Peristaltic modes of single vortex in U(1) and SU(3) gauge theories

Toru Kojo (Kyoto University)

Kyosuke Tsumura (Fuji film corporation)

Hideo Suganuma (Kyoto University)

in collaboration with

「 Exploring QCD 」 at Isaac Newton Institute, 2007. 8. 23

based on PRD75, 105015 (2007)

This work is supported by the Grand-in-Aid for the 21st Century COE.

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

Abrikosov vortex in U(1) theory

electric Cooper-pair condensation

B

squeezemagnetic field

A.A.Abrikosov, Soviet Phys.JTEP 5, 1174(1957)

B

quantization of the total magnetic flux

z periodicity of the phase of Cooper-pairwave function

(topologically conserved)

Dual superconducting pictureDSC picture connects the string picture and QCD.

color confinement (static level)

linear potential between quarks

Color flux tube in QCD

dualmagnetic monopolecondensation

squeeze color electric flux

E

Y.Nambu, PRD.122,4262(1974)

‘t Hooft , Nucl.Phys.B190.455(1981)

Mandelstam, Phys.Rep.C23.245(1976)

quantization of the color electric flux(periodicity of the phase of monopole )

Dynamics of color flux tubeIn most cases, we consider the moduli-dynamics of strings, i.e.,

rotation translation stringy vibration

simplification of the problem

infinite length 　（ neglect the boundary ）translational invariance along the vortex line,

and cylindrical symmetry.“Peristaltic” mode

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)

D.O.F for the dual string ‘t Hooft, Nucl.Phys.B190.455(1981)

SU(3) gauge theory (QCD) U(1)3×U(1)8 gauge theory

fix the gauge of the off diagonal elements (Abelian projection)

8-gluon2 -gluon related to t3 ,t8 generators

6 -gluon related to other generators

“monopoles” with U(1)3×U(1)8 magnetic charge

remaining U(1)3×U(1)8 sym.

“ photon ”

“ charged matter ”topological configuration

(electric charge)

Merit:

Further, we rewrite U(1)e×U(1)e photon part in terms of the U(1)m×U(1)m dual photon.

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

Ezawa-Iwazaki, PRD25(1982)2681 Maedan-Suzuki, PTP81(1989)229Model – dual Ginzburg-Landau model

After the Abelian gauge fixing, we get the D.O.F, especially magnetic monopole, necessary to construct the dual strings.

dual photon field monopole field

same form as the Ginzburg-Landau type action

Next question: Do monopoles really condense? Do the effects of off-diagonal gluon fluctuations make theory untractable?

In low energy, QCD can be effectively described by “dual-photons” and monopoles (& quarks) degrees of freedom.

lattice resultsmonopole condenses !off-diagonal gluons become heavy (~1.2 GeV)

Amemiya-Suganuma, PRD60(1999)114509

Static solution (n = 1 vortex) z

G-L parameter:

minimize static energy with B.C for finite energy vortex solution

monopole

color electric field

energy density

color electric field

energy density

monopole

=(under rescaled unit)

We search for the solution with cylindrical symmetry & topological charge = n.

Excitation modes under the static vortex background

neglect 3rd and 4th order terms of fluctuations becausewe focus on the case where the quantum fluctuation is not so strong:

Consider only the axial symmetric fluctuation around the static vortex solution

Euler-Lagrange equation at 2nd ordervariation of the action

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.

Remark:

eqs. for fluctuations in the radial direction

dispersion relation:

Peristaltic modes of the vortex

“radial mass”

propagation 　 with “radial mass” mj 　

ω 、ｋｚ

conserved total color electric flux

EZ

monopole

EZ

EZ

Vortex-induced potential for fluctuations

= Mmonopole2

2 = Md-photon2

energy “threshold” for continuum states

V(r) for α (monopole)

V(r) for β (dual photon)

V(r) for α-β mixing

ex ）　 Type-II case

(independent of κ2)

Only the radial direction of the potential is nontrivial.

Energy spectrum ( the effect of the diagonal potential )

Type-IImonopole

gauge fieldV(r)

monopole

gauge fieldV(r)

Type-I

BPS

Around BPS saturation, characteristic discrete pole appear as a result of monopole – dual photon corporative behavior

1st excited state – wavefunction in the radial directionfluctuation of electric field

large

small

monopole

dual-photon

Type-I

Type-II

BPS

fluctuations of φ 、 Aθ

corporative

corporativeoscillation ~ eipr / r1/2

oscillation ~ eipr / r1/2

→ resonant scattering

r r

squeezed by monopole

squeezed by monopole

( total flux is conserved to 0)

( total flux is conserved to 0)(around)

Summary:

excitation energy ~ 0.5 GeV

κ2 ~ 3 → Type-II

resonant scattering type of vibrations appear.

DGL parameters are taken to fit the QQ potential results.

monopole self-coupling: λ ~ 25

dual-gauge coupling: gdual ~ 2.3

value of monopole cond.: v ~ 0.126 GeV

flux-tube in the vacuum:

We found the characteristic discrete pole around BPS value of GL parameter.

→ coherent vibration of Higgs and photon fields.

For the application to QCD:

We consider the vortex vibration with changing its thickness.

For the general vortex case:

Outlook and speculation:

For the application to hot QCD: temperature

Then, if the strength of effective 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.

1.0 fm 2.0 fm 3.0 fm

0.25 GeV2.0 GeV

1.0 GeV

1.5 GeV

0.1 GeV

Type-II (DGL case)

Type-I

BPS

Vortex – vortex “potential” per unit length ( for DGL, per 1 fm )

R R

R R

String picture of hadronsString picture of hadrons gives natural explanation for: Duality of the hadron reactions

= =s-channel t-channel string reaction

Regge trajectories of hadrons constant string tension

(hadron mass)2 angular momentum

The string picture may share important part of QCD.

Linear potential between quarks lattice studies for QQ, 3Q potential

universality of the string tension

Creutz, PRL43, 553 (1979) T.T.Takahashi et al, PRL86, 18 (2001)

Static solution z

G-L parameter

minimize with B.C for finite energy vortex solutionat vortex core, φ ＝ 0

in asymptotic region, sym. is restored

gauge fixing:

We search for the solution with cylindrical symmetry & topological charge = n.

topological quantization 　 (topological charge n)

Higgs

photon mixing Importance of mixing

BPS

Type-II

Type-I

around BPS

static vortex – Higgs bound state

bound state of static vortex and - mixed state

Same threshold leads the corporative behavior of Higgs & photon

then, lowest excitation energy is considerably decreased.

around the core,Higgs & photon are mixed

The property of the potential in the radial direction

V(r) potential induced by static vortex

= Mmono2

2 = Md-photon2

energy “threshold” for continuum states

V(r) for α (monopole)

V(r) for β (dual-photon)

V(r) for α-β mixing

ex ）　 Type-II case

0

2

(independent of κ2)

“central” region ( r → 0 ) “asymptotic” region ( r → ∞ )

α(r) → 　 rm × const.

β(r) → 　 rm× const.α(r) → 　 0

β(r) → 　 0

α(r) → 　 eipr

α(r) → 　 eipr

state below threshold state above threshold

(m 2 )≧

for all states

electric field

energy density

monopolesmonopoles

electric field

energy density

Static profile for Type-I & IIκ ＝ δ/ξ( ＝ λ1/2/ e) : G-L parameter

δ

ξ

pure metalex) high Tc SC, metal with inpurity

penetration depth δ: ～ 500 A, 0.3 - 0.4 fm

ex)

coherence length ξ: 25 – 104 A, 0.16 fm

(usually not considered)

condensed matter DGL (QCD effective theory)

G-L parameter κ: 0.05 － 20, ~ 1.6 － 2.0

=1/2

BPS

finite thickness string like

for Abrikosov vortex

To discuss the color flux linking specific charges, we have only to consider this part.

R

RAbelian dominance

off-diagonal gluon is heavy ( ~ 1.2 GeV )Amemiya-Suganuma, PRD60(1999)114509

same form as the Ginzburg-Landau type action

Higgs (Cooper-pair) field photon fieldGinzburg-Landau action:

We will consider color flux tube linking specific charges.

dual photon field monopole field

We have only to consider the GL-type action.

1st excited state – wavefunction in the radial directionfluctuation of electric field Ez

large

small

monopoles

d-photon

Type-I

Type-II

BPS

fluctuations of φ 、 aθ

corporative

corporativeoscillation ~ eipr / r1/2

oscillation ~ eipr / r1/2

→ 　 long tail

r r

squeezed by monopoles

squeezed by monopoles

( total flux is conserved to 0)

( total flux is conserved to 0)(around)

Ez

Ez

Ez

Thermodynamical Stability

exact solution de Vega-Schaposnik, PRD14,1100(1976)

no interaction between vortices

Type-I vortices system is

thermodynamically unstable

B B B B B

vortex lattice with topological charge n=1

( thermodynamically stable ）

vortex-vortex interactionattractive

vortex-vortex interaction

repulsive

BUsually, Type-I vortex is not considered,

but we consider the external magnetic fieldsqueezed enough to generate only one vortex

( Mn ： vortex mass with topological charge n )

(at least in tree level)

We study not only Type-II vortex but also Type-I vortex

not uniform

n > 1 vortex, classical profiles & potentials(κ ２ = 1/2 case )

n=1 n=2 n=3

increasing n total magnetic flux ( =2πn ) increases

Cooper-pair around core is suppressed

Cooper-pair around core is suppressed

& fluc. of Hz is enhanced

profile

potential

large potential around core for φ

“surface” between Cooper-pair and magnetic flux shifts outward

“mixing” potential shifts outward

n = 2, 3 energy spectrum n=2 n=3

The topological defect of Cooper-pair condensation is enlarged, then photon can easily excite around the core.

nThe property as static vortex + photon excitation becomes strong.

the lowest excitation becomes softer onegiant vortex

The threshold is unchanged, then continuum states behave like n =1 continuum states.

Summary for single Abelian vortex

We have discussed the “peristaltic” modes of single vortex.

We found, new discrete pole around κ2 ~ 1/2.

This discrete pole is characterized by the corporative behavior

of the Higgs and photon fields. As κ2 is increased, the low excitation modes change from the Higgs dominant modes to the photon dominant modes.

As n is increased, photon can excite more easily, and lowest excitation becomes softer one.

ω 、ｋｚr

zHiggs

rHZ

r

Summary for single color electric flux

excitation energy ~ 0.5 GeV.

(mass of color electric flux per 1fm ~ 1.0 GeV/fm.)

r

DGL gives κ2 ~ 3 - 4. → Type-II

Only resonant scattering type of excitations appear.

profile in radial direction

electric flux vibrate with long tail.

RR We can directly apply the previous arguments to

the color flux linking specific color charge.

4, Calculation in 2 - D (Preliminary)Motivation: We would like to discuss:1, excitation modes around 1-vortex without cylindrical symmetry.

2, the dynamics of the multi-vortices system, for example,vortex-vortex fusion into the giant vortex,

the giant vortex fission to the small vortices,

vortex - anti vortex annihilation and production,

bearing in mind the future application to the hadron physics:ex) meson-meson reaction: scattering

production of the resonance, especially exotic hadrons etc.

In this talk, we show only the static profile, vortex- vortex potential, and vortex- anti vortex potential.

Vortex – antivortex “potential” per unit length ( for DGL, per 1 fm )

Bz

Bz

same topological charge

no vortex

total magnetic flux is zero .

1.0 fm 2.0 fm 3.0 fm

2.0 GeV

1.0 GeV

0.5 GeV

1.5 GeV1.5 GeV sudden annihilation

of the flux

When d < 1.0 fm, our B.C, |ψ| ２ = 0 at the core is no more applicable.

R

R

R

R

Energy spectrum: New-type discrete pole

Type-IImonopoles

dual photon field

V(r)

dual photon field

Type-I

BPS

Around BPS saturation, characteristic discrete pole appear as a result of monopoles – d-photon corporative behavior.

monopoles

Sudden annihilation of fluxes (in DGL unit)

1.2 fm 1.0 fm

Around d = 1.0 - 1.2 fm, the fluxes suddenly annihilate.

This critical distance dcr is related with the penetration depth δ

dcr ~ (1.5 - 2.0) × 2δ

This value seems to be considerably large.

SU(3) gauge theory (QCD) U(1)×U(1) gauge theory

Abelian projection ‘t Hooft, NPB190,455(1981)

(fix the gauge of the off diagonal elements)

8-gluon2-gluon (“photon”)

6-gluon (“charged matter”)

Abelian monopole (topological object)

Abelian projection and monopoles

Usually magnetic monopole does not appear in U(1) gauge theory,but if theory includes SU(N) ( N>1) gauge fields,their specific topological configuration constructsU(1) point like singularity as a topological object.

R3 in physical space SU(2) variables in internal space

mapping

Two vortices system ( static case )field degrees of freedom ：　 Reψ, Imψ, Ax, Ay (without cylindrical sym.)

Starting from the case where the distance between two vortices is large, adopt the product ansatz for initial B.C:

ψ1+2 ＝ ψ1 ψ2 /const. A1+2 ＝ A1+ A2

Step 1)

As the previous 1-vortex system, we first search for the static profiles which minimize the static energy.

Step 2) Fix the |ψ| ２ = 0 at the vortices cores, and minimize the static energy with checking that the total magnetic flux is quantized appropriately.

(We need this B.C. only at the beginning of the calculation)

Step 3) After convergence, change the distance of the vortex core.

Step 4) Adopt the previous profile as I.C. and return to the step 2.

Then we acquire the static profile and the potential between two vortices .

Summary:We have discussed the “peristaltic” modes of single vortex.

We found, new discrete pole around κ2 ~ 1/2.

This discrete pole is characterized by the corporative behavior of the Cooper-pair and photon fields.

To discuss the dynamics of color flux, we need more careful treatments to retain the confinement property.We have also discussed the potential between vortices as a preparation for the dynamics of multi-vortices system.

Future work:Non axial symmetric excitation of single vortex.Dynamics of two vortices.Careful treatment of color flux with projection.

IntroductionAbrikosov vortex in U(1) theory

Cooper-pair condensation

B

Meissner effect

squeezed magnetic field

A.A.Abrikosov, Soviet Phys.JTEP 5, 1174(1957)

Color confinement in QCD

dual

B

magnetic monopolecondensation

dual Meissner effect

squeezed color electric fluxE

Y.Nambu, PRD.122,4262(1974)‘t Hooft , Nucl.Phys.B190.455(1981)Mandelstam, Phys.Rep.C23.245(1976)

SU(3) gauge theory (QCD) U(1)×U(1) gauge theory

Abelian projection ‘t Hooft, NPB190,455(1981)

(fix the gauge of the off diagonal elements)

8-gluon2-gluon (“photon”)

6-gluon (“charged matter”)

Abelian monopole (topological object)

Static U(1) vortex solution (n=1 case)

magnetic field penetrateswith small kinetic energy

magnetic field is stronglysqueezed by Higgs field

finite thickness string

Regge trajectories of hadrons constant string tension

(hadron mass)2 angular momentum

gauge fixing:

1) Abelian gauge fixing → U(1)2 monopole DOF naturally appear as topological objects.

2) include the auxiliary field U(1)m ｘ U(1)m dual photon B which couples with the monopole current. (Zwanziger, PRD3(19 ７ 0)880 )

4) integrate out the U(1)e ｘ U(1)e photon field A

3) sum up the monopole trajectory → 　 leading order ：　 kinetic term of monopole field correction ：　 monopole self

interaction ( Bardakci, Samuel, PRD18(1978)2849 )

(this is included phenomenologically)

Ezawa-Iwazaki, PRD25(1982)2681 Maedan-Suzuki, PTP81(1989)229Model – dual Ginzburg-Landau model

Static solution zG-L parameter

minimize static energy with B.C for finite energy vortex solution

at vortex core, φ ＝ 0

in asymptotic region, sym. is restored

gauge fixing:

We search for the solution with cylindrical symmetry & topological charge = n.

Ansatz for topological charge n

monopoles

electric field

energy density

electric field

energy density

monopoles