Fortschritte der Physik / Progress of Physics: Volume 31, Number 6 [Reprint 2021 ed.] 9783112497463, 9783112497456

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FORTSCHRITTE DER PHYSIK PROGRESS OF PHYSICS

Volume 31 1983 Number 6

Board of Editors F. Kaschluhn A. Lösohe

l ì . Rompe

Editor-in-Chief

F. Kaschluhn

Advisory Board

A. M. Baldin, Dubna J. Fischer, Prague 0 . Höhler, Karlsruhe K. Lanius, Berlin F. Lopuszaiiski, Wroclaw A. Salarn, Trieste 1). V. Shirkov, Dubna A. N. Tavkhelidze, Moscow 1. Todorov, Sofia J. Zinn-Justin, Saclay

CONTENT: B. M i l e w s k i Four-Dimensional Theories with Composite Gauge Fields and Their Supersymmetric Generalizations

313

338

J. K t t p s c h 339 -356

Strong Coupling Problems of S-Matrix Equations

AKADEMIE-VERLAG LS,SN 0015-8208

BERLIN

Fortschr. I'hys., Berlin 31 (1983) 6, 3 1 3 - 3 5 6

EVP 10,- 11

Instructions to Authors 1. Only papers n o t published and n o t submitted for publication elsewhere will be accepted. 2. Manuscripts should be submitted in English, with an abstract in English. Two copies are desired. 3. Manuscripts should be no less t h a n 30 and preferably no more t h a n a b o u t 100 paged in lenght. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuskript sheets should be n u m e r a t e d consecutively from " 1 " onwards. Foot, notes should be avoided. 5. The titel of t h e paper should be followed b y t h e authors n a m e (with first name abbreviated), b y t h e institution a n d its address f r o m which t h e manuscript originates. 6. Figures and tables should be restricted to t h e minimum needed to clarify t h e t e x t . They should be numbered consecutively and m u s t be referred t o in t h e t e x t and on t h e margin. Figures and tables should be added t o t h e manuscript on separate, consecutively n u m e r a t e d sheets. The tables should have a headline. Legends of figures should be s u b m i t t e d on a separate sheet. All figures should bear t h e author's name and number of figure overleaf. Photographs for half-tone reproduction should be in t h e form of highly glazed prints. Line drawings should be in a f o r m suitable for reproduction. The lettering should be sufficiently large and bold t o permit reduction. If requested, original drawings and photographs will be returned to the a u t h o r upon publication of t h e paper. 7. Formulae should not be written t o small a n d n o t with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary u p r i g h t typeface. Underlining to denote special typefaces should be done in accordance with t h e following code: Italics: wavy underlined with pencil (only necessary for t y p e written symbols in t h e text) Boldface italics (vectors): w a v y underlined twice Upright letters (all abbreviations like all units (cm, g, ...), all elements and particles (H, He, ..., n, p, ...), elementary mathematical functions like Re, I m , sin, cos, exp, . . . ) : green underlined Greek letters: red underlined Boldface Greek letters: red interlined twice Upright Greek letters (symbols of elementary particles): red and green underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters t h a t do not differ in shape, as c C, k K, o O, p P, s S, u U, v V, w W, x X, y Y, z Z). I t will help t h e printer if the position of subscripts and superscripts is marked with pencil in the following w a y : at, b[, J l f j ; , Mtj, W^ Please differentiate between following symbols a, a ; a, a , oo; a, d; c, C, c ; e, I; £, t, k, K,x; *, x, X, x, X ; I, 1; o, 0, a, 0 ; p, g; u, U, \J;v,v, V; [eiA^\ki

ipi

where Aki(x) are the elements of the SU(2) algebra, one has to use the covariant derivative in the Lagrangian yj(iy" 8„ — m) y> -> y>k(iy»(dftdk> +

igA/')

— mdk>) ipi.

(1)

The gauge fields Ahk' transform unhomogeneously under the adjoint representation of

SU(2)

A„ - > e^Ape-W

+ — g

(2)

so as to compensate the noncovariance of the derivate : = d/dx^1. The free Lagrangian for the gauge fields has, in analogy with electrodynamics, the form I

= - 1 Tr { V ' ) ;

F% =

S^A*

+ ±

[A„

Av]V.

(3)

The physical interpretation of the model gives rise to serious problems. The SU(2) symmetry was not sufficient for the classification of all strong interacting particles. Besides, the quantization procedure for such theories was not yet developed. And finally, the choice of proton and neutron as a fundamental dublet of elementary particles has not led to a realistic model. In 1 9 6 4 G E L L - M A N N [ 2 ] and Z W E I G [ 3 ] brought into consideration a new hypothetic triplet of particles christened by Gell-Mann quarks. The idea has proved to be very efficient in the classification of all known hadrons. The attempts to describe weak interactions in a framework of gauge theories encountered at first one serious difficulty. I t appeared that the Yang-Mills theory is renormalizable only when the gauge fields are taken to be massless. Meanwhile the short-ranged weak interactions should be mediated by massive bosons. The possibility of overcoming this difficulty was revealed in 1964 when the Higgs mechanism was invented [4] and then generalized for the nonabelian case [5]. The Higgs mechanism makes it possible to generate masses for gauge fields in the initially massless theory. I t has enabled S A L A M and W E I N B E R G [6] to formulate" the unified theory of weak and electromagnetic interactions. The technical problems connected with the quantization of gauge theories were gradually overcome. In 1967 F A D D E E V and P O P O V [ 7 ] formulated the quantum theory of nonabelian gauge fields using the method of path integral. The invariant procedure for the renormalization of the massless Yang-Mills theory was proposed in 1971, 72 by T A Y L O R [5] and S L A V N O V [9] whereupon T ' H O O F T [10] has generalized it for the case of broken symmetry. All this made possible the formulation of new quantum theory of strong interactions the quantum chromodynamics (QCD) — a theory of quarks and gluons with the gauge groupSU(3) — colour. This theory possesses a rare feature: the asymptotic freedom. I t means that the effective coupling constant tends to zero for large momenta. However at small momenta the behaviour of the theory is not fully understood. For instance the quark confinement is not yet explained. The proliferation of quarks is still another problem of strong interaciton theories. As far back as in 1974 the Gell-Mann trinity was destroyed by the discovery of the fourth flavour-charm, and subsequently fifth and sixth flavours — beauty and truth. This was the reason to develop theories

Four Dimensional Theories

315

(e.g. [II]) in which quarks are not elementary bricks of matter. But the model alternative to the Yang-Mills theory should be asymptotically free. We know a whole family of such models — the so-called sigma models. 1.2. Sigma Models

Nonlinear sigma models have been introduced in many-body theories as Heisenberg models. In the field theory they first appeared as the 0(4) model of pion dynamics, and afterwards as a family of 0(N) models [72]. In the 70's the theory of sigma models was developed in the direction of investigation of their quantum properties [IS] as well as their generalization to other manifolds [14,15]. But the main incovenience of sigma models was that they were usually considered in two dimensions. There were attempts to construct sigma models in the Minkowski space [16], but by simply increasing the number of dimensions one -looses conformai invariance. This difficulty was overcome by D U B O I S - V I O L E T T E and G E O R G E L I N [ 1 7 ] who have proposed a generalized sigma model on the Grassmann manifold, which, on the other hand, is the Yang-Mills model with composite gauge fields. But the attempts to quantize such model meet one great difficulty which is the fourlinearity of the Lagrangian. In this paper we try to overcome this problem in the framework of the 1 ¡N expansion for the simplified abelian model. 1.3. Supersymmetry

Supersymmetry was a new great discovery of the 70's. At first it was introduced in two-dimensional dual models [IS] and soon generalized to four dimensions by W E S S and Z U M I N O [29]. This new idea helped to resolve the problem of the so called no-go theorem of C O L E M A N - M A N D U L A [20] which states that in the framework of quantum field theory the nontrivial mixing of the Poincaré and internal symmetries is impossible. However, if one introduces a new notion of "supergroup" instead of "group", the nontrivial mixing becomes possible. Supersymmetry not only improves the renormalizability of the theories, but also is a good starting point to unification of all interactions including gravitational ones (supergravity). Many supersymmetric generalizations of already known theories were developed in the 70's. The supersymmetric version of the 0(N) model [22] and the CP(N — 1) model [23] were constructed in two dimensions. Also the super-Yang-Mills model was introduced [25, 26] but with one great disadvantage — the Fermi fields were in the adjoint representation of the gauge group and it made difficult the identification with physical multiplets for particles. In this paper we present a supersymmetric version of the fourlinear sigma model with composite gauge fields in which the fundamental fermions transform with respect to the fundamental representation of the gauge group. 1.4. Hierarchies of Elementarily

Up to the 60's all the hadrons were supposed to be elementary particles. This situation seemed alarming because their number (including resonances) was much bigger than the number of chemical elements in the periodic table. Although they were grouped according to SU(2) and SU(3) symmetries there were no physical motivation of such group-theoretical ideas. The breakthrough was done when the idea of compositeness of hadrons came into consideration. First on the ground of group theory but soon enriched by the Yang-Mills type of dynamics. 1*

316

B. MILEWSKI

Today the particles which have a status of elementarity are quarks, leptons and intermediate bosons (along with hypothetical Higgs particles responsible for the phenomenon of symmetry breaking). Quarks and leptons form generations, each of them being described by a Yang-Mills theory, with SU(3) — colour for quarks and SU(2) X £/(l) for leptons. Up to-day we know three such generations which are summarized in tablel. Table I generations

leptons

I II III

e fi. T

quarks

ve v^ v,

u c t

d s b

We can see that there arise several new problems. The problem' of unification of strong and electroweak interactions in the framework of one generation and the problem of the origin of generations are still open. There are two main directions of investigation of these problems. First tends to unite the interactions on the level of a larger group (in the same way as the electromagnetic and weak interactions were united in the SU(2) X U( 1) scheme) and the second is trying to construct composite models (as it was done with hadrons). There arise also a problem of naturalness of gauge theories. The bigger is the group we are to break, the larger is the number of Higgs particles with heuristical Lagrangians and coupling constants. DIMOPOULOS and STTSSKIND [27] tried to overcome this difficulty introducing additional "techniquarks" in their technicolor scheme. The bound states of techniquarks would act as Higgs particles. Simultaneously the models with composite quarks and leptons were being constructed [28, 31]. In some models, for instance, flavour and colour were separately carried by flavons and chromons. In other models colour and flavour arose only on a composite level [29]. In this paper we present a new idea of composite gauge fields. Recently such models have appeared in supergravity [30]. This is a new opportunity to solve the problem of naturalness of gauge theories. Particles which play the role of higgsons are now the subconstituents of the gauge bosons. Their Lagrangian is determined by geometry and the symmetry breaking occurs as a consequence of sigma model like constraints. 1.5. 1/iV E x p a n s i o n

Practically all computations in the quantum field theory are based on the perturbation expansion. One usually believes that few first terms of the series give good approximation to a physical quantity. But such assumption is not always justified. For instance in quantum electrodynamics the effective coupling constant (a parameter of the expansion) is smaller than one in the whole physically available range of energies. In the asymptotically free chromodynamics the effective coupling constant is a good expansion parameter for large momenta. However for small momenta, because of the infrared catastrophe, the results are not compatible with experiment. In the theories with internal symmetry, as an alternative parameter of expansion one can take the inverse of the number of components of the field: l/N. The method of 1/N expansion was succesfully applied to the GKOSS-NEVEU model [32], O(N) sigma model [33], 04 [34] and GP(N — 1) [i5] models. The most interesting were the attempts to apply the l/N expansion to chromodynamics [35]. Many interesting results, mainly qualitative, were obtained, especially in the region of small momenta.

Four Dimensional Theories

317

In this paper we apply the methods used by D'Adda, LIischeb, Di Vecchia [15] in the two-dimensional CP(N — 1) model to the four dimensional generalization of this model. A straightforward generalization meets several problems [16\ so one is forced to use fourlinear Lagrangians rather then bilinear. 1.6. Contents of the Work

Section 2.1. containts a short introduction into the field of generalized sigma models and in section 2.2. we describe quantization and renormalization procedure for the fourlinear CP{N — 1) model with composite gauge fields. In section 3.1. we briefly review the main features of the supersymmetric models with special emphasis on the superYang-Mills model, and in section 3.2. we present the supersymmetric version of the nonabelian fourlinear sigma model. Sections 2.2. and 2.3. were based on the joint publications of J . Lttkierski and the author. The Lagrangian of supersymmetric fourlinear sigma model was independently introduced in 1 The Lagrangian (1) then takes the form:

The natural extension of this procedure is to construct models with fields taking values in more general Riemann spaces, real as well as complex and quaternionic. Notice that the field n(x) in 0(4) model parametrizes the quotient space 0(4)/0(3). This is an example

318

B . MILEWSKI

of a wider class of so called Stiefel manifolds V(N, n) to which belong e.g. BV(N, n) = 0(N)/0(N

-n),

GV(N, n) = U(N)IU(N - n), HV(N, n) = Sp (N)/Sp (N -

n).

The point of such a manifold may be parametrized by means of NXn matrix vKj, with additional conditions: i, 7 = 1,..., n,

K=l,...,N.

(5)

In the sequel we shall be interested in the class of manifolds which arise from the Stiefel manifolds by identification of matrices connected by a "gauge transformation" from the groups, respectively; 0(n), U(n) and Sp (n). These manifolds, called Grassmann manifolds, may be represented as quotient spaces, resp.: RG(N, n) = 0(N)I0(N -

n)xO(n),

CO(N, n) = U(N)IU(N

-n)xU{n),

HO(N, n) = Sp (iV)/Sp (N — w)XSp(m). In particular for n = 1 we obtain a family of projective spaces of which the complex projective space GP(N — 1) = U(N)/U(N — 1)X?7(1) will be of our special interest. This brief revue does not contain all the possibilities. There are sigma models on Kahler manifolds, group manifolds (chiral models) etc. The Grassmann manifolds GG(N, n) may be parametrized in various ways: by means of gauge invariant projection operators, by means of unconstrained variables (the theory becomes then manifestly nonlinear) and finally, similar to the Stiefel manifolds, by means of N X n rectangular matrices with identification of U(n) gauge equivalent quantities. Covariant quantities may be constructed using covariant derivative: (V„v)Ki = a„vKi + ¿ » « 4 /

(6)

where the natural connection on a Stiefel manifold has the form: Ah = iv+ dhv.

(7)

In physics, this kind of connection is called a composite gauge field. Such objects were considered in physics in course of investigation of the Yang-Mills theory. It was stated [40] that any gauge field A^x) of U(n) group may be induced by a canonical connection [7] on the appropriate Stiefel boundle. Representation (7) was also used for the construction of instanton solutions of the Yang-Mills theory [4I\. It is then natural to try to formulate the Yang-Mills theory interms of more elementary quantities, such as the Stiefel manifold coordinates vKj. First let us construct the simplest generalization of the Lagrangian (1) using Stiefel coordinates vKj: £ =

Tr {(V„v)+ Ph) + oc(v+v - 1)}.

(8)

It is interesting to notice that if we treated the fields v and A^ (the latter appearing in the covariant derivative) as independent, we should obtain, as a result of the equations

319

Four Dimensional Theories

of motion, the relationship: (v+ dfl — d„v+v)

Aft =

¿1

v+

which is, due to the constraints, equivalent to (7). Let us notice also t h a t if A^ is to have a canonical dimension of a vector field equal to one, then the fields v have dimension zero and the Lagrangian (8) has dimension two. I t is obvious then, t h a t the Lagrangian (8) is conformal invariant only in two dimensional space-time. I t turns out t h a t it is possible also in four dimensions to construct a conformal invariant Lagrangian [17\ Moreover such Lagrangian appears to be a Yang-Mills Lagrangian with composite gauge fields. Let us begin with the introduction of a gauge covariant antisymmetric tensor: i

•FV =

\-v+ 8»v> v +

+

(9)

which enables us to construct the invariant Lagrangian: 2 = j

Tr [ F ^ F i " +

- 1)} •

(10)

The analogy with the Yang-Mills theory becomes evident when we use the composite gauge fields defined by (7). (9) takes then the form: JVv = 8^AV] -

A,}.

(11)

Another possibility of writing down the tensor F ^ emphasizes its close relationship with sigma models: F^ = iV[ltv+Vv] v. (12) The classical theory based upon the Lagrangian (10) becomes, for sufficiently large N , more general t h a n the Yang-Mills theory. I t turns out t h a t a n y solution of the YangMills equations of motion solves, taking into account relation (7), the equations of motion of the model (10). The opposite is not true. The main difficulty connected with such a formulation is the fourlinearity of the Lagrangian (10). The quantization procedure needs new techniques, which will be introduced for the case of the simplified (abelian) model in the next section. 2.2. 1 ¡N Expansion in the Fourlinear CP(N

— 1) Model

The Lagrangian which we shall use in this section will be a slight extension of the abelian version of the fourlinear GG(n, N) model (1.10). The extension lies in the addition of the invariant mass t e r m :

*

=

T (T

f f>

"' "

+

i(V7z

V Z)

" T

(1)

where zx(x) is an w-component complex isovector field, the covariant derivative is defined in the following way: i *> V,,z = 8,,% + iA,,z; All— — z8liz (2) z and the antisymmetric tensor has the form : Fp, = dpA, a

d,Ah.

) Throughout this section we use the Euclidean metric

(3) = 8^,.

320

B . MILEWSKI

The gauge transformations of the Z7(l) group act on the fields z, z in the following way: eio• -y— oc

J DzDzDoc exp j j " d*x

we get + /**) I.

The elastic unitarity equation with the cut off (4.3) is A(x H- iO, t) — A(x — iO, t) — 2iA(x 4- i'O, t) (x) A(x — iO, t)

if

x > 4,

(6.2)

where the convolution (x) has been defined in eq. (5.4). It is convenient to decompose (6.1) as A(s, t) = y + B(t, u) + B(st t) + B(s, u)

(6.3)

with the t — u symmetric function Bit, u) = Gy(t, u) + FAs, t) + FAs, u) = — f ^ l l l dx 71 J X — S

(6.4)

and its absorptive part b(x, t) = ax(x, t) + bN(x, t), t)=E

N-l

*„{x) (f + w"), (6.5)

dy k , *t = — cJ — , -\ «> + r— by{x, y) jiJ [ y -1 y - w (y + T)* 1

t

w = 4 — x — i. The fixed point problem which follows from eq. (6.2) can be written as b(x, t) = Ty[b(x, i)]

(6.6)

with the nonlinear mapping Tr[b(x, 0 the Cauchy integral

(to + ¿V1 + r)2i C (x + ¿0) dx n J x - w (x + ¿y 1 _ T)2i

=

— 00

is majorized by s

v

+

.

^

g const. \F\t,

r

.

n

[ /

+ l o g !

v

+

¿

±

1

Iw + i ] / 1 + r\2i

i y

1/2

(C.3)

uniformly for Im w 1 and e < arg w 5S n — s, e > 0. Moreover, the derivatives of 4>(w) n!

»

/ 2d ] /

.

l

/

T

—

+

¿0)

dx

— CO

are bounded by

\™(w)\ ^const. ,„|i-| a

uniformly for I m w S 1 and 4

Fortschr. Phya. Bd. 31, Heft 6

Re w = 0

if

0

a

t

JU-^j Tj.

(C.4)

354

J. KUPSCH

The bound (C.3) gives immediately an estimate for F(t) = (y< — l ) |J(QI ^ const.,,, (1 - t)' if t > 0. The derivatives of F(t) are calculated as the sum F(«\t)

=

X

»=i

c-nA'-'Ait

-

1) • (t ~

(C.5)

l)"n+W2>

with some constants cB>, and (C.4) yields the majorization |F«(