Regge Trajectories of Quark Gluon Bags
Using an exactly solvable statistical model, we discuss the equation of state of large/heavy and short-living bags of the quark gluon plasma (QGP).We argue that the large width of the QGP bags explains not only the observed deficit in the number of hadronic resonances, but also clarifies the reason...
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Відділення фізики і астрономії НАН України
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Regge Trajectories of Quark Gluon Bags / K.A. Bugaev, G.M. Zinovjev // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 586-592. — Бібліогр.: 25 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860180746738597888 |
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| author | Bugaev, K.A. Zinovjev, G.M. |
| author_facet | Bugaev, K.A. Zinovjev, G.M. |
| citation_txt | Regge Trajectories of Quark Gluon Bags / K.A. Bugaev, G.M. Zinovjev // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 586-592. — Бібліогр.: 25 назв. — англ. |
| collection | DSpace DC |
| container_title | Український фізичний журнал |
| description | Using an exactly solvable statistical model, we discuss the equation of state of large/heavy and short-living bags of the quark gluon plasma (QGP).We argue that the large width of the QGP bags explains not only the observed deficit in the number of hadronic resonances, but also clarifies the reason why the heavy QGP bags cannot be directly observed even as metastable states in the hadronic phase. Also the Regge trajectories of large and heavy QGP bags are established both in vacuum and in a strongly interacting medium. It is shown that, at high temperatures, the average mass and width of the QGP bags behave in accordance with the upper bound of the Regge trajectory asymptotics (the linear asymptotics), whereas, for temperatures below Tн=2 (Tн is the Hagedorn temperature), they obey the lower bound of the Regge trajectory asymptotics (the square root one). Thus, for T < Tн=2; the spin of the QGP bags is restricted from above, whereas, for T > Tн=2, these bags demonstrate the standard Regge behavior consistent with the string models.
Використовуючи точний розв’язок статистичної моделi, обговорено рiвняння стану великих/важких i короткоживучих мiшкiв кварк-глюонної плазми (КГП). Наведено аргументи того, що велика ширина мiшкiв КГП не тiльки пояснює дефiцит кiлькостi адронних резонансiв, але й причину того, що важкi мiшки КГП не можуть безпосередньо спостерiгатися навiть як метастабiльнi стани в адроннiй фазi. Також знайдено Редже траєкторiї великих i важких мiшкiв КГП як у вакуумi, так i в сильновзаємодiючому середовищi. Доведено, що за високих температур середня маса i ширина мiшкiв КГП пiдкорюються верхнiй границi асимптотики траєкторiї Редже (лiнiйна асимптотика), тодi як для температур, нижчих за Tн=2 (Tн – температура Хагедорна), вони пiдкорюються нижнiй границi асимптотики траєкторiй Редже (асимптотика кореня квадратного). Таким чином, для T < Tн=2 спiн мiшкiв КГП обмежено зверху, тодi як для T > Tн=2 цi мiшки демонструють стандартну Редже поведiнку, яка узгоджується з моделями струн.
|
| first_indexed | 2025-12-07T18:02:13Z |
| format | Article |
| fulltext |
K.A. BUGAEV, G.M. ZINOVJEV
REGGE TRAJECTORIES OF QUARK GLUON BAGS
K.A. BUGAEV, G.M. ZINOVJEV
Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
(14b, Metrolohichna Str., Kyiv 03680, Ukraine; e-mail:
bugaev@ th. physik. uni-frankfurt. de; Gennady. Zinovjev@ cern. ch )
PACS 25.75.-q,25.75.Nq
c©2010
Using an exactly solvable statistical model, we discuss the equation
of state of large/heavy and short-living bags of the quark gluon
plasma (QGP). We argue that the large width of the QGP bags ex-
plains not only the observed deficit in the number of hadronic reso-
nances, but also clarifies the reason why the heavy QGP bags can-
not be directly observed even as metastable states in the hadronic
phase. Also the Regge trajectories of large and heavy QGP bags
are established both in vacuum and in a strongly interacting medi-
um. It is shown that, at high temperatures, the average mass
and width of the QGP bags behave in accordance with the up-
per bound of the Regge trajectory asymptotics (the linear asymp-
totics), whereas, for temperatures below TH/2 (TH is the Hage-
dorn temperature), they obey the lower bound of the Regge tra-
jectory asymptotics (the square root one). Thus, for T < TH/2,
the spin of the QGP bags is restricted from above, whereas, for
T > TH/2, these bags demonstrate the standard Regge behavior
consistent with the string models.
1. Introduction
Regge poles have been introduced in particle physics be-
fore the QCD era. Since the beginning of the 1960s [1],
they are widely used to describe the high-energy interac-
tions of hadrons and nuclei. The Regge approach estab-
lishes an important connection between the high-energy
scattering and the spectrum of particles and resonances.
It served as a starting point to introduce the dual and
string models of hadrons. Up to now, a rigorous deriva-
tion of Regge poles in QCD remains an unsolved prob-
lem, since it is related to the nonperturbative effects in
QCD and the problem of confinement.
Nowadays, the Regge trajectories are widely under-
stood as a linear relation between the resonance mass
squared and the resonance spin or the radial quantum
number, whereas the Regge trajectory α(Sr) contains
information about the resonance mass Mr and width
Γr. Indeed, the resonance spin J is defined in the com-
plex energy plane as J = α
(
(Mr − i
2Γr)2
)
. Moreover,
the linear trajectories, i.e. α(Sr) ∼ Sr, which follow
from the string models, are often believed to be the only
Regge trajectories of hadrons.
However, it was shown long ago that, under the plau-
sible assumptions, the linear Regge trajectories corre-
spond to the upper bound of the asymptotic behavior,
whereas its lower bound is given by a square-root tra-
jectory, i.e. αl(Sr) ∼ [−Sr]1/2 [2, 3]. Moreover, there
were some indications [3] that the square-root trajectory
should give the asymptotic behavior of excited hadronic
resonances. The latter means that, for each family of
hadronic resonances, the Regge poles do not go beyond
some vertical line in the complex spin plane, i.e. the res-
onances should become infinitely wide in the asymptotic
limit S → +∞.
Since the linear Regge trajectories of hadrons gener-
ate the Hagedorn mass spectrum [4], the square-root
ones should lead to a weaker growth of the hadronic
mass spectrum. At first glance, it seems that the ex-
perimental mass spectrum of hadrons [5] does not show
an exponential increase at hadron masses above 2.5 GeV
and, hence, it evidences against the linear Regge trajec-
tories of heavy hadrons. Moreover, the best description
of particle yields observed in a very wide range of colli-
sion energies of heavy ions is achieved by the statistical
model which incorporates all hadronic resonances not
heavier than 2.3 GeV [6]. Again it looks like that heav-
ier hadronic species, except for the long living ones, are
simply absent in the experiments [7]. Thus, we are con-
fronted with a serious conceptual problem between a few
theoretical expectations and several experimental facts.
Recently, this conceptual problem was resolved within
the finite-width model (FWM) [8]. The FWM intro-
duces the medium dependent finite width of QGP bags
into an exactly solvable statistical model. It shows that
the large width of the QGP bags explains not only
the observed deficit in the number of hadronic reso-
nances, but also clarifies the reason why the heavy QGP
bags and strangelets cannot be directly observed even as
metastable states in the hadronic phase. In addition, the
FWM allows one to establish [9] the Regge trajectories
of large and heavy QGP bags both in vacuum and in a
strongly interacting medium. As will be shown below,
the average mass and width of the QGP bags behave at
586 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
REGGE TRAJECTORIES OF QUARK GLUON BAGS
high temperatures in accordance with the upper bound
of the Regge trajectory asymptotics (the linear asymp-
totics), whereas they obey the lower bound of the Regge
trajectory asymptotics (the square-root one) at low tem-
peratures. Thus, at low temperatures, the spin of the
QGP bags is restricted from above, whereas these bags
demonstrate the typical Regge behavior consistent with
the string models at high temperatures.
The work is organized as follows. In Section 2, we
discuss the main ideas and results of FWM. The Regge
trajectories of QGP bags are established in Section 3,
while our conclusions are given in Section 4.
2. FWM of QGP Bags
The main object of the FWM is the mass-volume spec-
trum of heavy and large QGP bags at a temperature T
(V0 ≈ 1 fm3, M0 ≈ 2.5 GeV [8, 9])
FQ(s, T ) =
∞∫
V0
dv
∞∫
M0
dm ρ(m, v) exp(−sv)φ(T,m) , (1)
where the thermal density of bags of mass m reads
φ(T,m) =
∞∫
0
p2dp exp
[
− (p2 + m2)1/2
T
]
. (2)
In (1), s denotes the variable of the isobar ensemble
which is dual to the system volume. An exponential
exp(−sv) in (1) describes the hard core repulsion be-
tween the bags [8], while the density of states ρ(m, v) of
bags of mass m and volume v has the form
ρ(m, v) =
ρ1(v) NΓ
Γ(v) ma+ 3
2
exp
[
m
TH
− (m−Bv)2
2Γ2(v)
]
, (3)
ρ1(v) = f(T ) v−b exp
[
−σ(T )
T vκ
]
. (4)
As one can see from (3), the density of states has a
Hagedorn-like parametrization with respect to the mass
and the Gaussian attenuation around the bag mass Bv
(B is the mass density of a bag of a vanishing width)
with the volume-dependent Gaussian width Γ(v) or the
width hereafter. We will distinguish it from the true
width defined as ΓR = αΓ(v) (α ≡ 2
√
2 ln 2 ). As was
shown in [8, 9], the Breit–Wigner attenuation of a reso-
nance mass cannot be used in spectrum (3) because, in
this case, the finite width leads to a divergency of the
mass integral in (1) above TH .
The normalization factor obeys the condition
N−1
Γ =
∞∫
M0
dm
Γ(v)
exp
[
− (m−Bv)2
2Γ2(v)
]
. (5)
The constants a > 0 and b > 0 are discussed in [8].
Moreover, the volume spectrum in (4) contains the
surface free energy (κ = 2/3) with the T -dependent sur-
face tension which can be parametrized in a general way
[10–12] as
σ(T, µ) =
σ− > 0 , T → TΣ(µ)− 0 ,
0 , T = TΣ(µ) ,
σ+ < 0 , T → TΣ(µ) + 0 .
(6)
For T ≤ TΣ(µ), such a parametrization is justified by
the usual cluster models like the FDM [13] and SMM
[14–17], whereas the general consideration for any T can
be driven by the surface partitions of the Hills and Dales
model [10].
The recent results obtained within the exactly solvable
models [11,12] also justify parametrization (6) and show
that the only physical reason for the degeneration of the
first-order deconfinement phase transition at low bary-
onic densities into a cross-over is the negative surface
tension coefficient in this region. Moreover, the exis-
tence of negative surface tension at the cross-over region
was directly demonstrated from the lattice QCD data
very recently within a new phenomenological model of
confinement [18].
An actual choice of the continuous functions σ± of the
temperature T and the baryonic chemical potential µ
along with parametrization of the nil line of the surface
tension coefficient TΣ(µ) for the tricritical and critical
endpoint of the QCD phase diagram can be found in
[11] and [12], respectively.
Spectrum (3) has a simple form, but is rather general
since both the width Γ(v) and the bag mass density B
can be medium-dependent. It clearly reflects the fact
that the QGP bags are similar to ordinary quasipar-
ticles with the medium-dependent characteristics (life-
time, most probable values of mass and volume). In prin-
ciple, one could consider various v-dependences Γ(v),
but it was shown in [8, 9] that the only square-root
dependence of the resonance width on its volume, i.e.
Γ(v) = γv1/2, does not lead to the problems with the
existence of large QGP bags.
For large bag volumes (v � M0/B > 0), factor (5)
can be found as NΓ ≈ 1/
√
2π. Similarly, one can show
that, for heavy free bags (m � BV0, ignoring the hard
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 587
K.A. BUGAEV, G.M. ZINOVJEV
core repulsion and thermostat),
ρ(m) ≡
∞∫
V0
dv ρ(m, v) ≈
ρ1(mB )
B ma+ 3
2
exp
[
m
TH
]
. (7)
It originates in the fact that, for heavy bags, the Gaus-
sian in (3) acts like a Dirac δ-function for Γ(v) = γv1/2.
Similarly to (7), one can estimate the width of heavy
free bags averaged over bag volumes and get
Γ(v) ≈ Γ(m/B) = γ
√
m
B
. (8)
Thus, the mass spectrum of heavy free QGP bags is a
Hagedorn-like one with the property that the heavy reso-
nances must have the large mean width. Hence, they are
hardly observable. This resolves the conceptual problem
of the deficit of observed heavy hadronic resonances as
compared with the Hagedorn mass spectrum. Applying
these arguments to the strangelets, we conclude that, if
their mean volume is a few cubic fermis or larger, they
should survive for a very short time, which is similar
to the results of [19] predicting an instability of such
strangelets.
An analysis of the full spectrum (1) allows one to de-
termine the pressure of the QGP bags [8] which shows
two drastically different regimes depending on the value
of the most probable mass of a bag (v � V0)
〈m〉 ≡ Bv + Γ2(v)β , with β ≡ T−1
H − T−1 . (9)
For temperatures below (above) T± = c± TH (0 < c± <
1), the most probable mass is negative (positive) and
spectrum (1) defines the low (high) temperature pressure
p− (p+)
p =
p+ ≡ T
[
βB + γ2
2 β
2
]
, 〈m〉 > 0 ,
p± ≡ BTβ
2 , 〈m〉 = 0 ,
p− ≡ −T B2
2 γ2 , 〈m〉 < 0 .
(10)
There are two remarkable facts concerning the low-
temperature pressure. First, for 〈m〉 ≤ 0, the result-
ing mass attenuation of the integrand in (1) decreases at
a fixed bag volume so rapidly that the only vicinity of
M0 contributes to the mass-volume spectrum. In other
words, all heavy QGP bags are extremely suppressed in
this regime, and, as a result, only the smallest bags with
the mass M0 and the width about Γ(V0) can contribute
to spectrum (1) [8]. Consequently, such QGP bags would
not be distinguishable from the usual low-mass hadrons.
Such a regime leads to the subthreshold suppression of
the QGP bags at low temperatures even in finite systems
and, hence, is able to explain the absence of heavy/large
QGP bags and strangelets for T < T±.
Second, for the non-vanishing functions γ and B at low
temperatures, i.e. γ0 = γ(T = 0) > 0 and B0 = B(T =
0) > 0, the QGP pressure at such temperatures should
be negative and linear in temperature p−(T → 0) ≈
−T B2
0
2 γ2
0
. Such a linear T -behavior of the QGP pressure
at low temperatures, pQGP = σpT
4 −A1T , is known for
a long time [21] and was reported by several groups (see
[9] for details). Using this fact and matching p+ with
pQGP, it was possible to estimate the resonance width
coefficient γ and the mass density B from the lattice
QCD data [9]
γ2(T ) = 2β−1[σpTHT (T 2 + TTH + T 2
H)−B(T )] , (11)
B(T ) = σpT
2
H(T 2 + TTH + T 2
H) , (12)
where 3σp is the Stefan–Bolzmann constant of the QGP.
Equations (11) and (12) allow one to determine T± =
0.5TH and show that the resonance width at the zero
temperature very weakly depends on the number of el-
ementary degrees of freedom in QGP, but strongly de-
pends on the cross-over temperature Tco
ΓR(V0, T = 0) ≈ Cγ V 1/2
0 T 5/2
co α , (13)
where the constant Cγ , depending on the number of
color and flavor states of QGP, varies between 1.22
and 1.3 [9]. The minimal width of the QGP bags
strongly increases with temperature. For instance, at the
Hagedorn temperature, one obtains ΓR(V0, T = TH) =√
12 ΓR(V0, T = 0). Therefore, for Tco ∈ [170; 200] MeV,
the minimal width of the QGP bags is ΓR(V0, T = 0) ∈
[400; 600] MeV and ΓR(V0, T = TH) ∈ [1400; 2000] MeV.
These estimates clearly show us that, even without the
subthreshold suppression, the heavy/large QGP bags
cannot be directly observed at any temperature due to
a very short life-time.
We note that one of the most remarkable features of
the FWM is that its mass dependence of a mean reso-
nance width can help to determine the Regge trajectories
of QGP bags and to resolve a few problems related to
them.
3. Asymptotic Trajectories of QGP Bags
Nowadays, there is a great interest in the behavior of
the Regge trajectories of higher resonances in the con-
text of the 5-dimensional string theory holographically
dual to QCD [20] which is known as the anti-de-Sitter
588 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
REGGE TRAJECTORIES OF QUARK GLUON BAGS
space/conformal field theory (AdS/CFT). However, as
was mentioned earlier, the Regge trajectories are widely
understood only as the linear relation between the res-
onance mass squared and the resonance spin or the ra-
dial quantum number. We would like to determine the
full Regge trajectories of QGP bags. In our analysis,
we follow Ref. [3] based on the following most general
assumptions: (I) α(S) is an analytical function, having
only the physical cut from S = S0 to S = ∞; (II) α(S)
is polynomially restricted at the whole physical sheet;
(III) there exists a finite limit of the phase trajectory
at S → ∞. Using these assumptions, it was possible to
prove [3] that, for S →∞, the upper bound of the Regge
trajectory asymptotics at the whole physical sheet is
αu(S) = −g2
u [−S]ν , with ν ≤ 1 , (14)
where the function g2
u > 0 should increase slower than
any power in this limit, and its phase must vanish at
|S| → ∞.
On the other hand, it was also shown in Ref. [3] that,
if one requires in addition to (I)–(III) that the transition
amplitude T (s, t) is a polynomially restricted function of
S for all nonphysical t > t0 > 0, then the real part of
the Regge trajectory does not increase at |S| → ∞, and
the trajectory behaves as
αl(S) = g2
l
[
− [−S]1/2 + Cl
]
, (15)
where g2
l > 0 and Cl are some constants. Moreover, (15)
defines the lower bound for the asymptotic behavior of
the Regge trajectory [3]. The expression (15) is a gener-
alization of a well-known Khuri result [2]. It means that,
for each family of hadronic resonances, the Regge poles
do not go beyond some vertical line in the complex spin
plane. In other words, it means that the resonances be-
come infinitely wide in asymptotics S → +∞, i.e. they
are moving out of the real axis of the proper angular mo-
mentum J and, therefore, there is only a finite number
of resonances in the corresponding transition amplitude.
To compare the FWM results with trajectories (14)
and (15), we need to relate the mass and the width of
QGP bags, since they are independent variables in this
model. Nevertheless, this can be done for their averaged
values.
To illustrate this statement, we recall our result on
the mean Gaussian width of the free bags averaged with
respect to their volume (8) by spectrum (7). Using the
formalism of [3], it can be shown that, at zero tempera-
ture, the free QGP bags of mass m and mean resonance
width αΓ(v)|T=0 ≈ αγ0
√
m
B0
precisely correspond to
the Regge trajectory
αr(S) = g2
r [S + ar(−S)3/4] with ar = const < 0 .
(16)
Indeed, substituting S = |S|ei φr into (16), then expand-
ing the second term on the right-hand side of (16), and
requiring Im [αr(S)] = 0, one finds the phase of the phys-
ical trajectory (one of four roots of one fourth power in
(16))
φr(S)→
ar sin 3
4π
|S|1/4
→ 0− , (17)
which vanishes in the correct quadrant of the complex S-
plane. Considering the complex energy plane E =
√
S ≡
Mr−iΓr
2 , one can determine the mass Mr and the width
Γr,
Mr ≈ |S|1/2 , Γr ≈ −|S|1/2φr(S) =
|ar|M1/2
r√
2
, (18)
of a resonance belonging to trajectory (16).
Comparing the mass dependence of the width in (18)
with the mean width of free QGP bags (8) taken at T =
0, it is natural to identify them,
afree
r ≈ −αγ0
√
2
B0
= −4 γ0
√
ln 2
B0
, (19)
and to deduce that the free QGP bags belong to the
linear Regge trajectory (16). Such a conclusion is sup-
ported and justified by the well-established results on the
linear Regge trajectories of hadronic resonances [22] and
by theoretical expectations of the dual resonance model
[23], the open string model [24], the closed string model
[24], and the AdS/CFT [20]. Moreover, the most direct
way to connect the FWM bags with the string models
is provided by the recently suggested model of the con-
finement phenomenon [18] which allows us to relate the
string tension of a confining color tube and the surface
tension of QGP bags.
We now consider the second way of averaging the
mass-volume spectrum over the resonance masses
m(v) ≡
∞∫
M0
dm
∫
d3k
(2π)3 ρ(m, v) m e−
√
k2+m2
T
∞∫
M0
dm
∫
d3k
(2π)3 ρ(m, v) e
−
√
k2+m2
T
, (20)
which is technically simpler than averaging over the reso-
nance volume, but we will make the necessary comments
on the other way of averaging in the appropriate places.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 589
K.A. BUGAEV, G.M. ZINOVJEV
Using the results of the preceding section, one can find
the mean mass (20) for T ≥ 0.5TH (or for 〈m〉 ≥ 0 ) to
be equal to the most probable mass of a bag, from which
one determines the resonance width:
m(v) ≈ 〈m〉 and (21)
ΓR(v) ≈ 2
√
2 ln 2 Γ
[
〈m〉
B + γ2β
]
= 2 γ
√
2 ln 2 〈m〉
B + γ2β
. (22)
Two last equations lead to a vanishing ratio ΓR
〈m〉 ∼
〈m〉−1/2 in the limit 〈m〉 → ∞. Comparing (21) and
(22) with the mass and the width (18) of the Regge tra-
jectory (16) and applying absolutely the same logic we
used for the free QGP bags, we conclude that the loca-
tion of the FWM heavy bags in the complex energy plane
is identical to that of one of the resonances belonging to
trajectory (16) with
〈m〉 ≈ |S|1/2 and ar ≈ −4γ
√
ln 2
B + γ2β
. (23)
The most remarkable output of such a conclusion is that
the medium-dependent FWM mass and the width of the
extended QGP bags obey the upper bound for the Regge
trajectory asymptotic behavior obtained for point-like
hadrons [3]!
It is also of interest that the resonance width formula
(22) follows from that for the most probable volume,
vE(m) ≈ m√
B2 + 2γ2s∗
=
m
B + γ2β
, (24)
of heavy resonances of mass m�M0 that are described
by the continuous spectrum FQ(s, T ) (1). This result
can be easily found by maximizing the exponential in
FQ(s, T ) with respect to the resonance volume v at a
fixed mass m [9].
The extracted values of the resonance width coefficient
along with relation (12) for B(T ) allow us to estimate
ar as
ar ≈ −4
√
2T TH
2T − TH
ln 2 . (25)
This expression shows that, for T → TH/2 + 0, the
asymptotic behavior (16) breaks down since the reso-
nance width diverges at fixed |S|. We hope for that such
a behavior can be experimentally observed [25] at NICA
(Dubna, Russia) and FAIR (Darmstadt, Germany) en-
ergies.
Now we can find the spin of the FWM resonances
J = Reαr(〈m〉2) ≈ g2
r 〈m〉
[
〈m〉 − a2
r
4
]
, (26)
which has a typical Regge behavior up to a small correc-
tion. Such a property can also be obtained within the
dual resonance model [23], the models of open [24] and
closed [24] strings, and the AdS/CFT [20]. These mod-
els support our result (26) and justify it. Note, however,
that, in addition to the spin value, the FWM determines
the width of hadronic resonances. The latter allows us
to predict the ratio of widths of two resonances having
spins J2 and J1 and appearing at the same temperature
T :
ΓR
[
〈m〉
∣∣
J2
(B+γ2β)
]
ΓR
[
〈m〉
∣∣
J1
(B+γ2β)
] ≈
√
v
∣∣
J2√
v
∣∣
J1
≈
√
〈m〉
∣∣
J2√
〈m〉
∣∣
J1
≈
[
J2
J1
]1/4
, (27)
which, perhaps, can be tested at LHC.
We now turn to the analysis of the low temperature
regime, i.e. to T ≤ 0.5TH . Using the previously ob-
tained results from (20), we find
m(v) ≈ M0 , (28)
i.e. the mean mass is volume-independent. Taking the
limit v → ∞, we get the ratio Γ(v)
m(v) → ∞ which closely
resembles the case of the lower bound of the Regge tra-
jectory asymptotics (15). Similarly to the analysis of the
high temperature regime, relation (15) yields the tra-
jectory phase and then the resonance mass Mr and its
width Γr
φr(S)→ −π +
2|Cl|| sin(argCl)|
|S|1/2
, (29)
Mr ≈ |Cl|| sin(argCl)| and Γr ≈ 2|S|1/2 . (30)
Again comparing the averaged masses and widths of
FWM resonances with their counterparts in (30), we
find a similar behavior in the limit of the large width
of resonances. Therefore, we conclude that, at low tem-
peratures, the FWM obeys the lower bound of the Regge
trajectory asymptotics of [3].
The other way of averaging, i.e. over the resonance
volume, results, in the leading order, in an infinite value
of the most probable resonance width [9] defined in this
way. Note that such a result is supported by the high-
temperature mean width behavior if T → TH/2 + 0.
As one can see from (25) and (18), in the latter case,
590 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
REGGE TRAJECTORIES OF QUARK GLUON BAGS
trajectory (16) also demonstrates a very large width as
compared with a finite resonance mass.
The FWM suggests that the Regge trajectories of
QGP bags have a statistical nature. The above estimates
demonstrate that, at any temperature, the FWM QGP
bags can be regarded as the medium-induced Reggeons
which belong, at T ≤ 0.5TH (i.e. for 〈m〉 ≤ 0), to the
Regge trajectory (15). Otherwise, they are described by
trajectory (16). Of course, both trajectories (15) and
(16) are valid in the asymptotic |S| → ∞, but the most
remarkable fact is that, to our knowledge, the FWM
gives us the first example of a model which reproduces
both of these trajectories and, thus, obeys both bounds
of the Regge asymptotics. Moreover, since the FWM
contains the Hagedorn-like mass spectrum at any tem-
perature, it shows that such a spectrum is not exclu-
sively related to the linear Regge trajectories. At low
temperatures (i.e. for 〈m〉 ≤ 0), the large/heavy QGP
bags have the square-root trajectory (15) which is in line
with expectations of Ref. [3].
4. Conclusions
Here, we briefly describe an exactly solvable statistical
model, the FWM, of the QGP equation of state. It ac-
counts for the Hagedorn mass spectrum and for a fi-
nite medium-dependent width of large QGP bags. The
inclusion of the Gaussian attenuation of the resonance
mass leads not only to the partition function convergent
at high temperatures, but also it allows us to explain
a huge deficit of the experimentally observed hadronic
resonances with masses above M0 ≈ 2.5 GeV compared
with the Hagedorn mass spectrum.
The FWM allows us to establish the full Regge tra-
jectories of large/heavy QGP bags both in vacuum and
in a strongly interacting medium. The free QGP bags in
vacuum have the linear Regge trajectory and, thus, obey
the upper bound of the Regge trajectory asymptotics. A
linear Regge trajectory is also found for the in-medium
QGP bags at temperatures above 0.5TH , whereas, for
temperatures below 0.5TH , QGP bags obey the lower
bound of the Regge trajectory asymptotics (the square-
root one) which is in line with expectations of Ref. [3].
The FWM allows us to connect the statistical de-
scription of the QGP equation of state with the Regge
poles method and show that the Regge trajectories of
large/heavy QGP bags have a statistical nature. These
findings bring forward the statistical models of the QGP
equation of state to a qualitatively new level of realism.
The research made in this work was supported in
part by the Program “Fundamental Properties of Physi-
cal Systems under Extreme Conditions” of the Division
of of Physics and Astronomy of the National Academy
of Science of Ukraine. K.A.B. acknowledges the par-
tial support by the Fundamental Research State Fund
of Ukraine, Agreement No F28/335-2009 for the Bilat-
eral project FRSF (Ukraine) – RFBR (Russia).
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K.A. BUGAEV, G.M. ZINOVJEV
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Received 29.10.09
РЕДЖЕ ТРАЄКТОРIЇ КВАРК-ГЛЮОННИХ МIШКIВ
К.О. Бугаєв, Г.М. Зiнов’єв
Р е з ю м е
Використовуючи точний розв’язок статистичної моделi, обго-
ворено рiвняння стану великих/важких i короткоживучих мi-
шкiв кварк-глюонної плазми (КГП). Наведено аргументи то-
го, що велика ширина мiшкiв КГП не тiльки пояснює дефiцит
кiлькостi адронних резонансiв, але й причину того, що важ-
кi мiшки КГП не можуть безпосередньо спостерiгатися навiть
як метастабiльнi стани в адроннiй фазi. Також знайдено Ре-
дже траєкторiї великих i важких мiшкiв КГП як у вакуумi,
так i в сильновзаємодiючому середовищi. Доведено, що за ви-
соких температур середня маса i ширина мiшкiв КГП пiдкорю-
ються верхнiй границi асимптотики траєкторiї Редже (лiнiйна
асимптотика), тодi як для температур, нижчих за TH/2 (TH –
температура Хагедорна), вони пiдкорюються нижнiй границi
асимптотики траєкторiй Редже (асимптотика кореня квадра-
тного). Таким чином, для T < TH/2 спiн мiшкiв КГП обмежено
зверху, тодi як для T > TH/2 цi мiшки демонструють стандар-
тну Редже поведiнку, яка узгоджується з моделями струн.
592 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
|
| id | nasplib_isofts_kiev_ua-123456789-56202 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 2071-0194 |
| language | English |
| last_indexed | 2025-12-07T18:02:13Z |
| publishDate | 2010 |
| publisher | Відділення фізики і астрономії НАН України |
| record_format | dspace |
| spelling | Bugaev, K.A. Zinovjev, G.M. 2014-02-13T20:39:43Z 2014-02-13T20:39:43Z 2010 Regge Trajectories of Quark Gluon Bags / K.A. Bugaev, G.M. Zinovjev // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 586-592. — Бібліогр.: 25 назв. — англ. 2071-0194 PACS 25.75.-q,25.75.Nq https://nasplib.isofts.kiev.ua/handle/123456789/56202 Using an exactly solvable statistical model, we discuss the equation of state of large/heavy and short-living bags of the quark gluon plasma (QGP).We argue that the large width of the QGP bags explains not only the observed deficit in the number of hadronic resonances, but also clarifies the reason why the heavy QGP bags cannot be directly observed even as metastable states in the hadronic phase. Also the Regge trajectories of large and heavy QGP bags are established both in vacuum and in a strongly interacting medium. It is shown that, at high temperatures, the average mass and width of the QGP bags behave in accordance with the upper bound of the Regge trajectory asymptotics (the linear asymptotics), whereas, for temperatures below Tн=2 (Tн is the Hagedorn temperature), they obey the lower bound of the Regge trajectory asymptotics (the square root one). Thus, for T < Tн=2; the spin of the QGP bags is restricted from above, whereas, for T > Tн=2, these bags demonstrate the standard Regge behavior consistent with the string models. Використовуючи точний розв’язок статистичної моделi, обговорено рiвняння стану великих/важких i короткоживучих мiшкiв кварк-глюонної плазми (КГП). Наведено аргументи того, що велика ширина мiшкiв КГП не тiльки пояснює дефiцит кiлькостi адронних резонансiв, але й причину того, що важкi мiшки КГП не можуть безпосередньо спостерiгатися навiть як метастабiльнi стани в адроннiй фазi. Також знайдено Редже траєкторiї великих i важких мiшкiв КГП як у вакуумi, так i в сильновзаємодiючому середовищi. Доведено, що за високих температур середня маса i ширина мiшкiв КГП пiдкорюються верхнiй границi асимптотики траєкторiї Редже (лiнiйна асимптотика), тодi як для температур, нижчих за Tн=2 (Tн – температура Хагедорна), вони пiдкорюються нижнiй границi асимптотики траєкторiй Редже (асимптотика кореня квадратного). Таким чином, для T < Tн=2 спiн мiшкiв КГП обмежено зверху, тодi як для T > Tн=2 цi мiшки демонструють стандартну Редже поведiнку, яка узгоджується з моделями струн. The research made in this work was supported in part by the Program “Fundamental Properties of Physical Systems under Extreme Conditions” of the Division of of Physics and Astronomy of the National Academy of Science of Ukraine. K.A.B. acknowledges the partial support by the Fundamental Research State Fund of Ukraine, Agreement No F28/335-2009 for the Bilateral project FRSF (Ukraine) – RFBR (Russia). en Відділення фізики і астрономії НАН України Український фізичний журнал Загальні питання теоретичної фізики Regge Trajectories of Quark Gluon Bags Редже траєкторії кварк-глюонних мішків Article published earlier |
| spellingShingle | Regge Trajectories of Quark Gluon Bags Bugaev, K.A. Zinovjev, G.M. Загальні питання теоретичної фізики |
| title | Regge Trajectories of Quark Gluon Bags |
| title_alt | Редже траєкторії кварк-глюонних мішків |
| title_full | Regge Trajectories of Quark Gluon Bags |
| title_fullStr | Regge Trajectories of Quark Gluon Bags |
| title_full_unstemmed | Regge Trajectories of Quark Gluon Bags |
| title_short | Regge Trajectories of Quark Gluon Bags |
| title_sort | regge trajectories of quark gluon bags |
| topic | Загальні питання теоретичної фізики |
| topic_facet | Загальні питання теоретичної фізики |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/56202 |
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