Ⅱ-5. Hadron Physics (Theory)

RIKEN Accel. Prog. Rep. 47 (2014)

General conditions ensuring relativistic causality in an effective field
theory based on the derivative expansion†
Y. Minami∗1 and Y. Hidaka∗1We discuss the general conditions ensuring relativistic causality in an effective field theory based on thederivative expansion. Relativistic causality implies
that the Green function vanishes in a space-like region. It is known that a naive derivative expansion
violates causality in some cases such as the first-orderrelativistic dissipative hydrodynamics. We note thatthe Lorentz covariance and time and space derivativesof equal order do not ensure causality. We derive thegeneral conditions for causality that should be satisfied
by any effective theory consistent with special relativity.
Derivative expansion is a useful tool at a low-energy
scale and widely used in effective theories. Chiralperturbation theory is a good example of a successful low-energy effective theory in hadron physics1) .
From a modern perspective, hydrodynamics is also a
low-enegy effective theory; the leading-order hydrodynamic equations are called Euler equations, and thefirst-order hydrodynamic equations are called Navier–Stokes equations.
Causality is an important concept in physics. In relativistic systems, the propagation of any information
cannot exceed the speed of light (relativistic causality).
However, it seems that a low-energy effective theory
in medium is incompatible with relativistic causality.
For example, in first-order relativistic hydrodynamics,
shear and heat flows violate causality because the firstorder equation has the form of a diffusion equation2,3) .
We note that the first-order hydrodynamic equation isLorentz covariant, i.e., the covariance does not ensure
the causality. It is argued that the acausality of thefirst-order hydrodynamics originates from the difference between the order of time- and space-like derivatives in the equation of motion. In the diffusion equation, the time-like derivative is of the first order, while
the space-like one is of the second order. However, wenote that equality in the order of time- and space-likederivatives does not ensure the causality. For example,
let us consider the following equation:
µ2µµνµ ντ (u ∂µ ) +u ∂µ +Γ(η −u u )∂µ ∂ν n(xµ ) = 0,(1)where n(x) is a scalar density, uµ is a constant timelike vector, η = diag(−1, 1, 1, 1),τ is the relaxation
time, and Γ is the diffusion constant. In this equation, the time-like derivative is of the same order as
the space-like one. This equation has the form of
the telegraphic equation such that the propagation is†∗1

restrictedin the region definedby vxt > xs , wherev = Γ/τ , xt = u·x, and xs = −(η µν − uµ uν )xµ xν .
If Γ < τ , causality is satisfied because the velocity issmaller than the speed of light, i.e., v < 13) . However,if Γ > τ , the propagation speed exceeds the speed oflight. Furthermore, if Γ = 0, causality is not violatedeven though the order of the time and space derivatives are different. Therefore, the equal order of space
and time derivatives in the equation of motion does
not ensure in itself that the Green function is causal.
What ensures the causality in general?
The purpose of this paper is to derive the conditions ensuring relativistic causality in an effective theory based on the derivative expansion. We will consider the retarded Green function in a scalar theory attree level, i.e., thermal and quantum fluctuations will
not be taken into account. In this case, the retarded
Green function in the derivative expansion is generally
written as a rational function in the momentum space:
GR (ω, k) =Q(ω, k),P (ω, k)(2)where P (ω, k) and Q(ω, k) are polynomials in ω andk:
P (ω, k)= pn (k)ω n + pn−1 (k)ω n−1 + ... + p0 (k), (3)Q(ω, k)= qm (k)ω m + qm−1 (k)ω n−1 + ... + q0 (k). (4)Here, n > m, and pj (k) and qj (k) are the polynomials
in k. Because we assumed isotropy, the Green function
turns becomes a function of k ≡ |k|. The relativistic causality implies that the retarded Green function
must vanish in the space-like region. Therefore, we derive the general condition ensuring Eq. (2) vanishes inthe space-region, which is given by ω(k)  < 1 and lim Im ω(k)  < ∞,lim Re(5)k→∞k→∞kk and the condition that pn (k) must not depend on k.
Here, ω(k) is a pole of Eq. (2). These conditions ensurecausality in effective theories based on the derivative
expansion, and they are the main results of our paper.
References

1) S. Weinberg, Phys. Rev. Lett. 17, 616 (1966); Physica A 96, 327 (1979).
2) W. Israel and J.M.Stewart, Ann. Phys. (N.Y.) 118(1979), 341.
3) S. Pu, T. Koide, D. H. Rischke, Phys. Rev. D 81, 114039(2010).

Condensed from the article in arXiv:1401.0006 [hep-ph]
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