有限耦合对Bjorken流体动力学的全息约束资源-CSDN文库

Open

Access

153 浏览量 2020-03-27 08:50:26 上传评论收藏 1.04MB PDF 举报

资源推荐

资源详情

资源评论

JHEP10(2017)110

Published for SISSA by Springer

Received: September 14, 2017

Accepted: October 7, 2017

Published: October 17, 2017

Holographic constraints on Bjorken hydrodynamics at

ﬁnite coupling

Brandon S. DiNunno,

a,b,e

Saˇso Grozdanov,

Juan F. Pedraza

and Steve Young

Theory Group, Department of Physics, The University of Texas at Austin,

2515 Speedway, Stop C1608, Austin, TX 78712, U.S.A.

Helsinki Institute of Physics, University of Helsinki,

P.O. Box 64, Helsinki FIN-00014, Finland

Instituut-Lorentz for Theoretical Physics, Leiden University,

Niels Bohrweg 2, Leiden 2333 CA, The Netherlands

Institute for Theoretical Physics, University of Amsterdam,

Science Park 904, 1090 GL Amsterdam, The Netherlands

Theory Group, Maxwell Analytics LLC,

600 Sabine Street, Austin, TX 78701, U.S.A.

E-mail: bsd86@physics.utexas.edu, grozdanov@lorentz.leidenuniv.nl,

jpedraza@uva.nl, scyoung@zippy.ph.utexas.edu

Abstract: In large-N

conformal ﬁeld theories with classical holographic duals, inverse

coupling constant corrections are obtained by considering higher-derivative terms in the

corresponding gravity theory. In this work, we use type IIB supergravity and bottom-up

Gauss-Bonnet gravity to study the dynamics of boost-invariant Bjorken hydrodynamics

at ﬁnite coupling. We analyze the time-dependent decay properties of non-local observ-

ables (scalar two-point functions and Wilson loops) probing the diﬀerent models of Bjorken

ﬂow and show that they can be expressed generically in terms of a few ﬁeld theory pa-

rameters. In addition, our computations provide an analytically quantiﬁable probe of the

coupling-dependent validity of hydrodynamics at early times in a simple model of heavy-

ion collisions, which is an observable closely analogous to the hydrodynamization time of

a quark-gluon plasma. We ﬁnd that to third order in the hydrodynamic expansion, the

convergence of hydrodynamics is improved and that generically, as expected from ﬁeld

theory considerations and recent holographic results, the applicability of hydrodynamics is

delayed as the ﬁeld theory coupling decreases.

Keywords: Holography and quark-gluon plasmas, AdS-CFT Correspondence, Quark-

Gluon Plasma

ArXiv ePrint: 1707.08812

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP10(2017)110

JHEP10(2017)110

applicability of hydrodynamics to the infrared (IR) dynamics of various systems without

quasiparticles has been ﬁrmly established much more recently through the advent of gauge-

gravity duality (holography) [25–28]. In inﬁnitely strongly coupled CFTs with a simple

holographic dual, the mean-free-time is set by the Hawking temperature of the dual black

hole, t

mft

∼ ~/k

T .

In a CFT in which temperature is the only energy scale, this implies

that hydrodynamics universally applies to the IR regime of strongly coupled systems for

ω/T  1, where the frequency scales as ω ∼ 1/t

hyd

(and similarly for momenta, q/T  1).

A natural question that then emerges is as follows: how does the range of applicability

of hydrodynamics depend on the coupling strength of the underlying microscopic quantum

ﬁeld theory? Qualitatively, using simple perturbative kinetic theory arguments (see e.g. a

recent work by Romatschke [29] or ref. [30]), one expects the reliability of hydrodynamics to

decrease (at some ﬁxed ω/T and q/T ) with decreasing coupling constant λ. The reason is

that, typically, the mean-free-time increases with decreasing λ. From the strongly coupled,

non-perturbative side, the same picture recently emerged in holographic studies of (inverse)

coupling constant corrections to inﬁnitely strongly coupled systems in [31–34],

which we

will further investigate in this work.

In holography, in the limit of inﬁnite number of colors N

of the dual gauge theory,

inverse ’t Hooft coupling constant corrections correspond to higher derivative gravity α

corrections to the classical bulk supergravity. In maximally supersymmetric N = 4 Yang-

Mills (SYM) theory, dual to the IR limit of ten-dimensional type IIB string theory, the

leading-order corrections to the gravitational sector (including the ﬁve-form ﬂux and the

dilaton), are given by the action [37–41]

IIB

2κ

√

−g



R −

(∂φ)

−

4 ·5!

+ γe

−

W + . . .



, (1.1)

compactiﬁed on S

, where γ = α

ζ(3)/8, κ

∼ 1/N

and the term W is proportional to

fourth-power (eight derivatives of the metric) contractions of the Weyl tensor

W = C

αβγδ

µβγν

ρσµ

ρσδ

αδβγ

µνβγ

ρσµ

ρσδ

. (1.2)

The ’t Hooft coupling of the dual N = 4 CFT is related to γ by the following expression:

γ = λ

−3/2

ζ(3)L

/8, where L is the anti-de Sitter (AdS) length scale. For this reason,

perturbative corrections in γ ∼ α

are dual to perturbative corrections in 1/λ

3/2

Another family of theories, which have been proven to be a useful laboratory for

the studies of coupling constant dependence in holography, are curvature-squared theo-

ries [31–34, 42, 43] with the action given by

2κ

√

−g



R − 2Λ + L



+ α

µν

+ α

µνρσ



. (1.3)

We will henceforth set ~ = c = k

= 1.

Aspects of the coupling constant dependent quasinormal spectrum in N = 4 theory were ﬁrst analyzed

in refs. [35, 36].

– 2 –

JHEP10(2017)110

Although the dual(s) of (1.3) are generically unknown,

one can treat curvature-squared

theories as invaluable bottom-up constructions for investigations of coupling constant cor-

rections on dual observables of hypothetical CFTs.

From this point of view, it is natural

to interpret the α

coeﬃcients as proportional to α

. Since the action (1.3) results in

higher-derivative equations of motion, the α

need to be treated perturbatively, i.e. on the

same footing as the γ ∼ α

corrections in N = 4 SYM. The latter restriction can be lifted

if one instead considers a curvature-squared action with the α

coeﬃcients chosen such

that α

= −4α

= α

. The resulting theory, known as the Gauss-Bonnet theory

2κ

√

−g



R +



− 4R

µν

+ R

µνρσ





, (1.4)

results in second-derivative equations of motions, therefore enabling one to treat the Gauss-

Bonnet coupling, λ

∈ (−∞, 1/4], at least formally, non-perturbatively.

Even though

this theory is known to suﬀer from various UV causality problems and instabilities [47–64],

one may still treat eq. (1.4) as an eﬀective theory which can, for suﬃciently low energy

and momentum, provide a well-behaved window into non-perturbative coupling constant

corrections to the low-energy part of the spectrum. This point of view was advocated

and investigated in [31, 34, 42, 43] where it was found that a variety of weakly coupled

properties of ﬁeld theories, including the emergence of quasiparticles, were successfully

recovered not only from the type IIB supergravity action (1.1) but also from the Gauss-

Bonnet theory (1.4).

An important fact to note is that these weakly coupled predictions

follow from the theory with a negative λ

coupling (increasing |λ

|).

We can now return to the question of how coupling dependence inﬂuences the validity of

hydrodynamics as a description of IR dynamics by using the above two classes of top-down

and bottom-up higher derivative theories. The ﬁrst concrete holographic demonstration of

the failure of hydrodynamics at reduced (intermediate) coupling was presented in [31]. The

same qualitative behaviour was observed in both N = 4 and (non-perturbative) Gauss-

Bonnet theory. Namely, as one increases the size of higher derivative gravitational couplings

(decreases the coupling in a dual CFT), there is an inﬂow of new (quasinormal) modes along

the negative imaginary ω axis from −i∞. Note that at inﬁnite ’t Hooft coupling λ, these

modes are not present in the quasinormal spectrum. However, as λ decreases, the leading

new mode on the imaginary ω axis monotonically approaches the regime of small ω/T .

In the shear channel,

which contains the diﬀusive hydrodynamic mode, the new mode

collides with the hydrodynamic mode after which point both modes acquire real parts in

In some cases, such terms can be interpreted as 1/N

corrections rather than coupling constant correc-

tions [44, 45]. See also [34] for a recent discussion of these issues.

It is well known that curvature-squared terms appear in various eﬀective IR limits of e.g. bosonic and

heterotic string theory (see e.g. [46]).

Note that through the use of gravitational ﬁeld redeﬁnitions, the action (1.3) and any holographic

results that follow from it can be reconstructed from corresponding calculations in N = 4 theory at inﬁnite

coupling (α

= 0) and perturbative Gauss-Bonnet results. See e.g. [42, 47].

We refer the readers to ref. [34] for a more detailed review of known causality problems and instabilities

of the Gauss-Bonnet theory.

See [65] for conventions regarding diﬀerent channels and the connection between quasinormal modes

and hydrodynamics.

– 3 –

JHEP10(2017)110

their dispersion relations. Before the modes collide, to leading order in q, the diﬀusive and

the new mode have dispersion relations [31, 34]

= −i

ε + P

+ ··· , (1.5)

= ω

+ i

ε + P

+ ··· , (1.6)

where the imaginary gap ω

, the shear viscosity η and energy density ε, and pressure P

depend on the details of the theory [31, 34]. Note also that both the IIB coupling γ and

the Gauss-Bonnet coupling −λ

have to be taken suﬃciently large in order for this eﬀect

to be well described by the small-q expansion (see ref. [34]). In the sound channel,

1,2

= ±c

q − iΓq

+ ··· , (1.7)

= ω

+ 2iΓq

+ ··· , (1.8)

where c

= 1/

√

3 is the conformal speed of sound and Γ = 2η/3 (ε + P ). In both channels, it

is clear that the IR is no longer described by hydrodynamics. To quantify this, it is natural

to deﬁne a critical coupling dependent momentum q

(λ) at which Im |ω

)| = Im |ω

in the shear channel, and Im |ω

1,2

)| = Im |ω

)| in the sound channel. With this

deﬁnition, hydrodynamic modes dominate the IR spectrum for frequencies ω(q), so long

as q < q

(λ). To leading order in the hydrodynamic approximation, in N = 4 theory, q

scales as q

∼ 0.04 T/γ ∼ 0.28 λ

3/2

T , while in the Gauss-Bonnet theory, q

∼ −3.14 T/λ

Even though these scalings are approximate, they nevertheless reveal what one expects

from kinetic theory: the applicability of hydrodynamics is limited at weaker coupling by a

coupling dependent scaling whereas at strong coupling, hydrodynamics is only limited to

the region of small q/T , independent of λ  1.

Understanding of hydrodynamics has been important for not only the description of

everyday ﬂuids and gases, but also a nuclear state of matter known as the quark-gluon

plasma that is formed after collisions of heavy ions at RHIC and the LHC. Hydrodynam-

ics becomes a good description of the plasma after a remarkably short hydrodynamization

time t

hyd

∼ 1 − 2 fm/c measured from the moment of the collision [66–71]. In holography,

heavy ion collisions have been successfully modelled by collisions of gravitational shock

waves [72–79], including the correct order of magnitude result for the hydrodynamization

time (at inﬁnite coupling). Coupling constant corrections to holographic heavy ion colli-

sions were studied in perturbative curvature-squared theories (Gauss-Bonnet) in [32], which

found that for narrow and wide gravitational shocks, respectively, the hydrodynamization

time is

hyd

= 0.41 − 0.52λ

+ O(λ

) ,

hyd

= 0.43 − 6.3λ

+ O(λ

) ,

(1.9)

where T

hyd

is the temperature of the plasma at the time of hydrodynamization. For

= −0.2, which corresponds to an 80% increase in the ratio of shear viscosity to entropy

density, we thus ﬁnd a 25% and 290% increase in the hydrodynamization time [32]. Thus,

In kinetic theory (within relaxation time approximation), the hydrodynamic pole does not collide with

new poles, but rather crosses a branch cut, which on the complex ω plane runs parallel to the real ω axis [29].

– 4 –

剩余47页未读，继续阅读

评论收藏

内容反馈

weixin_38500948

粉丝: 3
资源: 915

有限耦合对Bjorken流体动力学的全息约束

全息N = 4 SYM等离子体流体动力学对有限耦合校正的敏感性

有限能量尺度下全息流体动力学的输运系数

Bjorken和Gubser流动的高阶各向异性流体力学

相对论流体动力学远非局部平衡

大D全息全息Bjorken流

随机流体动力学和扩展的保形带电流体的长时间拖尾

重离子碰撞中热磁化的Bjorken流的横向膨胀

高斯-邦尼全息术中的回潮和流体动力吸引子

关于杨米尔等离子体的流体动力吸引子

Bjorken情景中（1 + 1D）横向流的非相对论近似数值理想磁流体动力学

从自由流到热化再返回的流体力学

松弛时间近似中动力学理论的非平衡吸引子

Bjorken极化求和规则和红外安全QCD耦合

HERA数据和共线改善的BK动力学

pQCD中极化Bjorken和规则的优化确定

QCD强耦合的最大保形原理的含义

在未来的电子对撞机中对包括截面在内的库仑校正

大动量传递时三环非单重味对结构函数g1（x，Q2）的贡献

质子的自旋结构函数g1p和Bjorken和规则的检验

RHIC能区的Landau流体力学模型研究 (2010年)

红外安全QCD中的非单极性核子结构功能

横极化质子上唯一的ω介子muo产生

在Tevatron能量下质子-核碰撞中产生的粒子的高斯伪快速分布观察

远非平衡膨胀等离子体的非扰动流变行为

最新资源