Momentum anisotropy effects for quarkonium in a weakly-coupled quark-gluon plasma below the melting temperature

1.2 MB·

by  Simone Biondini

TUM-EFT 88/16
Momentum anisotropy effects for quarkonium in a weakly-coupled quark-gluon
plasma below the melting temperature
S. Biondini,1 N. Brambilla,2,3 M. A. Escobedo,4,5 and A. Vairo2
1Albert Einstein Center, Institute for Theoretical Physics,
University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
2Physik-Department, Technische Universit¨at Mu¨nchen,
James-Franck-Str. 1, 85748 Garching, Germany
3Institute for Advanced Study, Technische Universit¨at Mu¨nchen,
Lichtenbergstrasse 2 a, 85748 Garching, Germany
4Department of Physics, P.O. Box 35, 40014 University of Jyv¨askyla¨, Finland
5Institut de Physique Th´eorique, Universit´e Paris Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, France
(Dated: January 25, 2017)
In the early stages of heavy-ion collisions, the hot QCD matter expands more longitudinally
7
than transversely. This imbalance causes the system to become rapidly colder in the longitudinal
1
directionandalocalmomentumanisotropyappears. Inthispaper,westudytheheavy-quarkonium
0
spectrum in the presence of a small plasma anisotropy. We work in the framework of pNRQCD at
2
finite temperature. We inspect arrangements of non-relativistic and thermal scales complementary

n to those considered in the literature. In particular, we consider temperatures larger and Debye
a massessmallerthanthebindingenergy,whichisatemperaturerangerelevantforpresentlyrunning
J
LHCexperiments. Inthissettingwecomputetheleadingthermalcorrectionstothebindingenergy

4 and the thermal width induced by quarkonium gluo-dissociation.
2

] I. INTRODUCTION e.g., [11]). At weak coupling this longitudinal expansion
h
causes the system to quickly become much colder in the
p
- In present day experiments at the Large Hadron Col- longitudinal than in the transverse direction, moreover
p the anisotropy can persist for a long time [12–16]. Re-
lider (LHC) and at the Relativistic Heavy Ion Collider
e
cently the properties of an anisotropic QGP have been
h (RHIC) a rich and broad program is ongoing to inves-
the subject of several investigations carried out in the
[ tigate QCD at finite temperature. The establishment
framework of viscous hydrodynamics [17–21].
of a hot QCD medium, dubbed as quark-gluon plasma
1 So far the effect of a local anisotropy on a quark-
(QGP),hasbeeninferredthankstotheobservationofat
v
antiquark bound state has been taken into account via
6 least two striking signatures: jet quenching and quarko-
hard thermal loop (HTL) resummation of the gluon self
5 nia suppression. In particular the latter, which has been
energy, where a finite momentum anisotropy is assigned
9 proposedsincelongasaprobeoftheQGPformation[1],
6 will be the subject of the present investigation. to the degrees of freedom entering the loops [22–24].
0 Numerical solutions of the Schr¨odinger equation for the
Together with the experimental activity also the theo-
. bound state show that the anisotropy tends to decrease
1 retical understanding of heavy quarkonia in medium has
0 progressed significantly in the last years. A key to it the effect of Landau damping and thus to increase the
7 has been the study of the heavy quark-antiquark po- quarkonium melting temperature [25, 26], whereas ana-
1 lytical estimates are found in [24].
tential in a thermal environment. The heavy quark-
: In this work, we assume the quarkonium to be a
v antiquark potential has been derived at high tempera-
i tures (T (cid:29) 1/r (cid:38) m , where r is the quark-antiquark Coulombicsystem,sothatitsinversesizescaleslikemαs,
X D and its typical binding energy like mα2, where m and α
r distance and mD the Debye mass) in [2–4], and further are the heavy-quark mass and strongs coupling respecs-
a computed in a wider range of temperatures in an effec-
tively. This is realized when mα is much larger than
tivefieldtheoryframeworkofQCDinthestaticlimit[5] s
the temperature scale (moreover we consider negligible
and for a large but finite heavy-quark mass [6]. The
the effects of the hadronic scale Λ ). In particular,
real part of the potential shows at high temperatures QCD
we aim at investigating the heavy-quarkonium spectrum
Debye screening, which is a source of quarkonium disso-
whentherelevantscales,thenon-relativisticandthermal
ciation. The potential has also an imaginary part that
ones, satisfy the following hierarchy
stemsfromtwofurtherdissociationmechanisms: Landau
damping [7, 8] and gluo-dissociation [9, 10]. m(cid:29)mα (cid:29)πT (cid:29)mα2 (cid:29)m ,Λ , (1)
A complete understanding of quarkonium in medium s s D QCD
has to account for realistic QGP features. Among these and in the presence of a finite momentum anisotropy of
isthemomentumanisotropyofthethermalmediumcon- the QGP constituents. In a weakly-coupled QGP, the
stituents. IndeedhighlyLorentzcontractednucleicollide Debye mass, m , scales like m ∼ gT and provides
D D
along the beam-axis, so that the longitudinal expansion the inverse of an electric screening length. The hierar-
of the hot QCD medium is more important than the chy of scales (1) may be relevant for the Υ(1S), whose
radial expansion perpendicular to the beam axis (see, mass, inverse radius and binding energy are respectively
2
m ≈ 5 GeV, mα ≈ 1.5 GeV and mα2 ≈ 0.5 GeV [27]. zero temperature. The corresponding Lagrangian den-
s s
In an expanding and then cooling QGP, the regime (1) sityreadsasfollows(weshowonlytermsrelevantforthe
is met at some point, say for T (cid:46) 2T ≈ 0.3 GeV for present work) [31–33]:
c
bottomonium. Note that this temperature is below the
bottomonium melting temperature [28]. In so doing we L =−1Fa Faµν +(cid:88)nf q¯iD/q
partly generalize the study carried out in [6] for the pNRQCD 4 µν i i
i=1
isotropic case. (cid:90)
Since the quarkonium is assumed to be a Coulombic + d3r Tr(cid:8)S†(i∂ −h )S+O†(iD −h )O(cid:9)
0 s 0 o
system, we do not include in the real part of the po-
tential any term to model a (screened) long-range inter- +Tr(cid:8)O†r·gES+S†r·EO(cid:9)+... , (3)
action (as done, e.g., in [26]). Such an inclusion would
where r is the heavy quark-antiquark distance vector,
√ √
not be supported by the hierarchy of energy scales (1). S=S1 / N andO=OaTa/ T aretheheavyquark-
c c F
The spectrum has also an imaginary part that provides
antiquarkcolor-singletandcolor-octetfieldsrespectively,
the quarkonium width. In the situation of interest for
q aren lightquarkfieldstakenmassless,N isthenum-
this work, mα2 (cid:29) m , gluo-dissociation is the domi- i f c
s D ber of colors, TF = 1/2, and traces are understood over
nant mechanism producing the thermal width. Such a
colorandspinindices. Wehavetakenthematchingcoef-
mechanism has been reinterpreted as and connected to
ficientsatleadingorder. Thedotsstandforhigher-order
the singlet-to-octet break up in potential non-relativistic
termsinthemultipleexpansionandforoctet-octettran-
QCD (pNRQCD) at finite temperature in [29].
sitions that we do not need in the following. The singlet
Following a common choice in the literature we im-
and octet Hamiltonians read
plement a momentum anisotropy via distribution func-
p2
tions (B for Bose–Einstein, F for Fermi–Dirac) that h = +V(0)+... , (4)
read [17, 18] s,o m s,o
where p = −i∇ and the dots stand for higher-order
fB,F(q,ξ)≡N(ξ)fB,F(cid:16)(cid:112)q2+ξ(q·n)2(cid:17) , (2) terms in the 1/mr expansion. The singlet and octet
iso
static potentials are at leading order in α : V(0) =
√ s s
where ξ is the anisotropy parameter, N(ξ) = 1+ξ is −C α /r and V(0) = α /(2N r) respectively; C =
F s o s c F
a normalization factor that guarantees the same number (N2−1)/(2N ) is the Casimir of the fundamental repre-
c c
of particles for the anisotropic and isotropic distribution sentation of SU(N ).
c
functions and fB,F(q) is understood to be either a Bose– Thecomputationsthatwearegoingtoperforminthis
iso
Einstein or a Fermi–Dirac isotropic distribution for glu- and in the next section share similarities with the ones
onsandquarksrespectively. HencefB,F(q,ξ)isobtained done for quarkonium in a hot wind in the same temper-
from the corresponding isotropic distribution by remov- ature regime [34, 35]. In both cases we are dealing with
ing particles with a large momentum component along a problem in which the distribution of particles in the
the anisotropy direction n, and accordingly ξ > 0 pa- medium has a preferred direction.
rameterizes the anisotropy strength. The normalization Thermalcontributionstotherealandimaginaryparts
factor N(ξ) is often put to one in the literature, though of the heavy-quarkonium spectrum come from consid-
its origin and impact have been discussed in [30]. As far ering self-energy diagrams in pNRQCD and integrating
asthepresentworkisconcerned, wekeepthenormaliza- themovermomentumregionsscalingrespectivelylikethe
tion factor in the following calculations. temperatureandthebindingenergy. Integratingoverthe
The outline of the paper is the following: in Sec. II momentum region scaling like the temperature amounts
we compute the thermal modification of pNRQCD, at matching pNRQCD to another effective field theory,
pNRQCDHTL, by integrating out the scale πT in the dubbed pNRQCDHTL in [5, 6], where only modes with
presence of a momentum anisotropy. At this stage and energy and momentum smaller than πT are dynamical.
at our accuracy thermal effects are encoded in the sin- Thermal contributions are then encoded in the color-
glet potential. In Sec. III we compute in pNRQCDHTL singlet potential of pNRQCDHTL. We will consider inte-
the temperature-dependent real and imaginary parts of grating over the momentum region scaling like the bind-
thequarkoniumspectrum. Thelattercorrespondstothe ing energy in the next section.
quarkonium thermal width. Conclusion and discussion The leading thermal contribution to the color-singlet
are found in Sec. IV. potential comes from the self-energy diagram in Fig. 1,
wheretheloopmomentumissettobeq ∼πT. Byusing
theverticesandpropagatorsofthepNRQCDLagrangian
II. MATCHING PNRQCD TO PNRQCDHTL we obtain
(cid:104)Ω|TS(t,r,R)S†(0,0,0)|Ω(cid:105)=−4πα C
According to (1), one has to integrate out the heavy- s F
(cid:90) i i
quark mass and the typical momentum transfer before × e−iP0t+iP·R(cid:104)r| r I r |0(cid:105),
dealing with any thermal effect. Hence our starting P P0−hs+i(cid:15) i ij j P0−hs+i(cid:15)
point is pNRQCD, whose coefficients can be obtained at (5)
3
We comment briefly about the result: first, at this
order no imaginary part, and hence no thermal width,
arises; second, for ξ → 0 the result in (9) agrees with
the isotropic case derived in [6]. Finally, we notice that
the term in the second line in (9) is of order ξ when ex-
panding for a small anisotropy parameter, signaling that
FIG. 1. Color-singlet self-energy diagram in pNRQCD. Sin-
its origin is entirely due to the breaking of the spherical
gle lines stand for quark-antiquark color-singlet propagators,
symmetry of the parton momentum distribution.
doublelinesforcolor-octetpropagators,curlylinesforgluons
and a circle with a cross for a chromoelectric dipole vertex.
III. THERMAL CORRECTIONS TO THE
where T stands for time ordering, (cid:82) ≡ (cid:82) d4P/(2π)4, SPECTRUM
P
Pµ = (P0,P) and |Ω(cid:105) is the ground state of the theory.
The thermal part of the self-energy loop integral, Iij, is Inoursettingthenextrelevantscaleafterthetemper-
given by ature is the quarkonium binding energy. The process we
(cid:90) i(q )22πδ(q2) (cid:18) q q (cid:19) are looking at is again a singlet-to-octet transition, how-
I = 0 δ − i j fB(q,ξ). (6) ever with energy and momenta scaling like mα2 rather
ij P −q −h +i(cid:15) ij |q|2 s
q 0 0 o than πT. This contribution is not part of the potential
Wehavetoseparatetermsthatgointothewave-function but comes as a low-energy correction to the spectrum of
renormalization from those that go into the color-singlet pNRQCDHTL. Itmaybecomputedatleadingorderfrom
potential of pNRQCD . To this end we rewrite the one-loop diagram in Fig. 1, where now, however, the
HTL
P0 − h = P0 − h − ∆V, where ∆V = (h − h ) = typical loop momentum is selected to be of order mα2.
o s o s s
(N α )/(2r) + ..., and, due to the condition q ∼ πT Toensurethatwearecomputingonlycontributionsfrom
c s
that sets the loop momentum to be much larger than the momentum region q ∼ mα2 (cid:28) πT, we need to ex-
s
theenergyoftheheavyquark-antiquarkpair,weexpand pand the anisotropic distribution function
the octet propagator in (6). After dropping terms that √
(cid:16) (cid:17)−1 T
go into the wave-function renormalization, the part of fB(q,ξ)= e|Tq| 1+ξλ2 −1 ≈ (cid:112) , (13)
r I r that contributes to the color-singlet potential of |q| 1+ξλ2
i ij j
pNRQCD reads
HTL
whereλ=q·n/|q|isthecosineoftheanglebetweenthe
(cid:18) 2 (cid:19)(cid:90)
r I r |contr.toVs =i +∆V r2 2πδ(q2)fB(q,ξ) gluon momentum and the anisotropy direction. We keep
i ij j q∼πT m q onlytheleadingterminthe|q|/T expansion. Differently
(cid:90) q q from the calculation in Sec. II, we cannot expand the
−i(∆Vr r ) 2πδ(q2) i jfB(q,ξ). (7)
i j |q|2 octet propagator. Then the contribution from the mo-
q
mentum region q ∼ mα2 to the self-energy diagram in
To match onto pNRQCD we compute the correla- s
HTL Fig. 1 reads
tor (cid:104)Ω|TS(t,r,R)S†(0,0,0)|Ω(cid:105) in pNRQCD and re-
HTL
quirethisexpressiontobeequalto(5). Thecolor-singlet δΣ=−i4πα C r I r | , (14)
s F i ij j q∼mα2
potential of pNRQCD turns out to be the same as in s
HTL
pNRQCD plus a thermal correction δV that reads
s where
δVs =−i4παsCF riIijrj|cqo∼nπtrT.toVs . (8) r I r | =Tr (cid:90) i(q0)22πδ(q2) (cid:18)δ − qiqj(cid:19)
The integral can be easily evaluated and the final result i ij j q∼mα2s i q P0−q0−ho+i(cid:15) ij |q|2
r
for the anisotropic potential at finite temperature is × j . (15)
(cid:112)
|q| 1+ξλ2
2πα C T2 πα2C N T2r
δV = s F F (ξ)+ s F c F (ξ)
s 3m 1 12 2 Theintegral(15)hasavanishingimaginarypart. This
πα2C N T2(r·n)2 means that there is no contribution coming from δΣ, as
+ s F c F (ξ), (9)
12r 3 defined in (14), to the real part of the spectrum. Hence,
the thermal shift in the binding energy is entirely due to
where the definitions of the functions embedding the
the shift in the singlet potential, δV , computed previ-
anisotropy parameter are s
√ ously in (9). We can write it as
arctan ξ
F (ξ)=N(ξ) √ , (10)
1 ξ δEbind =(cid:104)nlm|δVs|nlm(cid:105), (16)
√ √
(cid:18) (cid:19)
arctan ξ 1 arctan ξ
F (ξ)=N(ξ) √ + − √ , (11) where |nlm(cid:105) are eigenstates of the singlet Hamiltonian
2 ξ ξ ξ ξ
h , with quantum numbers n, l (orbital angular momen-
√ √ s
(cid:18) (cid:19)
arctan ξ 3 3arctan ξ tum) and m (orbital angular momentum along the z
F (ξ)=N(ξ) √ − + √ .(12)
3 ξ ξ ξ ξ direction). Since, according to our hierarchy of energy
4
scales, the potential entering h is the Coulomb poten- whereas both F (ξ) and G (ξ) vanish linearly in ξ. In
s 3 2
tial, the states |nlm(cid:105) are just Coulombic bound states. this limit both the binding energy (17) and the thermal
At leading accuracy, δE then reads width(19)reducetopreviouslyknownexpressionsfound
bind
in [6]. In Tab. I we show some benchmark values of the
2πα C T2
δE = s F F (ξ) anisotropy functions.
bind 3m 1
+παsNcT2 (cid:2)3n2−l(l+1)(cid:3)(cid:18)F (ξ)+ F3(ξ) 45
12m 2 3
40
(cid:19)
2
+ F (ξ)Cl0 Clm , (17) 35
3 3 2l00 2l0m
V (cid:76)
e 30
M
wwhitehrethtehneotCalteibonscCh–jJ1GMj2omrd1man2 (cCojJe1ffiMj2cmie1mnt2s=ar0eifuJnd>erjs1t+oojd2 ∆Ebind 25 (cid:72)
or J <|j1−j2|). 20
15
ξ F (ξ) F (ξ) F (ξ) G (ξ) G (ξ)
1 2 3 1 2
10
0.1 1.016 1.346 0.026 1.032 0.009 160 180 200 220 240 260 280 300
0.3 1.043 1.367 0.072 1.089 0.026 TMeV
0.5 1.067 1.383 0.114 1.141 0.041
FIG. 2. (Color Online) Binding-energy shift of a 1S (n =
1 1.110 1.414 0.200 1.246 0.077 (cid:72) (cid:76)
1,l =0) bottomonium state according to (17). We show the
binding-energyshiftfortheisotropiccase,blacksolidline,and
for two different values of the anisotropy parameter ξ = 0.5
TABLEI. Theanisotropyfunctionsdefinedin(10)-(12),(20)
and ξ =1 in orange and red solid (dashed) lines respectively
and (21) for some values of ξ. √
when the normalization factor is N(ξ)= 1+ξ (N(ξ)=1).
For all the figures (here and in the following) we have taken
The integral (15) has a non-vanishing real part that
α (2πT) and considered it running at one loop with three
s
contributes to the imaginary part of δΣ. The imaginary quark flavours. The bottom-quark mass has been chosen to
part of the self energy gives rise to a thermal width: be half of the Υ(1S) mass, i.e., 4730 MeV.
Γ=−2(cid:104)nlm|Im(δΣ)|nlm(cid:105)
=8π2α C T (cid:104)nlm|r (cid:90) δ(En+q0−ho)q02 400
s F i (cid:112)
|q| 1+ξλ2
q
(cid:18) (cid:19) 350
q q
× δ − i j (2π)δ(q2)r |nlm(cid:105),(18)
ij |q|2 j
V 300 (cid:76)
where E = −m(C α )2/(4n2) is the energy of the Me
n F s (cid:72)
bound state. The final result reads (cid:71) 250
4 (cid:18)C N2 C2N C3 (cid:19)
Γ= α3T F c + F c + F G (ξ) 200
3 s 4 n2 n2 1
(cid:18)C N2 C2N C3 (cid:19) 150
+α3T F c − F c + F G (ξ)Cl0 Clm , 160 180 200 220 240 260 280 300
s 4 2n2 n2 2 2l00 2l0m
TMeV
(19)
FIG.3. (ColorOnline)Thermalwidthofa1S (n=1,l=0)
where the anisotropy functions are in this case bottomonium state according t(cid:72)o (1(cid:76)9). The different curves
(cid:0)√ (cid:1) are defined as in Fig. 2.
arcsinh ξ
G (ξ)=N(ξ) √ , (20)
1
ξ
√
(cid:0) (cid:1) (cid:112)
(1+2ξ/3)arcsinh ξ − ξ(1+ξ)
G2(ξ)=N(ξ) (cid:112) .(21) IV. CONCLUSION AND DISCUSSION
ξ3
The appearance of a thermal width follows from the fact Inanearlystage,heavy-ioncollisionsarecharacterized
thatthesinglet-to-octettransitionbecomesarealprocess bypartonmomentumanisotropies. Accordinglytheevo-
if the emitted gluon has an energy of the order of the lution of the fireball is described in terms of viscous and
binding energy. anisotropichydrodynamicalmodels. Duetothefactthat
The limit ξ →0 corresponds to the isotropic case. For hardprobes,likeheavyquarkonia,getformedinsuchan
ξ →0,wehavethatF (ξ)→1,F (ξ)→4/3,G (ξ)→1, earlystageoftheheavyion-collisionsandexperiencethe
1 2 1
5
medium until late times, their dynamics has to account
for an anisotropic momentum distribution of the QGP 0.010
constituents. In this paper, we have derived for the hi-
erarchy of scales (1) and at leading order the real and
imaginary thermal parts of the quarkonium spectrum in (cid:61)0(cid:71)(cid:123) 0.005
an anisotropic QGP. The imaginary part originates from (cid:61)1 (cid:144)
(cid:123)(cid:71)
the quarkonium gluo-dissociation in the medium. Our ∆
0.000
result complements previous studies for an anisotropic
plasma where the real and imaginary part of the quark-
antiquarkpotentialwereobtainedforatemperaturescale (cid:45)0.005
larger than the inverse radius of the bound state. In 0.0 0.2 0.4 0.6 0.8 1.0
so doing we extend the knowledge of a weakly-coupled
Ξ
quarkonium to temperature ranges that may be reached
during the QGP evolution at present day colliders. FIG. 5. (Color Online) Ratio of the differences between
the thermal corrections to the widths of 1P (n = 2,l = 1)
0.25 and 2S (n = 2,l = 0) bottomonium states and the 2S-state
thermal width, respectively second and first line in (19), as
a function of ξ. The blue (lower) line refers to the m = ±1
0.20
states whereas the green (upper) line to the m=0 one.
isoE
iso∆E 0.15 (cid:200)(cid:144)
ani(cid:45)∆E 0.10 vthaeluebipnrdoivnigdeesntehrgeyr.eaTlphaeyrtaorfethgeivtehnerimna(l1c7o).rreIcntiFonigs.to2
∆
(cid:200) we show the binding-energy shift for a 1S bottomonium
0.05
state in the isotropic case, ξ = 0, and in the case of a
finite momentum anisotropy, ξ = 0.5 and ξ = 1. We
0.00
0.0 0.2 0.4 0.6 0.8 1.0 see that the impact of an anisotropic plasma crucially
Ξ depends on the normalization factor, either N(ξ) = 1
√
0.25 or N(ξ) = 1+ξ, respectively shown in dashed and
solid lines. For N(ξ) = 1 the anisotropy reduces the
0.20 thermal correction to the binding energy, whereas for
√
N(ξ)= 1+ξ it increases it.
isoiso(cid:71)(cid:71) 0.15 (cid:200)(cid:144) leaTdhsealcsoomtpoutaantiiomnagoifnathrye psparetctcruomminign fproNmRQthCeDsHeTlfL-
(cid:45)
ani(cid:71) 0.10 energydiagramofFig.1evaluatedatthebinding-energy
(cid:200) scale. The imaginary part may be understood as a ther-
0.05 mal width, whose explicit expression is in (19). In Fig. 3
we show the thermal width for a 1S bottomonium state
0.00 in the isotropic case, ξ = 0, and in the case of a finite
0.0 0.2 0.4 0.6 0.8 1.0
momentum anisotropy, ξ = 0.5 and ξ = 1. Also here
Ξ
the size and sign of the thermal corrections strongly de-
pend on the normalization factor, either N(ξ) = 1 or
FIG. 4. Relative change in the binding energy (upper plot) √
N(ξ) = 1+ξ, respectively shown in dashed and solid
and thermal width (lower plot) due to the presence of a mo-
mentum anisotropy. δEani is the binding-energy shift in (17) lines. Althoughthedependenceontheanisotropyisqual-
evaluatedforξ(cid:54)=0,whereasδEiso isthebinding-energyshift itativelysimilarinthebindingenergyandthermalwidth,
in (17) evaluated at ξ=0, for a 1S bottomonium state. In a wefindthattheeffectoftheanisotropyismoreimportant
similarwaywehavedefinedthethermalwidths,ΓaniandΓiso, forthebindingenergywithrespecttothethermalwidth
taken from (19). For so√lid (dashed) lines the normalization when N(ξ)=1 (see dashed line√s in Fig. 4), whereas the
has been taken N(ξ)= 1+ξ (N(ξ)=1). opposite is true when N(ξ) = 1+ξ (see solid lines in
Fig. 4).
The real thermal part of the spectrum comes from Finally, we comment on the effect of an anisotropic
thermal corrections to the potential defined in the con- QGPonthebound-statepolarization. InFig.5weshow
text of pNRQCD . They are encoded in the self- the differences between the thermal corrections to the
HTL
energy diagram of Fig. 1 evaluated at the temperature widths of 1P and 2S bottomonium states. For ξ ≤ 1
scale. The result is given in (9). Thermal corrections such differences are typically of the order of few per mill
to the potential are proportional to the square of the (at most 1%) with respect to the corresponding 2S state
temperature and, as discussed elsewhere, do not show thermalwidth. Thissuppressionisduetovariouseffects:
Debye screening [5, 6]. The corresponding expectation theratiobetweentheanisotropyfunctionsG andG ,see
2 1
6
the benchmark values in Tab. I, the combination involv- ACKNOWLEDGEMENTS
ingthecolorfactorsN andC ,andtheClebsch–Gordan
c F
coefficients. We conclude that for small anisotropies the The work of S.B. was partly supported by the Swiss
effectofananisotropicQGPonthebound-statepolariza- National Science Foundation (SNF) under grant 200020-
tion is tiny and possibly phenomenologically irrelevant. 168988. N.B. and A.V. acknowledge support from the
DFGgrantBR4058/1-2“Effectivefieldtheoriesforhard
probes of hot plasma” and the DFG cluster of excellence
“Origin and structure of the universe” (www.universe-
cluster.de). The work of M.A.E. was supported by the
European Research Council under the Advanced Inves-
tigator Grant ERC-AD-267258 and by the Academy of
Finland, project 303756.
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