SCREENING OF THE QUARK-QUARK INTERACTION

G. A. Sardanashvili and A. E. Mikula UDC 530.12

We calculate the screening of a linearly increasing quark--quark potential in a many- quark system and explore whether such a system can form a condensate of the Cooper type.

An integral element of the majority of g a u g e models of the interaction of elementary particles is the noninvariant Higgs vacuum. It is described by introducing an external classical Higgs field and then the interaction of quantum fields and particles with this ex- ternal field represents their interaction with the vacuum and leads to effects like the ap- pearance of mass for matter and gauge fields and the mixing angle~ However, the physical nature of the Higgs field and the noninvariantvacuum remain unclear.

A widely held view at the current time is that such a vacuum can be represented by some kind of condensate. It is possible that it is a gluon condensate. But such a condensate would not possess "flavor," whereas the Higgs vacuum in models of the grand unification is not invariant to "flavor~ A different view is that the Higgs vacuum is a condensate of the Cooper pair type. For example, several years ago the so-called "technicolor" model was ac- tively discussed. In this model a Higgs boson was represented as a pair of "techniquarks" interacting via "technicolor" gluons rather than "color" gluons.

The condensate could be of a different type: a pion condensate formed by pairs of par- ticles and antiparticles; a Cooper condensate formed by pairs of particles; with short-range forces causing pair-formation; with long-range pairing forces. In order for pairs to occur there must be attractive forces between their component particles and these forces must be long-ranged (their magnitude is not the important issue) for the condensate to have char- acteristics like an energy gap and a phase transition temperature when the energy gap becomes equal to zero. This type of condensate can serve as a model of the Higgs vacuum. The forma- tion of this condensate at a certain temperature (energy) is the key to describing the hier- archy of spontaneous broken symmetries in models of the grand unification. As a first step it is natural to assume that this condensate consists of pairs of quarks, still remaining within the pre-Onnes models.

There exists an attractive force between quarks which increases with distance and leads to the formation of quark--antiquark pairs (quarkonium). This heavy meson is a special bound state of a quark and antiquark [I], unlike q-mesons, for example. The interaction between quarks, although carried by massless particles (gluons) is apparently not a long-range inter- action, like the electromagnetic interaction, because of the interaction of gluons. But even if the quark-quark potential increases with distance in an unbounded manner, it will be screened in a many-quark system and therefore it becomes short-ranged. We show this in the present paper.

We consider a system of nonrelativistic (heavy) quarks in the approximation of a poten- tial quark--quark interaction. The interaction potential is invariant to rotations in "color" space and has the form

~',~ (r) = ~ (r) X~ I",l "~ (1)

where 2I~ = are the Gell-Mann operators of the color spin of quarks. In the field-theoretic picture o~ potential (i) corresponds to one-gluon exchange between quarks with an effective (taking into account the gluon interaction) interaction charge. The exact form of the poten- tial ~ (r) is unknown, but taking into account the property of asymptotic freedom (at small distances) and direct numerical calculations in the lattice models of chromodynamics and the spectrum of quarkonium (at large distances), one obtains a generally accepted approximate Cornell potential [I]

M. V. Lomonosov Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh gavedenii, Fizika, No. 2, pp. 82-86, February, 1988. Original article submitted May 24, 1985.

154 0038-5697/88/3102-0154512.50 �9 1988 Plenum Publishing Corporation

1 b ( 2 ) - - ~ ( r ) - - a t . 3 ' r

To describe screening it is sufficient to limit ourselves to a system of one-color quarks in a background of uniform compensating charge. We use the method of temperature-dependent Green's functions as represented by continuum integrals [2]. The generating functional of the system has the form

Z = N - ~ " [d~l [dQ] e x p S ,

0

1

0

d'~xc~y~(x, ~) ,;-(y, ~) v ( x - y) 0, (y, -) 9 (x , ,),

(3)

where ~ is the wave function of the quarks; X is the chemical potential; I is the source. Applying the mean-field method, Z can be rewritten as:

X 6 (~%, p~) B (p, -'7 Pa - - P'-'), ~ = , t~ = - - 3

1 d:~pz(% p) V - ' ( p ) z ( - - t o , - - p ) , w = S ~ - - 2

Z = N -~ S IdOl [@1 [dzl exp {S, + S,= -i- So};

u~

p2 (2n - - 1) r. % = ~ m - - ) , , w= , [ n l = O , 1 . . . . . oc;

(2~.)3;'-'3 ~,'~ ~ J oJ 1 - cv~ - I -o~ 3

(2n - - 1) r~ 2n=

2it=

(4)

The compensating charge is taken into account by carrying out the integration in S o for p~0.

The functional Z in the form (3) can be obtained from Z in the form (4) by integrating the latter with respect to the mean field o and using the formulas for Gaussian integrals.

The generating functional gator is given by

the mean field propagator by

(4) leads to the following diagram procedure. The quark propa-

- - ( i a ) l - a p : ) - ; g ( p , - - p ~ ) g ..... =, ( 5 a )

V ( p , ) g ( p , - r - p2) B .......

and the ~-~ interaction vertex by

We integrate the functional Z with respect to the fields ~. into account vacuum polarization of the fields $ by the field o. sian we find

Z = ~ ' - - I f [d~] exp {S, - tr In 3,I -~- ]MI} ,

where M is an operator in the function space of the fields ~ and has the form

~M (,;~, p~; ~2, P~) = - - (iu, - - sp) & (p, - - p ~ ) B~_~,~ - - i (2=)-3n~-~;2~ (%, P3) g (P, - - P 2 + P3) g,,,, . . . . . . . . . .

(Sb)

(5c)

This corresponds to taking Since the integral is Gaus-

Expanding tr in M in a series of o, we obtain the renormalization of the functional S o by adding to it the term in this expansion quadratic in ~ (the other terms in the expansion describe the spectrum of excitations of the field o). In the diagram procedure defined by (5), it is represented by the diagram (Fig. I) where the solid lines denote the field ~ and

155

V'-I(~, p) = V-~(p)-~

= V-1(p)---

where n(e) = (exp(Be) + i) -x.

Fig. I

the dashed lines denote the field o. The potential V' resulting from this renormalization is (for ~ = 0) the desired screened potential in the many-quark system. The screening occurs becausesof polarization by the quarks. This method of constructing V' corresponds to the random phase approximation, which is normally used in discussing screening, such as the screening of the Coulomb potential in an electron gas [3].

We calculate V' for T = 0. From the renormalized action S~ we find

(2~) ~ ,

1 ~ d ~ , n ( ~ p + p , - - i ~ ) - - n ( a p ' ) , (2~) ~ i~ + ~p+p, - - ~p~

where % = CF is the Fermi energy.

At T = 0

n (-:) =/(p) =

p2 o, ~-m > ),,

p~ 1, - - < ) , ,

2m

We then obtain at m = 0, T = 0

v ' - ~ ( p ) = v - ' ( p ) - I ~ d3 p, ( 2 : p

/ (p + p') - / (p')

Epq-p, -- Ep,

= v-~{p) (.~)i~ ~ P~-/ ' T , T

(6)

where the last term exists only for p < 2PF.

The polarization corrections to the potential V-1(p) in (6) correspond exactly to the analogous term describing the screening of the Coulomb potential in an electron gas [3]. The principal factor determining the damping of the field at large distances is the constant (2~)-ImPF , which is multiplied by a factor of two when the logarithmic term is expanded in a

series in p.

In the potential (2) we keep the part varying linearly with r, which dominates at large

I distances V (p) = -if ? (P) ---- 8r'aP -4,

and in (6) we keep only the constant term 2(2~)-2mPF . Then the calculation must be done us- 8ha

ing a regularized potential V(p)-~-lira V(p, ~) (for example; V(p, ~)- ):

( )_i . V ' (p )=8r .a p 4 + 4 m a p p ,

V ' ( r ) = 2a exp - - 4 r sin r - - ] / ~ - 1 : 4 (7 ) r k ~ \ } j

4ma ~ - - p F .

When we i n c l u d e t h e t e r m b r -~ i n ~ and t h e l o g a r i t h m i c c o r r e c t i o n i n (6 ) we f i n d t h a t t h e d a m p i n g a n d o s c i l l a t i n g t e r m s i n v o l v e d i f f e r e n t c o n s t a n t s i n ( 7 ) , b u t t h e q u a l i t a t i v e f e a t u r e s o f t h e s c r e e n i n g o f t h e p o t e n t i a l r e m a i n s u n c h a n g e d . I t c a n b e shown t h a t s c r e e n - i n g must take place for any polynomial potential ~ (r) and will be of the form rne -~r if the

156

quantity (~-~(p) + a)-~ has a complex pole and r-~sin(~r + d) if it has a real pole.

Screening will also exist in a system of three-color quarks and antiquarks. In this case the term describing the quark-quark interaction in the functional (3) is replaced by

(x - y) ( ~ (x) Ira%# b (x)) ~ (y) I ~ d (y)),

where a, b, c, d are "color" indices, and the Lagrangians L~ and L~ in (4) are replaced by

L~ = -- i ~ (~) t~%~ ~ (~) ~m (X), Lo = -- ! ~ (x) ~-~ (x -- y) ~ (y). 2

Hence the polarization term in (6) is multiplied by the quantity I = Imablmba (no summation with respect to m). We have I = 3 for a system of quarks and I = 6 for a system of quarks and antiquarks.

We have calculated the screening of the potential at T = 0. Screening also takes place when the temperature is nonzero. In the approximation p << 2W~mT, 0 << mT=/e 2 (9 is the den- sity) se can obtain an expression analogous to (7) with ~ = 8~e0T -I.

Hence in a many-quark system the quark-quark interaction may not provide the formation of a condensate of the necessary type. In several models the quark condensate is constructed from the same type of four-fermion interaction ~ as the Cooper pair condensate [4]. How- ever the question arises as to the nature of this interaction. In the case of a Cooper con- densate it is produced by interactions between electrons due to phonons in the lattice of ions. Therefore to model the Higgs field as a quark condensate it is necessary to assume the existence of an analogous "phonon" interaction between quarks, and also a '~edium" in which these phonons arise. The role of such a medium could be played either by the gluon vacuum (condensate), since in this case it is not required to have flavor, or any type of pre-Onnes vacuum.

.

4.

l)

2.

LITERATURE CITED

A. A. Bykov, I. M. Dremin, and A. V. Leonidov, Usp. Fiz. Nauk, 143, 3 (1984). V. N. Popov, Continuum Integrals in Quantum Field Theory and Statistical Physics [in Russian], Atomizdat, Moscow (1976). J. R. Schrieffer, Theory of Superconductivity, Bemjamin, New York (1964). M. Bailin and A. Love, Phys. Rep., 107, 327 (1984).

ENERGY SPECTRUM OF THE DIRAC EQUATION FOR THE SCHARZSCHILD AND KERR FIELDS

I. M. Ternov and A. B. Gaina L~C 531.51:524.8;531.51.51;52-i/-8

We consider the effect of relativistic corrections and rotation of the central body on the structure of the energy spectrum of a particle with spin in the Schwarzchild and Kerr fields. A splitting of levels is obtained, which corre- sponds to the classical shift of the perihelion of the orbit and precession of the plane of the gravitational spin-orbit interaction and several nonlinear spin effects are calculated.

The discrete spectrum of resonance states in the case of finite motion (E < ~.c 2) of spinless and spinning particles in the Schwarzschild and Kerr fields was considered in [I, 2]. In the case of particles with spin one-half, if

�9 I GM ~ o % = 2 G ~ N I l~c ~ 1, a ~ ~ ~ - - (1)

M c c ~

t h e n t h e s e s t a t e s a r e c h a r a c t e r i z e d by the n o n r e l a t i v i s t i c h y d r o g e n i c s p e c t r u m [ 2 ] . However it is obvious that relativistic and spin effects, and also the rotation of the central body, must lead to a more complicated spectrum. This question is considered in the present paper.

Although separation of variables can be done exactly for the Dirac equation in the Kerr field [3], the nonlinear effects that arise when a + GM/c =, are difficult to take into ac-

M. V. Lomonosov Moscow State University. Kishenev Polytechnic Institute. Translated from Izvestiya Vysshykh Uchebnykh Zavedenii, Fizika, No. 2, pp. 86-92, February, 1988. Orig- inal article submitted May 24, 1985.

0038-5697/88/3102-0157512.50 �9 1988 Plenum Publishing Corporation 157