Master's Thesis Presentation 2013/02/19

理学研究科物理学専攻修士研究発表会

有限アイソスピン化学ポテンシャルにおける
強結合クォーク物質の基底状態

理学研究科物理学専攻鈴木克彦研究室
岩田裕平

2013年2月19日 @ Tokyo University of Science (kagurazaka)

目的原子核の5∼15倍の密度

相対論的重イオン衝突実験中性子星
http://www.rarf.riken.go.jp/rarf/rhic/acc/rhic.html http://spaceinfo.jaxa.jp/ja/supernova_explosion_neutron_star.html

■ 高密度な領域で実現される物質の状態を調べること

用いる力学：量子色力学（Quantum Chromodynamics : QCD）

■ 電磁気学

e
電子光子（電磁場）

■ 量子色力学 (QCD)
g
q
クォークグルーオン ■ 特徴
低エネルギースケールで非摂動論的な強結合をする

u c t +2/3
電荷本研究で扱うのは低エネルギー・強結合の物理
d s -1/3
b
軽重

多体に拡張すると ― ‘‘More is different’’ in QCD

■ 電磁気学

プラズマ超伝導ボーズ・アインシュタイン凝縮
■ 量子色力学 (QCD)
低エネルギーにおけるQCD相図

■クォーク・グルーオンプラズマ（高温）

□カイラル凝縮（低温・低密度）
クォーク・反クォークの凝縮
陽子・中性子の質量の起源

■カラー超伝導（高密度）
クォーク・クォークの凝縮

実験観測 ▷ QCD相図
陽子数・中性子数に差荷電中性条件より
＝uクォーク数とdクォーク数に差 uクォーク数とdクォーク数に差

陽子中性子
電荷：2/3 -1/3

u d
加速器実験高密度天体

u,d非対称性の軸
が必要

研究の概要 YI, H. Abuki and K. Suzuki, [arXiv:1206.2870];
YI, H. Abuki and K. Suzuki, [arXiv:1209.1306].

■ u, dクォークの非対称性（有限アイソスピン化学ポテンシャル）が加わった時、
臨界点近傍の基底状態がどのように変化するか調べる

■ モデルに依存しない方法としてGinzburg-Landau理論を用い、定性的に評価する

温度 T
高密度QCDでは第一原理計算不可能
第一原理計算可能臨界点モデルでアプローチされてきた領域
カ D. Toublan and J. B. Kogut, PLB (2003);
D. T. Son and M. A. カイラル凝 A. Barducci, R. Casalbuoni, G. Pettini and L. Ravagli, PRD (2004).
イ

Stephanov, PRL (2001). 縮
ラ

¯
u u d d
ル相

¯ この領域にアイソスピン化学ポテンシャルが
子
転移

間及ぼす影響を、より一般的に理解したい
中ャルクォー
イ ¯
u d ク化学
パ縮
電凝
シ
ン d µq = ( ポテン
荷テ µ µu + µ シャル
d )/2
ポ
¯
d u 化
学
ピ
ン µu
ソ
ス =
アイ µI

なぜGinzburg-Landau (GL) 理論？
■ Landau理論を使えばモデルに依存しない議論をすることができる
Landau理論微視的なモデル

◎ 対称性よりポテンシャルを導出 ◎ Lagrangian密度（＋平均場近似）
L
2 (µq , T ) 4 (µq , T ) 6 (µq , T )
Landau ( )= + +
2 4 6
2 4 6
（：秩序変数） ◎ 熱力学ポテンシャル (µq , T )

{ 2 [ 6 ], 4 }
◎ スケール変換して平面で基底状態
1

を決定 ◎ ポテンシャルを最小化する基底状態を決定
4 (µq , T ) =0 モデル依存性がある
無秩序相 150
無秩序相
2[ 6 ](µq , T )
1

T MeV
100
温度秩序相（凝縮）
=0 モデルを使えば 50
秩序相（凝縮） { 2, 4} {µq , T } 0
の写像を探せる 200 0 100 300 400
クォーク化学ポテンシャル q MeV
モデルに依存しない議論を行える D. Nickel, PRL 103, 072301 (2009).

Ginzburg-Landau理論
空間に依存
■ 秩序変数：カイラル4元ベクトルの平均場 ¯
u u ¯
d u
=( , 1, 2, 3) ¯
qq , ¯
qi 5
q
中性荷電 c 中性カイラル凝縮 π中間子凝縮（3成分）

■ ポテンシャルを秩序変数で展開（Ginzburg-Landauポテンシャル）
2 4 6 D. Nickel, PRL 103, 072301 (2009).
GL [ , ]= + ( ) + ( )
2 2 2 2 3
2 4 6 カイラル対称性
+
4,b
( )2 +
6,b
( , )2 （4成分を等価に扱う対称性）
秩序変数の微分： 4 6
6,c 6,d
SU (2)L SU (2)R
空間非一様を促進 + [ 2
( ) 2
( , ) ]+2
( )
2
6 6
対称性を‘‘あらわに’’破る
2 4 4,b 4,c
+ 2
+ 4
+ ( 2 2
) c
2
+ ( c)
2
2 c
4 c
4 c
4 U (1)I3 ;V U (1)I3 ;A
アイソスピン対称性を破る項を加えるさらに自発的
に破れると
■ 対称性より, 1 = , 2 = 3 =0 荷電πが凝縮 σが凝縮

■ GL係数 , （温度・化学ポテンシャルの関数）を関係付けていく

GL係数間の関係
YI, H.Abuki, K.Suzuki, arXiv:1206.2870 [hep-ph].
■ パラメータ ) を µI の2次まで展開
(µI
S : u, dクォークのプロパゲータ
2 cµ2I O(µ4 )
I tr[Su Sd ]
0 dµI2
4 (µI = 0) u u
>
4
=
4b 0 db µ2I 6 (µI = 0)
.
>
0 dc µ2 2 (0)
u

>
d

>
4c I
> 2 µI > 4

■ 展開係数を微視的に（右のようなループ d d
計算を用い）決定
tr[Su Sd Su Sd ]
c= 1/2, d = db = dc = 1.
µI u u d
係数が負＝有限で荷電π凝縮が存在すると
ポテンシャルを下げる
> >
> (0)
2
>

>
d d µI d d
>

>
4 >> 6
■ スケーリングして、 2 , 4 }
{ 平面でポテ
>
u d u
ンシャルを最小化する基底状態を探す .
.
.

µI = 0
アイソスピン対称（）な基底状態
点線：二次相転移
実線：一次相転移

D. Nickel, PRL 103, 072301 (2009).
■ ゼロアイソスピン化学ポテンシャル D. Nickel, PRL 80, 074025 (2009).

モデルによる解析
無秩序相
150
− 無秩序相
α2 [α6 1 ] モデルを使えば

T MeV
100 空間非一様
マッピングできる
空間一様カイラル凝縮相
50 カイラル凝縮相

温度
空間一様
0
カイラル凝縮相 0 100 200 300 400
クォーク化学ポテンシャル q MeV
空間非一様
カイラル凝縮相

■ カイラル凝縮は高密度相転移近傍でドメイン構造を作る
変分して決定
GLポテンシャルのEuler-Lagrange方程式 (z) = k sn(kz; )
z
=0

A.I. Buzdin, H. Kachkachi PLA 225 341 (1997). Euler-Lagrange方程式の解の一つ：1次元方向zへの変調を持つ

µI = 0
アイソスピン非対称（）な基底状態実線：一次相転移

■ 有限アイソスピン化学ポテンシャル
非一様性まで加えたモデルによる先行研究はない
一様
荷電π中間子凝縮相空間非一様
モデルを使えばカイラル凝縮相

温度
マッピングできる

無秩序相空間一様
空間一様凝縮相

空間非一様
クォーク化学ポテンシャル
空間非一様荷電π中間子凝縮相空間非一様
荷電π中間子凝縮相

■ 凝縮の形は？変分して決定
(z) = k sn(kz; ) (z) = k sn(kz; )
Euler-Lagrange方程式を連立

= 0, =0
z
or z

カイラル凝縮がドメイン構造を持つ解荷電π凝縮がドメイン構造を持つ解
2つの凝縮が混ざる場合は解析的に解くのは困難（数値計算によって別の形が現れる可能性がある）

まとめと結論
■ アイソスピン非対称性によって、臨界点近傍でも荷電π中
間子凝縮が誘起される一様
荷電π中間子凝縮
■ 臨界点近傍より低温度・高密度な領域では、荷電π中間子
無秩序

温度
凝縮が空間非一様なドメイン構造を作った方がエネルギー
的に優位である
非一様
一様カイラル凝縮
■ 構造が複雑化したことにより、（多重）臨界点が増加するカイラル凝縮
非一様荷電π中間子
凝縮
■ 解析的な手法によって臨界点近傍の基底状態を予言した点
で、QCD物質分野への貢献をした
本結果から予測される有限
アイソスピン密度での相図
Speculation
■ 荷電π中間子凝縮が天体で実現したら、星内部の散乱現象
に効く可能性がある

ntum Chromodynamics are presented the years, too. Including vaccum polarisation ofterms, similar but
Including β1 and higher order three light
single measurement.
books and articles, asexclusive[2,3,4,5, only quark flavours and extended and volume effects, as overall
As can be seen, the values of e.g. means vary complicated relationsmeans αsunderstand and cor-
rect for finite2
lattice spacing
for to (Q2 ), the a function of α
e between a minimum of 0.11818 and a maximum 0.11876. error ofln 2 as in equation 4, emerge. They can be s
following, only a brief summary of and of these Q results significally decreased over time, while
ativethat in the casethecalculating theirmeans and ac- the value ofµαs (MZ )today quote the smallest overall aver-
Note
QCD ”rules” of these exclusive overall errors, age. Lattice results gradually approached the world error
and of running coupling 0

cording to the numerically, such that for a given value of αs (µ2 ), cho
くりこみ群によるQCDの解析
given.out of the eight cases small error scaling factors on αs (MZ ); it is, however, ensuring to see and note that
in four
a suitable reference lattice results is the marginally the Z0 b
0

of g = 1.06...1.08 had to be applied, while in the other the world average without scale like only mass of
µ = MZ while s (Q2 ) size be total uncertainty on
cases, overall correlation factors of about 0.1, and in one different, 0 , α the smallcan of the accurately determined a
case of 0.7, had to be applied to assure χ2 /ndf = 1. Most the world average 2 naturally, largely influenced by the
is,
0 ) changes 0Z )=energy result. Q ≥ 1 GeV2 .
nce of thesaverage value αs (MZ result fromto αs (MQCD. lattice scale
notably, α
0.1186±0.0011 when omitting the lattice With S. Bethke (2009).

αs known at a specific energy scale 0.5 µ2
Λ2 = 1/(β α (µ2July 2009

有効結合定数
))
,
5 Summary and Discussion
dence is given by the renormalisation α s(Q) e 0 s

In this review, new results and measurements of αs are dimensional parameter Inelasticintroduced such that
a Deep Λ is Scattering
summarised, and the world average value of αs (MZ ), as 0.4 e+e– Annihilation
2
0

previously given in [7,28,6], is updated. Based on eight
tion 4 transforms into Heavy Quarkonia
(Q )measurements, which partly use new and improved
srecent
2
= β αs (Q2 ) . (1) 1
∂Q NNLO and lattice QCD predictions, the new av-
N3LO, 2
αs (Q ) = .
erage value is 2 /Λ2 )
0.3 β0 ln(Q
■ カラー量子数を持つ± 0.0007 ,
ansion of the β function is calculated
αs (MZ ) = 0.1184
0

pproximation to [8]:
■ 低エネルギーで非摂動論的な強結合をする
which corresponds
Hence, the Λ parameter is technically identical to th
ergy scale Q where αs (Q2 ) diverges to infinity. To g
β0 α2 (Q2 ) − βMS =(Q2 ) 9 ) MeV .
s
(5) 3
Λ 1 αs (213 ± 0.2
numerical example, Λ ≈ 0.1 GeV for αs (MZ0 ≡ 91.2
βThis4result ) − β3 α5 (Q2 ) +one obtained in(2) pre- 0.12 and Nf = 5.
2 αs (Q2 is consistent with the O(α6 ) , the =
s s
viuos review three years ago [28], which was αs (MZ ) =
0 In complete 4-loop approximation and using t
本研究で扱ったのは低エネルギーの物理
0.1189±0.0010. The previous and the actual world average
have been obtained from a non-overlapping set of single
parametrisation, the running coupling is given [9] by
0.1
（温度でいうと0∼1兆[K]）
results; their agreement therefore demonstrates a large de- QCD α s (Μ Z) = 0.1184 ± 0.0007
gree of compatibility between the old and the new, largely
2 1 1 110
improved set of measurements. αs (Q ) = − 3 2 β1 [GeV] 100
Q ln L
The individual mesurements, as listed in table 1 and β0 L β0 L
エネルギースケール
displayed in figure 5, show a very satisfactory agreement Fig. 6. Summary of measurements of αs as a function of the
, each other and with the overall average: only one respective energy1 Q. β1 curves are QCD predictions for
with scale
2
The 2 β2

第一原理計算と符号問題 - OUT IN H EAVY-I ON C OLL
P HASEDIAGRAM : F REEZE
Lattice QCD スパコンによる第一原理計算が可能
Simulations

ﬁgure taken from Blaschke

our interests
P D
A −M
NIC

分配関数 Z = Tre SQCD
= DA [detMq (µq )]e Sg
Ratios

- + - + - *0 - *0 -
p /p Λ /Λ Ξ /Ξ µ /π K /K K /π- 0 /π- K /h K /h
p
Dirac op. Mq (µq ) = i Dµ m + µq
1
π
Statistical model describes composi
det Mq (µq )† =with Mq (freeze
det few µq )
Mq (µq )† = i µ Dµ m + 0 µq = 5
Mq ( µq ) Heavy-Ion Collisions
5
det Mq complex (µq = 0) ∞
gi
T =1
174 MeV ln Z[T, V, {µ}] = ±V dp
期待値
-1
O = f.o.
µf.o.=Z MeV
46
DA [det Mq ] e Sg
complex! i
2π 02

QCD臨界点(2 ﬂavor) H. Fujii, M. Ohtani (2004)

静的臨界性→Landau理論
■ chiral limit (m = 0) ■m≠0

SU (2) SU (2) O(4)
( , 1, 2, 3)

二次

T
二次

c hi chiral-symmetric h≠0
sym
me ral- qq = 0
tr y -
bro
qq = 0 ke n 一次
一次

µq

凝縮が空間一様と仮定した基底状態実線：一次相転移

■ ゼロアイソスピン化学ポテンシャル ■ 有限アイソスピン化学ポテンシャル

一様荷電π中間子凝縮相
無秩序相

無秩序相

一様カイラル凝縮相一様カイラル凝縮相

6次までのLandau理論と等価（例 : spin1-Ising）

温度
150
T MeV
温度

100

50

0
0 100 200 300 400
クォーク
q MeV クォーク化学ポテンシャル
化学ポテンシャル

凝縮は一様としていいのか
๏ 非一様カイラル凝縮からの示唆@カイラルリミット
☞[fm]スケールの変調がある相 E. Nakano and T. Tatsumi, PRD (2005).
D. Nickel, PRD (2009);
S. Carignano, D. Nickel and M. Buballa, PRD (2010).
☞低次元（一次元）変調が好まれる H. Abuki, D. Ishibashi and K. Suzuki, PRD (2012);
S. Carignano and M. Buballa, arXiv (2012).
温度
!→0 σ(x)

amp
臨界点（リフシッツ点）
z
二次 -amp
二次
二次
!=1 σ(x)
amp sin(kz) ( 0)
z
一様非一様 0<!<1
σ(x)
-amp amp
tanh(kz) ( 1) z

(x) = k sn(kz; ) -amp

Why inhomogeneous?
123706 L ETTERS K. M. SUZUKI e

■ 物性理論においても示唆 FFLO state
2 1=2
locity and vF0 ¼ hv ik where hÁ Á Áik 0
(a)

free energy difference [10 ]
-2
rface ■ フェルミ面の”ズレ”によって生じる
average. We assume a magnetic
-axis. The Eilenberger units of R0 for -0.1
magnetic ﬁeld are used.19,20) The order
atsubara frequency !n are normalized Abrikosov
L=30
L=27
-0.2 L=200 L=25
L=100 L=23
L=75 L=21
L=50 L=20
conditions, the order parameter is L=41 L=19
L=17
L=35
-0.3
0.97 0.98 0.99 H/Hc21
picture taken from Machida
X
カイラル凝縮でも
yÃ
0 N0 T h f þ f ik ð2Þ (b)

T/Tc=0.1
1
化学ポテンシャルの増加→フェルミ面の形成
0<!n !cut

H/Hc2
P
→反クォーク・クォーク間のフェルミ面ミスマッチング
þ 2T !À1 . We use ! ¼
0<!n !cut n cut HLO
0.95
self-consistently determined by
2T X Hcr
r Â Mpara ðrÞ À 2 hv Im gik ;
~ ð3Þ 0.9
0! K. SUZUKI, et. al. JPSJ (2011)
n
0 0.1 0.2 0.3 0.4 0.5
oth the diamagnetic contribution of T/Tc2

st term and the contribution of the

20
基底状態のAnsatz

chiral Real Kink Crystal
σ(x),π(x)
(x) = k sn(kz; ) , = 0. m
z
-m
pionic Real Kink Crystal

= 0, (x) = k sn(kz; ) .

deformed chiral spiral T. Tatsumi, arXiv:1102.0064 [hep-ph].

(z) = mcn(qz; ), (z) = msn(qz; ) . σ(x),π(x)
m
0 z
(z) = mcos(qz; ), (z) = msin(qz; )
E. Nakano and T. Tatsumi, Phys. Rev. D 71, 114006 (2005). -m

21
非一様凝縮で計算するダイヤグラム（詳細）
๏ GLポテンシャル
2 4 6
GL [ (x), (x)] = + ( ) + ( )
2 2 2 2 3
h
2 4 6
4,b 6,b
+ ( ) +
2
( , )2
4 6
6,c 6,d
+ [ 2
( ) 2
( , ) ]+2
( )2
6 6
2 4 4,b 4,c
+ 2
+ 4
+ ( 2 2
) c
2
+ ( c )2
2 c
4 c
4 c
4

๏ 新しく考慮すべきダイアグラム
σ σ π π
p σ p π π
(0)
α α (0)
β µI2 α

4,b 4 4,c 6
σ
p π p
π
σ σ π π
D. Nickel, Phys. Rev. Lett. 103, 072301 (2009).

22
凝縮が非一様な場合の結果
๏ 有限アイソスピン密度 ๏ ゼロアイソスピン密度

µI = 0
二次

（リフシッツ点） −1
二重臨界点三重臨界点 α2 [α6 ]
二次 =0
4 2
µ
(3 I /32, -3 I /2)
三重臨界点（リフシッツ点） µ =0
σ(
x) 二次
二重臨界点（リフシッツ点） =
0
非一様カイラル凝縮相二次二次
非一様荷電パイ
臨界点一次
4 2 中間子凝縮相非一様カイラル凝縮相
(0.21µ I, -2.22µ I)
YI, H.Abuki, K.Suzuki, arXiv:1206.2870 [hep-ph]

有限質量の場合
カレント質量mに比例(h∝m) explicit chiral
๏ GLポテンシャル
☟ symmetry breaking
GL [ , ] =
2 2
+
4
( ) +
2 2 6
( )
2 3
h SU (2)L SU (2)R
2 4 6
SU (2)V
2 4 4,b
+ 2
+ 4
+ ( 2 2
) c
2
other explicit breaking
2 c
4 c
4 c
SU (2)V U (1)I3 ;V
☝
アイソスピン密度により
๏ 相図有限アイソスピン密度・有限カレント質量
自発的に破れて
温度実線：1次

？
パイ中間子凝縮が起きる
点線：2次
臨界終点
π

σ
？
σ(x)
π(x)
クォーク密度 23

凝縮一様・有限アイソスピン密度・有限質量
๏ カイラルリミット(h=0) ๏ 有限カレント質量 (h≠0)
3/2
h[ 6 ]
1 = 0.5
µI 6 ]
2[

一様荷電パイ中間子凝縮相一様荷電パイ中間子凝縮相

2[ 6 ]
1
2[ 6 ]
1
二重臨界点

(0, 0) カイラル対称相三重臨界点

三重臨界点

(0, −µI )
2
一様カイラル凝縮相臨界終点

4/5 2/5 2
(2.28h , −2.25h −µI )
YI, H.Abuki, K.Suzuki, arXiv:1206.2870 [hep-ph] 24

location of CEP
2-ﬂavor QCDにおいて、臨界点の位置は未だ不確定

温度T 高fugacity領域 μqT に存在する場合、正当化

µq/T=1

臨界点
対称相
quarkからの寄与 gluonの寄与
カイ
ラル
凝縮相
縮相 μI
凝ャル
間子ン
シ
クォ
中ポ
テーク
電パイ化
学化学
ポテ
ンンシ
荷ス
ピャル
ソ μ q
アイ

quark loop contribution

the feedback from quark loops

S0 = diag(Su , Sd )
T Nc 1 n (x) = (x)1 + i 5 (x) 1
= Tr (S0 (x)) .
V n=2
n

☞σとπの係数をμIで摂動展開し、比例係数を決定 2 1 aµ2
I O(µ4 )
I
(0)
2
4 = 0 1 bµ2
I
(0)
4
,
a = 0, b = 1, c = 1/2, d = db = dc = 1. 6 0 0 1 (0)
6
2 cµ2I O(µ4 )
I
(0)
4 0 dµI2
= 4 .
4b 0 db µ2I
(0)
6
4c 0 dc µ2I

if the dy!ðMðyÞÞ ¼is restricted andwhere k and M that, A and Following constants.1 Following the C
modulation 0:
closely follow the discussion in [37]to 1D, Jacobi’s 0ellipticwhere k and Mbeingthe discussionnumbers. dis
demonstrate are constants.1 B 0 are arbitrary in
with
MðxÞthe modulation is restricted to 1D,[37], we canelliptic [37], we can show that this function obeys a fo
WS
if function gives in fact an adequate solution to the problem. this function we see that Msn constitutes a
Jacobi’s show that and (11), obeys a fourth-order
Eqs. (10)
results inus start with thean adequate solution toequation for thedifferentialEL equation (10) when the follo
Let the following Euler-Lagrange differential equation
function gives in fact fourth-order ordi- (EL) the problem. solution to the equation
why elliptic function
Let us start with the Euler-Lagrange (EL) equation for algebraic Mð4Þ00þ ðAk2 1Þk2 ð2 þ 1ÞM00 À k2 ð12 À BÞ
inhomogeneous chiral condensate,
near differential equation (when 4 0): 0 ¼ Mð4Þ þ ðA þ 1Þk2 ð20þ 1ÞM À þ are À BÞthe ¼ equations all satisfied: In the
sn sn ð12
Mk4 ½Að2 þ 1Þ2 À B2 Š; conde
sn sn 2
inhomogeneous chiral condensate, Z2 00 62 ¼ 0
2 M0
00 02 soidal
ðzÞ þ 3M À 10½MðM Þ þ M M dy!ðMðyÞÞ ¼ 0: Š 0 00
0 Þ2 þ M2 M00 Š ½Msn ðMð2 þ 1Þ2 À Bþ ½Ak4 ð2 þ 1Þ2 À B
Â þ ½Ak4 sn Þ2 þ Msn Msn2Š 2 Š2
2
Z
MðxÞ WS
Â ½Msn ðMsn sn sn À3 ¼ Àð1 þ AÞk ð þ 1Þ; forms
2 M À 6M3 þ 6MMðxÞ 5
: dy!ðMðyÞÞ ¼ 0:
(9) 2k 4 2k4 24 3k4
3 þ Fig. 2
WS k2 3k
Â Msn À 2 ð1 þ 2 Þð3 Â Msn À M2sn þ 4Þð3 À B þ AÞMsn of 42
À B þ AÞM3ð1 þ ð4 À BÞMsn ; 5
M0
ð4
The condition results in the following fourth-order ordi- M0 10 ¼ 0 2 ð12 0À BÞ;
M
olve this equationresultsdifferential equation (when 0):
The 非線形微分方程式 athe following fourth-order ordi-
nary, but nonlinear in suitable 1D
condition to find M0 meter
4 (11) over t
herenary, but nonlinear way to find equation (when 4 0):
is no systematic differential out the 4
00 2 00 with A and À 2k ð1 arbitrary numbers.bei
À6 B being Comparing
ð4Þ nonlinear differential 0 with A 2 and B being arbitrary¼numbers. þ 2 Þð3 À B þ AÞ; C K
n to a fourth-order ðzÞ þ 3M À 10½MðM Þ þ M M Š
0¼M Eqs. (10) and (11), we M02 see that M constitutes a
ð4Þ 00 0 2 Eqs. (10)00and (11), we see that M constitutes a sufficient
2 sn
wever it 0 ¼ M so difficult À 310½MðM Þ þ M M Š
is not ðzÞ þ 3M to see that 5 solution to sn EL equation (10) when the follo
the
þ 62 M À 6Msolution :to solution to the EL equation (10) when 4the following five
þ 6M (9)
ic function is a particular 3 5 algebraic 6 ¼ 3k ð4 À BÞ:
equations are all satisfied:
þ 62 M À 6M þ 6M : (9)are all satisfied: M4 4
Let snðz; Þ be to solve this equation to algebraic suitable 1D
We need Jacobi’s elliptic function find a equations 62 ¼ k ½Að2 þ 1Þ2 À B2 Š;
0
CRYSTALLINE CHIRAL CONDENSATES OFF6 ¼ . .4. 2 þ 1Þ2 À B2
対応 THE ½Að PHYSICAL
he elliptic modulus.There is no systematicfind a to find 2out ktheEqs. (12c)À3 ¼ Š; þ we 2see þ 1Þ;=k; BÞ ¼
We then set
We need to solve this equation to way suitable From
modulation. 1D and (12e), AÞk ð
Àð1
(12a)
2 ðM
0
E is
e where k and M0 are constants.1 Following finddifferential þ AÞk2 ð2 À28Þ, k2 thelim(12b) sn ðz;in al
modulation.snðkz; Þ; no fourth-order way to
There is a systematic nonlinear out the
general solution to と仮定した時に従う非線形微分方程式 À3 ¼ Àð1 in
the discussion BÞ ¼ ð2; þ 1Þ; but
ðM0 =k; latter M We c
results AÞ=
q¼2
e Msn ðzÞ ¼ M0 (10) 2. Then we 10 ¼ theð12 2 !II À0 =k; BÞ ¼
À BÞ; ðM0
d generalwe can showa that is not so difficult differential ð12 À BÞ; take M0 choice (12c) phase
[37], solution to fourth-order nonlinear to fourth-order
equation. However it this function obeys a 10 ¼ that seeand k 2 2

¼ equation. However function isPhys.difficult to solutionMto case we see Eqs.2k4
differential equation Kachkachi, soa Lett. A 225, 341see that 0
Jacobi’s Buzdin and H. is not
A.I.
elliptic it particular (1997). this in 2 (12b) and (12d) are d
2 Þð3 À B þ AÞ;
Fin
d Jacobi’s elliptic Let snðz; Þ be Jacobi’s elliptic function 4 we are À6 ¼with 0only algebraics
Eq. (9) [37]. function is a particular solution so that
2 to 2k left À 2 ð1 þ two
In À B þM we still have ½ to
2 Þð3the latter limit, A goes 2
;
k
Eq.¼(9) [37]. Let 1Þk2 ð2 þ 1ÞMsn À then setBÞ ¼ À Mand þ condensate jMðzÞj vanisheson
ð4Þ
þ ðA þ
0 Msnbeing the elliptic
with be Jacobi’s ð12 À À6 (12a) 2
snðz; Þmodulus. We elliptic function 0 ð1 (12b), whereas
00 4
AÞ; (12d) three
the
w
o- 2
M0 set parameters, , k, 6 ¼ 3k4 ð4 À BÞ: these two equ
and A. From
d with being the elliptic modulus. We then 3k4solve soidal. M0 a functionlimits are
These two of the c
Msn ðzÞ ¼ M0 snðkz; Þ;
0 Þ2 þ M2 M00 Š þ ½Ak4 ð2 þ 1Þ2 À B2 Š 4 6 ¼ (10) À BÞ: and k as
can ð4 (12e)
Â ½Msn ðMsn ðzÞ ¼ Msn
Msn sn snðkz; Þ; M
(10) 0 From A (and and Denoting see ðMin the ¼
parameter Eqs.forms2as (12e), we these0 =k; BÞ
(12c) ). we assumed functi
0
2k 4 From Eqs.4(12c)and ðM0wewe Fig. at0 =k;the¼ ð1; 2Þ or solution
3k and (12e), BÞ seeð2;2(a) one-parameter of ma
A,
¼ À28Þ, but
=k; arriveðM a BÞ the latter results in c
amplitude
and . Then we take the choice ðM0 =k; BÞ ¼
Â Msn À 2
ð1 þ 2 Þð3 À B þ AÞMsn=k; BÞ ¼ ð2; À28Þ, but the latter . Alsocomplex k2 togethe
3
ðM0 þ
4
the EL ; 5
ð4 À BÞMsnequation: results in depicted
of
in this case we 2 Eqs. (12b) and (12d) are d
see
M0 and . M0 we take the choice ðM0 =k; BÞ ¼ ð1; 2Þ, and
Then
so that we sn ðz; left ave kA A snðkA z;algebraic
M are AÞ ¼ defined by A Þ: roo
meter m withdegenerate the

, |M(z)|
74002-4 only two
in this case we see Eqs. (12b) and (12d) are we still have three
(12a) and (12b),the elliptic
whereas
4002-4 (11) only over algebraic of A,modulation
so that we are We stress that for any value equations
left with two as long as 0

ground state ansatz
chiral Real Kink Crystal σ(x),π(x)
m
(x) = k sn(kz; ) , = 0. z
-m

pionic Real Kink Crystal

= 0, (x) = k sn(kz; ) .

deformed chiral spiral T. Tatsumi, arXiv:1102.0064 [hep-ph].

(z) = mcn(qz; ), (z) = msn(qz; ) . σ(x),π(x)
m
0 z
(z) = mcos(qz; ), (z) = msin(qz; )
E. Nakano and T. Tatsumi, Phys. Rev. D 71, 114006 (2005). -m

Amplitude of condensates
1
CSNdataTABLE13.dat u 1:(sqrt($2*$3))
PSNdataTABLE13.dat u 1:(sqrt($2*$3))
homoCtab13.dat

一様カ
0.9
homoPtab13.dat

0.8
イラル
凝縮

非一様カ
0.7

0.6
σ,π

0.5

0.4

イラル凝
0.3

0.2

0.1

縮
0
-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2
1
一様カイ
1
CSNdataTABLE22.dat u 1:(sqrt($2*$3)) CSNdataTABLE28.dat u 1:(sqrt($2*$3))

非
ラル凝縮
0.9
一様カイ homoCtab22.dat 0.9 homoCtab28.dat

ラル凝縮一
homoPtab22.dat homoPtab28.dat

0.8
様 0.8

0.7 カ 0.7

イ非一
ラ様π
σ,π

0.6 0.6

0.5
非ル凝 0.5
中間
一子凝
様縮
0.4 0.4

0.3 π中 0.3 縮
0.2 間 0.2

子
0.1
凝 0.1

縮
α2 α2
0 0
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.5 0 0.5 1 1.5

(a) ing minM !ðMconst Þ ¼ minM;k h!ðM0
1 FF (spiral)
worked out numerically and the lo
LO (sinoidal)
point was found as
0.8 SN (solitonic)
2 ðLO $ SBÞ ffi 0

振幅
Homogeneous
0.6

mave
which is larger than I $ 0:1389 fo
2
0.4
reflecting the fact that the LO phase
the SN phase. Crossing the critical
phase to the LO phase, the magnitu
黒
0.2
drops by about 20%. Also the ratio
0 mass to the magnitude of the wave v
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
just at the critical point has been fo
2

(x) = k sn(kz; )
These can be summarized as
(b)
1.4 M0 ðLOÞ M0
ffi 0:81;
1.2 Mconst ðSBÞ kð

赤 1 We note that all these ratios are th
associated with the first-order phase

波数
0.8
SB and LO phases, at the sixth ord

q
(x) = msin(kz)
0.6 We also remark that the first Ansa
FF (spiral)
0.4
k as the same order in magnitude
LO (sinoidal)
transition. At the onset of the cond
SN (solitonic)
0.2 amplitude M0 vanishes while k rem
Homogeneous
derivative terms are more important
青
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 geneous terms. In contrast, at the ons
2 formation, the derivative terms play

(x) = mexp(ikz)
(c)
B. Most general condensate with
0
We here try to see if Jacobi’s ellipt
favorable solution among 1D modula
note that Jacobi’s elliptic function j
H. Abuki, D. Ishibashi and K. Suzuki, PRD 85, 074002 (2012). -0.1
[x10]

meter subgroup of solutions to the E

FF (spiral wave)
エネルギー
be stressed that the original EL equa
fourth-order differential equation, w
-0.2 LO (sinoidal wave) cally the sum of three differential eq
SN (solitonic wave) can be obtained from the second-or
Homogeneous state tion which the elliptic function obeys
-0.3 that it really covers all the solution
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 equation. Keeping this in mind, we
2
assumption for the spatially modula

37,38] where it is shown that the solution
other hand, from the coefficients of quartic terms, we see
ese two2 situationsÀ 1Þ=4d, we see conden- energy is an
mensionality, ð2d is the solitonic that the
0 ¼ Á function of 3ÁMðxÞ À 10½MðrMÞ2 . In 2 ÁMŠ
creasing MðxÞ þellipticdfunction.2 HereIIweM fact, after
near ¼ 2 þ that the free energies are on the order of the dimensionality
the extremal of energy density 0 of modulation. The above formulas for energy density can
ed by Jacobi’s
þ 62 M À M0 þ demonstrate that,
inimizing over [37],3and6M5 ; (32) 0 be easily generalized0.15 case with 0.25 condensate in
he discussion in 6M
ecomes restricted to 1D, Jacobi’s elliptic
0.05 0.1 to the 0.2 the LO 0.3 0.35 0.4
on is
dimension of condensates
2 ¼ @ the @2 þ @2

solution to 2 þ problem. is the 3 three- an arbitrary dimension, d, defined by 2
n where the operator3 Á dx
fact an adequate2 y
z 3 sffiffiffi
þ for 2
À (EL) equation O the À
LO;dD ¼ À Laplacian. This is a fourth-order nonlinear :
dimensional 2
the Euler-Lagrange 16 2d À 1 2 16 2 (b)
chiral condensate, equation. The solution space for the
partial differential MLO;dD ðxÞ ¼ M0 ðsinðkx1 Þ þ Á Á Á þ sinðkxd ÞÞ; (35)
(37) d
partial differential equation is much wider than that for the
Z
that the higher dimensional Therefore it less favor-
ordinary differential equation.LO phases areis not trivial
ote dy!ðMðyÞÞ ¼ 0: with x ¼ ðx1 ; x ; . . . ; xd Þ being the d-dimensional vector.
MðxÞJacobi’s elliptic function stays as the most favorable
that WS 0 In this case the 2thermodynamic potential looks like
ble than the 1D one, yet they have smaller energies than
estructure(chiral spiral) state. to 1D modulations is taken
1D-FF when the restriction
esults in thewe do not search for the formal solution, but
away. Here following fourth-order ordi- 2 3 2 2d À 1 4
We remark that the ordering of energies with the dimen- LO;dD ¼ À M0 þ M0
equation (when 4 0):
ear differential specific crystals having the simple square or 2 16 4d
onality is some
only try not universal, but rather specific to the LO type.
o cubic symmetry. In order to the 00 Š
see 3M00 fact, we firstÞdefine demonstrate how the dimen-
this À 10½MðM0 2 þ M2 M d-dimensional FF-type 5ð4d2 À 16d þ 11Þ 6
zÞsionality of a crystal structure affects the free energy,-0.1
þ

[x10]
we þ 2
M0 : (36)
omplex condensate as
3 on the LO-type phase which has a simpler
5
48d 1D (FF)
2 M À 6M þ 6M :
concentrate
sffiffiffi (9)
form than the elliptic function. We set multidimensional From the dependence of the quartic coefficient on the
3D (LO)
1 dimensionality, ð2d À 1Þ=4d, we see that the energy is an
lve FF;dD ðxÞ ¼ condensates þaeikx2 þ Á Á Á1Deikxd Þ: (38)
LO-type equation to ikx1 as suitable þ
M this real M0 ðefind
d
ere is no systematic way to find out the increasing function of d near 2 ¼ II . 2D (LO) after
2 In fact,
pffiffiffi -0.2 minimizing over M0 , the extremal of energy density
n MLO;1D ðxÞ ¼ 2M0 sinðkzÞ; differential
to a fourth-order nonlinear 1D (LO)
ever it isnotand HH , the values of that
densities sn so difficult to see Fourier strengths becomes
MLO;2D ðxÞ ¼ M0 ðsinðkxÞ þ sinðkyÞÞ; (33)
crepresentativeissffiffiffi particularand D.
function points A, B, C, solution to
a 1D SN3
2 3 2 d 2 3
Let sn snðz; Þ Hbe 2 sn
Jacobi’s elliptic function LO;dD ¼ À À þO À :
MLO;3D ðxÞ 1
M¼ MH sn H 2 16 2d À 1 Homogeneous 2 16
MWe then setsinðkyÞ þ sinðkzÞÞ: M5
MðsinðkxÞ þ 3 M5
e elliptic modulus.
1
3 03
-0.3 (37)
0.0111 0.0111 6 Â 10À8 6 Â 10À8 Oð10À13 Þ Oð10À13 Þ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Msn ðzÞ ¼
0.5291 ðxÞ M0 snðkz; Þ; the ‘‘egg-carton’’ Ansatz in [34].
(10) Note that the higher dimensional LO phases are less favor-
MLO;2D 0.5291 0.0077 to 0.0077
is equivalent 0.0001 0.0001
0.8673 condensate0.0563
Each 0.8665 0.0559 0.0039 0.0038
is characterized by two real parameters able than the 1D one, yet they 2 have smaller energies than
1.0672 0.9781 0.1951 are0.1077 determined via0.0132
M and q whose values to be 0.0455 minimi- the 1D-FF (chiral spiral) state.Ishibashi and K. Suzuki, PRD 85, 074002 (2012).
H. Abuki, D.
0
zation of . In Figs. 6(a) and 6(b) we show effective order We remark that the ordering of energies with the dimen-
■ 低次元的な凝縮の方がエネルギー的には優位（空間等方性の破れ？）type.FIG. 6 (colorsionality is not universal, but effective orderLO
online). (a) The rather specific to the parameters fo
parameters and free energies. One can see that the energy is
the 1D-LO To see this fact, we first2D-LO d-dimensionalgreen), 3D-LO
an increasing function of the dimensionality. Also it is (dashed, red), define the (dotted, FF-type
0 (dot-dot-dashed, magenta), as the solitonic state (solid) as a
notable that the critical points at which the transitions complex condensate
sffiffiffi
and
from the Wigner to the crystal phases take place are
common among all three states. To see this analytically, 2 . MFF;dDcomparison, 1the ikx2 þ Á Á Á þ eikxd Þ:the FF phase i
function of For
ðxÞ ¼
1
M0 ðeikx þ e
quantity for
(38)

he1homoge- 2n ¼ þ 4Nc Nf T ;
point (E) on the critical
; 2G n;1
m X Z dp
ð2Þ3 ðði!m þ Þ2 À p2 Þn 1
þ Þ phase the homoge-
-order 2 À p2 Þn
us phase and 2n ¼ þ 4Nc Nf T 3 ðði! þ Þ2 À p2 Þn
;
he location 2G m ð2Þ m(54)
nt the second-order phase
ð2 ; 4 Þ ¼(54) (54)
order one. Thewhere ! ¼ Tð2m þ 1Þ is the fermionic Matsubara fre-
location
The mapping from GL space onto (μq-T) space
m
ally found as ð2 ; 4 Þ ¼ summation over the frequency can be done,
Matsubara frequency. The
where !m ¼ Tð2m þ for and :
cy can be done, resulting in the following expressions 1Þ is the fermionic Matsubara fre-
: (51)
2 6and 4 : a2 quency. The summation over 2the frequency can be done,
4
ja j3 resulting in the following expressions for 2 and 4 :
; 0:16 マッピングの時にカットオフや結合定数等、モデルパラメータに依存してしまう
6
: (51) Z dp 1 À fF ðp À

6 a8 2 0 is the PHYSICAL REVIEW D 85, 074002 (2012)Þ À fF ðp þ Þ ;
a8 2 ¼ À 4Nc Nf 1
Àpoint, þ Þ
fF ðp three PHYSICAL REVIEW ð2Þ3 074002 Z
2G D 85, (2012) 2p
; 0 is theZ dp
n;1 X
t appears for þ 4Nc Nf T X Z 23 ¼ À 4N2 Nf 2 n ;dp 1 À fF ðp À Þ À fF ðp þ Þ ;
1 1
2n ¼ is
e 南部-Jona-Lasinioモデルを用いた場合の 3 H. Abuki, D. (55a)
locationn;1 6 dp 2G 1 c Ishibashi and K. Suzuki, PRD 85, 074002 (2012).
figure.which þ 4Nc Nf T m ð2Þ3 ðði!m þ Þ2 À p2 Þn ; CRYSTALLINE CHIRAL CONDENSATES OFF THE . . .
0:27Þ
2G
2n At this point, three
¼
(55a)
ð2Þ 2p
2G ð2Þ ðði! þ Þ À p Þ(54)
et at once. The location m パラメータ m
is
Z dp D À f074002 Þ À fF ðp þ Þ
0.5 (55a)
2 ðÀ0:016; 0:27Þ ¼ N N REVIEW 1 85, F ðp À (2012)
¼ which
PHYSICAL
4 þc 1Þfis the fermionic Matsubara fre-
(54) 60 60 60
þ Þ (52) ¼ Tð2mX Z dp
6 where !m ð2Þ3 p3 80 80 80
: 1Z
quency. The 4Nc Nf T þ 1Þ is the fermionic ðpcanÞ þ;f0 ðp þ ÞÞ À fF ðp þ Þ
¼ n;1 ¼ summation over Z frequency 2Àdp2 n1fre- F ðp À0.4
where !mþ Tð2m the Matsubara À f
3 ðði! þ0 Þ À p Þ
be done,
jaresultingThethe followingover the ¼ NforF2 can be4 :done,
ð2Þ dp f
8 2n 3
6j
2G a2
quency. in 6 summation cexpressionsmc Nf and 3 F
þ N Nf 4 frequency ð2Þ 2
m :p3 (55b)
6 resulting in the following expressions 3 and p
Þ 2 with
ompete ; 0:27 : (52) ð2Þ for 2 :
a8
: a8 Z Þ 4 (54)
(55b) Z dp 1 À f ðp À Þ À f ðp þdp f0 ðp À Þ þ f0 ðp þ Þ
mensionless 4N N 1 F 0.3 F
2 ¼ À c f Z F þ Nc Nf F
;3 2
: (55b)
1
ap- with
1Þ 2p F ð2Þ
3 1 À f ðp À Þ À f ðp þ Þ
intwhere !m À 4N We can derivefermionic expressions; for higher p
T is 2G compete ð2Þ
hiral phase ¼ Tð2m þ dp is the F similar Matsubara fre- order coef-
2 ¼ Nf 8=0
T/
c 3
gher orderThe summation ð2Þ
point. Any dimensionless momentum can be done,
g values:2Gcoef- ficients.over the frequency integral in 2 (4 ) has quadratic
The 2p (55a)
6=0
quency.
T is ap- 0.2
We can 2 and similar expressions Ã. higher
tant when in the following expressions for derivethat: we need a cutoff for As a order coef-
resulting point(logarithmic) divergence so 4 (55a)
4 ) has quadratic 4=0
y the following consequence, Þ À f functional forms become22 ¼ has quadratic
a cutoff Ã. As avalues: fF ðp Àficients.ðp þ Þ
Z dp 1 À their The momentum integral in (4 )
4 ¼1 c f Z Ã2 Zf dp 1 ÀT=Ã;À Þand F þ Þ ð=Ã; T=ÃÞ with g
F
fF ðp3 gÞ À f ðpdivergence so that we need a cutoff Ã. As a 2=0:
become NÀ 4Nc Ndp 3 1 ð=Ã; À(logarithmic) 4 ¼ f;2
2 ¼
N ¼
2 ð2Þ 1 À f ðp Þ À fF ðp þ Þ
p 0.1
¼ Nc g Z 3ð2Þ F
f
Á with Nf GÃ2 as03 the dimensionless NJL coupling and ffn g being g=2.0
:369 (53) ð2Þ f ðp À Þp3 f0 2p þ Þ their functional forms become 2 ¼
T=ÃÞ 4 Á Á ; 2G
dp F consequence,
þ F ðp g=2.5
þ Nc Nf Z 2 f ð=Ã; T=Ã; Higher ¼ f ð=Ã; T=ÃÞ with g
some3 dimensionless functions. gÞ and order coefficients,
and ffn g being ð2Þ f0 ðp À Þp2 f0 ðp þ Þ Ãþ 1 : (55b)
(55a) 4 2 g=3.5
dp F
der coefficients, 2n (n ! 3), no 2longer have the(55b) divergence so that ffn g being
þ Nc Nf 3
2F : UV
GÃ as the dimensionless NJL0coupling and
:308 Á Ácan Z dpð2Þfunctional p ðp þ Þbe summarized as 2n ð; TÞ ¼ coefficients, 0.6
We 1
(53)
ergenceÁ ;soderivetheirÀ fF ðp À Þ À ffor higher order coef-
some can
forms
that similar expressions F dimensionless functions. Higher order 0 0.2 0.4 0.8 1 1.2
as ¼ Nc NTÞ ¼ fn ð=TÞ=2ðnÀ2Þ . for ! 24) is nofunction have the UV (two)
ficients. f
We ð; derive similar expressions f12 (f 3), order coef- having three divergence so that
4
2n can Theð2Þ3momentum integral2n (n higher a quadratic

p3 in ( ) has longer /Λ
period. The arguments, while F for4 (n !forms a
ving three (two)divergence integralweffunctional quadratic only a single argu- 2n ð; TÞ ¼
:001 Á Á Á ;All Zmomentum so Þtheir ðp2þ Þ)cutoff 3) As can be summarized as
(logarithmic) dp f0 ðp À that f0 need a has Ã. has
ficients. F þ in n (
they 2 þ Nc Nf ð2Þ3 =T. Thus forms cutoff of vanishing (color ! 3
(logarithmic) ment, functional2 ð=TÞ=:2ðnÀ2Þ .(55b)¼ ) is function online). Mapping of the conditions of vanishing
y a consequence, divergence so that nwe need abecomeÃ.2 (fFIG. a102n for n having three (two)
single argu- their
do not f the condition f1 a2
pf ð=Ã; T=ÃÞ with As
e wouldð=Ã; T=Ã; gÞ into a straight line in while fgfor (n if f3) has onlyto single argu-
g Ã2 f21 say! 3 turns and 4 ¼ 2 forms become TÞ ¼
consequence, their functional
2n for nthe dimensionless arguments, the ð; 2 plane ! n ðxÞ ¼ 0 hasa the (, T)-phase space. Solid (red) lines
ng the felliptic T=Ã; gÞ and NJLfcouplingT=ÃÞ ffn g being
period. All ¼ ð=Ã; and with ng 本研究ではα60と仮定しているので、青い領域に相当
GL coefficients
GÃ 1 ð=Ã; a solution for x ! 0.
Ã ¼In has as
if WeðxÞ dimensionless functions.ment, =T. Thus the condition of vanishing ¼2n for n ! 3
e fsome derive similar not
sense 0
(T)nitself.that they do expressions
can
2
4 2for higher order coef-
Higher order coefficients, correspond to 0 for several different values of the NJL

荷電パイ中間子凝縮と散乱現象
D. Blaschke,N.K. Glendenning,A. Sedrakian
”Physics of Neutron Star Interiors”

■ 中性子ーパイ中間子凝縮相互作用により
ニュートリノの放出率上昇パイ中間子凝縮
を考えた場合

ニュートリノ放出による中性子星冷却率
■ 中性子星の冷却に寄与

■ 空間的に非一様な構造だと？

星の質量/太陽質量

Master's Thesis Presentation 2013/02/19

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Master's Thesis Presentation 2013/02/19