Non-relativistic particles in a thermal bath

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by  Antonio Vairo

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(cid:13)
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Non-relativistic particles in a thermal bath
n
a
J
4 AntonioVairoa
1
1Physik-Department,TechnischeUniversitätMünchen,

James-Franck-Str.1,85748Garching,Germany
]
h
p
Abstract.Heavyparticlesareawindowtonewphysicsandnewphenomena. Sincethe
-
p late eighties they are treated by means of effective field theories that fully exploit the
e symmetriesandpowercountingtypicalofnon-relativisticsystems. Morerecentlythese
h
effectivefieldtheorieshavebeenextendedtodescribenon-relativisticparticlespropagat-
[
inginamedium.Afterintroducingsomegeneralfeaturescommontoanynon-relativistic

1 effectivefieldtheory,wediscusstwospecificexamples: heavyMajorananeutrinoscol-
v lidinginahot plasmaof StandardModel particlesintheearlyuniverseand quarkonia
4 producedinheavy-ioncollisionsdissociatinginaquark-gluonplasma.
0
2
3
1. 1 Introduction
0
4 Heavyparticlesareawindowtonewphysicsfortheymaybesensitivetonewfundamentaldegreesof
1 freedom.Someofthesenewdegreesoffreedommaybethemselvesheavyparticles(like,forinstance,
:
v the heavy neutrinosthat we will discuss in section 2). Heavy particles can also be clean probes of
i new phenomena emerging in particularly complex environments. Examples are heavy quarks and
X
quarkoniaasprobesofthestateofmatterformedinheavy-ioncollisions.
r
WecallaparticleheavyifitsmassMismuchlargerthananyotherscaleEcharacterizingthesys-
a
tem. ThescaleE mayincludethespatialmomentumoftheheavyparticle,massesofotherparticles,
Λ , symmetry breaking scales, the temperature T of the medium and any other energy or mo-
QCD
mentumscale thatdescribestheheavyparticleanditsenvironment. Underthisconditiontheheavy
particleisalsononrelativistic. ThehierarchyM Ecallsforalowenergydescriptionofthesystem
≫
intermsofasuitableeffectivefieldtheory(EFT)whosedegreesoffreedomareafieldHencodingthe
low-energymodesoftheheavyparticle,andallotherlow-energyfieldsofthesystem. Inareference
frame where the heavy particle is at rest up to fluctuations that are much smaller than M, the EFT
Lagrangianhasthegeneralform
1
= H iD H+higher-dimensionoperators powersof + . (1)
† 0 lightfields
L × M L
In the heavy-particle sector the Lagrangianis organizedas an expansion in 1/M. Contributions of
higher-dimensionoperators to physical observables are counted in powers of E/M. It is crucial to
notethattheEFTLagrangiancanbecomputedsettingE =0. Henceitsexpressionisindependentof
thelow-energydynamics.TheprototypeoftheEFT(1)istheheavyquarkeffectivetheory[1,2].
ae-mail:[email protected]
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In this contribution,we will concentrateon the case of a heavyparticle of mass M propagating
inandinteractingwitha mediumcharacterizedbyatemperatureT muchsmallerthan M. Thisisa
special case of the previousone. The EFT Lagrangianthat describesthe system at an energyscale
muchlowerthanMhasagainthestructure(1). Aspointedoutbefore,thetemperaturedoesnotenter
in the computation of the Lagrangian, which is fixed by matching at T = 0. This means that the
Wilsoncoefficientsencodingthecontributionsofthehigh-energymodescanbecomputedinvacuum.
ThetemperatureisintroducedviathepartitionfunctionoftheEFTandaffectsthecomputationof
theobservables.Contributionsofhigher-dimensionoperatorsarecountedinpowersofT/M. Inorder
tostudythereal-timeevolutionofphysicalobservablesandinparticulardecaywidths,itisconvenient
tocomputethepartitionfunctionintheso-calledreal-timeformalism. Thisconsistsinmodifyingthe
contour of the partition function to allow for real time (see e.g. [3]). In real time, the degrees of
freedomdouble. However,intheheavy-particlesectortheseconddegreesoffreedomdecouplefrom
thephysicaldegreesoffreedomsothat,aslongasloopcorrectionstolightparticlescanbeneglected,
theonlydifferencewithT =0EFTsconsistsintheuseofthermalpropagators [4].
In the following, we will consider heavy particles interacting weakly with a weakly-coupled
plasma. We will computefor them the correctionsto the width inducedby the medium, which we
calltheirthermalwidth, Γ. Inparticular,insection2wewillcomputethethermalwidthofaheavy
MajorananeutrinointeractingweaklywithaplasmaofmasslessStandardModel(SM)particlesinthe
primordialuniverse.Whereasinsection3wewillcomputethethermalwidthofaquarkonium,which,
liketheΥ(1S),isheavyenoughtobeconsideredanon-relativisticCoulombicboundstate,andispro-
ducedinheavy-ioncollisionsofsufficienthighenergythattheformedmediumisaweakly-coupled
plasmaoflightquarksandgluons.
2 Heavy Majorana neutrinos
We consider a heavy Majorana neutrino, described by a field ψ of mass M much larger than the
electroweakscale, M M ,coupledtotheSMonlythroughHiggs-leptonvertices:
W
≫
1 M
= + ψ¯i∂/ψ ψ¯ψ F L¯ φ˜P ψ F ψ¯P φ˜ L , (2)
L LSM 2 − 2 − f f R − ∗f L † f
where istheSMLagrangianwithunbrokenSU(2) U(1) gaugesymmetry,φ˜ = iσ2φ ,withφ
LSM L× Y ∗
theHiggsdoublet,L areleptondoubletswithflavor f,F isa(complex)YukawacouplingandP =
f f L
(1 γ5)/2,P =(1+γ5)/2aretheleft-handedandright-handedprojectorsrespectively.Thisextension
R
−
oftheSMprovidesamodelforneutrinomassgenerationthroughtheseesawmechanism[5,6].Italso
providesamodelforbaryogenesisthroughthermalleptogenesis[7,8]. Forarecentreviewsee[9].
Letusconsiderbaryogenesis.DifferentlyfromtheSM,themodel(2)hasthepotentialtooriginate
a large baryonasymmetry. The mechanism is the following. At a temperatureT <M the neutrino
falls out of equilibrium.1 This happens because, as the temperature decreases, re∼combination pro-
cessesbecomeless andless frequentwhile theneutrinodecaysin theplasma. Sincethe neutrinois
aMajoranaparticle,anetleptonasymmetryisgenerated. Thisistransferredtoabaryonasymmetry
throughsphalerontransitions. ThephasesoftheYukawacouplingsF intheLagrangian(2)provide
f
extrasourcesofCandCPviolationsbesidesthoseintheSM.Finallythegeneratedbaryonasymmetry
isprotectedfromwashoutaftersphaleronfreeze-outatatemperatureT M . Hencethemodelmay
W
∼
fulfillthethreenecessarySakharovconditionsforbaryonasymmetry[10]inastrongerwaythanthe
SMdoesandaccountfortheobservedbaryonasymmetryintheuniverse.
1 In(2)we havesimplified the realistic case with moreneutrino generations byconsidering only oneheavy Majorana
neutrino.
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Figure1.HeavyMajorananeutrinodecayingintheearlyuniverseplasma.
Animportantquantityforleptogenesisistherateatwhichtheplasmaoftheearlyuniversecreates
Majorana neutrinos with mass M at a temperature T. This quantity is in turn related to the heavy
Majorana neutrino thermal width in the plasma, see figure 1. We consider the temperature regime
M T M . TheheavyMajorananeutrinosarenon-relativistic,withmomentum
W
≫ ≫
pµ = Mvµ+kµ, kµ M. (3)
≪
Inkineticequilibriumtheresidualmomentumkµisoforder √MT;faroutofequilibriumitisoforder
T. IftheneutrinoisatrestuptofluctuationsthataremuchsmallerthanM,thenvµ =(1,0).
At an energy scale much smaller than M the low-energy modes of the Majorana neutrino are
describedbyafieldNwhoseeffectiveinteractionswiththeSMparticlesareencodedintheEFT[11]:2
= + , (4)
L LSM LN
where
iΓ (1) (2) (3) 1
= N¯ i∂ T=0 N+ L + L + L + . (5)
LN 0− 2 ! M M2 M3 O M4!
TheLagrangian(5)isofthetype(1),theonlydifferencebeingthattheMajorananeutrinoisagauge
singletwithafinitewidthatzerotemperature,Γ ,duetoitsdecayintoaHiggsandlepton.
T=0
Figure2.One-loopmatching
conditionfortheneutrino-Higgs
coupling.
The powercountingof the EFT indicatesthatthe leadingoperatorsresponsibleforthe neutrino
thermaldecayaredimension5operatorscontributingto (1). ThesymmetriesoftheEFTallowonly
L
foronepossibledimension5operator,whichis
(1) =a N¯Nφ φ. (6)
†
L
This describes the scattering of Majorana neutrinos with Higgs particles. Scatterings of Majorana
neutrinoswith gaugebosons, leptonsor quarksare subleading. TheWilson coefficienta is fixed at
oneloopbythematchingconditionshowninfigure2. Theleft-handsidestandsforan(in-vacuum)
2AnEFTforheavyMajoranafermionshasbeenconsideredalsoin[12].
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diagraminthefundamentaltheory(2),whereastheright-handsideforan(in-vacuum)diagraminthe
EFT. Double linesare neutrinopropagators,single lines lepton propagatorsand dashed lines Higgs
propagators.Forthedecaywidthonlytheimaginarypartisrelevant;itreads
3
Ima= F2λ, (7)
−8π| |
whereλisthefour-Higgscoupling.
Figure3.Higgstadpolecontributingtotheneutrinothermal
width.
Thethermalwidthinducedby(6)maybecomputedfromthetadpolediagramshowninfigure3,
wherethedashedlinehastobeunderstoodnowasaHiggsthermalpropagator. Theleadingthermal
widthreads[13,14],[11],[15]
Ima F2M T 2
Γ=2 M hφ†(0)φ(0)iT =−| 8|π λ(cid:18)M(cid:19) , (8)
where φ (0)φ(0) standsforthethermalcondensateofthefieldφ. NotethatintheEFTthecalcu-
† T
h i
lationhassplit intoa one-loopmatching,shownin figure2, whichcan be donein vacuum,and the
calculationofa one-looptadpolediagram,showninfigure3, whichisdonein thermalfield theory.
Theresultingsimplificationofthecalculationwithrespecttoafullyrelativistictreatmentinthermal
fieldtheoryistypicaloftheEFTapproach.
In a similar fashion one can calculate T/M suppressed corrections to the thermal decay width.
Alsoforthemthecalculationsplitsintoanin-vacuummatchingofhigher-dimensionoperatorsinthe
expansion(5) and in the calculation of one-looptadpole diagramsin thermalfield theory. Only di-
mension7operatorscontributetothewidthatnextorderinT/M. Theseareeightoperatorsbelonging
to (3). TwoeachdescribecouplingsoftheMajorananeutrinotoHiggsparticles,leptons,quarksand
L
gaugebosonsrespectively. Finally,thethermalwidthatfirstorderintheSMcouplingsandatorder
T4/M3 reads[14],[11]
F2M T 2 λk2T2 π2 T 4 7π2 T 4
Γ= | 8|π "−λ (cid:18)M(cid:19) + 2 M4 − 80(3g2+g′2) (cid:18)M(cid:19) − 60 |λt|2 (cid:18)M(cid:19) # , (9)
wheregistheSU(2)coupling,g theU(1)coupling,andλ thetopYukawacoupling.
′ t
3 Heavy quarkonia
Heavy quarkonia are bound states of heavy quarks. A quark is considered heavy if its mass M is
much larger than the typical hadronic scale Λ . Quarkonia include bound states of charm and
QCD
bottom quarks. They are a probe of the state of matter made of gluonsand light quarksformed in
high-energyheavy-ioncollisions[16]. Thereasonsarethatheavyquarksareformedearlyinheavy-
ioncollisions,1/M 0.1fm 1fm,henceheavyquarkoniumformationissensitivetothemedium,
∼ ≪
andthatthequarkoniumdileptondecayprovidesa cleanexperimentalsignal. Thedissociationofa
heavyquarkoniuminaplasmaofquarksandgluonsissketchedinfigure4.
InternationalConferenceonNewFrontiersinPhysics2013
Figure4.Heavyquarkoniumdissociatinginaplasmaoflightquarksand
Q gluons.
Q
UndertheconditionM T andforquarkoniaformedalmostatrestinthelaboratoryrestframe,
≫
themassMoftheheavyquarkisthelargestscaleinthesystem;asinthegeneralframeworkdiscussed
inthe introduction,we mayconsiderquarkonianon-relativisticparticlessuitableto bedescribedby
non-relativistic EFTs of the type (1). Differently from the Majorana neutrino case, quarkonia are,
however,compositesystemscharacterizedbyseveralinternalenergyscales,whichinturnmayprobe
thermodynamicalscalessmallerthanthetemperature. Hencethesituationismorecomplexthanthe
one discussed in section 2. The energy scales characterizing a non-relativistic bound state are the
typicalmomentumtransferintheboundstate,whichisoforder Mv,andthetypicalbindingenergy,
which is of order Mv2. The parameter v 1 is the relative heavy-quarkvelocity. This is of order
≪
α foraCoulombicboundstate. Wecallthesescalesthenon-relativisticscales. Thenon-relativistic
s
scales are hierarchically ordered: M Mv Mv2. Effective field theories exploiting the non-
≫ ≫
relativistichierarchyinvacuumhavebeenreviewedin[17]. Foraweakly-coupledplasma,arelevant
thermodynamicalscale,whichissmallerthanthetemperature,istheDebyemass,m ,i.e. theinverse
D
of the screening length of the chromoelectricinteractions. This is of order gT, hence we have that
parametricallyT m . We callthescalesT,m andpossiblysmallerscalesthethermodynamical
D D
≫
scales. For a discussion on the energyscale hierarchy in the case of Υ(1S) producedin heavy-ion
collisionsattheLHCwereferto[18,19]. Forfirstexperimentalevidenceofsuppressionofexcited
bottomoniumstatesattheLHCwereferto[20].
non−relativistic thermodynamical EFTs
scales scales
M
Figure5.ScalesandEFTsforquarkoniumina
thermalbath.
M v T NRQCD
mD
M v2 pNRQCD
The existenceof a hierarchyof energyscales calls for a descriptionof the system in terms of a
hierarchyofEFTsofQCD,whichisthefundamentaltheoryinthiscase. ManyEFTsarepossiblein
dependenceofthespecificorderingofthethermodynamicalscaleswithrespecttothenon-relativistic
ones. These are schematically shown in figure 5. For temperatureslarger than those consideredin
figure5quarkoniumdoesnotform.
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We call generically non-relativistic QCD (NRQCD) [21, 22] the EFT obtained from QCD by
integratingoutmodesassociatedwith thescale M andpossiblywith thermodynamicalscaleslarger
thanMv. TheLagrangianreads
D2 D2
=ψ iD + +... ψ+χ iD +... χ+ + , (10)
† 0 † 0 light
L 2M ! − 2M ! ··· L
whereψ(χ)isherethefieldthatannihilates(creates)theheavy(anti)quark.
We call generically potentialNRQCD (pNRQCD) [23, 24] the EFT obtained from NRQCD by
integratingoutmodesassociatedwiththescale Mvandpossiblywiththermodynamicalscaleslarger
than Mv2. The degrees of freedom of pNRQCD are quark-antiquarkstates (cast conveniently in a
coloursingletfield SandacolouroctetfieldO),lowenergygluonsandlightquarkspropagatingin
themedium.TheLagrangianreads
p2 p2
= d3r Tr S i∂ V +... S+O iD V +... O
† 0 s † 0 o
L Z ( − M − ! − M − ! )
1
+Tr O r gES+H.c. + Tr O r gEO+c.c. + + , (11)
† † light
· 2 · ··· L
n o n o
where E is the chromoelectricfield and g is now the SU(3) coupling. Both EFT Lagrangians(10)
c
and (11) are of the form (1); only the field S is a gauge singlet. The pNRQCD Lagrangian is also
organizedasanexpansioninr: risthedistancebetweentheheavyquarkandantiquark,whichisof
order1/(Mv). At leadingorderin r, the singlet field S satisfies a Schrödingerequation. Hence the
Wilson coefficientV may be interpretedas the singlet quarkoniumpotential. Similarly the Wilson
s
coefficientV may beinterpretedasthe octetquarkoniumpotential. Theexplicitexpressionsof the
o
potentials depend on the version of pNRQCD that is considered; in particular, if thermodynamical
scaleshavebeenintegratedout,V andV maydependonthetemperature. Wehavesetequaltoone
s o
possibleWilsoncoefficientsappearinginthesecondlineof (11).
Akeyquantityfordescribingtheexpectedquarkoniumdileptonsignalisthequarkoniumdisso-
ciationwidth.Atleadingorderitmaybeusefultodistinguishbetweentwodissociationmechanisms:
gluodissociation, which is the dominant mechanism for Mv2 m , and dissociation by inelastic
D
≫
partonscattering,whichisthedominantmechanismfor Mv2 m . Beyondleadingorderthetwo
D
≪
mechanismsare intertwined and distinguishingbetween them would becomearbitrary, whereas the
physicalquantityisthetotaldecaywidth.
mv2
Figure6.QuarkoniumgluodissociationinpNRQCD.
Gluodissociationisthedissociationofquarkoniumbyabsorptionofagluonfromthemedium[25,
26]. The gluon is lightlike or timelike (if it acquires an effective mass propagating through the
medium). Gluodissociationisalsoknownassinglet-to-octetbreakup[4, 27]. Theprocesshappens
whenthegluonhasanenergyoforder Mv2. HencegluodissociationcanbedescribedinpNRQCD.
Inparticular,itcanbecalculatedbycuttingthegluonpropagatorinthepNRQCDdiagramshownin
figure6,wherethesinglelinestandsforaquark-antiquarkcoloursingletpropagator,thedoubleline
InternationalConferenceonNewFrontiersinPhysics2013
foraquark-antiquarkcolouroctetpropagatorandthecirclewithacrossforachromoelectricdipole
vertex.
80
60
DM
Α2(cid:144)Es1 40 Figure7.Γ1S duetogluodissociation(continuousblackline)
CF vstheBhanot–Peskinapproximation(dashedredline).
G(cid:144)@1S
20
0
0 1 2 3 4 5
T(cid:144)ÈE1È
Cuttingrulesatfinitetemperature[28–31]constrainthegluodissociationwidthofaquarkonium
withquantumnumbersnandl,(QQ) ,whichisatrestwithrespecttothemedium,tohavetheform
nl
d3q
Γ = n (q)σnl (q), (12)
nl Z (2π)3 B gluo
qmin
where σnl is the in-vacuumcross section (QQ) +gluon Q+ Q, and n is the Bose–Einstein
gluo nl → B
distribution. The explicit leading order (LO) expression of σ1S for a Coulombic 1S state like the
gluo
Υ(1S)is[27,32]
αC E4 exp 4ρ arctan(t(q))
σ1glSuoLO(q)= s3F210π2ρ(ρ+2)2Mq15 t(q)2+ρ2 (cid:16)t(q)2πρ (cid:17), (13)
(cid:16) (cid:17) et(q) 1
−
whereρ = 1/(N2 1),t(q) = q/E 1, E = MC2α2/4andC = (N2 1)/(2N ). Thecorre-
c − | 1|− 1 − F s F c − c
spondingwidthis shownin figpure7. Note thatthe gluodissociationformulaholdsfortemperatures
suchthatT Mvandm Mv2 E . Infigure7wealsocomparewithapopularapproximation,
D 1
≪ ≪ ∼| |
theso-calledBhanot–Peskinapproximation[33,34]. ThisisthelargeN limitofthefullresult(13)
c
(butkeepingC =4/3intheoverallnormalization). TakingthelargeN limitamountsatneglecting
F c
finalstateinteractions,i.e. therescatteringofaQQpairinacolouroctetconfiguration.
Figure8.Quarkonium
dissociationbyinelastic
partonscatteringinNRQCD.
Dissociation by inelastic parton scattering is the dissociation of quarkonium by scattering with
gluons and light-quarks in the medium [35, 36]. The exchanged gluon is spacelike. Dissociation
by inelastic parton scattering is also known as Landau damping [37]. Because external gluons are
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transverse, according to NRQCD each external gluon is suppressed by T/M, see (10). At leading
order,wemaythereforejustconsidercontributionstothewidthcomingfromcuttingdiagramswitha
self-energyinsertioninonesinglegluonexchange[38]. Iftheexchangedgluoncarriesamomentum
oforderMv,thentherelevantdiagramsmaybecomputedinNRQCD,seefigure8. Iftheexchanged
gluon carriesa momentummuch smaller than Mv, then the relevantdiagramsmay be computedin
pNRQCD,seefigure9. Inbothfiguresthedashedcirclestandsforaone-loopself-energyinsertion.
Figure9.Quarkoniumdissociationbyinelasticpartonscattering
inpNRQCD.
Cuttingrulesatfinitetemperatureconstrainthewidthbypartonscatteringofaquarkoniumwith
quantumnumbersnandl,whichisatrestwithrespecttothemedium,tohavetheform
d3q
Γ = f (q) 1 f (q) σnl(q). (14)
nl Z (2π)3 p ± p p
Xp qmin h i
Thesumrunsoverthedifferentincomingandoutgoingpartons(pstandsforparton,itmaybeeither
a lightquark,q, for whichthe minussign holds, or a gluon, g, forwhich the plussign holds), with
f = n and f = n (n is the Fermi–Diracdistribution). The quantityσnl is the in-mediumcross
g B q F F p
section (QQ) + p Q+ Q+ p. The convolutionformula correctly accounts for Pauli blocking
nl
→
in the fermionic case (minus sign). Note that (14) differs from the correspondinggluodissociation
formula(12)inthefactthatitaccountsforthethermaldistributionsofboththeincomingandoutgoing
partons. Moreover,thecrosssectionσnl isnotanin-vacuumcrosssection. Explicitexpressionsfor
p
the cross section in the case of a Coulombic 1S state like the Υ(1S) can be found in [38]. These
arevalid fortemperaturessuchthatT m Mv2 E . Thecorrespondingwidth isshownin
D 1
≫ ≫ ∼ | |
figure10,wherewehaveassumedthreelightquarksinthemedium.Notethedifferentnormalization
ofthewidthwithrespecttofigure7.
800
600
DM
2(cid:144)E1 Figure10.Γ duetoinelasticpartonscattering.Wehave
Α2s 1S
ΠCF 400 takenmDa0 =0.5and|E1|/mD =0.5,wherea0istheBohr
G(cid:144)@81S radius.
200
0
2 4 6 8 10
T(cid:144)ÈE1È
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4 Conclusions
Inaframeworkthatmakesclosecontactwithmoderneffectivefieldtheoriesfornon-relativisticparti-
clesatzerotemperature,onecancomputethethermalwidthofnon-relativisticparticlesinathermal
bath in a systematic way. In the situation M T one may organizethe computationin two steps
≫
andcomputethephysicsatthescale M asinvacuum. IfotherscalesarelargerthanT,thenalsothe
physicsofthosescalesmaybecomputedasinvacuum.Wehaveillustratedthisontheexamplesofa
heavyMajorananeutrinodecayingintheearlyuniverseplasmaandaheavyquarkoniumdissociating
inaweakly-coupledquark-gluonplasma.
Acknowledgements
I thank Simone Biondini, Nora Brambilla and Miguel Escobedo for collaboration on the work presented in
section2andNoraBrambilla,MiguelEscobedoandJacopoGhiglieriforcollaborationontheworkpresented
insection3. Iacknowledge financial support fromDFGandNSFC(CRC110), andfromtheDFGcluster of
excellence“Originandstructureoftheuniverse”(www.universe-cluster.de).
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