[PDF] - In tro duction Spin correlations for the and pairs PDF Document

SLIDE 1

SLIDE 2

In

tro duction Spin correlations for the

and
pairs

generated in hea vy ion colli sions and resp ectiv e angular correlations at the join t registration

f

hadron ic deca ys

f

t w

h

yp erons with space parit y nonconserv ation giv e imp

rtan

t information ab

ut

the c haracter and mec hanism

f

m ultipl e pro cesses The adv an tage

f

the

and
systems
v

er

ther
nes

is due to the fact that the P

dd

deca ys

p
and
p
serv

e as eectiv e analyzers

f

spin state

f

the

and
particles

In connection with this spin corre lations in the

and
systems

can b e rather easily distinguished and studied exp erimen tall y b y the metho d

f

momen ts

v

er the bac kground

f

a large amoun t

f

pro duced secondary particles This fact is esp ecially meaningful for the in v estigations

f

m ultipl e generation at mo dern and fu ture ion colliders lik e RHIC LHC NICA since the p

larization

parameters

esp

ecially for the

pair
are

v ery sensitiv e to the scenario

f

pro cess after the act

f

collision

f

relativistic hea vy ions

General

structure

f

the spin densit y matrix

f

the pairs

and
The

spin densit y matrix

f

the

and
pairs

just as the spin densit y matrix

f

t w

spin

particles in general can b e presen ted in the follo wing form

I
I
P
I
I
P
X

i

X

k

T

ik

i
k
in

doing so tr

Here
I

is the t w

ro

w unit matrix

x
y
z
is

the v ector P auli

p

erator x y

z
P
and

P

are

the p

larization

v ectors

f

rst and second particle

P
h
i

P

h
i
T

ik

h
i
k

i are the correlation tensor comp

nen

ts

In

the general case T ik

P

i P k

The

tensor with comp

nen

ts C ik

T

ik

P

i P k describ es the spin correlations

f

t w

particles
The

resp ectiv e

neparticle

densit y matrices are as follo ws

I
P
I
P

SLIDE 3 The trace

f

the correlation tensor is T

T

xx

T

y y

T

z z

h
i
The

eigen v alues

f

the

p

erator

equal
t
for

three triplet states total spin S

and
s
for

the singlet state total spin S

Let

us in tro duce the

p

erators

f

pro jection

n

to the triplet states and

n

to the singlet state

P

t

P

s

The

follo wing matrix equalities are satised

P
t
P

t

P
s
P

s

P

t

P

s

P

s

P

t

F
r

all the purely triplet states the trace

f

the correlation tensor T

whereas

for the purely singlet state T

No

w let us in tro duce the

p

erator

f

p erm utation

f

spin pro jections ha ving the form

P
I
I
Then

w e

btain
P
P

t

P

t

P
P

t

P
P

s

P

s

P
P

s

The

eigen v alues

f
P
equal
for

the three triplet states and

for

the singlet state

The

t w

particle

spin densit y matrix

ma

y b e decomp

sed

in to the triplet singlet and the mixed singlettrip let parts

t
s
ts
st
in

doing so

t
P

t

P

t

s
P

s

P

s

ts
P

t

P

s

st
P

s

P

t

The

relativ e fraction

f

the triplet states amoun ts to W t

tr
t
tr
P

t

P

t

tr
P

t

T
and

the relativ e fraction

f

the singlet state amoun ts to

SLIDE 4 W s

tr
s
tr
P

s

P

s

tr
P

s

T
in

doing so W t

W

s

T
W

t

W

s

W

s

Due

to the

rthogonalit

y

f

the pro jection matrices

P

t and

P

s

the

fol lo wing relation holds tr

ts
tr
st
tr
P

s

P

t

There

are also some matrix equalities con taining the

p

erator

f

p erm u tation

f

spin pro jections

P
P
t
P
t
P
s
P
s
P
ts
P
ts
P
st
P
st
It

should b e noted that the matrices

t

and

s

incorp

rate
nly

symmetric com binations

f

spin

p

erators

f

t w

particles

whereas the matrices

ts

and

st

include

nly

an tisymmetric com binations

f

spin

p

erators

The

explicit form ulas for the triplet

t
singlet
s
and

mixed

ts
st
comp
nen

ts

f

the spin densit y matrix are as follo ws

t
h
T
I
I
I
I
P
P
X

i

X

k

i
k
T

ik

T

k i

T
ik
i
s
T
I
I
ts
st
h
I
I
P
P
X

i

X

k

i
k

T ik

T

k i

i
i
k

SLIDE 5

ts
st
i
h
X

i

X

k

X

l

ik

l

i
k

P l

P

l

X

i

X

k

X

l

ik

l

i
I
I
i
P

k P l i

where
ik

l is the totally an tisymmetric tensor

f

the rd rank In doing so the matrices

t
s

and

are

Hermitian and the matrix

is

an tiHermitian

If

the rst particle and second particle ha v e dieren t relativistic momen ta the p

larization

v ectors P

P
and

the correlation tensor comp

nen

ts with left and righ t indices are sp ecied resp ectiv ely

in

the rest frames

f

the rst and second particle

in

the unied co

rdinate

axes

f

the cm frame

f

t w

particles
Spin

correlations and angular correlations at join t registration

f

deca ys

f

t w

particles

in to the c hannel

p
An

y deca y with the space parit y nonconserv ation ma y serv e as an ana lyzer

f

spin state

f

the unstable particle

The

normalized angular distribu ti

n

at the deca y

p
tak

es the form d w n d

n
P
n
Here

P

is

the p

larization

v ector

f

the

particle

n is the unit v ector along the direction

f

proton momen tum in the rest frame

f

the

particle
is

the co e cien t

f

P

dd

angular asymmetry

The

deca y

p
selects

the pro jections

f

spin

f

the

particle
n

to the direction

f

proton momen tum the analyzing p

w

er equals

n
No

w let us consider the double angular distributi

n
f

!igh t directions for protons formed in the deca ys

f

t w

particles

in to the c hannel

p
normalized

b y unit y

the

analyzing p

w

ers are

n
n
It

is describ ed b y the follo wing form ula

SLIDE 6 d

w

n

n
d
n
d
n
P
n
P
n
X

i

X

k

T

ik n i n k

where

P

and

P

are

p

larization

v ectors

f

the rst and second

particle

T ik are the correlation tensor comp

nen

ts n

and

n

are

unit v ectors in the resp ectiv e rest frames

f

the rst and second

particle

dened in the common

unied
co
rdinate

axes

f

the cm frame

f

the pair

i

k

f
g
fx

y

z

g

By

using the metho d

f

momen ts the comp

nen

ts

f

p

larization

v ectors and correlation tensor ma y b e determined as a result

f

a v eraging com bi nations

f

trigonometric functions

f

angles

f

proton !igh t

v

er the double angular distribut ion

P

i

hn

i i P k

hn

k i T ik

hn

i n k i

Here

hi

Z
d
w

n

n
d
n
d
n
d
n
d
n
n

x

sin
cos
n

y

sin
sin
n

z

cos
n

x

sin
cos
n

y

sin
sin
n

z

cos
where
and
and
are

the p

lar

and azim uthal angles

f

emission

f

protons in the rest frames

f

the rst and second

particle

resp ectiv ely

with

resp ect to the unied system

f

co

rdinate

axes

d
n
sin
d
d
and

d

n
sin
d
d
are

the elemen ts

f

solid angles

f

proton emission

The

double angular distribut ion ma y b e in tegrated

v

er all angles except the angle

b

et w een the v ectors n

and

n

cos
n
n
cos
cos
sin
sin
cos
A

t this in tegration the solid angle elemen t d

n
can

b e dened without losing generalit y

in

the co

rdinate

frame with the axis z b eing parallel to the v ector n

and

the solid angle elemen t d

n
is

dened in the co

rdinate

frame where the p

larization

parameters are sp ecied d

n
sin
d

d d

n
sin
d
d

SLIDE 7 here

is

the azim uthal angle

f

rotation

f

the v ector n

around

the v ector n

So

the angular correlation b et w een the proton momen ta at the deca ys

f

t w

particles

is expressed as follo ws d w cos

Z

d

w

n

n
d
n
d
n
dd
n
sin
d
In

doing so n

Z

n

d

d

n
n
Z

n

d

d

n
n

i n k

Z

n i n k d d

n
cos
ik
The

angular correlation b eing describ ed b y the form ula

d

w cos

T

cos

sin
d
is

determined

nly

b y the trace

f

the correlation tensor T

W

t

W

s

and

it do es not dep end

n

the p

larization

v ectors

singleparti

cl e states ma y b e unp

larized
So

nally w e ha v e d w cos

W

s

W

t

cos
sin
d
W

s and W t are relativ e fractions

f

the singlet state and triplet states resp ectiv ely

Correlations

at the join t registration

f

the deca ys

p
and
p
Due

to C P in v ariance the co e cien ts

f

P

dd

angular asymmetry for the deca ys

p
and
p
ha

v e equal absolute v alues and

pp
site

signs

The

double angular distribu tion for this case is as follo ws

d
w

n

n
d
n
d
n
P
n
P
n
X

i

X

k

T

ik n i n k

SLIDE 8 here

and
Th

us the angular correlation b et w een the proton and an tiproton momen ta in the rest frames

f

the

and
particles

is describ ed b y the expression d w cos

T

cos

sin
d
W

s

W

t

cos
sin
d
where
is

the angle b et w een the proton and an tiproton momen ta

Mo

del

f
neparticle

sources F

r

describing the momen tumenergy correlations and related spin cor relations

f

iden tical particles generated in pro cesses with high m ultipli cit y

the

mo del

f
neparticle

sources

constituen

ts

is

widely applied

In

the framew

rk
f

this mo del the sources emitting particles do not

v

er lap in space and time

it

is supp

sed

that the sizes

f

the sources themselv es are small as compared with the distances b et w een them In accordance with this eac h source is c haracterized b y the co

rdinate

x i

fr

i

t

i g In doing so the spatial region

ccupied

b y all the sources is v ery small as compared with the sizes

f

detectors measuring the particle momen ta p

and

p

and

the duration

f

the generation pro cess is v ery small as compared with the time parameters

f

detectors According to the mo del

f
neparticle

sources

Kop

ylo v P

dgorets

ky

particles

are emitted b y the sources indep end en tl y and incoheren t ly

Th

us at the early stage

f

particle formation

at

the stage

f

hadroniza tion

f

quarks and gluons

the

momen tumenergy and spin correlations are absen t

Correlations

for iden tical particles pro duced in the same ev en t

f

collision arise

n

accoun t

f

the eects

f

quan tum statistics

Bose

statis tics for particles with in teger spin and F ermi statistics for particles with halfin teger spin

and

nalstate in teraction

A

t presen t the mo del

f

sources is successfully used as w ell for the description

f

pair momen tum energy correlations

f

noniden ti cal particles conditioned exclusiv ely b y the nalstate in teraction

It

is essen tial that the pair momen tumenergy correlations and spin cor relations connected with the particle iden tit y and the nalstate in teraction dep end up

n

the momen tum dierence in the cm frame

f

the pair and as up

n

the parameters

f

the pro cess up

n

the spacetime c haracteris tics

f

the region

f

m ultiple generation

f

particles corresp

nding

to the

SLIDE 9 socalled freezeout

In

accordance with the mo del

f

sources the corre lations reac h the maxim um at relativ e momen ta b eing small as compared with the in v erse spacetime dimensions

f

the generation region whereas in the limit

f

large relativ e momen ta they disapp ear

This

fact has serv ed as a basis for elab

ration
f

the correlation metho d

the

socalled correlation fem toscop y

allo

wing

ne

to in v estigate exp erimen tall y the spacetime de v elopmen t

f

the pro cesses

f

m ultiple generation

f

leptons photons and hadrons

Since

the momen tumenergy correlations and spin correlations in the framew

rk
f

the mo del

f
neparticle

sources are substan tial

nly

in a suf cien tly narro w range

f

small relativ e momen ta in most

f

real ev en ts the particle densit y in phase space is small enough so that it w

uld

b e p

ssible

to consider the pair correlations

nly
disregarding

the triple correlations moreo v er the correlations

f

higher

rders
as

w ell as neglecting their in!uence up

n

the pair correlations

In

the mo del under consideration the sources ha v e a v ery broad momen tum sp ectrum as compared with the relativ e momen ta b eing c haracteristic for pair correlations th us the emission probabiliti es for eac h

f

the

ne

particle sources c hange insignican tl y under the v ariation

f

momen ta p

and

p

within

the correlation eect

that

is the socalled smo

thness

condition

The

metho d

f

correlation fem toscop y

based
n

the source mo del has b een used successfully enough for studying the correlations in pro cesses

f

collision

f

elemen tary particles

But

b y its essence this approac h is the most adequate

ne

namely for pro cesses

f

m ultipl e generation

f

particles in collisions

f

hea vy n uclei

Let

us remark that when describing the correlations

f

pions generated in m ultiple pro cesses the mo del

f

b

son

formation in socalled c haotic and coheren t states is used as w ell

In

the framew

rk
f

this mo d el the in terference correlations

f

iden tical b

sons

are partly suppressed in connection with the fact that the b

sons

b eing pro duced in the same quan tum state are already initial l y symmetrized just at their generation Analogous results follo w from the mo del incorp

ratin

g b

th

the

neparticle

sources and m ultiparticl e sources with the xed n um b er

f

particles

Ho

w ev er

n

accoun t

f

the P auli principl e these mo dels are inapplicabl e to fermions

SLIDE 10

Spin

correlations at the generation

f
pairs

in m ultiple pro cesses a The F ermistatistics eect leads not

nly

to the momen tumenergy

correlation

s at small relativ e momen ta

correlation

fem toscop y

but

to the spin correlations as w ell

The

follo wing relation holds in consequence

f

the symmetrization

r

an tisymmetrizati

n
f

the total w a v e function

f

an y iden tical particles with nonzero spin

b
sons

as w ell as fermions

S

L

Here

S is the total spin and L is the

rbital

momen tum in the cm fra me

f

the pair A t the momen tum dierence q

p
p
the

states with nonzero

rbital

momen ta die

ut

and

nly

states with L

and

ev en total spin S surviv e

Since

the particle spin is equal to

at

q

the
pair

is gener ated

nly

in the singlet state with S

Mean

time at the momen tum dierence q

there

are also triplet states generated together with the singlet state

Within

the con v en tional mo del

f
neparticle

sources emitting unp

larized

particles the triplet states with spin pro jections

and
are

pro duced with equal probabiliti es

If

correlations are neglected the singlet state is generated with the same probabilit y

the

relativ e w eigh ts are f W t

f

W s

When

taking in to accoun t the F ermi statistics and sw a v e nalstate in teraction whic h is essen tial at close momen ta

at
rbital

momen ta L

the

con tribution

f

nalstate in teraction is suppressed

the

fractions

f

triplet states and the singlet state b ecome prop

rtional

to the quan tities

W

t q

hcos

q xi W s q

hcos

q xi

B

int q

here

q

p
p
is

the dierence

f

momen ta x

x
x
is

the dierence

f

co

rdinates
f

t w

sources
In

the ab

v

e form ula

hcos

q xi

Z

W x cos q x d

x

SLIDE 11 is the F ermistatistics con tribution here W x is the distribu ti

n
f

dier ence

f

co

rdinates
f

t w

sources

B int q

is

the con tribution

f

sw a v e nalstate in teraction

f

t w

particles

In doing so Rq

W

t q

W

s q

hcos

q xi

B

int q

is

the correlation function describing the momen tumenergy correlations

f

t w

particles

with close momen ta

The

correlation function Rq

represen

ts the ratio

f

the t w

particle

sp ectrum to the noncorrelated bac kground whic h is constructed usually as a pro duct

f
neparticle

sp ectra from dieren t ev en ts at the same v alues

f

momen ta In terms

f

inclusiv e cross sections w e ha v e

d
d
p
d
p
Rq
tot

hnn

i

hni

d
d
p
d
d
p
where

n is the m ultipli ci t y and

tot

is the total in teraction crosssection

for

the P

isson

distributi

n
f

m ultipli ci t y w e ha v e hnn

ihni
b

The spin densit y matrix

f

t w

particles

with close momen ta at the emission

f

unp

larized
particles

has the follo wing structure

W

s q

s
W

t q

t

W s q

W

t q

Rq
hcos

q xi

B

int q

s
hcos

q xi

t
Here
s
I
I
is

the densit y matrix

f

the singlet state and

t
I
I
is

the densit y matrix

f

the unp

larized

triplet state a v eraged

v

er the spin pro jections

t
t
t
t
s
t
I
I
It

is easy to see that Eq

for
can

b e rewritten in the form

SLIDE 12

I
I
hcos

q xi

B

int q

Rq
The

correlation tensor comp

nen

ts

T

ik

C

ik

hcos

q xi

B

int q

hcos

q xi

B

int q

ik
dep

end up

n

the momen tum dierence as w ell as up

n

the spacetime parameters

f

the generation region the trace

f

the correlation tensor amoun ts to T

X

i T ii

hcos

q xi

B

int q

hcos

q xi

B

int q

Th

us

n

accoun t

f

the eects

f

quan tum statistics and nalstate in teraction at small relativ e momen ta t w

iden

tical particles initial ly unp

larized

P

P
and

noncorrelated b y spins remain unp

larized

as w ell but their spins b ecome correlated

A

t q

w

e

btain

hcos q xi

T

ik

ik
singlet

state

On

the

ther

hand in the limit

f

large q

hcos

q xi

B

int q

Rq
T

ik

ie

b

th

the momen tumenergy and spin correlations v anish

c

No w let us consider the emission

f
particles

with equal p

larization

v ectors e P

e

P

e

P

It

should b e noted that at the stage

f

emission b y sources correlations are absen t The fraction

f

the triplet state with the total spin pro jection

n

to the direction

f

e P and the resp ectiv e constituen t

f

the spin densit y matrix are as follo ws

f

W t

e

P

t
I
l
I
l
here

e P

j

e Pj and l is the unit v ector directed along e P

Analogously
w

e ha v e the follo wing fractions and spin densit y matrix constituen ts for the triplet state with total spin pro jection

SLIDE 13 f W t

e

P

t
I
l
I
l
for

the triplet state with total spin pro jection

f

W t

e

P

t
I
I
l
l
and

for the singlet state f W s

e

P

s
I
I
In

doing so the fractions

f

spin states f W t

f

W t

f

W t

f

W s

b

ey the normalization condition f W t

f

W t

f

W t

f

W s

and

the primary spin densit y matrix is describ ed b y the expression

f

W t

t
f

W t

t
f

W t

t
f

W s

s
I
e

P

I
e

P

A

t lo w relativ e momen ta

n

accoun t

f

F ermi statistics and nalstate in teraction the fractions

f

triplet states and singlet state c hange and b e come prop

rtional

to the follo wing quan tities W t

q
e

P

hcos

q xi

W

t

q
e

P

hcos

q xi

W

t

q
e

P

hcos

q xi

W

s q

e

P

hcos

q xi

B

int q

The

inclusiv e crosssection

f

generation

f

the

pair

with close mo men ta is prop

rtional

to the correlation function describing the momen tum energy correlations

SLIDE 14 Rq

W

t

q
W

t

q
W

t

q
W

s q

e

P

hcos

q xi

e

P

B

int q

In

doing so the renormalized densit y matrix is determined b y the relation

Rq
W

t

q
t
W

t

q
t
W

t

q
t
W

s q

s
In

accordance with this the p

larization

parameters

f

the

particles

renormalized due to the eects

f

F ermi statistics and sw a v e nalstate in teraction tak e the form P

P
Rq
hcos

q xi e P

T

ik

Rq
hcos

q xi e P i e P k

e

P

hcos

q xi

B

int q

ik
Irresp

ectiv e

f

the primary p

larization

e P

at

the momen tum dierence q

nly

the singlet state

f

the

pair

is realized and the renormalized p

larization

v ectors P

P
tend

to zero

The

sw a v e nalstate in terac tion amplies the predominan t role

f

the singlet state If e P

then

in the limit q

the

generation

f
pairs

is forbidden

in

full accordance with the P auli principl e

d

In the cm frame

f

the

pair

w e ha v e q

f

kg where k is the momen tum

f
ne
f

the particles In doing so the momen tum k is connected with the relativ e momen tum q in the lab

ratory

frame b y the Loren tz transformation

w

e use the unit system with

c
k
q
qv

v jv j

v

q

here

v

p
p
is

the v elo cit y

f

the

pair

in the lab

ratory

frame

v
is

the Loren tz factor q

p
p
and

q

The

Loren tz transformations

f

co

rdinates

are giv en b y the expres sions

SLIDE 15 r

r
rv

v jv j

v

t t

t
vr
where

r

x
x
and

t

t
t
The

in terference term connected with iden tit y quan tum statistics is determined b y the expression hcos q xi

hcos

kr

i
Z

W v r

coskr
d
r
where

W v r

Z

W x d t

Z

W r

t
d

t

is

the distribut ion

f

co

rdinate

dierence b et w een t w

sources

in the cm frame

f

the

pair
Mean

time the con tribution

f

sw a v e nalstate in teraction is expressed as follo ws

at

the sizes

f

the generation region in the cm frame exceeding the eectiv e radius

f

in teraction

f

t w

particles
B

int q

B
k

v

Z

W v r

bk

r

d
r
where

the function bk r

has

the structure

bk

r

jf
k

j

r
Re
f
k
e

ik r

cos

kr

r
jf
k

j

d
r
Here

k

jkj

r

jr
j

f

k
is

the amplitude

f

lo wenergy

scattering

In the framew

rk
f

the eectiv e radius theory

f
k
a
d
a
k
i

k a

where

b y denition a

is

the length

f

sw a v e scattering and d

k

d d k

Re
f
k
is

the eectiv e radius

The

in tegral

with

expression

inside

appro ximately tak es in to accoun t the dierence

f

the true w a v e function

f

t w

in

teracting

particles

SLIDE 16 with the momen ta k and k at small distances from the asymptotic w a v e function

f

con tin uous sp ectrum

The

information ab

ut

the parameters

f
scattering

is con tained in the w

rks

studying double h yp ern uclei and pair correlations in the reactions with formation

f

t w

particles
see

for example

Analysis
f

the exp erimen tal data leads to the conclusion that the length

f
scattering

is comparable b y magnitude

fm
with

the length

f

neutronneut ron scattering

In

the case

f

Gauss distribu ti

n
f

co

rdinates
f

t w

indep

end en t sources in the lab

ratory

frame with the meansquare radius p hr

i
p

hr

i
p

r

and

the meansquare emission time p ht

i
p

ht

i
w

e

btain

for the function W v r

W

v r

r
p

r

v
exp
r
r
n
r
r
n
r
v
In

doing so hcos kr

i
exp
k
r
v
kn
r
and

the con tribution

f

sw a v e nalstate

in

teraction at the momen tum k

maxim

um v alue

is

as follo ws

B
v
a
r
A
d
p
r
p
C

a

where
q

r

v
A
u

arcsin u C

u

ln

u
u
u
v

p r

Spin

correlations at the generation

f
pairs

in m ultiple pro cesses In the framew

rk
f

the mo del

f

indep end en t

neparticle

sources spin correlations in the

system

arise

nly
n

accoun t

f

the dierence b et w een the in teraction in the nal triplet state

S
and

the in teraction in the nal singlet state

A

t small relativ e momen ta the sw a v e in teraction pla ys

SLIDE 17 the dominan t role as b efore but con trary to the case

f

iden tical particles

in

the case

f

noniden ti cal particles

the

total spin ma y tak e b

th

the v alues S

and

S

at

the

rbital

momen tum L

In

doing so the in terference eect connected with quan tum statistics is absen t

If

the sources emit unp

larized

particles then in the case under consid eration the correlation function describing momen tumenergy correlations has the follo wing structure

in

the cm frame

f

the

pair
Rk

v

B
t

k v

B
s

k v

The

spin densit y matrix

f

the

pair

is giv en b y the form ula

I
I
B
t

k v

B
s

k v

Rk

v

and

the comp

nen

ts

f

the correlation tensor are as follo ws T ik

B
t

k v

B
s

k v

B
t

k v

B
s

k v

ik
here

the con tributions

f

nalstate triplet and singlet

in

teraction are determined b y the expression

analogously

to Eqs for the

in

ter action

with

the replacemen t cos kr

e

i k r

in

Eq

wing

to the noniden ti t y

f

the particles

and
B
st

k v

jf
st

k j

h
r
i
Re
f
st

k

h

e ik r

e

i kr

r
i
k

jf

st

k j

d

d k

Re
f
st

k

A

W v

where

f

st

k

is

the amplitud e

f

the sw a v e lo wenergy singlet

triplet
scattering
Some

information

n

the

in

teraction at lo w energies ma y b e

btained

b y in v estigating for example the annihilati

n

pro cess p

p
A

t su cien tly large v alues

f

k

ne

should exp ect that

B
s

k v

B
t

k v

SLIDE 18 In this case the angular correlations in the deca ys

p
p
connected

with the nalstate in teraction are absen t

T

ik

T
Angular

correlations in the deca ys

p
and
p
and

the mixed phase Th us at su cien tly large relativ e momen ta

for

k

m
ne

should exp ect that the angular correlations in the deca ys

p
and
p
connected

with the in teraction

f

the

and
h

yp erons in the nal state ie with

neparticle

sources

are

absen t

But

if at the considered energy the dynamical tra jectory

f

the system passes through the socalled mixed phase then the t w

particle

sources consisting

f

the free quark and an tiquark

start

pla ying a noticeable role

F
r

example the pro cess s

s
ma

y b e discussed

In

this pro cess the c harge parit y

f

the pairs s

s

and

is

equal to C

LS
where

L is the

rbital

momen tum and S is the total spin

f

the fermion and an tifermion

Mean

time the C P parit y

f

the fermion an tifermion pair is C P

S
In

the case

f
negluon

exc hange C P

and

then S

ie

the

pair

is generated in the triplet state in doing so the trace

f

the correlation tensor T

Ev

en if the frames

f
negluon

exc hange are

v

erstepp ed the quarks s and

s
b

eing ultrarelativi sti c in teract in the triplet state

S
In

so doing the primary C P parit y is C P

and

due to the C P parit y conserv ation the

pair

is also pro duced in the triplet state Let us denote the con tribution

f

t w

quark

sources b y x

Then

at large relativ e momen ta T

x
Apart

from the t w

quark

sources there are also t w

gluon

sources b eing able to pla y a comparable role Analogously with the annihilati

n

pro cess

in

this case the trace

f

the correlation tensor is describ ed b y the form ula

the

pro cess g g

is

implied

T
sin
sin

SLIDE 19 where

is

the v elo cit y

f
and
in

the cm frame

f

the

pair
is

the angle b et w een the momen ta

f
ne
f

the gluons and

in

the cm frame

see
A

t small

the
pair

is pro duced in the singlet state

total

spin S

T
whereas

at

in

the triplet state

S
T
Let

us remark that at ultrarelativi sti c v elo cities

ie

at extremely large relativ e momen ta

f
and
b
th

the t w

quark

and t w

gluon

mec hanisms lead to the triplet state

f

the

pair
T
In

the general case the app earance

f

angular correlations in the deca ys

p
and
p
with

the nonzero v alues

f

the trace

f

the correlation tensor T at large relativ e momen ta

f

the

and
particles

ma y testify to the passage

f

the system through the mixed phase

Summary

It is advisable to in v estigate the spin correlations

f
and
pairs

pro duced in relativistic hea vy ion collisions

see

also

The

spin correlations are studied b y the metho d

f

angular correlations

metho

d

f

momen ts

The

spin correlations as w ell as the momen tumenergy

nes

mak e it p

ssible

to determine the spacetime c haracteristics

f

the generation region and b esides the parameters

f

lo wenergy scattering

f
n
and
n
The

spin correlations should b e in v estigated join tly with the momen tum energy correlations The authors are grateful to AV Efremo v AO Kec hec h y an R Led nic ky

Y

uA P anebratsev OV T ery aev and MV T

k

arev for the in terest in this w

rk

and useful discussions

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