クォーク・グルーオン・プラズマ中での粒子生成と熱化の諸問題

福嶋健二（東京大学）
非平衡系の物理学ー階層性と普遍性ー!12010年~2015年くらいでQGP分野の皆が考えていたことをレヴューします
熱化の諸問題について：RoPP (arXiv:1603.02340)

2015年〜皆の関心はカイラル物理へ
外部電磁場・回転効果：PPNP (arXiv:1812.08886)

このトークではQCD特有の問題には言及しません!2

こんな物理を考えたい
高密度「場」“グルーオン飽和”
した原子核

高密度
「場」

衝突
時間発展

熱的状態
「粒子」
量子補正の強い非線型性に

流体力学と状態方程式
でよく記述される状態“QGP”

より原子核の個性が失われる

古典と量子を時間発展でつなぐ物理
Dec. 28, 2018 @ YITP

!3

熱化へのステップ
Fukushima, RoPP (2016)

流体化：理論的には面白い問題だが現象としてwell-definedではない
粘性項をたくさん入れる（再和する）とより早く流体化（ここではあまり議論しません）
Dec. 28, 2018 @ YITP

!4

初期条件 = 負の圧力
カラー電磁場の伸びた状態

GLASMA: Lappi-McLerran (2006)
Flux-tubes: Fujii-Itakura (2008)

「縦」方向
負の圧力
「横」には正の圧力

引き伸ばされると場のエネルギーが増加
膨張を抑える力がはたらく（負の圧力）
Dec. 28, 2018 @ YITP

!5

bative estimate (for the coupling g = 0.5 that was a
なぜ「負の圧力」が問題なのか？
n).
an equivalently rewrite (80), using s ⇠ @P/@T ⇠ 4llows:1st-order hydro + boost inv. + conformal sym.
PL
3⌧ T 16(⌘/s) 大きな「ずれ粘性」=,
を取り込む必要PT
3⌧ TEvolution+ 8(⌘/s)to the Quark-Gluon Plasma

Dec. 28, 2018 @ YITP
Figure11.

Viscous Hydro

GlasmaBoltzmann-Vlasov Transport

lies PLずれ粘性について再和/PT ⇡ 0.5 for initial ⌧0 ⇠Anisotropic
0.5 fm/c
and T0 ⇠ 0Hydrodynamics
1d, at した流体理論を作ればRHIC energy. The reduction of PL /PT sug
全て解決すると信じて
a motiveto pursue for an improved hydrodynami
いる人たちもいる
te PL /P(desirable)
pictureisT ⌧ 1. The schematic
0
Strickland,
etc…
adapted
fromHeinz,
Strickland’s
picture. It is usuall!6

Schematic picture of the evolution of PL /PT a

「場」の古典近似
初期状態は高密度の「場」で与えられるので古典場の運動方程式を解いたらいいのでは？
(cf. TDHF)
2000年〜 Venugopalan, Krasnitz, Nara, Lappi, etc…

PLはせいぜいゼロに近づく（相互作用が切れる）
古典軌道のまわりに量子揺らぎを手で入れる
2005年〜 Venugopalan, Romatschke, Gelis, Fukushima, etc…
2007年〜 Berges, Sexty, Scheffler, Schlichting, etc…
Dec. 28, 2018 @ YITP

!7

「場」の半古典近似
古典統計シミュレーション
Classical Statistical Simulation (CSS)

初期量子

古典軌道

揺らぎ
初めは皆けっこう適当に白色ノイズとかやってました
そのうち理論的な導出法がわかってきました
Dec. 28, 2018 @ YITP

!8

2
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. For
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wha
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eouldvariables
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oned,
however,that
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could thetakeideainquantumfield
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mple
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heclassical
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e useful W
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cal particle
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H(x, p) = d x[53].
hx 12 x|Ĥ|x + 12 xi e ip· x/~ .
pureof the one-dimensional
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wave-function
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unctionisdetermined
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h
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! p ). of= H(p,
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sina probability@p @x @xdistribution@p W (x, p; fort) , classica(43)
thebrackets:
Wignerfunction
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sson
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e variables
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e classical Hamiltonian
appearingabove
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+
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oned, however,that
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Z
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1
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when
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naiv
x|
Ĥ|x
+
xi
e
.
H(x,
p)
=
d
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ere is no term of O(~). Therefore,
at
least
at
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O(~)
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古典時間発展ation
as a probability distribution needs
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It canbe proved
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function has all the quantum e↵ects
andtheclassical
equationsofmotionDec. 28, 2018 @ YITP
hen expand
the aboveequationof motionin termsof ~ tofind that, !9atwhic
theWingerfunction(calledtheHusimifunction)isalwaysnon-negative,
intact. This observation is the theoretical foundation of the classical statistica「場」の半古典近似

「場」の半古典近似
古典統計シミュレーション
2

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初期量子

古典軌道

揺らぎ
初期量子揺らぎは初期状態におけるWigner関数で与えられる → 基底状態ならゼロ点振動
Dec. 28, 2018 @ YITP

!10

量子トンネル
古典統計シミュレーションは量子トンネルを記述できるのか？
一般的な枠組みできちんと調べた人は(案外?)いない
Schwinger機構を正しく記述することは知られている
Gelis-Tanji (2013)

一様な電場中で粒子・反粒子が対生成する（負エネルギーから正エネルギーへのトンネル）
Dec. 28, 2018 @ YITP

!11

（話は逸れるが）Schwinger機構
負エネルギーから正エネルギーへのトンネル
ほとんど１次元ポテンシャル散乱問題と一緒

⇠e

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電場
時間
単なるアナロジーではなく確率過程量子化を使うと量子力学の散乱問題にマップされます
Fukushima-Hayata (2014; see v1)
Dec. 28, 2018 @ YITP

!12

the present case. A constant A3 amounts to a shif
nterpreted as a different frame thechoice.
It is most
momentum
p3 , which is interpreted as a differe
which the particles and antiparticles
naturalmeasured
to sit in at
the frame in which the particles an
p3 is zero. The gauge potential
the areelectric
x0 and= +1
at rest if their p3 is zero. The gaug
field are sketched in fig. 4.

（話は逸れるが）Schwinger機構

場の量子論的な取り扱い
x0

out-state

1/ω

A3

x0

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E

E

in-state

A3

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0

x0

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Figure 7: ofProfileof thetaken
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Ref.
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n gauge potential
andassociated
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違う状態
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粒子の定義が異なる
Figureelectric
4: Sketchof the chosen gauge potential andelectric field has a peak at x0 =as0 awhose
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Dec. 28, 2018 @ YITP
!13

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場の量子論的な取り扱い
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一様な電場の問題ではSchwinger-Keldysh形式, represents
the spin-magnetic coupling.
より簡便な計算方法がある
d to reexpress [A] in terms of the proper-time integration,or relation,
ループ計算で出てくるtrlnを書き直す (固有時間積分)✓ ◆ Z 1A
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（話は逸れるが）Schwinger機構 2場の量子論的な取り扱い

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z
<
0.
for
z
<
0.
et
al.
tion
in
a
reduced3<
3
ong-axis.thefrom
Ref.from[57].
Ref. [57].integration should go along c (dashed line),¯3-axis.
proper-timehin| (x) (y)|ini.
y3 > 0 (and x3 y3 < 0), > (and < , respectively) is chosen. The
ht-handedparti- For x3cfor Sintheis complexified
for zare0
3 >presented
paths
look
di↵erent
fromcontour
the contours
in Ref. [56]asbut>theseequivalent.
Figure taken
net chirality
aremay while
あたかも有限温度の場の量子論のようにand < for z3 < 0.
from Ref. [57].

µfor an operator
for an operator
O(t) in theO(t)Heisenberg
in the Heisenberg
representation.
representation.
In
where
Fµ for
F
F
=
F
8EB
=
8EB
µ
any mforincludinganym including
the m ! the
0 limit.
m !This0 limit.
resultThis
is astonishing,
result is astonishing,though thisthoughis correct!
this is「固有時間」を虚軸方向にずらす
quantumquantumfield
theoryfield
calculus
theorycalculusthegenerating
thegenerating
functional
functiohet setup.
present
After
setup.
takingAfter taking
obviously
obviously
contradicts
contradicts
an expectedanexpectedrelation(82).
relation
Second,(82).theSecond,lastInterm
the last/ mtermin Eq.
/ (78)m incaE
for an operator
O(t)
in
the
Heisenberg
representation.
i
F
=
8EB
represents
anamplitudehout|ini,
andwe mustandFirst,
weclearly
must
Dec.
28,an
2018amplitude@ hout|ini,
YITP!15 (84)
1 cleaany
m 0includingthem!
0represents
limit.
This
resultisBecauseastonishing,thoughthisis behavescorrect!
Eq.
h0forhjpresumed
presumed(whichis(whichis
droppedindroppedthechiralinthelimitchiral
of
m
limit
!
0.
of
m
!
0.
the
Becausecondensate
thecondensate
likebehaves
/
m
like
, the
/
5i =
quantum distinguish,
field theory
calculusthegenerating
functionalp.
Aftertakingdistinguish,
obviously
an(82).
thethelast condensate
term/min Eq.
(78)
cananot
alwaysthetions
methodinthemethoddiscussedin contradicts
the discussed
term
in the/ expectedm
term
cancels/relation
moutcancelswith Second,out
the with
condensate
and
such
and
a combinationsuchcombination
maybesurvim

s. If the theory has one Dirac fermion and Abelian gauge bacviolationis formulated on the level of the operator identity as（話は逸れるが）Schwinger機構

2
e
µ
µ⌫↵
¯i 5
@
j
=
✏
F
F
+
2m
µ
µ⌫
↵
5
カイラル量子異常への応用
16⇡ 2
光子はヘリシティ１持ってるので対生成はフェルミオン
of parallel
E and B, therefore, the expectation values takenのカイラリティを ２変化させる（磁場が 対称性を破る）

2
e
EB
0
¯i 5 i ,@t hj5 i =
+
2mh
2⇡ 2
= 0 (which canbe checkedby explicit calculations). This
カイラル量子異常は質量にかかわらず
0
hout|@
j
|ini
=
0
t
5
ation. Let us assume that 振幅では全く見えない
we can drop the last term in the chir from trivial as we will
Then, we can derive the a2 argue later).
2eEB0⇡m /eE
Schwingerの対生成率
hin|@
j
|ini
=
e
ideration. t 5
22⇡
Copinger-Fukushima-Pu,(2018)and thetrong background of B, the dimensional
reduction PRL
occursn relations belong to the right-handed
and the left-handed
heliDec. 28, 2018 @ YITP
!16
<latexit sha1_base64="rZcvuC9jU2lhRLrsVLnAvOrpAlQ=">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</latexit><latexit sha1_base64="mJxJ6fdEWtZXRC95oqtUpmCtbDw=">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</latexit>

equavalue
or. Ine bother thelongity byw,

is apexists
o zero.
lationn-zeroero.

still. Alntertm cans for ✏
trains
We fitin themmed

2（話を戻して）圧力の等方化

density decreases approximately as ⌧ 1 . Moreover, theresult are always very close to the LO results, suggestingthat at such small couplings the Weibel instability does
not play an important role over the timescales we haveconsidered.

Gelis-Epelbaum, PRL (2013), see also a plenary talk @ QM2013

0.01

0.1

1

2

3

4

0.01

+1

+1

1/2
1/3

1/2
1/3

0

0

0.1

1

2

3

4

PT / εPT / εPL / εPL / ε
-1

0.1

-1

LO

1.0

going
not vathem
large
fieldsτ [fm/c]

τ [fm/c]

(8)

entum

L

expected value at the LHC (g ⇡ 2).
To provide more intuition on the relevant timescales,
we also provide the time in fermis/c on the upper horizontal scale of the figures 3 and 4. The calibration of thisscale requires that one chooses the value of Qs in GeV,
here taken to be Qs = 2 GeV, a reasonable value fornucleus-nucleus collisions at LHC energies. In order tohighlight the e↵ect of the quantum corrections, we also
show the Leading Order results (dotted curves).

10.0

20.0

30.0

40.0

LO

0.1

1.0

FIG. 3. Evolution of the ratios PT ,L /✏ for g = 0.1 (↵s =8 · 10 4 ). The bands indicate statistical errors, estimatedas the result obtained before any subtraction divided by thesquare root of the number of samples. The dotted curves
represent the LO result.

20.0

30.0

40.0

Qs τ

Qs τ

weak coupling

10.0

strongcoupling
FIG. 4. Evolution of the ratios P /✏ for g = 0.5 (↵ =T ,L

2 · 102s).

古典統計近似に量子揺らぎを入れると説明できる？
Even though g = 0.5 is still a very weak coupling in

QCD, there is drastic increase in the ratio PL /PT , that
now approaches 0.60 at times of the order of one fm/c.
Dec. 28, 2018 @ YITP
!17
In both cases, ✏ = PT = PL at ⌧ = 0+ . After a time
As a consequence, ✏ falls faster than in free streamingof order Qs 1 , the longitudinal pressure turns positive and because of the energy reduction due to the work done

In
resum
work
heavy
settle
comp
How
reach
after
1 fm/
This
ues o
the p
nicallconje
exper
ion cois stroresum
Mosess h（話を戻して）圧力の等方化
古典統計シミュレーションで問題は解決したのか？

答えは「ノー」
古典統計シミュレーションには原理的制約がある
低運動量モードしか見てはいけない！
圧力やエネルギーは高運動量モードからの寄与を足し合わせると紫外発散している！
Dec. 28, 2018 @ YITP

!18

µµ @ G (k, x) = G (⌃2ik
(GarGra )G.ra ) .
rr ar ar ⌃ra⌃)raaaar
2ik @µµGrrrr(k, x) = Grr (⌃+)⌃+aa⌃(G(54) (54)対応する運動論で考える

f isis large
large now,
now,GG(p,x)x)is isdominant,dominant,
then
Dyson
equation(p,andandthenthe theDyson
equationleadsleads
to to
rrrr
equation withwiththethecollisioncollisionterm
term
approximation
given
equation
in inthisthis
approximation
given
by by
✓✓ ◆◆classicalclassical⇣ ⇣ 1 ⌘ ⌘i⌃
@f
i(⌃
⌃
)
@f
i(⌃
⌃ra )
1 aai⌃ar ar ra==f +f + + + . aa .
(55) (55)@t
2k2k
2 2 2k0 2k0
0 0@t coll
場の古典運動方程式からBoltzmann方程式を導く
coll-energies canto tothetheordinary
Feynmanrule rulein thein the-energies
can bebecomputed
computedaccording
accordingordinary
FeynmanMueller-Son,
PLB(2004) / Jeon,
PRC (2004)
ory. Because
writtenin terms
of Gofrr G/ (1/2+ f ), weory.
Because all
allself-energies
self-energiesareare
writtenin terms
rr / (1/2 + f ), we
erstand that4 the collision term for the 2 $ 2 scattering is modified from theerstandthat理論で2
the collision
term for the 2 $ 2 scattering is modified from theφ
2散乱積分を求めると
onal form of (30) into
onal formof (30)◆into✓Z
classical
2✓@f (k1 ) ◆classicalZ2=(2⇡)4 (4)4(k(4)
k3 k4 )@f (k1 )1 + k2@t=4!(k1 ) k2 ,k3 ,k4 (2⇡)(k1 + k2 k3 k4 )2$2n@t
4!(k1 ) k2 ,k3 ,k4
2$2⇥ n(f3 + 12 )(f4 + 12 )(f1 + 12 ) + (f2 + 12 )(f3 + 12 )(f4 + 12 )⇥ (f3 + 12 )(f4 + 12 )(f1 + 12 ) + (f2 + 12 )(f3 + 12 )(f4 + o12 )o(f1 + 12 )(f2 + 12 )(f3 + 12 ) (f4 + 12 )(f1 + 12 )(f2 + 12 ) , (56)
(f1 + 12 )(f2 + 12 )(f3 + 12 ) (f4 + 12 )(f1 + 12 )(f2 + 12 ) , (56)e cubic terms and the quadratic terms reproduce the correct ones in (30),
ise cubic
above terms
form haslinear terms,/ freproduce
f2the fcorrect
andextrathe quadratic
terms
ones in (30),
3 + f4
1 . Surprisingly,
enceof theselinearstructureregionis aboveform
has terms
extra change
linear theterms,/ f3 of+ theory
f4 f2in the
f1 .UVSurprisingly,
Dec. 28, 2018 @ YITP

!19

ence of these linear terms change the structure of theory in the UV region

ecause f is large now, Grr (p, x) is dominant, and then the Dyson equation leads tokinetic equation with the collision term in this approximation given by◆classical
Evolution✓to@fthe
Quark-Gluoni(⌃Plasma⌃ra ) ⇣
1 ⌘ i⌃aa
ar
=
f+
+
.
(55)
@t coll2k0
22k0
which the collision
term should takea conventional expression,✓◆ can be computed
Z according to the ordinary Feynman rule in thehe self-energies
@f
(k
)
1
1
4 (4)
4
=
(k
+ k2 ofk3Grrk4/
) |M
(k)|
theory.
Becauseall self-energies are (2⇡)
writtenin1 terms(1/2
f ),2 we
普通のBoltzmann方程式と見比べてみよう
2$2+@t!(k1 ) k2 ,k3 ,k4
2$2the 4⌫an understand thatcollisionn term for the 2 $ 2 scattering is modified
o from theonventional
form of (30) into
⇥ f3 f4 (1 + f1 )(1 + f2 ) f1 f2 (1 + f3 )(1 + f4 ) .
(3
量子Boltzmann✓◆classical
Z R
R 3
2
3
@f
(k
)
1
Here we introduceda=compact
notation;⌘dk/[2!(k)(2⇡)] withk2!(k)
=
4
(4)
f3 f=
f
+
f
f
f
f
f
f
f
f
f
+
f
f
k 1 (2⇡)
4 1
3 4 2
2 3
1 (k
2 14 + k23 4 k3 f1 f
4)@t
for massless bosons,fi ⌘ f (k4!(kthe degeneracy factor associat1 ) ⌫k2representsi ), and,k3 ,k4
2$2nwith internal quantum
When1 the system g
1 number
1 (such 1as spin degeneracy).
1
1(f3 + 2 )(f4 + 2 )(f1 + 2 ) + (f2 + 2 )(f3 + 2 )(f4 + 2 )古典近似 ⇥df /dt
equilibrated,= 0 and so the detailed balance is realized, from which the Boo
Einstein distribution functionisderivedasfollows.
Let
usrequirethatf(f1 + 12 )(f2 + 12 )(f3 + 12 ) (f4 + 12 )(f1 + 12 )(f2 + 12 )3 f4, (1+f(56)1 )(
f2 ) f1 f2 (1+f3 )(1+f4 ) = 0 for arbitrary k1 , k2 , k3 , and k4 , which is a sufficient (bnot cubic
necessary)tothe collisionterm
here thetermscondition
quadratic
in logarithm
(30),
=and
f3 f4thef1 +flet
f1 f2 f3reproduce
f1 fvanish.
+ fThen,
f4 by
f1onesftaking
3 f4 f2 terms
2 f4 the3correct2
we canshow,
hile thisaboveform has1 extra linear terms, / f3 + f4 f2 f1 . Surprisingly,◆f✓ f )◆✓◆✓◆+✓ 1 (f+fhe presence of these linear
terms
changethestructureoftheory
in
the
UV
3
4
1
2
+ f1
1 + f2
1 + f3
1 + f4 region
4
ln
+ ln
=Epelbaum-Gelis-Wu,
ln+ ln PRD (2014)
.
(3
f
f
f
f
諸悪の根源
1
2
3
4

対応する運動論で考える<latexit sha1_base64="FrLhpOnVV0x3A0niNPHELPUc/WA=">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</latexit><latexit sha1_base64="FSaISlPVrKlYOKfLEUM1k4JEHdE=">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</latexit>

This means that ln[(1 + f )/f ] should
be a conserved quantity, and so!20 should
Dec. 28, 2018 @ YITP
expressed as a linear combination of basic conserved quantities; 1, k, and k 0 = !(対応する運動論で考える
平衡解 → 散乱積分が消える分布（詳細釣り合い）
量子Boltzmann

f (k) =
<latexit sha1_base64="YO8obb87IwHEJQiUKFhjiQ4tN/I=">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</latexit>

古典近似

1e[!(k) µ]/TTf (k) =
!(k)

µ

1

1
2

<latexit sha1_base64="rk4kuFDjEmpAtjaxER+Iebn5TZQ=">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</latexit>

古典近似はPlanck分布に対するRayleigh-Jeans近似
Einsteinが指摘したようにRJ近似はエネルギーが紫外発散
高密度でよい近似を高運動量領域まで使ってはいけない
Dec. 28, 2018 @ YITP

!21

HSK ( ± , ⇡± ) ⌘ H( + , ⇡+ ) H( , ⇡ )HSK ( ± , ⇡± ) ⌘ H( + , ⇡+ ) H( , ⇡ )˙ L( )
H(
,
⇡)
⌘
⇡
˙
改良しようとした人たちはいた
H( , ⇡) ⌘ ⇡L( )the Hamiltonian
Hamiltonianininthe
theSchwinger-Keldysh
Schwinger-Keldysh
formalism.
ss theformalism.
henecessity
havingtwo
twofields
fields + +and
and is ais consequencea consequenceof quSchwinger-Keldysh形式で考えよう
Gelis of quantu
e necessity
ofofhaving
Zevolution
anics:
becausethetheevolutionofofthe
the
field
is notdeterministic
to qunics: because
field
is notdeterministic
duedueto quantu
ations,hO[mayfollowpathforfor
wavefunction]i =follow
D 0±⇢[di↵erent
0± ]ations,itit maya adi↵erent
paththethewavefunction
andandforex conjugate.
conjugate.
theclassicalclassical
limit
~!
these
fieldswould
star
Z InInthe
Z limit
ex
~R!
0, 0,
these
twotwofields
would
start
fr
i ⇡+ ˙ along⇡ ˙identicalH( ± ,⇡trajectories.
+ along± ) trajectories
mon value,
value,
andevolvedeterministically
identicalmonandevolvedeterministicallyT⇥
D ± D⇡± e
O[ + ](t0 )=the0± classicalmentsuggests±that
that
theclassicallimit
limit
more
easilyassessedby intro
ent suggests
is is
more
easilyassessedby introduc
llowing
llowing new
newfields,
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22

⇡+
+⇡
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+⇡
⇧⇧
''⌘⌘ +22, ,, ,⌘ ⌘+ +⇧ ⇧⌘ ⌘⇡+⇡+ ⇡ ⇡. .

classical
thethefield
smallerthanclassicallimit,
limit,oneonethusthusexpects
expectsthat
thatfield is muchis muchsmallertht
Dec. 28, 2018
@ YITP
!22
'.
After
this
redefinitionofthefields,theLagrangiandensity
becomes'. After this redefinition of the fields, the Lagrangian density becom

改良しようとした人たちはいた
Schwinger-Keldysh形式で考えよう
hO[ ]i =Z⇥

D 0± ⇢[ 0± ]ZRi ⇡+ ˙ + ⇡D ± D⇡± e˙H( ± ,⇡± )± (t0 )= 0±<latexit sha1_base64="dkbpqL+NVo2HLq5Izs5oAxudKbM=">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</latexit>⇡+ ˙ +<latexit sha1_base64="5/HkSoqotdc7pOhPu3UJvLK9jHQ=">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</latexit>

Z

Gelis

· · · = ⇧' ˙ + ⇧ '˙⇧' ⇧ + r · r'

6'3

O[ + ]

24' 3

この項さえなければσを積分できる！
Dec. 28, 2018 @ YITP

!23

改良しようとした人たちはいた
Schwinger-Keldysh形式で考えよう

Gelis⇡+ ˙ +'3<latexit sha1_base64="5/HkSoqotdc7pOhPu3UJvLK9jHQ=">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</latexit>⇧· · · = ⇧' ˙ + ⇧ '˙⇧' ⇧ + r · r'を積分すると'˙<latexit sha1_base64="6DKitDqW8nOCFE9e5GJste/XfxE=">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</latexit><latexit sha1_base64="owlWPAdMvachyskHDS9UouSvOXE=">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</latexit><latexit sha1_base64="mQR9aAOSHAIelJ/OZHcTzGCl4Ag=">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</latexit>

を積分すると

˙'
⇧

6

24

' 3

⇧' = 0'+<latexit sha1_base64="WuNfRQLFhan/f/juel2r9QD3HVU=">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</latexit>

3

6

' =0

これはHamiltonの古典運動方程式に他ならない
初期揺らぎは密度行列で与えられる → CSSの導出
Dec. 28, 2018 @ YITP

!24

改良しようとした人たちはいた
Schwinger-Keldysh形式で考えよう

Gelis⇡+ ˙ +'3<latexit sha1_base64="5/HkSoqotdc7pOhPu3UJvLK9jHQ=">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</latexit>· · · = ⇧' ˙ + ⇧ '˙⇧' ⇧ + r · r'

6

24

' 3

この項を入れた数値計算の枠組みはむかし
複素ランジュバン方程式みたいな形で作ってみたんですが、数値計算がすぐに不安定になって全く使い物になりませんでした・・・
腕に覚えのある人は頑張ってみてください。
Dec. 28, 2018 @ YITP

!25

熱化の話の続き
積分量をみるのはよくないので分布をみよう

Inverse Particle Cascade

Energy Cascade

ボルツマン方程式の「固定点」は熱分布だけではない(Kolmogorov-Zakharov spectrum)Dec. 28, 2018 @ YITP

!26

+
=0,
k.
(90)
@t k @ksmall
and energy damping
iven collision kernel of, for example, 2 $ 2 scattering, we can identify theven collision kernel of,for example,2an$ 2flow
scattering,canidentify
the
value
where
the energy
flux iskwe
givenbyinterval,
⌧
k
⌧
k
to
de
+
nd
there
may
sometimesbemultiplesolutions;onecorresponds
to
the
e
✓
◆
Z one
Z
d there may sometimes be multiple solutions;corresponds
totheenergyk
Kolmogorov-Zakharov
(KZ)
spe
@f
and
the
other
corresponds
totheparticlecascade.
d1
and the other corresponds to the particle
cascade.
"k =dq d⌦ q!(q)easierfind⌫ ⌫using
using
continuity
equationrathersolvinisis easierfindthethecontinuity
equationratherthan thansolvingk+tion
to thetoto
Quark-Gluon
Plasma1 @tthe3
ann equation.
equation.
Letususconsider
considera scattering
process
involving
p and
(in
f
(k)
⇠
.
Kolmogorov-Zakharov
spectrumannLetascattering
process
involving
p
(in
⌫ The
Let us explain a bit more on this exponent.
karticleswaves,
andsuppose
supposefollowingscaling
properties
forenergy
the ed represents
theandnumberof spatial
dimensions.
The
energyconservation
rticlesororwaves,thethefollowingscaling
properties
fortheterm gives 2 , and the p 1 distribution fuctions (as
ssed
by
the
energycontinuity
equation:
ion
relationandthescattering
amplitude
ason relationand the scattering
For asa givencollision kernel of, fo
スケール指数を導入(amplitude
p 粒子の散乱)
together with the phase space integrals (including q@Ek ==µ@"↵ k↵of 1⌫,
may
!(µk)
!(k)
,, ,
M(µk
, µk
, due
. .2.),there.)2 ,. . . .)
(91)
!(µk)
!(k)
M(µk
,2µk
.=to
. .)µ energy-momentum
=M(k
µ
M(k
.c
1 , ksometimes
2 ,1.,.k1and+ µone= 0integral
(93
drops
o↵
@t
@k
cascadeandtheenergy
othercorrespon
from
!(q).
the
energycascade,asaresultofwhich
theenergyspectrum
becomestconsiderconsidertheenergycascade,asaresultofwhichthespectrumbec
the energy
flow
flux
is
given
by
エネルギーの流れ（エネルギーカスケード）
dependent.
ofZof
the
distribution
the theone-dimensionalItiseasierto find⌫energyusin✓ function◆function
When
the systemreachesa steady
state
in
te
Ztermsdependent. InInterms
thedistribution
one-dimensional
e
k
@f
d 1
(p 1)d p↵+2⌫(p 1)
miswrittendown
asrealized,
then
"
must
independentof .k.usF"k =down dq
d⌦ q and
!(q)
⇠ kbe
(94
m is writtenasBoltzmannequation.
LetkZ@t collk+Z
thecomponentbut
thezerothpowerofk)dzeroth1
out)particlesorwaves,
and
su
(t)
=
d⌦
kthis
(k,(k,t) .The(92)
d !(k)f1
kbit
s explain EaE
moreon
exponent.
amplitude
squaredinthecollisio
(t)
=
d⌦
k
!(k)f
t)
.
k
index from thedispersion
energy cascade:
スペクトルが定常的なとき
relationandthegives 2 , and the p 1 distribution fuctions (assumed
to be larger
than
thescattunity
2to ⌫(pp↵ 1)+(p 1)d wher
@Eintegralskk
her with the@"
phase
space
(including
q)
lead
↵
⌫E = d +
.
=
=0
!(µk)
=
µ
!(k
p 1 Finally p↵ appea
ntegral drops@ko↵ due to@tenergy-momentum conservation.
f (k) ⇠k⌫

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Dec. 28, 2018 @ YITP
!27
!(q).
For 2 $ 2 scatteringina(3+1)-dimensional
massleWe
firstconsidertheenergycasc
When the system reaches a steady state in terms of the energy, @E /@t = 0

k.
f (k)
⇠ capillary(90)
example,
the
waves
on
deep
water
has
⌫
=
17/4
(Z
E
⌫
k dimensional analysis,small
kis sensitiveandenergydamping⌫tothestructure
of
m
ntly
from
the
Kolmogorov
spectrum
that
is
fixed
by
the
E
iven
collision
kernelof,forexample,2$2
scattering,wecanidentify
thectrum),
theacoustic
turbulence⌫2Escattering,= 9/2 (Zakharov-Sagdeev
spven
collision
kernelof,forexample,2$
wecanidentify
thevalue
an solutions;interval,
k corresponds⌧⌧onkdeeptodein (95).
For
example,thecapillary
waves
wa
+
非熱的な固定点
nsitivetomay
the sometimes
structureofmicroscopic
interactions
askseennd
there
bemultiple
one
to
the
e
of sometimes
the Filonenko
solution
belongs
totheparticlecascade.
Inthidbranchthere maybe multiple
solutions;onecorresponds
totheenergyspectrum),
the=acoustic
turbulence(KZ)⌫E = 9/2
Kolmogorov-Zakharovspepillary
waves
on
deep
water
has
⌫
17/4
(Zakharovand
the
other
corresponds
totheparticlecascade.
E
andthetheparticle
other corresponds
tocontinuity
the particle equation:cascade.
ernumberetc.
Anotherbranchofthesolution
belongs
to thepa
iseasier
to
find
⌫
using
the
continuity
equationratherthansolvinustic
turbulence⌫=
9/2(Zakharov-Sagdeev
spectrum),is easier to find ⌫ Eusing the continuity equation rather than solving
the1
considertheparticlenumbercontinuity
equatio@µLet
kbelongs
kshould
ann @Nequation.
Let
usconsider
consideracascade.
scattering
process
involving
p and
(in
f
(k)
⇠
.
Kolmogorov-Zakharov
spectrumannequation.
usascattering
process
involvingp(inolutiontotheparticle
In
this
case
we
⌫
+
=0,
k
@N
@µ
articleswaves,
andsuppose
supposefollowingproperties
forenergy
the ekkscaling
rticles@t
ororcontinuity
waves,
thethefollowingscaling
properties
for
the
@kand
numberequation:+=0,
ion
relationandthescattering
amplitude
asonand the scatteringFor
a givenkernelof, fo@t as@kandcollision
スケール指数を導入(amplitude
p 粒子の散乱)therelation
one-dimensional
particlenumberµk is theparticle
↵ ↵
where
N
is theparticlenumberand= 0 ,in !(µk)(96)
ofone-dimensional
⌫,
and
there
may
sometimes==µ!(k), k, the
M(µk
,
µk
,
.
.
.)
=
µ
M(k
,
k
,
.
.
.)
.
(91)
!(µk)
µwas
!(k)
M(µk
,
µk
,
.
.
.)
=
µ
M(k
,
k
,
.
.
.)
.
fined
the
same
as
energycascadewith!(k)removed
1 12
1
21
2
2
which are definedcascadein the samewasas otherthe energy
cascadandthecorrespontconsider
afterparticle
all
leads
to
the
energycascade,as asaµresultofwhich
the energy
spectrum
becomestionalconsidertheenergycascade,aresultofwhich
theenergyspectrumbec
numberandistheparticle
flow
flux,
k
and
(94),
that
after
all
leads
to
粒子の流れ（パーティクルカスケード）
dependent.
terms
thethefunction
the theone-dimensional
Itfunction
is easierto (93)find ⌫energy
usin
dependent.
of
distribution
one-dimensionale2termsofcascade(p+ distribution
1)↵
was as theInIn
energywith !(k)removedfrom2(p + 1)↵misis⌫written
down
as
=
d
+
.
m
writtendown
asN
equation.
Letus
同様の議論から⌫Boltzmann
.
N =d+
to
ZZ p
1
p 1
d 1
out)
particles or waves, and
su
EE
=
d⌦
k
(k,
t)
.
(92)
d !(k)f1k (t)4/32対の散乱の場合(p+
1)↵
d⌦ kthe!(k)f(k,t)spectrum.
k (t) =Hence,
particlecascaderesultsin theKZkspectrufo
article cascade
results
in
the
KZ
with
f
⇠
.
(97)
dispersion
relation and the scatt
5/3
5/3
p
1
scatteringinamasslesstheory,
while
it
was
f
⇠
k
a massless エネルギーカスケード
theory, while it was f ⇠ kfrom the energy casc↵
Generally
speaking,
theparticlecascadeoccurs
in
4/3
!(µk)
=
µ
!(k
yresults
speaking,
theparticlecascadeoccursinadirection
towardin theKZ
spectrum
with
f
⇠
k
for
2
$
2
パーティクルカスケード
and the 5/3
energy cascadetowardlarger k. Let us!28 recall
Dec.
28,
2018
@
YITP
gy
cascadetowardk. Letrecallthat thewe consider
they, while
it wasf ⇠ larger
k
from
theus
energycascade.
We
firstconsiderenergy casccascade from k

to k+ (where k⌧ k+ ) and introdu

2(p + 1)↵with !(k) removed from (93)as the energy
cascade.
N =d+p 1 非熱的な固定点
4/3
1)↵
le+ cascade
results
inthe
KZ
spectrum
with
f
⇠
k
for 2
.
(97)
1
エネルギーカスケード
大きなkに流れるsslesstheory, while it was f ⇠ k 5/3 fromthe energy cascade.
4/3 a direction
eaking,thecascadets in theKZparticle
spectrumwith foccurs⇠ k infor小さなkに流れる
2 $ 2 toward smal
パーティクルカスケード
5/3 k. Let us recall that we consider the spe
scade
towardlargerhile
it wasf⇠k
the energy cascade.
運動量 k- および fromk+ における流れを考える
clekcascade
occurstoward thesmallerktok ⌧inka+ )direction
and introduce
one-particle
energy+ (where
粒子密度は(従って密度の流れも)k- theの方がずっと大きい
rger
k.
Let
us
recallthat
weconsiderspectraland the particle flow µ± at k = k± . Then,the total quantitie
大きなkから小さなkに向かって流れる(inverse
cascade)k+)andintroducethe
one-particle
energy
!
"⌧
=!
µ
+
!
µ
,
which
leads
to
±
+ +
e flow µ±1粒子エネルギー
at k = k± . Then,
the
total
quantities are
w
(k)
は
k
の方がずっと大きい+'µ ," ' !+ µ+ .
µ , which leads to
もし粒子の流れの差より大きければ k+ で大きくなる(98) implies
that the dynamical evolution of the spectrum gene
' !+ µ+ .小さなkから大きなkに向かって流れる (direct(98)ect energy cascade (from smaller k to k+ ) andcascade)the inverse part thedynamical
evolution of the spectrum generally
gerk tok ). パリティを破るとこの限りでない
N. Yamamotoade (from smaller k to k+ ) and the inverse particle
Dec. 28, 2018 @ YITP

!29

equation to the non-perturbative Dyson-Schwing
look for the scaling
solution. This question was add
非熱的な固定点
Figure 13. Speculated distribution speof the 2PI large-N expansionof O(N) scalarthef(without expansion).
Figureis adapted
analyticalsolution is either ⌫ = 4 or 5 and the num2体の散乱積分は摂動の最低次の解析
it hasbeen concluded in [73] that
非摂動的にはいろんな指数がありえる

5.2.2. Scenario with non-thermal fixed-point As e
that例えばa certain
amount =of 4particles
fall into the⌫non-thermal.
overpopulated andBerges-Rothkopf-Schmidt,
the particle number
is appro
PRL (2008)
A similar
analysisby themeansof theDyson-Schwisituation
happens
expected
spectrumshould bin あるいは初期分布の粒子数がものすごく大きい
a puretheory inThis[113],
andthat
it wasallowsYang-Millsfor 1 $ 2 scattering.
means
we s
because avalues
scalar condensate
does not scale with the(overpopulation)ときにはBECが作られる
intermediate
might be possible.
find, Berges,
2011年〜Blaizot, Gelis, Liao,
Venugopalan, etc
Later,
the scenario
hasMcLerran,
been summarizedas
3 confirmed by the clas
1 2散乱が可能になる
Berges,and this has⌫Ebeen= ,⌫N = 1 .
When f (k) . 1 quantum2fluctuations dissipate hi
Dec. 28, 2018 @ YITP
!30
In
this
case,
there
is
anotherpossibility,
which
is
re
k
the tail behaves as f ⇠ e. In the regime whereBergesのシナリオ
非熱的固定点＋overpopulationの場合

Inverse Particle Cascade

Energy Cascade

このような解析はしばしば「乱流」と言われることもあるが、これはwave-turbulence (fluidではない)
Dec. 28, 2018 @ YITP

!31

ng Propertiesスケーリングによる分類

arious scenarios out, it is quite useful to introduce aor theその後、このような分布(固定点)にどのように
gluon distribution as a function of time. The se
近づいていくのか、という考察から時間を含む
rmed in many classical statistical simulations implies
スケーリングが提案された
for the gluon distribution:↵f (k, ⌧ ) = (Qs ⌧ ) fS (Qs ⌧ ) k? , (Qs ⌧ ) kz ,s systematicallystudied in [92] (which nicely reviews
Qsは理論(物理系)の典型的なスケール
lattice非等方的なのでβとγは別の値をとってよいdiscretization). The exponents ↵, , and cm dynamical evolution, and these are reminiscent of th
ある種の「臨界指数」のようなもの
inity of IR fixed points on the renormalization flow. T（いわゆる動的臨界指数とは違う）
rsality out of equilibrium, Dec.
andunlike the static situatio
28, 2018 @ YITP
!32
on of the universality class according to the dimension

スケーリングによる分類

Evolution to the Quark-Gluon Plasma
Authors
BMSS [32]B [93]
BGLMV [50]
KM [94]

29

↵
2/3
3/4
(3

s )/7

7/8

0
0
(1 + 2 s )/7
0

1/3
1/4
(1 + 2 s )/7
1/8

粒子数保存（粒子が壊れない限り成り立つ）
Table 1.
Di↵erent exponents according to di↵erent scenarios with one

↵

2

=

1

dimensional expansion. If complete thermalization with isotropy is achieved↵ = 0, = = 1/3 should be realized.

エネルギー保存（膨張で外に仕事をすると不成立）↵ 3
3.1. Scaling Properties=

1

To sort熱平衡前にみえるスケーリングを
various scenarios out, it is quite useful toBerges-Boguslavskiintroduce a scaling form of thesolution for the gluon distribution as a function of -Schlichting-Venugopalan,time. The self-similarity that ha
PRL (2015)
非熱的固定点と呼んでいる
been confirmed
in many classical statistical simulations implies the following scaling!33
properties for the gluon distribution: Dec. 28, 2018 @ YITP↵

結論
我々QGP研究者は5~10年間ですごく賢くなったが肝心のQGPの熱化の問題はぜんぜん解けていない
熱力学的にバルクな量を扱うには、高密度から低密度まで正しい理論的な枠組みが必要だが、今のところ高密度（古典的）領域と低密度（運動論的）領域とが、うまくつなげられていないKS形式で書くことはいつでもできるのだが数値計算に便利な形で理論が整理されていない
Dec. 28, 2018 @ YITP

!34