pp annihilation into four charged pions at rest and in flight

13 Pages·2004·0.58 MB·English

Eur. Phys. J. C 35, 21–33 (2004) THE EUROPEAN
Digital Object Identifier (DOI) 10.1140/epjc/s2004-01811-8
PHYSICAL JOURNAL C
pp
annihilation into four charged pions at rest and in flight
The OBELIX Collaboration
P. Salvini1, V.I. Tretyak2, V. Filippini1, A. Fontana3, P. Montagna3, A. Panzarasa1, A. Rotondi3,
M. Bargiotti4, A. Bertin4, M. Bruschi4, M. Capponi4, A. Carbone4, S. De Castro4, L. Fabbri4, P. Faccioli4, D. Galli4,
B. Giacobbe4, F. Grimaldi4, U. Marconi4, I. Massa4, M. Piccinini4, N. Semprini Cesari4, R. Spighi4, S. Vecchi4,
M. Villa4, A. Vitale4, A. Zoccoli4, A. Bianconi5, M. Corradini5, A. Donzella5, E. Lodi Rizzini5, L. Venturelli5,
A. Zenoni6, C. Cical`o7, A. De Falco7, A. Masoni7, G. Puddu7, S. Serci7, G. Usai7, O.E. Gorchakov2, S.N. Prakhov2,
A.M. Rozhdestvensky2, M.G. Sapozhnikov2, M. Poli8, P. Gianotti9, C. Guaraldo9, A. Lanaro9, V. Lucherini9,
C. Petrascu9, D. Panzieri10, F. Balestra11, M.P. Bussa11, L. Busso11, P. Cerello11, O. Denisov11,14, L. Ferrero11,
A. Grasso11, A. Maggiora11, F. Tosello11, E. Botta12, T. Bressani12, D. Calvo12, F. De Mori12, A. Feliciello12,
A. Filippi12, S. Marcello12, M. Agnello13, F. Iazzi13
1 INFN Sezione di Pavia, Pavia, Italy
2 Joint Institute for Nuclear Research, Dubna, Russia
3 Dipartimento di Fisica Nucleare e Teorica dell’Universit`a di Pavia and INFN Sezione di Pavia, Pavia, Italy
4 Dipartimento di Fisica dell’Universit`a di Bologna and INFN Sezione di Bologna, Bologna, Italy
5 Dipartimento di Chimica e Fisica per l’Ingegneria e per i Materiali, Universit`a di Brescia, Brescia, Italy and INFN gruppo
collegato di Brescia, Brescia, Italy
6 Dipartimento di Meccanica, Universit`a di Brescia, Brescia, Italy and INFN Sezione di Pavia, Pavia, Italy
7 Dipartimento di Scienze Fisiche, Universit`a di Cagliari and INFN Sezione di Cagliari, Cagliari, Italy
8 Dipartimento di Energetica dell’Universit`a di Firenze, Firenze, Italy and INFN Sezione di Bologna, Bologna, Italy
9 Lab. Naz. di Frascati dell’INFN, Frascati, Italy
10 DipartimentodiScienzeeTecnologieAvanzate,Universit`adelPiemonteOrientaleandINFNSezionediTorino,Torino,Italy
11 Dipartimento di Fisica Generale “A. Avogardo” dell’Universit`a di Torino and INFN Sezione di Torino, Torino, Italy
12 Dipartimento di Fisica Sperimentale dell’Universit`a di Torino and INFN Sezione di Torino, Torino, Italy
13 Dipartimento di Fisica del Politecnico di Torino and INFN Sezione di Torino, Torino, Italy
14 on leave ob absence from Joint Institute for Nuclear Research, Dubna, Russia
Received: 2 December 2003 /
Published online: 23 April 2004 – (cid:1)c Springer-Verlag / Societa` Italiana di Fisica 2004
Abstract. The spin-parity analysis of the data on the p¯p → 2π+2π− annihilation reaction at rest in
liquid and in gaseous hydrogen at 3 bar pressure and in flight at p momentum of (cid:3) 50 MeV/c, collected
by the Obelix spectrometer at the LEAR complex of CERN, is presented. The relative branching ratios
(a1(1260) → σπ)/(a1(1260) → ρπ) = 0.06±0.05 and (π(1300) → σπ)/(π(1300) → ρπ) = 2.2±0.4 are
obtained. It is also shown that the inclusion of the exotic meson π1(1400), JPC= 1−+, mass and width
M =1.384±0.028, Γ =0.378±0.058 GeV/c2, in its decay to ρπ, improves the fit and some implications
of these results are briefly discussed. The relative S and P-wave annihilation percentages in four charged
pions at two target densities are obtained.
1 Introduction ofwhatpresentlyarecalledtheρ(770),a (1260),f (1270)
1 2
anda (1320)mesons,togetherwithaππS-wavewithpoles
2
One of the most interesting aspects of the at0.8and1.1GeV/c2andazeroat0.94GeV/c2.Thespin-
parity analysis was performed by assuming only 1S and
0
p¯p→2π+2π− (1) 3S p¯pprotoniuminitialstates.However,fromtheensem-
1
ble of the existing data on pp annihilations at rest now it
annihilation at low energy is the possibility of producing turnsoutthatthepercentageofprotoniumannihilationsin
many light mesons, according to the list of Table 1. P-waves in liquid is larger than 10% [2–4]. We note that
Thefirstanalysisofthereaction(1)goesbacktoabout theinclusionofthefourP-wavesofTable1makestheanal-
30yearsago,whenDiazetal.[1]analyzed6665annihila- ysisdifficult,becausetheintermediatestatesproducedby
tionsatrestinaliquidhydrogen(LH2)bubblechamber.A theannihilationcontainmanylarge-widthresonancesthat
satisfactoryfitwasachievedbyconsideringtheproduction overlap or interfere each other.
22 The OBELIX Collaboration: pp annihilation into four charged pions at rest and in flight
Table 1. Possible intermediate states (channels) in the p¯p→2π+2π− reaction (no-
tations are from (5) and (6)). The exotic state 1−+ is indicated as π1, σ stands for
the (ππ) S-wave interaction and ρ(cid:1) for the ρ(1450) meson. Only the lowest angular
momenta L are considered
pp channel I L l1 l2 pp channel I L l1 l2
1S0 ρρ 0 1 1 1 3P0 σσ 0 0 0 0
a2(1320)π 0 2 2 1 ρρ 0 0 1 1
π1π 0 1 1 1 σf2(1270) 0 2 0 2
3S1 ρσ 1 0,2 1 0 a1(1260)π 0 1 0 1
ρ(cid:1)σ 1 0,2 1 0 [π(1300)]σππ 0 0 0 0
ρf2(1270) 1 0 1 2 [π(1300)]ρππ 0 0 1 1
a1(1260)π 1 0 0 1 3P1 ρρ 0 0 1 1
a2(1320)π 1 2 2 1 a1(1320)π 0 1 0 1
[π(1300)]σππ 1 1 0 0 a2(1320)π 0 1 2 1
[π(1300)]ρππ 1 1 1 1 σf2(1270) 0 2 0 2
π1π 1 1 1 1 π1π 0 0 1 1
1P1 ρσ 1 1 1 0 3P2 ρρ 0 0 1 1
a1(1260)π 1 1 0 1 σσ 0 2 0 0
a2(1320)π 1 1 2 1 a1(1260)π 0 1 0 1
ρf2(1270) 1 1 1 2 a2(1320)π 0 1 2 1
π1π 1 0 1 1 σf2(1270) 0 0 0 2
[π(1300)]σππ 0 2 0 0
[π(1300)]ρππ 0 2 1 1
Toovercomethisdifficulty,wedecidedtostudythere- (see Sect. 3) and with the inclusion in Table 1 of a set of
action(1)inflightwithlowmomentump¯andatrestwith intermediate states more complete than that considered
targets of different density. The annihilation in flight at in [10].
very low momentum proceeds mainly from S-wave, with One of the points of our investigation is the system
a P-wave contribution of about 4%, for a p momentum (π+π−) with the two pions in relative S and P waves.
of about 50 MeV/c [5–7]. On the contrary, the annihila-
TheππisoscalarS-wave,knowninthepastastheσme-
tion at rest takes place after the electromagnetic cascade
sonortheσinteraction(andnotedinthiswayinTable1),
of the antiprotonic atom and the percentage of annihila-
has been studied extensively during the last decades. An
tionsfromtheSandP-atomicstatesvarieswiththetarget
updatedreanalysisofthetopiccanbefoundin[11],wherea
density[2–4].Inthiswaythemixtureofthesixp¯ppartial
fullsetofreferencestoearlierworksisalsogiven.Atpresent,
wavesorinitialstatesischanged,togetherwiththecorre-
thiswaveisparametrizedasthecoherentsumoffourpoles:
sponding meson production from the intermediate states. the broad f (600) (now called the σ, Γ (cid:2) 600 MeV),
0
In principle, the combined spin-parity analysis of these the narrow f (980) (Γ (cid:2) 50 MeV), the broad f (1370)
0 0
data should help to disentangle the contributions of the (Γ (cid:2)300MeV)andthenarrowf (1500)(Γ (cid:2)100MeV).
0
various mesons.
In the previous work [8] we verified that the data in flight
In 1997 our Collaboration published the spin-parity were compatible with a three pole S-wave; here in Sect. 3
analysis of a sample of 31157 annihilation events taken in wecheckthispointbyfittingthethreedatasamples,com-
flightatverylowp¯incidentmomentum((cid:2)50MeV/c)[8]. paringthisresultalsowiththatobtainedwithsomerecent
Thebestfitsolutionrequiredalsothepresenceofaρ(1450) four pole amplitudes.
state and of a heavy pion π(1300). However, these contri- For what concerns the (π+π−) P-wave, in addition to
butions were small and only the S-wave was considered, thehugeρ(770)signal,thereissomeuncertaintyaboutthe
omitting the 4% of P-wave contribution. parametersoftheρ(1450)andρ(1700)radialexcitations[8,
Thisstudycontinuedwiththecombinedanalysisoftwo 11].Inourcaseweconsideredonlythelowermassρ(1450)
samples obtained from the annihilation at rest in gaseous case, listed as ρ(cid:1) in Table 1. Here we investigate further
andinliquidhydrogen(LH2)targets[9]andwiththevery the influence of the ρ(cid:1) mass on our best fit solution, since
preliminary analysis of all the three samples [10]. in our previous analysis we found a mass lower than that
Herewepresentthefinalresultsofthespin-parityanaly- quoted in the Particle Data Group (PDG) [8].
sisofthesethreesamples,obtainedthroughthecontinuous Anotherimportantreasonforourinterestinthep¯p→
refinementofouralgorithmsandmodelsoftheamplitude 2π+2π− channelconcernstheexoticmesonπ (1400)with
1
The OBELIX Collaboration: pp annihilation into four charged pions at rest and in flight 23
quantumnumbersJPC =1−+,I =1,whichisacandidate
for a hybrid meson or a multiquark state, with quantum
numbers not accessible to the qq system. The evidence of
its existence comes from two different experiments that
analysed the decay mode into ηπ [12–15].
Inthesedatawesearchforthismesonbyanalyzingthe
π → ρπ decays, that are allowed from the 1S , 3S , 1P
1 0 1 1
and3P states.Forthefirsttimeweconsiderherealsothe
1
π production from S-states and perform a grid search to
1
find the best fit mass and width values.
Finally,weconsideralsotheπ(1300)meson(indicated
alsoasπ(cid:1))productionanditsdecaybranchingratiointoρπ
and σπ, which is still not very well known experimentally
(theoreticalpredictionscanbefound,forexample,in[16]).
The amplitude obtained taking into account all these
physicsaspectsandtheintermediatestatesofTable1gives
much better χ2 values than in [10]. The results will be
discussed in detail in Sect. 4.
2 Event selection
Fig.1.Missingmasssquareddistributionofthe4-prongevents
thatpasstheselectioncriteriadescribedinthetext.Thepeak
We consider a total of 90000 annihilation events from re-
corresponds to the p¯p → 2π+2π− channel, the shoulder at
action 1, divided into three different data sets, each one
its right contains mostly p¯p → 2π+2π−π0 events, whereas
with (cid:2) 30000 annihilation events, obtained at rest on a
the underlying flat background comes mainly from the p¯p →
LH2 target, at rest on a gaseous hydrogen target (at 3 π+π−2π0 reaction
barpressure)andinflightatverylowincidentantiproton
momentum (around 50 MeV/c).
The data were taken until the end of 1996 with the
Thecompletedescriptionoftheapparatuscanbefound
Obelix spectrometer at the Low Energy Antiproton Ring
in [21].
at CERN. Foreachdatasetwecollectedabout4106 annihilation
Thewholeapparatuswasimmersedinamagneticfield,
events with a four prong TOF multiplicity trigger. About
reaching the maximum value of (cid:2) 0.5 T along its axis. In one tenth of these events((cid:2) 3.5105) passed the following
ordertooptimizethenumberoftheannihilationevents,the
“4-prong” selection criteria:
inner part of the apparatus was different for the different
data samples. For the annihilation events at rest in the - fourtrackssuccessfullyconnectedtothevertexintothe
gaseoushydrogentargetandinflightatlowmomentum,a fiducial target volume;
20cmradiusand(cid:2)50cmlengthcylindricaltargethasbeen - zero total charge;
used, containing gaseous hydrogen at 3 bar (at rest data) - tracklengths>20cm,tohaveamomentumresolution
or0.8bar(inflightdata)pressure.Adetailedexplanation <2% [20].
oftheexperimentallayoutandofthein-flightannihilation
selectionisgivenin[17,18].Fortheannihilationeventsat Thetypicalmissingmassdistribution(forinflightannihi-
rest in LH2 the target was the same used for the n¯ data lation) of the 4-prong events obtained with this selection
taking [19]. procedure is shown in Fig. 1. Finally, the p¯p → 2π+2π−
Thetargetsweresurroundedbyafirstlayerof30thin(1 eventswereextractedfromthe4-prongsetsbyapplyinga
cm)plasticscintillators,arrangedinabarrelandproviding four-constraintkinematicalfitwith95%C.L.Attheendof
thestartforthetimeofflightsystem(TOF).Theincident theselection,thethreefinaldatasetshad(cid:2)30000events
antiprotons were selected by requiring the coincidence of each. All this procedure was carefully repeated with the
the signals of the beam scintillator and this scintillator Obelix Monte Carlo code, and the resulting background
barrel.Theouterscintillatorbarrel(90elements,9×4×350 contamination resulted less than 1%.
cm3 each) allowed the first level trigger on multiplicities The contamination from the p¯p → K0K0 → 2π+2π−
S S
andonparticleidentification.AJetDriftChamber(JDC), reactionisnegligible,sincetheK0K0 intermediatestateis
S S
placed between the two scintillator barrels, was devoted forbiddenfromS-waveinitialstates,sothatthemeasured
to the tracking of charged particles, with a momentum frequency in LH2 is consistent with zero ((4 ± 3)10−6,
resolution of ∼ 2% for pions around 300 MeV/c [20]. The from [22,23]) and in a NTP gas target is very low ((3±
JDC allowed the determination of the annihilation vertex 1)10−5, from [24]). Comparing this last value with the
and contributed with the TOF to the identification of the frequency (4) and taking into account the K0 → π+π−
S
event topology and to the particle identification through branchingratio,oneobtainsanupperlimitcontamination
the dE/dx measurement. of 7 K0K0 events in a sample of 30000 events.
S S
24 The OBELIX Collaboration: pp annihilation into four charged pions at rest and in flight
Fig. 2. Bidimensional invariant mass distributions for p¯p → 2π+2π− experimental data taken in hydrogen targets at three
different conditions. The number of combinations per event is 4 for M(πa+πb−) versus M(πc+πd−) (top) and 8 for M(πa+πb−πc±)
versus M(πa+πb−) (bottom)
Thep¯p→2π+2π−reactionbranchingratioshavebeen production of the f (1270) meson increases with P-wave.
2
previously measured and are: Finally, the scatter plot of the three pion invariant mass
versusthetwopiononeofFig.2andthespectrad,e,and
B4π(in flight)=(7.61±0.35)10−2 , (2) fofFig.3showacleara2(1320)production,associatedto
B4π(liquid)=(6.00±0.24)10−2 , (3) a more complicated dynamics.
B4π(NTP gas)=(6.4±0.9)10−2 , (4)
3 The amplitude
wherethevalue(2)comesfromourpreviousanalysisofthe
inflightdata[8],thevalue(3)istheweightedaverageofthe
valuesatrestinLH2(6.9±0.6)10−2[1],(5.8±0.3)10−2[22], Weassumethattheamplitudefortheppannihilationinto
(5.9±0.5)10−2 [23]andthelastvalue(4)comesfromthe four pions is well described taking into account the two
following decay chains :
annihilations at rest in gaseous hydrogen at NTP of [24].
is aWgeooadsseusmtiemaintethaelsofoflolorwoiunrg3thbaatrtghaesNtaTrPgetgaisnvsapliutee p¯p→[AB]L→[(ππ)l1(ππ)l2]L (5)
ofthedifferenttargetpressures,becausetheinterpolation
(cid:1) (cid:2) (cid:1) (cid:2)
pofretshseurPes-wshaovwesptehrcaetntthaegebsrabnectwhienegnr1ataionsdta1k5enbaartt3arbgaetr p¯p→[Aπ]L→ [Bπ]l1π L→ [(ππ)l2π]l1π L (6)
target pressure can differ from the NTP ones by no more
where A and B are resonant states and L, l , l are the
than 5% [2,25]. 1 2
twoparticleorbitalangularmomentumquantumnumbers.
Asdiscussedbefore,thethreedatasetsareexpectedto
The possible intermediate states (channels) are summa-
be very different for what concerns the mixture of partial
rized in Table 1.
waves involved in the pp annihilation. ThedecayintensityJ fortheannihilationatrestcanbe
This behaviour is confirmed by our data, as reported
writtenasanincoherentsumoverthesixlowestppatomic
in Figs. 2 and 3. In the π+π− scatter plots of Fig. 2 a states of Table 1 with JPC (or 2S+1LJ) fixed quantum
strong ρρ signal is present in each sample; a system made
numbers. Indicating with the index J these atomic states
byaρrecoilingagainstaππcouple(theσ)withanenergy
we can write:
around 1.1 GeV is also evident in the in flight and liquid (cid:3)
target samples, that is in association with a high fraction J(q1,q2,q3,q4)= |AJ |2 where (7)
of S-waves. Figures 3a, b, and c show clearly that the
J
The OBELIX Collaboration: pp annihilation into four charged pions at rest and in flight 25
Fig. 3. Comparison between experimental data (points with error bars) and theoretical amplitude (histogram) from fit D. a,
b, c: M(π+π−), d, e, f: M(π+π−π±) respectively for the three different data sets. The number of combinations per event is 4
for all the plots
(cid:3) (cid:3) (cid:3)
AJ = [aJkCIZJk(q)Fk(q)]p ≡ aJk FIkp , usually assumes aJ ≡ aJM, |M| ≤ J, so that any term
Ikp k Ip aJλpλp in(8)correspondstoatermaJM withM =0,±1.
Inthisway,fortheinflightdatathefreeparametersarea
where qi, i=1,2,...,4 are the four-momenta of the final subset of the aJ coefficients of the at rest amplitude, and
pions,CIk aretheisospinc(cid:4)oefficients,aJk arethecomplex theconnectionbetweentheinflightandatrestamplitudes
parameters to be fitted, stays for the permutations is assured during the fit.
p
over identical particles, k ≡ (L,l ,l ) identifies the set of ThespinfunctionsZarewrittenintermsofthecovari-
1 2
quantumnumbersofthestateslistedinTable1,ZJk(q)are ant Zemach tensors in the formulation of [27], extending
thespinfunctions,F describestheenergybehaviourofthe the formalism to the case of the (ρρ)l system in a relative
decaychainandqdenotesthesetofbreak-upmomentaof angularmomentuml=1(seetheAppendix).Alltheseten-
theintermediatestatesofthereaction.Thenormalization sorsgivethesamespinamplitudesofthecovarianthelicity
is such that |aJk|2 represents the fraction of events due formalism of [28].
to the corresponding FIkp isobar amplitude, ignoring the As in our previous paper [8], the complex functions
interference terms. Fk(q) describe the energy-dependent part of the ampli-
Thedecayintensityforthein-flightannihilationevents tudes. In most of the cases of Table 1 they are writ-
is written as an incoherent sum over the four possible ten as a product of Breit-Wigner functions describing the
(λp,−λp) helicity states of the pp system, corresponding resonances present in the decay chain, parametrized as
to the three spin component values MJ =0, ±1: in [29,30]:
(cid:3) 1 m Γ
J(q1,q2,q3,q4)= |Aλpλp |2 where (8) F(q)= (qR)l Wl(q)m20−m2−0 i0m0Γ(m) (9)
λp,λp
(cid:3)
Aλp,λp = IJp[aJ,λpλpkCIZJ,λpλpk(q)Fλpλpk(q)]p . Γ(m)=Γ0 mm0 qq0 WWl2l2((qq0))
We note that in (7) states with different J ≡ JPC add where R = 1 fm (a typical value [30]), q0 is the break-up
momentumcalculatedattheresonancemassvalue,listhe
incoherently because they are in principle distinguishable
relative angular momentum of the resonance decay prod-
(astheprotoniumcascadeX-raysareobservable),whereas
in (8) the states with different JPC values interfere [26]. ucts, and Wl are the Blatt-Weisskopf centrifugal barrier
factors. The first three coefficients with l=0,1,2, are:
Since in (7) the complex coefficients aJk parametrize
theisobardecayprobabilityfromacertainJPC state,one W (q)=1 ,
0
26 The OBELIX Collaboration: pp annihilation into four charged pions at rest and in flight
(cid:5) (cid:6)
2 1/2 Λ =−1, corresponding to the usual K-matrix scattering
3
W (q)=qR , (10)
1 1+(qR)2 theory [29].
(cid:5) (cid:6) Duringthedataanalysisdescribedbelow,wecompared
13 1/2 the N/D approach to the P-vector one [29] and verified
W (q)=(qR)2 .
2 9+3(qR)2+(qR)4 that the two parametrizations give consistent results, as
in [33].
Thefactor1/(qR)l multiplyingtheBreit-Wignerfunction Finally,wetreattheoverlapoftheρandρ(cid:6)resonances,
eliminates the terms (qR)l in (10), since they are already whichhavethesamespin-parityandthesamepartialwaves,
presentinthecovariantZemachtensors.Onthecontrary, bywritingthefunctionFkof(7,8)asa1×1K-matrix[29]:
this correction is absent in the definition of the widths
Γ(m). Fk(m)=[1−iρ(m)Kρρ(cid:1)]−1Pρρ(cid:1) , (16)
The break-up momenta q(m) f(cid:7)or the decays into two where
pions are calculated as q(m)=m 1−(4m2/m2)/2 and
π (cid:5) (cid:6)
tphaertItinwalotwh-beidoctdhaysmepahosafssqdeuesappseainctdeweinos-cbgeiowvdeeynevdbayelcuρaay(tmse)ttoh=edp2eqhs(camrsieb)s/epmtah.cee Kρρ(cid:1) =(cid:3)j=21 ρm(miΓjj) WW11((qqj)) 2 m2j −1 m2
factor integrating the squared decay amplitude over the
(cid:9)
(cid:5) (cid:6) (cid:10) (cid:11)
three-pion phase spρajc(ema)v=a(cid:8)ilab|Alej|Φ2d3Φ3, (11) Pρρ(cid:1) =(cid:3)j=21βj ρm(mjΓjj) WW11((qqj)) qq0jj m2j −1 m2
andmj,Γj,ρ(mj)andqj aremass,width,two-bodyphase
where A is the decay amplitude of a1(1260), a2(1320), space factor and break-up momentum of the two ρ’s. The
π(1300) or π1 and j = (ρπ), (σπ) stands for the decay factor W1 is defined in (10). We drop also in this case, by
mode. For instance to describe the spin-0 resonance de- multiplying by (qj/q), the factor q from the P-vector, be-
cays π(1300) → σπ , π(1300) → ρπ, with nominal mass causeitisalreadypresentintheZemachtensors.Thecom-
and width m0 and Γ0 and the coupling coefficients γj2 = plex production parameters βj are left free during the fit.
(Γj/Γ0)/ρj(m0),weusetheBreit-Wignerfunction[29,31]: The data have been fitted with the maximum likeli-
hoodmethod.UsingtheMinuitprogram[34]weminimize
[Fk(q)]j = m2−m2m−0Γi0mγjΓ (m) , (12) Lthe q=uaLntity −Lln(LtLot),.wOhnereeaLchtotdaistathseett,otthael lliikkeelliihhoooodd
0 0 tot tot flight liquid gas
is computed as
with partial and total widths given by:
N(cid:12)ev (cid:13)(cid:14)(cid:3) (cid:15)
Γj(m)=Γ0γj2ρj(m) (13) L= Jj JMC , (17)
(cid:3) j=1
Γtot(m)= j Γj(m) . where Nev=30000, Jj is the event by event decay proba-
bilityof (7)and(8)andthedenominatoristhelikelihood
The decay branching ratio is defined in this case as: normalization,evaluated(foreachdatasample)overaset
of 90000 Monte Carlo events.
Γ(π(cid:1) →σπ) = γπ2(cid:1)→σπρπ(cid:1)→σπ(m0) (14) Theannihilationfractionsfromoneofthesixp¯pinitial
Γ(π(cid:1) →ρπ) γπ2(cid:1)→ρπρπ(cid:1)→ρπ(m0) states J are defined from the aJK coefficients of (7) as:
(cid:4)
for AthseincasoeurofptrheveioπuπsSp-awpaevre[8in],tetrhaectfiuonnct(itohneFσkteorfm(7i)n, PJ = (cid:4)Jkk||aaJJkk||22 . (18)
Table1),isparametrizedintermsoftheK-matrixformal-
These fractions can be multiplied by the branching ratio
ism[29].FollowingtheN/Dmethod,itispossibletowrite (2–4), giving the p¯p → 2π+2π− branching ratio from the
the σ interaction in the form [32]:
state J, that usually is written in the form [2–4]:
(Λ +Λ s)K +iρ Λ (K K −K2 )
Fk=1−iρ1K 2−iρ11K −2ρ3ρ (1K1 2K2 −1K22 ) (15) PJ(δ)B4π(δ)=fJ(δ)EJ(δ)gJBJ , (19)
1 11 2 22 1 2 11 22 12
where the explicit dependence of the terms on the target
where s is the invariant mass squared of the ππ system, density δ has been put in evidence. Here fJ is the S-wave
and ρ and ρ are the two-body phase space factors for (or P-wave) fraction in the case of the 1S , 3S (or 1P
1 2 0 1 1
the decay into ππ and KK¯. The Λ production complex 3P0 3P1 3P2) states, EJ is the enhancement factor that is
parametersaredifferentforeachofthesixinitialp¯pstates equaltoonewhenthepopulationoftheJ statefollowsthe
ofTable1andareleftfreeduringthefitwhenσisproduced spin statistics, gJ is the known spin statistical weight and
in p¯p annihilation. For the σ produced by the resonance BJ is the density independent hadronic branching ratio
decays, we use the Λ parameters fixed at Λ =1, Λ =0, p¯p→2π+2π− from the J channel.
1 2
The OBELIX Collaboration: pp annihilation into four charged pions at rest and in flight 27
Table 2. Results of fit A (see the text). The decay fractions are normalized to 100 for
each p¯p state, whose relative weight within the S and P-waves is also given. The π(cid:1)
indicates the π(1300) pion and σ the (ππ) S-wave interaction. Some of the channels of
Table1,suppressedfordynamicalreasonsandrefusedbythefit,areomittedhere.The
interference terms for the amplitude of (8) are very different from those of (7), so for
thein-flightdatawegiveonlythetotalP-wavepercentage.Theχ2isthepseudo-χ2 per
bin calculated over a set of 15 histograms (see the text)
pp channel LH2 and gas pp channel LH2andgas
1S0 all 24.5±0.5 3P0 all 25.4±0.9
3S1 all 75.5±0.5 3P1 all 9.4±1.0
1P1 all 7.6±1.3 3P2 all 57.6±1.2
1S0 ρρ 31.8±0.9 3P0 σσ 60.5±1.5
a2π 68.2±0.9 ρρ 16.9±1.2
3S1 ρσ 53.4±0.6 σf2 8.7±1.0
ρf2 23.0±0.5 a1π 12.3±1.2
a1π 18.0±0.5 π(cid:1)π 1.6±0.6
a2π 2.3±0.2 3P1 ρρ 36.2±3.4
π(cid:1)π 3.3±0.3 σf2 23.7±4.1
1P1 ρσ 56.5±4.0 a1π 10.1±3.4
ρf2 10.3±0.7 a2π 30.0±6.2
a1π 0.7±0.2 3P2 σσ 2.4±0.4
a2π 32.5±4.8 σf2 48.3±1.3
a1π 25.1±1.3
a2π 20.2±1.0
π(cid:1)π 4.0±0.8
P-wave percentages χ2
flight LH2 gas 2lnLTOT flight LH2 gas
10.2±0.9 28.2±0.8 64.6±0.9 61758 1.3 1.3 1.3
4 Results less constant also when significant changes in the log like-
lihood are observed. For these reasons in the following we
The most important fits will be referred to as follows: will refer to this quantity as the pseudo-χ2.
In Figs. 3 some of the histograms of the previously
A) All the channels from Table 1 are considered, except
thoseconcerningtheπ (1−+)production.Theannihi- mentioned set are shown. In Tables 2 and 3 the results
1 from fits A and D are presented.
lationfractionsdefinedin(18),(19)fromSandP-waves
The (ππ) S, P and D-waves considered in the fit are
(different L states) vary with the target pressure, but
listed in Table 1.
the atomic S and P-sublevel fractions are the same in
Themorecomplicatedisthe(ππ)S-wave(indicatedasσ
liquid and at 3 bar hydrogen target pressure.
inTables1,2and3)sinceintheenergyregionavailablefour
B) The same channels as in fit A are included. The anni-
resonant states are present: the very broad f (400–1200),
hilationfractionsfromatomicsublevelscanvaryfreely 0
thef (980),f (1370)andf (1500).Thespin-parityfithas
as a function of the target pressure. 0 0 0
beenperformedusingdifferentformsforthe(ππ)S-wave:a
C) Same as fit A, but with the π production.
1 4-poleK-matrixfrom[35]anda3-poleK-matrixfrom[36],
D) Same as fit B, but with the π production.
1 using(15)andtheN/Dmethod[32].Westressthat,inthe
The different fits are obtained by minimizing the log like- caseof3-poleK-matrix,theN/Dmethodisequivalentto
lihood(17)andcanbealsoqualitativelycomparedonthe the production vector approach, as both introduce three
basisofaχ2perbinfromasetof15histograms,empirically complex parameters (for each initial state) in the fit and
chosen to give a χ2 tracking adequately the log likelihood it is possible to write the analytical relations between the
changes and assuming a value (cid:2) 1 over a set of Monte parametersinthetwoapproaches.Onthecontrary,theuse
Carlo data when the right amplitude is used. However, in ofthe4-pole(ππ)S-waveintheP-vectorframeintroduces
some cases, in the vicinity of the best fit solution, this χ2 fourcomplexparameters:onemorethantheN/Dapproach.
behaves as an inconsistent estimator, remaining more or Despite the different number of fit parameters, the two
28 The OBELIX Collaboration: pp annihilation into four charged pions at rest and in flight
Table 3. Results of fit D and percentages of the intermediate states. Description as in Table 2, only
the % of atomic sublevels are different for annihilation in gas (3 bar pressure) and in liquid
pp channel LH2 gas pp channel LH2 gas
1S0 all 25.3±0.7 23.4±1.2 3P0 all 41.2±1.8 23.8±0.9
3S1 all 74.7±0.7 76.6±1.2 3P1 all 15.6±1.6 8.2±0.8
1P1 all 3.2±1.5 9.6±0.8 3P2 all 40.0±2.3 58.4±1.1
1S0 ρρ 33.1±0.8 3P0 σσ 56.0±1.4
a2π 58.4±2.0 ρρ 16.3±2.5
π1π 8.5±1.8 σf2 8.7±0.9
3S1 ρσ 52.0±0.7 a1π 15.6±1.5
ρf2 24.6±0.5 π(cid:1)π 3.4±0.7
a1π 17.8±0.5 3P1 ρρ 34.1±0.7
a2π 1.9±0.2 σf2 16.2±3.0
π(cid:1)π 2.8±0.2 a1π 0.6±0.6
π1π 0.9±0.1 a2π 2.0±2.0
1P1 ρσ 42.5±4.4 π1π 47.1±5.2
ρf2 14.8±3.8 3P2 σσ 2.1±0.4
a1π 4.3±1.9 σf2 51.6±1.0
a2π 34.7±5.2 a1π 22.3±1.3
π1π 3.7±2.0 a2π 20.4±0.9
π(cid:1)π 3.6±0.7
P-wave percentages χ2
flight LH2 gas 2lnLTOT flight LH2 gas
10.3±0.9 27.8±0.8 63.5±0.9 62240 1.2 1.2 1.2
different (ππ) S-waves in the two approaches give similar ]d
a
results, both for histogram shape and likelihood values: [r
0δ0
the fact can be explained taking into account that in our
channel only the low-energy part of the (ππ) S-wave is 6
important,whichisverysimilarforthetwo(ππ) S-waves
used. In the following we will refer to the results obtained
with the (ππ) S-wave from [36] as it requires less free 4
parameters in the fit. The corresponding phase motion is
compared in Fig. 4 with the ππ → ππ scattering data,
averaged from [37].
2
The(ππ)P-wavewasparametrizedbytheρ(770)Breit-
Wignerfunction,whilefortheρσstatecomingfromthe3S
1
and1P initialstates,wefoundnecessarytoincludeinthe
1 0
(ππ) P-wave also the ρ(1450) to improve the fit quality. 0.5 1 1.5
In this case we describe the ρ(770) and its first radial Mππ[GeV/c2]
excitation ρ(1450) in the same partial wave by means of Fig. 4. ππ S-wave amplitude phase motion (solid line) com-
the two-pole K-matrix of (16). Since the ρ(1700) state is pared to the data from [37]
veryfarfromthephase-spaceavailableforour4πchannel,
it was neglected in our analysis.
The Fk pole of (16) corresponding to the ρ(1450) is change significantly the low best fit value of the ρ(cid:1) mass.
found at a mass M = (1.360±0.010) GeV/c2 and width Thisresultcouldbeduetotherescatteringoftheρ(cid:1)daugh-
Γ =(0.270±0.020) GeV/c2. These values are lower than terpionswiththeothertwospectatorpions,thatthesym-
the nominal mass and width values quoted for this reso- metrized decay amplitudes based on the isobar model do
nance,buttheyresultinfullagreementwiththeweighted not take into account [38].
average of the world data for the ρ(1450) → ππ decay: Fortheresonancesdecayinginto3pionsthroughρπor
M =(1.376±0.007)andΓ =(0.290±0.008)GeV/c2[11]. σπwehaveconsideredthefollowingwellestablishedstates:
The use of different (ππ) S-wave descriptions does not a (1320), a (1260) and π(1300). Of these three mesons,
2 1
The OBELIX Collaboration: pp annihilation into four charged pions at rest and in flight 29
onlytherelativelynarrowa (1320)revealsaclearlyvisible For the π(1300) meson the best fit gives M =1.200±
2
peak in the 3π invariant mass distributions (see Fig. 3). 0.040GeV/c2andΓ =0.470±0.120GeV/c2.Byusing(14)
Weusethea (1320)massandwidthfromtheparticle we find the π(1300) decay branching ratio
2
data book [11], since they are known with rather good
accuracy. For the a1(1260) meson we use the values M = Γ(π(1300)→σπ)
1.230 GeV/c2 and width Γ = 0.400 GeV/c2 from [11], =2.2±0.4 , (21)
Γ(π(1300)→ρπ)
or we leave these parameters free in the fit; the best fit
result is obtained with M = 1.330±0.024 GeV/c2 and
Γ =0.580±0.041GeV/c2,amassvalueincloseagreement a value not in good agreement with our previous result of
with the CLEO results [39]. 5.2±0.7 [8] but in full agreement with the old analysis of
Wecannotexcludethatthelargewidthobservedforthis AaronandLongacreonthethree-pionsystemsproducedin
mesonisduetotheinterferencewiththeheaviera (1640), theπ−+p→(3π)+pandπ−+p→(3π)+preactions[43].
1
whichisatthelimitofourphasespace.Thereistentative However,wenotethatarecentspin-parityanalysisofthe
evidence for this meson in the τ → ντ3π decay [39,40] p¯d→π−4π0pspectator reactiongivesacompletelydifferent
and,morerecently,itsexistencehasbeensuggestedinthe value <0.15 for this ratio [44].
analysis of the p¯p→4π0 reaction in flight [41] and of the Also the presence of the exotic JPC = 1−+ π1(1400)
p¯p→ω3π0annihilationatrest[42].Therefore,ifthiseffect state proceeding through the ρπ decay has been investi-
is present in the data, our broad a can be regarded only gated. When the 1−+ production is included it improves
1
as an effective parametrization of the 1++ amplitude. sensiblythefit,undermanydifferentconditions:withdif-
The fit assigns a negligible percentage to the a decay ferent(ππ) S-waveandP-waveparametrizations,withthe
1
in σπ: a1 andπ(1300) mass and width values left free or fixed to
Γ(a →σπ) themeanvaluesof[11],withtheannihilationfractionsfrom
1 =0.06±0.05 , (20)
Γ(a →ρπ) antiprotonic atom sublevels fixed (fit A and C) or consid-
1
ered as free parameters varying with the target pressure
in agreement with the previous results [11]. (fit B and D).
1.6 1.6
a) fit B (no exotic) 3 b) fit D (exotic) 3
1.4 1.4
2]c 2]c
eV/ 1.2 2.5 eV/ 1.2 2.5
G G
π-) [ 1 2 π-) [ 1 2
+ +
πM( 0.8 1.5 πM( 0.8 1.5
0.6 0.6
1 1
0.4 0.4
0.5 0.5
0.2 0.2
0 0 0 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
M(π+π-) [GeV/c2] M(π+π-) [GeV/c2]
1.8 1.8
2π±] ) [GeV/c 111...246 23.5 2π±]) [GeV/c 111...246 23.5
π- 2 π- 2
+ 1 + 1
π π
M( 0.8 1.5 M( 0.8 1.5
0.6 0.6
1 1
0.4 0.4
0.5 0.5
0.2 0.2
c) fit B (no exotic) d) fit D (exotic)
0 0 0 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
M(π+π-) [GeV/c2] M(π+π- ) [GeV/c2]
Fig. 5. Absolute differences (in standard deviation units) between the bidimensional invariant mass distributions from fits B
and D (see the text) and the gas target data (3 bar pressure). For clarity the differences <1 are set to zero. The number of
combinations per event is as in Fig.2
30 The OBELIX Collaboration: pp annihilation into four charged pions at rest and in flight
)max200 2]V/c
nL Ge 0.5
−l [h
L dt
−2(ln 150 π wi1
0.4
100
0.3
50
0.2
0 1.3 1.35 1.4 1.45
π mass [GeV/c2]
1.2 1.4 1.6 1
π mass [GeV/c2] Fig. 7. 1σ,2σ and 3σ error ellipses for the estimate of the π1
1 mass and width from the value of lnL
Fig. 6. Dependence of (−2lnL) on π1 mass for the set of π1
width values
Wetrieddifferentfits,creatingagridofmassvaluesfor
theexoticwithstepsof50MeV/c2 andevaluatingthefits
It is also important to note that the fit results for the onthebasisofthedifferencein∆(−2lnL).Thebehaviour
masses and widths of a and π(1300) do not change sig- of(−2lnL)asafunctionoftheπ massvalueisshownin
1 1
nificantly also when the π contribution is considered. Fig. 6, and the minimum of −2lnL obtained by varying
1
Theimprovementinthefitduetothe1−+contribution both the π1 mass and width is shown in Fig. 7. The best
can be seen in Table 4 where the fits A and C (A–C) and estimate of the π1 parameters is
thefitsBandD(B–D)arecompared.Sinceaunitchange
inourlikelihoodcorrespondstoonestandarddeviationfor M =1.384±0.020 (stat)±0.035 (sys) GeV/c2 (22)
one degree of freedom, a five standard deviation improve-
ment corresponds, in Table 4, to a ∆(−2lnL) change of Γ =0.378±0.050 (stat)±0.050 (sys) GeV/c2 ,
(cid:2) 50 for each data set (in flight, liquid and gas). There-
wherethesystematicerrors(writtenas±∆/2,100%cover-
fore,wededucethattheintroductionoftheexoticgivesa
age)havebeenfoundbyvaryingthea (1260)andπ(1300)
definitive improvement in the data description. Since the 1
parameterswithintheirPDGconfidenceintervalsandcon-
absenceoftheexoticinthe3P amplitudeiscompensated
1 sideringtheshiftofourbestfitρ(1450)masswithrespect
by an increase in the a and a production (see Tables 2
1 2 to the PDG value. This result is in agreement with the
and 3), we observe only slight improvements in the scat-
present π world data [11].
ter plots after the inclusion of the exotic. However, the 1
improvement is present in all the spectra, with the best Theexoticmesonisproducedmainlyfromthe1S0 and
3P initial states, with a dominant production from the
evidence from the gas data (see Fig. 5). In particular, by 1
3P one, which is the only triplet state where there is no
comparing Figs. 5c and d, one sees a significant improve- 1
ment in the M(π+π−π±) invariant mass spectra around orbital angular momentum between the produced π1 and
therecoilingπ (L=0inTable1).Theinclusionoftheπ
the π mass. 1
1
productionfromthisstategivesbestfitvaluesforthea and
Asafurthercheck,weverifiedthattheintroductionof 1
a non-exotic state with JPC = 1+− and mass around 1.4 a2productionconsistentwithzero(seeTable3).Ifthesetwo
mesonsareexcludedfromthe3P state,theπ production
GeV/c2 doesnotimprovethefitquality,sinceinthiscase 1 1
∆(−2lnL)<10 for all the three data sets. increases from (47.1±5.2)% up to (53.5±5.3)% with a
negligiblelikelihooddecrease.Thedifferencebetweenthese
two percentages can be considered as an estimate of the
systematic error of the analysis.
Table 4. Changes in the log likelihood ∆(−2lnL) when the Tocheckwhethertheπ presencecouldmimicpossible
1−+ exotic is removed and the remaining contributions are 1
effectsduetotheapparatusacceptance,werepeatedthefit
re-optimized in the case of fixed (A–C) and free (B–D) S and
Dwithouttheinflightdata.Alltheresultingintermediate
P atomic sublevel fractions
state percentages have uncertainties very similar and are
inagreementwithinerrorstothedatareportedinTable3.
fits flight liquid gas total
The π fractions without the in flight data, for the 1S ,
1 0
A–C 180 142 100 442 3S1, 1P1, 3P1 waves are 6.9 ± 1.2, 1.1 ± 0.3, 21.8 ± 8.2
B–D 181 160 101 422 and 44.1±5.7 respectively, to be compared with the data
of Table 3. The larger difference is observed in the 1P
1