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УДК 531.9+539.12.01
SU(3) ГЛЮОННЫЙ КОНДЕНСАТ ГЛЮБОЛА
Джунушалиев Владимир Джумакадырович д.ф.-м.н., профессор
al-Farabi Kazakh National University, Almaty, Kazakhstan
Вычислено распределение глюонного конденсата для глюбола в скалярном приближении. В этом приближении SU(3) калибровочная группа разделяется на две части: 1) подгруппа
SU(2)^SU(3) .. „
v ' v ' ; 2) левый смежный класс
SU(3)/SU(2) я
v ' v ' . Для описания степеней
свободы SU(2) и смежного класса применялась техника непертурбативного квантования в скалярном приближении. При таком подходе двухточечные функции Грина являются билинейными комбинациями скалярных полей и 4-точечных функций Грина
Ключевые слова: ГЛЮОННЫЙ КОНДЕНСАТ, ГЛЮБОЛ, НЕПЕРТУРБАТИВНОЕ КВАНТОВАНИЕ, СКАЛЯРНЫЕ ПОЛЯ
UDC 531.9+539.12.01
SU(3) GLUEBALL GLUON CONDENSATE
Dzhunushaliev Vladimir Dzhumakadyrovich Doctor of Physics-Mathematical Sciences, professor al-Farabi Kazakh National University, Almaty, Kazakhstan
In a scalar approximation the distribution of a gluon condensate in a glueball is calculated. In this approximation the SU(3) gauge fields are separated on
two parts: (1) is the
SU(2) с SU(3)
subgroup,
SU(3)/SU(2)
(2) is the coset v y v y . Using an
approximate nonperturbative quantization technique two scalar fields are applied for the description of the SU(2) and coset degrees of freedom. In this approach 2-point Green's functions are a bilinear combination of scalar fields and 4-point Green's functions are the product of 2-points Green's functions
Keywords: GLUON CONDENSATE, GEUEBAEE, NONPERTURBATIVE QUANTIZATION, SCAEAR FIELDS
Introduction
One of the problems in a nonperturbative QCD is the determination of condensates. It is well known that the condensates can only be determined in a nonperturbative formulation of the QCD. There is a long history of attempts to determine the gluon condensate from first principles [1-2].
Glueball is (thought to be) bound state of gluons. The nonlinear properties of gluons create the possibility of a color-neutral state made of gluons only - glueball. Glueball properties cannot be computed with perturbation theory, and these remain very mysterious objects over thirty years after QCD was understood in its workings. For review, see [3].
In [4] the distribution of a gluon condensate in a flux tube stretched between quark and antiquark is calculated using approximate nonperturbative quantization technique. It is shown that a longitudinal chromoelectric field is confined with a surrounding coset chromomagnetic field. Such picture presents the concrete realization of dual QCD model in a scalar model of the flux tube. In the scalar model the SU(3) gauge fields are separated on
two parts: (1) is the c SU(3) subgroup5 (2) is the coset SU(3)/ SU(2) SU(2)
degrees of freedom are almost classical and the coset degrees of freedom are quantum ones.
A nonperturbative approach for the quantization of the coset degrees of freedom is applied. In this approach 2-point Green's functions are a bilinear combination of scalar fields and 4-point Green's functions are the product of 2-points Green's functions. The gluon condensate is an effective Lagrangian describing the SU(2) gauge field with broken gauge symmetry and coupling with the scalar field. Corresponding field equations give us the flux tube.
Here we continue the research in this direction and will calculate the distribution of gluon condensate in the glueball. In [5] the model of the glueball based on the calculations similar to [4]. Only one difference is that coset degrees of freedom are quantized as well. In this notice we would like to obtain the distribution of gluon condensate for the glueball using results from [5] and compare obtained results with the results of [4].
Scalar model of glueball
Following to [5] we separate SU(3) degrees of freedom on two parts: the first is the SU(2) a SU(3) su|3group5 ^ second is the coset SU(3)/SU(2) ^ average SU(3) Lagrangian using some assumptions and approximations. The main idea in this approximation is that 2 and 4-points Green's functions can be approximately described by
two scalar fields ^ :
(G2 r/v (X,, x2) = (a; (Xl )Avb (x2)) * c;bj(Xl )* * (*2)+<,
(G4)tdo (*1 > *2) = {K (*! Hb (x2 )A; (x3 )Ada (x4)) -[a;(x^(x2))(a;(x3)a^x 4)
(i)
(G2 )Z (Xl ,x2) = (a; {Xl K (x2))» c;nv * (*, )z * (*2) +m;nv,
(2)
(3)
(g<);™(x19x2) = (a;(Xi)a;(x2)az(x3)A2(xa)
(xx)Avn(x2))(App (x3)A% (x4)),
where a’b’c,d - 1,2,3 are 3^2) indexes, ~ 4,5,6,7,8 are cose^ indexes and
C in (ha
are some constants. Let us note that in [5] we used more than one fields: v instead
of ^ and ^ instead of ^ . This distinction is unessential. We use fields ^ in order to be close as soon as it is possible to the designation of the paper [4]. After that the effective Lagrangian is
Leff - {L
1 2
1 2
-Vj
1
ф: +
X
(5)
where
and
are some parameters. We use here (+,-,-,-) metric signature. It is
necessary to emphasize: The effective Lagrangian (5) is some approximation for the nonperturbative quantization of the SU(3) gauge theory.
The physical significance of this approach is following:
• The scalar fields describe nonperturbative quantized SU(2) and coset SU(3) / SU(2) degrees of freedom correspondingly.
The terms
and
in the effective Lagrangian (5) appear after the
(V A8)2
nonperturbative quantum averaging of the terms fl in initial SU(3) Lagrangian.
1 4 4
The terms ^ L appear after the nonperturbative quantum averaging of the term
j-ABC
it)2 V 2
The term ^ ^ appears after the nonperturbative quantum averaging of the term
jr Aab jr Amn ^ a m'
I 2 j 2 1 4 2 2 4
The terms ^ ^°°’ ^co’ % and ^°° are some additional assumptions.
The field equations describing glueball in the scalar model are
д^ф = -ф[х2 + \(ф2 -ф')]
=~х[ф2 +к2(х2 -%l)\
Special feature of equations is that they have regular solutions with special choice of
parameters. One can say that these equations are eigenvalue problem with eigenvalues.
A spherically symmetric regular solution of (6)-(7) equations describes a ball filled with fluctuating quantum SU(3) gauge fields. We interpret this ball as the glueball. In the spherically symmetric case equations (6)-(7) have following form
ф"+-ф'=ф[х2+\{ф2-<L2)]
JC
x"+- x‘' = x [<P + К {x2 - %l )]
(8)
(9)
where x = = <¡>/<¡>(0), X = X/H0\ I. = 1^(0) and ^ = ^ . Below we
will omit tildes over. The boundary conditions are
0(0) = 1, <|>'(0) = o, (10)
X(0) = Xo> X'(0) = 0. (11)
The regular solution for equations (8)-(9) with boundary conditions (10)-(11) does exist for
a special choice of <^co,^coparameters only. The profiles of 0(r)’X(r) Fig.l are presented.
Figure 1: The profile of - upper curve, X(x)_ lower curve; \ = 0.1, A2 = 1., (j)^ = 1.6171579, Xoo = 1.49273856.
/pA pA^v \
The distribution of the gluon condensate ' ‘lv ' in the glueball can be found from the effective Lagrangian (5). The gluon condensate is
G = -Lefr=(H?H-il)-(E?EAl) (l2)
ea ha
where ' ’ ' are chromoelectric and chromomagnetic fields; the signature of 3D
metric ^lJ is positive ^1J (+,+,+)’ ^-2,3 are the space indexes. We see that if
G(x) < 0 t]ien jn this area the chromoelectric field is predominant but if G(x) > 0 ^]ien t]ie
chromomagnetic field is predominant. The substitution of the fields ^ into the gluon condensate (12) gives us following
Gmm = -У2~х'2Л^2-Ы)1 +Н2 -xll -Ь-xl -\xV
2 2 4 4 4 2 (13)
The profile of G(x) is presented in Fig. 2. We see that in the glueball the quantum fluctuations of chromoelectric field are predominated.
X
Figure 2: The profile of the SU(3) gluon condensate.
It is interesting to take a look on the distribution of a SU(2) gluon condensate SU(2) in this model (here the group SU(2) is considered as the subgroup of SU(3) group). One can show that
Gsum =--f2 +—U2-^f
sm2) 2r 4 \r T./ ^
The corresponding profile in Fig. 3 is presented. We see that in this case the
Научный журнал КубГАУ, №93(07), 2013 года fluctuating chromomagnetic field is predominant.
Figure 3: The profile of SU(2) gluon condensate in the glueball.
Discussion and conclusions
Thus we have calculated the distribution of the SU(3) gluon condensate in the glueball. In the presented nonperturbative model the glueball is not a cloud of gluon-quanta but this object is more similar to a turbulent liquid with fluctuating velocities and consequently fluctuating lines of fluid flow. In this connection one can say that an object presented in the painting “Castagna” of famous surrealist P. Tchelitchew - see Fig. 4, is a schematic presentation of the glueball with fluctuating force lines of chromoelectric fields.
This result can be compared with the results [4]: the difference is that in the glueball does not exist a core with predominant chromoelectric field in contrast with the flux tube where there does exist a core filled with a longitudinal chromoelectic field stretched between quark and antiquark and confined with a fluctuating chromomagnetic field. In our opinion the reason for this is that in the core of the flux tube (considered in [4]) there exists almost classical chromoelectric field. Whereas in the glueball we have fluctuating chromofields with zero expectation values only.
Figure 4: “Castagna” of P. Tchelitchew as a schematic presentation of the glueball with fluctuating force lines of chromoelectric fields.
Acknowledgements
I gratefully acknowledge the Research Group Linkage Programme of the Alexander von Humboldt Foundation, the Department of Physics of the Carl von Ossietzky University of Oldenburg and, specially, J. Kunz for his interest in this research. The work was
supported by the Ministry of Education and Science of Kazakhstan grant No. 1626/GF3. Bibliography
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2. M. A. Shifman, A. I. Vainshtein, V. I. Zakharov. QCD and Resonance Physics. Sum Rules//Nucl. Phys. В147, 385-447, 1979.
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