Particle production by Schwinger mechanism

Naoto Tanji
Univ. of Tokyo
ATHIC2008@tukuba October 14

Non‐perturbative particle production mechanism
in heavy‐ion collisions

color flux tube model• Treat gluon fields as classical electric fields.
• The electric field decays into particles
by Schwinger mechanism.
The momentum distribution of created particles
and its time‐evolution are not fully understood.

We investigate• Dynamical view of pair creation

Non‐perturbative particle production mechanism
in heavy‐ion collisions

color flux tube model• Treat gluon fields as classical electric fields.
• The electric field decays into particles
by Schwinger mechanism.
The momentum distribution of created particles
and its time‐evolution are not fully understood.
ElectricMagnetic

Color glass condensate(Lappi,McLerran 2006)

Both electric and magnetic field exist

We investigate• Dynamical view of pair creation• Effects of a magnetic fieldSchwinger mechanism

Schwinger 1951

Non‐perturbative pair creation mechanism in an uniform and static classical electric field

Vacuum persistence probability
20, out | 0, in = exp(−2 Im L) = exp(− ∫ wd 4 x)
Pair creation probability per unit volume and unit timesemi‐classical picture of
Schwinger mech.

⎛ nπm 2 ⎞2(eE ) 2 ∞ 1⎟⎟ (fermion)exp⎜⎜ −w=3 ∑ 2(2π ) n =1 n
⎝ eE ⎠

no direct information about

particle number
time evolution

Schwinger mechanism

Schwinger 1951

Non‐perturbative pair creation mechanism in an uniform and static classical electric field

Vacuum persistence probability
20, out | 0, in = exp(−2 Im L) = exp(− ∫ wd 4 x)
Pair creation probability per unit volume and unit timesemi‐classical picture of
Schwinger mech.

⎛ nπm 2 ⎞2(eE ) 2 ∞ 1⎟⎟ (fermion)exp⎜⎜ −w=3 ∑ 2(2π ) n =1 n
⎝ eE ⎠

direct information about

particle mean number

particle number
time evolution

Need to define particle picture specificallysudden‐switch‐on electric field
To get dynamical view of pair creation,
we need to deal with non‐steady electric fields.

In a static electric field,
a distribution is also static.

tｔ0“observation”

E≠0

E=00z

switch‐onCanonical quantization in a constant electric field
Klein‐Gordon field and Dirac field interacting withthe classical electric fieldclassical field,

In the case of non‐Abelian electric field
gluon field and quark field obey the equations of
the same form as in Abelian case, up to first order.

,Ambjorn,Nielsen,Olesen 1979
Gyulassy,Iwazaki 1985

But in the case of gluon, magnetic moment causes a difficulty

Quantization

Particle pictureis established.

ambiguity in an external fieldInstantaneous particle picture
tｔ0

A=00

evolveA z = − Etz

positive frequency

Instantaneous particle picture
tｔ0

A z = − Etevolve

A=00

mixture of positive and negative frequency

zconsequence of pair creation

positive frequency

Instantaneous particle picturetinstantaneous positive and negative frequencyｔ0(t )(t )φ(t)φ+ p, − p (t )0

A z = − Eta solution under the gauge A z = − Et 0evolve

A=000zpositive frequency

Instantaneous particle picture
decompose

into positive and negative frequency instantaneously

Field expansion

Instantaneous particle picture (creation and annihilation operator) is introduced.
particle pair distribution function

boson distribution

accelerated according toclassical eq. of motion

p z = eEt

a = 0.01

a = 0.3
Particles are created firstwith about 0 momentum

fermion distribution

a = 0.01

a = 0.3

m T2m 2 + p T2=
a =2 eE
2 eE

boson distribution

accelerated according toclassical eq. of motion

m T2m 2 + p T2=
a =2 eE
2 eE

p z = eEt

a = 0.01

a = 0.3
Particles are created firstly
with about 0 momentum

fermion distributionn p (t ) ≤ 1 because ofPauli’s exclusion principal

a = 0.01

a = 0.3

Larger a , longer timeto create first particles.

Back reaction(numerical , 1+3 dim)

Kluger et al. 1991

Couple the field eq. andMaxwell eq. dE = − jdt

oscillating distribution
in momentum space

The time‐evolution of the electric field

Bose enhancement

Pauli blocking

bosonfermion

Time scale of back reactiontV : the time the electric field strength becomes 0 at firstt ~>V( 2π ) 3 π m 2 2 gE 0e≈
3N d g E0

3 e π m 2 gE 0tV2

N d g 2 ⋅ gE 0 [GeV 2 ][fm]roughly estimatedfrom the current withoutback reaction

N d : a number of inner
degrees of freedom

Time scale of back reactiontV : the time the electric field strength becomes 0 at firstt ~>Vm=0( 2π ) 3 π m 2 2 gE 0e≈
3N d g E0

gE0 = Q ≈ 1 GeV2S

23 e π m 2 gE 0tV

2N d g 2 ⋅ gE 0 [GeV 2 ]

t ≈V

3Nd g

2[fm]

[fm]

roughly estimatedfrom the current withoutback reaction

N d : a number of inner
degrees of freedom

If g 2 > O (1) , t V is the order or less than 1 fm.
~

Time scale of back reactiontV : the time the electric field strength becomes 0 at firstt ~>Vm=0( 2π ) 3 π m 2 2 gE 0e≈
3N d g E0

gE0 = Q ≈ 1 GeV2S23 e π m 2 gE 0tV

2

N d g 2 ⋅ gE 0 [GeV 2 ]

t ≈V

3Nd g

2

[fm]

[fm]

roughly estimatedfrom the current withoutback reaction

N d : a number of inner
degrees of freedom

If g 2 > O (1) , t V is the order or less than 1 fm.
~
Furthermore, a longitudinal magnetic field
makes t V smaller.

Time scale of back reactiontV : the time the electric field strength becomes 0 at firstt ~>Vm=0( 2π ) 3 π m 2 2 gE 0e≈
3N d g E0

gE0 = Q ≈ 1 GeV2S

23 e π m 2 gE 0tV

2

N d g 2 ⋅ gE 0 [GeV 2 ]

t ≈V

3Nd g

2

[fm]

[fm]

roughly estimatedfrom the current withoutback reaction

N d : a number of inner
degrees of freedom

If g 2 > O (1) , t V is the order or less than 1 fm.
~Furthermore, a longitudinal magnetic field
makes t V smaller.
This decay mechanism of an electric field plays an important rolein the initial stage of heavy‐ion collisions.

The effect of a magnetic fieldLongitudinal magnetic field• Landau level• Zeeman effect

E = ( 0, 0, E ) , B = ( 0, 0 , B )ElectricMagnetic

pT2 = px2 + p y2 → (2n + 1)eB (n = 0,1,2,L)± 2 seB

B=0

B≠0continuous spectrum

spin 0spin 1/2

lowest level

eB0753
10

The effect of a magnetic fieldLongitudinal magnetic field• Landau level• Zeeman effect

E = ( 0, 0, E ) , B = ( 0, 0 , B )ElectricMagnetic

pT2 = px2 + p y2 → (2n + 1)eB (n = 0,1,2,L)± 2 seB

B=0B≠0continuous spectrum

spin 0spin 1/2

lowest level

eB0753
10

The strong magnetic field
makes particles “heavy”
and suppresses pair creation.

The effect of a magnetic fieldLongitudinal magnetic field• Landau level• Zeeman effect

E = ( 0, 0, E ) , B = ( 0, 0 , B )Electric
Magnetic

pT2 = px2 + p y2 → (2n + 1)eB (n = 0,1,2,L)± 2 seB

B=0

B≠0

75310

independent of B

continuous spectrum

spin 0spin 1/2

lowest leveleB0

The strong magnetic field
makes particles “heavy”
and suppresses pair creation.

Pair creation isnot suppressed.

The number of modes degenerating
in one Landau level is proportional to B

The effect of a magnetic field
Total particle number and current of fermionsare enhanced by the magnetic field.
The time‐evolution becomes fasterdue to the magnetic field.

total particle number density(without back reaction)

time‐evolution of the electric field(with back reaction)Instability in spin‐1 case
• Landau level• Zeeman effect

pT2 = px2 + p y2 → (2n + 1)eB (n = 0,1,2,L)± 2 seB

B=0B≠0continuous spectrum

spin 0spin 1/2

lowest level

eB0

753
10
spin 1− eBnegative!!

Instability in spin‐1 case
• Landau level• Zeeman effect

pT2 = px2 + p y2 → (2n + 1)eB (n = 0,1,2,L)± 2 seB

B=0

B≠0continuous spectrum

spin 0spin 1/2

lowest level

eB0

75310

classical instability
open problem

spin 1− eBnegative!!

Nielsen‐Olesen instability
ωp =p z2 + m T2 can be pure imaginary.

exp(m iω p t ) become unstable.

Summary• We have revealed the momentum distribution of created particles
and its time‐evolution in uniform electric fields
by defining particle picture instantaneously.
• The time scale of the decay process of the initial electric field
due to back reaction has been estimated.
• We have shown that a magnetic field speeds up
the time‐evolution of the fermion system.

Remaining problems• More realistic configuration of an electromagnetic field,
especially the case that a field exists only inside the light‐cone• Interaction through quantum gauge field (collision)• Instability in gluon case