\ga\ga and g g decay rates for equal mass heavy quarkonia

0.13 MB·English

by  James T. Laverty

γγ and gg decay rates for equal mass heavy quarkonia
James T. Laverty,1,∗ Stanley F. Radford,2,† and Wayne W. Repko1,‡
1Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
2Department of Physics, The College at Brockport, State University of New York,
Brockport, New York 14420, USA
(Dated: June 16, 2009)
9 Wepresentacalculation ofthetwo-photonandtwo-gluon widthsfortheequalmassquarkonium
0 states 1S0, 3P0 and 3P2 of the charmonium and upsilon systems. The approach taken is based
0 on using the full relativistic qq¯ → γγ amplitude together with a wave function derived from the
2 instantaneous Bethe-Salpeter equation. Momentum space radial wave functions obtained from an
earlier fit of thecharmonium and upsilon spectra are used to evaluate thenecessary integrals.
n
u
PACSnumbers: 13.20.Gd,12.39.Pn,13.25.-k,13.40.Hq
J

6
1

]
h 1. INTRODUCTION
p
-
p Equal-massquarkoniaareeigenstatesofthechargeconjugationoperatorC witheigenvaluesC =( 1)L+S.
e As such, the 1S , 3P and 3P levels of charmonium and the upsilon system can decay into two p−hotons.
h 0 0 2
[ These same states can also decay into two gluons, which accounts for a substantial portion of the hadronic
decaysforstatesbelowthe cc¯orb¯bthreshold. Oneofus(WWR) waspreviouslyinvolvedinaninvestigation
3
of these decays [1], but, as more data become available, there has been a renewed interest in this subject
v
[2, 3, 4, 5, 6, 7, 8].
7
1 In the calculation of the two gamma widths that follows, we wish to include the relativistic and QCD
9 effects in the wave functions that are used to compute the two photon decay amplitudes. We do this by
3
making use of the variational wave functions obtained in [9]. These wave functions were computed using
.
1 a semi-relativistic model containing both the v2/c2 and one-loop QCD corrections to the potential and
0 optimized to provide an accurate description of the cc¯and b¯b quarkonium spectra. One of the parameters
9
determined in this process is the renormalization scale appropriate to the particular quarkonium system.
0
We use the resulting radial functions to construct the individual 1S , 3P and 3P wave functions with the
: 0 0 2
v proper spin dependence obtained from a decomposition of the instantaneous Salpeter equation. One of the
i objectives of taking this approach is to investigate how the inclusion of the full wave function information
X
compareswiththe practiceofusingthe squareofthe radialwavefunction R (0)2 fors-statesor R′ (0)2
r | n0 | | n1 |
a for p-states. This much can be accomplished by using the expressionfor the invariantamplitude, , which
M
can be derived using the instantaneous Salpeter wave function, φ(p~) [2, 3],
=e2 d3pTr C−1/ε′∗S (p k)/ε∗φ(p~)+C−1/ε∗S (p k′)/ε′∗φ(p~) . (1)
F F
M − −
Z
(cid:2) (cid:3)
Here, C is the charge conjugation matrix, k and k′ are the photon momenta, ε and ε′ are the photon
polarization vectors, S (p) is the quark propagator and p = (p~,i p~2+m2) = (p~,iE), with m denoting
F µ
the quark mass. The wave function φ(p~) is the Fourier transform of the instantaneous position space wave
p
function ψ(~x) and its radial portion is obtained from the variational calculation in Ref.[9].
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
2
TherearealsoQCDcorrectionstothetwogammawidths. Becauseofcolorconservation,thesecorrections
arefiniteattheone-looplevel. Theycould,inprinciple,becalculatedbyextendingEq.(1)asaperturbative
series in the QCD coupling α , but the one-loop corrections are known [10, 11, 12, 13] and will be included
S
in the final results.
Inthenextsection,weconstructthenecessarywavefunctions,evaluatethetraceandcalculatethewidths
Γ for the 1S , 3P and 3P states of the cc¯and b¯b systems. We conclude with a numerical evaluation of
γγ 0 0 2
the two-photon widths and some comments on the two-gluonwidths Γ .
gg
2. EVALUATION OF THE BOUND-STATE DECAYS
The wave functions for the solution of the instantaneous Salpeter equation can be decomposed into spin
singlet and spin triplet forms, which are [14]
1 i m
1φ(p~)= p~ ~γγ + γ +1 γ P(p~)C, (2)
4 4 5
2√2 −E · E
(cid:20) (cid:21)
for the singlet states and
1 1 p~p~ V~(p~)
3φ(p~) = i p~ V~(p~)+~γ V~(p~) ·
2√2"− E · · − E(E+m)!
m ~pp~ V~(p~) 1
+iα~ V~(p~)+ · +i ~γγ p~ V~(p~) C, (3)
5
· E E(E+m)! E · × #
(cid:16) (cid:17)
for the triplet states. These states have the conventional normalization
d3pP∗(p~)P(p~)=1 d3pV†(p~) V(p~)=1, (4)
·
Z Z
where
P(p~)= 1 φ (p) for 1S
√4π n0 0
V~(p~)= √14πpˆφn1(p) for 3P0 (5)
V (p~)= 3 ξ(M)pˆ φ (p) for 3P .
i 4π ij j n1 2
q
The spin-two polarization vectors ξ(M) satisfy ξ(M) =ξ(M) and ξ(M) =0.
ij ij ji ii
The explicit form of the qq¯ γγ amplitude in Eq.(1) is
→
/ε′∗[m i(/p k/)]/ε∗ /ε∗[m i(/p k/′)]/ε′∗
=e2e2 − − + − − , (6)
A q 2p k 2p k′
(cid:20) − · − · (cid:21)
whereeistheprotonchargeande thefractionalquarkcharge. WiththeaidofthewavefunctionsinEq.(5),
q
thetracesinEq.(1)canbeevaluatedratherstraightforwardlyforthespin0∓ states. Inthequarkoniumrest
frame, including the factors 1/2ω from the photon normalization and 1/(2π)3/2 from the Fourier transform
of the position space wave function, the results are,
1 m kˆ (ε′∗ ε∗)
(1S )=e2e2 d3p · × φ (p), (7)
M 0 q8π2 ω [(p~ kˆ)2 E2] n0
Z · −
3
for the 1S state, and
0
1 m 1 [ω(p~ kˆ)2ε′∗ ε∗+2E~p ε′∗~p ε∗]
(3P )=e2e2 d3p · · · · φ (p), (8)
M 0 q8π2ω2 pE [(p~ kˆ)2 E2] n1
Z · −
for the 3P state. The evaluation of (3P ) is somewhat more tedious, yielding
0 2
M
√3 1 ξ(M)
(3P ) = e2e2 d3p ij E(ε′∗p ε∗ ~p+ε′∗ ~pε∗p ) ωε′∗ ε∗p~ kˆkˆ p
M 2 q8π2ω2 p[(p~ kˆ)2 E2] − i j · · i j − · · i j
Z · − h
[ω(p~ kˆ)2ε′∗ ε∗+2Eε′∗ ~pε∗ ~p]p p
i j
+ · · · · φ (p), (9)
n1
E(E+m)
#
where we have dropped a term involving p p p that vanishes upon solid angle integration.
i j k
To complete the calculation of the amplitude, we need to evaluate the angular integrals
1
dΩ = A , (10)
p~[(p~ kˆ)2 E2] 0
Z · −
p p
dΩ i j = A kˆ kˆ +A δ , (11)
p~[(p~ kˆ)2 E2] 1 i j 2 ij
Z · −
p p p p
dΩ i j k ℓ = B kˆ kˆ kˆ kˆ +B (δ kˆ kˆ +δ kˆ kˆ +δ kˆ kˆ +δ kˆ kˆ +δ kˆ kˆ +δ kˆ kˆ )
p~[(p~ kˆ)2 E2] 1 i j k ℓ 2 ij k ℓ ik j ℓ iℓ j k jk i ℓ jℓ i k kℓ i j
Z · −
+B (δ δ +δ δ +δ δ ), (12)
3 ij kℓ ik jℓ jk iℓ
notingthatangularintegralsofthis type with anoddnumberofp ’sinthe numeratorvanish. The values of
i
the coefficients A through B are given in the Appendix. In terms of these coefficients, the 0∓ amplitudes,
0 3
including a factor of √3 for color, are
e2e2√3kˆ (ε′∗ ε∗) ∞ kˆ (ε′∗ ε∗)
(1S )= q · × dpp2mA (p)φ (p) √3αe2 · × I , (13)
M 0 8π2 ω 0 n0 ≡ q ω 0
Z0
for the 0− state and
√3e2e2ε′∗ ε∗ ∞ mp ε′∗ ε∗
(3P )= q · dp [ωA +(ω+2E)A ]φ (p) √3αe2 · J , (14)
M 0 8π2 ω2 E 1 2 n1 ≡ q ω2 0
Z0
for the 0+ state. As with the traces, the 3P integration over dΩ is more complicated since it involves four
2 p~
p ’s in the numerator. The result is [19]
i
3 ξ(M) ∞ 2
(3P ) = e2e2 ij dpp E A + B (ε′∗ε∗+ε′∗ε∗)
M 2 q8π2 ω2 − 2 E(E+m) 3 i j j i
Z0 (cid:20) (cid:18) (cid:19)
ω(B +5B +2B )+2EB
+ ω(A +A )+ 1 2 3 2 ε′∗ ε∗kˆ kˆ φ (p)
1 2 i j n1
− E(E+m) ·
(cid:18) (cid:19) (cid:21)
ξ(M)
3αe2 ij (ε′∗ε∗+ε′∗ε∗)I +ε′∗ ε∗kˆ kˆ J . (15)
≡ q ω2 i j j i 2 · i j 2
h i
The calculation of the widths can be performed using the formula for the decay of a particle with spin J
into two photons,
J
1 1
Γ = ω2 dΩ 2. (16)
γγ 16π2 2J +1 kˆ|M|
M=−J pol Z
X X
4
We sum over the photon polarizations using the Coulomb gauge
ε∗(kˆ)ε(kˆ)=δ kˆ kˆ , (17)
ij i j
−
pol
X
and over the spin-two projections using
2
1 1
ξ(M)∗ξ(M) = (δ δ +δ δ ) δ δ . (18)
ij mn 2 im jn in jm − 3 ij mn
M=−2
X
The resulting two-gamma widths are
3α2e4 α π2 20
Γ (1S ) = q I 2 1+ S , (19)
γγ 0 0
2π | | π 3 − 3
(cid:18) (cid:18) (cid:19)(cid:19)
3α2e4 α π2 28
Γ (3P ) = q J 2 1+ S , (20)
γγ 0 2πω2| 0| π 3 − 3
(cid:18) (cid:18) (cid:19)(cid:19)
3α2e4 16α
Γ (3P ) = q 6I 2+ I J 2 1 S , (21)
γγ 2 5πω2 | 2| | 2− 2| − 3 π
(cid:18) (cid:19)
(cid:0) (cid:1)
where conservation of energy requires that the photon energy ω satisfies ω = M /2. The last factors in
qq¯
Eqs.(19)-(21) include the one-loop QCD correction. The two-loop QCD correction to the decay rate has
been examined in [15] and appears to be large. We have not attempted to include this correction in our
evaluation of the decay amplitudes.
3. RESULTS AND CONCLUSIONS
Theintegralsthatremain,Eqs.(19)-(21),wereevaluatedusingboththeperturbativeandnon-perturbative
wave functions from [9] with the aid of Mathematica. Our results for the charmonium system are shown in
Table I [16, 17] and those for the upsilon system system are given in Table II.
TABLE I: The data denoted by ∗ were taken from Ref.[16] and those denoted by † were taken from Ref.[17]. The
column labeled ΓNR+QCD is the value of the width calculated using |Rn0(0)|2 or |Rn′1(0)|2 along with the QCD
correction.
Decay ΓPert Γnon−Pert ΓNR+QCD ΓExp
ηc →γγ 5.09 keV 3.48 keV 13.1 keV 7.2 ±0.9 keV∗
χc0→γγ 2.02 keV 2.12 keV 5.35 keV 2.53 ± 0.45 keV†
χc2→γγ 0.46 keV 0.19 keV 1.55 keV 0.60 ± .08 keV†
ηc′ →γγ 2.63 keV 1.96 keV 10.5 keV < 7.0±3.5 keV∗
For charmonium there is reasonable agreement between our calculations and the experimental results,
withsomepreferencefortheperturbativecalculation. Theresultsaremuchsmallerthanthenon-relativistic
widths including the one-loopQCDcorrection. The two-gammadecayratesfor the upsilonsystemhavenot
beenmeasuredandtheη groundstatehasonlyrecentlybeenobserved[18]. Thetrendofthewidthsobtained
b
usingthenon-relativisticresultsmodifiedbyQCDcorrectionsisagaintobelargerthanourcalculatedwidths.
Note that, in this formalism, knowledge of the 1S masses is not necessary for the calculation of their γγ
0
widths since the only mass that occurs in Eq.(19) is the quark mass, which we obtain from Ref.[9]. This
fact could account for the close agreement of our Γ (1S ) results with those of Ref.[3], even though these
γγ 0
authors use a very different potential to get their radial wave functions.
5
TABLE II:
Decay ΓPert Γnon−Pert ΓNR+QCD
ηb →γγ 0.30 keV 0.32 keV 0.55 keV
χb0 →γγ 32.9 eV 94.4 eV 58.4 eV
χb2 →γγ 7.19 eV 5.48 eV 9.85 eV
′
ηb →γγ 0.14 keV 0.15 keV 0.20 keV
′
χb0 →γγ 34.1 eV 94.5 eV 68.3 eV
′
χb2 →γγ 7.59 eV 5.57 eV 11.5 eV
′′
ηb →γγ 0.10 keV 0.11 keV 0.22 keV
Finally, we can convert our results from two-photon decays to two-gluon decays by replacing 3α2e4 by
q
2α2/3 in Eqs.(19)-(21). This decayprocess accountsfor a substantialportionof hadronicdecays for states
S
below cc¯or b¯b threshold. There are, however, significant radiative corrections as well as contributions from
three-gluon decays and, thus, the two gluon mode does not tell the whole story. Our results for these two-
gluon decays are given in Tables III and IV. From Table III, it can be seen that the two-gluon widths are
smaller than the hadronic widths of the charmonium states.
TABLE III:
Decay ΓPert Γnon−Pert ΓExp
ηc →gg 15.70 MeV 10.57 MeV 26.7±3.0 MeV
χc0→gg 4.68 MeV 4.88 MeV 10.2±0.7 MeV
χc2→gg 1.72 MeV 0.69 MeV 2.03±0.12 MeV
′
ηc →gg 8.10 MeV 5.94 MeV 14±7 MeV
TABLE IV:
Decay ΓPert Γnon−Pert
ηb →gg 11.49 MeV 12.39 MeV
χb0 →gg 0.96 MeV 2.74 MeV
χb2 →gg 0.33 MeV 0.25 MeV
′
ηb →gg 5.16 MeV 5.61 MeV
′
χb0 →gg 0.99 MeV 2.74 MeV
′
χb2 →gg 0.35 MeV 0.26 MeV
′′
ηb →gg 3.80 MeV 4.11 MeV
Acknowledgments
This work was supported in part by the National Science Foundation under Grant PHY-0555544.
APPENDIX A: ANGULAR INTEGRAL COEFFICIENTS
The coefficients A through B occurring in Eqs.(10)-(12) are presented below [20].
0 3
2π E p
A = ln − , (A1)
0
Ep E+p
(cid:18) (cid:19)
6
π E p
A = (3E2 p2)ln − +6Ep , (A2)
1
Ep − E+p
(cid:20) (cid:18) (cid:19) (cid:21)
π E p
A = m2ln − +2Ep , (A3)
2
−Ep E+p
(cid:20) (cid:18) (cid:19) (cid:21)
π E p
B = (105E4 90E2p2+9p4)ln − +(210E3p 110Ep3) , (A4)
1
12Ep − E+p −
(cid:20) (cid:18) (cid:19) (cid:21)
π E p
B = (15E4 18E2p2+3p4)ln − +(30E3p 26Ep3) , (A5)
2
−12Ep − E+p −
(cid:20) (cid:18) (cid:19) (cid:21)
π E p
B = 3m4ln − +(6E3p 10Ep3) . (A6)
3
12Ep E+p −
(cid:20) (cid:18) (cid:19) (cid:21)
[1] S. N.Gupta, J. M. Johnson, and W.W. Repko,Phys. Rev.D 54, 2075 (1996).
[2] D. Ebert, R.N. Faustov and V.O. Galkin, Mod. Phys.Lett. A 18, 601 (2003) [arXiv:hep-ph/0302044].
[3] C. S.Kim, T. Lee and G. L. Wang, Phys.Lett. B 606, 323 (2005) [arXiv:hep-ph/0411075]
[4] O. Lakhinaand E. S. Swanson, Phys.Rev. D 74, 014012 (2006) [arXiv:hep-ph/0603164].
[5] J. P.Lansberg and T. N. Pham, Phys. Rev.D 74, 034001 (2006).
[6] J. J. Dudekand R.G. Edwards, Phys.Rev. Lett. 97, 172001 (2006).
[7] J. P.Lansberg and T. N. Pham, Phys. Rev.D 75, 017501 (2007) [arXiv:hep-ph/0609268].
[8] F. Giannuzzi, arXiv:0810.2736 [hep-ph].
[9] S. F. Radford and W. W. Repko,Phys.Rev.D 75, 074031 (2007) [arXiv:hep-ph/0701117].
[10] R. Barbieri, G. Curci, E. d’Emilio and E. Remiddi,Nucl. Phys.B154, 535 (1979).
[11] R. Barbieri, M. Caffo, R. Gatto and E. Remiddi,Phys. Lett. 95B, 93 (1980); Nucl. Phys. B192, 61 (1981).
[12] W. Kwong, P.B. Mackenzie, R. Rosenfeld and J. L. Rosner, Phys.Rev.D 37, 3210 (1988).
[13] B.-Q. Li and K.-T. Chao, [arXiv:0903.5506].
[14] L. C. Hostler and W. W. Repko,AnnalsPhys. 130, 329 (1980).
[15] A. Czarnecki and K. Melnikov, Phys. Lett. B519, 212 (2001).
[16] C. Amsler, et al.,Phys. Lett. B667, 1 (2008).
[17] K. M. Ecklund et al. [CLEO Collaboration], arXiv:0803.2869 [hep-ex].
[18] C. Aubert,et al.,arXiv:0807.1086.
[19] The definitions of I2 and J2 contain an additional factor of E in the numerator missing in the corresponding
definitionsofI3andI4inRef.[1].Thisismerelyatypo-theintegralswereevaluatedwiththecorrectexpressions.
[20] These results were independentlychecked by us,but can be obtained from those in Ref.[1].