Table Of ContentSqueezed and entangled collinear gluon states
at the non-perturbative stage of QCD jet evolution
V.I. Kuvshinov, V.A. Shaparau
Institute of Physics, National Academy of Sciences of Belarus
F. Skaryna av. 68, 220072 Minsk, Belarus
E-mail: [email protected]; [email protected]
Abstract
At the non-perturbative stage of jet evolution, fluctuations of collinear gluons
arelessthanthoseforcoherentstatesthatisindicationofgluonsqueezedstates. We
show that gluon entangled states which are closely related with two-mode squeezed
states of gluon fields can appear by analogy with corresponding photon states in
4
quantum optics.
0
0
2
1 Introduction
n
a
J
Analogies between multiple hadron in HEP and photon production in quantum optics
2
2 (QO) discussed long ago [1]-[4]. It was shown, in particular, that the general distribution
that characterizes e+e−,pp¯, neutrino-induced collisions is a k-mode squeezed state distri-
1
v bution [5, 6], the multiplicity distribution of the pions has been explained by formalism
1
of the squeezed isospin states [7].
6
1 Squeezed states (SS), introduced by Stoler [8] and named by Hollenhorst [9], pro-
1 voke great interest in connection with their uncommon properties: they can have both
0
sub-Poissonian (forcoincide phases) andsuper-Poissonian (forantiphases) statistics corre-
4
0 sponding to antibunching and bunching of photons [10]-[13] and characteristic behaviour
/
h of the factorial and cumulant moments [14]. Moreover oscillatory behaviour of the mul-
p
tiplicity distribution of the photon SS [11] is different from Poissonian and Negative
-
p binomial distributions (NBD). The squeezed light is generated from the coherent one by
e
h nonlinear devices and is pure quantum non-perturbative phenomenon [11]-[13].
: Studying non-perturbative evolution of gluon states prepared by perturbative cascade
v
i stage in jets [15] we have proved within Quantum chromodynamics (QCD) that this stage
X
of jet evolution can be a source of gluon SS by analogy with nonlinear devices in QO for
r
a
photon [16]-[18]. In these papers we investigated single-mode squeezing effect for virtual
gluons with defined colour andvector components during small temporalevolution. Using
the Local parton hadron duality it is easy to show that in this case behaviour of hadron
multiplicity distribution in jet events is differentiated from the negative binomial one that
is confirmed by experiments for pp,pp-collisions [19]-[21].
At the same time two-mode photon SS [12] in the limit of infinite squeezing are iso-
morphic to the Bell states [22] which have been introduced in relation to the Einstein-
Podolsky-Rosen (EPR) paradox [23] and they are one of the examples of the entangled
states for two polarizations [24]. The states
Φ+ = ( + )/√2, Φ− = ( )/√2,
| i |li1|li2 |↔i1|↔i2 | i |li1|li2 −|↔i1|↔i2 (1)
Ψ+ = ( + )/√2, Ψ− = ( )/√2
| i |li1|↔i2 |↔i1|li2 | i |li1|↔i2 −|↔i1|li2
1
are the basis of the Bell states. Each of these entangled states, for example, Φ± , has a
| i
uncommon property: if one photon is registered with defined polarization (for example,
with polarization ), the other photon immediately becomes opposite polarized (longitu-
l
dinal polarization). Thus a measurement over one particle has an instantaneous effect on
the other, possibly located at a large distance. In particular, recently it was shown that
it is possible for two photons to travel a total of 600 metres through free space and still
remain ”entangled” [25].
At finitesqueeze factorr (r ispositive) a continuous variables entangled stateisknown
from quantum optics as a two-mode squeezed state [12, 24]
1 ∞
f = S(r) 0 0 = (tanhr)n n n , (2)
1 2 1 2
| i | i | i coshr | i | i
n=0
X
where S(r) = exp r(a+a+ a a ) is the operator of two-mode squeezing. Rewriting the
{ 1 2 − 1 2 }
expression (2) at small squeezing parameter r
f 0 0 + r 1 1 , (3)
1 2 1 2
| i ≃ | i | i | i | i
it is easy to show that the state vector f describes the entangled state. Indeed, the
| i
entanglement condition of considering photon state can be verified by investigation of the
conditional probability P (Y /X )(i,j = 1,2) [26, 27]
j i
δ
Y X f 2 ij
j i
P (Y /X ) = kh |h | ik = or (4)
j i
f X X f
h | iih i | i 1 δij,
−
where X = 0 , X = 1 , Y = 0 , Y = 1 .
1 1 2 1 1 2 2 2
| i | i | i | i | i | i | i | i
The dimensionless coefficient
1/2
a a+ 2 + a a 2
y = | 1 2| | 1 2| (5)
2(a+a +1/2)(a+a +1/2)
1 1 2 2
is the measure of entanglement for two-mode states [28], 0 y < 1 (entanglement is
≤
not observe when y = 0). Here a a+ = a a+ a a+ , a ,a+ are the annihilation and
i j h i j i−h iih j i i j
creation operators correspondingly. Averaging the annihilation and creation operators in
the expression (5) over the vector f (3) at small squeeze factor we have
| i
y √2r. (6)
≃
It is obvious that maximal coefficient of the entanglement is defined by maximal possible
value of the squeeze factor r1: y 0.462 at r = 0.327.
max
≈
Two-mode gluon states with two different colours can lead to qq¯-entangled states.
Interaction of thequark entangled states withstochastic vacuum (quantum measurement)
has a remarkable property, namely, as soon as some measurement projects one quark onto
a state with definite colour, the another quark also immediately obtains opposite colour
that leads to coupling of quark-antiquark pair, string tension inside qq¯-pare and free
propagation of colourless hadrons. Therefore the investigation of the gluon entangled
states is an issue of the day.
1Here calculations is restricted by first order on the squeeze factor.
2
2 Two-mode squeezed gluon states
In order to verify whether the gluon state vector describes the two-mode squeezed state
on colours h and g, it is necessary to introduce the phase-sensitive Hermitian operators
(Xh,g) = [bh + bg + bh+ + bg+]/(2√2) and (Xh,g) = [bh + bg bh+ bg+]/(2i√2) by
λ 1 λ λ λ λ λ 2 λ λ − λ − λ
analogy with quantum optics [12] and to establish conditions under which the variance
of one of them can be less than the variance of a coherent state. Here bh ,bg(bh+,bg+)
(λ) λ λ λ
are the operators annihilating (creating) of gluons of colours h,g = 1,8 and polarization
index λ = 1,3.
Mathematically, the condition of two-mode squeezing on colours h,g is expressed in
the form of the inequality
2
N ∆(Xh,g)1 < 0. (7)
λ
2
D (cid:16) (cid:17) E
Here N is the normal-ordering operator such as
2 1 2 2 2 2 2 2
N ∆(Xh,g)1 = bh bh + bg bg + bh+ bh+
λ 2 ±8( λ − λ λ − λ λ − λ
D (cid:16) (cid:17) E D(cid:16) (cid:17) E D E D(cid:16) (cid:17) E D E D(cid:16) (cid:17) E D E
2 2
+ bg+ bg+ +2 bhbg bh bg + bh+bg+ bg+ bg+
λ − λ λ λ − λ λ λ λ − λ λ
D(cid:16) (cid:17) E D E (cid:20)D E D ED E D E D ED E(cid:21) (8)
2 bh+bh bh+ bh + bg+bg bg+ bg + bh+bg bh+ bg
± λ λ − λ λ λ λ − λ λ λ λ − λ λ
(cid:20)D E D ED E D E D ED E D E D ED E
+ bg+bh bg+ bh .
D λ λE −D λ ED λE(cid:21))
The expectation values ofthe creationand annihilationoperatorsfor gluonswith specified
colour and polarization index are taken over the vector f which describes the evolution
| i
of the virtual gluon field during a small time interval ∆t within interaction representation
f in i∆tH (t ) in , (9)
I 0
| i ≃ | i− | i
(3) (4) (3)
where H (t ) = H (t )+H (t ) is the Hamiltonian three-gluon (H ) and four-gluon
I 0 I 0 I 0 I
(4)
(H ) self-interactions the explicit form of which are given in Appendix in momentum
I
representation, in is an initial state vector of the virtual gluon field, ∆t = t t (below
0
| i −
we will assume t = 0 and consequently ∆t = t).
0
We choose a product of the coherent states of the gluons with different colour and
3 8
polarization indexes as the initial state vector, that is in α = αb because
| i ≡ | i | λi
λ=1b=1
any vector may be decompose on these basis vectors and coherent statQes aQre widely used
in QO [12, 13]. By analogy with QO gluon coherent state vector αb is the eigenvector of
| λi
the corresponding annihilation operator bb with the eigenvalue αb which can be written
λ λ1
in terms of the gluon coherent field amplitude αb and phase γb of the given gluon field
| λ1| λ
αb = αb eiγλb1. In each gluon coherent state αb the gluon number with fixed colour b
λ1 | λ1| | λi
and polarization λ is arbitrary (the average multiplicity of given gluon is equal to square
of the gluon coherent field amplitude nb = αb 2) and phase of considering state γb is
h λi | λ| λ
fixed.
3
Averagingtheannihilationandcreationoperatorsbh,bg,bh+,bg+ in(8)overtheevolved
λ λ λ λ
vector f which is defined according to (9) and taking into account of chosen initial state
| i
vector we write the two-mode squeezing condition in the form
2 it
N ∆(Xλh,g)12 = ± 8 hα|[[HI(0),bhλ+],bhλ+]|αi−hα|[bhλ,[bhλ,HI(0)]]|αi
(cid:26)
D (cid:16) (cid:17) E
+ α [[H (0),bg+],bg+] α α [bg,[bg,H (0)]] α
h | I λ λ | i−h | λ λ I | i
+2 α [[H (0),bh+],bg+] α 2 α [bg,[bh,H (0)]] α < 0.(10)
h | I λ λ | i− h | λ λ I | i
(cid:27)
It is easy to show that the three-gluon self-interaction (as for single-mode squeezing of
the gluons [18]) does not lead to squeezing effect since
[[H(3)(0),bh+],bh+] = 0, [[H(3)(0),bg+],bg+] = 0, [[H(3)(0),bh+],bg+] = 0,
I λ λ I λ λ I λ λ (11)
[bg,[bh,H(3)(0)]] = 0, [bh,[bh,H(3)(0)]] = 0, [bg,[bg,H(3)(0)]] = 0. )
λ λ I λ λ I λ λ I
Thus only the four-gluon self-interaction can yield a two-mode squeezing effect. Indeed,
the two-mode squeezing condition can be written in the explicit form as
N ∆(Xλh,g)12 2 = ±i8tg2(2π)3 dk˜1dk˜2 ×(hα|bbλ+1(k1)bcλ+2(k2)−bbλ1(k1)bcλ2(k2)|αi
D (cid:16) (cid:17) E Z λX1,λ2
δ(2~k k~ k~ ) δ(2~k +k~ +k~ ) (f f +f f +2f f ) ελ1(k )εµ (k )
× − 1 − 2 − 1 2 ahb ahc agb agc ahb agc µ 1 λ2 2
εhλ(k)εν(k) ελ1(k )εµ(k)ελ2(k )ενi(hk) 2f f ελ1(k )εµ(k)ελ2(k(cid:16) )εν(k)
× ν λ − µ 1 λ ν 2 λ − ahg abc µ 1 λ ν 2 λ
+2 α bb+(k )bc (k ) α δ(2~k k~ +k~(cid:17)) δ(2~k +k~ k~ ) (f f +f fi
h | λ1 1 λ2 2 | i× − 1 2 − 1 − 2 ahb ahc agb agc
h i
+f f +f f ) ελ1(k )εµ (k )ελ(k)εν(k) ελ1(k )εµ(k)ελ2(k )εν(k) < 0.
ahc agb ahb agc µ 1 λ2 2 ν λ − µ 1 λ ν 2 λ )
(cid:16) (cid:17)
(12)
d3k
Here g is a self-interaction constant, dk˜ = , k is a gluon energy, εµ is a polar-
(2π)32k 0 λ
0
ization vector, f are structure constants of SU (3) group.
ahb c
For visualization, let us investigate the obtained two-mode squeezing condition (12)
for collinear gluon. In this case corresponding squeezing condition is
N ∆(Xλh,g)12 2 = ±t α4ksπ(fahbfahc +fagbfagc +fahbfagc +fagbfahc)
0
D (cid:16) (cid:17) E
αb αc sin(γb +γc ) < 0. (13)
× | λ1|| λ1| λ1 λ1
λX16=λ
Here we have taken into account that αλb1 = |αλb1|eiγλb1 and αλc1 = |αλc1|eiγλc1, αs = g2/(4π)
is a coupling constant2. The two-mode squeezing condition is fulfilled for any cases apart
from γb + γc = 0,π. In particular, if all initial gluon coherent fields are real or imag-
λ1 λ1
inary then the two-mode squeezing condition is not fulfilled as in the single-mode case.
Obviously, the larger are both amplitudes of the initial gluon coherent fields with different
colour and polarization indexes and coupling constant, the larger is two-mode squeezing
effect.
Thus non-perturbative gluon evolution (at large coupling constant) is very significant
under investigation of the squeezing effect.
2Here we also take into consideration the gluons with momenta directed toward jet development.
4
3 Entangled collinear gluon states
By analogy with QO we assume that two-mode gluon SS with fixed colours h,g is closely
connected with corresponding entangled states of the gluons.
At small value of the squeeze factor we have
2
r = 2 N ∆(Xh,g)1 . (14)
λ
2
(cid:12) (cid:12)
(cid:12)D (cid:16) (cid:17) E(cid:12)
(cid:12) (cid:12)
As initial states we can take a vector i(cid:12)ncluding 0h 0g (cid:12)evolution of which at small time
| λi | λi
can be written in terms of the squeeze factor as
f 0h 0g + r 1h 1g , (15)
| i ≃ | λi| λi | λi| λi
where the squeeze factor for the collinear gluons is defined according to (14) and (13).
The entanglement condition of considering gluon states with colours h,g and polariza-
tion λ can be verified by investigation of the conditional probability P (Y /X )(i,j = 1,2)
j i
by analogy with corresponding condition for photons (4) assuming that X = 0h ,
| 1i | λi
X = 1h , Y = 0g , Y = 1g . It is not complicated to make sure that the
| 2i | λi | 1i | λi | 2i | λi
condition (4) is fulfilled for the cases as for the two-mode squeezing for the state vector
f (15) which describes the non-perturbative evolution of the gluon fields during small
| i
time.
Rewriting the expression (5) for the entanglement coefficient in case two-mode gluon
states as
1/2
bhbg+ 2 + bhbg 2
y = | λ λ | | λ λ| (16)
2(bh+bh +1/2)(bg+bg +1/2)
λ λ λ λ
we can write the condition of the entanglement with taking into account (14) and (13)
α π
0 < t s (f f +f f +f f +f f ) αb αc sin(γb +γc ) < 1.
(cid:12) √2k ahb ahc agb agc ahb agc agb ahc | λ1|| λ1| λ1 λ1 (cid:12)
(cid:12)(cid:12) 0 λX16=λ (cid:12)(cid:12)
(cid:12) (cid:12) (17)
(cid:12) (cid:12)
Th(cid:12)us by analogy with quantum optics as a result of four-gluon self-interact(cid:12)ion we
obtain two-mode squeezed gluon states which are also entangled.
4 Conclusion
Investigating of the gluon fluctuations we have proved theoretically the possibility of
existence of the gluon two-mode squeezed states. The emergence of such remarkable
states becomes possible owing to the four-gluon self-interaction. The three-gluon self-
interaction does not lead to squeezing effect.
We have shown that QCD jet non-perturbative evolution leads both to squeezing and
entanglement ofgluons. It should benotedthat thegreater areboththeamplitudes ofthe
initial gluon coherent fields with different colour and polarization indexes and coupling
constant, the greater is squeezing effect of the colour gluons. We have demonstrated
that entanglement condition of the gluon states with fixed two colours and polarization
5
is defined by the corresponding squeeze factor which is dependent on the amplitudes and
phases of initial coherent gluon fields.
Two-mode gluon states with two different colours can lead to qq¯-entangled states role
of which could be very significant for explanation of the confinement phenomenon.
Appendix
H(3)(t) = ig(2π)3f dk˜ dk˜ dk˜ ~k ~ε (k )εν (k )ελ3(k )δ(~k +~k +~k )
I abc 1 2 3 × 1 λ2 2 λ1 1 ν 3 1 2 3
λ1X,λ2,λ3Z (cid:26)
[ba (k )bb (k )bc (k ) e−2i(k01+k02+k03)t ba+(k )bb+(k )bc+(k )e2i(k01+k02+k03)t]
× λ1 1 λ2 2 λ3 3 − λ1 1 λ2 2 λ3 3
+ba+(k )bb (k )bc (k )e2i(k01−k02−k03)t δ(~k ~k ~k )
λ1 1 λ2 2 λ3 3 1 − 2 − 3
~k ~ε (k )εν (k )ελ3(k )+~k ~ε (k )εν (k )ελ2(k ) ~k ~ε (k )εν (k )ελ3(k )
× 3 λ1 1 λ2 2 ν 3 2 λ3 3 λ1 1 ν 2 − 1 λ2 2 λ1 1 ν 3
h i
+ba+(k )bb+(k )bc (k )e2i(k01+k02−k03)t δ(~k +~k ~k )
λ1 1 λ2 2 λ3 3 1 2 − 3
~k ~ε (k )εν (k )ελ2(k )+~k ~ε (k )εν (k )ελ3(k ) ~k ~ε (k )εν (k )ελ3(k ) , (A.1)
× 1 λ3 3 λ1 1 ν 2 3 λ1 1 λ2 2 ν 3 − 1 λ2 2 λ1 1 ν 3
h i(cid:27)
g2
H(4)(t) = (2π)3 dk˜ dk˜ dk˜ dk˜ εµ (k )ελ3(k )εν (k )ελ4(k )f f
I 4 1 2 3 4 λ1 1 µ 3 λ2 2 ν 4 abc ade
λ1,λX2,λ3,λ4Z (cid:26)
δ(~k +~k +~k +~k ) bb (k )bc (k )bd (k )be (k )e−2i(k01+k02+k03+k04)t
× 1 2 3 4 λ1 1 λ2 2 λ3 3 λ4 4
h (cid:16)
+bb+(k )bc+(k )bd+(k )be+(k )e2i(k01+k02+k03+k04)t
λ1 1 λ2 2 λ3 3 λ4 4
(cid:17)
+4bb+(k )bc (k )bd (k )be (k )e2i(k01−k02−k03−k04)tδ(~k ~k ~k ~k )
λ1 1 λ2 2 λ3 3 λ4 4 1 − 2 − 3 − 4
+4bb+(k )bc+(k )bd+(k )be (k )e2i(k01+k02+k03−k04)tδ(~k +~k +~k ~k )
λ1 1 λ2 2 λ3 3 λ4 4 1 2 3 − 4
i
+2bb+(k )bc+(k )bd (k )be (k )e2i(k01+k02−k03−k04)tδ(~k +~k ~k ~k )
λ1 1 λ2 2 λ3 3 λ4 4 1 2 − 3 − 4
εµ (k )ελ3(k )εν (k )ελ4(k )(f f +f f )
× λ1 1 µ 3 λ2 2 ν 4 abc ade abe adc
h
+εµ (k )ελ2(k )εν (k )ελ4(k )f f . (A.2)
λ1 1 µ 2 λ3 3 ν 4 abd ace
i(cid:27)
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