Excited states of bosons and fermions in a four-Fermi quantum field theory

ANNALS

OF PHYSICS

149, 296-334

(1983)

Excited States of Bosons and Fermions in a Four-Fermi Quantum Field Theory* P. FURLAN International delfliniversita’

School for Advanced Studies (SISSA di Trieste, Italy, and Istituto Nazionale

), Trieste, Italy, Istituto di Fisica Teorica di Fisica Nucleare, Serione di Trieste, Itab!

AND

R. R~CZKA+ Inernational

School

for

Advanced

Studies

(SISSA), Warsaw,

Received

Trieste, Poland

January

Italy

and Institute

of Nuclear

Research.

12, 1983

A scalar-pseudoscalar four-Fermi quantum field model in four dimensional space-time is considered. Introducing the scalar and pseudoscalar collective bosons, the path-integral representation for the generating functional for Green’s functions in terms of the effective total action integral containing only collective bosons is expressed. Using a classical ground-state solution for collective bosons, a new formula for the generating functional for collective boson and fermion Green functions in terms of the effective propagators is derived. It is shown by a partly nonperturbative analysis that the excited states of collective bosons do exist and form finite trajectories in the plane mass-square-spin. These trajectories for bosom are approximately linear in J. as the experimental trajectories. The existence of fermion bound or excited states depend on the value of the dynamical parameters of the model. For snme values of dynamical parameters there are bound states for J= 4 and i. However, for most of other values bound or excited fermion states do not exist.

1. INTR~OUCTI~N One of the most interesting features of elementary particle physics is the existence of trajectories of excited mesons and baryons [ 11. These trajectories, in the plane mass-square-spin, are approximately linear for mesons as well as for baryons [ I]. This striking experimental fact must be explained in any successful model of a quantum field theory. In our previous works we presented an idea that the existence of boson and fermion trajectories is typical for boson-fermion field theories in which, in the pure boson sector, the nonlinear classical field equations admit a so-called ground-state solution 12, 31: the existence of this solution implies that the one-particle boson and fermion propagators possess isolated singularities in the plane mass* Partially supported + Permanent address:

by INFN Sezione di Trieste. Institute for Nuclear Research,

296 0003m4916183 Copyright All rights

$7.50

62 1983 by Academic Press. Inc. of reproduction in any form reserved.

00-68

1 Warsaw,

Hoia

69. Poland.

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square-spin: it turns out that these singularities form finite, approximately linear trajectories of excited states of the original bosons and fermions 131. In this work we carry out this programme for the pure spinor quantum field theory with the four-Fermi scalar-pseudoscalar interactions. We are convinced that the nonlinear spinor field theory provides a framework for a conceptually simpler formulation of elementary particle physics for fermions as well as for bosons. The recent progress in understanding of renormalizability properties of four-Fermi interactions (41 encourages us to think that a pure-fermion field theory is an interesting candidate for a model describing the physics of strong interactions 151. For a scalar-pseudoscalar model with four-Fermi interactions, we introduce as a basic tool for our analysis the concept of collective scalar and pseudoscalar bosons and we express the generating functional for time-ordered Green’s functions for collective bosons and original fermions as the Feynman-type integral over collective boson fields. In Section 2 we present the four-Fermi scalar-pseudoscalar Lagrangian describing our model; then, inserting composite collective boson fields, and integrating over Fermi fields, we obtain the generating functional for fermion and composite boson Green’s functions. In Section 3 we investigate the properties of Green’s functions for collective bosons. We show that the effective total action integral has a ground-state solution which minimalizes the total energy functional for collective bosons. Using this ground-state solution as an expansion point for the total action we derive a new exact representation for the generating functional for Green’s functions of collective bosons. In this representation the ordinary bare propagators for collective bosons are replaced by new effective propagators which contain some nonperturbative pieces of information on the considered dynamics. We analyse in detail the mass-square-singularities of the one-particle collective boson propagator in Section 4. First applying the h-expansion to the exact propagators we show that up to O(h’) they are proportional to the effective propagators. Next, using the spectral representation for the effective propagators we determine their mass-square-spin singularities. We show that the trajectories of excited states for bosons are finite and approximately linear and that the excited states with higher spins are unstable: all these properties are in a qualitative agreement with experimental data for boson resonances. We derive in Section 5.1 a general formula for 2m-point Green’s functions for the original fermion field. We show that these Green’s functions can be represented as a series of collective boson Green’s functions with known coefficients. In Section 5.2 we derive an approximate expression for the one-particle fermion propagator in the form of h-expansion. We show that up to O(fi312) this propagator is proportional to the effective fermion propagator: using this fact we show that the singularities of oneparticle propagator in the plane mass-square-spin depend on the values of the dynamical parameters of the model: for some values of these parameters we have one bound state for J = $ and $, respectively: however, for most values of dynamical parameters we have neither bound nor excited fermion states.

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FURLAN AND R&ZKA

In the conventional quantum field theory the lack of a mass-square singularity in one-particle propagator is interpreted as the evidence for the nonexistence of the corresponding asymptotic particle states. Consequently our analysis provides an illustration of a possible confinement mechanism for fermions in pure fermion models of field theory.

2. COLLECTIVE

BOSONS IN FOUR-FERMI

THEORY

We shall derive in this section the form of the generating functional for timeordered bosons and fermions Green’s functions 5(x, ,....x,) using the technique of collective boson field. In this formulation bosons appear in pure fermion field theories as composite objects constructed from the original fundamental fermion fields. We are convinced that this method of analysis of boson-fermion field theories by starting from a pure fermion fundamental field theory is conceptually simplest and deserves a careful study [5]. Let us consider the model described by the following total Lagrangian density for Fermi fields:

where fiO has dimension (mass), while 6 and G have dimension (mass) P2. The renormalizability of four-Fermion interaction (2.1) was proven by Eguchi using the conventional perturbation theory and by Tamvakis, Guralnik, and others using the mean-field expansion [4]. It was shown there that although the bare coupling constants are cutoff dependent the resulting renormalized coupling constants are cutoff independent. The generating functional for the fermion Green’s functions has the form J(% ii) = Z-‘j

exp i j d4x(Y(y)

where q and q are anti-commuting given by

+ VW+ W}(x) Diy DI,?,

c-number sources, and Z is a normalisation z = J(0, 0).

(2.2)

factor, (2.3)

The following Gaussian integral over the bosonic fields oc and u,

2

-(f

G

i

uc-igh@sy

(xl Dv,, Da, 3

(2.4)

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where a is a constant to be determined later, g, and $,, are dimensionless constants and cl0 and &, have dimension of mass, is a constant. Inserting (2.4) into (2.2) we obtain J(v, fl) = 2-l

.I’ exp i \ d4x{Y’(y,

pc, a,.) + riy + y/v}(x)

Dyi D@ Dq,. DO,.,

(2.5)

where

(2.6)

once we relate G and G with 2,. g,, j&, pu, as follows:

The Euler-Lagrangian equation for the Lagrangian density (2.6) give us the composite, collective boson fields 9,. and uC, in terms of the fundamental fermion fields w, i.e., (2.81

We can perform now the integration J(q, f) = 2-l

over Fermi fields in Eq. (2.5) and we obtain

1 exp iJ‘ d4x I-nG,(p,,

- i Tr In[K,(q,,

oc)q

a,)] - $ (qC - a)’ - $0:

1 (s) Dq, Da,,

(2.9)

where for a general scalar field (o and pseudoscalar field u we define

G,(x, Y; v, a) = [K,((o, a) + k] --’ (x, y), KJfi4 0) = (ia - MO + i, v + ige ys a),

(2.10)

with the new, effective mass defined by

MO= A, + g,a.

(2.11)

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AND

RljCZKA

By absorbing a constant factor into the redefinition we can rewrite the generating functional (2.9) as

+iTr

f -!-[(ia-MJ’ tl=l n

of the normalization

constant,

(x)Dq~,Da,.

(2.12)

(-&~c-igoy5ac)]”

Let us denote by Y(n)(~C, o,) the nth contribution given by the series appearing in (2.12), i.e.,

I

to the effective action integral

.i”“‘(O,,o,)=~!‘d’xTr[(iB-M,)-’

(-&PC-ig,y,a,)j”

X . . . G,

Tx1 N-t?, vc - ig, y5u,>(x, I1

(2.13)

where .W, Y) = G,(x, ~)/~~=~~=o = x(x -Y>.

(2.14)

It is easy to see, by power counting, that divergent terms appear in each .Y ~~),.o,.j when n < 5; for n > 5, instead, all .YCn)(p,, u,) are finite. Let us denote, then, by ,Y’~‘(u)~, a,) the divergent contribution to each .Y““‘(cP,, u,.). We get

.Y’g)(&,a,>=-J‘d4xaZ,{[g~cpf+g~u~]}(x), where I, and I,, are quadratically I2 = 4i

and logarithmically -d4p (2n)4p2

J

I, = -4i

. d4p 1 (2n)4

(2.15d)

divergent integrals, respectively,

1 -M;’

(2.16a)

1 (p’ -M;)’

’

(2.16b)

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We obtain then the following only:

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effective action integral for the collective bosons

The term S?(9,, a,)(x) contains both the finite contributions to .P’~~~,.o,,-CU’i”,‘,c.D<,) and the contribution of all YCn) (e,,o,) with n 2 5. We fix now the undetermined constant a in such a way that no term linear in the boson fields appears in the effective action (2.17), i.e., ^

a = iETr{Z(O)). Remark.

(2.18)

From the definition

:W,(x) w&):= V,(x)w+>+ iL’,,(O)

(2.19)

we see that choice (2.18) provides a new definition of the collective boson fields (2.8) ^ G oc= i- :IJiy,ty:. 9,=4+. go

From Eq. (2.11) we see that Eq. (2.18) implies the following relation between the bare fermion mass fro and the effective fermion mass MO: Mo=~0+idTr{~(0)J=~o+&VoZ2.

(2.20)

Let us introduce now the resealed boson fields 90,= z, ‘129,,

ur = z, “*u c-7

(2.21)

= (4 I,) - ‘12,

(2.22)

and the common resealed coupling constant g, zz z’/*o go = zyg,

where z,’

= f&z,,

Z,‘=fg;z,.

(2.23)

Let us stress that resealed fields and coupling constant are different from fully renormalized fields and coupling constants.

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FURLAN AND R.+XKA

Since (2.24) the effective bosonic action integral (2.17) becomes

(2.25)

+ 2Mo .wJ,ccPf+ 4, - f ‘g(pf t a:)2 + &&, u,))(x), where we have introduced ^ ,u;,= (G-’ -Z,)gf

t 6M;I

(2.26a)

,u;,= (G-’ - Z,)gf t 2M;

(2.26b)

and

and 2(o,, ur), becauseof Eq. (2.24), has the same functional dependenceon (D,and CT,as 2(cp,, or) had on cp, and uC. The generating functional becomesthen

J(Qj)=z-’

je-’

~iiG~(o,.o,)~~iS(rp,,rr,)

Dq,

Da,,

(2.27)

where G,((o,, u,) is given by Eq. (2.10). It was shown in [4i] using the conventional renormalization theory and independently in [4ii] using the technique of effective propagators that the original fourFermi field theory (2.1) is equivalent to a renormalized Yukawa theory. Therefore there exists a bare Yukawa Lagrangian 9”(p,, u, v/) which provides the renormalized Yukawa field theory equivalent to the original four-Fermi field theory (2.1). Indeed, using the Eguchi analysis [4i] we obtain: PROPOSITION.

The generating functional

where Z, = J’(O, 0) and 2’;’ is a general Yukawa Lagrangian %;‘(a, u. w) = rj?(+Z- MO) w t g’ :Wv: rpf ig” :y?y5ty: u t +(a,#

+ #,a)’

-f&p)’

-f&J2

t K2’pU2- f(l,fp4 + A2qJ2u2+ L,uJ),

+ K,$ (2.29)

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after integration over fermion fields and appropriate resealing, has the same structure as the generating functional (2.27) if g’ =g”

4

xg,

=+a,

(2.30)

=g2.

The proof of this proposition is given in the Appendix. As follows from this proof the dimensional Fermi coupling constants G and d and the bare fermion mass a0 in (2.1) are traded for the masses pi and ,u: of collective bosons and the effective fermion mass M,. It is therefore not surprising that the dimensionless Yukawa coupling constant g is not fixed by the equivalence condition. It would seem therefore that the obtained Yukawa theory possesses more free dynamical parameters than the original theory. However, the analysis carried out in Sections 4 and ‘5 indicates that, properly resealing the boson field we eliminate the coupling constant g from the dynamical equation for one-particle propagators for Bose as well as Fermi particles. Thus, the obtained dynamical properties of composite bosons and original fermions such as mass-square-spin trajectories of excited states do not depend on the value of the Yukawa coupling constant g. Since the theory defined by the generating functional (2.27) is equivalent to the corresponding Yukawa theory, the S-matrix elements and the Green functions in both theories will coincide in the noncutoff limit 141. Consequently we shall construct the generating functional J(?j, v. f,,f,) for collective boson and fermion Green functions using (2.29) and (2.30). Denoting by y’(o, 0, v) the Yukawa Lagrangian (2.29) with coupling constants given by (2.30) we obtain

where Z = J(0, 0, 0,O) and

Scaling down the fermion fields v/’ = h-“‘~l/. over fermion fields I#, W’ we obtain

4’ = &‘/‘IJ

and integrating

in (2.31)

(2.32)

304

FURLAN

AND

RJCZKA

where

GP, 0)=cd4xP(p, u)(x) and

P(q9,u)= ~(a,fp)2 + ~
3. COLLECTIVE

(2.33)

of the A-expansion

BOSON GREEN FUNCTIONS

3.1. Construction of Generating Functional We shall now derive a new exact representation for Green functions for collective bosons. The generating functional J(f, ,f,) for collective boson Green functions is obtained from (2.32) by setting f = q = 0

x

e(i/l)Sd4x[-(ro/2)(0+r~)~-(u/2)(O+u~)ot

Y&(P.o)I(x)

(3.1.1)

@D,,,

where

%P>0)= 2M,gp((02 + a’) - f gyyl*+ u2)2.

(3.1.2)

The time-ordered Green’s function r (k41)for collective bosons are defined by the formula $k.l) (x

I,..., xk+,)= z-’

I1*I, (D(Xi>jj

+,+j)det[&,(@

U)]

e(i'h)S'oD.o)

Dy,

Do. (3.1.3)

The rigorous analysis of path-integrals is carried out usually in the framework of the theory of Fresnel integrals [7]. In this work we shall use recent far-reaching results on a classical nonlinear field theory for the derivation of a new representation for the generating functional J(f, ,f,). Equation (3.1.1) can be written in the form J(f, ,f,) = Z - ’ ( F((o, a& ,f,) e(i’h)b(09rr) Dq Da,

(3.1.1’)

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which is often considered in stationary-phase-approximation analysis, where F(rp, a; f,,f,) is treated as a slowly varying function of (D and o. The analysis carried out in [ 7ii] indicates that indeed the determinant det [K,(rp, a)] in cases in which it could be explicitly calculated, is a polynomial in the fields, i.e., slowly varying function. by expanding S(o, u) around a Consequently, we shall calculate .I(‘&&) distinguished stationary point (ground-state solution) (o,,, u,,), solution of the classical field equations (3.1.4) implied by the interaction Lagrangian (3.1.2). The detailed properties of these solutions will be specified in Section 3.2. Now we shall suppose that they do exist and we shall use them for the derivation of a new formula for J(f,,f,). Expanding S(p, (I) around (o,,, uO), using the fact that S,, = 0, S,, = 0, and setting Zt”‘o’ = (o- (pOand Zt”*u’ = u - u. as the new integration variables, we obtain J(f,,f,)=

Z-’

t p,,, h”‘u

j F(h”*p

+ u,:f,,f,)

where

= w;y(Do, and P”:k.“((o,,

(3.1.7)

a,) $!I0 $

a,) is defined by the formula

hdx)

&4z2)

*‘*

&tzk)

6u(zk+

1) “’

d”.(Zk+l)

W,,.uo

ktl =

Lz!qk“‘(po)

a,)

n i=2

cY4’(x

-

Zi).

One could consider formula (3.1.5) as a new representation for the generating functional with a new effective interaction Lagrangian (3.1.7) and new effective propagators G,(x,y)

= [Km - i&J-’

(x,y),

G,(x, y) = IK,

- ie] -I (x, y). ’

(3.1.8)

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This would be allowed if the propagators (3.1.8) would be well defined: however, it turns out that&-due to the existence of the so-called zero translation modes (ZTM)-the propagotors (3.1.8) are not well defined. In order to see this in the simplest case and anticipating results of Section 3.2, let us supposethat p,, is a timeindependent radial solution which is square-integrable together with its derivatives ako,, and u,, = 0. Then the vectors

satisfy the equation

consequently the inverse (3.1.8) of K, does not exist. In order to define the propagators (3.1.8.) properly one has to remove the ZTM from the domain D of the path integral (3.15). The technique for the removal of ZTM was described in detail in IS]. It consists of the introduction into the path integral (3.1.5) of the following identity (3.1.10) Here Z7’ is the set of one-parameter subgroups of the Poincare group 17 which do not enter into the stability subgroup L7, of the solution (qO, u,, = 0), rf’ is the subset of one-parameter subgroups in U1 whose generators provide ZTM, d’= dim Z?‘. a = (a, ,..., a&), a,, a = I,...) d’, are the parameters labelling the one-parameter subgroupsin fi’

where U=, is a representation of a one-parameter subgroup of EL, and da = nf’ 1 da,. u:(z), a = l,..., d’, are ZTM produced by the generators 2” of 2’, i.e., U: = Z’(o,. The Jacobian ,P(o) has the form (3.1.11)

-P(q) = det IIMT;; II, with llw;:lI( = jJ”f u:(z)(Zbd(z)

d”z

/I

,

a, b = l,..., d-.

‘Here Zb are generators of one-parameter subgroups in fiL. The meaning of identity (3.1.10) becomes intuitively more clear if we set F”(a) = J’ u~(z)(U,p)(z) dz4; then (3.1.10) reduces to well-known identity

_d1 I a-,r1 dPS(P)

= 1.

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Inserting (3.1.10) into (3.1.1), carrying out the change of variables (D’ = U, (D, expanding S(p, o) around (qO, 6, = 0) and using new variables h”‘v)” = cp’ - q,, and /r’i2~rr = IS’3 we now obtain

zu;(z)(h”‘(p

+ cpJ(z)

I

.P(h”2~

+ cpo) D9 Do.

We shall now eliminate the ZTM from the domain D of this functional integral. Let U, be eigenfunctions of K,, i.e., (3.1.12)

K,u, = qnun.

It will follow from Eq. (3.1.6) and from the properties of the groundstate solution q,, that K, is self-adjoint in L2(M3”). Consequently the set {u,,} is complete: hence we may write the field cpin the form dl q?=

z Cl=1

P,u;+y’c,u,, ,I

where the second sum runs over the eigenvectors which are not ZTM. Now we may write D9, using this expansion, in the form DP =j fil

de, !,” dc,, .

According to the fact that a = l,..., d’,

6 J’d” z@z> 9(z) = @,), I I

and that for radial solutions i d4zz&(z) coo(z)= 0, we may eliminate ZTM from the domain D of path integral and we obtain .‘(f,J,>=

Z-’

i,_ da/

W+‘*9 + 9,,), (h”2a); D’ x eirl - $‘pK,p i OK Do+h-‘~~rfol~/~m.h~f~~)l(.~)d’x

UJ,,

all independent

U&l

2-@I”*9

+ 9”) D’9 Da,

(3.1.13)

D@‘,

(3.1.14)

with D’9 = n; dc,. Applying twice the identity [e-+b”&$‘](@)

=

z;-’

( F(@

+

@t)

@“-

$@‘K@‘lWd4x

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for the function

we finally obtain from (3.1.13)

Here 1 G;(x, y) = 2 n a-i&

%(X) 4%‘)~

where C’ runs over all eigenstates U, of K, except ZTM. One may consider formula (3.1.15) for J(f, ,f,) as a new quantization formula for a boson field theory possessing the ground-state (or soliton) classical solution (qO, u, = 0). Formula (3.1.15) is numerically equal to the original formula (3.1.1): consequently the global expressions for rtk*” Green’s functions are Poincare covariant. Hence, the group averaging over ii’ is not carried out for a recovery of the Poincare symmetry broken by the chosen expansion point (qO, ui, = 0), but for the considering of all ground-state solutions with the same value of the total action S(yl, u) and the same minimal energy P,(cp, u). One may utilize the final formula (3.1.15) for a new effective perturbation theory or a new &expansion. The perturbation theory will be described in terms of generalized Feynman diagrams in which lines will be represented by the effective propagators GG and G, and the vertices will be determined by monomials from Y’jff(h”*~, h”*u) and .P(h”*p + pO). Since the propagators CL and G, already contain via (cJ+,, u, = 0) some nonperturbative pieces of information, one might expect that presently even low order effective perturbation expansion might provide some nontrivial, partly nonperturbative description of the physical properties of the considered interaction. However, the experience with quantum mechanics ]9] and some models in quantum field theory [lo] teaches us that the most effective approximation for strongly interacting systems is provided by h-expansion. Therefore, in the further analysis we shall utilize formula (3.1.15) for calculating the finite A-expansion for various quantities of physical interest. In particular we derive in Section 4 the finite h-expansion for one-particle propagator r(*‘(x, y) for o and u bosons.

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3.2. Ground State Solution for a, and o Fields We have derived formula (3.1.15) for the generating functional .I(&&) supposing that there exists the distinguished ground-state solution (rp,,, o,) for the field equations determined by the interaction Lagrangian (3.1.2). We show now that for the considered interactions the ground-state solution indeed does exist. Strictly speaking the final formula (3,1.15) is valid for any expansion point (oO, u,,): only if we approximate it by a finite number of terms provided by Aexpansion, the expansion point must be selected carefully in order that the considered finite number of terms are significant for the exact formula. To better understand this problem, let us suppose that we work in the Euclidean formulation: in this case the generating functional (3.1.1) takes the form

where

and

(3.2.2.) Bearing in mind Lapiace’s method for evaluating the integral (3.2.1) it is clear that we should take the expansion point (oO, a,) which absolutely minimalizes the total action integral S,((o, a). Clearly this expansion point must necessarily satisfy the Euclidean field equations .:

-A,o,=~.

-A, p. = 2, 0

(3.2.3) 0

We now give the precise definition of a ground-state solution [ 111 and we show that it does exist for interaction (3.1.2). In order to see the general pattern we formulate the formalism for a multiplet cpE RN in IF?’ space. Definition

3.2.1.

A solution ‘p. to the field equations k = l,..., N

in R” is said to be a ground-state

(3.2.4)

one if

(1) (p. is radial, i.e., cpo(x)=cpo@), p= (cz=r x:)“~ and &,k E C’(R”); N = 1, then rpo@) is positive and decreasing: (p&+) > oo@2) for p, < pz;

if

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FURLAN AND R.jCZKA

(2) if cp(x) is any solution of (3.2.4) in L,Tc(R”) with p,(q)(x) E L’(R”), then 0 < ~,(cp,) ,< I,; (3) 1D4e& (x) < ceWsp, k = l,..., N, for every x E R”, IpI < 2, and some c, 6 > 0. Notice that (1) is usually satisfied for a ground-state solution in quantum mechanics: in turn, (2) says that a ground-state solution absolutely minimalizes the total Euclidean action. Finally condition (3) says that a ground state ‘p,, is localized together with its derivatives in the spatial as well as in the temporal directions. Theorem 3.2.1 provides the conditions for the interaction Lagrangian i/7;(p) = --@i/2) (p2+ Yi(cp) and the currentj,(cp) = iiyi(cp)/@, sufficient for the existence of a ground-state solution (p,, [ 111. THEOREM

3.2.1. Let n > 2 and Iet $,(cp) and j,(q)

be continuous f%nctionsfrom

R” into IF?satisfying (1) j,(O) = 0, (2)

lim,,,,,j,(cp)/l(pl’

(3) j,(q)

(4)

,< 0 with I = (n + 2)/(n - 2). 1 < k < N.

is dgferentiable

at cp= 0 and

i?;(q) > Ofor some cp in R”,

then there exists a ground-state solution cpO@)of Eq. (3.2.4) in the space Fi” One readily verifies that the interaction Lagrangian %$(rp,a) given by Eqs. (3.2.2) and (3.1.2) satisfies all conditions of Theorem 3.2.1 in IR’. Consequently there exists a ground-state solution ((D,,,o,) which absolutely minimalizes the total action integral S,(p, a) in the space of solutions of Eq. (3.2.4). Thus the solution (oO, a,,) could be taken as the expansion point for the Euclidean action S,.(q, u) of the quantum theory. The Euclidean formulation is now commonly used due to its simplicity and to the possibility of analytic continuation of global expressions for the Euclidean (Schwinger) Green’s functions to the Minkowski space. However, the physical aspects of a given quantum field model is more transparent when considered directly in Minkowski space-time. In addition although the global Schwinger functions may be analytically continued to global r-functions, it is not clear if a finite approximation given by h or a perturbative expansion also analytically continues and represents a main contribution to relativistic Green’s functions. Consequently we shall use from the beginning the relativistic formulation. However, in the relativistic framework the situation is much more complex. Indeed in this case the path integral presents a case of oscillatory integral. The stationaryphase approximation method developed for such integrals teaches us that in this case

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the essential contribution to J(f,,f,) is provided by all extremal points of the total action integral S(9, a), i.e., by minima, maxima, and saddle points (see [ 1,2] Chap. III] for a rigorous treatement of oscillatory integrals). This intuitively follows from the fact that in the neighbourhood of all these points the oscillations of exp[ihP’S(9, a)] are small and for a slowly variable integrand the contribution to the integral is significant. Consequently one should request first that the expansion point (cpO,u,,) should, be stationary for the total action integral S(9, a). i.e.,

SJ90,a,>= 03 SJ90.a,)=0. This leads to the field equations for (9,‘, uO) (3.2.5) Clearly there is an infinite number of distinct solutions (90, a,) of (3.2.5), which can be enumerated by distinct pairs of initial conditions 90@03xh ~O,(fO~ xl = Pr9oN0,

x)3

uo@o. XL ~o&oX) = (~,~OWO~ x)

at some time to. Therefore one needs some additional condition which may select out of the set of all solutions of Eq. (3.2.5) a distinguished one. This condition is provided by the energy stability condition: we request that the expansion point (opt, a,) should be such a solution of (3.2.5) for which the energy functional

Pof9d=j i1b: + P912+ 7~; +

(Vu)‘]

- ?;(9. u)}(L x) dx,

(3.2.6)

reaches the absolute minimum which is nonnegative. The physical meaning of the energy stability condition is better understood if we emphasize its connection with the problem of small vibrations around the point (90, a,). If (vlo, uo) is any solution of (3.25) then the small vibrations 69(r) = u(x) and 6u(x) = u(x) around (90, uo) satisfy the equations of motion K, u(x) = 0,

K, u(x) = 0.

(3.2.7)

We see therefore that the operators K, and Kc--which by (3.1.8) define the effective propagators G, and G,- coincide with those which determine the small vibrations of fields 9 and u around a solution (90, uo). On the other hand, from a physical point of view the small vibrations should be considered around a minimum of energy and if possible around an absolute minimum. This justifies the introduction of energy stability condition as the additional criterion for a selection of the expansion point ((po, a,) for the total action integral S(9, a). We show in Section 4 that the singularities in the mass-square of one-particle propagators r(‘*O’ and r(“*2’ for 9 and u field, respectively, are determined by the

312

FURLAN

AND

RACZKA

eigenvalues of energy for small vibrations: this gives a posteriori justification for the energy stability condition. We now derive the consequences of energy stability condition. First note that the energy functional P,((o, a)(t), as a constant of the motion, is time-independent and for any fixed time t is a functional of four conjugate variables o(t, x), ~,(t, x), o(t, x), and rr,(t, x). These variables can be considered as the initial conditions at t for a solution (p,, a,) of Eq. (3.2.5). A so 1ut’ion (qo(t, x), a,(t, x)) will be energetically stable if for any variation of initial conditions p(t, x), n,(t, x), a(t, x), and n,(t, x) at any fixed time t the energy functional P,(q,, u)(t) for (v)~, a,) is minimal: this implies for every t

@o(t) = 0, wt, x) “?o.*o

@o(t) =o, %o(t~xl 0”,00 @o(t) = 0. %K xl ‘po.~o

~@o(t) 6u(t,x)

=

0,

fJO.UlI

(3.2.8)

Equation (3.2.6) and (3.2.8) imply that for all t we have

~,,(L x) = 0,

T&

x>= 0,

-~,p,=c3 %o ’

-A u 32 3 O iiu,’

(3.2.9)

This implies the important conclusion that only static solutions can minimalize in the space of solutions of (3.2.5) the energy functional. We have obtained therefore that the energy stability condition implies that (oo, a,) must be static solutions of Eq. (3.2.9) which absolutely minimalize the energy functional in the space of solutions. We show now that this is indeed possible for the interaction Lagrangian (3.1.2). In fact we observe that it is sufficient to check for the Lagrangian P1(rp, a) the assumptions of the Theorem 3.2.1 in space R 3. In the present case n=3, 1=5, N=2 and j, =j,

= - p;rp + 2M, g(3$

+ d) - 2g2((p2 + u2) cp,

j, = j, = --,~~a + 4Mo gfpu - 2g2(p2 + a’) u. Consequently we see that assumption u = 0 and the condition

(l)--(3)

/I:, < 4M;

(3.2.10)

of Theorem 3.2.1 are satisfied. Setting (3.2.11)

we see that J&V, a) > 0 for some cp. Thus condition (4) is also satisfied. Consequently there exists a ground-state solution (qo, a,) which absolutely minimalizes functional (3.2.1). This functional, however, for static solutions in R3 coincides with the energy functional P,(rp,, a,) given by (3.2.6). Accordingly, the assertion of Theorem 3.2.1 implies that for the considered model there exists a ground-state radial solution ((am, uo(r)) which absolutely minimalizes in the space of solutions the energy functional.

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STATES

IN

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QUANTUM

Let us remark that condition (3.2.11) is just the bound-state mass condition for a fermion-antifermion composite system. It will be evident from the further analysis that the properties of ground-state solution (rpO,a,) depend crucially on the extrema of the interaction Lagrangian $,(o, a) in the plane (9, a). In order to determine these extrema we pass to the radial field variables R(x) and a(x) defined by the formula q(x) = R(x) sin a(x), In these variables the interaction

u(x) = R(x) cos a(x).

Lagrangian J&(p, u) takes the form 2

L$R, a) = - $

(Jo’~sin’a+p~cos’a)+2M,gR”sina-$R4.

The interaction Lagrangian as a functional of a has two isolated extrema for a = n/2 and a = 471, and the extremum curve defined by the equation sin a = 2M,gR@2, -pi)-‘.

(3.2.12)

The isolated extrema have the form (a, = oO, u = u0 = 0); consequently they reduce 21(o, a) to the form

This Lagrangian-once condition (3.2.1) is satisfied- has the shape as a function of cp as shown in Fig. 1. Since the ground state q0 must be radial, the field equation (3.2.9) for o,(r) reduces to the following Newton-like equation: (3.2.14) with the resistance force -2r-’ dp,/dr, the potential force -~~,(cpO)/~oO and with r playing the role of time. Since the factor at the first derivative is singular at Y = 0, the function rp,Jr) must satisfy the following boundary conditions: P,(O) = r > 0,

$

(0) = 0.

(3.2.15)

It is now evident from Fig. 1 that, if we begin the motion with a relatively low value of <, then, because of the resistance force, p(r) will oscillate around ‘p2 with a decreasing amplitude (see Fig. 2). If we increase the value of r we shall reach the moment when the initial potential energy q(l) will be so large that ~1reaches the point (o, with zero velocity. This will be just the ground-state solution. Let us note that the isolated extrema a = 7r/2 and a = 3n/2 correspond to the solutions (rpO= R.

314

FURLAN

FIG.

1.

The interaction

AND

RACZKA

Lagrangian

(3.2.2)

for u = 0.

u0 = 0) and (q,, = -R, u,, = 0): thus, since (4” must be positive, it is sufficient to restrict ourselves to one extremum, say a = 7$?. Clearly, it is an almost hopeless task to find the ground-state solution in the analytic form: the rather sophisticated computer analysis carried out for us by Mr. Kraskiewicz, using A. Skorupski and A. Senatorski programme, gives the numerical shape (Fig. 3) of the ground-state solution (for ,uV = 1.4 Gev, M, = 1.2 Gev. We have obtained in that manner the ground-state solution (q,,(r), a,(r) = 0) which absolutely minimalizes the energy functional. Theorem 3.2.1 admits in principle other ground-state solutions which could correspond to the extremum curve (3.2.12). However, the computer analysis of the resulting Lagrangian seems to indicate that there are no other ground-state solutions besides that corresponding to the isolated minima. The ground state (~7~.u0 = 0) implies the potentials (3.2.16) (3.2.17) which detine Green’s function

FIG.

G, and G, given by (3.1.8). Let us note the striking

2.

A solution

of Eq. (3.2.14).

EXCITED

FIG.

3.

STATES

IN FOUR-FERMI

The ground-state

solution

QUANTUM

315

of Eq. (3.2.14).

fact that the potentials V,, V, (as well as the fermion potential Vf, = -gq,, given by (5.2.5)) do not in fact depend on the coupling constant g. Indeed if we set in (3.2.14) o,, = gP ‘QO, then we obtain (3.2.18) Consequently the solution @,, is g-independent. Inserting this solution in (3.2.16) and (3.2.17) we obtain V,=p;-

12M,@,+6@;,

V,=p;-4M,@,+2@&

(3.2.19) (3.2.20)

Similarly (3.2.21)

v,. = -@().

Consequently the only free parameters in this model in the approximation considered below are the bare masses ,uV and ,u~ of collective bosons and the bare fermion mass M,. We see therefore that the cumbersome cutoff-dependent coupling constant g(A) disappeared from the dynamical consideration. The shape of V, and the effective potential Vzff(r; J) = V,(r)

+

J(J+ 1) rz

is depicted in Fig. 4. We see that these potentials are resonance-like, they will imply a finite number of excited states and that the higher excited states will be more unstable than the lower ones: one might expect therefore that these potentials will imply the existence of finite trajectories of resonances similar to those observed experimentally. The potential V,, has similar properties. Let us note that condition (2) of Definition 3.2.1 excludes those solutions of Eq. (3.2.4) which do not provide P&I)(X) integrable.

316

FURLAN AND RrjCZKA

FIG. 4. The shape of the effective potential Vz(r:J)

It may occur that there exist solutions of (3.2.5) which violate this condition and have energy PO(q) lower than the ground-state solution. For instance it is evident from Fig. 1 that (3.2.5) admits for (qO, u,, = 0) three constant solutions q, = 0, oz and, q3, defined by the condition

with

Pld(P2) = 09

P&P*)= a,

PO(%)= --co.

It is clear from the energy stability condition that the solution (qz, 0) cannot be a candidate for the expansion point: similarly it seemsthat the expansion around a solution (ox, 0), which has infinite negative energy, is not physically justified. Finally the expansion around pi, would lead to the conventional perturbation theory and would not reflect the existence of the particle-like structure in the nonlinear classical theory represented by the ground-state solution.

EXCITED

317

STATES IN FOUR-FERMIQUANTUM

It seems therefore that the ground-state solution, which among x-dependent solutions has the absolutely minimal energy, is the best candidate for the expansion point. We note that some authors subtract the infinite constant 2$(cpJ from K(q)(x): so they make PO(cpl) = 0. However, we do not see for a moment in four-dimensional Minkowski space-time convincing arguments for such rather drastic redefinition of the theory. Therefore we shall use as the expansion point the ground-state solutions ((00300 = 0). 4.1. TRAJECTORIES

OF COLLECTIVE

BOSONS EXCITED

STATES

4.1. The Structure of the One-Particle Propagator

We shall now analyse the properties of one-particle propagators for (D and (I fields. In general, according to the Kallen-Lehmann theorem, the spectrum of particles in a given field theory is determined by the discrete singularities in pi of the spectral measure of the one-particle propagator [ 131 (4.1.1)

+*yp, 7Pz) = 6(4)(P, + P2) r’*‘(p:),

where

One usually impose the following (see, e.g., [ 13i]) p$P: I-, 1

normalization

condition

on rC2’(pf)-function

- M:) 5’*‘(P:) = z,, 3

where Mf is the lowest discrete singularity on p: of t”‘(p:). In our case using (3.1.1.5) we obtain the following formula for the one-particle propagator for the (Dfield: r’2,0’(x,y)

=

P[(oo(x>

‘PO(Y)

+

~“‘cp0(x)(dlJ>)

+

~“‘cp,(Y)(lo(x))

+ wP(x) rp(Y))l,

(4.1.4)

where P=

dalJ, s‘71

(4.1.5)

and WP))

P ue-(i/*)ls,c;a,+s,G,s,lp((p) = z- 1zozo x

,~-(AI/z~

+

rpo)

e(i/~)(S;“f(h’/*,,hli2rr)

&t[K,(fi’12p /mzo=O.

+ po, fi”20)] (4.1.6)

318

FURLAN

We now calculate h((p(x) o(y)) in powers of h, we obtain

AND

R4CZKA

up to O(A2): expanding each factor of Eq. (4.1.6)

fi(cp(x) V(Y)> = fiC’G;(x,

Y) + O@*),

(4.1.7)

where C’ is a constant which will be determined from the normalization condition (4.1.3). In our case the effective Green’s function Gi(x, y) is given by the formula G;(x,y)

= z’ (q, - k-’ n

u,(x) u,*(y),

(4.1.8)

where, by (3.1.12), u,, satisfies co+

(4.1.9)

vo)%l=q,%l

with V, given by (3.2.16). We see that the maximal set of commuting which determine the complete set {u,) consists of the following operators a I,

H=-A,+

V,,

J2,

operators

J,.

Therefore Eq. (4.1.9) implies that n = (E, wi,, J2, J3). Since V, is time-independent, we can set 24Jt, x) = e-iEtty&), with u/dx) satisfying

z = {wfir, J*, J3},

the Schrodinger-like

equation

Hv,i(x) = (-A, + V,> w,-(x)= dy,(x). Setting u/,-(x) = 4Jr) d2 dr* Consequently,

(4.1.10)

(4.1.11)

Y;?(Q), we obtain, 2d r dr

the following

J(J$l) r*

+ V,(r)

representation

J

d&)

= wf,$J) d,(r).

(4.1.12)

for GI, holds: (4.1.13)

It is instructive

to integrate this formula over E. We get

We see now explicitly that if we would not remove ZTM the propagator G, would not be well defined. We now determine the operator P given by (4.1.5). In our case the stability

EXCITED

STATES

IN FOUR-FERMI

QUANTUM

319

subgroup fl,, of the solution (rp,, co = 0) is the T’ x SO(3) subgroup, where T’ is the time-translation subgroup: hence the set fll which via Mackey theorem completes Z7, to n (i.e. ZZ= nJ7’) consists of the space-translation subgroup T3 and the set of pure Lorentz transformations. Let Z, and Z,, be the generators of these oneparameter subgroups, then the vectors

and u;ts = z,,rp, = t a,&,

s= 1.2.3

(4.1.14)

satisfy the equation K,u = 0. Consequently they are candidates for ZTM. However, it is evident that u:+’ vectors (s = 1,2, 3) can be decomposed in the basis (u,(t, x) = e -iEt~n(~)}: consequently they do not represent independent ZTM. Thus the set fi’ is smaller than Z7’ and coincides with the subgroup T3 of spatial translations. Hence P=

. daU,. ! i-1

(4.1.15)

Notice that (4.1.15) is just the projection operator on a state with the total momentum P = 0. Indeed the operator (4.1.5) acting on any n-point function Fb , ,..., x,) gives (%(P

, )..., p,) = cF3) ‘l pi F((pl T***rPn)r ( ,T, 1

where F(p I ,..., p,) is the Fourier transform of F. Therefore the operator P in formula (4.1.8) will produce in the momentum representation the spatial momentum conservation law. This illustrates well the mechanism of the reconstruction of the original symmetry in every order of h-expansion by the removal of ZTM. Let us now take the Fourier transform of (4.1.4) in the rest frame p, = (M, O), M # 0. Since oO(x) is time-independent and we take M # 0, the first three terms in (4.1.4) have vanishing Fourier transform. Furthermore. from Eqs. (4.1.7) and (4.1.13) we obtain ~(*~~‘(p,,p2) = A6(M + E,) c?(~)(~Jy’

m’(o) ~-M*“(‘) _ iF + W’).

(4.1.16)

We see that up to O(h2) the one-particle propagator r (2*o’for o-particles will have the mass-squaresingularities whose location is determined by w:r(J). It is evident from the above derivative that a similar formula will be obtained for the one-particle propagator G,(x, y). In this case the location w:,(J) of mass-square singularities for o-particles will be determined from Eq. (4.1.12) with I/, replaced by V, given by Eq. (3.2.17).

320

FURLAN

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RACZKA

It seems that a physical interpretation of the obtained results is the following: according to the conventional theory the mass-square singularities of the one-particle dressed propagators represent the bound or the excited states of a given physical system. In our case, as follows from (4.1.16), the location of the mass-square singularities is determined by the eigenvalues o:,(J). In turn, these eigenvalues are determined by Eq. (4.1.12): if V, = 0, then (4.1.12) represents a motion of a free oneparticle system considered in the angular momentum representation; in this case propagator (4.1.16) will provide the standard Feynmann propagator for a free particle. If V, # 0, then it is natural to interpret Eq. (4.1.12) as the equation of motion of a meson in the potential generated by its own meson cloud of extended mesons: this cloud is represented by the potential V,(p,). The center of mass motion of the meson and the cloud was separated out: the origin of the reference frame for the relative motion is located in the center of the resting extended meson field o,(r) generating the cloud, for which the field momenta P,(qO) = 0. The invariant energy of the relative motion is given by the eigenvalues wf,, which might be discrete or continuous depending on the radial form in the meson-cloud potential VV(rpO). For radial potentials, wi, depend on the total angular momentum J of the meson-cloud system in its relative motion; consequently, J should be interpreted as the admissible spins of the observed physical dressed mesons. This implies that in the plane mass-square-spin we obtain for the physical meson certain trajectories MQJ) = oi,O, n, = 0, l,..., J = 0, I,..., depending on the form of the original interaction Lagrangian. It should be stressed that the potentials V, and V, are determined by the exact solution (qO, a,) of nonlinear field equations: consequently even in the considered approximation up O(fi2) the propagators r”,‘) and r’oT2) contain some nonperturbative pieces of information on the given dynamics. The experience with the hexpansion in quantum mechanics 191 and quantum field theory in two-dimensional space-time [lo] teaches us that h-expansion gives a very accurate approximation for physical quantities for strongly interacting systems. Hence, one might expect that also for the considered model in four-dimensional space-time one also obtains a reasonable approximation. 4.2. Trajectories

of Collective Boson Excited Slates

We shall now calculate the explicit form of the trajectories of the excited states for the v, particles. In order to do this, we have to find the eigenvalues wi, for the Schrodinger-like equation (4.1.12) for the potential V, given by Eq. (3.2.16). The inspection of the form of the effective potential Ve,ff(r; J) given in Fig. 4 shows that there will be a finite number of discrete eigenvalues w:,(J). In order to calculate these numbers we used a special computer programme “Reson” elaborated by Senatorski and Skorupski. It follows from formula (3.2.19) that the location of eigenvalues w:,(J) depends upon the bare mass ,u~ and the effective mass MO of the original fermion field only. If pI, increases, then from Fig. 4 and the numerical computations, the number of the

EXCITED

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IN

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321

QUANTUM

excited states increases. The slope of the trajectories depend on the value of Mi: if Mi increases then the slope decreases. We present in Fig. 5 the results of the numerical computation in the case of radial excitation n, = 0, for the following values of the masses pm and M, which define the potential (3.2.19): ,uw= 1.4 GeV, M, = 1.2 GeV. The stars in Fig. 5 represent the coordinates M2 and J of the observed isotopic meson singlets with I = 0 and the natural parity. We see that with the chosen parameters the present model gives a satisfactory description of experimental data for zero isotopic-spin mesons. Obviously, in order to have a meaningful comparison of predicted trajectories with those observed experimentally, we have to include isotopic spin: we will do this in our next work [ 141. For the considered values of parameters ,u,$ and M,, the trajectory corresponding to the radial excitation n, = 1 contains only one point with J = 0 and M(0) 2 2.1 GeV. However, if we decrease M,, then we get several points on the trajectory n = 1. It is evident from Fig. 4 that eigenstates for higher spins may become unstable. The numerical analysis shows that only the excited state with J= 4 becomes unstable with the width r= 16 MeV. Summarizing, the excited state trajectories for collective (o-mesons have the following properties: (1) All trajectories M:,(J) are approximately linear in J. (2) The number of trajectories is finite. (3) Each trajectory contains a finite number of excited states: the number of excited states decreases if the radial number n, increases, (4) Higher spin resonances are unstable with the increasing half-width f. One obtains similar conclusions for c-particles using the potential (3.2.17). All these features are in a qualitative agreement with experimental data [ 1 ]. This encourages us to think that the four-Fermi interaction after inclusion of isotopic spin may be a candidate for a description of strong interactions [ 141.

l=O

0 (783) --..A 1

2

I/ 3

4 M2

FIG.

5.

Trajectories

of one-particle

excited

[GeV2]

states of (o field.

322

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4.3. Implications

of Derrick’s

AND

RqCZKA

Theorem

Derrick’s theorem states that if pO(x) is a static solution with a finite energy, then the operator H given by (4.1.11) has at least one negative eigenvalue ~51, [ 151. The corresponding eigenstate will have an imaginary energy and is presumably unphysical. This state appears also on Fig. 5 for the lowest spin .I= 0. One may look at this problem from two points of view: (1) The calculation of r ‘2’0’-function up to O(h2) terms is a kind of WKB approximation adopted for the need of quantum field theory. By the Correspondence Principle this semiclassical approximation is asymptotic [ 91: it is reasonably good for the large quantum numbers (i.e., for the asymptotic spectra) of observed quantities and it is the worst one for the lowest eigenvalues. Thus one might expect that the deviation from the correct value of the energy for the lowest energy and angular momentum state is so big that it produces one negative eigenvalue UJ~ associated with J= 0. (2) One might think that the appearance of a lowest negative eigenvalue for the mass-square is a more fundamental problem, which should be resolved with full generality as the ZTM problem. This can be carried out by a method similar to that presented in Section 3.1: indeed an inspection of formula (3.1.10) instructs us that, in order to remove the Derrick mode v. satisfying .the equation

it is sufficient to replace (3.1.10) by the formula

Here we took into account that, for the ground-state solution ((Do, do = 0), I?’ = T”: the operator 17, represents a one-parameter rotation group O(2). The functions G,(z) and G,(z) have the form: i,(z) = cos(mi t) w,(z), G,(z) = sin(oi t) we(z). Repeating the analysis of Section 3.1 we now find

6

u:(z) (o(z) d4z

I

= 6(E,),

a = 1, 2, 3;

6LIjop(z) &z I=6(c$),

where Cb is the projection of q onto i,, i = 1, 2. Consequently the ZTM u;l, a = 1, 2, 3 and the Derrick modes 4, will be eliminated from the functional integral (3.1.13). The elimination of the Derrick modes from (3.1.13) implies the following changes in formulas (3.1.15) for J(f,,f,):

EXCITED

STATESIN

323

FOUR-FERMI QUANTUM

(i) The propagator Gi(x, y) given by (3.1.13) will not contain the term
p’= PP,

with

P, =

10 WJ, -

Since, however, all terms in (3.1.8) are S0(3)-invariant, constant 271.

the action reduces to the

(iii) The Jacobian (3.1.11) will be replaced by the new Jacobian with (& + 2) X (2’ + 2) entries. We seetherefore that the Derrick modes 4, may be removed completely from the effective propagator similarly to ZTM, and that the obtained final formulas have a satisfactory physical interpretation. A more general analysis of Derrick’s Theorem and its implications is presented in 121.

5. FERMION GREEN'S

FUNCTIONS

5.1. Properties of 2m-Point Fermion Green’s Functions We shall now derive a new formula for fermion Green’s functions r”““. (2.32) we obtain

Using Eq.

r’2m’(Y1,Zrr.....V,rZm) = (-0” (01WY,) lii(z,) **. v(.v,) F(zd IO) = hmZ-’

A 17 G,(yj, zj; cp,a)det[K,(o, jz I

a)] e’i’h)s’w*a’Dq Da,

(5.1.1)

where A is an operator which antisymmetrises fermion labels and G,,,(jj, z; o, o) is defined by Eq. (2.10). It follows from (5.1.1) that rt2”‘)-functions are representedby the path integrals of the form rC2”‘( y, ) z, ,...) ym, zm) zz z-

’ 1 F’2m)(q, o)e(i~*‘S“‘~“)Dp

D,J

(5.1.2)

324

FURLAN

AND

R4CZKA

with (5.1.3)

FcZrn’(~, 0)~ h”A ,B, G,(.Yj, zj; q,. 0) det[K,(yl, a)].

We see therefore that the fermion degrees of freedom disappeared and we remained with the path integral over Bose fields v, and (T. These integrals can be evaluated in the same way as that of the pure Bose field theory considered in Section 3.1. In order to get rid of bosons ZTM we insert into (5.1.1) the identity (3.1.10). Next we expand S(q, a) around the ground-state solution (qq,, c0 = 0), we perform the substitution !~“~a)’ = U,rp - p,,, A’l’a’ = Uao, and we obtain 7(2rn)

=

z-

1

da i i-3

x fi

a= I

‘F’2m’[U,‘(ii’*(19

+ c/lo),

u,‘(h”20)]

J

6 j u:(z) q(z) d4z ] .P(h “‘v, + cp,) e(i’h)Serc h”‘myh”20) Dq Da. I

(5.1.4)

Carrying out the same analysis as in Section 3.1 we obtain 7’2m’(y,,

z, q..., Y,,

=

Zm>

~m~-~~~~~~e-~i/2~[6,CI,~,tS,G,6,1~

x det[K,(h”‘y,

!J

t qo, h”‘o)]

m

.P(A”‘q

G,(.v~,

t qo) e

Zjt

fi’j2q

(i/n)s~“‘(~‘/z,,h”~o)

+ (~0,

fi”‘o)

lm~o~O.

(5.1.5)

Formula (5.1.5) is exact: since the dependence on the Planck constant fi is explicit, it may serve as a starting point for h-expansion. We illustrate the usefullness of formula (5.1.5) calculating 5‘*‘-functions up to O(fi3’2) terms. Expanding G,(A”‘y, + oo, A”‘cr) around (qo, u. = 0) and using (3.1.3) we obtain

A(k+‘)‘2[(k + /)!I-’ kfl-0

Jk+‘G,b- z; (00,001,7(k,/J , (5.1.6) dkfp6’u

where

~k++Xw~UlodJo), 7(k,,) dkfp 6’u Gk+‘G,(y,

= d4zi

&p(z,)

...

&p(z,)&7(z,+,)

z;

rp,,,

a,> 1..

7W) &7(z,+,)

(Z13-.Zk+/).

Using the fact that 7(k*‘) boson Green’s functions start their it-expansion with h” terms, we conclude from formula (5.1.6) that presently

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This shows that the probability amplitude for finding a one-particle fermion state in the classical limit is zero: this is in agreement with a general conviction that fermions do not exist in the classical sense. 5.2. Trajectories

of Fermion Excited States

Now we determine explicitly the trajectories of the excited fermion states which appear in the present model. As follows from the Kallen-Lehmann spectral representation the spectrum of fermions in a given theory is determined by the discrete singularities in p: of the one-particle propagator, which in general is given by the formula [ 131 “7

-MY

\ k-,

5’*yp1 ,p*> = 22s’4’(p, +p2)

. Pl

+ Mk)

pf-M:+ie

Pl(m2) y.p,--m+i&~y.p,fm-ie

P2(m2)

(5.2.1)

here ZZk are normalization constants and pl, and p2 are spectral measures depending on the dynamical properties of the considered theory. We shall now derive-using the h-expansion-an approximate formula for the oneparticle propagator t(*) and we shall determine the resulting mass-square singularities. Using formula (5.1.6) and the fact that r (k-l) boson Green’s functions start their Aexpansion with the ho term, we obtain ~(“(y, z) = hPG,(y,

z; (po, uo) + O(h-“‘).

(5.2.2)

The group-averaging operator P, as previously, is given by Eq. (4.1.15): consequently in the momentum representation we obtain +*‘(P1,

P2) = hs’4’(P,

+p*)

.qp,,

-p,)

(5.2.3)

+ o(fi”‘*),

where .VJp, ,p2) is the Fourier transform of PG, after factorisation of ~?‘~‘(p, +p2). We see that up to O(A3’2) the mass-square singularities of the one-particle fermion propagator are determined by the corresponding singularities of the effective propagator. This propagator in the coordinate representation satisfies K, G,(x, y) = [$+‘a, - MO - v~]

G,(x,

y) =

6’“‘(x

-

-v) 1,

(5.2.4)

where by Eqs. (5.2.2) and (3.2.21) v, = -grp, = -Qo.

(5.2.5)

Let x, be eigenfunctions of K, defined by the formula K,Xn = 4, YoXn.

(5.2.6)

326

FURLAN

Then G, can be written

AND

RACZKA

in the form

G,(-GY>= $ (9, + i&)-l x,(x> 0 Mv)~ n

Since VI: is radial we may set x,(x) = ePiE’y/+(x); we obtain for wa(x) the eigenenergy equation

2, =XntYO.

then multiplying

Eq. (5.2.6) by y,,

J-fvAx) = I-ia . V + 2dM, + vF)l vi(x) = u,,(J) v,-(x)~ with

(52.7)

(52.8)

qn = E - w,~. Consequently G, can be written in the form G&Y)

= 4, ,“_

-iE(t,-f,J w + iE ‘Y,-(x) 0 W,-(Y). n I.

(5.2.9)

Now using (5.2.3) and passing to c.m.s. p, = (M, 0) we obtain

+ O(k’*).

(5.2.10)

The physical interpretation of the obtained result seemsto be the following one: Eq. (5.2.8) representsthe relative motion of the original fermion in the potential I’, generated by its own meson cloud of extended mesons.The center of massmotion of the cloud and the fermion was separatedout. The origin of the reference frame for the relative motion is located in the center of the remaining extended meson field v,(v). The invariant energy of the relative motion is given by the eigenvalues w”,, which might be discrete or continuous depending on the radial form of the potential. Equation (5.2.8) can be reduced to the radial form: consequently the eigenenergies w,~ will depend on the total angular momentum J of the fermion-cloud system in its center of mass: accordingly, J should be interpreted as admissible spins of the physical dressedfermions. It follows from representation (5.2.10) that the mass singularities M,,r of the oneparticle fermion propagator r(*) coincide with the eigenvalues w,(J): therefore, the singularities Mnr(J) will form in the mass-square-spinplane certain ‘trajectories whose shapewill depend on the original interaction Lagrangian, which determines I’, . In order to find the localization of mn,(J) we have to solve the eigenvalue problem (5.2.8). This is performed using the following spinor radial representation (see. e.g., 116, Sect. 35 1:

“f(r) ~,,A~~ VI ‘Ax) = (-)‘I +‘-I’)‘* g(r) flJ,,,J6, q?) *

(5.2.11)

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It is well known

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that there are two

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kinds of solutions:

one with

I’ = I + 1 and

J = I+ $ and the other one with 1’ = I- 1 and J = I- i. Setting K=-(J+f)=-(l+

1)

=J+$=l

J=l+i. for J=l-$,

for

and inserting (5.2.11) into (5.2.8) we obtain the following system of radial equations for the functions F = rf and G = rg:

F’+zF-(M,+ r

V,-+u,)G=O

G’-+G-(MO+

v,,.-w,,~)F=o.

(5.2.12)

Eliminating G and setting F=exp

[-fjPdr]Z(r)

with P=-

V;.(.(M,+ vF+wn)-‘.

we obtain the following Schrodinger-like equation for 2:

-- d*Z + V&Z = wirz, dr2

(5.2.13)

where

Kdr; w,,,.J>=

K(K$ 1) r2 ++,+v*,

= IK/ (IKI- 1) -r IKI v, + v,, r2

K < 0,

(52.14)

with

v, = - $Avf! + &4*(V;.)*+ (M, + vg*, A=(M,+wnrf VP)-‘. We see that the effective potential is energy and angular momentum-dependent: we see also that the connection of the effective potential with the original one in (5.2.8) is highly nontrivial.

328

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AND

RqCZKA

(MO +‘4)[Gev]

FIG.

6.

The original

potential

V,(r)

+ M,, for fermion

fields.

The function M, + V,., as follows from Eq. (5.2.5) and Fig. 3, has the shape depicted in Fig. 6. Notice that by (3.2.21) the potential V!&(r; w,,, J) is g- (and therefore A-) independent. Consequently, the only free dynamical parameter is the fermion mass M,. However, even this parameter is already fixed by the condition that the boson effective potentials (3.2.19) and (3.2.20) fit reasonably well the experimental trajectories of meson resonances. Consequently, the potential V&.(v; o,,~, J) is completely tixed. The shape of V$, for M, = 1.2 GeV, w = 0.5: 1 and 1.5 GeV and J = 4 is presented in Fig. 7. V&

(,:a,

J)

[GeV’]

J=y2

r lfml , .2

F1c.7.

The effective

potential

I’&(r;

.3

.4

3

.6

.7

w,~, J) for ,uu, = 1.4 GeV and M, = 1.2 GeV.

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We see that for every fixed cc),, the potential I’!& has a singularity at some r = rO(o). This singularity is implied by the function A in (5.2.14) since V,(r) is negative and monotonically increasing. We see also from Fig. 7 that the bottom of the potential for o,~= 0.5 GeV is above oi,: hence for this value of energy it is impossible to have a bound or a resonance state. In turn, for w,, = 1 GeV, wfi, is in the potential well, but the computer analysis shows that the bound-state cannot be formed. Finally for w,,~ = 1.5 GeV 0: is much above the plateau of the potential, equal to A4: = 1.44 GeV’; thus, also in this case a bound- or resonance-state cannot be created. The part of the potential on the lefthand side of the potential singularity has the bottom much above the corresponding a;,: consequently here too bound-states cannot be formed. It was confirmed by the numerical analysis that the same conclusions hold in the interval 0 < w,~ < 10 GeV for spin J = f and i. The numerical analyses of the shape of I’:,, for J= i leads to the same conclusions. This implies that the one-fermion propagator has no isolated mass singularity for the chosen values of free parameters M, and ,LL~of this model. The presence of a mass singularity of the one-fermion propagator is interpreted in the conventional field theory as the proof of the existence of a one-particle asymptotic fermion state [ 13i]. Consequently, the lack of mass singularities in the one-fermion propagator should be interpreted as an argument that the one-fermion states do not exist asymptotically in the present model. This is a correct conclusion under the condition that, taking into account higher orders of &expansion in our analysis, they do not introduce new singularities into one-fermion propagator: the analysis of the influence of a higher order A-expansion on the location of mass singularities of oneparticle propagators in quantum mechanics and models of quantum field theory in two-dimensional space-time [ 181 would suggest that this is unlikely. If higher order h-expansion terms do not change our result, then the present model would provide an interesting dynamical mechanism of a fermion confinement without confining potential. This could in principle serve as an example of a possible confinement mechanism for quarks in QCD. One might ask the following question: if we take other values of dynamical parameters ,u~ and M, than those which lit the boson trajectories, then can we get fermion bound states? The numerical analysis of the spectral problem for potential (5.2.14) shows the following: for most of the values of the bare mass p. and the effective fermion mass M, the potential I’:,, has the form given in Fig. 7: however, for some values, e.g., for ,um= 3.7 GeV and M, = 2.37 GeV, I’:,, has no singularity and has the form given in Fig. 8. These potentials admit the fermion bound states w0 = 1.9 GeV for J = f and w0 = 1.8 GeV for J = i. For J = $ and higher the potential I’:,, is repulsive. If we now take the obtained values of rue and M, and calculate the meson trajectories, we find that we have only two meson excited states, namely, nzz’ g 0, my’ = 3.2 GeV and my’ E 0, mu*’ = 3.1 GeV. We see therefore that the problem of the generation of boson and fermion excited states is a subtle dynamical problem which strongly depends upon the numerical

330

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AND

RtjCZKA

FIG. 8. The effective potential V:,,(r; w,~,.,J) for pu, = 3.7 GeV and M, = 2.4 GeV.

values of bare masses and coupling constants. For one definite set of these parameters we have excited states of composite bosons which are close to experimentally observed bosons, whereaswe do not have at all excited or asymptotic fermions: for other sets of model parameters we have excited states of fermions as well as bosons, but the structure of boson trajectories changes significantly. 6.

DISCUSSION

We conclude our work with the following remarks: (i) The most important result of our work consists of a demonstration that in a pure fermion model of quantum field theory there exist trajectories of excited bosons which are similar to the trajectories observed experimentally. This noteworthy result was obtained by a partly nonperturbative analysis. This suggeststhat pure fermion models of quantum field theory may provided a conceptually simpler formulation of a theory of strongly interacting elementary particles. The further analysis of four-fermion interactions containing isotopic spin is presented in our work 1141. (ii) The method of the collective bosons allows for a complete elimination of fermion fields from the path-integral representation for generating functional. The obtained effective field theory for collective bosons possessesthe ground-state solution which generates the potentials V,, and V, for the effective one-particle propagators. It is remarkable that these potentials, which uniquely follow from the

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331

original four-fermion interaction, imply the existence of trajectories of excited states for bosons which are qualitatively similar to those observed experimentally. This should be contrasted with other methods, where the bag-like or confining potentials were introduced by rather ad-hoc arguments [ 171. (iii) The analysis of mass singularities of one-fermion propagator indicates that they strongly depend on the dynamical parameters of the four-Fermi model: for some values of these parameters we have bound states for J = 4 and 5 and for other values we have no mass singularities at all. The lack of mass singularities of the one-fermion propagator indicates that the corresponding fermion has no asymptotic states. This fact may provide an interesting dynamical mechanism for a fermion confinement which may also hold in some quark models, especially those formulated in terms of pure fermion fields [ 191. (iv) We draw attention to the fact that formula (3.1.15) for generating functional may serve as a basis for a new perturbation or h-expansion for Green’s functions. In this expansion each line in the generalized Feynmann diagram will be represented by the effective Green’s function G; or G, and each vertex by a monomial from LP;‘9 det [K&, a)] or the Jacobian .P(A “*u, + pO). Since Green’s functions contain singularities corresponding to several particles with positive masses and spins, an exchange of one line in the obtained generalized Feynman diagrams corresponds to an exchange of the whole trajectory of particles: in addition Green’s functions G,, G,, and G, already contain some nonperturbative pieces of information on the considered dynamics. Thus one might expect that in the present case even a low order perturbative or h-expansion might contain a significant amount of information on a considered physical system.

APPENDIX

Now we give the proof of the proposition from Section 2, based on the Eguchi analysis [4i]. Integrating (2.28) over Fermi fields and calculating det[K,(rp, o)], we obtain

where

- $c,lli - g*z, + 3M; g*z(J $2 - f@ - g’*z, + A4; g’ZZ,) u* + (lc, + M, g”Z,) p3 + (5 f M, &*I,)

f/m*

- 4 [(lb, + f g4Z,) p4 + (A, + g*g’*z,) c&7* + (4 + f g’4zo) a41 + L.&p,, u)}(x). where P(v, u) is functionally identical to ii’(p,, a,) of Eq. (2.17).

(A.21

332

FURLAN

Let us introduce the following

AND

RACZKA

resealed fields: (A.3)

resealed coupling constants (A.4a) (A.4b) K ,r =

IK,

+

&t’f,g(&’

-

I)]

g;‘=,

(A.4c)

K =,.=

[K,

+

2hf,,g(P;’

-

I)]

~;=&

(A.4d)

Air=

[A, +g=(P;‘-

l>P;,

(A.4e)

A,, = [/I, + 2gQ,

- I)] JvJO,

(A.4f)

ljr = [A, + g’=(P,’

- l)] -q,

(A.%)

and resealed masses ,+

[pi-g2Z2

+ 6M;(&-’

- l)] P,,

pi, = [pi - g’=z, t 2M&Y,

’ - 1)] F,,

(A.51

&Kc= (1 t ;g’=I,)-‘.

(‘4.6)

where SW= (1 ++g2z,)-‘,

Then we can express the generating functional tities, i.e.,

(A.1) in terms of the resealed quan-

where

and, since g9 = g,9,, g’a = giurr

G,(x,Y>= [K,(v,a)- ic]- ’ (x,Y>, K,(9,u)

= &(v)~, 0,) = @ - MO + g,9, t kLy,u,).

(A.9)

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333

By comparing the two generating functionals (2.27) and (A.7) we see that, in order that they have the same structure, the following conditions must be satisfied: s:

(A. 10a)

=gr,

2Mo

K 1r=K2r=

(A.lOb)

gr 3

++3rzg;.

(A. 10~)

Let us see what the implications of Eqs. (A.lOa), (A.lOc) are for the bare coupling constants. From Eqs. (A.4a). (A.4b), (A.6) (A.lOa) we get

g’ = g;

(A.ll)

then pm=$o=p

(A.12)

and therefore (K,, -

Kzr)

=

(K,

-

KZ)

P3’2,

(1, -A3)P2,

(l,r-IZ3r)=

(A,, - 2A,,) = (1, - 2n,> $2. Then from Eqs. (A. lob), (A.lOc),

(A.13)

and (A. 13) we get

KI = K,, Furthermore,

(A.14)

from Eqs. (A.4a),

(A.4b). (A.lOb),

[K,

-

+ 2kf,g(P-’

I)]

P3”

(A.lOc), =

(A.17).

and (A.12) we get

2h’f,&‘$“2,

i.e., K,=K,=&tf,g

(A.15)

and

i.e., (A.16)

334

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AND

RACZKA

We remark that, since ,u+:~and ,ut, are independent quantities (as appears from Eqs. (2.26a), (2.26b)), no condltlon is imposed on ,B: and ,LL~by Eq. (AS). ACKNOWLEDGMENTS

The authors thank Professors P. Budinich are grateful to Doctors A. Skorupski and A. for calculating the ground-state solution and calculations. One of us (R.R.) gratefully Professor P. Budinich and I.S.A.S.

and R. Rajaraman for inspiring discussions. The authors Senatorski for permission to use their computer programme energy levels and to Mr. J. Kraskiewicz for performing the acknowledges the warm hospitality extended to him by

REFERENCES

1. M. ROOS et al. (Eds.), Review of particle properties, Whys.Left. B IllB (1982). l-294. 2. R. R.+CZKA. “Second Quantization of Boson-Fermion Field Theories with Solitons and Instantons,” preprint, I.S.A.S., Trieste, Italy (1982). 3. R. R.+CZKA AND J. KRASK~EWICZ, Trajectories of Excited States of Mesons and Nuclcons in the Yukawa-like Models of Quantum Field Theory, preprint, I.S.A.S.. Trieste. Italy. 1982. 4. (i) T. EGUCHI, Phys. Rev. D 14 (1976). 2755; 17 (1978), 611. (ii) K. TAMVAKIS AND G. S. GURALNIK, ibid. 18 (1978), 4551. (iii) G. KONISHI AND W. TAKAHASHI. ibid. 23 (1981), 380. (iv) P. REMBIESA, ibid. 24 (1981). 1647. 5. R. R.+CZKA, A Local Spinor Quantum Field Theory as a Model for Strong Interactions, preprint, I.S.A.S., Trieste, Italy, 1982. 6. (i) L. FADEEV. “Methods in Field Theory,” Les Houches (R. Balian and J. Zinn-Justin. Eds.). North-Holland, Amsterdam. 1976. (ii) C. BENDER, F. COOPER, AND G. S. GURALNIK. Ann. Ph.vs. (N.Y.) 109 (1977), 165. 7. (i) S. ALBEVERIO AND R. J. HBEGH-KROHN. “Mathematical Theory of Feynmann Path Integrals,” Lecture Notes in Math., Voi. 523, Springer. New York, 1976. (ii) E. B. BOGOMOLNY. L. N. LIPATOV, AND V. A. FATEYEV. “Lecture Notes,” INP. Leningrad. 8. A. JEVICKI, Nucl. Phys. B 117 (1976), 365. R. R~CZKA AND L. ROSZKOWSKI,“Analysis of Green’s Functions and Stability Problems in Models of Quantum Field Theories with Solitons.” preprint, I.C.T.P., Trieste, Italy, 1983. 9. V. MASLOV AND M. V. FEDORIUK, “Asymptotic Methods in Quantum Mechanics.” trsanslated from Russian by I. Niederle, Reidel, Dordrecht. 1981. 10. (i) R. DASHEN, B. HASSLACHER, AND A. NEVEU, Phys. Rep. D 10 (1974). 4114, 4130: II (1975). 3424. (ii) A. LUTHER. Phys. Rev. B 14 (1976): 2153. 11. H. BERESTYCKI AND P. L. LIONS, in “Bifurcation Phenomena in Mathematical Physics and Related Phenomena” (C. Bardos and D. Bessis. Eds.), Reidel, Dordrecht. 1980. 12. M. V. FEDORIUK. “Saddle-Point Method,” Moscow. [Russian\ 13. (i) C. ITZYKSON AND J. B. ZUBER. “Quantum Field Theory. ” McGraw-Hill. New York. 1980. (ii) J. RAFELSKI, L. P. FULCHER. AND A. KLEIN. Phys. Rep. 38 (1978). 227. 14. P. FURLAN AND R. R~CZKA, PhJs. Lett. B 124 (1983), 527. 15. G. H. DERRICK. J. Math. Phys. 5 (1964), 1252. 16. V. BERESTETSKIi. E. LIFSHITZ. AND L. PITAEVSKli. “Relativistic Quantum Theory.” Vol. 4. Course of theoretical physics, Pergamon, Oxford. 197 1. 17. C. QUIGG AND J. L. ROSNER. ‘&Quantum Mechanics with Application to Quarkonium.” review; Phys. Rep. 56 (1979), 167-235; F. SCHC~BERL.A Non-perturbative Calculation of Meson and Baryon Masses with a One-parameter Potential, preprint, TH 3287, CERN. 1982. 18. R. RAJARAMAN, “Solitons and Instantons,” North-Holland. Amsterdam, 1982. 19. AMATI et al., PhJw. Lett. B 102 (1981). 408.