Monopole production and rapid decay of gauge fields

Aiichi IwazakiNishogakusha University

High energy heavy ion collisions

Generation of color electric and magnetic fieldsaccording to a model of color glass condensate

thermalizedquark gluon plasma

Quarks and gluons are producedby the rapid decay of the gauge fields

decay time< <1fm/c

（～ 0.5fm/c ?)

Hirano, Nara2004

We have not yet foundthe rapid decay mechanismof the gauge fields.

We wish to propose a rapid decay mechanism of the gauge fields.

High energy densityof the color gauge fields

~ 2

4

gQs

width offlux tube

Homogeneous in longitudinal direction

2~)( sQgBgE

field strength

GeVGeVQs 2~1

0 BE

sQ

1 sQ

saturation momentum

Ensemble ofZ2 vortices ?E

B

B

BE

E

EB

EB

The gauge fields are unstable.

Dumitru,NaraPetreska, 2013

( RHIC or LHC )

Characteristics of the gauge fields

|A(pL=p,t)/A(pL=p,t=0|2

x

FF xtBxtBD 222' ))),(()),(((

J. Berges, S. Scheffer and D. Sexty, 2008

Exponential growth of the distancebetween nearby gauge fields at t=0

B B’ t=0

FFD

Kunihiro, Muller, Ohnishi, Schafer, Takahashi, Yamamoto (2010, 2013)

Exponential growth offluctuations aroundthe gauge fields

Exponential growth oflongitudinal pressure offluctuations aroundthe gauge fields in expandingglasma (τ,η coordinates )

Romatschke and Venugopalan 2006

Fukushima and Gelis 2011

It has been found thatthese instabilities do notlead to sufficiently rapid decayof the gauge fields for QGP to berealized within 1fm/c.They have been discussed tobe Nielsen-Olesen instabilities.

Nielsen-Olesen instability

classical instability in SU(2) gauge theory

22

22241

)(4

)(

||2

1)(

4

1

gAAig

DDAAFLgluon

3 AA

2

21

iAA

（ Electromagnetic fieldsrepresent the backgroundgauge fields ）　

3,2,1,)(||2

1|| 22

jiAAigDDL jiijjiijjiit

The term can be positive or negativefor arbitrary magnetic field B0

,

00 Agauge

igAD iii

Iwazaki 2008,

Itakura, Fujii, 2008

（ charged vector fieldsare fluctuations aroundthe gauge fields ）　

Nielsen and Olesen 1978,

Homogeneous B )0,0,0( 1221321 AABBB

gBDL t 2|||| 22

221

i

negative potential

0),exp( 22 sQgBti

Nielsen-Olesen unstable modesoccupying lowest Landau level

Negative potentialfor homogeneous B -2gB

Potential forinhomogeneous B

)(~;0),exp( 22seffeff QOgBgBgBti

growth rate

sQgB

growth rate

effgB

gBD 222

02

),exp( t

gB2Bound states in the Lowest Landau level

Bound states exist with

We may represent these bound states by using effective magnetic field

Eq. of motion

Numerical results ( nonexpanding glasma )

sQgtA )25.0~17.0())(3.0~2.0();exp( 4/12

growth rate

saturationmomentum

cfmQ s

/2.117.0

11 A roughly estimated decay timeof the background fields

5.22

1

4

12

gg

GeVQ

s

s

J. Berges, S. Scheffer and D. Sexty, 2008

Kunihiro, Muller, Ohnishi, Schafer, Takahashi, Yamamoto (2010, 2013)

When we represent the growth rate by using effective homogeneous magnetic field such as ,we find

effgB

effgB2)17.0( seff QgB

effteff gBDL 2|||| 22

222 06.0)17.0(22 sseffeff QQgBm

small effective mass( imaginary )

Effective Lagrangian of describing the instabilityunder inhomogeneous magnetic fields is given such that using effective homogeneous magnetic field

seff QgB 17.0~

effgB

small growth rate

)exp( t

long decay time cfm /2.1~1

Using the effective Lagrangian, we calculate the back reaction of the unstable modes on the background gauge fields and show how fast the fields decay.

We show that the monopole production leads to much morerapid decay of the gauge fields than the production of Nielsen-Olesen unstable modes

Similarly, we wish to calculate the back reaction of magnetic monopoles on the background gauge fields by using an effective Lagrangian of the monopoles.

The monopoles are such objects whose condensation gives rise to “quark confinement” in QCD.

dimi

di

mmmd

mtmonopole

AigD

mDL

42222 ||||||||

Effective Lagrangian of magnetic monopolesdescribing quark confinement

m

di

m

A

gg

GeVOm

2)1(

larger than

magnetic charge

dual gauge fields

GeVQfor

GeVQfor

gg

g

s

s

sm

23

15.2

2

12

‘tHooftMandelstam1976

Koma, Suzuki 2003

GeVmeff 25.0~

describes dual superconductors

GeVQs 1for

monopole field

monopoleL

We calculate the decay time of background colorelectric ( magnetic ) fields by using the effective Lagrangian of Niesen-Olesen unstable modes ( magnetic monopoles ) in expanding glasma(τ,η coordinates )

Note thatthe monopoles occupy Landau levels underbackground electric field, while Nielsen-Olesen modes occupy Landau levels under magnetic field.

Our assumptions

Relevant monopoles occupy only the lowest Landau level

Their distribution is almost homogeneous in transverse plain so that magnetic field affected bythe monopole production is almost homogeneous.

The similar assumptions for Nielsen-Olesenunstable modes are adopted.

)||1

|(| 22

244 effteff gBDdxLdxg

igAD

)|),(||),(|1

|),((| 222

224

effeff gBDddLLdxg

2

0

22

0

|)(|;)(),(

n

TnnTn

Tnn xadxLxa

)4

||()(

2

21Teffn

n

xgBExpixx

Effective Lagrangian of Nilesen-Olesen (N-O) unstable modes in τ,ηcoordinates

)log(2

1,

30

3023

20 xx

xxxx

Assuming homogeneousdistribution of N-O modesin transverse plain, the dynamicalvariable is left.

2

0

|)(|n

Tnn xa

),(

( wave functions ofthe lowest Landau level )

)||||||1

|(| 22222

244mmeffmm

dmeff mEgDdxLdxg

dm

d AigD

)|),(||),(||),(|1

|),((| 22222

22

mmeffmmm mEgDddL 2

0,

22

0, |)(|;)(),(

n

TnmnTn

Tnmnmm xadxLxa

)4

||()(

2

21,Teffmn

nm

xEgExpixx

wave functions ofthe lowest Landau level under

Effective Lagrangian of magnetic monopoles in the lowest Landau level

effmEg

d

m

effmeff

AA

mm

EggB

00

Nielsen-Olesen monopole

Equations of motion of Nielsen-Olesen modes

222

22

0

2

|),(|))((2

2)()1

(

),())((1

1

),()1

(

gAd

ggA

gAgBeff

);(),,(2

),(

gAed i homogeneous in Txand

0

1

)(

effgBgB We assume that

background magnetic field decreases

with the expansion

The equations describe how the electric field

decays via the production of Nielsen-Olesen unstable modes )(

1 AE

Maxwell eq.

initial conditionsWe use the initial conditions given by Dusling, Gelis, and Venugopalan

jiijiij

j

ii

i

ii

CzWeCzWegE

i

atccCzWgE

ee

d

realed

i

,2/,2/)1(4/)1(

,2/,2/)1(4/)1(

202,1

1

,,20

4/)1(

21

)()(1

),(

0..)(2

2),(

),,(2

2/)(),(

2/,2/)1(,2/

),(20

,

iiigEz

zW

Whittaker function 0

)(

,,,,

,,,

jiji

jiji

CCCC

CC

2/122/,2/)1(

2/)1(22/,2/)1(

2/)1(

20|)(||)(|

1),( zWezWe

gEiiii

τ→0

Without taking average of initial values

after obtaining the time evolution of we take the initial value, CC )0,(

),(

τ→0

2011, 2012

That is, we include next toleading order of quantumeffects on the evolutionof the background gaugefields.

2/,2/)1(,2/

),(1

),(

20

,20

iiigEz

zWgE

),()2/)0((1

),())((1

1

),()1

(

222

22

0

2

gEgB

gAgB

eff

eff

Positive energy solutions of the equation with the parameters, )0(0 gEgEgBeff

asτ→0

Whittaker function

2/122/,2/)1(

2/)1(22/,2/)1(

2/)1(

20|)(||)(|

1),( zWezWe

gEiiii

τ→0

This initial condition comes from the average,

jiijiij

j CzWeCzWegE

i ,2/,2/)1(4/)1(

,2/,2/)1(4/)1(

202,1

1 )()(1

),(

2,1,,

22/,2/)1(

2/)1(

2,1,,

22/,2/)1(

2/)1(

20|)(|

|)(|1

),(),(

jjjii

jjjii

CCzWe

CCzWe

gE

with the use of the formulae,

τ→0

τ→0

2/20igEz

For simplicity, we take the simple initial conditions,

Similar procedures of initial conditions even in the case of the magnetic monopoles are assumed.

)0(

)(

Bg

Bg

m

m

)0(

)(

gE

gE

Decay of the electric fieldproducing Nielsen-Olesen modes

fm/c

fm/c

Decay of the magnetic fieldproducing magnetic monopoles

ten times more rapid decay

We should note howthe gauge field rapidlydecays producingthe magnetic monopoles.

GeVQ

GeVmGeV

gEgB

ss 2,3/1

7.0,34.0~

)0()0(

tentative results

0.1fm/c

1 fm/c

|)01.0,(| 0 gE

Initial amplitude of Nielsen-Olesen unstable modes

|)01.0,(| effmm Bg

Initial amplitude of magnetic monopoles

The initial amplitude is 10 times larger than theamplitude of Nielsen-Olesen unstable modes

2/1

22/,2/)1(

2/)1(

22/,2/)1(

2/)1(

20 |)(|

|)(|1

zWe

zWe

gE ii

ii

2/1

22/,2/)2.31(

2/)2.31(

22/,2/)2.31(

2/)2.31(

2 |)(|

|)(|1

zWe

zWe

Bg ii

ii

effm

Pair creations of magnetic monopoles under magnetic fields by Schwinger mechnism

production rate of monopoles

3

12101.3~)

)(exp( 2

2

sseffm

effm andGeVQforBg

Egm

103.2~)exp( eff

eff

gE

gB

2)34.0( GeVgBgE effeff

GeVm 7.0

Tanji and Itakura, 2012

The production rate of the monopolesis about 10 times larger than thatof Nielsen-Olesen unstable modes.

Compare the production rate of the monopoles with that of Nielsen-Olesen unstable modes

production rate of N-O modes

conclusions

We have shown that the gauge fields generated afterhigh energy heavy ion collisions decay much more rapidlyproducing magnetic monopoles than Nielsen-Olesenunstable modes.

Although our calculation does not properly take into account precise initial conditions so that the result ispreliminary, it shows that the role of the magnetic monopoles in the realization of thermalized QGP is important.

time

J. Berges, S. Scheffer and D. Sexty, 2008

Numerical simulations

2|)0,(/),(| tptp LL exponential growth ofthe fluctuations

Kunihiro, Muller, Ohnishi, Schafer, Takahashi, Yamamoto (2010, 2013)

Exponential growth of the distancebetween nearby gauge fields

x

FF xtBxtBD 222' ))),(()),(((

B B’ t=0

t

FFD

Numerical simulations

gBDxtB 2||)),(( 22

( in our notations )