21COE 出張報告
原子核理論研究室
D3 古城 徹
関空~シンガポール
約 6 時間
シンガポール~ロンドン
約 12 時間
ロンドン~ケンブリッジ
約 2 時間
ケンブリッジ ~ 31のカレッジからなる大学町
トリニティカレッジ
(ニュートン、ダーウィン、ディケンズ , ... を輩出)
~ニュートン輩出、ノーベル賞31人
宿舎
キングスカレッジ
会議のテーマ
量子色力学(QCD)に関連するあらゆるトピックを扱い、
QCDに対する総合的な理解を図る
10-10m
10-15m
カラーの閉じ込め
環境変化(有限温度、密度)に対するQCD真空の応答
有限温度、密度における素励起の性質の変化
Topics
関心ある現象はいずれもQCDの非摂動効果と関連
→ 摂動論を超えた様々な手法を開発していく必要性
強相関 QGP の現象論~重イオン衝突実験の現象論 (6 talks)
格子QCDの大規模数値解析 (QCD相図、QGP中における素励起スペクトル、
QGP流体の輸送係数~ずれ粘性、熱伝導度)
(6 talks)
モデル的アプローチ:
(11 talks)
QCDの対称性を反映した現象論的モデルの解析( PNJL, dual-GL model )
次元を落とし簡単化したQCDの解析 ( (2+1)-D Yang-Mills )
内部対称性を落とし簡単化したQCDの解析 ( SU(2) color QCD )
Holographic QCD, Gauge-Gravity duality ( AdS/CFT or AdS/QCD 対応)
(20 talks)
超弦理論を介して量子効果が強いQCDを、双対な重力理論の古典計算で解析
→ 真空中のハドロンの性質、QGPへの適用
Peristaltic modes of single vortex
in U(1) and SU(3) gauge theories
based on PRD75, 105015 (2007)
Toru Kojo
(Kyoto University)
in collaboration with
Hideo Suganuma
(Kyoto University)
Kyosuke Tsumura
(Fuji film corporation)
This work is supported by the Grand-in-Aid for the 21st Century COE.
「Exploring QCD 」 at Isaac Newton Institute, 2007. 8. 23
Contents
I, Dual superconductor (brief review)
I-1, Dual superconducting picture for string
I-2, Dual Ginzburg Landau model
II, Peristaltic modes (Main results)
II-1, Static vortex solution and classification of vortex
II-2, Fluctuation analysis ~ Peristaltic modes
III, Summary and outlook
String picture of hadrons
String picture of hadrons gives natural explanation for:
Duality of the hadron reactions
=
s-channel
=
t-channel
string reaction
Regge trajectories of hadrons
(hadron mass)2
angular momentum
constant
string tension
Linear potential between quarks
universality of the string tension
lattice studies for QQ, 3Q potential
Creutz, PRL43, 553 (1979)
T.T.Takahashi et al, PRL86, 18 (2001)
The string picture may share important part of QCD.
Dual superconducting picture
DSC picture connects the string picture and QCD.
Abrikosov vortex in U(1) theory
Color flux tube in QCD
A.A.Abrikosov, Soviet Phys.JTEP 5, 1174(1957)
Y.Nambu, PRD.122,4262(1974)
‘t Hooft , Nucl.Phys.B190.455(1981)
Mandelstam, Phys.Rep.C23.245(1976)
B
electric
Cooper-pair
condensation
squeeze
magnetic field
z
dual
B
periodicity of
the phase of Cooper-pair
wave function
magnetic monopole
condensation
E
squeeze
color electric flux
quantization of the color electric flux
(periodicity of the phase of monopole )
color confinement
quantization of the total magnetic flux
(topologically conserved)
(static level)
linear potential between quarks
Dynamics of color flux tube
In most cases, we consider the moduli-dynamics of strings, i.e.,
rotation
translation
stringy vibration
Instead, following the dual superconducting scenario,
we focus on the dynamics of internal degrees of freedom,
i.e., excitation of the flux tube with changing its thickness.
When we consider short strings, this type of excitation becomes important
instead of the stringy excitation because stringy excitation cost energy ~ π/L.
(typical length ~ 1 fm for usual hadrons)
simplification of the problem
“Peristaltic” mode
infinite length (neglect the boundary)
translational invariance along the vortex line,
and cylindrical symmetry.
D.O.F for the dual string
‘t Hooft, Nucl.Phys.B190.455(1981)
fix the gauge of the off diagonal elements (Abelian projection)
U(1)3×U(1)8 gauge theory
SU(3) gauge theory (QCD)
8-gluon
remaining U(1)3×U(1)8 sym.
2 -gluon related to t3 ,t8 generators
6 -gluon related to other generators
topological configuration
“ photon ”
“ charged matter ”
(electric charge)
“monopoles” with U(1)3×U(1)8 magnetic charge
Further, we rewrite U(1)e×U(1)e photon part
in terms of the U(1)m×U(1)m dual photon.
Merit: Squeezing of electric flux can be described in the same way
as the squeezing of magnetic flux (dual photon becomes massive).
Dual-photon couples to the monopole current with the dual gauge coupling,
strong coupling region can be treated as
the weak coupling regime
Model – dual Ginzburg-Landau model
Ezawa-Iwazaki, PRD25(1982)2681
Maedan-Suzuki, PTP81(1989)229
After the Abelian gauge fixing, we get the D.O.F, especially
magnetic monopole, necessary to construct the dual strings.
Next question: Do monopoles really condense? Do the effects
of off-diagonal gluon fluctuations make theory untractable?
lattice results
monopole condenses !
off-diagonal gluons become heavy (~1.2 GeV)
Amemiya-Suganuma, PRD60(1999)114509
In low energy, QCD can be effectively described by
“dual-photons” and monopoles (& quarks) degrees of freedom.
dual photon field
monopole field
same form as the Ginzburg-Landau type action
z
Static solution (n = 1 vortex)
(under rescaled unit)
G-L parameter:
=
We search for the solution with cylindrical symmetry & topological charge = n.
minimize static energy with B.C
for finite energy vortex solution
monopole
color electric field
energy density
monopole
color electric field
energy density
Excitation modes under the static vortex background
Consider only the axial symmetric fluctuation around the static vortex solution
neglect 3rd and 4th order terms of fluctuations because
we focus on the case where the quantum fluctuation is not so strong.
Euler-Lagrange equation at 2nd order
variation of the action
Remark:
Because of the translational (t, z) and rotational invariance of the static vortex background,
eqs for (t, z) directions are easily solved
axial symmetric fluctuations are completely decoupled from angular dependent modes.
Peristaltic modes of the vortex
eqs. for fluctuations in the radial direction
“radial mass”
dispersion relation:
conserved total color electric flux
ω、kz
EZ
EZ
propagation
with
“radial mass” mj
EZ
monopole
Vortex-induced potential for fluctuations
Only the radial direction of the potential is nontrivial.
渦糸付近では対称性が回復 → 粒子の質量は軽くなる
ex) Type-II case
V(r) for α (monopole)
V(r) for β (dual photon)
V(r) for α-β mixing
energy “threshold”
for continuum states
= Mmonopole2
2 = Md-photon2
(independent of κ2)
Energy spectrum
( the effect of the diagonal potential )
Type-II
monopole
V(r)
BPS
gauge field
Type-I
V(r)
gauge field
monopole
Around BPS saturation, characteristic
discrete pole appear as a result of
monopole – dual photon
corporative behavior
1st excited state – wavefunction in the radial direction
fluctuations of φ、 Aθ
fluctuation of electric field
small
Type-I
monopole
squeezed by monopole
( total flux is conserved to 0)
dual-photon
(around)
BPS
corporative
squeezed by monopole
( total flux is conserved to 0)
oscillation ~ eipr / r1/2
Type-II
large
oscillation
~ eipr / r1/2
corporative
r
→ resonant scattering
r
Summary:
For the general vortex case:
We consider the vortex vibration
with changing its thickness.
We found the characteristic discrete pole
around BPS value of GL parameter.
→ coherent vibration of Higgs and photon fields.
For the application to QCD:
flux-tube in the vacuum:
DGL parameters are taken to fit the QQ potential results.
monopole self-coupling:
dual-gauge coupling:
λ ~ 25
gdual ~ 2.3
value of monopole cond.: v ~ 0.126 GeV
κ2 ~ 3 → Type-II
resonant scattering type
of vibrations appear.
excitation energy ~ 0.5 GeV
Outlook and speculation:
For the application to hot QCD:
temperature
Then, if the strength of
the monopole self-interaction λ(T) becomes weak,
=
becomes weak, and
the property of color-electric flux approach to the Type-I vortex.
The monopole - dual photon coherent vibration can appear
in non-zero temperture.
質疑応答(要約)
Q,円柱対称な励起よりも歪んだ形の振動の方が重要では?
A,Type-II の場合はvortex は表面を減らす方がエネルギー的には得する
ので円柱対称なモードだけで十分でしょう。Type-I については逆なので
今の解析からはあまり確定的なことは言えません。
Q,実際のハドロンでこのような形で励起スペクトルは見えるのか?
A,今の解析では大分簡単化した場合を扱っているので、実際のハドロン
物理への応用を行なうにはもう少しrealistic な状況設定にする必要が
あると思ってます。これについてはいずれやろうと思っています。
Q,実際の超伝導体ではどうか?
A,今回見つけたdiscrete pole が出るのは BPS 付近の GL parameter
の場合ですが、対応する材料があれば見えると思います。
まとめ
今回の海外出張は、物理関係者のみならず、
外国の人と沢山話す機会があり、非常に良い経験となった。
→ 英語の練習は普段からもっとしておくべきだったが。。
質疑応答に関しては、もっと経験を積む必要性を痛感した。
会議中のトークに刺激を受け、
いくつか研究テーマの候補を思いつくことができた。
最後に、このような機会を与えて下さった
21COE 外国旅費補助委員会の方々に感謝したいと思います。
Vortex – vortex “potential” per unit length
( for DGL, per 1 fm )
1.0 fm
2.0 fm
3.0 fm
Type-II (DGL case)
R
R
2.0 GeV
0.25 GeV
1.5 GeV
BPS
R
R
0.1 GeV
Type-I
1.0 GeV
Thermodynamical Stability
vortex-vortex interaction
exact solution
attractive
B
B
( Mn:vortex mass with topological charge
B
de Vega-Schaposnik,
PRD14,1100(1976)
n)
vortex-vortex interaction
repulsive
B
B
no interaction
between vortices
Type-I vortices system is
thermodynamically unstable
not uniform
B
(at least in tree level)
vortex lattice with topological
charge n=1
( thermodynamically stable)
Usually, Type-I vortex is not considered,
but we consider the external magnetic field
squeezed enough to generate only one vortex
We study not only Type-II vortex but also Type-I vortex
n > 1 vortex, classical profiles & potentials
(κ2 = 1/2 case )
profile
n=1
increasing n
n=2
n=3
total magnetic flux ( =2πn ) increases
Cooper-pair around core is suppressed
potential
Cooper-pair around core is suppressed
large potential around core for φ & fluc. of Hz is enhanced
“surface” between Cooper-pair and magnetic flux shifts outward
“mixing” potential shifts outward
n = 2, 3 energy spectrum
n=2
giant vortex
n
n=3
the lowest excitation becomes softer one
The topological defect of Cooper-pair condensation
is enlarged, then photon can easily excite around the core.
The property as static vortex + photon excitation becomes strong.
The threshold is unchanged, then continuum states behave
like n =1 continuum states.
Vortex – antivortex “potential” per unit length
R
( for DGL, per 1 fm )
R
Bz
same
topological charge
Bz
R
R
1.0 fm
no vortex
total magnetic
flux is zero .
2.0 fm
3.0 fm
2.0 GeV
1.5 GeV
1.5 GeV
sudden annihilation
of the flux
When d < 1.0 fm, our B.C, |ψ|2= 0
at the core is no more applicable.
1.0 GeV
0.5 GeV
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