Quarkonium dissociation in a thermal bath

0.12 MB·English

by  Antonio Vairo

Quarkonium dissociation in a thermal bath
Antonio Vairo
Physik-Department,TechnischeUniversitätMünchen,James-Franck-Str.1,85748Garching,Germany
5 Abstract. Inaneffectivefieldtheoryframeworkwereviewthetwomainmechanismsofquarkoniumdissociationinaweakly
1 coupledthermalbath.
0
2 Keywords: Quarkonium,quark-gluonplasma,non-relativisticeffectivefieldtheories
PACS: 12.38.Mh,12.39.Hg,14.40.Pq

n
a
J
INTRODUCTION

6
2
Heavyparticlesmaybesensitivetonewfundamentaldegreesoffreedomandserveasprobesofnewphysics.Heavy

] particles themselves may appear as new fundamental degrees of freedom, an example discussed at this conference
h beingheavyMajorananeutrinos[1].Heavyparticles,however,mayalsoserveasprobesofnewphenomenaemerging
p
fromparticularlycomplexenvironments.Wewillfocusonthissecondaspecthereanddiscuss,insomespecialcases,
-
p howheavyquarkoniaareaffectedbythehotmediumformedinpresent-dayheavy-ioncolliders.Atthisconference,
e thishasalsobeendiscussedelsewhere,forinstance,in[2].
h
WeconsideraparticleofmassMheavy,ifMismuchlargerthananyotherenergyscaleEinthesystem.Inparticular,
[
ifMisalsomuchlargerthanthemomentumoftheparticle,thisqualifiestheparticleasnon-relativistic.Thehierarchy
1 ofenergyscales,M E,allowsaneffectivefieldtheory(EFT)treatmentatascalem suchthatM m E.TheEFT
v describesthedynam≫icsoftheeffectivedegreesoffreedomthatexistatthescalem .Thesearethe≫fieldH≫representing
7 thelow-energyfluctuationsoftheheavyparticle,andallotherlow-energyfields.Inareferenceframewheretheheavy
6
particleisatrestuptofluctuationsoforderE orsmaller,theEFTLagrangianhastheform
3
6
1
0 L =H†iD H+[operatorsofdimension d>4] +L , (1)
1. 0 × Md−4 light
0 whereL describesalllowenergyfieldsbesidesH.Intheheavy-particlesector,theLagrangianisorganizedasan
light
5 expansionin1/M.Contributionsofhigher-orderoperatorstophysicalobservablesaresuppressedbypowersofE/M.
1
TheEFTLagrangianmaybecomputedsettingE =0.
:
v A special case is the case of a heavy particle of mass M in a medium characterized by a temperature T such
i that M T. The system is described at a scale m such that M m T by an EFT Lagrangianthat has the same
X
≫ ≫ ≫
structure as (1) if the heavy particle is at rest up to fluctuations of order T or smaller. This follows from the fact
r
a thatthe LagrangianmaybecomputedsettingT =0. Inparticular,theWilsoncoefficientsencodingthehigh-energy
modesmaybecomputedinvacuum.TheLagrangianisagainorganizedasanexpansionin1/M,whosehigher-order
contributionstophysicalobservablesaresuppressedbypowersofT/M.
Temperature is introduced via the partition function. Sometimes it is useful to work in the real-time formalism.
Despitethefactthatinrealtimethedegreesoffreedomdouble(“1”and“2”),theadvantagesarethattheframework
becomesverycloseto the oneof T =0 EFTsand that,in the heavy-particlesector,the seconddegreesof freedom,
labeled“2”,decouplefromthephysicaldegreesoffreedom,labeled“1”.Thisoftenleadstoasimplertreatmentwith
respecttoalternativecalculationsinimaginarytimeformalism[3].
Inthe following,we willconsiderthe specialcase ofa heavyquarkonium,i.e., a boundstate ofa heavyquarkQ
and a heavy antiquark Q¯, weakly bound, interacting with a weakly coupled plasma. This means that we consider
a quarkonium sufficiently tight to be described as a Coulombic bound state (e.g. the bottomonium ground state)
interactingwithamediumsufficientlyhottobehaveasaweaklycoupledquark-gluonplasma[4,5].Wewillcompute
thermalcorrectionstothewidthinducedbythemedium.Wewillcallthese,thequarkoniumthermalwidth,G .
HEAVYQUARKONIAIN ATHERMAL BATH
Lattice data suggest a crossover from hadronic matter to a plasma of deconfined quarksand gluons happeningat a
critical temperature T =154 9MeV [6]. High energydensities with temperatureslargerthan T are producedin
c c
±
heavy-ioncollisionsatRHICandLHC.High-energyprobesareneededtoidentifyandqualifythestateofmatterthat
isformedthere.Heavyquarkoniumhasbeensuggestedasoneofthoseprobes[7].Thisisbecauseheavyquarksare
formedearly in heavy-ioncollisions, heavyquarkoniumformationwill be sensitiveto the mediumand the dilepton
signalmakesanideallycleanexperimentalprobe.
Quarkonium,beingacompositesystem,ischaracterizedbyseveralenergyscales[8]:M,Mv(momentumtransfer,
typicalinversedistance),Mv2 (kineticenergy,bindingenergyE ,potentialV),...,wherevistherelativeheavy-
binding
quarkvelocity;v a foraCoulombicboundstate.Theseinturnmaybesensitivetothermodynamicalscalessmaller
s
than the tempera∼ture. The thermodynamical scales are: p T, m (Debye mass, i.e. inverse screening length of the
D
chromoelectricinteractions),....Thenon-relativisticscalesarehierarchicallyordered:M Mv Mv2.Inaweakly-
coupled plasma one assumes this also to be the case for the thermodynamical scales: p≫T m≫ gT. A weakly
D
coupledquarkoniumisthebottomoniumgroundstate,¡ (1S);producedinheavy-ionexperim≫entsat∼theLHCitmay
possiblyrealizethehierarchyofenergyscales[9,10]:M 5GeV >M a 1.5GeV >p T 1GeV >M a 2
0.5GeV m >L , fora temperatureofthe plasmabo≈f about320MbeVs≈ 2T. Theexisten≈ceof a hierarbchys o≈f
D QCD c
∼ ≈
energyscales ca∼lls for a descriptionof the system (quarkoniumat rest in a thermalbath) in terms of a hierarchyof
EFTs.ThesearelistedontherightarrowofFigure1.
non−relativistic thermodynamical EFTs
scales scales
M
M v T NRQCD
m
D
M v2 pNRQCD
FIGURE1. EnergyscalesandEFTsrelevantforaheavyquarkoniuminaweakly-coupledquark-gluonplasma.
Non-relativistic QCD (NRQCD) is obtained by integrating out modes associated with the scale M [11, 12] and
possiblywiththermodynamicalscaleslargerthanMv.TheLagrangianisorganizedasanexpansionin1/Mandisof
theform(1):
D2 D2
L =y † iD + +... y +c † iD +... c + +L , (2)
0 0 light
2M −2M ···
(cid:18) (cid:19) (cid:18) (cid:19)
wherey (c )isthefieldthatannihilates(creates)the(anti)fermion.
Potentialnon-relativisticQCD(pNRQCD)isobtainedbyintegratingoutmodesassociatedwiththescaleMv[13,
14] and possibly with thermodynamical scales larger than Mv2. The degrees of freedom of pNRQCD are quark-
antiquarkstates(colorsingletS,coloroctetO),lowenergygluonsandlightquarkspropagatinginthemedium.The
Lagrangianisorganizedasanexpansionin1/Mandr,thedistancebetweentheheavyquarkandantiquark,
(cid:209) 2 (cid:209) 2
L = d3rTr S† i¶ + r V +... S+O† iD + r V +... O
0 s 0 o
Z ( M − ! M − ! )
1
+Tr O†r gES+H.c. + Tr O†r gEO+c.c. + +L , (3)
light
· 2 · ···
(cid:8) (cid:9) (cid:8) (cid:9)
whereV andV arethequark-antiquarkcolour-singletandcolour-octetpotentialsrespectively,andEisthechromo-
s o
electric field. The term L in the NRQCD and pNRQCD Lagrangiansdescribes the (in-medium)propagationof
light
gluonsandlightquarks.
Dissociation
Aquantitythatmayberelevantfordescribingtheobservedquarkoniumdileptonsignalemergingfromheavy-ion
collisions is the quarkoniumthermaldissociation width. Two distinctdissociation mechanismsmay be identified at
leadingorder:gluodissociation,whichisthedominantmechanismfortemperaturessuchthatm Mv2,dissociation
D
by inelastic parton scattering, which is the dominant mechanism for temperatures such that m≪ Mv2. Beyond
D
≫
leadingorderthetwomechanismsbecomeintertwinedandadistinctionbetweenthemlesspractical.
FIGURE2. pNRQCDcuttingdiagramresponsibleforgluodissociationatleadingorder.InapNRQCDFeynmandiagram,single
linesstandforquark-antiquarkcoloursingletpropagators,doublelinesforcolouroctetpropagators,curlylinesforgluonsanda
circlewithacrossforachromoelectricdipolevertex.
Gluodissociationisthedissociationofquarkoniuminducedbytheabsorptionofagluonfromthemedium[15,16].
The exchanged gluon is lightlike or timelike. The process happens when the gluon has an energy of order Mv2,
thereforeitmaybedescribedatthelevelofpNRQCDbythecuttingdiagramshowninFigure2.Foraquarkoniumat
restwithrespecttothemedium,thedissociationwidthcanbewrittenas
d3q
G = n (q)s nl (q), (4)
nl Zqmin (2p )3 B gluo
wheres nl isthein-vacuumcrosssection(QQ) +g Q+Q,andn theBose–Einsteindistribution.Gluodissoci-
gluo nl → B
ationisalsoknownassinglet-to-octetbreakup[3].
Dissociation by inelastic parton scattering is the dissociation of quarkoniumby scattering with gluons and light-
quarksinthemedium[17,18].Theexchangedgluonisspacelike.IntheNRQCDLagrangianeachtransversegluonis
suppressedbyT/M (see(2)),henceatleadingorderonejustneedstoconsiderdiagramswithoneexchangedgluon.
IftheexchangedmomentumisoforderMv,thendissociationbyinelasticpartonscatteringisdescribedatthelevelof
NRQCDbythecuttingdiagramsshowninFigure3.IftheexchangedmomentumisoforderMv2,thenitisdescribedat
thelevelofpNRQCDbythecuttingdiagramshowninFigure4.Foraquarkoniumatrestwithrespecttothemedium,
thethermalwidthhastheform[19]
G =(cid:229) d3q f (q)[1 f (q)]s nl(q), (5)
nl p Zqmin (2p )3 p ± p p
wherethesumrunsoverthedifferentincomingandoutcominglightpartons p(p=gforgluonswiththeplussign,
and p=q for quarks with the minus sign), f (q)=n (q)=1/(eq/T 1) and f (q)=n (q)=1/(eq/T+1). The
g B q F
−
FIGURE3. NRQCDcuttingdiagramsresponsiblefordissociationbyinelasticpartonscatteringatleadingorder.InaNRQCD
Feynmandiagram,singlelinesstandforquarkorantiquarkpropagators.Theshadedblobsrepresentgluonself-energydiagrams.
crosssections nl maybeidentifiedwiththein-mediumcrosssection(QQ) +p Q+Q+p.Weobservethatthe
p nl →
formuladiffersfromthegluodissociationformula(4),apointnotalwaysappreciatedbytheliteratureonthesubject.
DissociationbyinelasticpartonscatteringisalsoknownasLandaudamping[20].
FIGURE4. pNRQCDcuttingdiagramresponsiblefordissociationbyinelasticpartonscatteringatleadingorder.
We define the quarkonium dissociation temperature as the temperature at which G E and the screening
binding
∼
temperature as the temperatureat which 1/m is of the orderof the quarkoniumBohr radius. At weak coupling,it
D
holdsthatp T Mg4/3 andp T Mg.ThisimpliesthatT T .Inthespecialcaseof
dissociation screening screening dissociation
the ¡ (1S),T h∼asbeenestimated to be∼about450MeV [21], whichimpl≫iesthatT 320MeV isbelow the
dissociation
≈
dissociationtemperature.
Inthefollowing,wewillconsiderdifferenttemperatureregimesbelowthedissociation(andhencethescreening)
temperatureandabovethecriticaltemperature[19].Fortheseregimeswewillprovidethedissociationcrosssections
for a quarkonium that is a Coulombic 1S bound state. The wave function is r1S = 1/(√p a3/2)exp( r/a ),
h | i 0 − 0
a =2/(MC a )theBohrradius,C =(N2 1)/(2N )=4/3andN =3thenumberofcolours.
0 F s F c − c c
ThetemperatureregionT Mv m
D
≫ ≫
InthetemperatureregionT Mv m the1Sdissociationcrosssectionbypartonscatteringis
D
≫ ≫
s 1S=s f(m a ), (6)
p cp D 0
where
s 8p C n a 2a2, (7)
cq≡ F f s 0
s 8p C N a 2a2, (8)
cg≡ F c s 0
2 (mDa0)4 16+8(mDa0)2ln 4/(mDa0)2
f(m a ) 1 4 − , (9)
D 0 ≡ (mDa0)2" − ((mDa0)2−4)3(cid:0) (cid:1)#
andn isthenumberoflightquarks.Thecrosssectionisnotrelated,evenatleadingorder,withazerotemperature
f
process. The underlyingreasonis the infrared(IR) sensitivity of the cross section at the momentumscale Mv. This
IRsensitivityiscuredbythehard-thermalloopresummationatthescalem andsignaledbythelogarithmln(m a ).
D D 0
Thecrosssectionismomentumindependent.
Thequasi-freeapproximationamountsatreplacings nl by2s Q,wheres Qisthein-vacuumcrosssection p+Q
p p p →
p+Q.Hencethequasi-freeapproximationisequivalenttoneglectinginterferencetermsbetweenthedifferentheavy-
quark lines in the amplitude square. Interference terms are the ones sensitive to the bound state. This corresponds
in approximating f(m a ) 2/(m a )2. The approximation holds only for m 1/a , but at these temperatures
D 0 D 0 D 0
≈ ≫
quarkoniumis dissociated. For lower temperaturesthe approximationis largelyviolated by bound-stateeffects, see
Figure5.
100 full result
quasi−free approximation
10
Σ1S
p
Σ
cp
1
0.1
0.1 0.2 0.5 1.0 2.0 5.0 10.0
mDa0
FIGURE5. Thecrosssections p1S/s cpasafunctionofmDa0.Thecontinuouslinecorrespondstotheexactresult,whereasthe
reddashedlinecorrespondstothequasi-freeapproximation.
ThetemperatureregionT Mv m
D
∼ ≫
InthetemperatureregionT Mv m the1Sdissociationcrosssectionbypartonscatteringis
D
∼ ≫
s 1S=s h (m a ,qa ), (10)
p cp p D 0 0
where
(m a )2 3 (qa )2 1
h (m a ,qa ) ln D 0 +ln 0 ln(1+(qa )2), (11)
q D 0 0 ≡− 4 −2 1+(qa )2 −2(qa )2 0
(cid:18) (cid:19) (cid:18) 0 (cid:19) 0
(m a )2 3 (qa )2 1 1
h (m a ,qa ) ln D 0 +ln 0 + ln(1+(qa )2). (12)
g D 0 0 ≡− 4 −2 1+(qa )2 2(1+(qa )2)−(qa )2 0
(cid:18) (cid:19) (cid:18) 0 (cid:19) 0 0
Thecrosssectionismomentumdependent.
ThetemperatureregionMv T m Mv2
D
≫ ≫ ≫
InthetemperatureregionMv T m Mv2 the1Sdissociationcrosssectionbypartonscatteringisgivenby
D
≫ ≫ ≫
4q2
s 1S(q)=s ln 2 . (13)
p cp m2 −
(cid:20) (cid:18) D(cid:19) (cid:21)
5
4
3
Σ1S
q m a =0.1
D 0
Σ
cq T Mv m
2 D
∼ ≫
Mv T m Mv2
D
≫ ≫ ≫
T Mv m
1 D
≫ ≫
0
0 2 4 6 8 10
qa0
FIGURE6. Dissociationcrosssectionsduetoscatteringwithlightquarks,s q1S/s cq,asafunctionofqa0.Thedashedbluecurve
showsthecrosssectionforMv T m Mv2,thecontinuousblackcurveshowsthecrosssectionforT Mv m ,andthe
D D
≫ ≫ ≫ ∼ ≫
dot-dashedredcurveshowsthecrosssectionforT Mv mD;forallthecurvesmDa0=0.1.
≫ ≫
5
4
3
Σ1S
g m a =0.1
D 0
Σ
cg
T Mv m
2 ∼ ≫ D
Mv T m Mv2
D
≫ ≫ ≫
T Mv m
1 ≫ ≫ D
0
0 2 4 6 8 10
qa0
FIGURE 7. Dissociation cross sections due to scattering with gluons, s g1S/s cg, as a function of qa0. The dashed blue curve
showsthecrosssectionforMv T mD Mv2,thecontinuousblackcurveshowsthecrosssectionforT Mv mD,andthe
≫ ≫ ≫ ∼ ≫
dot-dashedredcurveshowsthecrosssectionforT Mv mD;forallthecurvesmDa0=0.1.
≫ ≫
Thecrosssectionsduetoinelasticscatteringswithquarksandgluonsinthethreedifferentregimesaresummarizedin
Figures6and7respectively[19].Thethermalwidthfollowsfrom(5)andisdisplayedinFigure8[22].
ThetemperatureregionMv T Mv2 m
D
≫ ≫ ≫
InthetemperatureregionMv T Mv2 m ,gluodissociationbecomessuchacompetitivedissociationprocess
D
≫ ≫ ≫
todominate,atleastparametrically,overthedissociationbypartonscattering.
800
600
DM
(cid:144)
21
E
Α2s
F 400
C
Π
8
(cid:144)@
S
1
G
m a =0.5
D 0
200
E /m =0.5
1 D
| |
n =3
f
0
2 4 6 8 10
T(cid:144)ÈE È
1
FIGURE 8. Thermal width due to inelastic parton scattering. We have integrated over the continuous black curves shown in
Figures6and7andsummedoverallpartonsaccordingto(5).
Infact,the1Sdissociationcrosssectionbypartonscatteringisinthisregime
4q2
s 1S(q)=s ln +ln2 2 , (14)
p cp m2 −
(cid:20) (cid:18) D(cid:19) (cid:21)
whichinducesathermalwidthofordera T (m /Mv)2.
s D
×
The1Sgluodissociationcrosssectionisatleadingorder
s 1S (q)= a sCF210p 2r (r +2)2 E14 t(q)2+r 2 exp t4(qr)arctan(t(q)) , (15)
gluoLO 3 Mq5 (cid:16) 2pr (cid:17)
et(q) 1
(cid:0) (cid:1) −
where r 1/(N2 1), t(q) q/E 1 and E = MC2a 2/4 [23, 24]. This cross section induces a thermal
≡ c − ≡ | 1|− 1 − F s
widthofordera T (Mv2/Mv)2, i.e.,largerbya factor(Mv2/m )2 thanthedissociationwidthbyinelastic parton
s × p D
scattering.Inthisregime,gluodissociationisindeedthedominantdissociationmechanism.
Ofthesameorderasthedissociationbypartonscatteringisthegluodissociationatnext-to-leadingorder,whichhas
beencalculatedin[19].Thegluodissociationcrosssectionuptonext-to-leadingorderreads
s 1S (q)=Z(q/m )s 1S (q), (16)
gluo D gluoLO
m2
Z(q/m )=1 D ln(8q2/m2) 2 . (17)
D −4q2 D −
(cid:2) (cid:3)
Theresultfollowsfromaddingagluonself-energydiagramtothegluonpropagatorbeforeorafterthecutinFigure2.
Acomparisonbetweenthegluodissociationcrosssectionsuptoleadingandnext-to-leadingorderisinFigure9.
TheBhanot–Peskinapproximationisalargelyusedapproximationoftheexactgluodissociationformula(15)that
consistsinneglectingfinalstateinteractions,i.e.therescatteringofaQQpairinacoloroctetconfiguration[25,26].
Becausethequark-antiquarkcolour-octetpotentialisV =1/(2N ) a /r,itvanishesinthelargeN limit:
o c s c
×
29pa E 5/2(q+E )3/2
s 1S (q) 16 s| 1| 1 =16s 1S(q), (18)
gluoLO −N−c−→¥ 9 m q5 BP
→
d3q
G n (q)16s 1S(q)=G , (19)
1SLO −N−c→−→¥ Zq≥|E1|(2p )3 B BP 1S,BP
200
LO
150 NLOform Ma2=0.2
D× 0
NLOform Ma2=0.5
D× 0
Σ1S
gluo 100
a (cid:144)M
0
50
0
1 2 3 4 5 6 7
q
E
1
FIGURE 9. 1S gluo-dissociation cross section at leading order (continuous black line) and up to next-to-leading order for
mDMa20=0.2(dashedblueline)andformDMa20=0.5(dottedredline).
where we have keptC =4/3 in the overall normalization and E = MC2a 2/4. The cross section s 1S and the
widthG arerespecFtivelythe crosssectionandthe widthin the1Bha−not–PFesksinapproximation.A comBpParisonof
1S,BP
theexactwidthwiththeBhanot–PeskinwidthisinFigure10[23].Athightemperatures,theexactwidthisabout13%
largerthanitsapproximation,whichisconsistentwiththelargeN limit.Atlowtemperaturesthethermalwidthgoes
c
to zero,howeverthe exactandapproximatewidthsvanishwith differentfunctionalbehavioursandthe approximate
widthlargelyovershootstheexactone.
80
60
DM
(cid:144)
2E1
Αs 40
F
C
(cid:144)@
S
G1
20
0
0 1 2 3 4 5
T(cid:144)ÈE È
1
FIGURE10. Exactgluodissociationwidthatleadingorder(continuousblackline)vstheBhanot–Peskinapproximation(dashed
redline).
CONCLUSIONS
We have studied the dissociation of quarkonium in a thermal bath of gluons and light quarks in a framework that
makes close contact with modern effective field theories for non relativistic bound states at zero temperature. In a
weakly-coupledframework,ourconclusionsmaybeconciselysummarizedinthefollowingway.
(i) ForT <E thequark-antiquarkpotentialinthemediumcoincideswiththepotentialatT =0.
binding
(ii) ForT >E thepotentialgetsthermalcontributions.
binding
(iii) Form <E quarkoniumdecaysdominantlyviagluodissociation(akasinglet-to-octetbreakup).
D binding
(iv) Form >E quarkoniumdecaysdominantlyviainelasticpartonscattering(akaLandaudamping).
D binding
(v) ForT >T (<T ),quarkoniumdissociatesbeforeforming.
dissociation screening
∼
Inastrongly-coupledframework,thehierarchyofnon-relativisticscalesispreserved,whereasthethermodynamical
hierarchy may break down. This requires a non-perturbative definition and evaluation of the potential (real and
imaginary).
ACKNOWLEDGMENTS
I acknowledgefinancialsupportfromDFG andNSFC (CRC 110),andfrom the DFG cluster of excellence“Origin
andstructureoftheuniverse”(www.universe-cluster.de).
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