Electromagnetic Annihilation Rates of $χ_{c0}$ and $χ_{c2}$ with Both Relativistic and QCD Radiative Corrections

19 Pages·0.17 MB·English

by  Han-Wen Huang

PUTP-95-28
Electromagnetic Annihilation Rates of χ and χ with
c0 c2
Both Relativistic and QCD Radiative Corrections
6
9
9
Han-Wen Huang1,2 Cong-Feng Qiao1,2 Kuang-Ta Chao1,2
1

n 1 CCAST (World Laboratory), Beijing 100080, P.R.China
a
J
3 Department of Physics, Peking University, Beijing 100871, P.R.China
1
3

1
v
Abstract
0
8
3 We estimate the electromagnetic decay rates of χ γγ and χ γγ by taking into
c0 c2
1 → →
0 account both relativistic and QCD radiative corrections. The decay rates are derived in the
6
9 Bethe-Salpeter formalism and the QCD radiative corrections are included in accordance with
/
h factorizationassumption. Using a QCD-inspired interquarkpotential, we obtain relativistic BS
p
- wavefunctionsofχ andχ bysolvingBSequationsforthecorresponding2S+1L states. Our
p c0 c2 J
he numerical result for the ratio R = ΓΓ((χχcc02→→γγγγ)) is about 11−13 which agrees with the update
:
v
E760experimentaldata. Explicitcalculationsshowthatthe relativisticcorrectionsduetospin-
i
X
dependent interquark forces induced by gluon exchange enhance the ratio R substantially and
r
a
its value is insensitive to the choice of parameters that characterize the interquark potential.
Our expressionsfor the decay widths are identical with that obtained in the NRQCD theory to
the next-to-leading order in v2 and α . Moreover, we have determined two new coefficents in
s
the nonperturbative matrix elements for these decay widths.
1
1 Introduction
Charmonium physics is in the boundary domain between perturbative and non-perturbative
QCD. Charmonium decays may provide useful information on understanding the nature of in-
terquark forces and decay mechanisms. Both QCD radiative corrections and relativistic correc-
tions are important for charmonium decays, because for charmonium the strong coupling constant
α 0.3 [defined in the MS scheme (the modified minimal subtraction scheme)] and the velocity
s
≈
squared of the quark in the meson rest frame v2 0.3, both are not small. Decay rates of heavy
≈
quarkonium in the nonrelativistic limit with QCD radiative corrections have been studied (see,
e.g.,ref.[1, 2, 3, 4]). However, the decay rates of many processes are subject to substantial relativis-
tic corrections [4]. With this goal in mind, people have studied relativistic corrections to the decay
rates of S-wave charmonium η , J/ψ and their radial excited states [5, 6, 7]. These results show
c
that relativistic effects are significant in the cc¯systems especially for the hadronic decays of J/ψ.
In the present paper, we will investigate the relativistic corrections to the electromagnetic decays
of P-wave charmonium states χ γγ and χ γγ.
c0 c2
→ →
The P-wave charmonium decays are interesting. Now their experimental results are quite
uncertain. The Crystal Ball group (see [8, 16] and references therein) gives Γ(χ γγ) = 4.0
c0
→ ±
2.8keV. But for Γ(χ γγ), its central value differs significantly among various experiments
c2
→
[9, 10, 11], and the ratio of the photonic widths for χ andχ states measured by E760 is much
c0 c2
larger than that measured by other two groups. Theoretically, in the nonrelativistic limit, the ratio
is 15 [1], and it will increase to about 7.4 [8] if QCD radiative corrections are considered. Recently
4
a rigorous factorization formula which is based on NRQCD has been developed for calculations of
inclusivedecayratesofheavyquarkonium. Inthisapproachthedecaywidthsfactorintoasetoflong
distance matrix elements of NRQCD with each multiplied by a short distance coefficient. To any
given order of relative velocity v of heavy quark and antiquark, the decay rates are determined by
several nonperturbativefactors which can be evaluated usingQCD lattice calculations or extracted
byfittingthedata. Thestudyofthephotonicdecays ofχ andχ canalsopovideadetermination
c0 c2
for the nonperturbative factors in the decays of P-wave quarkonium.
In this paper, we will use the Bethe-Salpeter (BS) formalism [12] to derive the decay am-
plitudes and to calculate the decay widths of χ γγ and χ γγ. The meson will be treated
c0 c2
→ →
as a bound state consist of a pair of constitutent quark and antiquark (higher Fock states such as
1
QQ¯g > and QQ¯gg > are neglected because they don’t contribute to electromagnetic decays) and
| |
described by BS wavefunction which satisfies the BS equation. A phenomenological QCD-inspired
interquark potential will be used to solve for the wavefunction and to calculate the decay widths.
Bothrelativistic andQCDradiativecorrectionstonext-to-leading orderwillbeconsideredbasedon
the factorization assumption for the long distance and short distance effects. The remainer of this
paper is organized as follows. In Sec.2 we derive the reduced BS equation for any angular momen-
tumstate 2S+1L ofheavy mesons. InSec.3wegive outthedecay amplitudesofχ γγ(J = 0,2)
J J
→
and use the solved relativistic BS wavefunctions to calculate the numerical results of decay widths.
A summary and discussion will be given in the last section.
2 Reduced BS equations for any angular-momentum state 2S+1L
J
of heavy mesons
Define the Bethe-Salpeter wavefunction, in general, for a Q Q¯ bound state P > with overall
1 2
|
mass M and momentum P = ( P~2+M2,P~)
q
χ(x ,x )=< 0Tψ (x )ψ¯ (x )P >, (1)
1 2 1 1 2 2
| |
and transform it into momentum space
χ (p)= e−iP·X d4xe−ip·xχ(x ,x ). (2)
P Z 1 2
Here p (m ) and p (m ) represent the momenta(masses) of quark and antiquark respectively,
1 1 2 2
X = η x +η x , x= x x ,
1 1 2 2 1 2
−
P = p +p , p = η p η p ,
1 2 2 1 1 2
−
where η = mi (i= 1,2).
i m1+m2
We begin with the bound state BS equation [12] in momentum space
i
(p/ m )χ (p)(p/ +m ) = d4kG(P,p k)χ (k), (3)
1− 1 P 2 2 2π Z − P
where G(P,p k) is the interaction kernel which dominates the interquark dynamics. In solving
−
(3), we will employ the instantaneous approximation since for heavy quarks the interaction is
2
dominated by instantaneous potentials. Meanwile, we will neglect negative energy projectors in
thequarkpropagatorswhichareofevenhigherorders. DefiningthreedimensionalBSwavefunction
Φ (p~)= dp0χ (p),
P Z P
we then get the reduced Salpeter equation for Φ (p~)
P
(M −E1−E2)Φ(p~)= Λ1+γ0Z d3kG(P,p~−~k)Φ(~k)γ0Λ2−. (4)
Here G(P,p~ ~k) represents the instantaneous potential, Λ+(Λ−) are the positive (negative) energy
−
projector operators for quark and antiquark respectively
E +γ ~γ p~ +mγ
Λ1 = 1 0 · 1 0
+ 2E
1
E γ ~γ p~ mγ
Λ2− = 2− 0 · 2− 0
2E
2
E = p~ 2+m2, E = p~ 2+m2.
1 q 1 1 2 q 2 2
We will follow a phenomenological approach by using QCD inspired inter-quark potentials, which
are supported by both lattice QCD and heavy quark phenomenology, as the kernel in the BS
equation. The potentials include a long-ranged confinement potential (Lorentz scalar) and a short-
ranged one-gluon exchange potential (Lorentz vector)
V(r) = V (r)+γ γµV (r),
S µ V
⊗
(1 e−αr)
V (r) = λr − ,
S
αr
4α (r)
V (r) = s e−αr, (5)
V
− 3r
where the introduction of the factor e−αr is to regulate the infrared divergence and also to in-
corporate the color screening effects of dynamical light quark pairs on the QQ¯ linear confinement
potential. In momentum space the potentials become
G(p~) = G (p~)+γ γµG (p~),
S µ V
⊗
λ λ 1
G (p~) = δ3(p~)+ ,
S −α π2(p~2+α2)2
2 α (p~)
s
G (p~) = , (6)
V −3π2p~2+α2
3
where α (p~) is the quark-gluon running coupling constant and is assumed to become a constant of
s
O(1) as p~2 0
→
12π 1
α (p~)= .
s 27 ln(a+p~2/Λ2 )
QCD
The constants λ,α,a and Λ are the parameters that characterize the potential.
QCD
For any given angular-momentum state 2S+1L of mesons, its three dimensional wavefunc-
J
tion in the rest frame of mesons takes the following two forms:
(i)S=0, then J=L,
ΦLm(p~) = Λ1+γ0(1+γ0)γ5γ0Λ2−YLm(pˆ)φ(p); (7)
(ii)S=1, then J=L-1,L,L+1 for L=0 or J=1 for L=0,
6
ΦJM(p~)= X < JM|1lLm > Λ1+γ0(1+γ0)γlΛ2−γ0YLm(pˆ)φ(p) (8)
l,m
where Y (pˆ) is the spherical harmonic function and < JM 1lLm > is the Clebsch–Gordan coeffi-
Lm
|
cient. Substituting Eq.(7) and (8) in Eq.(4), one derives the equations for the scalar wavefunction
φ(p)
(i)S=0
(M E (p) E (p))g (p)φ(p)
1 2 1
− −
E (p)E (p)+m m +~p2
= 1 2 1 2 d3k(G (p~ ~k) 4G (p~ ~k))g (k)P (cosΘ)φ(k)
− 4E (p)E (p) Z S − − V − 1 L
1 2
E (p)m +E (p)m
1 2 2 1 d3k(G (p~ ~k)+2G (p~ ~k))g (k)P (cosΘ)φ(k)
− 4E (p)E (p) Z s − V − 2 L
1 2
E (p)+E (p)
+ 1 2 d3kG (p~ ~k)p~ ~kg (k)P (cosΘ)φ(k)
4E (p)E (p) Z S − · 3 L
1 2
m m
+ 1− 2 d3k(G (p~ ~k) 2G (p~ ~k))p~ ~kg (k)P (cosΘ)φ(k) (9)
4E (p)E (p) Z S − − V − · 4 L
1 2
where
(E (p)+m )(E (p)+m )+p~2
1 1 2 2
g (p) =
1
4E (p)E (p)
1 2
(E (p)+m )(E (p)+m ) p~2
1 1 2 2
g (p) = −
2
4E (p)E (p)
1 2
4
E (p)+m +E (p)+m
1 1 2 2
g (p) =
3
4E (p)E (p)
1 2
E (p)+m E (p) m
1 1 2 2
g (p) = − −
4
4E (p)E (p)
1 2
E (p) = p~2+m2
1 q 1
E (p) = p~2+m2
2 q 2
(ii)S=1
(M E (p) E (p))f (p)φ(p)
1 2 8
− −
1
= d3k[(2G (p~ ~k) G (p~ ~k))f (k)(m +m )
4E (p)E (p){Z V − − S − 1 1 2
1 2
G (p~ ~k)f (k)(E (p)+E (p)]P (cosΘ)φ(k)
s 2 1 2 L
− −
+ [ d3k(4G (p~ ~k)+G (p~ ~k))f (k)(E (p)E (p) m m +p~2)
Z V − S − 8 1 2 − 1 2
k
+ (G (p~ ~k) 2G (p~ ~k))f (k)(m E m E )]P (cosΘ) φ(k)
S V 7 1 2 2 1 J
− − − − p
+ d3k[(2G (p~ ~k) G (p~ ~k))f (k)(m +m )
Z V − − S − 5 1 2
k
G (p~ ~k)f (k)(E +E )]p~ ~kP (cosΘ) φ(k) , (10)
S 6 1 2 J
− − · p }
where
1
f (p) = ((E (p)+M )(E (p)+M )+p~2)
1 1 1 2 2
4E (p)E (p)
1 2
1
f (p) = ((E (p)+m )(E (p)+m ) p~2)
2 1 1 2 2
4E (p)E (p) −
1 2
2(E (p)+m )
1 1
f (p) = f (p) =
3 4
4E (p)E (p)
1 2
2
f (p) = f (p) =
5 6
− −4E (p)E (p)
1 2
1
f (p) = (E (p)+m E (p) m )
7 1 1 2 2
4E (p)E (p) − −
1 2
1
f (p) = (E (p)+m +E (p)+m ).
8 1 1 2 2
4E (p)E (p)
1 2
5
The normalization condition d3pTr Φ+(p~)Φ(p~) = 2M for the BS wavefunction φ(p) leads to
{ } (2π)3
R
(E (p)+m )(E (p)+m ) 2M
dpp3 1 1 2 2 φ2(p) = . (11)
Z 4E E (4π)3
1 2
Totheleadingorderinthenonrelativisticlimit, Eqs.(7)and(8)arejusttheordinarynonrelativistic
Schro¨dinger equation for orbital angular momentum L with simply a spin-independent linear plus
Coulomb potential. Solving equation (9) or (10), we can get the spectra and wavefunctions for
any given angular-momentum state 2S+1L of heavy mesons. With these wavefunctions we can
J
calculate hadronic matrix elements of the processes involving corresponding states, and the rela-
tivistic corrections due to interquark dynamics are included automatically in them. This approach
is different from convential ones which start from Schro¨dinger equation with all relativistic effects
considered perturbatively.
3 Decay rates of Γ(χ γγ) and Γ(χ γγ)
c0 c2
→ →
Electromagnetic decays of χ and χ proceed via the annihilation of cc¯ to two photons.
c0 c2
Hereonlyelectromagnetic interactionsareconsidered,andcolor-octetcomponentswhichcontribute
dominantly in hadronic decays of P-wave quarkonium do not contribute to electromagnetic decay
widths, because final states are the photons which can not be produced via the annihilation of
color-octet QQ¯ pair. So two photonic decays of χ for J = 0, 2 can be well expressed in the BS
cJ
formalism and relativistic corrections are incorporated systematicly in the decay rates. In the BS
formalism the annihilation matrix elements can be written as follows
< 0Q¯IQ P >= d4pTr[I(p,P)χ (p)], (12)
| | Z P
where I(p,P) is the interaction vertex of the QQ¯ with other fields (e.g., the photons or gluons)
which, in general. may also depend on the variable q0 (the time-component of the relative momen-
tum). If I(p,P) is independent of q0 (e.g., if quarks are on their mass-shells in the annihilation),
the equation can be written as
< 0Q¯IQ P >= d3pTr[I(p~,P)Φ (p~)], (13)
| | Z P
For process χ γγ or χ γγ with the momenta and polarizations of photons k ,ǫ
c0 c2 1 1
→ →
6
and k ,ǫ , the decay amplitude can be written as
2 2
T =< 0c¯Γ cχ > ǫµǫν (14)
| µν | cJ 1 2
for J=0,2, where p (p ) is the charm quark(antiquark) momentum, and
1 2
1 1
Γ = e2[γ γ +γ γ ]
µν ν µ µ ν
/p /k m /k /p m
1 1 1 2
− − − −
Since p0+p0 = M, as usual we take
1 2
M
p0 = p0 = . (15)
1 2 2
Therefore, the amplitude T becomes independent of p0. In terms of T, the decay rates can be
written as
1
Γ(χ γγ)= T 2dΩ (16)
cJ → 2! X X Z | |
spinpolar
for J = 0,2, where the factor 1/2! is needed because N! same graphs appear for N-photon final
states. The photon polariztion is summed over in the Feynman gauge,
εµ(k )εν∗(k )= gµν,
1 1
X −
helicity
Substituting BS wavefunction (8) into (16), after summing over final states and averaging over
initial states, we get
Γ(χ γγ) = 24e4 α2(c +3c +2c )2 (17)
c0 Q 1 2 3
→
12e4 α2
Γ(χ γγ) = Q (c2 2c c +7c2) (18)
c2 → 5 1− 1 3 3
where
1 ~p2 3(p~ kˆ)2
c = d3p [ E2 mE + · ]p~ ~k
1 Z (p~ ~k)2+m2{− − − 2 2 ·
−
p~4 3p~2(p~ kˆ) 5(p~ kˆ)4 φ(p)
+[ + · · ]
− 4 2 − 4 } p
1 p~ ~k 1 φ(p)
c = d3p · [p~2 (p~ kˆ)2]+ [p~2 (p~ kˆ)2]2
2 Z (p~ ~k)2+m2{ 2 − · 4 − · } p
−
1 E2 mE 1 φ(p)
c = d3p − − [p~2 (p~ kˆ)2]+ [p~2 (p~ kˆ)2]2
3 Z (p~ ~k)2+m2{ 2 − · 4 − · } p
−
7
In the nonrelativistic limit, (17) and (18) reduce to
24e4 α2
Γ(χ γγ)= Q d3ppφ (p)2
c0 → m4 |Z χc0 |
32e4 α2
Γ(χ γγ)= Q d3ppφ (p)2
c2 → 5m4 |Z χc2 |
Using the Fourier transformation of wavefunctios
3
d3ppφ (p)= R′ (0)
Z χcJ √8 χcJ
we derive the well known results in coordinate space, which is consistent with that given in [1]
27e4 α2
Γ(χ γγ) = Q R′ (0)2 (19)
c0 → m4 | χc0 |
36e4α2
Γ(χ γγ) = Q R′ (0)2, (20)
c → 5m4 | χc2 |
′
whereR (0) is thederivative of radial wavefunction at theorigin, andin the nonrelativistic limit,
χcJ
′ ′
R (0) = R (0), due to the heavy quark spin symmetry.
χc0 χc2
Recently, in the framework of NRQCD the factorization formulas for the long distance and
short distance effects were found to involve a double expansion in the quark relative velocity v
and in the QCD coupling constant α [13, 14]. To next-to-leading order in both v2 and α , as an
s s
approximation, we may write
α π2 28
Γ(χ γγ) = 24e4 α2(c +3c +2c )2[1+ s( )] (21)
c0 → Q 1 2 3 π 3 − 9
12e4α2 α 16
Γ(χ γγ) = Q (c2 2c c +7c2)(1 s ) (22)
c2 → 5 1 − 1 3 3 − π 3
where we have used QCD radiative corrections given in [15]. We must emphasize that above
factorization formula are correct only to next-to-leading order in v2 and α . If higher order effects
s
are involved, the decay widths can not be factored into a integral of wavefunction and a coefficient
that can be written as a series of α . NRQCD has applied a more general factorization formula for
s
quarkonium decay rates, which will be discussed in detail later.
8
For the heavy quarkonium cc¯ systems, m = m = m , Eqs. (9) and (10) become much
1 2 c
simpler. We take the following parameters which appear in potential (5),
m = 1.5Gev, λ= 0.23Gev2, Λ =0.18Gev,
c QCD
α = 0.06Gev, a= e = 2.7183.
With these values the mass spectrum of charmonium are found to fit the data well. In Fig.1 and
Fig.2 the solved scalar wavefunctions both in momentum and coordinate space for P-wave triplet
χ states are shown and we can see explicitly the differences between wave functions for J = 0, 2
cJ
but they are same in the nonrelativistic limit. Substituting φ (p) and φ (p) into (21) and (22),
χc0 χc2
we get
Γ(χ γγ) = 5.32keV,
0
→
Γ(χ γγ) = 0.44keV
2
→
their ratio is
Γ(χ γγ)
0
R = → = 12.1. (23)
Γ(χ γγ)
2
→
Our results are satisfactory. as compared with the Particle Data Group experimental values [16]
Γ(χ γγ) = 5.6 3.2keV, and Γ(χ γγ) = 0.32 0.1keV. Here in above calculations the
c0 c2
→ ± → ±
value of α (m ) in the QCD radiative correction factor in (21) and (22) is chosen to be 0.29 [4],
s c
which is also consistent with our determination from the ratio of B(J/ψ 3g) to B(J/ψ e+e−)
→ →
[5] .
Moreover, in order to see the sensitivity of the decay widths to the parameters, especially
the charm quark mass, we use other two sets of parameters
m = 1.4Gev, λ= 0.24Gev2,
c
m = 1.6Gev, λ= 0.22Gev2,
c
with other parameters keeping unchanged (the heavy quarkonia mass spectra are not sensitive to
a, α for α 0.06GeV), By the same procedure, we obtain
≤
Γ(χ γγ) = 5.82(4.85)keV,
0
→
Γ(χ γγ) = 0.50(0.39)keV,
2
→
9