Fortschritte der Physik / Progress of Physics: Band 27, Heft 9 1979 [Reprint 2021 ed.] 9783112522769, 9783112522752

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM A U F T R A G E D E R P H Y S I K A L I S C H E N

GESELLSCHAFT

DER DEUTSCHEN DEMOKRATISCHEN VON F. KASCHI.U IIN.

REPUBLIK

L Ö S C H E . R. R I T S C H L U N D R. R O M P E

H E F T 9 • 1 9 7 9 • B A N D 27

A K A D E M I E

-

V E R L A G

EVP 1 0 , - M 31728

•

B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu

richten

— in der D D R a n das Zeitungsverlriebsamt, a n eine B u c h h a n d l u n g oder an den A K A D E M I E - V E R L A G , DDK - 108 Berlin, Leipziger S t r . 3 - 4 — im sozialistischen Ausland an eine B u c h h a n d l u n g f ü r fremdsprachige L i t e r a t u r oder an den zuständigen Postzeitungsvertrieb — in der B R D und Westberlin an eine B u c h h a n d l u n g oder an die Ausliefcrungsstcllc K U N S T U N D W I S S E N , Erich Bieber, 7 S t u t t g a r t 1, Wilhelmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, H ö c h s t ä d t p l a t z 3 — in den übrigen westeuropäischen Ländern an eine B u c h h a n d l u n g oder an die Auslieferungsstelle K U N S T U N D W I S S E N , Erich Bieber G m b H , CH - 8008 Zürich/Schweiz, D u f o u r s t r a ß e 51 — im übrigen Ausland an den I n t e r n a t i o n a l e n B u c h - u n d Zeitschriftenhandel; den B u c h e x p o r t , Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, D D R - 701 Leipzig, P o s t f a c h 160, oder an den A K A D E M I E - V E R L A G , D D R - 108 Berlin, Leipziger Str. 3 - 4

Zeitschrift ,, Fortschritte der Physik** Herautgeber: Prof. Dr. Frank Kaschluhn, Prof. D r . Artur LAsche, Prof. Dr. Rudolf Ritsehl, Prof. D r . Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 22 36221 und 22 36229; Telex-Nr. 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R - 104 Berlin, Hessische StraOe 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V E B Druckhau» „Maxim Gorki", D D R - 74 Altenburg, Carl-von-Ossietzky-StraBe 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik 1 * erseheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je B a n d : 180,— M zuzüglich Versandspesen (Preis Tür die D D K : 120,— M). Preis je Heft 15,— M (Preis für die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/27/9. © 1979 by Akademie-Verlag Berlin. Printed in the German Democratio Republio. AIS (EDV) 57 618

ISSN 0 0 1 5 - 8 2 0 8 Fortschritte der Physik 27, 4 0 3 - 4 3 4 (1979)

Nonlocal Quark Model and Meson Decays A. Z. DUBNICKOVA, G. V. EFIMOV, M. A. IVANOV

Joint Institute for Nuclear Research, Laboratory of Theoretical Dubna,

Physics,

USSR

Abstract In the framework of nonlocal quantum field theory we have constructed a quantized field called the virton field satisfying the conditions: i) the field of free virton quanta is equal identically to zero, ii) the causal function of this virton field differs from zero. We consider this virton field can be a good candidate for the description of quark confinement. It is suggested that the physical particles are described by the ordinary quantized fields but they interact with each other through the quark-virton field only. Using the simplest form of the interaction Lagrangian invariant under transformations of the SU(3) X SUC(3) x U1 group, the mass corrections, some strong, weak and electromagnetic decays of the vector and pseudoscalar mesons are considered. A good agreement with experiment is obtained.

1. Introduction

The main problem of all quark models is that quarks are not discovered experimentally. A large number of models was suggested to explain the quark confinement (see, for example [2]). A common feature of these models is the assumption t h a t quarks do exist as physical particles but they cannot arise according to a certain dynamical mechanism. Such a mechanism could consist in an interaction with a gluon field, "bags" and so on. I n this paper the alternative hypothesis is proposed: quarks do not exist at all as usual physical particles and exist in the virtual state only. This hypothesis can be realized in the following way [2]. I n the framework of quantum field theory one succeeded in finding such "particles", called "virions", which possess the following properties. First, the field describing free virtons is identically zero, i.e. virtons do not exist in a free state, and, second, the causal Green function of the virton field is nontrivial, i.e. virtons exist in a virtual state only. The virton field can be constructed by methods of nonlocal quantum field theory [J]. The virton field is a good candidate to describe quarks and does not need any additional fields (like gluon fields) to ensure the quark confinement. I n the present paper we take the first step to apply this model to low energy meson physics. The interaction of the virton field with fields of physical particles (mesons, baryons, and so on) can be realized in two ways. First, one can looji for physical particles as bound states of an appropriate system of virtons. However, this possibility will not be consi32

Zeitschrift „Fortschritte der Physik", Hett 9

404

A . Z. D t j b n i c k o v a , G. V. E f i m o v , M. A. I v a n o v

dered in this paper because we cannot find any bound states in quantum field theory. The second possibility consists in that real physical particles are describe d as usual local quantum fields but they interact with each other through the virton field. For example, the interaction of the meson field n(x) with the virton field q(x) can be described by a Lagrangian of the type X^x) = gn{x) (q(x) y5q(x)). This possibility permits one to construct the finite unitarity ¿'-matrix by methods developed in [3] and to calculate any amplitudes of physical processes. I n this approach the quark-virton field is considered as a field carrying the interaction between hadrons. I n other words, hadrons which are described in the usual way, interact indirectly with each other by means of the quark-virton exchange. Two essential problems arise in the application of this model to the hadron physics: First, what is the interaction Lagrangian of hadrons with quarks? Second, is it possible to describe physical effects of strong interactions in the framework of perturbation theory, i.e. will effective coupling constants be less than one? In the given paper we consider some processes of meson physics of low energies (mass corrections and decays of pseudoscalar and vector mesons) in this nonlocal quarkvirton model with the simplest choice of the interaction Lagrangians. I t turned out that this model can describe quite well the processes under consideration and effective coupling constants are found to be less than unity.

2. Quark Confinement in the Nonlocal Quantum Field Theory We introduce a new quantized field describing particles which do not exist at all as usual physical particles, like electron, proton and sp on, and exist in the virtual state only. These nonexisting particles will be called "virions" and the field q(x) describing these particles the virton field. This field can be constructed in the following way. The fact that the usual elementary particles are observed in experiment is expressed mathematically in the quantum field formalism so that the fields of free particles satisfy the Dirac or Klein-Gordon equation. We will suppose that virtons are pure quantum field objects of such a kind that the field of free virtons is equal to zero identically, i.e., q(x) =

0.

(2.1)

We will use the Lagrangian formalism for describing elementary particles. Then our hypothesis means that in the Lagrangian of the virton field (x) =q(x)Z(p)q(x),

(2.2)

where p — ic = iy^ d/dx^, the operator Z(p) should be chosen in such a way that the unique solution of the equation Z(P) q(x) = 0 (2.3) should be (2.1). Thus, we postulate that the virton field is described by the Lagrangian (2.2) and satisfies the equation (2.3) the solution of which in the classical case is zero (2.1). On the other hand, we want the Green function of the field q(x) obeying the equation Z{P) G(x - y ) = id(x -

y)

(2.4)

405

Nonlocal Quark Model and Meson Decays

to be nontrivial G(x-y)

(2.5)

=iZ-i(P)ô(x-y).

I t means that the virton field which equals zero in the free state can exist nevertheless in the virtual state. I f this virton field is connected with fields of usual physical particles, this will lead to the nontrivial interaction of these particles. I n the framework of the standard methods of local quantum field theory it is impossible to satisfy the equation (2.3) with solutions (2.1) and the equation (2.4) with solution (2.5) at the same time. However, in the framework of the nonlocal field theory developed in [3] this problem can be solved. The idea consists in the following. We want to construct a regularized quantum field qô(x) defined on a Fock space which satisfies the imposed conditions lim q\x) = q(x) = 0, (2.6)

lim (0| T(q»(x) q6(y)) |0> = iZ-\p)

ô(x -

y).

Here we give a solution of this problem. 3. Equation for the Virton Field In the Lagrangian formalism the noninteracting fermion field q(x) is described by the Lagrangian density 20 (x)=q(x)Z(P)q(x), (3.1) where Z(P) is an operator. For instance, for Dirac and Klein-Gordon equation it has, respectively, the form Z{f>)=P-m, Z{P) = p2 - m?. B y the variational principle the field q(x) for Lagrangian (3.1) obeys the equation Ztf>)q(x)=

0.

(3.2)

The equation for the free Green function 0(x — y) of the field q(x) is written in the form Z(P)G(x-y)=id(x-y).

(3.3)

For the Dirac or Klein-Gordon equation the solution and quantization of eq. (3.2) is the well studied problem. Our first task is a follows: to find, within the standard Lagrangian formalism, classes of such operators Z(P) defining equations which could pretend to describe the virions in the framework of our hypothesis. As stated above, we proceed from the assumption that the virions do not exist as usual physical particles. This hypothesis can be realized in the following way. We suppose that the field describing the free virton field satisfying the equation (3.2) is identically equal to zero q(x) = 0 . (3.4) I n other words, in the Lagrangian (3.1) the operator Z(P) should be chosen in such a way that the unique solution eq. (3.2) should be the zeroth solution (3.4). 32*

406

A . Z . DUBNICKOVA, G . V . EFIMOV, M . A . IVANOV

Consider now what conditions on the form of Z(P) follow from our requirements. First, from the requirement that the Lagrangian should be real and the action 8 = /

dxl0{x)

should exist as a functional on rather smooth functions in the Minkowski and Euclidean spaces it follows that the function Z(z) should be an entire analytic function of variable z and [Z(z)]* = Z(z*). Second, the requirement that eq. (3.2) possesses the unique solution (3.4) implies that the function Z(z) has no zeros at any values, of z. The general, form of entire functions of a finite order satisfying these requirements is as follows Z{z)=Cep»W, (3.5) where -PA-(z) is a polynomial of degree N with real coefficients, C is a constant. We will use the methods of nonlocal quantum field theory. I t means that the Green functions (2.5) should decrease in the Euclidean region. This requirement leads to the following condition on the function Z(z): Z{z) = 0 (exp (-z 2 ) j V ' a )

when

z2

-oo.

(3.6)

Further, if we introduce a principle of minimum in the sense that we take the lowest degree of polynomial PN(z) in the exponential function (3.6) which allows all the above requirements to be satisfied, then we obtain Z{z) = —M exp \-lz -

^z2},

(3.7)

where M, I and L are constants. The constant M of the dimensionality of mass in this approach gives the scale of the virton field q{x). It is not independent variable because no physical characteristics depend on it directly. In fact, this constant will enter only into definite combinations with the coupling constants of interactions of our virtons with other elementary particles we will consider below. The constants I and L are fundamental in our approach. They will define the dynamics of all possible virton interactions. Thus our requirements permit us to find the operator Z{p) with two independent parameters I and L only and to avoid any functional arbitrariness. Finally, the operator Z(P) = —M exp | - I f -

(3.8)

obeys the above conditions. 4. Quantization of the Virton Field

q(x)

Our further problem is to quantize the system described by Lagrangian (3.1) with operator Z(p). This task is rather peculiar as the corresponding classical solution of eq. (3.2) is identically equal to zero. To solve the problem we use methods developed in the quantum field theory with nonlocal interaction [3]. The idea of our method of quantization is as follows: in Lagrangian (3.1) the operator Z(p) is changed by a regularized operator Zs(p) such that, first, the

Nonlocal Quark Model and Meson Decays

407

function Zs(z) has an infinite number of zeroes

(4I)

w - f f k - m ) a t points zj=Mj(d)>0

(j = 1 , 2 , . . . )

which in the limit of removing the regularization (5 MjU5) -> oo and, second,

0 (4.2)

lim Z6(z) = Z(z). 0

(4.3)

I n our case this can be achieved, for example, in the following way: n\

\ 4/

(z + [¿? n iMf '

¿o A . ! \ n\\±) 4 /

^

+

« j - l ^ W >±i - ^ '

(44)

Here = x (T -

{j =

2>

• ••0 '

(4 5)

-

H = 2ljL2, the parameter a > 0, and Aj(ô) are positive coefficients which can be determined easily. A parameter n0 defines the decreasing of the regularized function \Zi(z)Yl in the complex z-plane:

Let us introduce the system of fields qf{x) =

q»(x),

p

q>'

(/ = 1 , 2 , . . . )

(4.6)

j=i

and I0(x)=q(x)

Z(P)

q(x)^I0\x)

CO = q\x) Z>{$>) q>(x) = 2J (-1)'?,'(«) >=i

(i> - JW)

&'(*)•

(4-8)

These fields g^ix) correspond to the fictitious "ghost" nonphysical quanta with mass Mj{d) and have no real physical meaning. They play an auxiliary role and should disappear in the limit (5 - > 0.

408

A. Z. DTJBNT&KOVA, G. Y. EFIMOV, M. A. Ivanov

The solution of these equations can be written in the usual form qfix) =

/ dk[V%djke-^

+ W^h^]

q,s(x) =

/ « » [ F ^ e « * + W%hjke-ik*]

(4.9) (4.10)

where Fyjt and Wjk are the Dirac spinors and if0 =

= 1lM?{d) + k\

The Hamiltonian of this system of fields has the form oo H0* = £ (-1)'- / dkEjk(d) \djkdjk - hjkhfk1. j=i

(4.11)

As the energy of our system must be positive the spinor fields qflx) should be quantized according to the canonical procedure of quantization with the indefinite metrics: {djk, dyk'} = {hjk, hjf'v) = (—1)»' 6jrd{k — k'). The rest of anticommutators equals zero. The space of states J f containing all ghost particles is a vector space with the indefinite metrics. It consists of x)

the vacuum state |0), that is unique, defined by the conditions djk |0> = hjk |0) = 0

and normalized to (0 [ 0) = 1, ii) one particle and many-particle states which can be constructed in terms of the basic vectors |jn, im) = —L= djlhl • • • dJnK • htPl • • • ]/«,! ml

|0).

All the one-particle, many-particle states and vacuum generate a complete system of eigenstates in the vector J f . It is essential that vacuum |0) and operators djk and hjk are independent of the parameter of regularization d. We define the space J f (E) to consist of normalized physical states of this system with the energy nonexceeding an energy E: W(E) = £ j d » k j d ••;Pm) € Z2.

Let us define the space of test functions Z2. We say that the function of N variables ult...,uN __ _ / («!, ..., uN) £ Z2 if / (m1; . . U y )

Nonlocal Quark Model and Meson Decays

409

is differentiable and If(u u ..., Uy)I ^ a e x p

f

|«v|2|

for any e > 0 and constant Ct > 0. The space -Z2 which is the space of Fourier transformations of functions J C Z 2 consists of entire analitical functions /(f 1; ..., £A.) for which there exists Cs > 0 such that |/(f„...,c*)| ^ Oe exp | e ^

j

for any e > 0 and OO

00

/ dii ••• J

— 00

N

—00

|/(|] + irju

+ irjn)| < oo

for any r n , ..., rjN. Then for yi(E) € 3V(E) we have \y>(E), V(E)) = £ f d"k f dmp6(E - £ Ejk(d) - £ {jn.im} X(_ip+«|/{.Bj.m)(fe>p)|2o

(5.2)

Thus the quantized field q(x) of the virton is equal to zero. The Green functions in the limit

+ y» V

=

,

(6.14)

p2^—pE2.

fi->iî>E,

Later on it will not be important for us to know the explicit form of regularized expressions. I t is essential only that the regularization, first, does exist, second, provides the transition to the Euclidean metrics and, third can be removed. 6. Interaction of Virtons with Hadrons There exist two different possibilities to consider the interactions of our virtons with fields of stable particles (mesons, baryons). One way is the following. We do not need any gluon fields to "glue" virtons in our approach, therefore we can introduce the Lagrangian of virton field of the type I(x)

= q(x) Z(P) q(x) + A(q(x) r 3 (x))»

(6.1)

and look for bound states of systems of virtons. Along this way usual stable particles should be bound states of virtons. This idea deserves an independent research and is not simple because the problem of finding the bound states in quantum field theory is not solved yet. Another way consists in that the usual particles are considered as elementary particles and are described by standard quantized fields satisfying a Klein-Gordon or Dirac equation. However, fields (for example, a meson field n(x) or a baryon field B(x)) cannot interact with each other directly but through an intermediate virton field q(x). For example, the interaction Lagrangian can be as follows h(x)

= gn(x) (q(x) y5q(x)) + f\{B(x) y^x))

(q°(x) Yliq{x)) + h.c.].

(6.2)

J u s t this second possibility will be considered further. The ^-matrix for the interaction Lagrangian (6.2) can be constructed by methods of nonlocal quantum field theory. Instead of 'f j(x) in (6.2) let us introduce the regularized interaction Lagrangian ¥i s (x) which depends on the regularized field %iS(x) = grtx) (q>y5q>) + /[(B(z) y ^ ) (q^q*)

+ h.c.].

(6.3)

The regularized ^-matrix is defined in the usual way Ss = T exp {i f dxl/

(a;)}.

(6.4)

I n [3] it has been shown t h a t in the limit d 0 there exists a finite unitary causal (S-matrix in each perturbation order S = lim S" (6.5) 0

if the causal Green function is an entire analytical function and decreases in the Euclidean region. All details can be found in [3]. Thus we can consider any interactions of mesons and baryons through an intermediate virton field q(x). The essential point is that these virtons cannot be created because the field of free virtons is equal to zero.

412

Z. D t j b o t C k o v a ,

A.

G. V.

E f i m o v ,

M.

A.

I v a s t o v

7. Physical Meaning of Parameters I and L

Let us consider the physical meaning of the constants I and L in (3.8). For this aim we examine the potential corresponding to the causal Green function Gc(x) (5.11) as usual when the Yukawa potential is deduced. We introduce an interaction between the field q(x) and two fermion sources ip-, (x) and yi2(x) of the type XM

= g[fi(x) q{x) +

?(«) + h.c.]

(7.1)

and calculate the energy of interaction between them due to the exchange of the quanta q{x): W = g2 j j dxJ dx2 [ipi(xi) Oc{xx — x2) y>2{%2) + h.c.]. (7.2) Let us suppose that these sources are point-like, i.e., Vj(x)

= uSi3)(x - r,),

(# = 1,2),

(7.3)

where u is the Dirac spinor describing the spin of our source at rest so that uu = 1 and uy^u = 0. Then we obtain W(r) = g2uGc(r) u — const ( l - -Lj exp [ - ( r + lfjL*\ + ( l + i j exp [ - ( r -

If/L*] (7.4)

where r =

— r2\ and

Gc(r) = J

*« = i

dpeiprGc(p),

j , sin l i p * cos I }'p2 — /p ,2

(7.5)

The potential W(r) decreases with r -> oo as exp {—r2/_L2) in contrast to the Yukawa potential 1 \r exp {—mr). Let us define the average value (r2) of the distribution described by the function W{r), then one can obtain f drr2 W(r) 3 r2 (7"6) < > = Jr dr J W(r)\ = & T (2P + L2)' This means that the Green function Gc(x) considerably decreases at distances of order ~ ]/2Z2 -f- L2. Let us compare the behaviour of this Green function with that of the Green function of usual particles. We can see that the latter noticeably decreases at distances of an order of the particle Compton wave length A = 1/m (the Yukawa potential). Now we may suppose that the "mass" of our virton described by eq. (3.2) with operator (3.8) is defined by the expression

I t should be noted that is is not real mass of the virton because our virtons do not exist as real particles.

Nonlocal Quark Model and Meson Decays

413

8. Meson Mass Correction Here we want to give an example of calculation methods in our model in the case of the meson mass correction appearing due to the interaction (2.1) between mesons and virions. According to (6.3) the regularized Lagrangian can be written in the form = ihn(x) (?«(*) y,qs{x)).

I/(x)

(8.1)

The ^-matrix element corresponding to the diagram Fig. 1 a is as follows S2 = - j j j d x dyn(x) 2y (x - y) n{y),

(8.2)

where Z%\x - y ) = -i(ih)* = -ih* Sp {ys G*{x - y) y5 G\y - x)}.

—

0

(8.3)

—

Fig. 1. The diagrams contributing to the mass operator in the second (a) and fourth (b) perturbation order

The Fourier transformation of the regularized mass operator is defined by the following way £2s{p2) = / dxe^x S2\x). (8.4) The representation (8.4) can be understood as a distribution defined on Z2. Let us go to the Euclidean metrics and, further, put 0.5,

A(x)

q\x) Z»[p + ej^x))

q>(x).

(10.2)

Now let us introduce the system of fields

and Am + dj,

q6

gW

the fields qf are transformed in the same way q/{x) -> qf(x) eie°f = ( - 1 ) ' â/fS^x = (-1)'

iff

y) rA(y -

1

-

x') f

(2n)*iJ

f

dpp-'T'1*-1'' Mj(d) — P — ie'

Tig. 3. The vacuum polarization in the virton electrodynamics

The vacuum polarization contains the ultraviolet divergences. In order to remove them, we will use the gauge invariant Pauli-Villars regularization procedure with additional conditions (see [3]). Then we obtain nl(p)

ns(p2)

= {g^-p*

= f dxe-tp* nl(x)

=

r 4M,\ò)

E

This series converges well because for 6

-p^pv)II*(p2),

du j/l — it (l + J ' V

u — le

0 M}(d) ~ fjd • 1/L (a > 1) and we get (when (10.7)

i r

The function IIs(p2) tends to zero as

V—*py(c)

we obtain M{a)

= i f dqE S p {d^kE)

M(b) = i f dqE S p {d^(k2E) Mic)

d^k^)

= i f dqE S p {[d^(kE) G(iqE)]

rE\,

G(iqE)

G(iqE)

rEG{iqE

(11.6)

rE\,

+ ipE)

rE\.

Now the problem is to calculate the functions d^JJCn) ••• d^ky) G(p) because the direct use of the definition (11.2) is quite difficult. We proceed in the following way. The propagator G(fi) is the entire analytical function (5.13). We can represent it in the form (11.7)

where the contour C envelops the positive real axis as shown in Fig. 7 because we integrate in the Euclidian region where G{i$E)

1

= jf

f

e x p UlqE

L^

— — q

1

(11.8)

A

and

Making use of due representation (11.7) and formulas (11.3) we obtain d^{knE)

••• d^k

1E)

G{i(iE)

= ^ r JdCG(i£)

d^k^)

—

d^(k1B)

1

(11.9)

Nonlocal Quark Model and Meson Decays

421

This representation is very convenient for different calculations. For example, simple calculations produce easily: =

1

^

„

1 C

d t G j i Q

2

n i ]

£

+

-

X {