$η\to π^0 γγ$ decay within a chiral unitary approach revisited

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by  E. Oset

η π0γγ decay within a chiral unitary approach revisited
→
E. Oset1, J. R. Pel´aez2 and L. Roca3
1Departamento de F´ısica Te´orica and IFIC, Centro Mixto Universidad de Valencia-CSIC,
Institutos de Investigaci´on de Paterna, Aptdo. 22085, 46071 Valencia, Spain
2Departamento de F´ısica Te´orica II, Universidad Complutense. 28040 Madrid, Spain.
3Departamento de F´ısica, Universidad de Murcia, E-30071, Murcia, Spain
(Dated: February 2, 2008)
Inviewoftherecentexperimentaldevelopmentsontheexperimentalsideintheη π0γγ decay,
→
and the fact that the Particle Data Group in the on line edition of 2007 reports sizable changes of
the radiative decay widths of vector mesons used as input in the theoretical calculations of [1], a
reevaluation of the decay width of the η in this channel has been done, reducing its uncertaintyby
almost a factor of two. The new input of the PDG is used and invariant mass distributions and
8 total widths are compared with the most recent results from AGS, MAMI and preliminary ones of
0 KLOE.TheagreementofthetheorywiththeAGSandMAMIdataisverygood,bothforthetotal
0
rates as well as for theinvariant mass distributions of the two photons.
2

n PACSnumbers: 13.40Hq,12.39Fe
a
J
7 I. INTRODUCTION taintheO(p6)chiralcoefficientsbyexpandingthevector
1 meson propagators,leads [9] to results about a factor of
The η π0γγ reaction has been quite controversial two smaller than the ”all order” VMD term when one
] → keeps the full vector meson propagator. The lesson ob-
h given the large discrepancies between different theoret-
tained from these studies is that ChPT can be used as a
p ical approaches trying to match the scarce experimen-
- tal data. For a long time the standard experimental re- guiding principle but the strict chiralcounting has to be
p abandonedsince theO(p6) andhigher ordersinvolvedin
sults have been those of early experiments [2],[3], giving
e
thefull(“allorder”)VMDresultsarelargerthanthoseof
h Γ = 0.84 0.18eV. More recent experiments with the
± O(p4). Also these calculationshad severalsourcesofun-
[ Crystal Ball detector at AGS [4] reduced this value to
certainty,oneofthemostimportantwasthecontribution
Γ = 0.45 0.12eV. A new reanalysis of AGS data gives
v1 Γ=0.285± 0.031 0.049eV[5] and a more recent anal- ofthea0(980)resonance,forwhichnoteventhesignwas
± ± known. Thus, one is lead to rely directly on mechanisms
3 ysis with the Crystal Ball at MAMI provides the rate
for the reaction, leaving apart the strict chiral counting.
3 Γ = 0.290 0.059 0.022eV [5]. At the same time the
6 last two ex±perimen±ts have provided the much awaited
The theoretical situation improved significantly with
2 invariant mass distribution for the two photons, which
thethoroughrevisionoftheproblemin[1],wherethedif-
.
1 was thought to provide valuable information concerning
ferentsourcesofuncertaintywerestudiedandthea0(980)
0 the theoretical interpretation. Some preliminary results
contribution was reliably included by using the unitary
8 from KLOE at Frascati [6] are also available with values
extensions of ChPT [15, 16, 17, 18]. Within this chi-
0
around Γ=0.109 0.035 0.018 eV.
: ± ± ral unitary approach for the interaction of pseudoscalar
v Thetheoreticalmodelsshowalsoasimilardispersionof mesons the a0(980), as well as the f0(980) or the σ(600)
Xi theresults,fromlargevaluesobtainedusingmodelswith resonances,are dynamically generatedby using as input
quarkboxdiagrams[7,8]tomuchsmallerones,obtained thelowestorderchiralLagrangians[19]andresumingthe
r
a mostlyusingideasofchiralperturbationtheory(ChPT), multiplescatteringseriesbymeansoftheBetheSalpeter
which are quoted in [1]. equation. Anothersourceofcorrectionsin[1]wastheuse
The η π0γγ reaction has been traditionally consid- of the newest data for radiative decay of vector mesons
→
ered to be a border line problem to test chiral pertur- of the PDG 2002 [3]. It was noted in [1] that the rates
bation theory (ChPT). The reason is that the tree level had significantly changed from previous editions of the
amplitudes, both at O(p2) and O(p4), vanish. The first PDG, to the point that the η π0γγ widths calculated
non-vanishing contribution comes at O(p4), either from in [9, 14] would have changed→by about a factor of two
loopsinvolvingkaons,largelysuppressedduetothekaon should one have used the new data for radiative decay
masses, or from pion loops, again suppressed since they of vector mesons of the PDG 2002 instead of the former
violate G parity and are thus proportional to m m ones. Another improvement in [1] was the unitarization
u d
[9]. ThefirstsizablecontributioncomesatO(p6)bu−tthe of the pair of mesons of the VMD terms beyond the tree
coefficientsinvolvedarenotpreciselydeterminedandone level. Furthermore, to have a better control on the reac-
mustrecurtomodels. Inthis sense,either VectorMeson tion, the consistency of the model with the related reac-
Dominance (VMD) [9, 10, 11], the Nambu-Jona-Lasinio tionγγ π0η wasestablished. Finally,in[1]athorough
→
model (NJL) [12], or the extended Nambu-Jona-Lasinio analysisofthetheoreticalerrorswasdonebyconsidering
model(ENJL)[13,14],havebeenusedtodeterminethese allsourcesofuncertaintyandmakingaMonteCarlosam-
coefficients. However, the use of tree level VMD to ob- ple of results obtained with random values of the input
2
within the uncertainties. for the different radiative decays, together with the the-
The final result obtained in [1] was oretical results (using G = 69MeV and f = 93MeV)
V
and experimental [3, 20] branching ratios. In Table I
Γ=0.42 0.14 eV, (1) we quote the results of the PDG version of 2002, which
±
were used as input in the evaluation of the results in [1],
which is still in agreement with the present experimen-
together with the new results of the PDG 2007 on-line
tal results within uncertainties. Nevertheless, five years
edition [20] which are used in the present paper.
after the publication of these results some novelties have
The agreementofthe theoreticalresults with the data
appeared that call for a revision of the problem. In-
is fair but they canbe improvedby incorporatingSU(3)
deed,onceagainthedatafortheradiativedecayofvector
breakingmechanisms[23]. For thatpurpose, we normal-
mesonsofthe ”online”PDG2007[20]havesignificantly
ize here the C couplings so that the branching ratios in
changed with respect to the data of the PDG 2002 used i
Table I agree with experiment.
in ref. [1]. The correctiondue to these changes is impor-
Once the VPγ couplings have been fixed in this way,
tantanditproducesabouta 25%decreaseinthe central
we can use them in the VMD amplitude corresponding
value of the result of Eq. (1). At the same time, the
to the diagram of Fig. 1, for what we follow the details
theoretical uncertainty is reduced by almost a factor of
of [1]. Next we briefly describe the other mechanisms
two. On the other hand, the new experimental results
considered in [1].
regarding the two photon invariant mass distribution [5]
provide an extra challenge for the theoretical models.
In view of this, it has become necessary to update the
workof[1]toaccountforthenewestexperimentalresults III. OTHER MECHANISMS
of the PDG 2007 [20] and to compare with the most
recent experimental data of the η π0γγ decay. The
In[1]othermechanismswereconsideredwhicharenot
→
model used here is, hence, the same as the one of [1]
affectedbythemodificationsofthe previoussection. We
and the only changes are the use as input of the new
refreshthemgraphicallyandforwardthereaderto[1]for
vector mesons radiative widths. Thus, we refrain from
details.
providingdetailedexplanationsonthe modelandinthis
In Fig. 2 we show the diagrams that go through kaon
brief report we just concentrate on the changes.
loops. These diagrams, with the unitarization of the
meson-meson interaction depicted in Fig. 3, were shown
in [24] to be mostly responsible for the strength of the
II. VMD CONTRIBUTION 0
γγ π η reaction in the region of the a0(980) reso-
→
nance. It was also shown in [1] that the considerationof
Following[9]weconsidertheVMDmechanismofFig.1 the mechanisms of Fig. 1 and Fig.4 improvedthe agree-
which can be easily derived from the VMD Lagrangians ment with the data at low π0η invariant masses.
γ γ
γ
γ
K+ π0 γ K+ π0 K+ π0
η ρ,ω π0 K+ (cid:0)(cid:0)(cid:1)(cid:1) + K+ (cid:0)(cid:0)(cid:1)(cid:1) + (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) +
(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) γ K− η γ K− η γ K− η
FIG. 1: Diagrams for the VMD mechanism.
K+ π0 K+ π0
involving VVP and Vγ couplings [21] + A (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) + A (cid:0)(cid:0)(cid:1)(cid:1)
K− η K− η
G
= ǫµναβ ∂ V ∂ V P , = 4f2egA QVµ ,
VVP µ ν α β Vγ µ
L √2 h i L − h i FIG. 2: Diagrams for thechiral loop contribution
(2)
where V and P are standard SU(3) matrices for the
µ
vector mesons and pseudoscalar mesons respectively [1].
In Eq. (2) G= 3g2 , g = GVMρ [21] and f = 93MeV, K+ π0 K+ πK(cid:0)(cid:1)0Kη(cid:0)(cid:1) π0
with GV the cou4pπl2ifng of ρ−to√π2πf2in the normalization of (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) + (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) +...
[22]. From Eq. (2) one can obtain the radiative decay K− η K− η
widths for V Pγ, which are given by
→
2 FIG. 3: Resummation for γγ π0η.
3 2 2 GV 3 →
Γ = αC G k , (3)
V→Pγ 2 i (cid:18) 3MV (cid:19)
The vector meson exchange diagrams of Fig. 1 were
wherekisthephotonmomentumforthevectormesonat unitarized in [1] by including the resummation of dia-
restandC areSU(3)coefficientsthatwegiveinTableI gramsofFig.3,producingthediagramdepictedinFig.4,
i
3
i Ci Bith Biexp (PDG 2002 [3]) Biexp (PDG2007 [20])
ρ→π0γ q23 7.1×10−4 (7.9±2.0)×10−4 (6.0±0.8)×10−4
ρ→ηγ √23 5.7×10−4 (3.8±0.7)×10−4 (2.7±0.4)×10−4
ω π0γ √2 12.0% 8.7 0.4% 8.91 0.24%
→ ± ±
ω→ηγ 3√23 12.9×10−4 (6.5±1.1)×10−4 (4.8±0.4)×10−4
KKKK∗∗∗∗−+00→→→→KKKK0+−0γγγγ −√3√232((21−+MMMMωφωφ)) 1237..33××1100−−44 (9(.293±±02.)9)××101−0−44 ((293..91±±02..90))××1100−−44
TABLE I: SU(3) Ci coefficients together with theoretical and experimental branching ratios for different vector meson decay
processes.
wherethethick dotrepresentsthefullmeson-mesonuni- 0.031 0.049 eVandMAMIΓ=0.290 0.059 0.022 eV
± ± ±
tarized amplitude. Note that these mechanisms are also [5]. However,allthesedecaywidthsaremuchlargerthan
affected by the renormalizationof the VVP vertices dis- the preliminaryresults ofKLOEatFrascatiΓ=0.109
±
cussed in the previous section. 0.035 0.018 eV [6].
±
The mass distribution of the two photons provides ex-
tra information which was claimed to be relevant to fur-
η K+ ,K0
η η ther test theoretical models. In [1] the differential cross
ρ,ω + K* +,K*0 sectiondΓ/dMγγ wasgiven. Wepresentheretheupdated
results in Fig. 6, where the contribution of the different
π0 π0
π0 K− ,K0 mechanisms is shown. The new experiments reported in
[5]providemeasurementsofdΓ/dM2 whichcanbe con-
a) b) γγ
trasted with theoretical predictions.
FIG. 4: Loop diagrams for VMD terms. The diagrams with
thetwocrossedphotonsarenotdepictedbutarealsoincluded 1.4
in thecalculations.
1.2
Finally, a small term related to the three meson axial V] 1
e
anomaly,andshowndiagrammaticallyinFig.5,wasalso G
V/
tsamkaelnl binytihtseelcfaglciuvelastiaonnosnin-ncee,glaigsibnloetecdonintri[b9u],tiaolnthuopuognh M [eγγ 00..68
d
interference with the other terms. Γ/
d
0.4
η π π0 η K π0 0.2
0
0 0.1 0.2 0.3 0.4
π K Mγγ [GeV]
FIG.6: Contributionstothetwophotoninvariantmassdistri-
FIG. 5: Diagrams with two anomalous γ 3M vertices. bution. From bottom to top, short dashed line: chiral loops;
→
longdashedline: onlytreelevelVMD;dashed-dottedline: co-
herentsumofthepreviousmechanisms;doubledashed-dotted
line: idem butaddingtheresummed VMD loops; continuous
line: idembutaddingtheanomaloustermsofFig.5,whichis
IV. RESULTS thefullmodelpresentedinthiswork(wearealsoshowingasa
dottedlinethefullmodelbutsubstitutingthefulltK+K−,ηπ0
By considering all the modifications discussed in sec- amplitudeby its lowest order).
tion II, the integrated width that we obtain is
Note that in the experiments of [5] the magnitude
dΓ/dM2 is given, while in [1] and in Fig. 6 dΓ/dM is
γγ γγ
Γ=0.33 0.08 eV (4) evaluated. Although these distributions are equivalent,
±
in practice the first one is more useful to study the spec-
which should be compared to the result of [1] of Γ = trum at low invariant masses since it provides extra in-
0.42 0.14 eV. The new result comparesfavorablywith formationnotgivenbythesecondone. Indeed,dΓ/dM
γγ
±
themostrecentresultsofCristalBallatAGSΓ=0.285 is zero at the threshold of the γγ phase space. However,
±
4
dΓ/dM2 contains an extra 1/2M factor and leads to decay. The parallel advances in theory reflected by the
γγ γγ
afinite valueatzeroγγ invariantmass. This finite value work of [1] have alloweda detailed comparisonof results
and the shape of the distribution close to threshold offer whichhasgivenagoodagreementbothforthetotalrate
an extra test to the theory that would be missed had we as well as for the invariant mass distributions with the
simply taken dΓ/dM for comparison. This of course most recent finished results. The discrepancy with the
γγ
implies that the measurements can be done with good preliminarydata ofFrascatiis worrisome,but we should
precision at the threshold. On the other hand, for the waittillthesedataarefirmbeforeelaboratingfurtheron
high mass region of the spectrum the dΓ/dM distri- the discrepancies.
γγ
bution is more suited to reveal the effects of different
theoretical mechanisms, as we have shown in Fig. 6.
In Figs. 7(a) and 7(b) we compare the theoretical re-
sults that we obtain with the distributions obtained at
MAMI and AGS. The agreement is good, in shape and
size, and the theory provides indeed a finite value at
threshold compatible with experiment, which has nev-
ertheless large errors. It is interesting to see that the
AGS data show clearly an increase of the distribution at
low invariantmasseswhich is a feature ofthe theoretical
results. The data of MAMI, however, have too large er-
Acknowledgments
rorsatthresholdanddoesnotallowonetoseethistrend
of the results. At large values of the invariant mass the
agreement of the theory with the MAMI data is better
than with the AGS data. This work is partly supported by DGICYT con-
In order to offer a different perspective of the com- tract number FIS2006-03438, and the Generalitat Va-
parison of the results at the higher mass region of the lenciana. This research is part of the EU In-
distribution, we show in Figs. 7(c) and 7(d) our final re- tegrated Infrastructure Initiative HADRONPHYSICS
sults for dΓ/dMγγ compared to the data of [5] properly PROJECT Project under contract number RII3-CT-
transformed to these variables. 2004-506078. JRP’s research is partially funded by
Spanish CICYT contracts FPA2007-29115-E,FIS2006-
03438, FPA2005-02327, UCM-CAM 910309, as well
V. SUMMARY as Banco Santander/Complutense contract PR27/05-
13955-BSCH. L.R. akcnowledges further support from
In summary, we have witnessed an important exper- Fundaci´on S´eneca grant No. 02975/PI/05 and CICYT
imental advance in the recent years on the η π0γγ contracts FPA2004-03470and FPA2007-62777.
→
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10
6
this work this work
CB@MAMI CB@AGS
8 5
2V] 2V]
V/Ge 6 V/Ge 4
2M [eγγ 4 2 [eMγγ 3
Γ / d Γ / d 2
d d
2
1
0 0
0 0.04 0.08 0.12 0.16 0 0.04 0.08 0.12 0.16
M2 [GeV2] M2 [GeV2]
γγ γγ
(a) (b)
2 2
1.8 this work 1.8 this work
CB@MAMI CB@AGS
1.6 1.6
V] 1.4 V] 1.4
e e
V/G 1.2 V/G 1.2
e e
M [γγ 1 M [γγ 1
d 0.8 d 0.8
Γd / 0.6 Γd / 0.6
0.4 0.4
0.2 0.2
0 0
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
M [GeV] M [GeV]
γγ γγ
(c) (d)
FIG. 7: Two photon invariant mass squared (upper raw) and two photon invariant mass (lower raw) distributions. The data
are from [5] for the Crystal Ball at MAMI (left panels) and for the Crystal Ball at AGS (right panels). The shaded region
corresponds to the band of values of the present work considering the theoretical uncertainties.
6
(1995) 240.
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