Hadroni Physis in Peripheral Heavy Ion Collisions
A. A. Natale
Institutode FsiaTeoria, UniversidadeEstadual Paulista,
RuaPamplona145, 01405-900,S~aoPaulo,SP,Brazil
Reeivedon9Marh,2002
Wedisusstheprodutionofhadroniresonanesinveryperipheralheavyionollisions,wherethe
ionsollidewithimpatparameterlargerthantwiethenulearradiusandremainintatafterthe
ollision. Weomparetheresonaneprodutionthroughtwo-photonanddoublePomeronexhange,
showingthatwhenweimposetheonditionforaperipheralinterationtheproessdominates
overthePomeroninteration,duetotheshortrangepropagationofthislastone. Wealsodisuss
theobservationoflightresonanesthroughthesubproess !R !,whihisaleansignal
forglueballandidatesaswellasonewaytohektheexisteneofapossiblesalar meson.
I Introdution
Collisionsatrelativistiheavyionolliderslikethe
RelativistiHeavyIonColliderRHIC/Brookhavenand
the Large Hadron Collider LHC/CERN (operating in
itsheavyionmode)aremainlydevotedtothesearhof
theQuarkGluonPlasma. Inadditiontothisimportant
feature of heavy-ionolliders, wewill also have
ultra-peripheralollisions with impat parameterb > 2R
A ,
where R
A
isthenulearradius,andwhere theions
re-main intataftertheollision.
These interations will be mostly of
eletromag-netiorigin:two-photon()orphotonulearproesses
(A). Duetotheverystrongphotoneldofeahharge
Z aeleratedion,the photonluminositywill bequite
high. IntheaseofRHIC nalstatesproduedin the
two-photonproesswithaninvariantmassuptoafew
GeV will appear at large rates. Above this sale the
photonluminositydropsveryfast. AtLHCanalstate
withamassalmosttwoordersofmagnitudelargeran
still be produed at reasonable rates. The variety of
proessesthat an be studied in heavy ionperipheral
ollisionshavebeenextensivelyreviewedreently[1℄.
Thefatthathadroniresonanes(R )ouldbe
pro-dued at largerates in peripheral heavyion ollisions
wasalreadydisussedmanyyearsago[2℄. Perhapsthis
may be oneof the mostinterestingstudies to be
per-formed at RHIC, beause the mahine will serve asa
fatory of light hadronsin and A reations.
Ve-tor resonanes will appear at huge rates in
photonu-lear reations[3℄, as well as salar and pseudosalar
resonanesintwo-photonproesses[4℄. Wewill
parti-andpseudosalarresonanesthroughtheproess.
Two-photon physisat e +
e olliders provided for
alongtimealotofinformationonhadroniresonanes
[5℄. Thetwo-photonproessisveryimportantbeause
it involves the eletromagneti oupling of the
reso-nane, and its knowledge with high preision is very
useful,for instane, to unravelthe possible amount of
mixing in some glueball andidates [6℄,
omplement-ing the information obtained throughthe observation
of hadroni deays. Another interesting study is the
possible prodution of alight salarmeson () whose
existenehas been disussed for several years [7℄. We
stressagainthattheadvantageofrelativistiheavyion
olliders isthat thephotonluminosityfor two-photon
physisis orders of magnitude larger than the one at
availablee +
e mahines.
Wewilldisusstheprodutionoflighthadroni
res-onanesinultra-peripheralheavyionollisions.Wewill
showthattheproess!R!,see Fig. (1),an
beobservedformanyresonanes aboveorat thesame
level of thebakground. Themain bakgroundis the
ontinuumreation!,thisonewillbedisussed
aswellassomeotherbakgroundontributions.
Dou-ble Pomeron exhange may also ompete with the
physis,wewillpointoutthatthisontributionisnot
importantforveryheavyions.
Thedistribution ofthis review isthe following: In
SetionIIwepresentthedistributionfuntionsfor
pho-tonsandPomeronsin theion,anddisusssomeofthe
approximations to obtain realisti ross setions. We
ompare proesseswith the ones initiated by
ofsomeglueballandidatesandintheaseofapossible
salar meson. SetionIVontainsouronlusions.
Figure1. Diagramfor fusioninaperipheralheavy-ion
ollision. Theblobrepresentsaontinuumorresonant
pro-essleadingtoatwo-photonnalstate.
II Two-photon and double
Pomeron exhange proesses
A. Distributionfuntions
The photon distribution in the nuleus an be
desribed using the equivalent-photon or Weizs
aker-Williamsapproximationintheimpatparameterspae.
DenotingbyF(x)dxthenumberofphotonsarryinga
frationbetweenx andx+dx ofthetotalmomentum
ofanuleusofhargeZe,weandenethetwo-photon
luminositythrough
dL
d =
Z
1
dx
x
F(x)F(=x); (1)
where = ^s=s, s^is the square of the enter of mass
(.m.s.) systemenergyofthetwophotonsandsofthe
ion-ion system. Thetotal ross setion ofthe proess
AA!AAis
(s)= Z
d dL
d ^
(^s); (2)
where(^^ s)istheross-setionofthesubproess!
X.
Thereremains only to determine F(x). Inthe
lit-eraturethere are several approahes fordoing so,and
we hoose the onservative and morerealisti photon
distributionofRef. [8℄. CahnandJakson [8℄,usinga
presriptionproposed byBaur [9℄, obtainedaphoton
distribution whih is not fatorizable. However, they
were able to give a t for the dierential luminosity
whihisquiteusefulin pratialalulations:
dL
=
Z 2
2
16
(z); (3)
wherez=2MR p
,Misthenuleusmass,Ritsradius
and(z)isgivenby
(z)= 3
X
i=1 A
i e
biz
; (4)
whih isat resulting from thenumerialintegration
ofthephotondistribution,aurateto2%orbetterfor
0:05 < z < 5:0, and where A
1
= 1:909, A
2
= 12:35,
A
3
= 46:28, b
1
= 2:566,b
2
= 4:948,and b
3
= 15:21.
Forz<0:05weusetheexpression(seeRef. [8℄)
dL
d =
Z 2
2
16
3
ln( 1:234
z )
3
: (5)
Theonditionforrealistiperipheralollisions(b
min >
R
1 +R
2
)ispresentinthephotondistributionsshowed
above.
Theproessesthat we shalldisuss analso be
in-termediated by the dirative subproess PP ! X,
whereP isthePomeron.
In the ase where the intermediary partiles
ex-hangedinthenuleus-nuleusollisionsarePomerons
instead of photons, we an follow losely the work of
Muller and Shramm [10℄ and make a generalization
oftheequivalentphotonapproximationmethodtothis
newsituation. Sotherosssetionforpartile
produ-tionviatwoPomeronsexhangeanbewrittenas
PP
AA =
Z
dx
1 dx
2 f
P (x
1 )f
P (x
2 )
PP (s
PP
); (6)
where f
P
(x)is the distribution funtion that desribe
the probability for nding a Pomeron in the nuleus
withenergyfrationxand
PP (s
PP
)isthesubproess
ross setionwith energy squareds
PP
. Inthe aseof
inlusivepartile produtionweusethe formgiven by
Donnahieand Landsho[11℄
f
P (x)=
1
4 2
x Z
(xM) 2
1
dtj
AP (t)j
2
jD
P (t;s
0
)j 2
; (7)
whereD
P (t;s
0
)isthePomeronpropagator[12℄
D
P (t;s)=
(s=m 2
)
P (t) 1
sin( 1
2
P (t))
exp
1
2 i
P (t)
; (8)
with s thetotalsquared.m. energy. TheRegge
tra-jetoryobeyedbythePomeronis
P
(t)=1+"+ 0
P t,
where " = 0:085, 0
P
= 0:25GeV 2
and t is a small
exhanged four-momentum square, t = k 2
<< 1,
so the Pomeron behaves like a spin-one boson. The
term in the denominator of the Pomeron propagator,
[sin( 1
2
P (t))℄
1
, is the signature fator that express
thedierentpropertiesofthePomeronunder CandP
onjugation. Atveryhigh.m. energythisfatorfalls
veryrapidly withk 2
by 0
P ln(s=m
2
),mistheprotonmass,anditispossible
to negletthisk 2
dependene,
sin 1
2
(1+" 0
P k
2
)os( 1
2
")1: (9)
IfwedenethePomeronrangeparameterr
0 as r 2 0 = 0 P ln (s=m 2 ); (10)
thePomeronpropagatoran bewritten as
jD
P
(t= k 2
;s)j=(s=m 2 ) " e r 2 0 k 2 : (11)
Sineweareinterestedinthespatialdistributionofthe
virtual quanta in the nulear rest frame we are using
t= k 2
.
Thenuleus-Pomeronouplinghastheform [11℄
AP
(t)=3A
0 F
A
( t); (12)
where
0
=1:8GeV 1
isthePomeron-quarkoupling,
A is the atomi number of the olliding nuleus, and
F
A
( t) is thenulear form fator forwhih isusually
assumed a Gaussian expression (see, e.g., Dreeset al.
in [13℄)
F
A
( t)=e t=2Q 2 0 ; (13) where Q 0
=60MeV.
Performing the t integration of the distribution
funtion inEq.(7) weobtain
f P (x) = (3A 0 ) 2 (2) 2 x s 0 m 2 2" Z (xM) 2 1 dte t=Q 2 0 = (3A 0 Q 0 ) 2 (2) 2 x s 0 m 2 2" exp " xM Q 0 2 # :
Thetotal rosssetionfor ainlusivepartile
pro-dutionisobtainedwiththeabovedistributionandalso
withtheexpressionforthesubproessPP !Xas
pre-sribedinEq.(6).
B. Is double Pomeron exhange a bakground
for proesses?
The double Pomeron exhange produing a nal
state X have matrix elements with the same angular
strutureastheone. Thisiseasytoobservewithin
theDonnahieandLandshomodel [12℄. Forinstane,
in this model toompute therosssetionof the
sub-proessPP !R it isassumedthat thePomeron
ou-plesto thequarksoftheresonane(R )likeaisosalar
photon[12℄. Thismeansthatthesubproessross
se-tion of PP ! R anbeobtainedfrom suitable
modi-ations onthe ross setion for ! R , and it will
dier onlybytheoupling onstantand aform fator
desribing the phenomenologial Pomeron-quark
ou-pling. Therefore it is natural to ask if the Pomeron
proessisabakgroundtothetwo-photonproess,and
whenithastobeaddedtothealulationofaspei
proess.
In general the double Pomeron exhange is not a
bakgroundforthetwo-photonproessandthisiseasy
to understand. The Pomeron ontrarily to the
pho-tondoesnotpropagateat largedistanes, atually its
propagatorhasarangeparameterr
0
denedinEq.(10).
Whenweimposetheonditionforultraperipheral
ol-lisions (i.e. the nulei do not physially ollide) the
rosssetiondiminishesonsiderably.
InEq.(6)theaseswherethetwonuleioverlapare
not exluded. To enfore the realisti ondition of a
peripheral ollisionit is neessary to perform the
al-ulationtakinginto aounttheimpatparameter
de-pendene,b. Itisstraightforwardto verify thatin the
ollisionof twoidential nulei thetotal ross setion
ofEq.(6)ismodiedto [10℄
d 2 PP!X AA d 2 b = Q 02 2 e Q 02 b 2 =2 PP AA ; (14) where(Q 0 ) 2 =(Q 0 ) 2 +2r 2 0
. Thetotal rosssetion
for inlusive proesses is obtained after integration of
Eq.(14)with theondition b
min >2R
A
in thease of
identialions.
Another way of to exlude events due to inelasti
entralollisionsisthroughthe introdutionof an
ab-sortionfatoromputedintheGlauberapproximation
[14℄. This fatormodies the rosssetion in the
fol-lowingform d gl AA d 2 b = d PP!R AA d 2 b exp A 2 b 0 Z dQ 2 (2) 2 F 2 A (Q 2 )e iQb = d PP!R AA d 2 b exp A 2 b 0 Q 2 0 4 e Q 2 0 b 2 =4 ; (15) where 0
is the nuleon-nuleon total ross setion,
whose value for the dierent energy domains that we
shall onsider is obtaineddiretly from the t of Ref.
[15℄ 0 =Xs +Y 1 s 1 +Y 2 s 2 ; (16)
with X = 18:256, Y
1
= 60:19, Y
2
= 33:43, = 0:34,
1
=0:34,
2
=0:55, F
A (Q
2
)=e Q
2
=2Q 2
0
and we
ex-empliedEq.(15)fortheaseofresonaneprodution,
i.e., PP!R
AA
isthetotalrosssetionfortheresonane
produtionto bedisussed in thesequene. The
inte-grationinEq.(15)isoverallimpatparameterspae
We ompared the rates for double Pomeron
ex-hange and two-photon prodution of several nal
states like resonanes, a pair of pions and a hadron
luster of invariant mass M
X
. The details of the
al-ulations an be found in Ref. [16℄ and here we will
desribepartoftheresultsofthat work.
In Table I we ompare the ross setions for
Meson M
R
(R!)
RHIC
LHC
RHIC
PP
0
135 810 3
7.1 40 0.05
547 0.463 1.5 17 0.038
0
958 4.3 1.1 22 0.04
2979 6.6 0:3210 2
0.5 0:4710 4
0
3605 2.7 0:3610 3
0.1 0:3410 5
b
9366 0.4 0:1310 7
0:3710 3
0:1110 10
TableI.Crosssetionsforresonaneprodutionthroughphoton-photon()anddouble-Pomeron(PP)proesses.
ForR HIC, p
s=200GeV/nuleon,weonsidered 238
UionandforLHC, p
s=6300GeV/nuleon,thenuleusis
206
Pb. Therosssetionsareinmbarn. Ratesomputedwiththegeometrialutb>2R
A .
Meson gl
AA =
PP!R
AA
(LHC) gl
AA =
PP!R
AA
(RHIC)
0
3:5410 3
1:510 2
3:5810
3
1:4710 2
0
3:4610 3
1:510 2
3:4710 3
1:3210 2
0
3:6110 3
1:510 2
b
3:510 3
1:4510 2
TableII.RatiosofrosssetionsfordirativeresonaneprodutionalulatedwiththeGlauberabsorptionfator
totheonewiththegeometrialutintheollisionof 238
UforenergiesavailableatRHIC( p
s=200GeV/nuleon),
andollisionsof 206
PbforenergiesavailableatLHC ( p
s=6:300GeV/nuleon).
Nuleus
p
s
PP!R
AA
gl
PP
AA
Au(A=197) 100 0.044 0:5510 3
2.4
Ca(A=40) 3500 0.043 0:3910 3
0.14
Si(A=28) 200 0:3410 2
0:1510 3
0:6910 2
Si(A=28) 100 0:2210 2
0:1210 3
0:3910 2
TableIII:Crosssetionfor 0
produtionfordierentionsandatdierentenergies. TheenergiesareinGeV/nuleon
andtherosssetionsinmbarn. PP!R
istherosssetionomputedwiththegeometrialutand gl
istheone
withtheabsorptionfator.
TableIIshowstheratiosofrosssetionsfor
dira-tiveresonaneprodutionalulatedwiththeGlauber
absorption fatorto theonewith thegeometrialut.
TheexlusionofentralollisionsthroughtheGlauber
absorptionfatorisstrongerthantheonewiththe
ge-ometrial ut. Table III shows the 0
prodution for
dierentionsandatdierentenergies. Weobservethat
theratesfordoublePomeronexhangebeomesloser
tothetwo-photonrateswhen wegoto lighter ions.
Theprodutionofalusterofpartilesthrough
dou-blePomeronexhangeisalsodominatedbythe
pro-ess(seeFig. (1)ofRef. [16℄).
Ingeneralwemaysaythatfor veryheavyionsthe
double Pomeron exhange gives ross setionsone
or-derofmagnitude(or more)belowthetwo-photon
pro-ess. This hangeswhenweollidelighterions. Aswe
godownfrom largehargeionstosmallerones the
ef-fetofPomeronphysisinreasesanditdominatesthe
The fat that the Pomeron has ashort range
param-eter is notthe only fat that ountswhen we analyze
eahspeiproess. Itmustbealsorememberedthat
thePomeron ouplesto lightand heavyquarks
dier-ently. Apartfromkinematisthedierenesintherates
of resonaneprodution byphotonsand Pomerons
in-reases between those resonanes formed by lightand
heavyquarks,asseenin TableI.
III The reation + ! +
A. The ontinuumproess
Wearepartiularlyinterestedinthephoton-photon
sattering beause it an be a very lean signal for
hadroniresonaneslikeglueballsandthemeson. On
theotherhandthissatteringisimportantbyitself,and
The subproess ! up to energies of a few
GeVis dominatedby theontinuousfermion box
dia-gram,andisabakgroundfortheresonant!R!
proess. It wasrst alulatedexatlyby Karplus
and Neuman[17℄andDeTollis[18℄. Therearesixteen
heliityamplitudes fortheproessand,dueto
symme-try properties, the numberof independent amplitudes
will be only ve, that may be hosen to be M
++++ ,
M
++ , M
+ + , M
+ +
and M
+++
. Where the
+ or denotes the irular polarization values +1
and 1. The remaining heliity amplitudes may be
obtained from parity and permutation symmetry. Of
theseveheliityamplitudes,threearerelatedby
ross-ing, heneit is suÆient to give just three, whih are
presentedin detailin Ref. [19℄.
Thedierentialrosssetionofphotonpair
produ-tionfromphotonfusion,i.e. theboxdiagram,is
d
dos =
1
2
4
s (
X
f q
2
f )
4 X
jMj 2
: (17)
is the sattering angle, is the ne-struture
on-stant, q
f
is the harge ofthe fermion in the loopand
thesumisovertheleptonseandandthequarksu,d
ands,whiharetherelevantpartilesinthemasssale
that weshalldisuss. Anotherpossibleontributionto
this ontinuum proess omesfrom pion loops, whih,
apartfrompossibledoubleounting,wereshowntobe
negligibleomparedtotheaboveone[22℄. Theseond
sumisoverthesixteenheliityamplitudes,M
1234 ,
where
1 and
2
orrespondtopolarizationsof
inom-ing photons and
3 and
4
for the outgoing photons.
The matrix elements summed over the nal
polariza-tionsandaveragedovertheinitialpolarizationsisgiven
by
X
jMj 2
= 1
2 fjM
++++ j
2
+jM
++ j
2
+jM
+ + j
2
+ jM
+ +
j 2
+4jM
+++ j
2
g:
Weonsiderthesatteringoflightbylight,thatis,
the reation ! , in Au-Au ollisions for
ener-gies available at RHIC, p
s = 200 GeV/nuleon. We
heked our numerial odereproduing themany
re-sultsoftheliteraturefortheboxsubproess,inluding
asymptoti expressions for the low and high energies
omparedtothefermionmasspresentintheloop,and
the peak value of the ross setion (see, for instane,
Ref. [20℄).
InFig. (2)thedependene oftheion-ionross
se-tion with the osine of the sattering angle , in the
twophoton enter-of-masssystem, isshownfor an
in-variantphotonpairmassequalto500MeV.Itis
possi-bleto observethattherosssetionisstronglypeaked
in the bakward diretion, but is relatively at out
to os 0:4, where it starts rising very fast. It is
symmetri withrespetto and thesamebehavioris
It is possible to observe that the ross setions is
stronglyrestrited when weintrodue anangular ut.
Wewill impose in allthe alulationsthroughout this
workautinthesatteringangleequaltojosj=0:5.
This ut is onservative, but it will make possible to
omparetherosssetionoftheboxdiagramwithrival
proesses, that will be disussed in the following
se-tions,aswellasit is enoughto eliminatethe eet of
doublebremsstrahlung(whihdominatestheregionof
josj1). Finally, this kindofut istotally
onsis-tentwiththerequirementsproposedin Ref. [3℄
The fermions that ontribute in the box diagram
are the leptons e and and the quarks u, d and s,
whih are important for the mass range that we are
interested (heavier quarks will give insigniant
on-tributions and the same is true for the harged weak
bosons). Weassumedfortheirmassesthefollowing
val-ues: m
e
=0:5109MeV,m
=105:6584MeV,m
u =5
MeV,m
d
=9MeVandm
s
=170MeV.
Theeletrongivesthemajorontributiontothe
to-tal result. Theseond most importantontribution is
duetothemuon,but itisatleastoneorderof
magni-tudesmallerthantheeletronone. Thedandsquark
ontribution(uptoO(2GeV))aresmallerduetotheir
massesandharges,beausetheproessisproportional
to(q 2
f )
4
where q
f
istheirharge. Theirontributionis
alsoinsigniantomparedto theeletronresult.
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
1
Figure2. Angular distributionofZZ !ZZ sattering
ataninvariantmassof500MeV.Thesatteringangleis
inthephotonpairenter-of-masssystem.
AsdisussedinRef. [2℄thesatteringanindeed
bemeasuredinperipheralheavyionollisions. Theut
intheangular distributiongivesbak tobakphotons
wherethe!proessisoverwhelmedbythe
pres-ene ofresonanes likethe , 0 and others. Eventhe
broad resonaneouldbeoftheorder ofthe
ontin-uum proess. Just to give oneidea of the number of
events,withaluminosityof2.010 26
m 2
s 1
[3℄and
hoosingabinofenergyof200MeV,enteredatthe
en-ergyof700MeV(whihisfreeofanystrongresonane
deayinginto two-photons), wehave1532 events/year
assuming100%eÆienyinthetaggingoftheionsand
photondetetion.
B. The proess!R!: glueballs
Photon pair prodution via the box diagram is a
bakground to ! R ! proess (or vie versa),
bothhavethesameinitialandnalstates,andforthis
reasontheyaninterfereoneinanother. Normallythe
interferenebetweenaresonaneandaontinuum
pro-essis unimportant, beauseonresonanethetwoare
outofphase.
The total ross setion for the elementary
subpro-ess ! R ! assuming a Breit-Wigner prole
is
d
ZZ
dM
=16 dL
dM
2
(R!)
(M 2
m 2
R )
2
+m 2
R 2
total
; (18)
where M is the energyof thephotons reatedby the
ollisionof theions. (R!)(
)and
total are
thepartialandtotaldeaywidthoftheresonanewith
massm
R
initsrestframe.
WearegoingtodisussonlyJ =0resonanesmade
ofquarksaswellofgluons. Thereation! 0
!
was already disussed many years ago [21℄, where it
waslaimedthat theinterferenevanishes. This result
was ritiizedby De Tollis and Violini [22℄, aÆrming
(orretly)thattheinterfereneexists. However,aswe
willdisussafterwards,oresonanetheinterfereneis
negligible. Iftheinterfereneis negleted,Eq.(18)an
be used and we show in Fig. (3) the result for some
resonaneprodution(; 0
;(1440);f
0
(1710)),whose
invariantmassoftheproduedphotonpairisbetween
500MeVand2000MeV.Foromparisonwealsoshow
the urve of the ontinuum proess. It is possible to
seeinthatgurethewellpronounedpeaksofthe
res-onanes and 0
. We assumed for their masses the
valuesof547.3MeVand957.78MeV,respetively,the
totaldeaywidthisequalto1.18keVandthe 0
one
is equalto 0.203 MeV.Their partial deay width into
0
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
1
10
10
2
600
700
800
900 1000
Figure 3. Invariant mass distribution of photon
produ-tion (with the ut josj < 0:5). The solid urve is for
theboxdiagram,thedashedurvesaredueto theproess
!R!,whereRarethepseudosalarsresonanes
and 0
andtheglueballsandidates(1410)andf0(1710).
WerestrittheanalysistotheJ =0glueballs
an-didates(1440)andf
0
(1710). Forthe(1440)weused
the mass and total deay width values of Ref. [23℄,
m
R
= 1405 MeV,
total
= 56 MeV, for the deay
widthintophotonsweusethevaluegiveninRef. [24℄,
= 5:4 keV. We see in Fig. (3) that the peak for
this resonane is of the same order of the ontinuum
proess. Fortheotherglueballandidate,f
0
(1710),the
peakis learlyabovethebakground. Forthis onewe
assumedthevalueslistedinRef. [23℄ofmassandtotal
width, m
R
=1715MeV
total
=125MeV,andforthe
two-photondeaywidthweadoptedthevalue
enoun-teredbytheALEPHCollab. [25℄,
=21:25keV.In
all these asesthe resonanesan be easily studied in
peripheralheavyionollisions.
O resonanewe anexpet a negligible
ontribu-tionfortheproess!R!andonsequentlythe
same for its interferene with the ontinuum proess.
However, it is instrutive to present a more detailed
argumentaboutwhytheinterfereneanbenegleted.
Inordertodosoweareobligedtointrodueamodelto
alulatetheamplitudesfortheproess!R!.
These amplitudes will be omputed with the help of
an eetive Lagrangian for the pseudosalar
intera-tionwith photons(the salarasewill be disussedin
thenextsetion),whihisgivenbyg
p "
F
F
p ,
where g
p
is the oupling of the photons to the
pseu-dosalareld
p ,"
istheantisymmetritensorand
F
pseudosalarhadroniresonaneare [22℄:
M
++++ =
2
2
F(r
t );
M
+ + =
2
2
F(t
t );
M
+ +
= 2
2
F(s
t );
M
+++
= 0;
M
++ =
2
2
fF(s
t
)+F(t
t
)+F(r
t )g;(19)
where isthene-strutureonstant,=(m
f =m
R )
2
,
and
F(x)=16x 2
m
R
4x 1+i total
m
R
1
: (20)
The presene of the ne-struture onstant in
Eq.(19)isaonsequeneofthefatthattheamplitudes
M in these equations will be used in Eq.(17), so it is
neessaryto get the orretdependene of thepartial
rosssetionwiththisonstant.
A numerial evaluation of the ross setion using
Eq.(19) (for the sameresonanes present in Fig. (3))
showsa totally negligible eet o resonanein
om-parison with the box ontribution. On resonane the
two proessesare out of phaseand the interfereneis
absent. Weannowproeedwithanargumentshowing
that theinterferene isnot important. Let us assume
thatoresonanetheproessesareinphase,andfora
momentweforgetthetanduhannelsontributionin
Eq.(19). Thes hannelontribution anbewrittenas
M
2
=(s m 2
R +i
R m
R
), and denoting the ontinuum
ontributionbyM
1
weanwritethefollowing
interfer-ene term
2
s
(s m 2
R )
2
+ 2
R m
2
R [(R eM
1 R eM
2
+ImM
1 ImM
2 )
(s m 2
R
)+(R eM
1 ImM
2
ImM
1 R eM
2 )
R m
R ℄:
We an verify that the term proportional to s m 2
R
integratestozerowhenintegratedin abinenteredat
m 2
R
. Withtheseondtermthesituationisdierent. If
ImM
1
orImM
2
6=0(assumingR eM
1
andR eM
2 6=0)
then there is anontrivialinterferene. However,sine
wearedealingwithJ =0amplitudes,theonly
nonvan-ishingheliityamplitudesarethoseinwhihtheinitial
heliities and the nal heliitiesare equal. Inspetion
of theM
++++
and M
++
amplitudes ofthe box
di-agram (in the limit m
f
m
R
) shows that they are
purely real, and thesamehappenswith M
2
(obtained
from thes hannel ontribution of Eq.(19)), resulting
in avanishinginterferene!
Ofourse,thisanalysisismodeldependent. In
par-tiular,atthequarkleveltheouplingg
p
hastobe
sub-stitutedbyatrianglediagram,whihmayhavearealas
wellasanimaginarypart(see,forinstane, Ref. [26℄).
However,thisouplingisreal forheavyquarksandits
imaginarypartisquitesuppressediftheresonane
ou-plesmostlyto lightquarks(m
f
m
R
,what happens
forthe resonanes that we areonsidering in thease
oftheu andd quarks,forthesquarkthesuppression
isnotsostronginthease,butwestillhaveanextra
suppressiondueto itseletriharge). Finally,the
in-terferenedoesappearwhenweonsidertheamplitudes
withtheresonaneexhangedin thetand uhannels,
but it is easy to see that they are kinematially
sup-pressedandalsoproportionalto thesmallvalueofthe
totaldeaywidth. Ifthetotalwidthislarge(asweshall
disussin thenextsubsetion)theinterfereneannot
benegleted. Althoughtheaboveargumenthastorely
onmodelsfor thelowenergyhadroniphysis,we
be-lievethat the diret omparison betweentheresonant
andtheontinuumproesses,aspresentedin Fig. (3),
isfairlyrepresentativeoftheatualresult.
In Table IV we show the number of events above
bakgroundforthe, 0
andf
0
(1710)whih,asshown
in Fig. (3) are learly above the box ontribution.
In the ase of the f
0
(1710), as well as in the
me-son ase to be disussed in the next subsetion, the
deay of the resonaneinto a pair of neutral pions is
present. A pair ofneutralpions analsobe produed
in aontinuous twophoton fusion proess. The rates
forallthereationsdisussedinthisworkanbe
mod-iedifbothneutralpions,nomatteriftheyomefrom
aresonaneora ontinuous proess, are misidentied
withphotons. Thisaidentalbakgroundanbeeasily
isolatedmeasuringitsinvariantmassdistribution,and
makinga utthat disriminates asingle photon
om-ingfrom theproessesthat weare studying, fromone
thatprodued twoneutralmesons(mostlypions)
sub-sequentlydeaying into two photons. Forexample,in
theaseofthesigmameson(seenextsetion)eahone
oftheneutralpionsfromitsmuhlargehadronideay
should be misidentied. These pions would produe
pairs of photons with a large opening angle , where
os(=2)= p
1 4m 2
=m
2
. However,thealorimeters
alreadyinuse inmanyexperimentsare ableto
distin-guishbetweenthesetwoandsinglephotoneventswith
higheÆieny(see,forinstane,Ref. [27℄).
partile events/year
7:4410
5
0
2:6710 4
f
0
(1710) 42
TableIV.Numberofevents/yearabovebakgroundfor
the, 0
There is alsoanother aidental bakgroundwhih
may ause problems to the measurement of the
res-onane deay into photons. This is the ontribution
from Pomeron!V!X. Thevetormesons,V,
areproduedwithap
T
distributionsimilartothe
reso-naneprodutionandathigherratesthansomeofthe
proesses!R ,e.g. Pomeron!! rateis10Hz
atRHIC[28℄,threeordersofmagnitudehigherthanfor
asimilarmass mesonofspin0or2. The ! branhing
ratio to three photons is 8.5%. If a small p
T
photon
fromthisdeayisundeteted,oneisleftwithalowp
T
two-photonnalstatethatouldbetakenforalighter
resonane. Athighermasses,onealsohas!, 0
,
K
L K
S
! X as well as ! f
2
(1270)! 0
0
and
possiblyopious produtionof(1450)and(1700)by
Pomeron interations. Clearly a full simulation of
allthesebakgroundproessesshould bekeptin mind
whenmeasuringtwo-photonnal states.
C. The proess!R!: the meson
Thepossibleexistene oflightsalarmesons(with
masseslessthanabout1GeV)hasbeenaontroversial
subjetforroughlyfortyyears. Therearetwoaspets:
theextrationofthesalarpropertiesfromexperiment
and theirunderlying quarksubstruture. Beausethe
J = 0hannels may ontainstrong ompeting
ontri-butions,suhresonanesmaynotneessarilydominate
their amplitudes and ould be hard to \observe". In
suh an instane their veriation would be linked to
themodelusedto desribethem.
Part of themotivation to study thetwo-photon
-nalstatesinperipheralheavy-ionollisionswasexatly
toverifyifweanobservesuhsalarmesonsinits
deay. Although this deay mode is quite rare, it has
theadvantageofnotbeingontaminatedbythestrong
interation of the hadroni nal states. In partiular,
itmayallowtoinvestigatethepossibleexisteneofthe
sigmameson. This meson is expeted to haveamass
between400-1200MeVanddeaywidth between
300-500MeV, deayingpredominantly into twopions. Of
ourse,anotherdeayhannelisintotwophotons,with
thebakgrounddisussedin thebeginningofthis
Se-tion.
ReentlytheE791CollaborationatFermilabfound
a strong experimental evidene for a light and broad
salar resonane, that is, the sigma, in the D +
!
+
+
deay [29℄. Theresonantamplitudes present
inthisdeaywereanalyzedusingtherelativisti
Breit-Wignerfuntiongivenby
BW =
1
m 2
m 2
+im
0 (m)
;
with
(m)=
0 m
0
m
p
p
0
2J+1
J
F 2
(p
)
J
F 2
(p
0 )
;
where m is the invariant mass of the two photons
forming a spin-J resonane. The funtions J
F are
the Blatt-Weisskopf damping fators[30℄: 0
F = 1for
spin 0 partiles, 1
F = 1= p
1+(rp
) 2
for spin 1 and
2
F =1= p
9+3(rp
) 2
+(rp
) 4
forspin2. The
param-eterr istheradius oftheresonane(3fm)[31℄and
p
=p
(M)themomentum ofdeaypartilesat mass
M,measuredintheresonanerestframe,p
0 =p
0 (m
R ).
TheDalitz-plotofthedeayanhardlybetted
with-outa0 ++
()resonane. Thevaluesofmassandtotal
deaywidth found bythe ollaborationwith this
pro-edure are 478 +24
23
17 MeV and 324 +42
40
21 MeV,
respetively.
We will disuss if this resonane an be found in
peripheralheavy-ionollisionsthroughthe subproess
!!. Itisimportanttonote thatallthe
val-ues related to the, like massorpartial widths, that
an be found in the literature are very dierent and
model dependent. In partiular, we nd the resultof
theE791experimentveryompellingandamongallthe
possibilitieswewill restritourselvesto theirrangeof
massand totaldeaywidth, while wevary thepartial
widthintotwophotons. Forthedeaywidthintotwo
photonsweassumethevaluesobtainedbyPennington
andBoglione,3:81:5keVand4:71:5keV[32℄,and
thevalueof106keV[33℄.
Ithasbeenveriedintheaseofsatteringthat
the use of a onstant total width in the resonane
shape is not a good approximation [34℄. In our ase
wewill disussthe! proessabovethe two
pi-onsthresholdwherethepeuliaritiesofthebroad
reso-nane,basiallydue tothe deayintopions, arenot
soimportant. Ofourse, anotherreasontostayabove
300MeV isthat we arealso far from the pion
ontri-butionto sattering. Inanyasewealsoomputed
the rosssetionwith aenergy dependent totalwidth
(m) '
0 (p
=p
0 )
2J+1
, whih, as shown by Jakson
many years ago [35℄, is more appropriate for a quite
broadresonane. Theneteetisaslightdistortionof
therosssetionshapewithasmallinreaseofthetotal
rosssetion. Sine thisoneisanegligibleeet
om-pared to the one that wewill presentin thesequene
wedonotshall onsideritagain.
O resonanewe anexpet a negligible
ontribu-tion for the proess ! R ! and onsequently
the same for its interferene with the ontinuum
pro-ess. Thisistrueiftheresonanehasasmalltotal
de-aywidth[19℄, butthisisnotthe ase. Totakeinto
to alulatetheheliityamplitudesof the meson
ex-hange. Using the eetive lagrangian g
s F
F
s ,
where g
s
is the oupling of the photons to the salar
eld
s andF
istheeletromagnetieldtensorthe
followingamplitudesomesout[19,22℄:
M
++++ =
2
2
F(r
t );
M
+ + =
2
2
F(t
t );
M
+ +
= 2
2
F(s
t );
M
+++
= 0;
M
++ =
2
2
fF(s
t
)+F(t
t
)+F(r
t )g:(21)
where isthene-strutureonstant,=(m
f =m
R )
2
,
andr
t ,s
t andt
t
arerelatedwiththestandard
Mandel-stamvariabless, t,andubyr
t =
1
4 s
m 2
f ,s
t =
1
4 t
m 2
f and
t
t =
1
4 u
m 2
f .
These amplitudes and the ones desribing the
fermion (with mass m
f
) box diagram enter in the
expression for the dierential ross setion of
pho-ton pair prodution from photon fusion d=dos =
(1=2)( 4
=s)( P
f q
2
f )
4 P
jMj 2
, to give the total ross
setionof Fig. (4). Weveriedthat theinterfereneis
destrutive.
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
400
500
600
700
800
900 1000
Figure4.Invariantmassdistributionofphotonpair
produ-tion. Thesolidurveisduetoboxdiagramonly,thedashed
oneisduetotheproess!!intheBreit-Wigner
approximation,thedash-dottedisthesalarontributionof
Eq.(21),andthedottedoneistheontinuumPomeron
pro-essPP!. Inallases
=4:7keV,andtheangular
utisequalto 0:5<os<0:5 .
TheeetiveLagrangianmodelusedtoomputethe
largerrosssetionthanthealulationwiththe
Breit-Wigner approximation at energies above M 600
MeV. It is dominated by the s hannel ontribution.
We onsider the Breit-Wigner result as the best
sig-nalrepresentationfor theresonantproessbeausewe
are using the E791 data and this onewastted by a
Breit-Wigner prole. The eetive Lagrangian gives
a nonunitary amplitude that overestimates the sigma
prodution above 600 MeV and shows the model
de-pendene inthe analysis thatweommentedbefore.
TheBreit-Wignerproleisnotabadapproximationas
longaswestay abovethe twopions threshold and in
thefollowingweassumethatthe signalis givingbyit
(the dashed urveof Fig. (4)) and the bakground is
givingbytheboxdiagramresult(thesolidurveofFig.
(4)). Notethat,duetothedestrutiveinterferene,the
atual measurement will give a urve below the solid
urveofFig. (4).
Fromtheexperimental pointof viewwewould say
thatthereation! hasto beobservedandany
deviation from the ontinuum proess must be
are-fully modeled until a nal understanding omes out,
withtheadvantagethat the nalstateis notstrongly
interating. Notethatinthismodelingthemesonwill
ontributeto!in asmallregionofmomentum
[19℄,evensoithastobesubtratedinordertoextrat
theomplete signal.
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Figure5.Signianeasafuntionofdeaywidthintotwo
photons, , for a sigma meson with mass equal to 478
MeV and total deay width of 324 MeV. The solidurve
was obtained integrating the ross setions in the
inter-val 438 <M < 519 MeV, the dashedone inthe interval
300<M <800MeV.Thesignal and bakgroundare
Wehangedthevaluesofthemassandtotalwidth
aroundtheentralonesreportedbytheE791
Collabo-ration. Wedonotobservedlargevariationsin our
re-sult,butnotiedthatitisquitesensitivetovariationsof
thepartialdeaywidthintophotons. Itisinterestingto
lookatthevaluesofthesignianewhihiswrittenas
L
signal =
p
L
bak
and haraterizesthestatistial
de-viation ofthenumberof theobservedeventsfromthe
preditedbakground. Thesignianeasafuntionof
thetwophotonsdeaywidth ofthesigmameson,with
mass equalto 478MeV and total deay width of 324
MeV, is shown in Fig. (5), were we used a
luminos-ityof L =2:010 26
m 2
s 1
at RHIC and assumed
one year of operation. The signiane is above 2
95%ondenelevel limitfortwophotondeaywidth
greater than 4.7 keV, while for a 5 disovery
rite-riaanbeobtainedwith
>7:5keV.Thenumbers
in Fig. (5) were omputed with the signal given by
the Breit-Wigner prole, and the bakground by the
pure box diagram. The solid urve wasobtained
in-tegratingtherosssetionsin therangeof
experimen-tal mass unertainty438 <M <519 MeV,while the
dashed urve resulted from the integration in the
in-terval 300 < M < 800 MeV. Note that there is no
reason,apriori,torestritthemeasurementtoasmall
bin of energy. This hoie will depend heavily onthe
experimental onditions. Therefore, for values of
alreadyquotedintheliteraturethesigmamesonhasa
hane to beseenin itstwophoton deay mode. The
disoverylimitsdisussedabovereferonlytoa
statisti-al evaluation. Our work showstheimportane ofthe
omplete simulationof the signal and bakground
in-luding an analysis of possible systemati errors that
maydereasethesigniane.
IV Conlusions
Peripheralollisionsatrelativistiheavyion
ollid-ersprovideanarenaforinterestingstudiesofhadroni
physis. One of the possibilities is the observation of
lighthadroniresonanes,whihwillappearquite
sim-ilarlytothetwo-photonhadroniphysisate +
e
ma-hineswiththeadvantageofahugephotonluminosity
peaked at small energies [1℄. Due to this large
pho-tonluminosityitwillbeomepossibletodisover
reso-nanesthat oupleveryweaklyto thephotons [4℄.
The double Pomeron exhange may be a
bak-ground forthe two-photon proesses. We haveshown
thatingeneralthePomeronontributiondoesnot
om-pete with the one, in the ase of veryheavy ions,
afterimposingtheonditionforaultra-peripheral
olli-has to be performed for eah spei nal state, due
to the dierentPomeron ouplingto severalpartiles.
If the ollisioninvolves heavy ions and thenal state
wearelookingatdoesnothavealargeouplingtothe
PomeronweanaÆrmthat doublePomeronexhange
anbenegletedintheevaluation ofprodutionrates.
In the ase of light ions Pomeron proesses are
om-petitivewith the two-photons ones,and dominate the
rosssetionsforverylightions.
Wedisussed theperipheralreationAA!AA.
Thisproess isimportantbeauseitmayallowforthe
rsttimetheobservationoftheontinuoussubproess
! in a omplete olliderphysis environment.
Thispossibilityonlyarisesduetoenormousamountof
photonsarriedbytheionsattheRHICenergies.
The ontinuous subproess is desribed by a
fermioni boxdiagram alulatedmanyyearsago. We
omputedtheperipheralheavyionprodutionofapair
ofphotons,verifyingwhiharethemostimportant
on-tributionstotheloop,whihturnedouttobethe
ele-tron at the energies that we are working, and
estab-lishedutsthatnotonlyensurethattheproessis
pe-ripheral as well aseliminate most of the bakground.
Aftertheutisimposedwestillhavethousandsof
pho-tonspairsassuming100%eÆienyintaggingtheions
anddetetingthephotons.
Theontinuous!subproesshasan
interest-inginterplaywiththeoneresultingfromtheexhange
of a resonane. We disuss the resonaneprodution
anddeayintoaphotonpair. Thisisanieinteration
toobservebeauseitinvolvesonlytheeletromagneti
ouplingsoftheresonane.Therefore,wemaysaythat
it is a lean signal of resonanes made of quarks (or
gluons) and its measurement is important beause it
omplementstheinformationobtainedthroughthe
ob-servationofpurelyhadronideays. Itmayalsounravel
thepossibleamountofmixing insomeglueball
andi-dates [6℄. We disuss the interferene between these
proess and ompute the number of events for some
speiases.
Thepossibilityofobservingresonanesthat ouple
weaklyto thephotonsisexempliedwiththe meson
ase. This meson,whose existene hasbeenfor many
yearsontraditory,givesasmallsignalin thereation
! ! . Howeverits eets maybe seenafter
oneyearofdataaquisition,providingsomelueabout
thiselusiveresonane. Using valuesofmass, totaland
partial widths urrentlyassumed in theliterature, we
ompute the full rosssetion within aspei model
and disuss the signiane of the events. Our work
showstheimportaneoftheompletesimulationofthe
two photonnal states in peripheral ollisions anbe
observed andmayprovidealarge amount of
informa-tionabouttheeletromagnetiouplingof hadrons.
Aknowledgments
Most of this work has been done in ollaboration
with C.G. Rold~ao. I would like to thank I.Bediaga,
C.Dib,R.RosenfeldandA. Zimermanformany
valu-able disussions. This researh was supported by the
ConselhoNaionaldeDesenvolvimentoCientoeT
e-nologio (CNPq),by Funda~ao deAmparo aPesquisa
do Estado de S~ao Paulo(FAPESP) and byPrograma
deApoioaNuleosdeExel^enia(PRONEX).
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