Characterizing hydrodynamical fluctuations in
heavy-ion collisheavy-ions from effective field theory approach
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Citation
Lau, Pak Hang Chris, Liu, Hong, and Yin, Yi. "Characterizing
hydrodynamical fluctuations in heavy-ion collisions from effective
field theory approach." Nuclear Physics A 982 (February 2019):
923-926 © 2018 The Author(s)
As Published
http://dx.doi.org/10.1016/j.nuclphysa.2018.10.056
Publisher
Elsevier BV
Version
Final published version
Citable link
https://hdl.handle.net/1721.1/124623
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
XXVIIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions
(Quark Matter 2018)
Characterizing hydrodynamical fluctuations in heavy-ion
collisions from e
ffective field theory approach
Pak Hang Chris Lau
a, Hong Liu
a, Yi Yin
aaCenter for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139
Abstract
Hydrodynamic fluctuations are found to be important in many situations such as cosmology, condensed matter system and heavy-ion collisions. Recently, an effective field theory for fluctuating dissipative hydrodynamics has been devel-oped in 1511.03646 and 1701.07817. We apply this theory to investigate nontrivial consequences of hydrodynamic fluctuations on the Bjorken expanding quark gluon plasma (QGP). In particular, we explore nontrivial effects due to in-teractions among hydrodynamical variables and noises which are systematically incorporated in the present framework but are not fully captured in conventional approaches.
1. Introduction
Relativistic Heavy Ion Collider (RHIC) in Brookhaven National Laboratory provides a testing ground for QCD. Collisions of heavy ions at relativistic speed create quark gluon plasma (QGP) which is the high temperature, high density phase of the QCD phase diagram. QGP behaves in a surprising way in which its evolution can be modelled as a nearly ideal fluid [1]. Therefore, hydrodynamics provides a useful framework to study the collective motion and its fluctuations of QGP. One of the simplest models is the Bjorken flow [2]. The Bjorken flow describes a static fluid in a Boost invariant background which agrees well with experimental measurements.
In the QGP system, thermal fluctuation of the energy density can be important. It gives a significant correction to the background. One of the prominent effects of the thermal fluctuation is the non-analytic behaviour of the two point correlation function of the stress tensor G(ω) ∝ ω3/2. It generates a
”long-time tails” feature of the stress tensor [3, 4]. There exist many attempts in including the effect of thermal fluctuation [5], but a systematic approach to deal with this kind of stochastic noise at higher order is still lacking. The purpose of this paper is to apply an effective field theory (EFT) approach to study noise correction in a controlled and systematic way. The EFT approach allows one to write down an effective action satisfying all the symmetry constraints expected from a microscopic theory. The effective action is expressed order by order as a derivative and noise expansion.
Nuclear Physics A 982 (2019) 923–926
0375-9474/© 2018 The Authors. Published by Elsevier B.V.
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https://doi.org/10.1016/j.nuclphysa.2018.10.056
With the effective action, one can study noise correction of the hydro-system perturbatively with loop diagrams and reproduce the known result from the conventional approach. In this work, we attempt to study the effect not captured by the perturbative calculation. We propose a way to study potential non-perturbative effect in hydrodynamics by computing the one-particle irreducible effective action of the system by integrating out the noise.
2. Effective hydro action
Hydrodynamics is typically formulated at the level of conservation equations of the stress tensor and currents
∇μTμν= 0 , ∇μJμ= 0 , (1)
with a supplement of constitutive relations. The leading order describes an ideal fluid and higher order trans-port coefficients such as shear viscosity η and bulk viscosity ξ enter at the first order derivative expansion.
The goal of this approach is to formulate a Lagrangian description of hydrodynamics to treat non-equilibrium system by systematically incorporates both the hydrodynamical degrees of freedom and fluc-tuation. A natural framework for treating non-equilibrium physics is the Schwinger-Keldysh formalism (or Closed Time Path, CTP) [6, 7]. We will follow the EFT approach developed in [8, 9].
The basic idea of this formalism is to consider the generating functional W[A1μ, A2μ] of a many-body system where (Arμ, Aaμ) are the source fields for the conserved quantities. By demanding gauge invariance
and locality, Crossley et.al. [8] identified the corresponding hydrodynamical degrees of freedom for the conserved quantities. For the purpose of this work, we will only focus on the conservation of energy-momentum. It is sourced by the background metric (grμν, gaμν), the hydrodynamical degrees of freedom
Xμr(σ) are interpreted as a map from the fluid co-moving coordinates σAto the physical spacetime xμ. The other dynamical field Xμais interpreted as the noise. The generating functional can be expressed as a path integral over these degrees of freedom.
eW[grμν,gaμν]=
DXrDXaeiShydro(Xr,gr;Xa,ga), (2)
where Shydrois the effective action of hydrodynamics with a conserved energy-momentum.
With the identification ofσAas the fluid spacetime coordinates, one expects the coordinates enjoy the following re-parametrisation invariance which provides a definition of fluid. These symmetries are referred as spatial and time diffeomorphisms.
σi→ σi(σi
), σ0→ σ0 (3)
σ0→ σ0= f (σ0, σi
), σi→ σi (4)
These diffeomorphisms have a clear physical interpretation. Spatial diffeomorphisms corresponds to rela-belling the fluid elements while time diffeomorphisms corresponds to the freedom of individual fluid element having its own internal clock.
The effective action is constructed as an expansion of the noise field Xaand derivatives satisfying the symmetries above. For the purpose of this paper, we only keep the effective hydro action up to second order in the a-fields. Shydro= ddx√−g 1 2T μνG aμν+2iβ−1ηΔμαΔνβGaμνGaαβ , (5)
where Gaμν= gaμν+ ∇μXν+ ∇νXμ,Δμν= gμν+ uμuν,β is the inverse temperature and η is the shear viscosity. Note that we have set the bulk viscosity to zero for simplicity.
P.H.C. Lau et al. / Nuclear Physics A 982 (2019) 923–926
3. Brownian Motion
The computation in the effective hydro action is complicated, so we demonstrate the idea by using a simple model of Brownian motion here. The effective Lagrangian for the toy model is
L = xa(∂2t + ν∂t)xr− xaF(xr)+
iκ
2x
2
a (6)
where F(xr) is a polynomial of xr. The equations of motion for xrand xaare
f≡ (∂2t+ ν∂t)xr− F(xr)= 0 , (7)
xa= 0 . (8)
We expand the field to quadratic order in fluctuation and identify the corresponding components of the matrix elements as before.
A= −xaF , B=(∂2t+ ν∂t)− F ∗ = δ f δxr , C= iκ , (9) M= −i A B B∗ C , (10)
where the fluctuation of the Lagrangian at quadratic order is expressed using the matrix M as δL(2)= i
2δx
TMδx , (11)
withδxT= (δx
rδxa) and∗denotes the conjugate operator obtained by integrating by part and F
= ∂2F ∂x2 r x r=xr . Note that the inverse of the matrix M at zeroth order of xagives the symmetrised (DS), retarded (DR) and advanced (DA) Green’s functions.
0 B(0) B∗(0) C −1 = DS DR DA 0 , (12)
where the subscript denotes the part of the terms containing no xa.
3.1. Background field method
One can compute 1PI effective action by adding a source term with the current defined as
Jr/a= −δxδL
a/r xa/r=xa/r
, (13)
where x= x is the expected value of x (background field) after including all fluctuations. The action is then
S [x, J] =
dtL[x] + xrJa+ xaJr. (14) Next, we expand the field about a background value x= x+δx and the 1PI effective action Γ[x, J] is obtained by integrating outδx. eiΓ[x,J]= Dδx expiS [x+ δx, J] (15) = exp i dtL[x] + xr/aJa/r+2iln det M (16) = exp i dtΓ[x, 0] + xr/aJa/r . (17)
The equation of motion is then obtained by variation of xr/aand set Jr/a= 0. It is modified to
(∂2 t+ ν∂t) ¯xr− F( ¯xr)= Tr DSF( ¯xr) , (18)
4. 1PI effective action of hydrodynamics
The 1PI effective action can be computed similarly as in the toy model of Brownian motion. We start with (5) and the resulting equation of motion is
∇μTμν(Xr)= i 2 −DS αβAαβν(1) + D R αβBαβν(1) + D A αβB∗(1)αβν , (19)
where (DS, DR, DA) are as defined in (12). The matrix elements A, B, B∗are as in (10) and only the terms containing one Xaare kept, for example
Aαβν(1) = δA αβ δXνa X a=0 . (20)
The equation of motion receives contribution coming from the noise and is accompanied by the equations of the Green’s functions.
5. Conclusion
We presented a proposal to study non-perturbative effect in hydrodynamics. By using an effective action of hydrodynamics formulated in the Schwinger-Keldysh contour, we computed the 1PI effective action of this theory. The conservation equations of the system receive correction coming from integrating out the noise in the path integral. These modified conservation equations form a set of coupled equations with the differential equations of the Green’s functions.
6. Acknowledgement
P.H.C. Lau is supported by the Croucher Foundation. Y. Yin is supported by Office of Science, Office of Nuclear Physics of U.S. Department of Energy under grants Contract Number DE-SC0011090. H. Liu is supported by the Office of High Energy Physics of U.S. Department of Energy under grant Contract Number DE-SC0012567.
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