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Condensate Structure of the Vacuum of Quantum Chromodynamics.

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Annalen der Physik. 7. Folge, Band 44, Heft 5, 1987, S. 345-350
J. A. Buth, Gipzig
Condensate Structure of the Vacuum of Quantum
Chromodynamics
J.HERRMANN
Sektion Physik der Friedrich-Schiller-Universitiit
Jena, DDR
Abstract. The non-perturbative vacuum structure of quantum chi-omodynemics (QCD)is
studied with the help of methods which are generalizations of those used to describe conc1enc;ztion
effects and quasi-particles in superfluid and superconductive mediums. The gliio:i condensation is
explained by the introduction of a new vacuum state defuied by a Bogoljubov transform.ition,
leading to non-vanishing vacuum expectation values n s e.g. the gluon condenmtJionpariuneter, a
negative vacuum energy density, and to a gap in the energy spectrum which is connected with excited
quasi-particle sta,tes with a rest mass.
Kondensatstruktur des Vakuums in der Quanterichromody~~amik
I n ha 1t s u ber si c h t. Die nichtstiirungstheoretisclie Vakuumstruktur der Qwntenchromodynamik (QCD) wird mit Hilfe von verallgeineinerten Xethoden untersucht, die zur Besclweibung
von Kondensationseffekten und Quasi-Teilchen in superfluiden und supraleitenden Xedien benutzt
werden. Die Gluonkondensntion wird durch Einfuhrung eines neuen Va~kuumzustnndeserklkt,, der
durch eine Bogoljubov-Transforrmtiondefiniert ist und zii nicht verschwindenden Vitkuomerwartungswert,en,wie z. B. dem sogenannten Gluon-Koiidensat.ions~r~~meter,
einer neeltiven VtLkuumenergiedichte und eincr Lucke im Energiespekt,rum,fiihrt,, die mit einer endlichen Ruhnmsse der
angeregten Quasi-Teilchenzustiindeverbunden ist.
There is an increasing interest in the study of the lion-perturbative structure of the
QCD vacuum to understand, e. g., the pheiioniei!on of gluon condelisation and the general behavioar of QCD at moderate and lorn momenta. For the investigation of this probleni various approaches have been considered (see e. g. [1-6]).
In the present paper the attempt is made to employ techniques which are generalizations of those used to describe condensation effects and quasi-particles in supeifluid
and superconductive mediums. In this approach the theoretical untlerstandiiig of the
gluon condensation, leading to a non-vanishing expectation value of the operator
(I$)Z
in vacuuni [l], is provided by a transition of the “pert~rbative”vaciiiini 10)
to a new vacuuin I@) by nieaiis of the Bogoljubov transformation. The new ground
state has the property of being annihilated by new annihilation operators of boson~,
in the definition of which the main part of the gluoii self-interactionis taken into account.
The introduction of the new vacuum is connected with non-vanishing expectation values
+
<@ IaAa,A I @)
0, (@ 1 a d - ga 1 @> =F 0,
(1)
-<@ 1 ciacj<, 1 @> 0, <@ 1 C K ~ -ga
C
1 0‘ 0
for colour singlets, where c;~, clil. are the creation and the aiinihilatioii operators (with
respect to the “perturbative” vaciiuni 1 0 ) for the colour-neutral gauge field, u i 2 ,alia
and b&, bx, the correspontling operatois for the coloui~-chargetlgauge field.
+
+
Ann. Physik Leipzig 44 (1987)6
346
Let us start from the QCD Hainiltonian with a colour group SU(2) in the Coulomb
gauge div A0 = 0. R e iiitroduce the colour-charged gauge field
and t,he colour-neut.ra1gauge field
R’eglecting the quark interaction terms trhe Hamiltonian will decompos!e into three
parts :
H = Ho H,T + H I “ .
(4)
Ho is the unperturbed Hamiltonian
+
+
Ho = z;,n
C [1)1;(aiij.a1ii b&.bK>.
+ cii>.cIi>.),
(5)
with wli= lK1. With HI’ t,he self-interaction of the transverse glwns are described:
where me hare used the abbreviations
x
(16o,,m,m,+
All operators have to be taken .in the normal orderitig
am,;‘-
q
1- 112 .
J. HERRNANN,
Condensate Structure of the Vacuum in QCD
347
and
x
(c$+q,A'bkIi,A
- C-K-q,A.aKA)
1-
The transverse gluon contribution H,TI are most essential for the effect of gluon condensation. By a unitary transformation
1
H' = e S H e - 8
H 0 HI
rs, H01 [H,HI1 y
Holl . .. 9 (8)
1
+ +
+
+ rfufJ, +
one can transform the three-gluon coupling term into a corresponding four-gluon coupling term. The operator S in (8) must be chosen in such a way, that the first order terms
in (8) vanish :
+
rs, H01
HI =
0
(9)
After t,he reduction of a four-gluon Hamiltonian (which corresponds to a transition to
the Frohlich Hamiltonian in the theory of superconductivity) one can continue by means
of various equivalent methods. We use, e. g., the random phase approximation, whereby
the non-vanishing expectation values (1)has to be taken into account. After some intermediate calculations the effective Hamiltonian reduces to
He,,
=
2 (WIi + pI(K)) (a,fAaK,+ b
KJ
+
+
2 d - K ~
+ a2abtRa) + 1 +
+
+
[ ( ~ l i p z ( I i ) )( C i a C K a
91(K)(axAb-KA
C~KIC-KA)
(10)
C ~ C K- A ) ]
pqir)(CiaC+lia
Here the introduced real parameters p l ( K ) , p 2 ( K ) ,y 1 ( K ) ,q , ( K ) are related t o the
expectation values (1).The equations determining these quantities will be given further
below. The Hamiltonian (10) can be diagonalized by the Bogoljubov transformations
+ VKB?K,A,
(11)
+
+ zKyi
with the conditions u i - 4 = 1 and wg - zg = 1. One has t o choose the functions
uK, v,, w, and zK in such a way, that non-diagonal terms in the Hamiltonian vanish:
+
+
+ 4)= 0,
(12)
+
+ T!?.(K)(4 + '$1 O .
aK,a
=UK~K,A
bkK,A
= uKB2K,A
c K , l = wKYK$
~(wI<
P ~ ( K ) UIiVK
)
2(WK
/-%(I{))w K z S
From (12) we obtain
vKaK,As
K,A
Y
QI~(K)(u%
=
348
Ann. Phyaik Leipzig 44 (1987)6
J. HERRNANN,
Condensate Structure of the Vacuum in QCD
349
where 6 is the angle between p and K,and for El one has to choose El in the equations
for pi and pll and E, in the equations for y, and p12. I n the equations (15), a renormalization of the gluon energy was taken into account, which can be done self-consistently by
a new separation of the Hamiltonian (7): H = H,’
HI‘ with H,’ = Heff,HI‘ = HI Heff Ho. I n the equation (9) then the gluon energy oIi has to be substituted by its
. introduction of new creation and annihiliation operators
“renormalized” value E ( K )The
(11) is connected with a transition to a new vacuum state defined by the relations
+
+
The energy of this new vacuum state is lower than the perturbative vacuum energy,
which follows from (14) with wl(Io> El(K,.The excited state means a quasi-particle
with an energy Ep.I n deriving the equations (15) we assumed a polarization unit vector
eIiAfor circular polarization, therefore the excited state 11) is a spin-singlet state with
a rest mass m = E ( p --t 0). The most essential problem therefore is the question of the
existence of a solution of the equations (16) and (17) with the property
It should be emphasized that the existence of such a gap is seemingly forbidden by a
theorem of Hugenholtz and Pines [7], who generally proved a phonon-like (that means a
gapless) excitation spectrum for a condensed boson gas. But indeed this theorem is valid
only under the conditions of a regular selfenergy matrix. What was proven in [7] is
nothing but a general relation, which corresponds in our notation to the relation
p l ( p = 0) = y L ( p= 0), I = 1, 2. Indeed, this relation is also valid in our case, following from the equations (16) and (17). Then a gapless excitation spectrum follows, if
y l ( p )and yL(:l(l))
are regular functions at zero momentum. I n the case of the QCD Hamiltonian (6) one can show that the self-energy matrix elements y L ( pand
) y l ( p )are singular
for p equal to zero. Due to this fact one gets a gap & ( p = 0) = lim (2mpyl(p)) larger
n=O
than zero.
Let us give a brief discussion of the equations (16) and (17); a n approximative solution will be given in a forthcoming paper. The main contribution in equation (16) and
(17) comes from the transverse gluon part. Neglecting the Coulomb term Hf and H F
one gets the relation pi(p) = y l ( p ) ,y&) = y 2 ( p ) .For moments smaller than a critical
value the denominator in (16) is everywhere positive, that means that the vacuum is
i n its non-perturbativ phase with a mass gap A&,.,pc) > 0. At the critical momentum
pc the iritegrand in (16) becomes singulary a t E(pJ = E(KJ E(pc - Kc). For a
momentum larger than the critical value ( p > pc) by a phase transition the vacuum
goes over into its perturbative phase with a mass gap A ; ( p > pc) = 0.
Finally let us consider the gluon condensation parameter, introduced phenomenologically in [l], and the vacuum energy density. Calculating the expectation value
+
Ann. Physik Leipzig 44 (1987) 5
360
(Fa> we obtain
I n a n order of magnitude estimation we assume p l ( K ) = y l ( K ) = p 2 ( K )= y , ( K ) ,
collp,(K) = const for K < pc and p F ( K =
) v l ( K )= 0 for K > pc. Then for the gluon
condenaation parameter we obtain
With a glueball mass of A, w 1GeV, a8 FV 0,2 and a critical momentum pc w 0,2 GeV
this give (F8) w 3 10-8 (GeV)4.
For the vacuum energy density from eq. (14) one can derive the following expression
Eva,
1
1
Eva, = -7
[ E l ( K ) - W K - p l ( K ) f -;i- (%(a) - a I < -p 2 ( K ) ) ]
.
z
v K,rl
3
=16na [2%(22
+ 4:)3’2 - 2P:M + 43
(21)
-
With the same parameters as in (20) this gives eVacw 2 10“ (GeV)4.
References
[l] SHIFMAN,
M. A.; VAINSTEIN,
A. I.; ZAKHAEOV,
V. I.: Nucl. Phys. B 149 (1979) 385, 448, 519.
.[2] SAWIDY,G. K.: Phys. Lett. 71 B (1977) 133.
[3] NIELSEN,
N. K.; OLESEN,P.: Nucl. Phys. B 144 (1978) 376.
[4] FUKUDA,
R.: Phys.Rev. D 21 (1980) 485.
[5] FUKUDA,
R.; KAZAMA,
Y.: Phys. Rev. Lett. 45 (1980) 114.
[S] MILTON V. A.: Phys. Lett. 104 B (1981) 49.
[~~‘HUQENHOLTZ
N. M.; PINESD.: Phys. Rev. 116 (1959) 489.
Bei der Redaktion eingegangen am 27. August 1985.
Anachr. d. Verf. :Dr. J. HERBMANN
Sektion Physik der
Friedrich-Schiller-Universit5t
Jene
Max-Wien-Platz 1
Jena
DDR-6900
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