Some Characteristic Parameters of Proton from the Bag Model

0.1 MB·

by  Z. G. Tan

Some Characteristic Parameters of Proton from the Bag Model
Z. G. Tan,1,∗ L. Y. Huang,1 and C. B. Yang2
1Department of Electronics and Communication Engineering,
Changsha University,Changsha, 410003,P.R.China
2Institute of Particle Physics, Hua-Zhong Normal University, Wuhan, 430079, China
We treat the mass of a proton as the total static energy which can be separated into two parts
thatcomefromthecontributionofquarksandgluonsrespectively. Weadopttheessentialofthebag
modelofhadrontodiscussthestructureofaprotonandfindthatthecalculatedtemperature,proton
radius, the bag constant are compare well with QCD results if a proton is a thermal equilibrium
system of quarksand gluons.
PACSnumbers: 12.39.Ba,14.20.Dh
0
1
0 I. INTRODUCTION the interactions between quarks . Their effect can be re-
2 placed by the bag pressure which confines the quarks in
n Exploring proton structure [1–3] is still one of most a hadron. On the other hand, if all interactions among
a important subjects for more profound enhancement of partonsinthebagareneglected,weassumethatthepar-
J human knowledge on strong interactions. It is also very tonsmightbe treatedasathermallyequilibratedsystem

9 helpfulforpeopletosearchforanewmatterstate–quark withagivenvolume. Thenpropertiesofahadroncanbe
gluon plasma (QGP) which is the deconfined state of investigated and some characteristic parameters for the
h] strongly interacting matter. Most theoretical investi- hadron can be obtained in the bag model.
p gations focus on Quantum Chromodynamics (QCD) [4]. The organizationof this paper is as follows. In Sec.II,
- However,thereareafew phenomenologicalmodelsabout we will discuss some features about a thermally equili-
p
nucleon structure and interactions. The classical string bratedQGPsystem. TheninSec.III, wegetanestimate
e
h modeldescribesmesonsasstringsegmentsexecutinglon- of the maximum kinetic energy of a confined quark in a
[ gitudinal expansion and contraction [5]. The bag model spherical cavity of radius R. Combining the discussions
describes quarks being confined inside a hadron [6]. In insectionII andIII by assumingits originfromthe con-
3
high energy collisions, the string model represents the tribution of gluons, the magnitude of the hadronic bag
v
1 process of particle production in the fragmentation of pressure is discussed in Sec.IV. The last section is for
9 a stretching string by creating pairs of quark and anti- conclusions and discussions.
8 quark. It worked well [7] for elementary collisions where
0 strings can be formed among the few initial partons and
1. breakuptoformthesoftfinalstatehadrons. However,in II. FREE EQUILIBRATED QGP
0 relativistic heavy ion collisions (RHIC), there are thou-
0 sands initial partons. It is intractable to pair partons Let us first consider a thermally equilibrated quark-
1 and have a string for each pair. Even if the strings are gluon plasma (QGP) system at first. When its temper-
:
v formed, they must be modified by the presence of many ature T and volume V are given, the total energy and
i other color charges. The independent fragmentation ap- particle number can be calculated:
X
proach [7], though valid for high Q2 partonic processes
ar in the vacuum, cannot explain experimentally observed Nf g
largep/π ratioincentralAu+AucollisionsatRHIC [8]. E = (2π¯hi)3 Z fi(T)p0dΓ
Another possible hadronization mechanism, the decon- i=X−Nf
finedquark(fledoutfromthebag)recombinationmodel, Nf g V
has been able to reproduce spectra for almost all stable = i f (T)p0|p|2dp, (1)
particles for different colliding systems. It provides a i=X−Nf 2π2¯h3 Z i
natural explanation for the baryon/meson ratio and the
nuclear suppression factor observed at RHIC[9]. Nf gi
N = f (T)dΓ
A natural mechanism for quark confinement is given (2π¯h)3 Z i
by the bag model [6]. While the bag model has a few i=X−Nf
different versions, we shall in this paper keep the essen- Nf g V
tial characteristics of the phenomenology of quark con- = 2πi2¯h3 Z fi(T)|p|2dp, (2)
finement. The gluons are mediate bosons which transfer i=X−Nf
where N is the number of flavors and g = N N is
f i c s
the degeneracy number for a parton and equals to the
∗[email protected] product of quantum numbers of quark’s color and spin.
f isthedistributionfunctionwhichisofFermi-Diracfor
i
2
that a proton may break and quarks may flee out from
2
the bag.
R = 0.0524 T −4/3
III. QUARKS CONFINED IN A HADRON BAG
1.5
m) Let’sgivesomediscussionsaboutquarkwave-function
R (f from theory. We assume the quarks are confined in a
spherical cavity of radius R, they are free fermions in-
1
side it but cannot fly out because all contributions from
gluonsare attributedto the bag constant. Thenthe sur-
face of the spherebag becomes the maximum rangethat
quarks can arrive. On the bag boundary where the cur-
0.05.06 0.08 0.1 0.12 0.14 0.16 0.18 rent of fermion must be zero. A hadron’s wave-function
T (GeV) is then product of those for quarks.
The Dirac equation for a free massless fermion in the
FIG. 1: The relation between the radius of a proton and its bag is
temperature when take its mass as the total energy. The
numerical simulative formula is given on thelegend as well. (iγµ∂ −m)ψ =0 with m=0, (8)
µ
where ∂ = (p0,p). We will in this paper use the Dirac
µ
quarks and Bose-Einstein for gluons representation
1
fi = 1+e(p0∓µq)/T ’-’ for quark, (3) γ0 =(cid:18)I0 −0I (cid:19),
’+’for anti-quark
1 and
f = for gluon. (4)
i ep0/T −1 0 σi
γi = ,
Here µ is the quark’s chemical potential. For the case (cid:18)−σi 0 (cid:19)
q
when the number density of the quarks is the same as
whereI isa2×2unitmatrixandσi arethePaulimatri-
that of the anti-quarks, µ =0.
q
For a massless quark gas with zero chemical potential ces. We write the four-component wave function for the
massless fermion ψ as
µ =0,Eqs.(1,2)canbesolvedanalytically. Forexample
q
for the case with only two flavors, we get the densities
ψ
for energy and quark number as (h¯ =1) ψ = +
(cid:18)ψ (cid:19)
−
E 7 π2 π2
ǫ= = (g +g ) T4+g T4 where both ψ and ψ are two dimensional spinors.
V 4 q q¯ 30 g30 + −
Eq.(8) becomes
37
= π2T4, (5)
30 p0 −σ·p ψ+
=0 (9)
n= N = 3ζ(3)(g +g )T3+ ggΓ(3)ζ(3)T3 (cid:18)σ·p −p0 (cid:19)(cid:18)ψ−(cid:19)
V 2π2 q q¯ 2π2
34×1.202 The lowest energy solution for the above equation is the
= T3. (6)
π2 s1/2 state given by[10]
Now if we treat a proton as a free equilibrated QGP ψ (r,t) = Ne−ip0tj (p0r)χ
+ 0 +
systemasusetheaboveresults,wecangettherelationof
the proton radius and temperature. The result is shown ψ−(r,t) = Ne−ip0tσ·ˆrj1(p0r)χ−,
in Fig. (1).
where j is the spherical Bessel function which can be
From Fig.(1), we can see, the radius of a proton is l
expressed by an elementary function
decreased with its internal temperature. The numerical
simulative formula is
1 d l sinx
j (x)=(−1)lxl , (10)
RT4/3 =0.0524 (7) l (cid:18)xdx(cid:19) x
If the temperature is 100MeV, the correspondingradius χ aretwo-dimensionalspinors,andN isanormalization
±
ofprotonisabout1fm. Ifaprotoniscompressedtohave constant. The confinement of the quarks is equivalent
a radius of 0.6 fm, the inner temperature will be about to the requirement that the normal component of the
170 MeV, which is close to the critical temperature, so vector current J = ψ¯γ ψ vanishes at the surface. This
µ µ
3
condition is the same as the requirement that the scalar The energy carried by the valence quarks in a proton
density ψ¯ψ of the quark vanishes at the bag surface r = is then 3×E¯ . So that contributions from gluons and
q
R. This leads to sea quarks to the energy is M −3E¯ . The bag pressure
q
decreases with the radium. This may be used to explain
ψ¯ψ r=R =[j0(p0R)]2−σ·ˆrσ·ˆr[j1(p0R)]2 =0 why the resonance particle (which has large radius thus
(cid:12) smallbagpressure)isusuallyunstablebecausetheirbags
or (cid:12)
are more fragile.
[j0(p0R)]2−[j1(p0R)]2 =0 (11) 200
190
From Eq.(10), solutionof the above equation is given by
180 B1/4 = 0.17 R−0.65
2.04 170
p0 R=2.04, or p0 = . (12)
m m R V)160
e
Tbehiinsgrbesruolktenm,etahnesktihnaetticinenoredrgeyr toof akneeypqtuhaerkbacganfrnoomt 1/4B (M114500
larger than p0m determined by the radius of the bag. 130
120
IV. BAG PRESSURE 110
100
0.8 1 1.2 1.4 1.6 1.8 2
Takep0 astheupperlimit,wecanseparatetheenergy R (fm)
m
of quarks from the total of a proton
FIG. 2: The bag pressure change with the radium of proton.
(gq+gq¯)V p0m p3dp The open circle is calculated with Eq.(14), and the line is
Eq = 2π2¯h3 Z 1+ep/T(R). (13) its numerical simulation results, while the star is with the
0 scenario that three quarksare surrounded bygluons.
For simplicity, we have neglected the chemical poten-
tial, and treat the quarks as massless. Then g = g =
q q¯
NcNsNf =3×2×2=12. V. CONCLUSION
The energyfromgluoncontributionprovidesthe pres-
sure effect directed from the region outside the bag
In this work we have discussed some feature of a pro-
ton from the bag model. Especially, we got the relations
M −E
q
B = . (14) between temperature and the radius of the proton when
V
a proton is treated as a non-interacting thermal equilib-
Here M is the mass of the hadron. rium QGP system. The bag pressure comes from the
From Eq.(14), we can easily learn that the the bag contribution of gluons.
pressurewillchangewithradiusasshowninFig.(2). We
also give the numerical formula
B1/4 =0.17R−0.65. (15)
Acknowledgments
The average kinetic energy of each quark can be calcu-
lated as We are grateful to the financial support from China
ChangShaUniversityundergrantNo.SF080101. Wealso
E p0m p3dp thank Prof. A. Bonasera for stimulating discussions and
E¯ = q = 0 1+ep/T(R). (16) comments.
q Nq Rp0m p2dp
0 1+ep/T(R)
R
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