein, M.Yu. Khlopov, JETP Lett. 23, 338
0 1 2 3 4 5 6
k (GeV) (1976).
(b)
[26] K.Nakamuraetal.(ParticleDataGroup),J.Phys.G37,
075021 (2010).
FIG. 3: The reduced wave function ϕ0(k) in momentum [27] D. Asner et al. (Heavy Flavor Averaging Group), eprint
space. (a) is for B and Bs mesons. The solid curve is for arXiv:1010.1589.
B meson, the dashed one is for Bs. (b) is for D and Ds [28] K. Ikado et al. (Belle collaboration), Phys. Rev. Lett.
mesons. Thesolid curveisforD, andthedashedoneforDs. 97,251802 (2006).
[29] B. Aubert (BABAR Collaboration), Phys. Rev. D76,
052002 (2007).
[30] B. Aubert (BABAR Collaboration), Phys. Rev. D 77,
91, 102001 (2003); S.J. Lee and M. Neubert, Phys. Rev. 011107 (2008);eprintarXiv: 0912.2453; Phys.Rev.D81,
D72, 094028 (2005). 051101 (2010).
[4] H. Kawamura, J. Kodaira, C. F. Qiao and K. Tanaka, [31] A. J. Schwartz, eprint arXiv: 0909.4473.
Phys. Lett. B 523, 111 (2001) [Erratum-ibid. B536, 344 [32] M. Bona et al., eprint arXiv:0709.0415.
(2002)]; Mod. Phys. Lett. A18,799 (2003); Nucl. Phys. [33] S. Narison, hep-ph/0202200 (2002); S. Narison, Phys.
B (Proc. Suppl.) 116, 269 (2003); T. Huang, X.G. Wu, Lett. B 520 (2001) 115.
M.Z. Zhou, Phys. Lett. B611, 260 (2005); T. Huang, C. [34] A.A.PeninandM.Steinhauser,Phys.Rev.D65,054006
F. Qiao and X.G. Wu,Phys.Rev.D73, 074004 (2006). (2002), hep-ph/0108110.
[5] S. Descotes-Genon, C.T. Sachrajda, Nucl. Phys. B625, [35] Z.T. Wei, H.W. Ke, X.F. Yang, Phys.Rev. D80, 015022
239 (2002). (2009).
[6] V.M.Braun,D.Y.IvanovandG.P.Korchemsky,Phys. [36] J.Bordes,J.A.Pen˜arrocha,K.Schilcher,JHEP,0511,014
Rev.D 69, 034014 (2004). (2005), hep-ph/0507241.
[7] H.N.Li and H.S. Liao, Phys.Rev. D70, 074030 (2004). [37] Z.G.Wang,W.M.Yang,S.L.Wan,Nucl.Phys.A744,156
[8] B.GeyerandO.Witzel,Phys.Rev.D72,034023 (2005). (2004).
[9] C.W. Hwang, Phys.Rev.D81, 114024 (2010). [38] G. Cvetiˇc, C.S. Kim, G.L. Wang, W. Namgung, Phys.
[10] A. De Ru´jula, H. Georgi and S.L. Glashow, Phys. Rev. Lett, B596, 84 (2004).
D12, 147 (1975). [39] D. Ebert, R.N. Faustov, V.O. Galkin, Phys. Lett. B635,
[11] S.Godfrey and N.Isgur, Phys. Rev.D32, 189 (1985). 93 (2006).
[12] P. Cea, P. Colangelo, G. Nardulli, G. Paiano, and G. [40] X.-H.Guo, M.-H. Weng,Eur.Phys.J.C50,63 (2007).
Preparata,Phys.Rev.D26,1157(1982);P.Cea,G.Nar- [41] C. Aubin et al. (Fermilab Lattice, MILC, and HPQCD
dulli, and G. Paiano, Phys.Rev.D28, 2291 (1983). Collaborations), Phys. Rev.Lett. 95, 122002 (2005).
[13] P. Cea, P. Colangelo, L. Cosmai and G. Nardulli, Phys. [42] C. Bernard et al. (Fermilab Lattice and MILC Collabo-
Lett.B206, 691 (1988). rations), PoS LATTICE2008, 278 (2008), eprint arXiv:
[14] P. Colangelo, G. Nardulli, M. Pietroni, Phys. Rev. D43, 0904.1895.
3002 (1991). [43] E. Follana, C. T. H. Davies, G. P. Lepage, and J.
[15] P. Colangelo, F. De Fazio, M. Ladisa, G. Nardulli, P. Shigemitsu, Phys.Rev.Lett. 100, 062002 (2008).
Santorelli, A.Tricarico, Eur. Phys.J. C8, 81 (1999). [44] C. T. H. Davies et al. (HPQCD Collaboration), Phys.
[16] N.Isgurand M.B. Wise, Phys.Lett.B 232, 113 (1989); Rev. D82, 114504 (2010).
237, 527 (1990); E. Eichten and B. Hill, Phys. Lett. B [45] The definition of thedecay constant in Ref.[14] is differ-
234, 511 (1990); 243, 427 (1990); B. Grinstein, Nucl. ent from that in the current work by a factor √2. The
Phys. B339 253 (1990); H. Georgi, Phys. Lett. B 240, quoted results of Ref. [14] in Table III have been com-
447 (1990); A. F. Falk, B. Grinstein, and M. E. Luke, pensated by thisfactor.
Nucl.Phys. B357, 185 (1991).