Turning on the Charm

0.22 MB·

by  Andre Peshier

Turning on the Charm
Andr´e Peshier
SUBATECH, Universit´e de Nantes, EMN, IN2P3/CNRS
4 rue Alfred Kastler, 44307 Nantes cedex 3, France
(Dated: February 2, 2008)
We argue that the strong jet quenching of heavy flavors observed in heavy-ion collisions is to a
large extent due to binary scatterings in the quark-gluon plasma. It can be understood from first
principles: thecharm collision probability beyond logarithmic accuracy and Markov evolution.
PACSnumbers: 12.38Mh
In particle physics, heavy flavorsare suitable to inves- light quarks (i=g,q) in the thermalized plasma,
8
tigate properties of the strong interaction. Their large
00 mass also proves useful when studying aspects of many- dEi = (dv)−1 ni(k) n¯i(k′) 1
2 body QCD for heavy-ion phenomenology, since quarks dx 2E Zk 2k Zk′ 2k′ Zp′ 2E′
n with m ≫ T can be considered as test particles in a (2π)4δ(4)(P+K P′ K′) 2 ω. (1)
a heat bath. Given this simplification, heavy quarks (say × − − |Mi|
J charm with mc 1.2GeV, which might be sufficiently X
largefortempera≈turesreachedatRHIC)canhelptoclar- Here d = 6 for a charm quark; v = p/E is its velocity,
3 n (k)=1/(ek/T 1)andn¯ =1 n arethe distribution
ify the ongoing debate on the parton energy loss in the fuinctions ofthe c∓ollisionpairtner±s,aindω =E E′ is the
h] quark-gluon plasma. For light partons, it is common to energy transfer in the scattering. 2 is sum−med over
attribute jet quenching entirely to radiative energy loss, i
p color and spin states of all particle|sM. |
motivated by its parametric energy dependence [1, 2].
-
Evaluating Eq. (1) with the Born cross sections [13]
p One may note that plasma parameters inferred in such
e approaches can be hard to reconcile with general expec- wouldyielddivergentresultsbecauseoflong-rangegauge
h interactions. Therefore,screeningeffects(whichformally
tations,cf.[3]. Whatismore,thepurelyradiativepicture
[ arisefromthermalloopcorrections)havetobetakeninto
ofjetsuppressionhasrecentlybeenchallengedbytheob-
account already in a leading order calculation. These
1 servation that heavy quarks (which radiate less [4]) are
v quenchedalmostasmuchaslightquarks,asconcludedby thermal corrections come along with the vacuum fluc-
5 tuations, which diverge and need to be renormalized.
analyzing electron yields from heavy flavor decays [5, 6].
9 Renormalizability, as such, is obviously not affected by
Thishasrevivedaninterest[7,8,9,10]forthecollisional
5 the(UV-finite)thermalcontributions–whichmightbea
energy loss as an additional suppression mechanism in
0
reason why renormalization is often utterly disregarded
. the range of moderately large momenta.
1 in finite-temperature field theory. However, only by this
0 With regard to the importance of the understanding fundamentalconceptQCDcalculationscanbepredictive.
8 ofjet quenching,itseemsnecessarytopointoutthatex-
Rigorous renormalizationcan be tedious, especially in
0 isting approaches suffer from several fundamental short-
thermal field theory. Fortunately, for the observables of
: comings, which makes their conclusions evasive. A ma-
v interest here, the leading-order results can be inferred
i jorityofapproachesisbasedonFokker-Planckequations, by elementary reasoning. For gluon exchange processes
X
despitethefactthattheassumeddominanceofsoftscat-
(t-channel scattering) it has been argued previously [14]
r tering [11] is justified only at leading logarithmic accu-
a that resumming and renormalizing the loop corrections
racy [12]. As one consequence, detailed balance must
amounts to replacing, in the Born amplitudes, the bare
be imposed in some way, whereas equilibration should
coupling by the running coupling α(t), schematically
be a prediction of the formalism. A second severe is-
sue of existing approaches is the disregard of the mo- α α(t)
, (2)
mentum dependence of the strong interaction. Strictly
t → t Π (ω,q)
T
speaking, QCD calculations are not predictive without −
discussing the running coupling (opposed to QED with where q =(ω2 t)1/2. We will parameterizethe thermal
α 1/137 in the widely applicable Thomson limit). self-energy Π −by an effective cut-off, of the order of the
QED ≈ T
For quantitative estimates, a value of the coupling has Debye mass and evaluated with running coupling [15],
then to be assumed; common is α = α( (2πT)2).
fix
−
However, the energy loss probes also parametrically soft µ2(t)=κ 4π 1+ 1n α(t)T2. (3)
· 6 f
scales, √αT, which is not only decisive for its high-
energy∼limit [22]; it will also turn out to be crucial for The customary ad hoc ch(cid:0)oice for t(cid:1)his screening mass is
heavy ion phenomenology. µ2fix =4π 1+ 16nf αfixT2, which we adopt only for the
sake of comparison with existing estimates. In fact, the
(cid:0) (cid:1)
Letusstartbyconsideringtheaverageenergylossofa coefficient κ in Eq. (3) can be calculated by compar-
charmquarksubjecttobinarycollisionswithgluonsand ing dEt−channel/dx, evaluated with effective cut-off, to
2
the strict resultbeyond logarithmic accuracy [16], which 10
yields 1 T=0.3GeV
κ=(2e)−1 0.2. (4)
≈ p) 10-1 ( V2)
Theresultingcut-offissmall,whichwillbeamainsource , fixcoupling
of differences between our and prevalent estimates. P( 10-2
Complementing the arguments put forwardin [14], we
10-3
now turn to the s and u-channel contributions to charm
scattering. In order to quantify ‘the’ coupling, consider 10-4
the vacuum corrections to the Born amplitudes, which -2 0 2 4 6 8
have logarithms of characteristic momenta. By specify- /GeV
ing a renormalization scale µ , bare quantities are ex-
R FIG. 1: Probability density Pi=g,qPi(ω,p = 8GeV). The
pressed in terms of physical ones, such as the coupling band gives the sensitivity under variation ν ∈ [1,2] of the
α(µ2R). By renormalization flow equations, all choices scaleintherunningcouplingα(νV2). Shownforc2omparison
of µ are physically equivalent. This allows, if there is
R istheresultobtainedwiththecouplingαfix andcut-offµfix.
onlyonecharacteristicmomentumP,tochooseµ =P.
R
Then the logarithmic term vanishes, and the renormal-
ized result looks like the Born approximation – albeit for a test particle with momentum p to change, per unit
with α α(P2). Now, the collisional energy loss (1), as of time, its energy by ω. The condition H 0, where
→ ≥
athermalaverageof 2, isdominatedby interactions
with s ET V2 ,|Mwhi|ere V2 = t or u m2 [12]. For H = (4π)4E2 s−(E+k)2 t2+ (2Ek−s+m2c)2
otvhafintshtpemainro∼ttmiecreumnltaeurd≫mikai|tnseceasmtl|eaattdeiec,tsce,fr.tmh[1ei7nr]ei.niWgsαienwd–eiltelhdfi−exovntihlryteuocanoliuetpyrleiVnleg2- −−ω4k22(sp(cid:2)2−(cid:0)+m22cω)(2k(,s+m(cid:1)2c)−E(cid:0)(s−m2c))(cid:1)t (8)
alsoforthe remainingsub-dominantscatteringcontribu- determines not only the(cid:3) bounds t in Eq. (7), it also
±
tions at the respective virtuality, not without estimating constrains the integral over k, in particular such that
the arising uncertainty. For time-like contributions, we k+ω 0 is always fulfilled. Thus energy gain is expo-
continue the 1-loop coupling according to Ref. [18], nential≥ly suppressed for ω < T. For ω > 0, P ω−2
uptonearthekinematicth∼res−hold,seeFig.1,whic∼halso
4π L−1
α(Q2)= − for Q2 ><0, (5) shows the uncertainties arising from the scale setting.
β0 ( 12 −π−1atn(L+/π) In the following we consider an ensemble of test par-
whereβ =11 2n withn =3,andL =ln( Q2/Λ2). ticles. In order to describe its momentum distribution
0 −3 f f ± ± f(p) it is convenient to introduce the rate (p′,p) for
With regard to quantitative estimates it is worthwhile
transitions p p′. The spectrum changes byPscattering
recalling that perturbative approaches can be of use at
→
out and into a given state: for a sufficiently small time
surprisingly soft momentum scales [19]. May details of
the behavior of α(Q2) in the deep infrared be uncertain, interval f(q,δt) = (1 δtΓ(q))f(q)+δt dp (q,p)f(p),
− P
where Γ(q)= dp (p,q) denotes the interaction rate of
thereexistsarobustconstraint(universalityhypothesis),
P R
particles with momentum q. Introducing
R
α¯ =Q−1 dQα(Q2) 0.5, (6)
u ≃ (q,p)=(1 δtΓ(q))δ(q p)+δt (q,p) (9)
Z|Q2|≤Q2u T − − P
makes explicit that the time evolution of the spectrum,
where Q = 2GeV [19]. Accordingly, we impose an up-
u
per bound on the running coupling (5), α(Q2) 1.1 for
our preferred QCD parameter Λ=0.2GeV (ad≤justed to f(q,t+δt)= dp (q,p)f(p,t), (10)
T
latticeresultsfortheheavyquarkpotential,cf.[15]). We Z
have verified that the deep infrared regionas well as the is a first order Markov process (the transition depends
precisevalueofΛarenotveryimportantforourconcerns only on the state rather than on preceding history). We
– for s and u-channel terms due to phase space suppres- discussthe resulting Markovchainonadiscrete momen-
sion, and for t-channel contributions by screening. tum space, which arises naturally from binning [24] and
Returning to Eq. (1), the thermal 2-body phase space reduces the convolution (10) to a matrix multiplication,
can be expressed as an integral over ω and the invariant
f (t+δt)= f (t). (11)
momentum transfer t [12]. This allows us to write q Tqp p
In other words, the evolution in (discrete) time is de-
dE
dxi =v−1 dωPi(ω,p)ω, termined simply by powers of the transition matrix Tqp,
Z which is not only conducive to numerical studies, it also
with the probability density [23] brings instructive insight readily [25]. Indeed, essential
n (k) t+ dt properties of the evolution follow alone from
P (ω,p)= i n¯ (k+ω) 2 (7)
i Zk 2k i Zt− √H |Mi| qTqp =1 for all p, (12)
X
P
3
whichholdsbydefinitionofΓ(q). Itimpliesfirstthat This relation between long and short-time aspects of
qp
mapsthehyperplaneH characterizedby f fT= equilibration is worth emphasizing. More important,
k k≡ p p
constantontoitself. Putdifferently,the‘norm’ f isin- Fig. 3 reveals that conventionalcalculations would over-
variantunder ,whichinthepresentcontextkdPeskcribes estimatethiscrucialtimescalebyafactorK 5 κ−1.
Tqp ≈ ≃
particlenumberconservation. Fromthisweinfer λ 1
i
| |≤
for the eigenvalues of ; otherwise the normwould not
qp
T
be conserved in repeated mappings. Among λ is an
i
eigenvalue λ = 1 since the rank of the matri{x } 1 10
1 qp
T −
is dim( ) 1. These properties of the eigenvalue spec- 5
Tqp − m fixcoupling
trumareprovenmorerigorouslyinthePerron-Frobenius f
theorem, cf. [20], which also shows that λ is a simple / 2
1 p
eigenvalue. Consequently, the stationary state 1
loss
0.5 gain
fqeq =nl→im∞Tqnpfp(0), (13) total T=0.3GeV
0 2 4 6 8 10
is unique and approachedfor each initial distribution.
The corresponding eigenvectors e of form a basis p/GeV
i qp
with the peculiarity that only e1 has aTcomponent per- FIG. 3: Average time τ for a charm quarkto change(gain
pendicular to the hyperplane, i.e., ei H for i = 1, orlose)momentumby∆∆pp>0.4GeV,comparedtothefixed-
∈ 6
see Fig. 2. Hence, we can separate in the evolution coupling estimate (uncertaintyband as in Fig. 1).
f(nδt) = λnφ e = feq + λnφ e a part
i=1 i i i i=2 i i i
which proceeds entirely in H. Since λ < 1, the ap-
i6=1 Realizing that binary collisions are far more effective
P P| |
thanpreviouslyestimated,weconsiderinanexploratory
1 y study jet quenching in the mid-rapidity regionof central
e collisionswithintheBjorkenmodel. Tocomparewithex-
2
set T0 Tc τ0 tlife R dNini/dp2t
e
1 x I 0.42⋆ 0.18 0.6 7.4 5.0⋆ (pt+0.5)2(1+pt/6.8)−21
II 0.30 0.165 1.0 5.0 6.6⋆ (p2+1.82)−3.5
0 1 t
FIG. 2: Equilibration in 2 dimensions (where H is given by
TABLE I: Representative parameter sets for the Bjorken
x+y=1);theinitialstatesare(1,0)and(0,1),respectively.
model(units: GeVorfm;⋆adjustedtototalentropyS ≈104).
proach to equilibrium for large t is exponentially fast,
isting results [8, 9], we adopt parameterizations as sum-
f(t) feq exp( t/τmix), (14) marized in Tab. I. For transparency, we will not discuss
k − k ∝ − effects of hadronization (for heavy flavors the partonic
where the so-called mixing time suppression can be indicative for the observed R ).
AA
Consider first the evolution of f (p ) p dN /dp2
τmix =δt/ln(λ−1) (15) 0 t ∝ t ini t
2 at constant temperature. Fig. 4 shows the hard part of
is determined by the second largest eigenvalue [26].
It is instructive to illustrate these essential features of 1
thermalization by means of a toy model, namely 10-1 T=0.3GeV
Tij =(1 l g)δi,j +lδi,j+1 gδi,j−1, (16) t) 10-2
− − − p, 0fm/c
with constants l and g parameterizing the relevant loss f( 10-3 24ffmm//cc
wastniatdthiogtnahaienry‘rteadmtiseptsreiirbnautTtuiqorpen.’iΘsFie=rxspt1,o/nitlenni(stli/aegla,)sfybieeqtino∝gseedexeptte(h−ramit/inΘthe)de, 1100--45 18e2qfumfmil/ic/bc
by the logarithmic ratio of loss and gain [27]. Further- 0 2 4 6 8 10
more, for large dimension of T , one can readily derive p/GeV
ij
FIG. 4: Evolution of the initial charm spectrum I at fixed T
λ =1 (√l √g)2, (17)
2 − − (for uncertainty band cf. Fig. 1).
which showsa non-analytic behaviorofthe mixing time.
In QCD, for sufficiently fast charm quarks, the the spectrum being quenched markedly already after a
(binned) loss rate dominates over gain (cf. Fig. 3), thus few fm/c, as could be anticipated from τ∆p 1fm/c.
∼
Taking then into account the path length distribution
τmix Γ−1 Γ−1 . (18) dN/dl (1 (l/2R)2)1/2 in Bjorken’s heat bath, T =
∼ loss ∼ total ∝ −
4
T (τ/τ )−1/3, is straightforward in the Markov formal- the Bjorkenmodel. Radialexpansionwill reduce the life
0 0
ism. Fig. 5 shows the charm quenching ratio r = timeoftheplasmaphase,butalsomodifythepathlength
AA
(dN(t )/dp2)/(dN /dp2); to compare to r > 0.85 distribution dN/dl. The resulting compensation will be
life t ini t AA
withfixedcoupling,andtoRefs.[7, 8,9,10]. Give∼nthat quantifiedinaforthcomingstudywithmorerealisticcol-
lision dynamics and including hadronization [21].
1.2
1.0 setI To summarize, we have shown in a Markov formalism
that,formoderatelylargemomenta,binarycollisionsare
setII
0.8
akeymechanismforjetquenchingofheavyflavors. This
AA 0.6 conclusion requires taking into account – more carefully
r
than in existing approaches – essential features of QCD,
0.4
namely the momentum dependence of the strong cou-
STAR[5]
0.2 PHENIX[6] pling and relevant screening effects. Both aspects are
(stat.errorsonly)
interconnected and, as a matter of fact, mandatory for
0.0
2 4 6 8 quantitative estimates from thermal field theory, unless
p /GeV temperatures are asymptotically large.
t
FIG. 5: Comparison of partonic charm quenchingin Bjorken In closing, since jet quenching has significantly influ-
model to the observed RAA. For better visibility, the inter- enced the current picture of the strongly coupled quark-
section of uncertainty bands(cf. Fig. 1) was not hatched. gluonplasma(sQGP),itseemsworthwhiletakingamore
general point of view. Commonly used but rather crude
we have calculated the underlying mechanism from first estimates(asfortherelevantscreeningrange)canleadto
principles, the comparison with the RHIC data [5, 6] is interpretationsofa‘too’stronglycoupledplasma. While
striking. Depending on details of the initial spectrum forheavyionphenomenologytherelevantcouplingiscer-
and hadronization, there is room for radiative quench- tainly not small, we do see the possibility to understand
ing (with reasonable parameters), which becomes more essential features of the quark-gluon plasma from first
important at larger p . With regard to the increasing principles – perturbative QCD can work like a charm.
t
r at smaller p we underline that the Cronin effect
AA t
has not been taken into account in the calculation. The Acknowledgments: I thank J. Aichelin, P.B. Gos-
smallvalue ofr atp >4GeVis quite robust,as veri- siaux, A. Smilga and in particular S. Peign´e for fruitful
AA t
fied here with two rather∼different parameter sets within discussions.
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