SQUEEZED
PARTICLE–ANTIPARTICLE
CORRELATIONS∗
Sandra S. Padula, Danuce M. Dudek
IFT/UNESP, Rua Pamplona, 145, 01405-900 São Paulo, SP,
Brazil
Otávio Socolowski Jr.
IMEF/FURG, C.P.474, 96201-900, Rio Grande, RS, Brazil
(Received February 4, 2009)
A novel type of correlation involving particle–antiparticle
pairs was
found out in the 1990’s. Currently known as squeezed or
Back-to-Back
Correlations (BBC), they should be present if the hadronic masses are
modified in the hot and dense medium formed in high energy heavy ion
collisions. Although well-established theoretically, such hadronic
correlations
have not yet been observed experimentally. In this phenomenological
study we suggest a promising way to search for the BBC signal, by
looking
into the squeezed correlation
function of and K+K− pairs
at RHIC
energies, as function of the pair average momentum, K12 = (k1 + k2)/2.
The effects of in-medium mass-shift on the identical particle
correlations
(Hanbury–Brown and Twiss effect) are also discussed.
PACS numbers: 25.75.Gz, 25.75.–q, 21.65.Qr, 21.65.Jk
1. Introduction
The hadronic particle–antiparticle correlation was already
pointed out
in the beginning of the nineties.
However, the final formulation of
these
hadronic squeezed or back-to-back correlations was proposed only at the
end of that decade [1], predicting that such correlations were expected
if the
masses of the mesons were modified in
the hot and dense medium formed
in
high energy nucleus–nucleus collisions. Soon after that, it
was shown that
analogous correlations would exist in the case of baryons as well.
An
interesting
theoretical finding was that both the fermionic (fBBC) and the
bosonic (bBBC) Back-to-Back Correlations were very similar, both being
Presented at the IV Workshop on Particle Correlations and Femtoscopy,
Kraków,
Poland, September 11–14, 2008.
(1225)
1226 S.S. Padula, D.M. Dudek, O. Socolowski Jr.
positive and treated by analogous formalisms. In what follows, we will
focus
our discussion to the bosonic case, illustrating the effect for and
K+K−
pairs, considered to be produced at RHIC energies [3].
Let us discuss the case of -mesons first, which are their
own
antiparticles,
and suppose that their masses are modified in hot and dense medium.
Naturally, they recover their asymptotic masses after the system
freezesout.
Therefore, the joint probability for observing two such particles,
i.e.,
the two-particle distribution, N2(k1, k2) = !k1!k2 ha†
k1
a†
k2
ak2ak1i, can be
factorized as N2(k1, k2) = !k1!k2 hha†
k1ak1iha†
k2ak2i + ha†
k1ak2iha†
k2ak1i +
ha†
k1a†
k2ihak2ak1ii, after applying a generalization of Wick’s
theorem for locally
equilibrated systems [4, 5].
The first term corresponds to the
product of the spectra of the two
’s,
N1(ki) = !ki
d3N
dki
= !ki ha†
ki
akii, being a†
k and ak the free-particle creation
and annihilation operators of scalar quanta, and h. . .i means thermal
averages.
The second term contains the identical particle contribution and is
represented by the square modulus of the chaotic amplitude, Gc(k1, k2)
=
√!k1!k2 ha†
k1
ak2i. Together with the first term, it gives rise to the femtoscopic
or Hanbury–Brown and Twiss (HBT) effect.
The third term, the
square modulus of the squeezed amplitude, Gs(k1, k2) =
√!k1!k2 hak1ak2i,
is identically zero in the absence of in-medium mass-shift. However, if
the
particle’s mass is modified, together with the first term it
leads to the squeezing
correlation function.
The annihilation (creation) operator of the asymptotic, observed bosons
with momentum kμ =(!k, k), a (a†), is related to the
in-medium annihilation
(creation) operator b (b†), corresponding to thermalized
quasi-particles,
by the Bogoliubov–Valatin transformation, ak = ckbk + s
−kb†
−k ; a†
k =
c
kb†
k +s−kb−k, where ck = cosh(fk), sk = sinh(fk). The
argument, fi,j(x) =
1
2 log Kμ
i,j(x) uμ(x)
K
i,j (x) u(x), is the squeezing parameter.
In terms of the above
amplitudes,
the complete correlation function can be written as
C2(k1, k2) = 1 + |Gc(k1, k2)|2
Gc(k1, k1)Gc(k2, k2)
+ |Gs(k1, k2)|2
Gc(k1, k1)Gc(k2, k2)
, (1)
where the first two terms correspond to the identical particle (HBT)
correlation,
whereas the first and the last terms represent the correlation function
between the particle and its antiparticle, i.e., the squeezed part.
The
in-medium modified mass, m, is related to the asymptotic mass, m, by
m2(|k|) = m2 − M2(|k|), here assumed to be a constant
mass-shift.
Squeezed Particle–Antiparticle Correlations 1227
2. Results
The formulation for both bosons and fermions was initially derived for
a
static, infinite medium [1, 2].
More recently, it was shown [3] in the
bosonic
case that, for finite-size systems expanding with moderate flow, the
squeezed
correlations may survive with sizable strength to be observed
experimentally.
Similar behavior is expected in the fermionic case. In that analysis,
a non-relativistic treatment with flow-independent squeezing parameter
was
adopted for the sake of simplicity, allowing to obtain analytical
results. The
detailed discussion is in Ref. [3], where the maximum value of
Cs(k,−k), was
studied as a function of the modified mass, m, considering pairs with
exact
back-to-back momentum, k1=−k2=k (in the identical particle
case, this
procedure would be analogous to study the behavior of the intercept of
the
HBT correlation function).
Although illustrating many points of
theoretical
interest, this study in terms of the unobserved shifted mass and
exactly backto-
back momenta was not helpful for motivating the experimental search of
the BBC’s. A more realistic analysis would involve
combinations of the momenta
of the individual particles, (k1, k2), into the average momentum of the
pair, K= 1
2 (k1+k2).
Since the maximum of the BBC effect is reached when
k1=−k2=k, this would correspond to investigate the squeezed
correlation
function, Cs(k1, k2) = Cs(K, q), close to |K|=0.
For a hydrodynamical ensemble, both the chaotic and the squeezed
amplitudes,
Gc and Gs, respectively, can be written in a special form derived
in [5] and developed in [1, 3].
Therefore, within a non-relativistic
treatment
with flow-independent squeezing parameter, the squeezed amplitude
is written as in [3], i.e., Gs(k1, k2) =
E
1,2
(2)
3
2
c0s0nR3 exph−R2
2 (k1 + k2)2i +
2n∗0R3
∗ exph−(k1−k2)2
8mT iexph−imhuiR(k1+k2)2
2mT iexph− 1
8mT
+ R2
2 (k1 + k2)2io,
the spectrum, as Gc(ki, ki)= Ei,i
(2)
3
2 n|s0 |2R3 + n∗0 R3
∗(|c0 |2 + |s0 |2) exp− k2
i
2mTo,
where R∗ = RpT/T∗ and T∗ = T + m2hui2
m
[3, 6].
We adopt here ~ = c = 1.
Inserting these expressions into Eq. (1) and considering the region
where
the HBT correlation is not relevant, we obtain the results shown in
Fig. 1.
Part (a) shows the squeezed correlation as a function of 2K = (k1 +
k2),
for several values of q = (k1 − k2). The top plot shows
results expected
in the case of a instant emission of the correlated pair. If,
however, the
emission happens in a finite interval, the second term in Eq. (1) is
multiplied
by a reduction factor, in this case expressed by a Lorentzian (F(t) =
[1 +
(!1 + !2)2t2]−1), i.e., the Fourier transform of an
exponential emission.
The result is shown in the plot in the middle of Fig. 1(a). We
see that
this
represents a dramatic reduction in the signal, even though its strength
is
sizable for being observed experimentally. If the system expands with
radial
flow (hui = 0.5), the result is shown in the plot at the bottom of Fig.
1(a),
1228 S.S. Padula, D.M. Dudek, O. Socolowski Jr.
again considering that the ’s are emitted during a finite
period of time,
t = 2 fm/c.
We see that, in the absence of flow, the squeezed
correlation
signal grows faster for higher values |q| than the corresponding case
in the
presence of flow. However, this last one is stronger in all the
investigated
|q| region, showing that the presence of radial flow enhances the
signal.
The
sensitivity of the squeezed-pair correlation to the size of the region
where
the mass-shift occurs is shown in Fig. 1(b) for two values of radii, R
= 7 fm
and R = 3 fm, keeping |q12| = 2.0 GeV/c fixed. The differences are
reflected
in the inverse width of the curves, plotted as a function of 2|K|. In
case of
no in-medium mass modification, the squeezed correlation functions
would
be unity for all values of 2|K| in both plots.
0.0006 0.0128 0.0249 0.0371 0.5 1
1000
2000
0.0006 0.0128 0.0249 0.0371 0.5 1
2.5
5
0.0006 0.0128 0.0249 0.0371 0.5 1
2.5
5
2
4
6
8
10
12
14
16
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
2
4
6
8
10
12
14
16
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Cs(2*K,q) Cs(2*K,q)
Fig. 1. In part (a), the squeezed-pair correlations are shown,
illustrating the effects
of flow and finite emission times.
Part (b) shows the response of the
BBC function
to the size of the squeezing region, with R = 7 fm (top) and R = 3 fm
(bottom).
In the case of the squeezed correlations of K+K− pairs, we
show in
Fig. 2(a) results for the generated momenta of the pairs within the
narrow
interval |K12| ≤ 10 MeV/c, by plotting the squeezed correlation,
Cs(m, q12)
versus m and q12. For the kaons, we can fix the value of the shifted
mass
to be m ≈ 650 MeV, corresponding to one of the maxima in
Fig. 2(a),
and then proceed similarly to what was done in the case. The result
is
shown in Fig. 3 of Ref. [7].
Also in this case the intensity of the
squeezed
correlation would be large enough to be searched for experimentally.
Next, we investigate how the behavior of the identical particle
correlations
could be affected in case of in-medium mass modification, since
the femtoscopic correlation function also depends on the squeezing
factor,
fi,j(m,m). The HBT correlation function is obtained by inserting the
Squeezed Particle–Antiparticle
Correlations 1229
0.5
1
1
1 2
1.2
1.4
1.6
0.5
1
1
1 2
1.2
1.4
1.6
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
C(k1,k2)
Fig. 2. Part (a) shows the behavior of the correlation function with
the in-medium
modified mass and the relative momentum of the pair. Part (b) shows the
HBT
correlation function when squeezing and radial flow are either absent
or present.
chaotic amplitude, Gc(k1, k2)=
E
1,2
(2)
3
2 n|s0 |2R3 exph−R2
2 (k1−k2)2i+n∗0R3
∗(|c0 |2+
|s0 |2) exph−(k1+k2)2
8mT iexph−imhuiR
2mT (k21
−k22
)iexph− 1
8mT + R2
2 (k1 −k2)2io,
together with the expression for the spectrum, into Eq. (1). We use the
case
of identical K±K± pairs as illustration, as seen
in Fig. 2(b).
The investigation
is extended to both the cases of instant emission (t = 0) and finite
emission (t = 2 fm/c), in which case the third term in Eq. (1) is
multiplied
by F(t) = [1+(!1−!2)2t2]−1. In this figure, we
can see the well-known result
corresponding to the narrowing of the femtoscopic correlation function
with increasing emission times, as well as the broadening the curve
with
flow in the absence of squeezing, as expected.
However, if the
squeezing
originated in the mass-shift is present, its effects tend to oppose to
those of
flow (for large |K|, it practically cancels the broadening of the
correlation
function due to flow), another striking indication of
mass-modification, even
in HBT!
3.
Conclusions
In the present work we suggest an effective way to search for the
back-toback
squeezed correlations in heavy ion collisions at RHIC, and later at LHC
energies, by investigating the squeezed correlation function, Cs(k1,
k2) =
Cs(K, q), in terms of 2K12 = (k1+k2), for different values of q12 =
(k1−k2).
We showed that, in the presence of flow, the signal is stronger over
the
momentum regions analyzed in the plots, suggesting that flow may help
to
effectively discover the BBC signal experimentally.
Another important
point
1230 S.S. Padula, D.M. Dudek, O. Socolowski Jr.
that we find, within this simplified model and in the non-relativistic
limit
considered here, is that the squeezing would distort significantly the
HBT
correlation function as well, tending to oppose to the flow effects on
those
curves, practically neutralizing it for large values of |K|.
The analysis in terms of the variable 2K would not be suited for a
genuine
relativistic treatment. In this case, however, a momentum variable
could be constructed, as Qback = (!1 − !2, k1 + k2) = (q0,
2K). In fact,
it would be preferable to redefine this variable as Q2
bbc = −(Qback)2 =
4(!1!2−KμKμ), whose non-relativistic limit is
Q2
bbc → (2K)2, as discussed
in Ref. [6, 7].
Finally, it is important to emphasize that all the
effects and
signals discussed here would exist only if the particles analyzed had
their
masses modified by interactions in the hot and dense medium.
S.S.P. is very grateful to the Organizing Committee of the WPCF 2008
for the kind support to attend the workshop. D.M.D. thanks CAPES and
FAPESP for the financial support.
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