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Z. Phys. C 73, 541—546 (1997)
Hadronic decays of excited heavy quarkonia
Thomas Mannel, Res Urech
Institut fur Theoretische Teilchenphysik, Universitat Karlsruhe, D-76128 Karlsruhe, Germany(e-mail: [email protected], [email protected])
Received: 26 March 1996
Abstract. We construct an effective Lagrangian for thehadronic decays of a heavy excited s-wave-spin-one quar-konium (@ into a lower s-wave-spin-one state (. We showthat reasonable fits to the measured invariant massspectra in the charmonium and bottomonium systems canbe obtained within this framework. The mass dependenceof the various terms in the Lagrangian is discussed on thebasis of a quark model.
1 Introduction
Bound state systems consisting of a heavy quark anda heavy antiquark are fairly well described in terms of wavefunction models, where the potential among the two heavyquarks is determined from fitting a phenomenologicalansatz to the data [1]. In this way many static properties ofthese systems may be understood even quantitatively.
As far as decays of heavy quarkonia are concernedthere has been also some theoretical progress recently [2,3]. For inclusive decays in which the two heavy quarksannihilate an effective theory approach has been de-veloped allowing for a systematic treatment of these pro-cesses. This method puts the description of this type ofdecays on a model independent basis.
The theoretical framework of exclusive hadronicdecays is in general still quite model dependent. However,for a large class of decay modes of excited quarkonia (@we may use chiral symmetry arguments to construct aneffective Hamiltonian for decays such as (@P(n or(@P(nn where n is a member of the Goldstone bosonoctet. This ansatz has been discussed already in the litera-ture [4], however, without giving a systematic derivativeexpansion in the spirit of chiral perturbation theory.
Such an expansion may be performed starting fromthe infinite mass limit for the heavy quarkonium|. The
|The idea we are proposing here is along the lines as in [5], wherethis method is applied to / decays
momentum of the heavy quarkonium scales with theheavy quark masses and hence it may not be used as anexpansion parameter. In order to define a systematic de-rivative expansion we decompose the momentum p@(p) ofthe heavy quarkonium in the initial (final) state asp@(p)"(m
Q#mQM ) v#k@(k), where k@ and k are small resid-
ual momenta of the order of the binding energy of theheavy quarkonium, and m
Qand mQM are the masses of the
heavy quark and antiquark respectively. Note that unlikein Heavy Quark Effective Theory where systems witha single heavy quark are considered we cannot use thestatic limit for quarkonia states. Rather one has to employthe non-relativistic limit and the expansion in k/m
Qand
k@/mQM becomes an expansion in Dvrel
D/c, where vrel
is therelative (three) velocity of the two heavy quarks.
The velocity v is the velocity of the initial state quar-konium and hence the momenta of the final state heavyquarks will be of the order of the mass difference of theinitial and final state quarkonia, which we shall assume tobe small compared to the masses m
Qand mQM .
In the spirit of chiral perturbation theory one maynow formulate a derivative expansion, which after Fouriertransformation becomes an expansion in the pion mo-menta and the residual momenta k@ and k. In the nextsection we shall use chiral symmetry to write down theleading as well as the chiral symmetry breaking terms insuch an expansion. In Sect. 3 we show that one mayobtain a good fit to the data, total rates as well as invari-ant mass spectra of the two pions in the decay (@P(nn.We shall compare the charmonia and the bottomonia anddiscuss the dependence of the parameters in the effectiveLagrangian on the bottom and charm mass in Sect. 4.Conclusions are given in Sect. 5.
2 Effective Lagrangian for heavy quarkonia decays
We construct an effective Lagrangian for the decays ofa heavy excited s-wave-spin-one quarkonium (@ intoa lower s-wave-spin-one state (. Let Ak be the field of theinitial excited s-wave 1~ state and Bk the one of the final1~ state. We shall consider here only quarkonia where
mQ"mQM , hence we look at charmonium or bottomonium
systems.The momenta of the heavy quarkonia are split into
a large piece which scales with mQ
and a small residualpart k which depends only weakly on the heavy quarkmass
p@"2mQv#k@ for Ak
p"2mQv#k for Bk
In coordinate space this is achieved by a phase redefini-tion of the fields
Ak (x)"exp(!i2mQv ·x)A(v)k (x)
Bk (x)"exp(!i2mQv ·x)B(v)k (x) (1)
such that the derivative acting on the fields with super-script (v) yields now k (or k@) which is small compared tothe heavy mass. The fields A(v) and B(v) obey the equationsof motion corresponding to static fields and are transversewith respect to the velocity vector v
iv ·LA (v)k (x)"0, vkA(v)k (x)"0 (2)
iv ·LB (v)k (x)"0, vkB (v)k (x)"0 (3)
We are interested in an effective Lagrangian for the had-ronic decays of the type APBn or APBnn, and weconstruct this Lagrangian using chiral symmetry. Higherorder terms may be included by chiral perturbation the-ory, i.e. by a systematic expansion in the derivatives (thesecorrespond to pion momenta or residual momenta forquarkonia) and in the light quark masses. The heavyquarkonia are singlets under the chiral symmetry, andhence the decays APBn are forbidden in the chiral limit.
In the chiral limit only decays to an even number ofpions are allowed, and the leading term in the derivativeexpansion obeying chiral symmetry consists of three terms
L0"gA (v)k B (v)k*Tr[(Llº) (Llº)s]
#g1A(v)k B(v)k*Tr[(v · Lº) (v · Lº)s]
#g2A (v)k B(v)*l Tr[(Lkº) (Llº)s
#(Lkº)s (Llº)]#h.c. (4)
where º is a unitary 3]3 matrix that contains the Gold-stone fields
º"exp (i'/F0)
'"
J2 A1
J2n0#
1
J6g8
n` K`
n~ !
1
J2n0#
1
J6g8
K0
K~ K0 !
2
J6g8B(5)
and F0
is the pion decay constant| in the chiral limit. Inprevious considerations the second term in (4) has been
|We use the convention in which the pion decay constant isF0K93 MeV
often omitted, but since a consistent definition of thederivative expansion forces us to introduce the vector v,the second term is a perfectly allowed one. We shall showin the next section that the magnitude of the coupling g
1is
not small compared to g and that a reasonable fit to theinvariant mass spectra of the pions may be obtainedalready from (4).
The quark mass matrix breaks chiral symmetry, andthe leading terms of this kind are
LS.B.
"g3A (v)k B (v)k*Tr[M(º#ºs!2)]
#ig@eklab [vkA (v)l LaB (v)*b!(Lk A (v)l ) va B (v)*b ] Tr[M(º!ºs)]#h.c. (6)
where
M"Am
u0 0
0 md
00 0 m
sB (7)
The two operators proportional to the e tensor are rel-evant for decays involving an odd number of pions. Therelative sign of the these terms is fixed by reparameteriz-ation invariance [6].
The Lagrangian in (4) has been considered also byCasalbuoni et al. [7], as well as the part proportional to g@in (6) [8]. In the latter case the authors include in additionthe mixing and direct coupling to the singlet g@, whereaswe concentrate on the contributions from the octet ac-cording to [9].
3 Fit to the data
3.1 (@P(nn
Starting from the Lagrangians given in (4) and (6) we fitthe experimental data on the decays t(2S) PJ/tn`n~ inthe charmonium system, and ¶ (aS)P¶ (bS)n`n~ in thebottomonium system, where (a, b) takes the values(2, 1), (3, 2) and (3, 1) . The amplitude has the form
A ((@P(n`n~)
"!
4
F20GC
g
2(m2nn!2M2n)#g
1(v · pn`) (v · pn~)
#g3M2nD e*( · e({
#g2[pn`k pn~l#pn`l pn~k] e*(k e({
lH(8)
where m2nn"(pn`#pn~)2 and e(, e({are the polarization
vectors of the heavy spin-one quarkonia . In the rest frameof the decaying particle (@ we have vk"(1, 0) and theamplitude (8) reduces to the non-relativistic expressiongiven by Brown and Cahn [10] a long time ago. In thisframe the heavy final particle ( moves non-relativisti-cally. If we neglect the motion of the final state heavyquarkonium, the amplitude proportional to g
2describes
the d-wave part of the dipion system, which is found to besuppressed experimentally to a few percent in the char-monium decay [11] as well as in the bottomonium decays[12, 13, 14]. This observation is supported in addition by
542
Table 1 Values of the coupling constantsobtained from fits to the data as indicatedin the text. The 0* means that thisparameter is ignored in this fit
g g1
g3
s2/d.f.
t(2S)PJ/tn`n~ 0.30$0.01 !0.11$0.01 0* 25.3/240.38$0.03 !0.59$0.19 1.55$0.58 20.1/23
¶ (2S)P¶ (1S)n`n~ 0.25$0.01 !0.05$0.01 0* 25.3/19¶ (3S)P¶ (2S)n`n~ 1.19$0.04 0* 0* 26.2/21
1.53$0.33 !0.36$0.34 0* 25.0/20¶ (3S)P¶ (1S)n`n~ 0.026$0.001 !0.087$0.003 0.54$0.02 87.6/29
some qualitative arguments that we will give in Sect. 4.Thus we set g
2"0.
The remaining coupling constants g, g1
and g3
can befitted to the total decay rate and the invariant massspectra of the two pions. The data for the (normalized)differential rates for t (2S)PJ/tn`n~ and ¶ (2S)P¶ (1S)n`n~ are taken from Fig. 7 in [12], for the ¶ (3S)decays we consider the data from [14]. Our resultsare listed in Table 1 and the fits are shown in Fig. 1 andFig. 2 . It is worth discussing briefly the results decay bydecay.
Fig. 1. The curves show the fit as indicated in the text to thenormalized invariant mass spectra of the two pions with x"(mnn!2Mn)/(M2S
!M1S!2Mn) and mnn"(pn`#pn~) . The ex-
perimental data are taken from [12]
f t (2S) PJ/tn`n~: We obtain already a good fit work-ing in the chiral limit, i.e. by ignoring g
3. The ratio g
1/g is
then given by
Ag1g B
ccN"!0.35$0.03 (9)
where the error is given by the fitting program, includingthe experimental uncertainties.
Fig. 2. The dots represent the data from the exclusive deacys, thecircles mark the inclusive distributions. The latter are slightly shiftedto the right for clarity. The straight lines show the fit to the invariantmass spectra of the two pions, the dashed curve in the case of ¶ (3S)P¶ (1S)n`n~ represents the fit with n`n~ final state interactionstaken into account (see text). The experimental data are taken from[14]
543
Varying g3
in the range g3"02 2.1 and fitting again
g and g1
gives an approximate linear relation among thetriple (g, g
1, g
3),
(g, g1, g
3)"A0.30#
1
20x, !0.11!
3
10x, xB, x"02 2.1
(10)
where s2min
is found at g3"1.55. The fit for s2
minis shown in
the upper plot of Fig. 1 .f ¶(2S)P¶ (1S)n`n~: A good fit is again obtained byputting g
3"0. Moreover, allowing for all three parameters
to be free, this fit is included in the set of solutions for whichs24(s2
min#0.01) holds. However, s2 as a function of the
couplings g1
and g3
is very flat around s2min
and thus theerrors of g
1and g
3in the all-parameter-fit are an order of
magnitude larger than the values themselves. Therefore wedo not indicate this fit here. In the case of g
3"0, g is of the
same order as in the corresponding charmonium decay,whereas the ratio g
1/g differs roughly by a factor of 2,
Ag1g BbbM
"!0.19$0.04 (11)
The fit corresponding to g3"0 is shown in the lower plot
of Fig. 1.f ¶ (3S)P¶ (2S)n`n~: In this decay, the experimentaldata have rather large errors. A reasonable fit is obtainedby using the current algebra amplitude, i.e. by puttingg1"g
3"0. If we allow in addition for a variation of
g1the s2 decreases a little bit, however, the error of g
1is of
the same order as the value itself. Altough g is larger bya factor of \(5!6) in comparison to the ¶ (2S) decay, theratios g
1/g are close to each other (within a large error)
Ag1g BbbM
"!0.23$0.22 (12)
In the upper plot of Fig. 2 we show the fit correspondingto g
1O0.
f ¶ (3S)P¶ (1S)n`n~: Here we are confronted with twoaspects which are absent in the other decays. Firstly thephase space is considerably larger than in all the otherdecays discussed so far. Since chiral perturbation theory isvalid only for momenta small compared to the scale"sSB"4nF
0^1.2 GeV, the fits obtained with the ampli-
tude (8) should be considered with some caution. Secondlythe experimental data indicate that the dipion invariantmass distribution is double peaked [13, 14], whereas allthe other decays have a single peak only.
The three parameter fit is indeed rather poor, however,it implies at least that there is a double peak (shown in thelower plot of Fig. 2 as the solid line). In order to improvethe situation, we pick up the idea of [15] by taking n`n~final state interactions into account. Working in the ap-proach where both the heavy particles are at rest, the firstline of the amplitude in (8) represents the dipion system inan s-wave. The amplitude (for g
2"0) can therefore be
improved by introducing the Omnes function [16]
A((@P(n`n~) DgÈ/0
" ) (m2nn ) A((@P(n`n~) DgÈ/0
)(s)"expAs
n=:
4MÈn
ds@d00(s@)
s@(s@!s)B (13)
where d00(s) (denoted as d from here on) is the energy
dependent phase shift of the isospin I"0 s-wave nnscattering amplitude. We decompose d in a low energyphase shift d
LE(‘‘low’’ with respect to the integration range
of )) and in a part coming from a (sharp) scalar resonance,dR. For the low energy tail we use the parametrization of
Schenk [17], which reproduces the measured data up tos1@2"0.9 GeV,
d"dLE#d
R,
tan dLE
(s)"A1!4M2n
s B1@2 4M2n!s
0s!s
0
]Ca0!bI0A1!
s
4M2n B#c0A1!
s
4M2n B2
D,bI0"b
0#a
0
4M2n4M2n!s
0
#a30,
dR(s)"nh(s!M2
R) (14)
with the numerical values
a0"0.2, b
0"0.24, c
0"0, s1@2
0"0.865GeV (15)
and the resonance mass MR
is left as a free parameter. Werepeat the fit procedure and get the following result,
¶ (3S) P¶ (1S)n`n~:
g g1
g3
MR
[GeV] s2/d.f.
0.014$0.001 !0.040$0.002 0.21$0.01 1.17$0.07 26.6/28
(16)
The fit is in good agreement with the data, see lower plotof Fig. 2 (dashed line). The parameters g
1and g
3are larger
than g, which has decreased (compared to the ¶ (2S)decay) by a factor of \(10—15). The resonance mass M
Ris
around 1 GeV, as it is expected from experimental data ond [18] and from the corresponding scalar resonancesf0(980) and f
0(1300) (or a mixture of them) listed by the
PDG [19].In summary we find that the data on the decays
(@P(nn can be fitted in most cases with two parametersg and g
1to a good accuracy. In the charmonium system
we find the ratio g1/g"0.35, whereas for bottomonia we
have g1/g^0.2. The observation, that the data can be
fitted accurately with operators allowed in the chiral limit,has first been made by Novikov and Shifman [20]. Anexception to this rule is found in the transition ¶ (3S)P¶ (1S) n`n~, where the amplitude (8) does not givea good description. The situation can be improved bytaking nn rescattering into account using the parametriz-ation of [17] and a scalar resonance, the mass of which ispreferably around 1 GeV.
3.2 (@P(n
In the case of the decay amplitudes of APBn wheren"n0, g experimental values exist in the charmoniumsystem, upper limits are known for the ¶ (2S) decays andfor the transition ¶ (3S) P¶ (1S) g [19]. For the (@(2S)decays we consider the coupling g@ in the Lagrangian (6)
544
with respect to the same decay channel in the differentsystems,
Kg@bb1
g@ccN Kn"M
¶ (2S)Mt(2S) C
!(¶ (2S)P¶ (1S)n)
!(t(2S) PJ/tn) D1@2
ADpt DDp¶ DB
3@2
Dp( D"1
2M({
M[M2({!(M(#Mn)2]
][M2({!(M(!Mn) 2]N1@2 (17)
where the stronger limit is found in the g decay channel,
Kg@bbM
g@ccN KnÒ(3.08 , Kg@bbM
g@ccN Kg(0.57 (18)
For ¶ (3S)P¶ (1S)g the phase space is much larger thanin the other decays, i.e. the g is by no means soft. Thepresent upper limits for bottomonia in the g decay channelare similar to each other
!(¶ (3S)P¶ (1S)g)
!total
(2.2]10~3 ,
!(¶ (2S)P¶ (1S)g)
!total
(2]10~3 (19)
However, equal branching ratios lead to completely differ-ent g@,
Dg@¶(3S)
D^Dg@
¶ (2S)D
14(equal branching ratios) (20)
mainly due to the difference in phase space.
4 Discussion of the results
In this section we shall discuss our results, in particular weshall try to give some arguments concerning the massdependence of the coefficients g, g
1and g@ (neglecting
contributions driven by the pion mass, i.e. g3"0). We
achieve an at least qualitative understanding by represent-ing the coefficients in the Lagrangian by matrix elementsof heavy quark operators. Hence we consider a model inwhich the heavy quarks and the light pseudoscalar me-sons are the relevant degrees of freedom; in this respectthis resembles the chiral quark model of Georgi andManohar [21]. Note that we omit in the following dis-cussion the decay ¶ (3S) P¶ (1S)n`n~ which is some-what beyond the scope of our method due to the largephase space for the pions.
First of all, this model easily explains the fact that thepion-d-wave contribution is suppressed, since this at leastrequires a dimension-6-operator for the heavy quarks,which would be contained in the operator
Ld~wave
"
c2
"3(QM
vckQv
) (QMvclQv
) Tr[(Lkº) (Llº)s
#(Llº) (Lkº)s] (21)
while the leading contribution to the pion-s-wave is pro-portional to a dimension-3-operator for the heavy quarks,
Ls~wave
P (QMvQ
v) (22)
Here Qv
are the operators of heavy quarks which havebeen rescaled with a phase according to
Qv"exp(imvx)Q
and the scale " is at least of the order of the chiralsymmetry breaking scale "sSB or maybe even the heavyquark mass; in the latter case the d-wave contribution tothis decay would be more strongly suppressed in the¶ decays as compared to the J/t transitions.
We have argued that the d-wave contribution is sup-pressed and hence we have only two operators left in thechiral limit. Both operators contribute to the s-wave of thedecays (@P(n`n~. Employing the model discussedabove, we have the corresponding two operators on thequark level
Lquark
"c(QMvQ
v) Tr[(Lkº) (Lkº)s]
#c1(QM
vQ
v) Tr[(v · Lº) (v · Lºs)] (23)
The coupling constants g and g1
are related to the coup-lings c and c
1in the quark LagrangianL
quarkthrough the
relations
gS(DA(v)kB(v)*k D(@T"ge*( · e({"cS(DQM
vQ
vD(@T (24)
g1S(DA(v)kB(v)*k D(@T"g
1e*( · e({
"c1S(DQM
vQ
vD(@T (25)
From the combined chiral and heavy mass expansion onewould expect that both coupling constants c and c
1do not
depend on the heavy quark mass. As far as the massdependence of g and g
1is concerned, one has to take into
account the mass dependence of the matrix elementS(DQM
vQ
vD(@T. For the operators itself there exists a heavy
mass limit, while the mass dependence of the states maynot be accessed within the framework of the 1/m
Qexpan-
sion [3]. To this end we have to rely on simple dimen-sionality arguments. Assuming that the states are nor-malized to unity we may write the matrix elements as
S( DQMvQ
vD(@TPE3 (26)
where E is a scale related to the binding of the heavyquarkonia. Here we may consider two extreme cases. Thefirst one is the purely coulombic scenario, in which theparameter E is the heavy mass times the fine structureconstant
E"as(m
Q) m
Q(27)
and therefore one would expect a strong scaling with theheavy mass. This is, however, not supported by the energydifferences in the known quarkonia, where e.g. the levelspacing M
2S!M
1Sis practically the same in the J/t
system and the ¶ system. On the other hand, the leptonicdecay widths are determined by the value of the wavefunction at the origin and for this case one has a signifi-cant mass dependence.
Without considering the dynamics of the bindingmechanism in detail, one cannot say much about the massdependence of the matrix elements of the quark operators.For definiteness we consider the scenario in which E isindependent of the mass of the heavy quark; consequentlywe would expect that g and g
1are independent of the
heavy quark mass, since the parameters c and c1
in thequark level Lagrangian L
quarkare mass independent. At
545
least for the coupling g this is supported by data, sincegbbM /gccN is practically unity for t (2S) PJ/tn`n~ and¶ (2S) P¶ (1S) n`n~.
However, this scenario also implies that g1/g should
not scale with the heavy mass. From our fits we find thatthe coupling g
1is larger in the J/t system by a factor
between two or three compared to the ¶ (2S) decay, whileg1/g for the different excited ¶ states is (within the errors)
constant, see (11, 12).The only consistent interpretation of this factor be-
tween g1/g in the charm and bottom systems (given the
scenario we are considering) is that the difference is in-duced by the QCD running of the corresponding coupling
g1 bbM
g1 ccN
"Cas(m
b)
as(m
c)D
i, i^2!3 (28)
Consequently the second term of our effective Lagrangianmust be governed at the level of QCD by an operator witha sufficiently large anomalous dimension. However, if ourassumption that the matrix elements of the quark oper-ators are mass independent is not valid, the mass depend-ence of g
1/g can be also due to the mass dependence of
these matrix elements.Let us finally also consider the chiral symmetry break-
ing contribution relevant for the decays t (2S) PJ/tnand ¶ (2S)P¶ (1S)n. Interpreting this transition again interms of a dimension-3 operator for the heavy quarks onewrites
L@quark
"c@(QMvc5Q
v) Tr[(º!ºs)] (29)
The leading term in the heavy mass expansion will vanishfor this operator, since
QMvc5Q
v"O(1/m
Q)
and hence one expects the coefficients g@ to scale as
g@bbM
g@ccN"
mc
mb
^0.3 (30)
assuming again that the mass dependence of the matrixelements is weak.
5 Conclusions
We have formulated an effective theory approach for thehadronic decays of heavy quarkonia based on the chirallimit for the light degrees of freedom and on the 1/m
Qexpansion for the heavy ones. The leading term in boththe inverse of the heavy quark mass and the inverse of thechiral symmetry breaking scale consists of three termsmultiplied by coupling constants which we have fittedfrom experiment. The fit to the invariant mass spectrum ofthe two pions in the decays t (2S) PJ/tn`n~ and ¶ (2S)P¶ (1S)n`n~ shown in Fig. 1, as well as ¶ (3S)P¶ (2S)n`n~ depicted in the upper plot of Fig. 2, is quite satisfac-tory. In addition we have fitted the decay ¶ (3S) P¶ (1S)n`n~, where however, unlike in the other cases, the fit israther poor (solid line in the lower plot of Fig. 2). The
situation can be improved by taking nn final state interac-tions into account, see the dashed line in the lower plot ofFig. 2.
We also have discussed the mass dependence of thevarious terms in the Lagrangian which unfortunately maynot be considered entirely in the effective theory. Wemodel the leading terms of the effective Lagrangian byswitching to a description in which the relevant degrees offreedom are pions and heavy quarks. This model re-sembles the chiral quark model as proposed by Georgiand Manohar [21]. In this picture the coupling constantsof the effective Lagrangian for the quarkonia and thepions may be reexpressed in terms of the ones of theunderlying quark Lagrangian and certain matrix elementsof heavy quark operators between heavy quarkonia states.The mass dependence of the matrix elements may not beevaluated from chiral and heavy quark symmetry andthus requires additional input. Using a specific assump-tion on the mass dependence of these matrix elements wearrive at definite predictions for the ratios gbbM /gccN andg@bbM /g@ccN . The predictions are consistent with what is foundexperimentally, although for g@bbM /g@ccN only an experimentalbound exists and more data is needed to confirm or falsifyour values.
Acknowledgements. R.U. thanks H. Leutwyler for discussions andH. Genz for providing unpublished notes. This work is partiallysupported by the Graduiertenkolleg ‘‘Elementarteilchenphysik anBeschleunigern’’.
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