Finite size vertex corrections to the three-gluon decay widths of $J/ψ$ and $Υ$ and a redetermination of $α_s(μ)$ at $μ=m_c$ and $m_b$

12 Pages·0.11 MB·English

by  H. C. Chiang

Finite size vertex corrections to the
three-gluon decay widths of J/ψ and
4
9
9 Υ and a redetermination of α (µ)
1 s

n at µ = m and m
a c b
J

1
1 H.C. Chiang, J. Hu¨fner and H.J. Pirner†
∗ †

1 Institut fu¨r Theoret. Physik der Universit¨at Heidelberg
v
and
3
3
Max-Planck-Institut fu¨r Kernphysik, Heidelberg, Germany
2
1
0
4
9
/ Abstract
h
p
-
p We calculate the corrections to the three-gluon decay widths Γ of cc¯and b¯b
3g
e
h
quarkonia due to the finite extension of the QQ¯ 3g vertex function. The
:
v →
i
X widths computed with zero range vertex are reduced by a factor γ where γ =
r
a 0.31 for the J/ψ and γ = 0.69 for the Υ. These large corrections necessitate
a redetermination of the values α (µ) extracted from Γ . We find α (m ) =
s 3g s c
0.28 .01 and α (m ) = 0.20 .01.
s b
± ±
∗Permanent address: Institute of High Energy Physics, Academia Sinica, Beijing, China
†Supported in part by the Federal Ministry of Research and Technology under the contract
number 06 HD 729
1
Since the discoveries of the charmonium and bottonium states, the physics of
heavy quark systems has been of continuous interest [1, 2]. The bound state struc-
ture of heavy quarkonia becomes simpler with increasing mass, because the con-
stituents move more and more non-relativistically in a static potential which is
dominated for r < 0.1 fm by the Coulomb-like field and for r > 0.2 fm by the confin-
ing potential. The Schr¨odinger equation with this potential reproduces successfully
the energies of the bound states 1s,2p,.... The leptonic widths Γ(QQ¯ µµ) of
→
the 3S -states, modified by radiative corrections, can also be obtained by an expres-
1
sion which is proportional to ψ(0) 2, where ψ(~r) is the non-relativistic bound state
| |
wave function. The value ψ(0) 2 is rather sensitive to the potential [3, 4]. Prob-
| |
lems exist with the gluonic width Γ(QQ¯ 3g) and with the photo-gluonic width
→
Γ(QQ¯ γgg). One usually calculates these quantities in the approximation of a
→
zero-range vertex, i.e. Γ(QQ¯ 3g) ψ(0) 2 and analogous for Γ(QQ¯ γ2g).
→ ∝ | | →
From a comparison with experiment one extracts a value for the strong interaction
coupling constant α (µ), where µ is the scale (µ = m for J/ψ and µ = m for Υ)
s c b
[5, 7]. The analysis usually proceeds in three steps. In the first step, expressions for
the ratios R and R
1 2
Γ(QQ¯ γgg)
R = →
1 Γ(QQ¯ ggg)
→
Γ(QQ¯ 3g)
R = → (1)
2 ¯
Γ(QQ µµ)
→
are derived in leading order α . In this order and using the zero- range approxima-
s
tion for the vertex, the decay rates are obtained from analogous calculations for the
three-photon decay of the triplet state of positronium with the photon-electron cou-
pling replaced by the SU3 coupling of gluons to quarks. Because of the zero-range
c
approximation, the rate is proportional to ψ(0) 2, the square of the non-relativistic
| |
wavefunction, whichcancelsoutintheratiosR andR . Inthesecondstep,thelead-
1 2
2
ing order result for R and R is multiplied by the first order α corrections, which
1 2 s
represent a factor of the form (1+B(µ)α (µ)/π). The coefficient B(µ) depends both
s
onthescaleµandtherenormalizationscheme. Inathirdstep, oneusuallytakesinto
account corrections of order ~q2/m2 phenomenologically [5]. These corrections may
h i
come from the finite extension of the decay vertex and from relativistic corrections
in the wavefunction or from other nonperturbative physics like condensates. In this
paper we give the first ab initio calculation of the effect from the finite extension of
the decay vertex, which we take into account by a calculated factor γ, where γ = 1
represents the zero range situation. The status of theoretical calculations for R
1
and R is given in Table 1 for the J/ψ and Υ together with the experimental values
2
for the two ratios. Values for α (µ) extracted from a comparison of the theoretical
s
expressions (γ = 1) with the data are also shown in this table.
The following observations can be made for the zero-range case: (i) The ratios
R and R give rather consistent values for α (µ) for the same decaying system
1 2 s
(J/ψ or Υ). (ii) Using the values for R which have small error bars, one obtains
2
precise values for α (m ) and α (m ) from J/ψ and Υ decays, respectively. The two
s c s b
values are nearly equal. This result is unexpected, because in a simple-minded QCD
analysis, the scale dependence of α (µ) would predict
s
lnm2/Λ2
α (m ) α (m ) b 0.28 .02 (2)
s c s b
≃ lnm2/Λ2 ≃ ±
c
instead of α (m ) = 0.19 .01 where we used Λ = 200 MeV. This discrepancy has
s c
±
been observed before [5] and a correction factor has been parametrized but never
calculated.
We improve on the previous calculations by taking the quark-momentum de-
pendence (or finite size) of the transition operator QQ¯ 3g into account. This
→
correction cancels in the ratio R , since the rates in the numerator and denominator
1
are affected in the same way. In the ratio R , only the three-gluon state is modi-
2
3
fied, whereas the s-channel photon annihilation still samples ψ(0) 2 = d3qψ˜(~q) 2,
| | | |
˜ R
where ψ(~q) is the Fourier transform of ψ(~r). The three-gluon decay amplitude can
be written as
d4q
˜
T(k ,k ,k ) = ψ (p,q)iM(p,q,k ,k ,k ), (3)
1 2 3 (2π)4 BS 1 2 3
Z
where k ,i = 1,2,3 are the four momenta of the gluons, M is the expression for
i
the QQ¯ 3g vertex and ψ˜ (p,q) stands for the Bethe-Salpeter wavefunction of
BS
→
the quarkonium state. In eq. (3) p denotes the c.m. momentum and q the relative
˜
momentum of the two quarks. In our calculation, we replace ψ by the non-
BS
˜
relativistic wavefunction ψ(q) according to [8]
2π
ψ˜ (p,q) = δ(q0)u(p/2+q)ψ˜(~q)v¯(p/2 q). (4)
BS
i −
The reduction of the Bethe-Salpeter wavefunction to the nonrelativistic wavefunc-
tion is a necessary approximation to proceed with the calculation. At short dis-
tances, i.e. for high momenta p~ > m , it is expected that the relativistic wavefunc-
Q
| |
tion falls off more slowly than the nonrelativistic wavefunction, since the relativistic
transverse one-gluon exchange kernel is less damped. Only a part of this effect is
taken into account in the radiative corrections [6] 0(α ) which include transverse
s
∝
¯
gluon exchange between the heavy Q and Q. The amplitude M factorizes into an
expectation value for the colour part of the gluons with fixed colors i,j,k and the
rest
λ λ λ 1
i j k
M(p,q,k ,k ,k ) = M Tr
1 2 3 ijk
2 2 2 √3!
1
= (d Ms(p,q,k ,k ,k )+if Ma(p,q,k ,k ,k ). (5)
ijk 1 2 3 ijk 1 2 3
4√3
The first part Ms includes the symmetric sum of the six diagrams arising from
the interchange of the three gluons, while the antisymmetric part Ma sums the
4
same diagrams with a negative sign for the odd permutations. d and f are the
ijk ijk
structure constants of the group SU . For ~q = 0, Ma = 0, and one finds for the
3
three-gluon rate the three-photon decay rate of positronium with the modification
α3 5 α3. For ~q = 0, the contribution of Ma is numerically found non-zero but
→ 18 s 6
negligible in comparison to Ms. The vertex M is calculated from the expression
M (p,q,k ,k ,k ,ǫ ,ǫ ,ǫ ) =
123 1 2 3 1 2 3
1 1
g3Tr ǫ ǫ ǫ P (p,q) . (6)
(6 1 p/2+ q+ k1 mQ 6 2 p/2+ q k3 mQ 6 3 · sz )
− 6 6 6 − 6 6 − 6 −
Here, the vectors ǫ ,i = 1,3 describe the gluon polarization and P (p,q) is the spin
i sz
projection operator in the 3S state [8]. We calculate the decay width Γ(QQ¯ 3g)
1
→
including the full ~q-dependence in the matrix element M
Γ(QQ¯ 3g) = (7)
→
1 d3k d3k d3k
1 2 3 (2π)4δ3(Σk )δ(Σω 2m ) T(~k ,~k ,~k ,~ǫ ,~ǫ ,~ǫ ) 2,
3! 8ω ω ω (2π)9 i i − Q | 1 2 3 1 2 3 |
polZ 1 2 3
X
d3q
~ ~ ~ ~ ~ ~
T(k ,k ,k ,~ǫ ,~ǫ ,~ǫ ) = M(p,~q,k ,k ,k ,~ǫ ,~ǫ ,~ǫ ) ψ(~q). (8)
1 2 3 1 2 3 (2π)3 1 2 3 1 2 3 ·
Z
The extensive numerical calculation to determine Γ is performed in the following
way. First we use the symbolic program REDUCE to evaluate the trace in eq. (6).
The output of this symbolic program is a rather long function M (50 kbytes) of
~ ~ ~
~q,k ,k ,k ,~ǫ ,~ǫ ,~ǫ . The T-matrix (8) is obtained after integrating M ψ(~q) over
1 2 3 1 2 3
·
q ,q ,q with an adaptive routine which gives a precision of 5%. We use wave-
x y z
functions from a numerical solution of the Schr¨odinger equation with Richardson’s
potential [3] given in ref. [4]. This potential goes over into the Coulomb potential for
short distances and becomes a confining linear potential at large distances. Richard-
son’spotentialgivesawavefunctionψ(q)whichfallsoffasymptoticallyasα (~q2)/ ~q 4.
s
| |
The matrix element M(q) ~q (cf. fig. 1) together with the wavefunction leads
∝ | |
analytically to a very weakly diverging integral proportional to lnln ~q max, which
| |
5
is in agreement with our numerical calculations. These show no sensitivity to a
variation of the cutoff between ~q max = 3mQ and ~q max = 7mQ.
| | | |
Finally the square of the T-matrix is summed over the gluon polarization vec-
tors and gluon momenta ~k = (0,0,ω ), ~k = ( √1 x2ω ,0, ω xω ), ~k =
1 1 2 3 1 3 3
− − − −
(√1 x2ω ,0,xω ). Energy conservation in eq. (7) fixes x:
3 3
−
1
x = [(2m ω ω )2 ω2 ω2]. (9)
2ω ω Q − 1 − 3 − 1 − 2
1 3
The two polarization vectors of each gluon are chosen to be ~ǫ (1) = ~e and ~ǫ (2) =
i y i
~ ~
k ~e / k . We use a seven point two-dimensional integration [9] routine to do the
i y i
× | |
final phase space integration over the Dalitz triangle given by 0 ω m, 0
1
≤ ≤ ≤
ω m and ω +ω m (cf. eq. (7)). We estimate our total numerical accuracy
2 1 2
≤ ≥
to be 10%. For consistency we checked our routines by setting ~q = 0 in the matrix
element M, which leads to the zero-range approximation T(0). The corresponding
decay width Γ0(QQ¯ 3g) agrees with the well- known result for positronium decay
→
modified by the gluonic couplings (eq. (5)) and the strong interaction wavefunction
ψ(r = 0). We denote by
γ = Γ(QQ¯ 3g)/Γ(0)(QQ¯ 3g) (10)
→ →
the reduction factor for the three-gluon decay due to the finite size effect in the
matrix element M(~q). We obtain
0.31 0.03 for J/ψ
γ = ± (11)
0.69 0.07 for Υ .
(
±
At first sight this correction is surprisingly large, especially for charmonium. In
order to demonstrate the origin of the finite-size reduction, we plot in fig. 1 the two
factors in the integral eq. (8), the radial Gaussian wavefunction ~q2ψ˜( ~q ) for J/ψ
| |
and Υ as a function of ~q /m where m = m for the J/ψ and m = m for the Υ,
Q Q c Q b
| |
together with the matrix element M¯s(~q,k ,k ,k ,~ǫ ,~ǫ ,~ǫ ) which has been averaged
1 2 3 1 2 3
6
over the directions of ~q, but is still a function of the polarization directions. The
matrix element M¯s(q) has its maximum at q = 0 and falls off to a minimum at
q/m 1. The exact shape depends on the gluon polarization vectors and the spin
Q
≈
projection s . We have checked the shape of the q-dependence of Ms(q) for various
z
~
combinations of momenta k and polarizations~ǫ . The shown case is rather typical.
i i
It is evident from fig. 1 that M¯s(q) cannot be well approximated by a parabolic
function
M¯s(q) = M¯s(0)(1 a q2/m2) (12)
− Q
˜
over the whole domain of values q of interest. If we approximate ψ(q) by a Gaussian,
the form of eq. (12) would lead to a correction factor
< v2 >
γest 1 a , (13)
≈ − c2 !
where < v2 > is the expectation value of the velocity of the quarkonium state under
consideration [10]. The authors of ref. [5] assume the correction factor to be of the
form eq. (13) and fit the coefficient a. They find a between 2.9 and 3.5. For a = 3,
the correction amounts to γ = .28 for the J/ψ and γ = .78 for the Υ. These values
are not too far from our numerical values eq. (10). This agreement is gratifying,
since it suggests that the finite-size correction is probably the most important effect.
The correctionfactorγ calculatedinthispaperleads tonewvaluesofα (m )and
s c
α (m ) if γ is inserted into the theoretical expressions of R in Table 1. (The ratio
s b 2
R is unaffected by this correction.) The QCD corrections to the gluonic width for
1
heavy mesons have been calculated in ref. [6] for the Υ-meson. Special care is taken
in this reference to avoid double counting of the one gluon Coulomb exchange. The
contribution to radiative corrections due to the Coulomb exchange between Q and
Q¯ must be dropped (i.e. class f in fig. 1 of ref. [6]), because it is included already
in the wavefunction. Only transverse gluons contribute to this class of radiative
7
corrections. Ref. [5] provides a practical concise summary of all QCD corrections to
the decays ofquarkonia, also forcc¯-quarkonia. The values ofα (µ) deduced fromthe
s
ratio R are shown in Table 1. The new values of α calculated with γ = 1 have to
2 s
6
be compared with those from R and are consistent. Now the values for α (m ) and
1 s c
α (m ) are rather different from each other, as is expected for the running coupling
s b
constant. We can use the values for α (µ) to determine the QCD scale parameter
s
(4)
Λ = Λ , which is related to α (µ) by
s
MS
12π 6(153 19n )ln(lnµ2/Λ2)
f
α (µ) = 1 − +... . (14)
s
(33 2nf)lnµ2/Λ2 − (33 2nf)2 ln(µ2/Λ2) !
− −
We use n = 4 and µ = m = 1.5 GeV for J/ψ and µ = m = 4.9 GeV for Υ. We
f c b
take the values of α (µ) with the finite size corrections from Table 1 (for Υ we use
s
an average of the values deduced from R and R ), insert them into eq. (14) and
1 2
solve for Λ. The values of α (µ) and the calculated scale parameters Λ are
s
(4)
α (m ) = 0.28 .01 Λ = 200 20 MeV
s c
± MS ± (15)
(4)
α (m ) = 0.195 .007 Λ = 255 45 MeV.
s b
± MS ±
We have also investigated the dependence on the choice of the scale µ discussed
in ref. [11] by going from µ = m to µ = 2m /3. The radiative corrections for
Q Q
the strong decay widths calculated from the expressions of ref. [5] change rather
drastically for the two cases. The new values of α (2m /3) and the calculated
s Q
(4)
values of the scale parameter Λ are the following
MS
α (2m ) = 0.35 0.01 Λ(4) = 210 20
s 3 c ± MS ± (16)
α (2m ) = 0.24 0.01 Λ(4) = 300 40.
s 3 b ± MS ±
(4)
Eqs. (15,16) summarize the main results of our paper. The values for Λ agree
MS
(4)
withthepresent worldaverage [12]ofΛ = 238 30 60MeV, wherethefirst error
MS ± ±
is statistical and systematic and the second reflects the uncertainty in the scale.
In this paper, we have shown that the inclusion of the finite size of the QQ¯ 3g
→
vertex reduces significantly the3g decay widths oftheJ/ψ andtheΥ. Thefinitesize
8
modifications necessitate a redetermination of the values for α (µ). These new α -
s s
values are in agreement with the QCD evolution of the coupling constant α (µ) with
s
thescaleµandleadtoconsistent valuesofΛ. Thecorrections duetofinitesizeofthe
3g vertex can be calculated without free parameter. Because of the large size of the
reduction of the 3-gluon width of charmonium, the question arises whether there are
not other relativistic corrections. Indeed we believe that the use of a nonrelativistic
wavefunction is the most crucial approximation in our calculation. To improve on
this approximation, much more conceptual and numerical effort would have to be
spent both on the relativistic bound-state problem and the radiative corrections.
Acknowledgements: We are grateful to W. Buchmu¨ller, K. K¨onigsmann, H.J.
Ku¨hn and C. Wetterich for several discussions and M. Jezabek for providing us with
the programforthe calculation of the quarkonium wave functions. H.C.C. gratefully
acknowledges financial support from the Max-Planck Society within the exchange
program with the Academia Sinica.
References
[1] W. Kwong, J. Rosner and C. Quigg, Ann. Rev. Nucl. Part. Sci 37 (1987)
325
[2] W. Buchmu¨ller (ed.), Quarkonia, North-Holland Publ. Comp., Amsterdam
1992
[3] J.L. Richardson, Phys. Lett. 82B (1979) 272
[4] J.H. Ku¨hn and S. Ono, Z. Phys. C21 (1984) 395
[5] W. Kwong, P.B. Mackenzie, R. Rosenfeld and J.L. Rosner, Phys. Rev. D37
(1988) 3210
9
[6] P.B. Mackenzie and G.P. Lepage, Phys. Rev. Lett. 47 (1981) 1244
[7] M. Kobel, contribution to the XXVI Int. Conf. on High Energy Physics,
Dallas 1992
[8] J.H. Ku¨hn, J. Kaplan, E.G.O. Safiani, Nucl. Phys. B157 (1979) 125
[9] K.S. K¨olbig, CERN-DD-Division 64-32, internal communication
[10] W. Buchmu¨ller and S.H.H. Tye, Phys. Rev. D24 (1981) 132
[11] S.D. Brodsky, G.P. Lepage, P. Mackenzie, Phys. Rev. D28 (1983) 228
[12] Particle Properties, Phys. Rev. D45 (1992) June 1
Figure Caption
Fig. 1: The wavefunctions q2ψ˜(q) for the Υ (m = m , full line) and for the
0 Q b
J/ψ(m = m , dashed line) are given in arbitrary units as functions of q/m .
Q c Q
The matrix elements M(q) (eq. (5)) integrated over the angle of qˆ are indicated
for different spin directions of the 3S -quarkonia. Squares give M(q) for J = 1;
1 z
−
crosses for J = 1 and diamonds for J = 0. The z-axis is defined by the momentum
z z
of one fast gluon. The energies and polarizations (ω = ω = 0.97m , ω = 0.06m
1 2 Q 3 Q
and ~ǫ = ~ǫ = ~ǫ = (0,1,0)) of the gluons are chosen to show typical variations of
1 2 3
M(q) with q/m .
Q
R = Γ(3S1→γgg)
1 Γ(3S1→ggg)
Exp Theory α (µ)
s
.09
J/ψ 0.10 0.04 16α(1 3.0αs) 0.19
± 5αs − π ± .05
Υ 0.0274 0.0016 4α (1 2.5αs) 0.189 0.011
± 5αs − π ±
10