Proceedings of International Conference on Physics in ... - KEK

1
0.8
0.6
0.4
0.2
B 2 z
E 2 T
0.5 1 1.5 2 2.5 3
Qsτ
Figure 3: Evolution <strong>of</strong> the field strength as a function <strong>of</strong>
Qsτ. Evolution <strong>of</strong> a purely electric flux tube is obtained by
exchanging E and B.
τ = 0+ is found as diag(E, E, E, −E) with energy density
E, which is quite different from the equilibrium form
diag(E, E/3, E/3, E/3). Note that the longitudinal pressure
∝ E 2 T − B2 z is always negative and only approaches
to zero at later times, while the transverse pressure ∝ B 2 z is
positive.
How will this highly anisotropic configuration <strong>of</strong> strong
fields evolve into locally thermalized plasma — quark
gluon plasma (QGP)? Here we argue that the initial
glasma configuration with color magnetic fields has unstable
modes, which play a role in the system evolution.
NIELSEN-OLESEN INSTABILITY
Since late 70s it has been known that a uniform magnetic
field configuration in SU(2) gauge theory is unstable — the
Nielsen-Olesen (N-O) instability[1], which we review here.
We decompose the SU(2) gauge fields into diagonal and
<strong>of</strong>f-diagonal parts, and assume a homogeneous magnetic
field in the 3rd color direction:
L = − 1
4 fµνf µν − 1
2 |Dµφν − Dνφµ| 2 + igf µν φ ∗ µφν
+ 1
4 g2 (φµφ ∗ ν − φνφ ∗ µ) 2 , (1)
where Aµ ≡ A 3 µ, φµ ≡ (A 1 µ + iA 2 µ)/ √ 2 and fµν =
∂µAν − ∂νAµ. D = ∂µ + igAµ is the covariant derivative
w.r.t. Aµ. Regarding Aµ gauge theory, the <strong>of</strong>f-diagonal
parts φµ behave as charged massless vector fields. They
form the Landau levels EN = (2N +1)gB (N = 0, 1, · · · )
in a constant magnetic field background B within the linear
approximation with φ being small fluctuations. However,
the most important observation is that the third term
in Eq. (1) gives the Zeeman splitting ±2gB for spin ±, resulting
in the mode energy
ω 2 = p 2 z + (2N + 1)gB ± 2gB , (2)
where we see that the lowest mode has the mass term with
a wrong sign −gB. This is the N-O instability. The growth
rate <strong>of</strong> the instability depends on the longitudinal momentum
pz as γ = √ gB − p 2 z. The instability appears even at
pz = 0, which is qualitative difference from the pz dependence
<strong>of</strong> the Weibel instability known in plasma physics.
Figure 4: Schematic picture for glasma with fluctuations,
between two receding nuclei.
GLASMA INSTABILITIES
As previously mentioned, the boost-invariant background
never acquires positive longitudinal pressure,
which is necessary for system isotropization. We consider
here small fluctuations φ around the background field in the
τ-η coordinates. For simplicity, we neglect the transverse
structure <strong>of</strong> the collision event and assume transversely a
uniform magnetic background B, even though, more realistically,
the background field should have transverse structures
characterized by the scale Qs, and an electric field in
the background is also allowed.
By combining the fluctuating fields φµ(τ, η, x⊥) as
φ ± ≡ 1
√ 2 (φ1 ± iφ2) , (3)
we find the mode equation for φ ± [2, 3],
(
)
˜φ ± = 0 , (4)
1
τ ∂τ (τ∂τ ˜ φ ± ) +
EN ± 2gB + ν2
τ 2
where ˜ φ is the mode with N specifying the Landau level
and with ν the momentum conjugate to η. There is another
equation for φ ∗ which has the opposite charge to φ.
• First we note that the spin-+ mode with ν = 0 in the
lowest Landau level N = 0 has a negative mass −gB
just as in the previous section. There exists the N-O
instability in the glasma.
• Nonzero ν modes are stabilized by the term (ν/τ) 2 .
For ν is dimensionless, physical longitudinal momentum
corresponds to pz ∼ ν/τ[6], which is large at
small τ with fixed ν. The maximum value νmax for
the instability is determined by
νmax = τ √ gB ∼ τQs . (5)
In fact, these points were observed in a numerical simulation
for the glasma[7], although the proportionality constant
in the second point was found very small there. At
the same time, we notice that the expansion generally delays
the appearance <strong>of</strong> the instability.