in Boost-Invariantly Expanding Electric Fields

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16 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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Naoto Tanji



High Energy Accelerator Research Organization (KEK), Japan

Standard and novel QCD phenomena at hadron colliders

May 30, 2011 ECT*

Nonperturbative Particle Production

in Boost
-
Invariantly Expanding Electric Fields
and Two
-
Particle Correlations

Particle production in the initial stage of heavy
-
ion collisions

Introduction

classical color electric fields

in the longitudinal direction



the Low
-
Nussinov model



the color glass condensate

color flux tubes

Particle production in the initial stage of heavy
-
ion collisions

Introduction

classical color electric fields

in the longitudinal direction



the Low
-
Nussinov model



the color glass condensate

Glasma flux tubes

Nonperturbative particle production via the Schwinger mechanism

How is high energy matter produced from the electric fields?

1/
Q
s

Lappi, MacLerran (2006)

Schematics of semi
-
classical tunneling

Non
-
perturbative particle production in a strong classical electric field

Schwinger

mechanism

Schwinger 1951

at RHIC

Very strong fields

Schematics of semi
-
classical tunneling

HICs may be a promising playground to study the strong field physics.

i.e. Field intensity to create e+ e
-

pairs



Not yet directly observed by experiments

Non
-
perturbative particle production in a strong classical electric field

Schwinger

mechanism

Schwinger 1951



Boost
-
invariant expansion of the electric fields

Features of this study



Field theoretical treatment
(not semi
-
classical kinetic description)

multi
-
particle correlations

Electric fields exist only between

two nuclei receding in nearly

the speed of light.



Real
-
time evolution

Heavy
-
ion collisions are a dynamic system.

Back reaction of the pair creation

Outline



Uniform field



Quantization in a pair
-
creating background



Pair creation in a constant field



Back reaction



Isotropization of system



Boost
-
invariantly expanding electric field



Pair creation in a constant field



Two
-
particle correlations



Back reaction

Quantization in a pair
-
creating background field

z

t

0

evolve

positive frequency

switch
-
on

superposition of positive

and negative frequency

z

t

0

evolve

superposition of positive

and negative frequency

time
-
dependent

Bogoliubov transformation

is non
-
zero.

Quantization in a pair
-
creating background field

longitudinal momentum distribution

transverse momentum distribution

created with approximately

0 longitudinal momentum

accelerated according to

classical eq. of motion

approximately

Gaussian form



constant field



QED






The time
-
evolution of the momentum distribution (phase space density)


for extension to quark production,

see N.T. Ann.Phys. (2010)

Back reaction

charge current

charge current generated by

particles and antiparticle produced

from the electric field

Maxwell equation

electric field

longitudinal momentum distributions

a few

fm/c

plasma oscillation

energy density

Isotropization of the system

The initial state with the longitudinal electric field is quite anisotropic.

In locally thermalized QGP, pressure must be isotropic.

Pressure of particles

examine the role of the pair creation for the isotropization of pressure

energy
-
momentum tensor

L , particle

T , particle

L , field

T , field

pressure

pressure

L , total

T , total

Degree of anisotropy is moderated by pair creation

even though this system is collision
-
less plasma.

Created quarks generate

positive longitudinal pressure.

Due to the back reaction,

the field strength is weakened.

Negative longitudinal pressure is compensated
by pressure generated by particles.

total pressure

Pressure

nucleus

nucleus

boost
-
invariant configuration

boost
-
invariantly expanding electric field

boost
-
invariantly expanding fluid (QGP)

?

Bjorken’s flow (the scaling flow)

Pair creation in boost
-
invariantly expanding electric fields

How do particles emerge from the boost
-
invariant field ?

coordinates

Field quantization in a curved coordinates system

boost
-
invariance

translational invariance in

Particles are created as an eigenstate of momentum conjugate to

The relations between two kinds of the creation/annihilation operators

denoted as

longitudinal “momentum” distribution

transverse momentum distribution



constant field



QED




The time
-
evolution of the momentum distribution (phase space density)


accelerated by the field

according to the classical

EOM

approximately

Gaussian form

created with approximately

0 longitudinal “momentum”

Similar to the uniform field case

longitudinal “momentum” distribution

transverse momentum distribution



constant field



QED




The time
-
evolution of the momentum distribution (phase space density)


accelerated by the field

according to the classical

EOM

approximately

Gaussian form

created with approximately

0 longitudinal “momentum”

Similar to the uniform field case

What is the physical meaning of ?

longitudinal momentum distribution

in the moving frame

is a momentum observed in a moving frame with velocity of

accelerated by the field

according to the classical

EOM

created with approximately

0 longitudinal “momentum”



created with 0 momentum

created with the same velocity

distribution as the Bjorken’s flow



Due to acceleration by the field, the velocity


distribution deviates from the scaling one.

Rapidity distribution

The rapidity distribution is flat because now the boost
-
inv. is perfect.

independent of rapidity

Two
-
particle correlations

near
-
side ridge phenomena

observed at RHIC and LHC

long
-
range rapidity correlations



In the uniform electric field

(Fukushima
-
Gelis
-
Lappi, 2009)

The correlation is given by the Fourier transform of the momentum distribution.

correlated only if (Bose
-
Einstein correlation)



In the expanding electric field

correlated even if

The correlation length in the rapidity space is about the inverse of

the width of the function in the
-
space.

Short range correlation

in the transverse momentum

The velocity distribution is

exactly the scaling one.

Long range rapidity correlation

The rapidity correlation can be long
-
range,

but it depends on the dynamics.

The time
-
dependence of the longitudinal correlation


momentum distribution

rapidity correlation

If the life time of the electric field is short enough ,

the long range correlation can survive.

Back reaction
(preliminary)

1+1 dim scalar QED

Maxwell equation

current density

electric field

number density

longitudinal momentum distribution

damping

energy density

longitudinal pressure

Energy is not conserved in total,

but is locally conserved.

Summary



Real
-
time dynamics of particle pair creation in strong electric fields


has been revealed.




Particles created from boost
-
invariantly expanding electric fields have


the scaling velocity distribution from the first instant they are created.




A long
-
range rapidity correlation can be emerge from the Schwinger


mechanism in the boost
-
invariant field.




Negative longitudinal pressure of the electric field is compensated


by pressure of created particles.



Outlook



back reaction in 1+3 dim system




gluon pair production




production of photons and dileptons from quarks which are created


from the color electric field




finiteness in the transverse direction and transverse expansion




perturbative vs nonperturbative particle production





1/
Q
s

back up slides

longitudinal momentum distributions

scalar particles

Bose enhancement

total number density

continues to increase until

the field are weakened sufficiently

spinor particles

total number density

Number density is rather saturated.

longitudinal momentum distributions

Pauli blocking



Landau level



Zeeman splitting

Electric

Magnetic

“light”

“heavy”

Glasma

Effects of a magnetic field

Longitudinal magnetic field

A strong magnetic field makes
scalar particles “heavy”

and suppresses pair creation.

Pair creation is

not suppressed.

independent of B

The number of modes degenerating

in one Landau level is proportional to B

charge current density

electric field

Quark pair production is enhanced by the magnetic field

number density

Pair creation is enhanced

Time evolution gets faster

current density

conduction current

polarization current

pair creation production of an electric dipole

anomalous distribution

Pressure

real particles

virtual processes

Back reaction

electric flux tube

string breaking

rarefaction of the field

Hadron gas

Quark
-
Gluon plasma

e.g. Lund model



weak fields



short
-
range correlation



strong fields



long
-
range correlation ?

Abelianization

a constant vector indicating

the color direction of electric fields

diagonalize

rotated weight diagram

Each quark field couples to the Abelianized gauge field via coupling .

Relation between and

gauge (Casimir) invariant quantity

characterizing the color direction (Nayak 2005)


characterizes the relative weight

between two different U(1) directions.

Color direction dependence

Color direction dependence

color current density

electric field

quark number density

Color direction dependence is not zero but very small for field quantities.


-
dependence

comes from here.

These quantities are obtained by summing up all the color components.

-
dependence enters only through the elementary symmetric polynomials

independent of

e.g.

The Pauli blocking



Old particles suppress the subsequent pair creation.



In particular, if occupation exceeds 1/2 , pair annihilation occurs.

Pair creation with an initial distribution in a constant electric field.

accelerated initial particles

Pauli blocking factor

e.g.

particles created

from vacuum

Interference between matter fields

Actually, there is one more term in .

sensitive to the phases of the Bogoliubov coefficients

neglecting the interference term

The meaning of the “momentum”

number distribution

Expectation of the number operator in
-
basis.

c.f. the expectation by the eigenstate of


usual momentum

independent of

rapidity nor

The eigenstate of contains all rapidity states with equal weight.

investigate the expectation of the one
-
particle state

Energy
-
momentum tensor

Cartesian

Lorentz
-
boost with the velocity

Energy
-
momentum tensor in the
-
frame

The frame moving with

the velocity

The meaning of the “momentum”

is

energy

longitudinal momentum

longitudinal pressure

observed in the
-
frame.

the expectation with the one
-
particle state

is the longitudinal momentum observed in the
-
frame.

means that a particle has the scaling velocity distribution.

The meaning of the “momentum”