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Decay of PSI and upsilon into a lepton pair and gluons
Volume 95B, number 1
PHYSICS LETTERS
8 September 1980
DECAY OF PSI AND UPSILON INTO A LEPTON PAIR AND GLUONS
J.P. LEVEILLE and D.M. SCOTT Physics Department, University o f Wisconsin.Madison, Madison, WI 53706, USA
Received 2 May 1980
We calculate the decay of a heavy vector quarkonium state into a lepton pair and two gluon jets, in lowest order QCD. We present branching ratios, and the lepton pair spectrum as a function of its invariant mass, and of the missing mass recoiling against it.
Ore and Powell calculated the decay of orthopositronium into three photons [1]. The decays of a heavy vector quarkonium state V into three gluons ggg [2], and into a real photon and two gluons 3'gg [3] can be obtained from their result by the appropriate changes involved in going from QED to QCD (quantum chromodynamics). The use of perturbative QCD is justified by the high mass of the vector meson. The asymptotic freedom property of QCD then ensures that the strong coupling % is small. This QCD picture of the hadronic decays of the T has received some experimental support [4], though it seems that the ~ is too light for the QCD signatures to manifest themselves. The inclusive prompt photon decays of the ~ have been observed [5] at a rate of the same order of magnitude as predicted by QCD. However, the momentum spectrum may be different from that predicted by QCD [3], and it may be that the ~b is too light for lowest order perturbation theory to be appropriate. A model to describe this phenomenologically by giving the gluons a mass has been proposed in ref. [6]. It will be interesting to see the inelusive prompt photon spectrum from the 'I', and the evidence concerning the three gluon decay of the qf' makes us optimistic that the "P, and the much awaited (and as yet unnamed) toponium vector state will provide important tests of the QCD calculations. In this paper we extend these calculations to the decay of vector states into a lepton pair and two gluons ~+ ~-gg, finding non-negligible branching ratios. In addition, the dependence on the lepton pair invariant 96
mass q, and on the missing mass opposite the lepton pair are calculable, and can provide detailed tests of the theory. The importance of this process was mentioned by Brodsky et al. [3]. The diagram we calculate is shown in fig. 1, where momenta are defined. The vector meson has massMv and momentum P, the lepton has mass rn~, and we further define (we calculate in the rest frame of the vector meson) x r = 2Izr/Mv, r = q2/M2,
r = q, k, 2; y q = (x 2 - 4r) 1/2
(1)
Energy-momentum conservation implies as usual Xq + x k +x~ = 2 .
(2)
The usual route [1,2,3] leads to dP(V -+ ~+Q-gg) = ct2 c~2s eq2 o ( q 2 ) M 2 v [ ~ ( O ) } 2 dq2dxqdXk
1087r
q2
[A 12. (3)
In eq. (3), eq is the quark charge, ~(0) is the bound state wave function at the origin (note that 47r1 ~(0)l 2 = [R(0)[ 2 where R is the radial wave function), and o is a phase space factor from the lepton pair: o(q 2) -- (1 + 2 m 2 / q 2 ) ( 1 - 4 m 2 / q 2 ) 1/2 •
(4)
The squared matrix element IA 12 was evaluated with the help of the algebraic manipulation program ASHMEDAI, and
Volume 95B, number 1
PHYSICS LETTERS
8 September 1980 161
I)
'
F-.
VzP Fig. 1. Lowest order QCD diagram for V -+ Q+£- gg. The five crossed diagrams are not shown.
I
>
I
1
r-,
q ' k +_q'~ + £ ' k } 2 2 D22D2
IC) 4
+q4
{
I 2;
4
+
+ - -
3{±
1)2}
DqDkD£
4 4}
DkD
2 " DqDkD
r=q,k,~
(5)
.
(6)
2r <~Xq <~l + r ,
(7)
Phase space boundaries are 4rn2~
{ ( 2 - - X q - - yq)~
+yq).
I00
pairs as a function of q, for ~ and T decays. In particular we plot the distribution r 1/2 dN/dr 1/2, with dN/dr 1/2 normalized to unity. The spectra are dominated by small q, because of the virtual photon propagator (in fig. 3 the effect of the propagator has been removed by multiplying by rl/2), and away from the lepton pair threshold the curves become parallel. The invariant missing mass squared opposite the lepton pair is (9)
Note that at fixed MX, r may approach its lower limit 4m 2, and so the missing mass distribution will be dominated by low q2 lepton pairs. Consequently, the shape of an experimental distribution will depend on the ability to measure lepton pairs of low invariant mass. Let us define
P(V -+ £+ £ - gg)/P(V -+ ")'gg)
RM4
I 810
M 2 = (k + £)2 = M 2 ( 1 _ Xq + T) .
AS an alternative to eq. (3), we may divide by P (V -+ "),gg) to find
1536rr(rr 2 - 9)
i
Fig. 2. The ratio P(V --+Q+£- gg)/ P(V --+ygg) as a function of the vector meson mass MV, for £ = e , p., "r.
Here the denominators are defined by
Dr=r2-p'r,
I 60
M v (GeV)
1
DqOk
+
M2
+ _
I 4~0
D4Dk VqD
2 DqOkD
=
I 0
8q2 Oq-Dk-O~ +,~2 2
..C÷ -[--
/
td 3
>
IA 12/16 = 8 [ ( q ' k ) 2 + (q~" £)_2 + (_~, k)21 D~2D k2 Dq 2 D~2Dk2 j
-
f
f io.I ÷~
{
e ÷ e-
162
f d q 2 dxq dxk °-~(q q2-)- IAI 2 (8)
The advantage of eq. (8) is that the quark charge eq, the strong coupling a s and the wave function at the origin ~(0) have cancelled. Our results on this quantity are shown in fig. 2, as a function of the vector meson mass MV, for £ = e, ~, r. In partictdar, our results for and T are displayed in the table. The branching ratios for the inclusive lepton pair decay are small, but not unmeasurably so. Next, in fig. 3 we show the spectrum of lepton
p =Mx/M v .
(10)
Table 1 The ratio E(V ~ £+ £- gg)/F(V --*7gg) is shown for V = 4, T and ~ = e,~,r.
T
e
#
r
1.07 X 10 .2
2.44 x 10 .3
0
1.24X 10 .2
4.11X 10 .3
3.46X 10 .4
97
Volume 95B, number 1
PHYSICS LETTERS
8 September 1980
I0
lO
....
'
'
'
I
r
,
,
,
I ' T÷'[_ ' ' ,., ............
1
(a,.
..
'
I¢----.Q..<
........
"
1
~
\ Z -ID
1(51
ee-
F
! T-- e*e----
\,
\',q "g'+ T 1(5 2
J
0
I
i
d
3
0
~ 0.5
tl
4 m 2 / M 2 <~r<<. (1 - 0) 2 ,
(11)
where Xq = 1 +
r - 0 2
.
(12)
In fig. 4 we show dN/dp for T decays. We have integrated over all q2. The shapes of dN/dp for ~ decays are very similar (except of course for r + r - ) . We can estimate the effects of giving the gluons a mass Mg -~ 0.8 GeV to model non-perturbative effects, as in ref. [6]. The ratios P(V ~ ~+ ~- gg)/F(V -+ 7gg) will be little changed as the phase space effects will be similar for both rates (remember that low q dominates the lepton pair rate). In figs. 3 and 4 the phase space effects will make the spectra vanish for r 1/2 ~> 1 - (2Mg/Mv) and p ~< 2Mg/Mv, respectively. In conclusion, we have studied the decay of a heavy quarkonium state into a lepton pair and two gluons. This can provide detailed tests of QCD per-
98
I
0.5
J
,
,
1.0
dN/dp of lepton pairs from T decay as
a function of p (p is defined in eq. (10)).
Then for fixed p the phase space boundaries are
Xq+yq),
,
1.0
Fig. 3. The spectrum r 1/2 tiN~dr 1/2 of lepton pairs from t) and T decay as a function of r 1/2 (r is defined in eq. (1)).
k~<1(2
......... ,
p= Mx/M v Fig. 4. The spectrmn
r I/z= q / M v
½(2-Xq-yq)<~x
,
turbation theory, possibly on the ~ and hopefully on the q9. We look forward to its measurement on the awaited toponium vector state, whose high mass will aid the accuracy of perturbation theory, and whose constituent quark charge of 2/3 will enhance prompt photon and lepton pair branching ratios. This research was supported in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alunmi Research Foundation, and in part by the Department of Energy under contract DE-AC02 76 ER0881, COO-881-145. References
[1] A. Ore and J.L. Powell, Phys. Rev. 75 (1949) 1696. [2] T. Appelquist and [i.D. Politzer, Phys. Rev. Lett. 34 (1975) 43; Phys. Rev. D12 (1975) 1404. [3] K. Koller and T.F. Walsh, Phys. Lett. 72B (1977) 227; S.J. Brodsky, T.A. DeGrand, R.R. Horgan and D.G. Coyne, Phys. Lett. 73B (1978) 203; H. Fritzsch and K.-H. Streng, Phys. Lett. 74B (1978) 90. [4] See for example, G. Wolf, Lectures at 1979 CERN-JINR School of Physics (Dobog6k6, Hungary), DESY 80/13 (1980). [5] G.S. Abrams et al., Phys. Rev. Lett. 44 (1980) 114; M.T. Ronan et al., Phys. Rev. Lett. 44 (1980) 367. [6] G. Parisi and R, Petronzio, CERN-TH-2804 (1980).
PHYSICS LETTERS
8 September 1980
DECAY OF PSI AND UPSILON INTO A LEPTON PAIR AND GLUONS
J.P. LEVEILLE and D.M. SCOTT Physics Department, University o f Wisconsin.Madison, Madison, WI 53706, USA
Received 2 May 1980
We calculate the decay of a heavy vector quarkonium state into a lepton pair and two gluon jets, in lowest order QCD. We present branching ratios, and the lepton pair spectrum as a function of its invariant mass, and of the missing mass recoiling against it.
Ore and Powell calculated the decay of orthopositronium into three photons [1]. The decays of a heavy vector quarkonium state V into three gluons ggg [2], and into a real photon and two gluons 3'gg [3] can be obtained from their result by the appropriate changes involved in going from QED to QCD (quantum chromodynamics). The use of perturbative QCD is justified by the high mass of the vector meson. The asymptotic freedom property of QCD then ensures that the strong coupling % is small. This QCD picture of the hadronic decays of the T has received some experimental support [4], though it seems that the ~ is too light for the QCD signatures to manifest themselves. The inclusive prompt photon decays of the ~ have been observed [5] at a rate of the same order of magnitude as predicted by QCD. However, the momentum spectrum may be different from that predicted by QCD [3], and it may be that the ~b is too light for lowest order perturbation theory to be appropriate. A model to describe this phenomenologically by giving the gluons a mass has been proposed in ref. [6]. It will be interesting to see the inelusive prompt photon spectrum from the 'I', and the evidence concerning the three gluon decay of the qf' makes us optimistic that the "P, and the much awaited (and as yet unnamed) toponium vector state will provide important tests of the QCD calculations. In this paper we extend these calculations to the decay of vector states into a lepton pair and two gluons ~+ ~-gg, finding non-negligible branching ratios. In addition, the dependence on the lepton pair invariant 96
mass q, and on the missing mass opposite the lepton pair are calculable, and can provide detailed tests of the theory. The importance of this process was mentioned by Brodsky et al. [3]. The diagram we calculate is shown in fig. 1, where momenta are defined. The vector meson has massMv and momentum P, the lepton has mass rn~, and we further define (we calculate in the rest frame of the vector meson) x r = 2Izr/Mv, r = q2/M2,
r = q, k, 2; y q = (x 2 - 4r) 1/2
(1)
Energy-momentum conservation implies as usual Xq + x k +x~ = 2 .
(2)
The usual route [1,2,3] leads to dP(V -+ ~+Q-gg) = ct2 c~2s eq2 o ( q 2 ) M 2 v [ ~ ( O ) } 2 dq2dxqdXk
1087r
q2
[A 12. (3)
In eq. (3), eq is the quark charge, ~(0) is the bound state wave function at the origin (note that 47r1 ~(0)l 2 = [R(0)[ 2 where R is the radial wave function), and o is a phase space factor from the lepton pair: o(q 2) -- (1 + 2 m 2 / q 2 ) ( 1 - 4 m 2 / q 2 ) 1/2 •
(4)
The squared matrix element IA 12 was evaluated with the help of the algebraic manipulation program ASHMEDAI, and
Volume 95B, number 1
PHYSICS LETTERS
8 September 1980 161
I)
'
F-.
VzP Fig. 1. Lowest order QCD diagram for V -+ Q+£- gg. The five crossed diagrams are not shown.
I
>
I
1
r-,
q ' k +_q'~ + £ ' k } 2 2 D22D2
IC) 4
+q4
{
I 2;
4
+
+ - -
3{±
1)2}
DqDkD£
4 4}
DkD
2 " DqDkD
r=q,k,~
(5)
.
(6)
2r <~Xq <~l + r ,
(7)
Phase space boundaries are 4rn2~
{ ( 2 - - X q - - yq)~
+yq).
I00
pairs as a function of q, for ~ and T decays. In particular we plot the distribution r 1/2 dN/dr 1/2, with dN/dr 1/2 normalized to unity. The spectra are dominated by small q, because of the virtual photon propagator (in fig. 3 the effect of the propagator has been removed by multiplying by rl/2), and away from the lepton pair threshold the curves become parallel. The invariant missing mass squared opposite the lepton pair is (9)
Note that at fixed MX, r may approach its lower limit 4m 2, and so the missing mass distribution will be dominated by low q2 lepton pairs. Consequently, the shape of an experimental distribution will depend on the ability to measure lepton pairs of low invariant mass. Let us define
P(V -+ £+ £ - gg)/P(V -+ ")'gg)
RM4
I 810
M 2 = (k + £)2 = M 2 ( 1 _ Xq + T) .
AS an alternative to eq. (3), we may divide by P (V -+ "),gg) to find
1536rr(rr 2 - 9)
i
Fig. 2. The ratio P(V --+Q+£- gg)/ P(V --+ygg) as a function of the vector meson mass MV, for £ = e , p., "r.
Here the denominators are defined by
Dr=r2-p'r,
I 60
M v (GeV)
1
DqOk
+
M2
+ _
I 4~0
D4Dk VqD
2 DqOkD
=
I 0
8q2 Oq-Dk-O~ +,~2 2
..C÷ -[--
/
td 3
>
IA 12/16 = 8 [ ( q ' k ) 2 + (q~" £)_2 + (_~, k)21 D~2D k2 Dq 2 D~2Dk2 j
-
f
f io.I ÷~
{
e ÷ e-
162
f d q 2 dxq dxk °-~(q q2-)- IAI 2 (8)
The advantage of eq. (8) is that the quark charge eq, the strong coupling a s and the wave function at the origin ~(0) have cancelled. Our results on this quantity are shown in fig. 2, as a function of the vector meson mass MV, for £ = e, ~, r. In partictdar, our results for and T are displayed in the table. The branching ratios for the inclusive lepton pair decay are small, but not unmeasurably so. Next, in fig. 3 we show the spectrum of lepton
p =Mx/M v .
(10)
Table 1 The ratio E(V ~ £+ £- gg)/F(V --*7gg) is shown for V = 4, T and ~ = e,~,r.
T
e
#
r
1.07 X 10 .2
2.44 x 10 .3
0
1.24X 10 .2
4.11X 10 .3
3.46X 10 .4
97
Volume 95B, number 1
PHYSICS LETTERS
8 September 1980
I0
lO
....
'
'
'
I
r
,
,
,
I ' T÷'[_ ' ' ,., ............
1
(a,.
..
'
I¢----.Q..<
........
"
1
~
\ Z -ID
1(51
ee-
F
! T-- e*e----
\,
\',q "g'+ T 1(5 2
J
0
I
i
d
3
0
~ 0.5
tl
4 m 2 / M 2 <~r<<. (1 - 0) 2 ,
(11)
where Xq = 1 +
r - 0 2
.
(12)
In fig. 4 we show dN/dp for T decays. We have integrated over all q2. The shapes of dN/dp for ~ decays are very similar (except of course for r + r - ) . We can estimate the effects of giving the gluons a mass Mg -~ 0.8 GeV to model non-perturbative effects, as in ref. [6]. The ratios P(V ~ ~+ ~- gg)/F(V -+ 7gg) will be little changed as the phase space effects will be similar for both rates (remember that low q dominates the lepton pair rate). In figs. 3 and 4 the phase space effects will make the spectra vanish for r 1/2 ~> 1 - (2Mg/Mv) and p ~< 2Mg/Mv, respectively. In conclusion, we have studied the decay of a heavy quarkonium state into a lepton pair and two gluons. This can provide detailed tests of QCD per-
98
I
0.5
J
,
,
1.0
dN/dp of lepton pairs from T decay as
a function of p (p is defined in eq. (10)).
Then for fixed p the phase space boundaries are
Xq+yq),
,
1.0
Fig. 3. The spectrum r 1/2 tiN~dr 1/2 of lepton pairs from t) and T decay as a function of r 1/2 (r is defined in eq. (1)).
k~<1(2
......... ,
p= Mx/M v Fig. 4. The spectrmn
r I/z= q / M v
½(2-Xq-yq)<~x
,
turbation theory, possibly on the ~ and hopefully on the q9. We look forward to its measurement on the awaited toponium vector state, whose high mass will aid the accuracy of perturbation theory, and whose constituent quark charge of 2/3 will enhance prompt photon and lepton pair branching ratios. This research was supported in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alunmi Research Foundation, and in part by the Department of Energy under contract DE-AC02 76 ER0881, COO-881-145. References
[1] A. Ore and J.L. Powell, Phys. Rev. 75 (1949) 1696. [2] T. Appelquist and [i.D. Politzer, Phys. Rev. Lett. 34 (1975) 43; Phys. Rev. D12 (1975) 1404. [3] K. Koller and T.F. Walsh, Phys. Lett. 72B (1977) 227; S.J. Brodsky, T.A. DeGrand, R.R. Horgan and D.G. Coyne, Phys. Lett. 73B (1978) 203; H. Fritzsch and K.-H. Streng, Phys. Lett. 74B (1978) 90. [4] See for example, G. Wolf, Lectures at 1979 CERN-JINR School of Physics (Dobog6k6, Hungary), DESY 80/13 (1980). [5] G.S. Abrams et al., Phys. Rev. Lett. 44 (1980) 114; M.T. Ronan et al., Phys. Rev. Lett. 44 (1980) 367. [6] G. Parisi and R, Petronzio, CERN-TH-2804 (1980).