Published for SISSA by Springer
Received: August 20, 2012
Revised: December 16, 2012 Accepted: December 26, 2012
Published: January 28, 2013
Mariano Chernico ,a Daniel Fernndez,b David Mateosb,c and Diego Trancanellid,e
aDepartment of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WA, U.K.
bDepartament de Fsica Fonamental & Institut de Cincies del Cosmos (ICC), Universitat de Barcelona (UB),Mart i Franqus 1, E-08028 Barcelona, Spain
cInstituci Catalana de Recerca i Estudis Avanats (ICREA), Passeig Llus Companys 23, E-08010, Barcelona, Spain
dInstituto de Fsica, Universidade de So Paulo,
05314-970 So Paulo, Brazil
eDepartment of Physics, University of Wisconsin,
Madison, WI 53706, U.S.A.
E-mail: mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected]
Abstract: We compute the screening length for quarkonium mesons moving through an anisotropic, strongly coupled N = 4 super Yang-Mills plasma by means of its gravity dual.
We present the results for arbitrary velocities and orientations of the mesons, as well as for arbitrary values of the anisotropy. The anisotropic screening length can be larger or smaller than the isotropic one, and this depends on whether the comparison is made at equal temperatures or at equal entropy densities. For generic motion we nd that: (i) mesons dissociate above a certain critical value of the anisotropy, even at zero temperature; (ii) there is a limiting velocity for mesons in the plasma, even at zero temperature;(iii) in the ultra-relativistic limit the screening length scales as (1 v2)[epsilon1] with [epsilon1] = 1/2, in
contrast with the isotropic result [epsilon1] = 1/4.
Keywords: Gauge-gravity correspondence, Holography and quark-gluon plasmas
ArXiv ePrint: 1208.2672
c
Quarkonium dissociation by anisotropy
JHEP01(2013)170
[circlecopyrt] SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP01(2013)170
Web End =10.1007/JHEP01(2013)170
Contents
1 Introduction 1
2 Gravity solution 3
3 Preliminaries 6
4 Static dipole in an anisotropic plasma 8
5 Dipole in an anisotropic plasma wind 115.1 Unbound quark-antiquark pair 135.1.1 Ultra-relativistic motion outside the transverse plane 155.1.2 Ultra-relativistic motion within the transverse plane 165.2 Bound quark-antiquark pair 165.2.1 Ultra-relativistic motion outside the transverse plane 175.2.2 Ultra-relativistic motion within the transverse plane 195.3 Isotropic limit 205.4 Numerical results for generic velocities 21
6 Dissociation temperature and dissociation anisotropy 24
7 Discussion 33
1 Introduction
A remarkable conclusion from the experiments at the Relativistic Heavy Ion Collider (RHIC) [1, 2] and at the Large Hadron Collider (LHC) (see the contributions on elliptic ow at the LHC in [3]) is that the quark-gluon plasma (QGP) does not behave as a weakly coupled gas of quarks and gluons, but rather as a strongly coupled uid [4, 5]. This places limitations on the applicability of perturbative methods. The lattice formulation of Quantum Chromodynamics (QCD) is also of limited utility, since for example it is not well suited for studying real-time phenomena. This has provided a strong motivation for understanding the dynamics of strongly coupled non-Abelian plasmas through the gauge/string duality [68] (see [9] for a recent review of applications to the QGP). In general, a necessary requirement for the string description to be tractable is that the plasma be innitely strongly coupled, = g2YMNc ! 1. Of course, the real-world QGP is not innitely strongly
coupled, and its dynamics involves a complex combination of both weak and strong coupling physics that depend on the possibly multiple scales that characterize the process of interest. The motivation for studying string models is that they provide examples in which explicit calculations can be performed from rst principles at strong coupling, in particular
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in the real-time domain. The hope is then that, by understanding the weak and the strong coupling limits, one may be able to bracket the dynamics of the real-world QGP, which lies somewhere in between.
During the initial stage after the collision the plasma is far from equilibrium, and after a certain time a hydrodynamic description becomes applicable. If one thinks of hydrodynamics as a gradient expansion around a locally isotropic system, it is somewhat surprising that the hydrodynamic description actually becomes applicable when the longitudinal and transverse pressures are still signicantly di erent. This can be explicitly seen, for example, in holographic descriptions [1013] in which gravity provides a valid description all the way from the far-from-equilibrium phase to the locally isotropic phase, across the intermediate hydrodynamic-but-still-anisotropic phase. Thus, during most of the time that viscous hydrodynamics is applied, the plasma created in a heavy ion collision is anisotropic, with the level of anisotropy in fact increasing as one approaches the edge of the system. The fact that the range of time and space over which the QGP is anisotropic is larger than traditionally assumed has provided additional motivation for the study of anisotropic plasmas.
In this paper we will investigate the e ect of an intrinsic anisotropy on the screening length between a quark-antiquark pair in a strongly coupled plasma. As we will review below, the plasma is static because it is held in anisotropic equilibrium by an external force [14, 15]. We will discuss all the caveats in more detail below, but we emphasize from the beginning that there are several reasons why, in terms of potential extrapolations to the real-world QGP, our results must be interpreted with caution. First, the sources of anisotropy in the QGP created in a heavy ion collision and in our system are di erent. In the QGP the anisotropy is dynamical in the sense that it is due to the initial distribution of particles in momentum space, which will evolve in time and eventually become isotropic. In contrast, in our case the anisotropy is due to an external source that keeps the system in an equilibrium anisotropic state that will not evolve in time. Nevertheless, we hope that our system might provide a good toy model for processes whose characteristic time scale is su ciently shorter than the time scale controlling the evolution of a dynamical plasma.
The second caveat concerns the fact that, even in an static situation, di erent external sources can be chosen to hold the plasma in equilibrium, so one may wonder to what extent the results depend on this choice. We will provide a partial answer to this question in section 7, where we will explain that our qualitative results, for example the ultrarelativistic limit, do not depend on the details of our solution but only on a few general features. Nevertheless, it would still be very interesting to compute the same observables in other strongly coupled, static, anisotropic plasmas. Only then a general picture would emerge that would allow one, for example, to understand which observables are robust, in the sense that they are truly insensitive to the way in which the plasma is held in anisotropic equilibrium, and which ones are model-dependent. Obviously it is the rst type of observables that have a better chance of being relevant for the real-world QGP. Our paper should be regarded as a rst step in this general program.
We will consider the screening length in the case in which the quark-antiquark pair is at rest in the plasma as well as the case in which it is moving through the plasma. For this purpose we will examine a string with both endpoints on the boundary of an asymptotically
2
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AdS spacetime [14, 15] that is dual to an anisotropic N = 4 super Yang-Mills plasma. The
gravity solution possesses an anisotropic horizon, it is completely regular on and outside the horizon, and it is solidly embedded in type IIB string theory. For these reasons it provides an ideal toy model in which questions about anisotropic e ects at strong coupling can be addressed from rst principles. For the particular case of a quark-antiquark pair at rest, the screening length has also been computed [16] in a di erent model [17] of a strongly coupled, anisotropic plasma. The results exhibit some di erences with respect to those presented here. While this may indicate some model dependence of the screening length, it is important to note that the solution of [17] possesses a naked singularity. Although this is a rather benign singularity, its presence introduces a certain amount of ambiguity in the calculations, which can only be performed by prescribing somewhat ad hoc boundary conditions at the singularity. In any case, this discussion is another indication that it would be interesting to compute the screening length in a larger class of models in order to ascertain which of its features are model-independent.
To avoid any possible confusion, we clarify from the beginning that the quarks and antiquarks that we will consider are innitely massive, i.e. the bound states that we will consider are the analogue of heavy quarkonium mesons in QCD. Thus, the reader should always have the word quarkonium in mind despite the fact that we will often refer to these states simply as mesons, heavy mesons, quark-antiquark bound states, dipoles, etc. This is specially relevant in the ultra-relativistic limit of the screening length, to which we will pay particular attention since it can be determined analytically. We emphasize that our results correspond to sending the quark and antiquark masses to innity rst, and then sending v ! 1. In particular, this means that in any future attempt to connect
our results to the phenomenology of the QGP, this connection can only be made to the
phenomenology of heavy quarkonium moving through the plasma.
The screening length for quarkonium mesons at rest in the anisotropic plasma of [14, 15] has been previously studied in [16, 18]. Our section 4 has some overlap with these references and, wherever they overlap, our results agree with theirs. Other physical properties of the anisotropic plasma that have been calculated include its shear viscosity [19, 20], the drag force on a heavy quark [18, 21], the jet quenching parameter [16, 18, 22], and the energy lost by a rotating quark [23]. The phase diagram of the zero-coupling version of the model considered in [14, 15] has been studied in [24]. Dissociation of baryons in the isotropic N = 4 plasma has been analyzed in [25].
2 Gravity solution
The type IIB supergravity solution of [14, 15] in the string frame takes the form
ds2 = L2 u2
FB dt2 + dx2 + dy2 + Hdz2 +du2
F
[vector] = az , = (u) , (2.2)
where [vector] and are the axion and the dilaton, respectively, and (t, x, y, z) are the gauge theory coordinates. Since there is rotational invariance in the xy-directions, we will refer
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[parenrightbigg]
+ L2e
1
2 d 25, (2.1)
1.2
H
1.2
6
H
6
1
1
5
5
0.8
B
F
0.8
4
4
0.6
FB
0.6
3
3
2
F
2
0.4
0.4
1
0.2
0.2
FB
B
1
0
0
0
0
[Minus]1
[Minus]1
[Minus]0.2
[Minus]0.2
[Minus]2
0 0.2 0.4 0.6 0.8 1
[Minus][notdef]0 0.2 0.4 0.6 0.8 1
2
u/uH
u/uH
Figure 1. Metric functions for a/T [similarequal] 4.4 (left) and a/T [similarequal] 86 (right).
to these as the transverse directions, and to z as the longitudinal direction. F, B and H are functions of the holographic radial coordinate u that were determined numerically
in [14, 15]. Their form for two values of a/T is plotted in gure 1. The horizon lies at u = uH, where F = 0, and the boundary at u = 0, where F = B = H = 1 and = 0. The
metric near the boundary asymptotes to AdS5 [notdef] S5. Note that the axion is linear in the
z-coordinate. The proportionality constant a has dimensions of mass and is a measure of the anisotropy. The axion prole is dual in the gauge theory to a position-dependent theta parameter of the form / z. This acts as an isotropy-breaking external source that forces
the system into an anisotropic equilibrium state.
If a = 0 then the solution reduces to the isotropic black D3-brane solution dual to the isotropic N = 4 theory at nite temperature. In this case
B = H = 1 , [vector] = = 0 , F = 1
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u4u4H , uH =
1T (2.3)
and the entropy density takes the form
2 N2cT 3 . (2.4)
Figure 2 shows the entropy density per unit 3-volume in the xyz-directions of the anisotropic plasma as a function of the dimensionless ratio a/T , normalized to the entropy density of the isotropic plasma at the same temperature. At small a/T the entropy density scales as in the isotropic case, whereas at large a/T it scales as [14, 15, 26]
s = centN2ca1/3T 8/3 , [a/T 1] (2.5)
where cent is a constant that can be determined numerically. The transition between the two asymptotic behaviors of the entropy density takes place at a/T [similarequal] 3.7.
For later use we list here the near-boundary behavior of the di erent functions that determine the solution (2.2):
F = 1 +
11
24a2u2 + [parenleftbigg]F4
siso = 2
+ 7
12a4 log u[parenrightbigg]
u4 + O(u6) ,
4
1.2
1.0
log(s/s iso)
0.8
0.6
0.4
0.2
0.0 [Minus]4 [Minus]2 0 2 4
log(a/T )
Figure 2. Log-log plot of the entropy density per unit 3-volume in the xyz-directions as a function of a/T , with siso dened as in eq. (2.4). The dashed blue line is a straight line with slope 1/3.
B = 1
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11
24a2u2 + [parenleftbigg]B4
712a4 log u[parenrightbigg]
u4 + O(u6) ,
H = 1 +
14a2u2 [parenleftbigg]
27B4
5 4032a4
16a4 log u[parenrightbigg]
u4 + O(u6) . (2.6)
The coe cients F4 and B4 depend on a and T and are known analytically in the limits of
low, and high temperature and numerically for intermediate regimes [15].
A feature of the solution (2.2) that played an important role in the analysis of [14, 15] is the presence of a conformal anomaly. Its origin lies in the fact that di eomorphism invariance in the radial direction u gets broken in the process of renormalization of the on-shell supergravity action. In the gauge theory this means that scale invariance is broken by the renormalization process. One manifestation of the anomaly is the fact that, unlike the entropy density, other thermodynamic quantities do not depend solely on the ratio a/T but on a and T separately. Fortunately, this will not be the case for the screening length, as we will see below.
To facilitate a (rough) comparison of the anisotropy in our system to that in other anisotropic plasmas it is useful to consider the ratio
= 4E + P? PL
3T s , (2.7)
where E is the energy density and P?, PL are the transverse and longitudinal pressures,
respectively. In addition to being dimensionless, this ratio has the virtue that it does not depend on a and T separately, but only on the combination a/T . For the isotropic N = 4
super Yang-Mills plasma = 1, whereas for 0 < a/T [lessorsimilar] 20 the ratio is well approximated by the expression
[similarequal] 1 0.0036
[parenleftBig]
a T
2 0.000072[parenleftBig]
a T
4, (2.8)
as shown in gure 3.
At various points we will refer to the limit T = 0 of the anisotropic plasma. The zero-temperature version of the solution (2.2) was found in [26]. In this case the string-frame metric exhibits a naked curvature singularity deep in the infra-red, and the
5
1
[Minus]2
[Minus]4
[Minus]6
[Minus]8
[Minus]10
[Minus]12
5 10 15 20
Figure 3. Ratio (2.7) as a function of a/T . The blue dots are the actual values of the ratio, and the red curve is the t (2.8).
Einstein-frame metric exhibits innite tidal forces [27, 28]. However, we emphasize that, for any nite temperature, the singularity is hidden behind the horizon and the solution is completely regular on and outside the horizon, exhibiting no pathologies of any type. Thus we will think of the T = 0 results as those obtained by taking the limit T ! 0 of
the nite-temperature results. Moreover, regulating the infra-red geometry in this or any other way is actually unnecessary for most of the physics of quarkonium dissociation. The reason is that, as we will see, in the limit in which a/T becomes large the penetration depth into the AdS bulk of the string that is dual to the quarkonium meson becomes very small. As a result, the dissociation is entirely controlled by the metric near the boundary, which is insensitive to the infra-red behavior described above.
3 Preliminaries
In this paper we dene the screening length Ls as the separation between a quark and an antiquark such that for [lscript] < Ls ([lscript] > Ls) it is energetically favorable for the quark-antiquark pair to be bound (unbound) [29, 30]. Obviously this satises Ls Lmax, where Lmax is the
maximum separation Lmax for which a bound quark-antiquark solution exists. We will determine Ls by comparing the action S([lscript]) of the bound pair, which is a function of the quark-antiquark separation [lscript], to the action Sunbound of the unbound system, i.e. by computing:
S([lscript]) = S([lscript]) S
unbound . (3.1)
The screening length is the maximum value of [lscript] for which S is positive (since we will work in Lorentzian signature). This may correspond to the value of [lscript] at which S crosses zero, in which case Ls < Lmax, or the maximum value of [lscript] for which a bound state exists, in which case Ls = Lmax. In the Euclidean version of our calculations, this criterion corresponds to determining which conguration has the lowest free energy, which is therefore the conguration that is thermodynamically preferred. As shown in gure 4, for a meson moving through the isotropic plasma (2.3) one has Ls < Lmax for v < vtrans, whereas for
v > vtrans one nds that Ls = Lmax, where vtrans [similarequal] 0.45 is the transition velocity between the two behaviors [3133]. These qualitative features extend to the anisotropic case, as we have
6
a/T
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0.0
[Minus]0.5
E dipole/Tp
[Minus]1.0
[Minus]1.5
[Minus]2.0
[Minus]2.5
[Minus]3.0
[Minus][notdef]0.00 0.05 0.10 0.15 0.20 0.25
3.5
T [lscript]
Figure 4. Energy di erence, as dened in (3.2), between a bound and an unbound quark-antiquark pair moving through the isotropic plasma (2.3) with velocities (from the rightmost curve to the leftmost curve) v = 0, 0.35, 0.85, 0.996. The dipole is oriented orthogonally to its velocity. For v < vtrans one has Ls < Lmax, whereas for v > vtrans one nds L = Lmax, where vtrans [similarequal] 0.45 is the transition velocity between the two behaviors. At v = 0 the screening length and the maximum separation are Ls [similarequal] 0.24/T and Lmax [similarequal] 0.27/T , respectively.
illustrated in gure 5. The transition velocity decreases with the anisotropy, so for large a/T one has Ls = Lmax except for very low velocities. Similarly, if the ultra-relativistic limit v ! 1 is taken at xed a and T , then obviously v > v
trans and again Ls = Lmax.
All our calculations will be done in the rest frame of the quark-antiquark pair, to which we will refer as the dipole rest frame. Since any observable can be easily translated between this frame and the plasma rest frame, we will speak interchangeably of mesons in a plasma wind and of mesons in motion in the plasma. We emphasize however that all the physical quantities that we will present, e.g. the screening length, are computed in the dipole rest frame.
The actions are scalar quantities, so Sdipole = Splasma. Moreover, in the dipole rest
frame we have
Sdipole = T E
dipole , (3.2)
since the dipole is static in its own rest frame. In this expression Edipole is the energy
(as opposed to the free energy) of the conguration and T =
[integraltext]
dt is the length of the integration region in time. Thus we see that our criterion, which is based on comparing the actions, can also be thought of as a comparison between the energies of the bound and the unbound congurations in the dipole rest frame.
We will see that the ultraviolet divergences in the string action associated to integrating all the way to the boundary of AdS cancel out in the di erence (3.1), and neither the bound nor the unbound actions possess infrared divergences associated to integrating all the way down to the horizon. This can be veried explicitly and it also follows from their relation to the energy in the rest frame of the dipole: while the energy of the unbound string pair possesses an infrared logarithmic divergence in the plasma rest frame [34], no such divergence is present in the dipole rest frame (see e.g. the discussion in [33]).
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0
0.1
[Minus]2
0.0
[Minus]4
[Minus]0.1
E dipole/Tp
[Minus]6
E dipole/Tp
[Minus]0.2
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[Minus]8
[Minus]0.3
[Minus]10
[Minus]0.4
[Minus]12
[Minus]0.5
0.00 0.05 0.10 0.15 0.20 0.25
0.00 0.05 0.10 0.15 0.20 0.25
T [lscript]
T [lscript]
Figure 5. Energy di erence in an anisotropic plasma, as dened in (3.2), between a bound and an unbound quark-antiquark pair oriented along the transverse direction x and moving along the anisotropic direction z. All the curves on the left correspond to a/T = 12.2 and di erent velocities (from the rightmost curve to the leftmost curve) v = 0, 0.35, 0.85, 0.996. All the curves on the right correspond to the same velocity v = 0.25 and di erent anisotropies (from the rightmost curve to the leftmost curve) a/T = 0, 6.5, 43, 744. For these anisotropies the corresponding transition velocities are respectively given by vtrans = 0.45, 0.29, 0.19, 0.11.
4 Static dipole in an anisotropic plasma
In an anisotropic plasma the screening length depends on the relative orientation between the dipole and the anisotropic direction z. Given the rotational symmetry in the xy-plane we assume without loss of generality that the dipole lies in the xz-plane, at an angle with the z-axis. We thus choose the static gauge t = , = u and specify the string embedding as
x ! sin x(u) , z ! cos z(u) . (4.1)
The string action takes the form
S =
L2 2 [prime] 2 [integraldisplay]
dt
[integraldisplay]
umax
0 du
1 u2
qB 1 + FH cos2 z[prime]2 + F sin2 x[prime]2
, (4.2)
where the 2 comes from the two branches of the string and umax will be determined below. The conserved momenta associated to translation invariance in the x, z directions are
8
[parenrightbig]
given by
x = 1 sin
sin x[prime]
@L
@x[prime] = BF
, (4.3)
u2
qB 1 + FH cos2 z[prime]2 + F sin2 x[prime]2
[parenrightbig]
z = 1 cos
@L
@z[prime] = BFH
cos z[prime]
. (4.4)
u2
qB 1 + FH cos2 z[prime]2 + F sin2 x[prime]2
[parenrightbig]
Inverting these relations we nd
x[prime] = pH csc u2 x
pF
pBFH u4 ( 2z + H 2x)
, z[prime] = sec u2 z pFH pBFH u4 ( 2z + H 2x)
. (4.5)
Substituting back in the action we arrive at
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pFH
S =
L2 2 [prime] 2 [integraldisplay]
dt
[integraldisplay]
umax
0 du
1u2 B
pBFH u4 ( 2z + H 2x)
. (4.6) For a U-shaped string describing a bound quark-antiquark pair the turning point umax
is determined in terms of the momenta by the condition that x[prime](umax) = z[prime](umax) ! 1. This happens if umax = uH, in which case F(umax) = 0, or if
BFH u4 2z + H 2x [parenrightbig][vextendsingle][vextendsingle]
umax = 0 . (4.7)
The rst possibility is not physically relevant because the second possibility is always realized rst, meaning that the string turns around at umax < uH, before reaching the horizon.
The only exception is the case x = z = 0, but this corresponds to x[prime] = z[prime] = 0, namely to an unbound pair of strings that descend from the boundary straight down to the horizon.
The momenta are determined by the boundary conditions that require the string endpoints to lie a distance [lscript] apart from each other:
[lscript]2 =
[integraldisplay]
umax
0 du x[prime] = [integraldisplay]
umax
0 du z[prime] . (4.8)
These two equations, together with (4.7), can be solved numerically to express the momenta and umax in terms of [lscript]. In this way the on-shell action (4.6) for a bound pair becomes a function of [lscript] alone. In order to determine Ls we subtract from this action the action of a static, unbound quark-antiquark pair, which is described by two straight strings hanging down from the boundary to the horizon. The action of this unbound pair is equal to (4.6) with the momenta set to zero and the range of integration extended down to the horizon:
Sunbound =
L2 2 [prime] 2 [integraldisplay]
dt
[integraldisplay]
uH
0 du
pB
u2 . (4.9)
We obtain the screening length by numerically determining the value of [lscript] at which the di erence S([lscript]) S
unbound crosses zero, since in the static case we always have Ls < Lmax. The result for this di erence as a function of [lscript] in the isotropic plasma [29, 30] described by eq. (2.3) is plotted in gure 4, from which we see that the screening length is
Liso(T ) [similarequal]
0.24
T [static dipole] . (4.10)
9
1.0
1.2
0.9
1.1
1.0
L s/L iso(T)
L s/L iso(s)
0.8
0.9
0.7
0.8
0.7
0.6
0.6
0.5 0 5 10 15 20 25 30
0.5 0 5 10 15 20 25 30
a/T
aN2/3c/s1/3
Figure 6. Screening length as a function of the anisotropy for a static quark-antiquark dipole lying at an angle with the z-direction (from top to bottom on the right-hand side of the plot) = /2, /3, /4, /6, 0. The screening length is plotted in the appropriate units to facilitate comparison with the isotropic result for a plasma at the same temperature (left), or at the same entropy density (right). The isotropic result is given in eqs. (4.10) and (4.11).
0 0 p
8
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1
1.6
0.8
1.4
1.2
0.6
L s/L iso(T)
L s/L iso(s)
1
0.8
0.4
0.6
0.2
0.4
0.2
p
4
3 p 8
p
2
0 0 p
8
p
4
3 p 8
p
2
Figure 7. Screening length for a quark-antiquark dipole lying at an angle with the z-direction for anisotropies a/T = 12.2 (red, solid), 42.6 (maroon, coarsely dashed), 86 (violet, dashed), 744 (orange, dot-dashed). The corresponding values in units of the entropy density are (in the same order) aN2/3c/s1/3 = 6.2, 19, 35, 242. The screening length is plotted in the appropriate units to facilitate comparison with the isotropic result for a plasma at the same temperature (left), or at the same entropy density (right). The isotropic result is given in eqs. (4.10) and (4.11).
The scaling with the temperature is expected on dimensional grounds. In the isotropic case the temperature and the entropy density are related simply through (2.4), so this result can be recast as
Liso(s) [similarequal] 0.24
1/3
2N2c 2s
[static dipole] , (4.11)
which will be useful later.
The results in the anisotropic case are plotted in gures 6 and 7. Figure 6 shows the screening length, for several orientations of the dipole, as a function of the anisotropy measured in units of the temperature (left) and the entropy density (right). The reason
10
for working with both normalizations is that we wish to compare the screening length in the anisotropic plasma to that in the isotropic plasma, and this can be done at least in two di erent ways: the two plasmas can be taken to have the same temperatures but di erent entropy densities, or the same entropy densities but di erent temperatures. Figure 7 shows the screening length as a function of the dipole orientation for several values of the anisotropy.
We see from gure 6(left) that Ls decreases monotonically as a increases, for any dipole orientation, if the temperature is kept xed. We also see from gure 7(left) that this e ect is more pronounced for a dipole oriented along the anisotropic direction. In contrast, the behavior of the screening length at constant entropy density depends on the dipoles orientation, as shown in gures 6(right) and 7(right). For dipoles aligned su ciently close to the anisotropic direction the screening length decreases with the anisotropy, whereas for orientations su ciently close to the transverse plane the screening length increases with the anisotropy.
5 Dipole in an anisotropic plasma wind
In this section we will consider a static quark-antiquark pair in an anisotropic plasma that is moving with constant velocity with respect to the dipole a dipole in an anisotropic plasma wind. We will pay particular attention to the ultra-relativistic limit, which can be understood analytically.1 This limit, together with the static results from section 4, will allow us to understand qualitatively the results at any velocity 0 < v < 1.
We will rst rewrite the solution (2.2) in a boosted frame, and then place a dipole in it see gure 8. Given the rotational symmetry in the xy-plane we assume that the boost velocity is contained in the xz-plane, and that it lies at an angle v with the z-axis. Thus we rst rotate to a new coordinate system dened through
t = ~t,
x = sin v + ~x cos v ,
y =,
z = cos v ~x sin v , (5.1)
and then perform a boost along the-direction by setting
~t = t[prime] v z[prime] [parenrightbig]
, (5.2)
where = 1/p1 v2 is the usual Lorentz factor. Below we will consider a dipole with an
arbitrary orientation with respect to both the velocity of the plasma and the anisotropic
1We recall that we rst send the quark mass to innity and then v ! 1 (see section 1).
11
JHEP01(2013)170
,
~x = x[prime] ,
= y[prime] ,
= v t[prime] + z[prime]
[parenrightbig]
z
v
JHEP01(2013)170
'
y
x
Figure 8. Orientation of the dipole in an anisotropic plasma wind. The winds velocity lies in the original xz-plane (before the boost (5.2)) at an angle v with respect to the anisotropic direction z.
The quark lies at angles [vector]q = (x, y, z) = [lscript]2(sin sin ', sin cos ', cos ) with respect to the relabeled directions (after the boost (5.2)), and the antiquark lies at [vector]q.
direction z see gure 8. We parametrize the orientation of the dipole by two angles , ' so that the quark lies at
[vector]q = (x[prime], y[prime], z[prime]) = [lscript]2(sin sin ', sin cos ', cos ) (5.3)
and the antiquark lies at [vector]q.
For notational simplicity, below we will drop the primes in the nal set of coordinates. To avoid confusion, we emphasize that the direction v of the plasma wind is always measured with respect to the original (x, y, z) axes, i.e. before the rotation and the boost above. In particular, motion within (outside) the transverse plane refers to a dipole in a plasma wind with v = /2 ( v [negationslash]= /2). In contrast, the orientation of the dipole is measured
with respect to the nal set of coordinates (x[prime], y[prime], z[prime]). However, if instead of specifying the dipoles orientation through a pair ( , ') we specify it by saying that the dipole is aligned with the x-, y- or z-directions, then we are referring to the original directions. Just as an illustration, consider the case of a plasma wind blowing along the original x-direction,i.e. a plasma wind with v = /2. Then we see from (5.1) and (5.2) that (x, z) (z[prime], x[prime]).
Thus in this case by a dipole oriented along the x-direction we mean a dipole with = 0.
After dropping the primes from the nal set of coordinates in (5.2) the ve-dimensional part of the metric (2.2) takes the form
ds2 = L2 u2
gttdt2+gxxdx2+dy2+gzzdz2+gtxdt dx+gtzdt dz+gxzdx dz+ du2
F
, (5.4)
12
where
gtt = BF v2(sin2 v + H cos2 v)
1 v2
, (5.5)
gxx = cos2 v + H sin2 v , (5.6) gzz = sin2 v + H cos2 v v2BF
1 v2
, (5.7)
gtx = (H 1)v p1 v2
sin(2 v) , (5.8)
gtz = 2v(BF sin2 v H cos2 v)1 v2
, (5.9)
gxz = 1 H p1 v2
sin(2 v) . (5.10)
In order to determine the screening length for a generic velocity we need to compare the actions of a bound and an unbound quark-antiquark pair, as in the static case of section 4. However, in the ultra-relativistic this is not strictly necessary because Ls = Lmax (see
section 3). In other words, in this limit we only need to determine the maximum possible quark-antiquark separation for which a bound state exists. Nevertheless, for completeness we will briey present the analysis of the unbound conguration. Each of the strings in the unbound pair is one of the trailing strings studied in [21], so the reader is referred to this reference for additional details. Note, however, that [21] worked in the plasma rest frame. Here we will work in the dipoles rest and focus on the ultra-relativistic limit.
5.1 Unbound quark-antiquark pair
As in section 4 we x the static gauge t = , = u, and specify the embedding of the unbound string as
x ! x(u) , z ! z(u) . (5.11)
The embedding in the y-direction is simply y = 0 because of rotational symmetry in the xy-plane and because the string is unbound. As we will see below, in the case of a bound string (dipole) the boundary conditions will generically imply a non-trivial embedding y(u).
The action for the unbound string reads
Sunbound =
JHEP01(2013)170
L2 2 [prime] 2 [integraldisplay]
dt
[integraldisplay]
uH
0 du
1 u2
pF1K0 + Kxxx[prime]2 + Kzzz[prime]2 + Kxzx[prime]z[prime] , (5.12)
where
K0 = gtt ,
Kxx = BF(cos2 v + H sin2 v) Hv2
1 v2
,
Kzz = BF sin2 v + H cos2 v
[parenrightbig]
,
Kxz = BF(1 H) p1 v2
sin(2 v) . (5.13)
13
Introducing the conjugate momenta
x = @L
unbound
@x[prime] , z =
@L
unbound
@z[prime] (5.14)
and solving for x[prime], z[prime] we nd
x[prime] = u2
FpBH
Nx
pD , z[prime] =
u2
FpBH
Nz
pD , (5.15)
where
Nx = Kzz x
1
2Kxz z ,
Nz =
1
2Kxz x + Kxx z ,
D = BHFK0 u4 Kzz 2x + Kxx 2z Kxz x z
[parenrightbig]
. (5.16)
Substituting into the action we arrive at
Sunbound =
pBHK0
u2pD . (5.17)
The momenta are determined by the condition that (5.15) remain real for a string that extends all the way from the boundary to the horizon. Following [21] we analyze this condition by noting that D can be rewritten as
D = 2u4
Kxz NxNz b[bracketleftBig]
L2 2 [prime] 2 [integraldisplay]
dt
[integraldisplay]
uH
0 du
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x z c
[bracketrightBig][bracketleftBig]BF
v2(sin2 v + H cos2 v)[bracketrightBig]
(5.18)
where
b = Hu4
(1 H)p1 v2 sin v cos v
, c = BF(1 H) sin v cos v
u4p1 v2
. (5.19)
As in [21] we must require that the zeros of the second summand in (5.18) coincide with one another and with those of Nx and Nz. One of the zeros of the second summand occurs at a critical value u = uc such that
Bc
Fc
dc v2 = 0 , dc Hc cos2 v + sin2 v , (5.20)
where Bc = B(uc), etc. At this point we have
NxNz[notdef]uc =
v4 cos v sin v p1 v2
(Hc 1) dc
dc x + (Hc 1) cos v sin v p1 v2 z
2. (5.21)
Noting that Hc > 1 and that Kxz < 0, we see that D would be negative at uc unless the
momenta are related through
x = (1 Hc) cos v sin v
dcp1 v2
z . (5.22)
14
Assuming this relation and requiring that the other zero in the second summand of (5.18) coincide with uc yields
2z = BcFc dc
u4c , 2x = BcFc
(Hc 1)2 cos2 v sin2 v
u4c(1 v2)dc
. (5.23)
Note that z does not vanish for any value of v, whereas x vanishes if v = 0, /2. The reason is that for these two particular orientations the plasma wind blows along the original z- or x-directions and the string orients itself with the corresponding axis [21]. As a consequence, the momentum along the orthogonal axis vanishes. However, the changes of coordinates (5.1) and (5.2) always relabel the direction of motion as z, so after these changes the non-vanishing momentum is labelled z irrespectively of whether v = 0 or v = /2.
We will analyze in detail the ultra-relativistic limit. This is facilitated by explicitly distinguishing the case of motion outside the transverse plane ( v [negationslash]= /2) and motion
within the transverse plane ( v = /2).
5.1.1 Ultra-relativistic motion outside the transverse plane
In the ultra-relativistic limit uc approaches the boundary, i.e. uc ! 0, and we can use the
near-boundary expansion (2.6) to determine it. The condition (5.20) yields in this limit [21]
u2c [similarequal]
4(1 v2)
a2 cos2 v [ v [negationslash]= /2] , (5.24)
which when substituted in (5.23) gives the momenta
2z [similarequal]
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a4 cos4 v 16(1 v2)2
. (5.25)
In these expressions we have ignored subleading terms in an expansion in 1 v2, for example we have set v [similarequal] 1, Hc [similarequal] 1, etc. Note that in this expansion x is subleading
with respect to z.
For later use we must evaluate how Sunbound scales with 1 v2 in the limit v ! 1. For this purpose we split the integration region, and hence the action (5.17), as
Sunbound = S(1)unbound + S(2)unbound , (5.26)
where S(1)unbound is the action with the integral in u ranging between 0 and uc, and S(2)unbound
is the action with the integral in u ranging between uc and uH. The reason for this separation is that in the rst interval u is small and hence we will be able to use the near-boundary expressions (2.6), (5.24) and (5.25). In order to exhibit the dependence on 1 v2 of S(1)
, 2x [similarequal]
a4 cos2 v sin2 v 16(1 v2)
unbound explicitly, it is convenient to work with a rescaled variable r which remains nite in the v ! 1 limit, dened thoughu = r
p1 v2 , uc = rc p1 v2 . (5.27)
In terms of this variable we get
S(1)unbound =
L2 2 [prime]
2 p1v2 [integraldisplay]
dt
[integraldisplay]
rc
0 dr
1 14a2r2 cos2 v + . . . r2 q1 14a2r2 cos2 v 116a4r4 cos4 v+. . .
. (5.28)
15
The divergence near r = 0 will cancel out with that in the action for the bound string. The integrand is smooth across r = rc. The crucial point is that the result is O
(1 v2)1/2[bracketrightbig]in the counting in powers of 1v2, and we will nd this same scaling in the bound string action
(see below). In contrast, S(2)unbound scales as 1 v2 in the ultra-relativistic limit. The reason is that u is not small in units of 1 v2 in the corresponding region of integration, so all the
dependence comes from the fact that the action (5.17) scales as 1/ z 1v2 in this region.
5.1.2 Ultra-relativistic motion within the transverse plane
In this case v = /2 and hence we see from (5.23) that x = 0. The condition (5.20) now gives [21]
u2c [similarequal] [radicalbigg]
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1 v2
C , (5.29)
where
576a4 F4 B4 , (5.30)
and we recall that F4, B4 are the coe cients that enter the near-boundary expansion (2.6).
Substituting (5.29) into (5.23) and dropping subleading terms as before we obtain the momentum in the z-direction (recall that this corresponds to the original x-direction):
z [similarequal]
1u2c = [radicalbigg]
C = 121
C1 v2
. (5.31)
It is now convenient to work with a rescaled radial coordinate r dened through
u = r(1 v2)1/4 . (5.32)
Splitting the unbound string action as before, we nd
S(1)unbound =
L2 2 [prime]
2(1 v2)1/4 [integraldisplay]
dt
[integraldisplay]
rc
0 dr
1 Cr4 + . . . r2p1 2Cr4 + . . .
. (5.33)
Again, the divergence near r = 0 will cancel out with that in the action for the bound string, which will also be of O
(1 v2)1/4[bracketrightbig]in the counting in powers of 1 v2 (see below). In contrast, S(2)unbound scales as 1/ x p1 v2 in the ultra-relativistic limit, and is therefore subleading.
In summary, we nd that in the ultra-relativistic limit
Sunbound =
8
>
<
>
:
O
(1 v2)1/2[bracketrightbig]if v [negationslash]= /2 [outside the transverse plane]
O
(1 v2)1/4[bracketrightbig]if v = /2 [within the transverse plane] .
(5.34)
5.2 Bound quark-antiquark pair
We now consider a dipole with an arbitrary orientation with respect to both the velocity of the plasma and the anisotropic direction z see gure 8. As before we x the static gauge = t, = u and specify the string embedding via three functions (x(u), y(u), z(u)) subject to the boundary conditions
[lscript]2 sin sin ' =
[integraldisplay]
umax
0 x[prime]du ,
16
[lscript]2 sin cos ' =
[integraldisplay]
umax
0 y[prime]du ,
[integraldisplay]
[lscript]
umax
2 cos =
0 z[prime]du , (5.35)
where umax is the turning point of the U-shaped string. The integral in the action of the bound string extends only up to this point and now includes a term proportional to y[prime]2:
S =
L2 2 [prime] 2 [integraldisplay]
dt
[integraldisplay]
umax
0 du
1 u2
qF1K0 + Kxxx[prime]2 + Kyyy[prime]2 + Kzzz[prime]2 + Kxzx[prime]z[prime] . (5.36)
All the Ks were dened in (5.13) except for Kyy, which is given by
Kyy = BF v2(sin2 v + H cos2 v)
1 v2
. (5.37)
Inverting these equations we get
x[prime] = u2
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The momenta are dened as
x = @L
@x[prime] , y =
@L
@y[prime] , z =
@L
@z[prime] . (5.38)
FpBHpD [parenleftbigg]
Kzz x
1
2Kxz z
[parenrightbigg]
,
y[prime] = u2pBH pD y ,
z[prime] = u2
FpBHpD [parenleftbigg]
1
2Kxz x + Kxx z
[parenrightbigg]
, (5.39)
where
D = BHFK0 u4 Kzz 2x + BFH 2y + Kxx 2z Kxz x z
[parenrightbig]
. (5.40)
Substituting these expressions into the action (5.36) we get
S =
pBHK0
u2pD . (5.41)
As in the case of the unbound string, we will now distinguish between the cases of motion outside and within the transverse plane, focusing on the ultra-relativistic limit.
5.2.1 Ultra-relativistic motion outside the transverse plane
The turn-around point umax is dened by the condition D(umax) = 0. In the ultra-relativistic limit we expect that this point approaches the boundary for the string solution of interest, as in the isotropic case. Thus in this limit umax can be determined by using the near-boundary expansions of the metric functions (2.6).
In the limit u ! 0 we nd the following expansions:
Kzz [similarequal] 1 +
L2 2 [prime] 2 [integraldisplay]
dt
[integraldisplay]
umax
0 du
a2u2 cos2 v
4 + [notdef] [notdef] [notdef] , (5.42)
17
Kxz [similarequal] 0
a2u2 sin v cos v 2p1 v2
+ [notdef] [notdef] [notdef] , (5.43)
Kxx [similarequal] 1
a2u2 cos2 v 4(1 v2)
+ [notdef] [notdef] [notdef] , (5.44)
from which it follows that
D [similarequal] 1
a2u2 cos2 v4(1 v2)
u4( 2x + 2y + 2z) + [notdef] [notdef] [notdef] . (5.45)
Similarly, the boundary conditions (5.35) take the form
[lscript]2 sin sin ' [similarequal]
[integraldisplay]
umax
0 du
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u2pD x + [notdef] [notdef] [notdef] , (5.46)
[lscript]2 sin cos ' [similarequal]
[integraldisplay]
umax
0 du
u2pD y + [notdef] [notdef] [notdef] ,
[lscript]2 cos [similarequal]
[integraldisplay]
umax
0 du
u2 pD [parenleftbigg]
z + [notdef] [notdef] [notdef] ,
In the ultra-relativistic limit, all the terms that we have omitted in the equations above, in particular in (5.45) and (5.47), are subleading with respect to the terms that we have retained provided the radial coordinate and the momenta scale as
u = r
p1 v2 , i =
pi1 v2, (5.47)
where r and pi are kept xed in the limit v ! 1. In terms of these rescaled variables (5.47)
the boundary conditions (5.47) take the form
[lscript]2 sin sin ' [similarequal]
p1 v2 px I2(p, v) ,
1
a2u2 cos2 v 4(1 v2)
[lscript]
2 sin cos ' [similarequal]
p1 v2 py I2(p, v) , [lscript]
2 cos [similarequal]
p1 v2 pz I2(p, v)
a2 cos2 v4 I4(p, v)
[parenrightbigg]
, (5.48)
where the integral
In(p, v)
[integraldisplay]
rmax
0 dr
rn
q1 a2r24 cos2 v r4(p2x + p2y + p2z)
(5.49)
is of O(1) in the counting in powers in (1 v2), and is nite if n 0. Further noting that
K0 = 1
a2u2 cos2 v 4(1 v2)
a2r2 cos2 v
+ O(u4) [similarequal] 1
4 , (5.50)
we see that the bound action scales as
S [similarequal]
L2 2 [prime]
2p1 v2 [parenleftbigg]I2
(p, v)
a2 cos2 v
4 I0(p, v)
[parenrightbigg] [integraldisplay]
dt . (5.51)
18
Since both this bound action and the unbound action (5.28) scale as (1 v2)1/2, the divergence at r = 0 in the bound action coming from the I2(p, v) integral would exactly
cancel that in the unbound action in the di erence (3.1). Moreover, by comparing the two actions we would conclude that the momenta pi introduced in (5.47) are indeed of O(1) in the counting in powers of (1 v2) in the ultra-relativistic limit. It would then follow that
the integrals In(p, v) are also of O(1), and therefore that the screening length scales as
Ls (1 v2)1/2 in the ultra-relativistic limit. However, as explained below (5.10), in the
ultra-relativistic Ls = Lmax is simply the maximum possible separation between a bound quark-antiquark pair, so it can be determined by maximizing [lscript] in (5.48) with respect to the momenta. Since the integrals are bounded from above for any value of the pi, and the maximum is v-independent, it follows that Ls = Lmax (1 v2)1/2.
5.2.2 Ultra-relativistic motion within the transverse plane
In this case v = /2 and the expansions of D and of the boundary conditions (5.35) become
D [similarequal] 1
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Cu41 v2
u4( 2x + 2y + 2z) + [notdef] [notdef] [notdef] (5.52)
and
[lscript]2 sin sin ' [similarequal]
[integraldisplay]
umax
0 du u2
x
q1
Cu41v2 u4( 2x + 2y + 2z)+ [notdef] [notdef] [notdef] ,
[lscript]2 sin cos ' [similarequal]
[integraldisplay]
umax
0 du u2
y
q1
Cu41v2 u4( 2x + 2y + 2z)+ [notdef] [notdef] [notdef] ,
[lscript]2 cos [similarequal]
[integraldisplay]
0 du u2 [parenleftBig]
1
umax
Cu4 1v
2
[parenrightBig]
z
q1
Cu41v2 u4( 2x + 2y + 2z)+ [notdef] [notdef] [notdef] ,
where C was dened in (5.30). As in the previous section, in the ultra-relativistic limit all the terms that we have omitted in the equations above are subleading with respect to the terms that we have retained provided the radial coordinate and the momenta scale in this case as
u = r(1 v2)1/4 , i =
pi p1 v2
, (5.53)
where r and pi are kept xed in the limit v ! 1. In terms of the rescaled variables the
boundary conditions (5.53) become
[lscript]2 sin sin ' [similarequal] (1 v2)1/4 px J2(p) , [lscript]
2 sin cos ' [similarequal] (1 v2)1/4 py J2(p) ,
[lscript]2 cos [similarequal] (1 v2)1/4 pz (J2(p) CJ6(p)) , (5.54)
19
where the integral
is of O(1) in the counting in powers in (1 v2), and is nite if n 0. Further noting that
K0 = 1
we see that the bound action becomes
S [similarequal]
(1 v2)1/2 if v [negationslash]= /2 [motion outside the transverse plane]
(1 v2)1/4 if v = /2 [motion within the transverse plane]
irrespectively of the dipole orientation.
5.3 Isotropic limit
The results above reduce to the isotropic result of ref. [31, 32] in the limit a ! 0. This
limit is most easily recovered from the results for motion within the transverse plane, since some of the terms in the expansions in section 5.2.1 vanish if a = 0, thus invalidating the analysis. In contrast, setting a = 0 in section 5.2.2 boils down to simply setting C to its isotropic value, which from (5.30) and (2.3) is
C = F4 =
Jn(p) =
[integraldisplay]
rmax
0 dr
rn
q1 r4(C + p2x + p2y + p2z)
(5.55)
C1 v2
u4 + O(u6) [similarequal] 1 Cr4 , (5.56)
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L2 2 [prime]
2(1 v2)1/4 [parenleftBig]J2
dt . (5.57)
Since both this bound action and the unbound action (5.33) scale as (1 v2)1/4, the divergence at r = 0 in the bound action coming from the J2(p) integral would exactly
cancel that in the unbound action in the di erence (3.1). Moreover, by comparing the two actions we would conclude that the momenta pi introduced in (5.53) are indeed of O(1)
in the counting in powers of (1 v2) in the ultra-relativistic limit. It would then follow
that the integrals Jn(p) are also of O(1), and therefore that the screening length scales as
Ls (1 v2)1/4 in the ultra-relativistic limit. However, as explained below (5.10), in the
ultra-relativistic Ls = Lmax is simply the maximum possible separation between a bound quark-antiquark pair, so it can be determined by maximizing [lscript] in (5.54) with respect to the momenta. Since the integrals are bounded from above for any value of the pi, and the maximum is v-independent, it follows that Ls = Lmax (1 v2)1/4.
In summary, we conclude that in the dipole rest frame the screening length scales in the ultra-relativistic limit as
Ls
(p) CJ2(p)
[parenrightBig] [integraldisplay]
8
>
<
>
:
(5.58)
1u4H = 4T 4 . (5.59)
Since the value of C does not a ect the ultra-relativistic scaling of the screening length, we recover the scaling
Liso (1 v2)1/4 [isotropic plasma] (5.60)
20
0.25
0.20
TL iso
Figure 9. Screening length for a dipole moving through an isotropic plasma in a direction orthogonal (top, blue curve) or parallel (bottom, orange curve) to its orientation.
found in the isotropic case by the authors of [31, 32]. As in the anisotropic case, the ultra-relativistic scaling of the screening length is independent of the dipoles orientation. In fact, even for v < 1, the isotropic screening length depends only mildly on the dipoles orientation, as shown in gure 9.
5.4 Numerical results for generic velocities
Away from the ultra-relativistic limit the screening length must be obtained numerically. For this reason we have focused on a few representative cases, namely those in which both the direction of the plasma wind and the dipoles orientation are aligned with one of the original x, y, or z axes. Given the rotational symmetry in the xy-plane, there are only ve inequivalent cases to consider, because if the wind blows in the z-direction then orienting the dipole along x or y gives identical physics. In each case, we plot the screening length both as a function of the velocity v for di erent degrees of anisotropy a, and also as a function of the degree of anisotropy for di erent values of the velocity. In each case the result can be qualitatively understood combining the static results from section 4 and the ultra-relativistic behavior derived analytically in section 5. We recall that in all cases below, by a dipole oriented along x, y or z we are referring to the original directions before the rotation (5.1) and the boost (5.2).
Wind along z and dipole along z. The numerical results are shown in gures 10 and 11. The curves in gure 10 start at v = 0 with the same value as the = 0 static result shown in gure 7, and that they vanish as (1 v2)1/4 in the limit v ! 1, in agreement
with (5.58)(top line) and (5.60). The screening length decreases with the anisotropy, irrespectively of whether T or s are kept xed.
Wind along z and dipole along x. The numerical results are shown in gures 12 and 13. We see that the curves in gure 12 start at v = 0 with the same value as the = /2 static result shown in gure 7, and that they vanish as (1 v2)1/4 in the limit v ! 1, in agreement with (5.58)(top line) and (5.60). In this case the screening length
decreases with the anisotropy for any velocity provided the temperature is kept xed. The
21
0.15
0.10
0.05
0.00 0.0 0.2 0.4 0.6 0.8 1.0
v
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1.0
1.0
0.8
0.8
L s/L iso(T)
L s/L iso(s)
0.6
0.6
0.4
0.4
0.2
0.2
0.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.0 0.2 0.4 0.6 0.8 1.0
v
v
Figure 10. Screening length for a plasma wind along the z-direction and a dipole oriented along the z-direction, for four di erent values of the anisotropy (from top to bottom) a/T = 12.2, 42.6, 86, 744. The corresponding values in units of the entropy density are (in the same order) aN2/3c/s1/3 = 6.2, 19, 35, 242. The screening length is plotted in the appropriate units to facilitate comparison with the isotropic result for a plasma at the same temperature (left), or at the same entropy density (right). The isotropic result is plotted in gure 9, and its ultra-relativistic behavior is given in eq. (5.60). At v = 0 the curves agree with the = 0 values in gure 7. As v ! 1 they vanish as (1 v2)1/4, in agreement with (5.58)(top line) and (5.60).
0.0 0 5 10 15 20 25 30
1.0
1.0
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0.8
0.8
L s/L iso(T)
L s/L iso(s)
0.6
0.6
0.4
0.4
0.2
0.2
0.0 0 5 10 15 20 25 30
a/T
aN2/3c/s1/3
Figure 11. Screening length for a plasma wind along the z-direction and a dipole oriented along the z-direction, at ve di erent velocities (from top to bottom) v = 0.25, 0.5, 0.7, 0.9, 0.9995. The screening length is plotted in the appropriate units to facilitate comparison with the isotropic result for a plasma at the same temperature (left), or at the same entropy density (right). The isotropic result is plotted in gure 9, and its ultra-relativistic behavior is given in eq. (5.60).
same behavior is found at constant entropy density for high enough velocities, whereas for low velocities the screening length at constant s actually increases with a.
Wind along x and dipole along x. The numerical results are shown in gures 14 and 15. The curves in gure 14 start at v = 0 with the same value as the = /2 static result shown in gure 7, and that they approach a nite, non-zero value as v ! 1, in
agreement with (5.58)(bottom line) and (5.60). As in previous cases, the screening length decreases with the anisotropy for any velocity provided the temperature is kept xed. The opposite behavior is found at constant s.
22
1.0
1.5
L ani/L iso(T)
0.8
L ani/L iso(s)
0.6
1.0
0.4
0.5
0.2
0.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.0 0.2 0.4 0.6 0.8 1.0
v
v
Figure 12. Screening length for a plasma wind along the z-direction and a dipole oriented along the x-direction, for four di erent values of the anisotropy a/T = 12.2 (red, solid), 42.6 (maroon, coarsely dashed), 86 (violet, dashed), 744 (orange, dot-dashed). The corresponding values in units of the entropy density are (in the same order) aN2/3c/s1/3 = 6.2, 19, 35, 242. The screening length is plotted in the appropriate units to facilitate comparison with the isotropic result for a plasma at the same temperature (left), or at the same entropy density (right). The isotropic result is plotted in gure 9, and its ultra-relativistic behavior is given in eq. (5.60). At v = 0 the curves agree with the = /2 values in gure 7. As v ! 1 they vanish as (1v2)1/4, in agreement with (5.58)(top line) and (5.60).
1.0
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1.2
L ani/L iso(T)
0.8
1.0
L ani/L iso(s)
0.6
0.8
0.4
0.6
0.4
0.2
0.2
0.0 0 5 10 15 20 25 30
0.0 0 5 10 15 20 25 30
a/T
a/s1/3
Figure 13. Screening length for a plasma wind along the z-direction and a dipole oriented along the x-direction, at ve di erent velocities (from top to bottom) v = 0.25, 0.5, 0.7, 0.9, 0.9995. The screening length is plotted in the appropriate units to facilitate comparison with the isotropic result for a plasma at the same temperature (left), or at the same entropy density (right). The isotropic result is plotted in gure 9, and its ultra-relativistic behavior is given in eq. (5.60).
Wind along x and dipole along y. The numerical results are shown in gures 16 and 17. We see that the curves in gure 16 start at v = 0 with the same value as the = /2 static result shown in gure 7, and that they approach a nite, non-zero value as v ! 1, in agreement with (5.58)(bottom line) and (5.60). The qualitative behavior in as
in the case of motion and orientation along x.
Wind along x and dipole along z. The numerical results are shown in gures 18 and 19. We see that the curves in gure 18 start at v = 0 with the same value as the = 0 static result shown in gure 7, and that they approach a nite, non-zero value as
23
1.0
1.6
0.9
1.4
L s/L iso(T)
L s/L iso(s)
0.8
1.2
1.0
0.7
0.8
0.6
0.6
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
v
v
Figure 14. Screening length for a plasma wind along the x-direction and a dipole oriented along the x-direction, for four di erent values of the anisotropy (from top to bottom) a/T = 12.2, 42.6, 86, 744. The corresponding values in units of the entropy density are (in the same order) aN2/3c/s1/3 = 6.2, 19, 35, 242. The screening length is plotted in the appropriate units to facilitate comparison with the isotropic result for a plasma at the same temperature (left), or at the same entropy density (right). The isotropic result is plotted in gure 9, and its ultra-relativistic behavior is given in eq. (5.60). At v = 0 the curves agree with the = /2 values in gure 7. As v ! 1 they approach a nite, non-zero value, in agreement with (5.58)(bottom line) and (5.60).
0 5 10 15 20 25 30
1.00
1.20
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0.95
1.15
L s/L iso(T)
L s/L iso(s)
0.90
1.10
0.85
1.05
0.80
1.00
0.95 0 5 10 15 20 25 30
0.75
a/T
aN2/3c/s1/3
Figure 15. Screening length for a plasma wind along the x-direction and a dipole oriented along the x-direction, at ve di erent velocities v =0.25 (yellow, dot-dashed), 0.5 (green, short dashed),0.7 (brown, medium dashed), 0.9 (cyan, long dashed), 0.9995 (blue, solid). The screening length is plotted in the appropriate units to facilitate comparison with the isotropic result for a plasma at the same temperature (left), or at the same entropy density (right). The isotropic result is plotted in gure 9, and its ultra-relativistic behavior is given in eq. (5.60).
v ! 1, in agreement with (5.58)(bottom line) and (5.60). The screening length decreases
with the anisotropy for any velocity provided the temperature is kept xed. The same is true at large anisotropies if the entropy density is kept xed.
6 Dissociation temperature and dissociation anisotropy
In previous sections we have focused on computing the screening length in an anisotropic plasma, Ls(T, a), and on comparing it to its isotropic counterpart Liso = Ls(T, 0). The
24
1.0
1.6
0.9
1.4
L ani/L iso(T)
0.8
L ani/L iso(s)
1.2
1.0
0.7
0.8
0.6
0.6
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
v
v
Figure 16. Screening length for a plasma wind along the x-direction and a dipole oriented along the y-direction, for four di erent values of the anisotropy a/T = 12.2 (red, solid), 42.6 (maroon, coarsely dashed), 86 (violet, dashed), 744 (orange, dot-dashed). The corresponding values in units of the entropy density are (in the same order) aN2/3c/s1/3 = 6.2, 19, 35, 242. The screening length is plotted in the appropriate units to facilitate comparison with the isotropic result for a plasma at the same temperature (left), or at the same entropy density (right). The isotropic result is plotted in gure 9, and its ultra-relativistic behavior is given in eq. (5.60). At v = 0 the curves agree with the = /2 values in gure 7. As v ! 1 they approach a nite, non-zero value, in agreement
with (5.58)(bottom line) and (5.60).
0 5 10 15 20 25 30
JHEP01(2013)170
1.00
1.20
L ani/L iso(T)
L ani/L iso(s)
0.95
1.15
0.90
1.10
0.85
1.05
0.80
1.00
0.95 0 5 10 15 20 25 30
0.75
a/T
aN2/3c/s1/3
Figure 17. Screening length for a plasma wind along the x-direction and a dipole oriented along the y-direction, at ve di erent velocities v =0.25 (yellow, dot-dashed), 0.5 (green, short dashed),0.7 (brown, medium dashed), 0.9 (cyan, long dashed), 0.9995 (blue, solid). The screening length is plotted in the appropriate units to facilitate comparison with the isotropic result for a plasma at the same temperature (left), or at the same entropy density (right). The isotropic result is plotted in gure 9, and its ultra-relativistic behavior is given in eq. (5.60).
screening length characterizes the dissociation of a quark-antiquark pair for xed T and a: a pair separated a distance [lscript] < Ls forms a bound state, but if [lscript] is increased above
Ls then the bound state dissociates. Similarly, one may dene a dissociation temperature Tdiss(a, [lscript]) that characterizes the dissociation of a quark-antiquark pair of xed size [lscript] in a plasma with a given degree of anisotropy a: for T < Tdiss the pair forms a bound state,
but if T is increased above Tdiss then the bound state dissociates. Analogously, one may dene a dissociation anisotropy adiss(T, [lscript]) such that a bound state forms for a < adiss but
25
1.0
1.0
L ani/L iso(T)
0.8
0.8
L ani/L iso(s)
0.6
0.6
0.4
0.4
0.2
0.2
0.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.0 0.2 0.4 0.6 0.8 1.0
v
v
Figure 18. Screening length for a plasma wind along the x-direction and a dipole oriented along the z-direction, for four di erent values of the anisotropy (from top to bottom) a/T = 12.2, 42.6, 86, 744. The corresponding values in units of the entropy density are (in the same order) aN2/3c/s1/3 = 6.2, 19, 35, 242. The screening length is plotted in the appropriate units to facilitate comparison with the isotropic result for a plasma at the same temperature (left), or at the same entropy density (right). The isotropic result is plotted in gure 9, and its ultra-relativistic behavior is given in eq. (5.60). At v = 0 the curves agree with the = 0 values in gure 7. As v ! 1 they approach a nite, non-zero value, in agreement with (5.58)(bottom line) and (5.60).
0 5 10 15 20 25 30
1.0
1.0
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L ani/L iso(T)
0.9
0.9
0.8
L ani/L iso(s)
0.8
0.7
0.7
0.6
0.6
0 5 10 15 20 25 30
a/T
a/s1/3
Figure 19. Screening length for a plasma wind along the x-direction and a dipole oriented along the z-direction, at ve di erent velocities (from bottom to top) v = 0.25, 0.5, 0.7, 0.9, 0.9995. The screening length is plotted in the appropriate units to facilitate comparison with the isotropic result for a plasma at the same temperature (left), or at the same entropy density (right). The isotropic result is plotted in gure 9, and its ultra-relativistic behavior is given in eq. (5.60).
not for a > adiss. It is useful to think of the three-dimensional space parametrized by (T, a, [lscript]) as divided in two disconnected regions by a two-dimensional surface: in one region quark-antiquark pairs bind together, while in the other one they do not. The functions Ls(T, a), Tdiss(a, [lscript]) and adiss(T, [lscript]) are then simply di erent parametrizations of the dividing
surface. It is therefore clear that if a triplet (T, a, [lscript]) lies on the dividing surface then
T Ls(a, T ) = Tdiss(a, [lscript])[lscript] , aLs(T, a) = adiss(T, [lscript])[lscript] , etc. (6.1)
In this section we will focus on the qualitative form of Tdiss and adiss. As we will
see, most of the analysis follows from the asymptotic behavior of the screening length for
26
2.0
1.8
1.6
1.4
1.2
1.0
v proper/v z
0.0 0.2 0.4 0.6 0.8 1.0
u/uH
Figure 20. Proper velocity in the z-direction at a position u away from the boundary, as dened in (6.2), for di erent values of a/T . From right to left, a/T = 1.38, 33, 86, 249.
a T . This means that, at the qualitative level, most of the results that we will obtain
would also apply if we were to replace the temperature by the entropy density as one of our variables. The reason is that, by virtue of (2.5), the limit a T corresponds to the limit
a s1/3 and vice versa. In addition, we will see that for generic dipoles orientations and
velocities, the large-anisotropy limit is entirely controlled by the near-boundary behavior of the metric at O(u2), which depends solely on a and is therefore completely insensitive to the values of the temperature or of the entropy density.
The key point in the large-a analysis is the requirement that no point on the string can move faster than the local speed of light in the bulk. Consider a meson moving with a velocity v that has a non-zero component vz along the z-direction. Then we see from (2.2)
that the proper velocity along this direction of a point on the string sitting at a value u of the radial coordinate is
. (6.2)
The function H(u) increases monotonically from the boundary to the horizon, and is does
so more steeply as a/T increases, as illustrated in gure 1. The combination F(u)B(u)
has the opposite behavior, as expected from the fact that gravity is attractive: it decreases monotonically from the boundary to the horizon. In the isotropic case H = 1 and FB de
creases more steeply as T increases. This is thus the rst hint that increasing the anisotropy has an e ect similar to increasing the temperature: both make vproper(u) a more steeply in
creasing function away from the boundary. We have illustrated the e ect of the anisotropy in gure 20, where we see that vproper/vz becomes a steeper function of u as a/T increases.
It follows that, for xed vz [negationslash]= 0, there is a maximum value of umax beyond which v
becomes superluminal, so no string solution can penetrate to u > umax. As we will corrobo
rate numerically, this upper bound on umax translates into an upper bound on Ls. Moreover, umax decreases as a/T increases. This means that for su ciently large anisotropies we can use the near-boundary expansions (2.6) in order to determine Ls, in analogy to what we did in the ultra-relativistic limit. As in that case, for vz [negationslash]= 0 the analysis is controlled by
27
JHEP01(2013)170
vproper(u) = vz
s
gzz(u)
gtt(u) = vz[radicalBigg] H (u)
F(u)B(u)
proper
the O(u2) terms in (2.6). The key point is that these terms depend on a but not on T , so by dimensional analysis it follows that umax a1 and Ls a1 in the limit a/T 1. This limit can be understood as a ! 1 at xed T , or as T ! 0 at xed a. We thus conclude
that, even at T = 0, a generic meson will dissociate for a su ciently large anisotropy adiss.
Mesons at rest and mesons whose velocity is exactly aligned with the transverse plane constitute an exception to the argument above, since in this case vz = 0 and their physics is mostly insensitive to the function H(u) which characterizes the anisotropic direction.
Therefore in this case we expect that umax and Ls will remain nite as we send a ! 1 at
xed T , and hence that dimensional analysis will imply Ls T 1.In summary, the heuristic argument above suggests that in the limit a/T 1 we
should have
Ls(T, a)
JHEP01(2013)170
8
>
<
>
:
const.[notdef]T 1 if the meson is static or in motion within the transverse plane,
const.[notdef]a1 otherwise. (6.3) The constants may depend on all the dimensionless parameters such as the velocity and the dipoles orientation. We will refer to the behavior in the second line as generic and to that in the rst line as non-generic, since the latter only applies if the velocity is exactly zero or if the motion is exactly aligned with the transverse plane. The generic behavior is of course consistent with the analysis of section 5.2.1. Indeed, we saw in that section that for motion outside the transverse plane the ultra-relativistic behavior of Ls is entirely controlled by the O(u2) terms in the metric, which depend on a but not on T .
Figure 21 shows our numerical results for umax, in units of T 1 and a1, as a function of a/T , for the ve physically distinct cases discussed in section 5.4. From the continuous, magenta curves in the rst two rows we see that umax goes to zero at large a/T in the cases of motion along z, irrespectively of the dipoles orientation. In contrast, we see that umax
does not go zero for a static meson (dashed, blue curves) or for a meson moving along the x-direction (continuous, magenta curves in the last three rows).
Recalling that the isotropic screening length is of the form Liso / 1/T , we see that the quantity plotted on the vertical axes in gures 6, 11, 13, 15, 17 and 19 is precisely proportional to T Ls(T, a). However, the asymptotic behavior (6.3) is not apparent in these plots because in most cases the horizontal axes do not extend to high enough values of a/T . For this reason we have illustrated the two possible asymptotic behaviors of Ls in
gure 22, where we have extended the horizontal axes to larger values of a/T . We see from the continuous, magenta curves in the rst two rows that Ls 1/a for motion along the
z-direction. For motion within the transverse plane we see from the same curves in the last three rows that Ls 1/T . This approximate scaling relation seems to hold quite precisely
for a dipole oriented within the transverse plane (3rd and 4th rows), whereas for a dipole oriented in the z-direction the product T Ls seems to retain a slight (perhaps logarithmic)
dependence on a/T at large a/T . We can draw similar conclusions from the dashed, blue curves in the gure, which correspond to static mesons. We see that for mesons oriented within the transverse plane (2nd, 3rd and 4th rows) the relation T Ls constant holds
28
0.5
50
z z
z z
0.4
40
0.3
30
Tu max
au max
0.2
20
0.1
10
0.0 0 100 200 300 400
0 0 100 200 300 400
a/T
a/T
0.5
14
z x
0.4
12
z x
10
0.3
8
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Tu max
au max
0.2
6
4
0.1
2
0.0 0 100 200 300 400
0 0 100 200 300 400
a/T
a/T
0.5
100
x x
0.4
80
x x
Tu max
0.3
60
au max
0.2
40
0.1
20
0.0 0 100 200 300 400
0 0 100 200 300 400
a/T
a/T
0.5
100
0.4
x y
80
x y
Tu max
0.3
60
au max
0.2
40
0.1
20
0.0 0 100 200 300 400
0 0 100 200 300 400
a/T
a/T
0.5
100
0.4
x z
x z
80
Tu max
0.3
60
au max
0.2
40
0.1
20
0.0 0 100 200 300 400
0 0 100 200 300 400
a/T
a/T
Figure 21. Value of umax in units of 1/T (left) or 1/a (right), as a function of the ratio a/T , for a dipole at rest (dashed, blue curve) and for a dipole moving with v = 0.45 (continuous, magenta curve). The rst letter on the top right corner of each plot indicates the direction of motion, and the second one indicates the orientation of the dipole.
29
0.25
50
z z
z z
0.20
40
0.15
30
TL s
0.10
aL s
20
0.05
10
0.00 0 100 200 300 400
0 0 100 200 300 400
a/T
a/T
0.25
50
z x
0.20
40
z x
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0.15
30
TL s
0.10
aL s
20
0.05
10
0.00 0 100 200 300 400
0 0 100 200 300 400
a/T
a/T
0.25
50
x x
0.20
40
x x
0.15
30
TL s
0.10
aL s
20
0.05
10
0.00 0 100 200 300 400
0 0 100 200 300 400
a/T
a/T
0.25
50
0.20
x y
40
x y
0.15
30
TL s
0.10
aL s
20
0.05
10
0.00 0 100 200 300 400
0 0 100 200 300 400
a/T
a/T
0.25
50
0.20
x z
40
x z
0.15
30
TL s
0.10
aL s
20
0.05
10
0.00 0 100 200 300 400
0 0 100 200 300 400
a/T
a/T
Figure 22. Screening length in units of 1/T (left) or 1/a (right), as a function of the ratio a/T , for a dipole at rest (dashed, blue curve) and for a dipole moving with v = 0.45 (continuous, magenta curve). The rst letter on the top right corner of each plot indicates the direction of motion, and the second one indicates the orientation of the dipole.
30
0.25
12
x
z
z
x
8
x
z
0.20
10
0.15
[lscript]T diss
0.10
[lscript]a diss
6
4
0.05
2
z
x
0.00 0 2 4 6 8 10 12
0
0.00 0.05 0.10 0.15 0.20 0.25
a[lscript]
T [lscript]
Figure 23. Dissociation temperature (left) Tdiss(a, [lscript]) = [lscript]1f(a[lscript]) and dissociation anisotropy
(right) adiss(T, [lscript]) = [lscript]1g(T [lscript]) for a dipole at rest (dashed curves) and for a dipole moving along
the z-direction with v = 0.45 (continuous curves). The orientation of the dipole is indicated by a letter next to each curve.
JHEP01(2013)170
1.0
0.8
0.6
v lim
0.4
0.2
0.0 0 5 10 15 20
a[lscript]
Figure 24. Limiting velocity, for xed anisotropy and T = 0, beyond which a meson oriented along the x-direction and moving along the z-direction will dissociate.
quite precisely, whereas for mesons oriented in the z-direction (1st and 5th rows) there seems to be some slight residual dependence on a/T at large a/T .
Combining the two plots on the left and the right columns of gure 22 we can eliminate a/T and obtain T Ls as a function of aLs and vice versa. Recalling (6.1) we see that we can interpret the result in the rst case as Tdiss(a, [lscript]) = [lscript]1f(a[lscript]), whereas in the second case we
get adiss(T, [lscript]) = [lscript]1g(T [lscript]). The functions f and g are the curves shown in gure 23(left) and gure 23(right), respectively. The right plot is of course the mirror image along a 45 degree line of the left plot. We see in gure 23(left) that the dissociation temperature decreases monotonically with increasing anisotropy and vanishes at a[lscript] [similarequal] 9.75 (for the chosen velocity
and orientation). On the right plot this corresponds to the dissociation anisotropy at zero temperature. As anticipated above, even at zero temperature, a generic meson of size [lscript] will dissociate if the anisotropy is increased above adiss(T = 0, [lscript]) / 1/[lscript]. The proportionality constant in this relation is a decreasing function of the meson velocity in the plasma. This implies that for a xed anisotropy there is a limiting velocity vlim above which a meson will dissociate, even at zero temperature. The form of vlim(a[lscript]) for T = 0 is plotted in gure 24.
31
1.0
1.0
0.8
0.8
T diss(v)/T diss(0)
T diss(v)/T diss(0)
0.6
0.6
0.4
0.4
0.2
0.2
0.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.0 0.2 0.4 0.6 0.8 1.0
v
Figure 25. Dissociation temperature for a meson moving along the x-direction and oriented along the z-direction (left) or along the x-direction (right). Each curve corresponds to a xed value of the product a[lscript] = 0 (blue curve), 1.4 (green curve), 25 (red curve).
The existence of a limiting velocity for quarkonium mesons is well known in a strongly coupled isotropic plasma [35, 36], in which case the dissociation at v = vlim is caused by
the temperature. What we see here is that in our anisotropic plasma this behavior persists as T ! 0 for generic motion. In this limit it is the anisotropy that is responsible for the
dissociation. In the case of ultra-relativistic motion the relation between adiss or Tdiss and
vlim can be obtained by combining the scalings (5.58) and (6.3). For generic motion these relations yield
adiss(T, [lscript])
1 [lscript] (1 v2lim)1/4 . [a T , vlim [lessorsimilar] 1] (6.5)
The scaling (6.5) agrees with the isotropic result [31, 32] and illustrates the fact that, for motion within the transverse plane, the limiting velocity in our anisotropic plasma approaches unity as T ! 0. This behavior is the same for a meson at rest, as illustrated
in gure 23, where we see that a su ciently small meson will remain bound in the plasma for any value of the anisotropy provided the plasma is cold enough. In fact, the form of the dissociation temperature for all anisotropies and all velocities within the transverse plane is qualitatively analogous to that of the isotropic case, as shown in gure 25. The fact that the curves in this gure approximately overlap one another signals that the dependence of the dissociation temperature on v and a[lscript] can be approximately factorized over the entire range 0 v 1.
In contrast, for generic motion we saw above that the limiting velocity is subluminal even at T = 0, vlim(T = 0, a[lscript]) < 1. Increasing the temperature simply decreases the value of the limiting velocity, vlim(T [lscript], a[lscript]) < vlim(T = 0, a[lscript]). Turning these statements around we
see that, at a xed anisotropy, the dissociation temperature is a decreasing function of the velocity that vanishes at v = vlim(T = 0, a[lscript]). This is illustrated in gure 26, where we see that vlim(T = 0, a[lscript]) decreases as the anisotropy increases, in agreement with gure 24. In
32
v
JHEP01(2013)170
1
[lscript] (1 v2lim)1/2 , [a T , vlim [lessorsimilar] 1] (6.4)
whereas for motion within the transverse plane we obtain
Tdiss(a, [lscript])
1.0
1.0
0.8
0.8
T diss(v)/T diss(0)
T diss(v)/T diss(0)
0.6
0.6
0.4
0.4
0.2
0.2
0.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.0 0.2 0.4 0.6 0.8 1.0
v
Figure 26. Dissociation temperature for a meson moving along the z-direction and oriented along the x-direction (left) or along the z-direction (right). Each curve corresponds to a xed value of the product a[lscript]. From right to left, a[lscript] = 0, 1, 5.4, 25.
order to facilitate comparison with the isotropic results of [3133], in gure 26 we have chosen to normalize the dissociation temperature by its value at v = 0 instead of by the dipoles size [lscript]. Our numerical results suggest that as v approaches vlim the dissociation temperature may vanish as
Tdiss(v, a[lscript])
Tdiss(0, a[lscript]) v2lim v2
v
JHEP01(2013)170
" . (6.6)
In this equation vlim = vlim(T = 0, a[lscript]) and " = "(a[lscript]) > 0 is an anisotropy-dependent
exponent. Unfortunately, the limit v ! vlim is di cult to analyze numerically, so our
results are not precise enough to allow us to establish (6.6) unambiguously. To emphasize this point, in gure 26 we have plotted as discontinuous the part of the curves between the last two data points. The last point lies on the horizontal axis at (v, T ) = (vlim, 0), and
the penultimate point lies at a certain height at (v [lessorsimilar] vlim, T > 0). Since this last bit of
the curves is an interpolation between these data points, it is di cult to establish whether the slopes of the curves diverge as they meet the horizontal axis, as would be implied by the scaling (6.6). Presumably, this scaling could be veried or falsied analytically by including the rst correction in T/a to the scaling in the second line of (6.3).
7 Discussion
We have considered an anisotropic N = 4 SYM plasma in which the x, y directions are
rotationally symmetric, but the z-direction is not. In the context of heavy ion collisions the latter would correspond to the beam direction, and the former to the transverse plane. The screening length of a quarkonium meson in motion in the plasma depends on the relative orientation between these directions, on the one hand, and the direction of motion of the meson and its orientation, on the other. This dependence can be parametrized by three angles ( v, , '), as shown in gure 8. We have determined the screening length for the most general geometric parameters and for any anisotropy. Our results are valid in the strong-coupling, large-Nc limit, since we have obtained them by means of the gravity dual [14, 15] of the anisotropic N = 4 plasma. The anisotropy is induced by a
position-dependent theta term in the gauge theory, or equivalently by a position-dependent
33
axion on the gravity side. One may therefore wonder how sensitive the conclusions may be to the specic source of the anisotropy. In this respect it is useful to note that the gravity calculation involves only the coupling of the string to the background metric. This means that any anisotropy that gives rise to a qualitatively similar metric (and no Neveu-Schwarz B-eld) will yield qualitatively similar results for the screening length, irrespectively of the form of the rest of supergravity elds.
An example of a rather robust conclusion is the ultra-relativistic behavior2 of the screening length (5.58), which for motion not exactly aligned with the transverse plane is Ls (1 v2)1/2. The 1/2 exponent contrasts with the 1/4 isotropic result [31, 32],
and follows from the fact that the near-boundary fall-o of the metric (2.2) takes the schematic form
g[notdef] = L2 u2
[notdef] + u2g(2)[notdef] + u4g(4)[notdef] + [notdef] [notdef] [notdef] [parenrightBig]
. (7.1)
As v grows closer and closer to 1 the point of maximum penetration of the string into the bulk, umax, moves closer and closer to the AdS boundary at u = 0. As a consequence, the physics in this limit is solely controlled by the near-boundary behavior of the metric. For generic motion the behavior is in fact governed by the O(u2) terms alone, and a simple scaling argument then leads to the 1/2 exponent above. In the isotropic case the O(u2)
terms are absent and the same scaling argument leads to the 1/4 exponent.
In fact, a similar reasoning allowed us to determine the large-anisotropy limit. Since the metric component gzz / H(u) grows as one moves from the boundary to the horizon,
a subluminal velocity of the meson at the boundary would eventually translate into a superluminal proper velocity (6.2) at a su ciently large value of u.3 This sets an upper limit on the maximum penetration length umax of the string into the bulk and hence on Ls. Moreover, gzz becomes steeper as a/T increases, so in the limit a/T 1 the point
umax approaches the AdS boundary (unless the motion is aligned with the transverse plane), just as in the ultra-relativistic limit. In this limit the physics is again controlled by the O(u2) terms in the metric, which depend on a but not on T . Therefore dimensional analysis implies that Ls = const. [notdef] a1, were the proportionality constant is a decreasing
function of the velocity. This led us to one of our main conclusions: even in the limit T ! 0, a generic meson of size [lscript] will dissociate at some high enough anisotropy adiss [lscript]1.
Similarly, for xed a and T , even if T = 0, a generic meson will dissociate if its velocity exceeds a limiting velocity vlim(a, T ) < 1, as shown in gure 24 for T = 0. As explained
in section 6, the conclusions in this paragraph would remain unchanged if we worked at constant entropy density instead of at constant temperature, since in the limit a s1/3 the physics would again be controlled only by the O(u2) terms in the metric.
The above discussion makes it clear that, at the qualitative level, much of the physics depends only on a few features of the solution: the presence of the g(2)[notdef] term in the near-
boundary expansion of the metric, the fact that the metric (7.1) be non-boost-invariant at order u2 (i.e. that g(2)[notdef] not be proportional to [notdef] ), and the fact that gzz increases as
2We recall that we rst send the quark mass to innity and then v ! 1 (see section 1).
3Note that the overall conformal factor 1/u2 in (2.2) plays no role in this argument, since it cancels out in the ratio (6.2).
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JHEP01(2013)170
0.08
0.06
0.04
0.02
0.00 [Minus]4 [Minus]2 0 2 4 6 8
log(a/T )
Figure 27. Log-log plot of the entropy density per unit 2-area in the xy-directions on a constant-z slice as a function of a/T , normalized to the isotropic result s2Diso = 2 N2cT 2.
a function of both u and a/T .4 The second condition is necessary because otherwise the physics of a meson in motion would be equivalent to that of a meson at rest, and we have seen that the latter is very similar to that of a meson in an isotropic plasma. The third condition ensures that umax moves close to the boundary as a/T increases. Note that adding temperature to an otherwise boost-invariant metric will only a ect g(4)[notdef] , and thus
this is not enough to make g(2)[notdef] non-boost-invariant. This conclusion is consistent with the fact that g(2)[notdef] is only a function of the external sources which the theory is coupled to.
From the gauge theory viewpoint, some heuristic intuition can be gained by recalling that the anisotropy is induced by dissolving along the z-direction objects that extend along the xy-directions [14, 15, 26]. The number density of such objects along the z-direction, dn/dz, is proportional to a. On the gravity side these are D7-branes that wrap the ve-sphere in the metric (2.2), extend along the xy-directions, and are homogeneously distributed in the z-direction. Increasing a has a large e ect on the entropy density per unit 3-volume in the xyz-directions, in the sense that s/T 3 ! 1 as a/T ! 1, as
shown in gure 2. In contrast, the entropy density per unit 2-area in the xy-directions on a constant-z slice, s2D/T 2, approaches a constant in the limit a/T ! 1. This is
illustrated in gure 27, which is based on our numerical calculations, but it can also be proven analytically following the argument in section 2.5 of ref. [26]. In view of these di erences, it is perhaps not surprising that the anisotropy has the largest e ect on the physics of mesons moving along the z-direction, and the smallest e ect on the physics of mesons moving within the transverse plane. Mesons at rest are also more sensitive to the anisotropy if they extend along the z-direction than if they are contained within the transverse plane. Presumably, the correct intuition behind this physics is that moving against the D7-branes is harder than moving along them.
We close with a few comments on existing weak-coupling results on the physics of quarkonium dissociation in the real-world QGP. In the isotropic case the velocity dependence of the heavy quark potential has been studied using perturbative and e ective eld theory methods, see e.g. [3740]. These analyses include modications of both the
4Again, up to possible overall conformal factors.
35
iso)
log(s2D/s2D
JHEP01(2013)170
real and imaginary parts of the potential, which are related to screening and to the thermal width of the states, respectively. They nd that meson dissociation at non-zero velocity results form a complex interplay between the real and the imaginary parts of the potential. However, the general trend that seems to emerge is that screening e ects increase with the velocity, while the width of the states decreases. The behavior of the real part is thus in qualitative agreement with the isotropic limit of our results. However, the extraction of a screening length from these analyses is not immediate due to the fact that the real part of the potential is not approximately Yukawa-like [39, 40], in contrast with the holographic result. In any case, an interesting consequence of the dominance of the real part of the potential is that, at su ciently high velocities, dissociation is caused by screening rather than by Landau damping [39, 40]. In the holographic framework, the thermal widths of our mesons could presumably be computed along the lines of [41].
To the best of our knowledge no results at non-zero velocity exist in the presence of anisotropies, so in this case we will limit ourselves to the static situation. We emphasize though that any comparison between these results and ours should be interpreted with caution, because the sources of anisotropy in the QGP created in a heavy ion collision and in our system are di erent. In the QGP the anisotropy is dynamical in the sense that it is due to the initial distribution of particles in momentum space, which will evolve in time and eventually become isotropic. In contrast, in our case the anisotropy is due to an external source that keeps the system in an equilibrium anisotropic state that will not evolve in time. We hope that, nevertheless, our system might provide a good toy model for processes whose characteristic time scale is su ciently shorter than the time scale controlling the time evolution of the QGP.
A general conclusion of refs. [4244] is that, if the comparison between the anisotropic plasma and its isotropic counterpart is made at equal temperatures, then the screening length increases with the anisotropy. This e ect occurs for dipoles oriented both along and orthogonally to the anisotropic direction, but it is more pronounced for dipoles along the anisotropic direction. The dependence on the anisotropy in these weak-coupling results is the opposite of what we nd in our strongly coupled plasma. In our case the screening length in the anisotropic plasma is smaller than in its isotropic counterpart if both plasmas are taken to have the same temperature, as shown in gure 6(left). We also nd that the e ect is more pronounced for dipoles extending along the anisotropic direction, as illustrated in gure 7(left).
Refs. [44, 45] argued that if the comparison between the anisotropic and the isotropic plasmas is made at equal entropy densities, then the physics of quarkonium dissociation exhibits little or no sensitivity to the value of the anisotropy. This is again in contrast to our results since, as shown in gure 6(right) and gure 7(right), the screening length in this case is just as sensitive to the anisotropy as in the equal-temperature comparison. The di erence in the equal-entropy case is simply that the screening length may increase or decrease with the anisotropy depending on the dipoles orientation.
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Acknowledgments
It is a pleasure to thank M. Strickland for helpful discussions. MC is supported by a postdoctoral fellowship from Mexicos National Council of Science and Technology (CONACyT). We acknowledge nancial support from 2009-SGR-168, MEC FPA2010-20807-C02-01, MEC FPA2010-20807-C02-02 and CPAN CSD2007-00042 Consolider-Ingenio 2010 (MC, DF and DM), and from DE-FG02-95ER40896 and CNPq (DT).
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39
JHEP01(2013)170
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SISSA, Trieste, Italy 2013
Abstract
(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)
We compute the screening length for quarkonium mesons moving through an anisotropic, strongly coupled ... = 4 super Yang-Mills plasma by means of its gravity dual. We present the results for arbitrary velocities and orientations of the mesons, as well as for arbitrary values of the anisotropy. The anisotropic screening length can be larger or smaller than the isotropic one, and this depends on whether the comparison is made at equal temperatures or at equal entropy densities. For generic motion we find that: (i) mesons dissociate above a certain critical value of the anisotropy, even at zero temperature; (ii) there is a limiting velocity for mesons in the plasma, even at zero temperature; (iii) in the ultra-relativistic limit the screening length scales as (1 - v ^sup 2^)^sup ^ with = 1/2, in contrast with the isotropic result = 1/4.
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Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer