Fortschritte der Physik / Progress of Physics: Volume 34, Number 1 [Reprint 2022 ed.] 9783112613740, 9783112613733

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FORTSCHRITTE DER Volume 34-1986 PHYSIK Number 1 PROGRESS OF PHYSICS Board of Editors

F. Kasehluhn A. Lösche R. Rompe

Editor-in-Chief F. Kasehluhn

Advisory Board

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

CONTENTS: T . I . KOPALEISHVILI

Quantum Field Theory Approach to the Pion-Two-Nucleon Interaction Problem

1 — 10

H . DORN

Renormalization of Path Ordered Phase Factors and Related Hadron Operators in Gauge Field Theories

AKADEMIE-VERLAG • BERLIN ISSN 0015-8208

Fortschr. Phys., Berlin 34 (1986) 1, 1 - 4 8

11—48

Instructions to Authors 1. Only papers not published and not 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 than 30 and preferably no more than about 100 pages in length. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuscript sheets should be numerated consecutively from " 1 " onwards. Footnotes should be avoided. 5. The titel of the paper should be followed by the authors name (with first name abbreviated), by the institution and its address from which the manuscript originates. 6. Figures and tables should be restricted to the minimum needed to clarify the text. They should be numbered consecutively and must be referred too in the text and on the margin. Figures and tables should be added to the manuscript on separate, consecutively numerated sheets. The tables should have a headline. Legends of figures should be submitted on a separate sheet. All figures should bear the author's name and number of figure overleaf. Photographs for half-tone reproduction should be in the form of highly glazed prints. Line drawings should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawings and photographs will be returned to the author upon publication of the paper. 7. Formulae should not be written to small and not with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary upright typeface. Underlining to denote special typefaces should be done in accordance with the following code: Italics: wavy underlined with pencil (only necessary for type written symbols in the text) Boldface italics (vectors): straight forward and wavy underlined Upright letters (all abbreviations like all units (cm, g , . . . ) , all elements and particles (H, He, . . . , n, p, ...), elementary mathematical functions like Re, Im, sin, cos, e x p , . . . ) : black underlined Greek letters: red underlined Boldface Greek letters: red interlined twice Upright Greek letters (symbols of elementary particles): red and black underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters that do not differ in shape, as cC,kK,oO, pP,sS,uU,vV,wW,xX,yY,z Z). I t will help the printer if position of subscripts and superscripts is marked with pencil in the following way: at, Mil, Mtj, Wnt Please differentiate between following symbols: a, a; a, + %r0rig°uvv]

(22 a)

taG0Ua.

+ ¿ 7 (sa, + t0G0)

tfau*

(22b)

where G0 and g°NN = g0 are given by the formulae (3), and g\d is defined analogously to g%N. The matrices tj (j = 1, 2) are again defined by the potentials Vf (5a) according to eq. (4a). As to the NN-scattering i-matrix in the presence of free pion (spectator) t3 for the state 3Sx + 3 D i it contains only the non-pole part of the total iVAT-scattering ¿-matrix. The transition potentials between two-particle states are defined by the expression wv

= MUnv)

+ rndgd°rdv

+ r,0(G0

+ G0t0G0)

(23)

r0,

where gd° is a propagator for a free deutron with mass md. The vertices rirj appearing in (22a, b) and (23) are defined by r0NN = L rSM) i r, On i

VNd

+ n f ,

rNm

= z

+

F0v, Fnd and

r®

(24 a)

+ -^idi)

(24b)

«'

1

+ rlSi>

^roio — FdNN

1

where r [ f ( j ) , r$(i), J 1 ^' and r^ are represented by the diagrams (7a, b); d3 is the pion propagator (3) and the vertices r{^Nd, r f ^ , j r f d > 0 are given by the diagrams 1r

NNd =—

0 >

• r"> ' dNN - =,

-scattering ¿-matrix tj constructed analogously to the previous approach; 2) the 2ViV-scattering ¿-matrix ¿3 for which in the state + 32)j only the non-pole part must be taken in contrast to the approach of ref. [4] where a total ¿-matrix is used; 3) the effective r/v-interaction potentials WNN,NN W„d,NN{WNNi7xd) and W„di„d defined by

wnv = m$(Vv) + rnigiriv

+ r,0G0r0,

(26)

and 4) the vertices

r0NN = z J?0>(7), rNm'= j; r$(i), r0„d = rfNid,-\r»d0 ) >

= r™^

(27)

where the vertices J 1 }^(j) and rfflli) are given by the diagrams (24a). As to the NNdvertex rNNd{rdNN) defined by diagram (24d), it is related to the state vector of bound state of the NN-system (deuteron) |yjd) by = g^i/lfd) where g NN is the none-oneparticle-pole part full Green operator of this system. From (26) one can see that, apart from the OBE potential, the effective ArAr-interaction potential WNNtNN = WNN (15) includes the deuteron pole term (as to how to eliminate the nonconnected term appearing in (26), this was discussed above). Unlike the approach of ref. [4], in the approach of ref. [5] there appeared additional effective transition potentials WNN,nd( Wnd,NN) and Wndnd which are defined by the amplitudes M™(NN, 7rd) (Jfg>(TCd, NN)),

nd).

(28)

However, the approach of ref. [5] has some advantage, namely, the system of integral equations (22a, b) or (25a, b) is written directly for the amplitudes of the reactions under study (la, b) and the amplitudes of the reactions red 7id and rcd ^ NN are two-body amplitudes, whereas in the approach of ref. [4] to determine such amplitudes one needs an additional integration of the solution of the eqs. (2a, b) or (14a, b) by deuteron wave function. It is worthwhile to note that the eqs. (2a, b) (as well as the eqs. (22a, b) explicitly have no crossing-symmetry although they are obtained in the framework of the quantum field theory. The cause of such a result is that only a s-channel cut has been used to derive the system of equations (2a, b), while u and ¿-channel cuts have not been considered. This is evident from the eq. (4a) for the rciV-scattering ¿-matrix. On the other hand, the importance of the allowance of crossing-symmetry for the 7tZV-scattering

Fortsohr. Phys. 34 (1986) 1

9

¿-matrix is well known from the Chew-Low theory [18]. As for the importance of allowing for this symmetry to the understanding of pion-nucleus dynamics, it was demonstrated in ref. [19], taking the 7t12C-scattering at low energies as an example. Thus there arose the problem of derivation of a system of equations in the framework of the quantum field theory for transition operators of all the processes under study (la, b) ensuring a crossing-symmetry for the amplitudes. The first attempt in this direction was made in refs. [20, 21] where a Low type equation was derived separately for the reactions 7rd —> NN and NN -> TZNN. In ref. [22] an attempt was made to derive a Low-type system of equations for all the reactions studied by using the LSZ reduction technique. I t turned out that in order to derive a closed set of equations it is necessary to consider the ¿-matrix for the transition 7VNN -> TZNN along with the transition ¿-matrix of the reactions (la, b). Owing to the complexity of this system of equations, the equations involved are not given here. It should be noted, however, for the given form-factors (vertices) this system of equations allows to obtain all the amplitudes which take into account not only the crossing of the initial and final pions (in appropriate channels) but the crossing of the other particles, taking part in the processes under study (la, b). Note that the system of Low-type equations, derived in ref. [22] for the coupled NNN and NN systems, are three-dimensional unlike the eqs. (2a, b; 4a, b) or (22a, b; 26a, b), which are four-dimensional and to reduce them to a set of threedimensional equations the use of a quasipotential procedure is additionally needed.

References [1] T. I. KOPALEISHVILI, SOV. J . Part. Nucl. 10 (1979) 167. [2] A . W . THOMAS a n d R . H . LANDAU, P h y s . R e p o r t s 5 8 (1980), 121.

[3] I . R . AFKAN a n d B. BLANKLADER, P h y s . R e v . C 22 (1980) 1638.

[4] Y. AVISHAI and T. MIZUTANI, Orsay preprint 1 PHO-TH 80-25, France, 1980 and Phys. Rev. C 2 7 (1983) 3 1 2 .

[5] A. I. MACHAVARIANI, Nucl. Phys. A 403 (1983) 480. [6] T . MIZTJTANI a n d D . S. KOLTUN. A n n . P h y s . ( N . Y . ) 1 0 9 (1977) 1.

[7] Y. AVISHAI and T. MIZUTANI, Nucl. Phys. A 326 (1979) 352; ibid. A 338 (1980) 377; c.ibid. A 352 (1981) 399. [8] M. STINGLE and A. T. STELBOVICS, J . Phys. 64 (1978) 1371,1389; Nucl. Phys. A 299 (1978) 391. [9] A. W . THOMAS a n d A. S. RINAT, P h y s . R e v . C 20 (1979) 216. [10] J . G. TAYLOR, N u o v o Cim. Suppl. 1 (1963) 857; P h y s . R e v . 150 (1966) 1321. [11] E . O . ALT, O. GRASSBERGER a n d W . SANDAS, N u c l . P h y s . 1 3 2 (1967) 1 6 7 ;

A. A. KHELASHVILI, J I N R Preprint P2-3371, Dubna, 1967. [12] A . A . LOGUNOV a n d A . N . TAVKHELIDZE, N u o v o C i m . 2 9 (1963) 3 8 0 ; R . B . BLANKENBERGER a n d R . SUGAR, P h y s . R e v . 142 (1966) 1051.

[13] S. OKUBO, Progr. Theor. Phys. 12 (1954) 603; H. FESHBACH. Ann. Phys. N.Y. 5 (1958) 357; ibid. 19 (1962) 287. [14] T. I. KOPALEISHVILI, "Some problems of the pion-nucleus interaction theory" ENERGOATOMIZDAT. Moscow, 1984. [15] T . MIZUTANI, C. FAYARD, G. H . LAMOT a n d S. NAHABETIAN, P h y s . R e v . C 2 4 (1981) 2 6 3 3 . [16] M . IDA, P h y s . R e v . B 1 3 5 (1964) 4 9 9 ; Y . S. J I N a n d S. W . MACDOWELL, i b i d . B 137 (1965) 68.

[17] K . HUANG a n d H . A. WELDON, P h y s . R e v . D 11 (1975) 257.

[18] G. E. CHEW and F. E. Low, Phys. Rev. 101 (1956) 1570. [19] J . B. CAMARATA and M. K. BANERJEE, Phys. Rev. Lett. 31 (1974) 610. [20] M. A. ALBERY, E . M. HENLEY, G . A . MILLER a n d J . F . WALTER. N u c l . P h y s . A 3 0 6 (1978)

447. [21] R. H. HACKMAN, Phys. Rev. C 19 (1979) 1873; ibid. C 25 (1982) 2602. [22] T. I. KOPALEISHVILI, A. I. MACHAVARIANI, Contributions. Advance Copy for participants of the 10th International Conference on Few Body Problems in Physics, Karlsruhe, Germany, August 21 — 27, 1983 (edit. B. Zeitnitz), p. 72 and to be published (Ann. Phys.).

10

T. I. KOPALEISHVILI, Quantum Field Theory Approach

Erratum W.

LTTCHA

Proton Decay in Grand Unified Theories Fortschr. Phys. 33 (1985) 10 On page 569, third line, read MW2 =

G2V/4

instead of ™w2 =

sr22^2/4

Fortschr. Phys. 34 (1986) 1, 11—56

Renormalization of Path Ordered Phase Factors and Related Hadron Operators in Gauge Field Theories H . DORN Sektion Physik der Humboldt-Universität zu Berlin, DDR

Abstract We give a review of the renormalization and short distance properties of path ordered phase factors in nonabelian gauge field theories. I t includes nonlocal gauge invariant meson, baryon and gluonium operators constructed with the help of such phase factors. Furthermore, the renormalization properties of functional derivatives of phase factors as they are needed in dynamical equations are considered. The discussion is based on an one dimensional auxiliary field formalism which enables the application of the usual language of local Green's functions.

Contents 1.

Introduction

12

2. 2.1. 2.2. 2.3. 2.4.

Path dependent phase factors in classical gauge field theory Conventions and elementary properties Gauge invariant coupling of phase factors Wilson functional and constraints on it Functional derivatives of path ordered phase factors

13 13 15 15 16

3.

3.4. 3.5. 3.6. 3.7.

Path ordered phase factors in quantum field theory and their perturbative renormalization . The problem Auxiliary z-field formalism General discussion of the renormalization of path ordered phase factors for smooth simple curves Modifications for closed, piecewise smooth, nonsimple curves Renormalization factors in 1-loop approximation Renormalized Wilson functionals in 1-loop approximation Consequences of the renormalization group

21 24 27 31 33

4. 4.1. 4.2. 4.3. 4.4. 4.5.

Gauge invariant nonlocal hadron operators The problem Renormalization of nonlocal meson and baryon operators Renormalization of the operators gzF^z and gF^,z a Renormalization of nonlocal gluonium operators Short distance behaviour of nonlocal hadron operators

37 37 38 40 42 43

5. 5.1. 5.2. 5.3.

Functional derivatives of renormalized path ordered phase factors First functional derivative Second functional derivative Dynamical equations for phase factors

46 46 48 50

6.

Conclusions

52

3.1. 3.2. 3.3.

. .

18 18 20

12

H. DORN, Renormalization of Path Ordered Phase Factors

].

Introduction

Nowadays gauge field theories are the most promising candidates for a theory of fundamental interactions. This is due to a lot of nice features. On one side, they yield the fascinating possibility of unification of all interactions, reaching from the experimental established unification of weak and electromagnetic phenomena in the Glashow-Weinberg-Salam theory up to unified theories including gravitation. On the other side, only gauge field theories can be asymptotic free thus explaining the success of the quark parton picture. Finally, the present status of lattice gauge theory raises hopes in sensible tests of nonperturbative effects in the near future. The gauge fields A ^ x ) interact among themselves and with matter fields according to the principle of local gauge invariance. Only gauge invariant quantities are observable. From the simplest case of electromagnetism we know the field strength tensor F^ = — SyA^ as a local gauge invariant field. B u t due to the A H A R O N O V - B O H M effect [1] the physical situation cannot be described by means of the local F ^ completely. One has to add the nonlocal gauge invariant quantities exp ig (j) A^ dx*. First investigations of these phase factors and attempts to formulate the theory in terms of them a r e d u e t o MANDELSTAM [2, 3].

Concerning their mathematical structure gauge field theories are fibre bundles with space-time as basis, compare e.g. [4]. The path ordered phase factor corresponds to the generalized parallel transport along the curve under consideration. Therefore, it is a fundamental geometrical quantity. This aspect is visible also in the lattice version [5]. The gauge field is described by the socalled link variables corresponding to phase factors connecting neighbouring lattice points. The up to now not successful search for quarks has found its theoretical description in the confinement hypothesis. Due to characteristic features of the gauge field ground state (vacuum) the field lines between two static quarks q, q are compelled to a flux tube connecting both particles. Thus a linear attracting potential is produced. A separation of both partners over macroscopic distances is possible only via the spontaneous creation of a new q, q pair. The path ordered phase factor yields an order parameter for confinement. D u e to Wilson's criterion confinement is realized if the functional ( t r P exp (ig (j) AM d x ' ' ^ for large enclosed area F(G) behaves like exp ( — k F ( C ) ) , k string tension [5]. Hence the phase factors play an essential role in the mathematical formulation of the confinement problem, too. Independently, and before the development of the above picture, so called string theories as relativistic field theories of one dimensional extended objects have been constructed. They emerged in connection with the Dual models in Regge phenomenology [6]. If one searches for a relation between nonabelian gauge theories and string theories the flux tube and the path ordered phase factor connected with it is the natural partner for the string. The papers of N A M B U [ 7 ] , P O L Y A K O V [ 8 ] , G E R V A I S and N E V E U [ 9 ] as well as M A K E E N K O and M I G D A L [ 1 0 ] led to a renewed and intensive discussion of this topic and of attempts to formulate gauge field theories in terms of path ordered phase factors [11, 12, 13]. Concerning a general solution of the dynamical equations for the phase factors no final break-through has been reached till now. In connection with the multicolour limit the loop space formalism of M A K E E N K O and M I G D A L [ 1 0 , 1 1 , 14] is the most developed one in what concerns both a technique of iterative solution as well as a connection to string theories with additional fermionic degrees of freedom [15].

The dynamical equations just mentioned are mainly devoted to the study of nonperturbative effects like confinement, condensates etc. However, as a first step one

13

Fortschr. Phys. 84 (1986) 1

has to get complete control over their perturbative part and over their renormalization. Due to the fundamental role of path ordered phase factors their renormalization is of interest also independent of the success of present attempts to use them as basic variables. The study of renormalization of the nonlocal object P exp (ig J AM da/) has been initiated by POLYAKOV [ 1 2 ] and GERVAIS, N E V B U [ 1 6 ] and continued by several authors. For the phase factors itself, their first functional derivative and for nonlocal hadron operators constructed with the help of phase factors the problem is completely solved. Questions remain with respect to the renormalization of the dynamical equations involving second order functional derivatives. The present paper is based on a series of studies [17, 29, 37, 45, 55, 58] devoted to perturbative renormalization of path ordered phase factors and related objects. In this context special attention is directed to dynamical equations for phase factors and to the investigation of nonlocal gauge invariant composite operators. This aspect distinguishes our review from other ones covering parts of the same subject [28, 40, 13, 72], Section 2 fixes notation and summarizes some relevant properties of classical phase factors. Section 3 is devoted to a systematic discussion of renormalization of phase factors and their peculiarities for non-smooth and non-simple curves in the framework of the auxiliary z-field formalism introduced in this context by GERVAIS, N E V E U [ 1 6 ] and AREEYEVA [ 1 3 , 1 8 ] . Section 4 studies renormalization and short distance properties of nonlocal gauge invariant hadron operators constructed with the help of phase factors. Section 5 applies to functional derivatives and dynamical equations. After the summarizing section 6 some appendices yield technical details.

2.

Path Dependent Phase Factors in Classical Gauge Field Theory

2.1.

Conventions and elementary properties

In view of applications to QCD we choose a theory of interacting quarks and gluons with gauge groups $i7(iV)-colour. Greek indices are Lorentz, latin indices from the first part of the alphabet are adjoint colour and from the second part fundamental colour indices. Ta are the generators of the SU(N) Lie algebra in fundamental representation: [Ta,Tb]

= ifabcTc,

tr (TaTb) = TÒat„

tr T a = 0 .

(2.1)

Gauge field, field strength tensor and covariant derivative in fundamental representation are A„ = A/Ta,

Dp —

t\v = 8MAv -

— igAp

or

-

ig[AM,

A,],

D^ = d^ — ig[Ap, •],

respectively.

(2.2)

Under a gauge transformation Q(x) we have f(x)

Q{x) f(x)

A # > - A N F^x)

for the quark field,

W

Q(x) FM,(X)

W

+ Q-\X).

I

O

*

™

.

(2.3)

14

H. Dokn, Renormalization of Path Ordered Phase Factors

Besides the fundamental representation of the Lie algebra valued gauge field AM we use the adjoint representation, too: A^* = AM«ea,

(ea)bc = ifbac.

(2.4)

Then tr

= Y

TbQ-\x))

(2.5)

and

FfJ -> Q^FfJ(Q^)-1 F%. ->

(2.6)

F%.

In the geometrical interpretation A^ are the coefficients of the Lie algebra valued connection form AM dx1*. Integrating along a given path in space-time (basis) we get the generalized parallel transport, in physical language the path ordered phase factor. These factors allow the-gauge invariant coupling of fields located at different points in space-time. For a piecewise smooth curve C = {x(a) | 0 ^ Q{x2) U(X2, Xu C) Q-1^).

(2.9)

Eqs. (2.7) to (2.9) are valid for phase factors in the adjoint representation in an analogeous way. Comparing the defining equations (2.7) one gets UlV(x2, xu G) = 1 tr (TaU(x2, xu G) TbU(x

1;

x„ (7"1)).

(2.10)

Of particular interest are phase factors for closed contours G, U(x, x, C); x 6 C. They depend on the reference point x and transform as D{x, x, G) -> Q(x) U{x, x, G) Q~\x).

(2.11)

For the decomposition of a contour G = C2 • Cj into two parts Glt C2 we have (xx 6 Gu x2 € C2; x0 junction point) U(x2, xit C2 • Ox) = U(x2, x0, C2) U(x0, xlt C,).

(2.12)

15

Fortsehr. Phys. 34 (1986) 1

For disjoint Gx, C2 we use (2.12) as a definition. An important special case is U(x2, xlt C) = U~1(x1, x2, C" 1 ).

(2.13)

Finally, since AM+ = AM V is unitary and in the adjoint case orthogonal (2.14)

V^{x2, xu G) = U+(x2, xu G). 2.2.

Gauge invariant couplings of phase factors

Besides the trivial gauge invariant coupling of two phase factors (compare (2.12)) and of a phase factor with a local field as e.g. U(y, x, C) ip(x) also phase factors corresponding to different curves terminating in a common point can be coupled among themselves as well as to local fields in a gauge invariant manner. Later on we use such couplings to construct gauge invariant nonlocal hadron operators. Because of £mi-"mji^mi«i "'

=

®

B

£

SU(N)^j

the coupling (2.15)

CN) is gauge invariant. From {Ta)mn {Ta)op

=

T

jy ^mn^Opj

(2-16)

and (2.5) we get (2.17)

{Q-^TaQ)mnQffi={Tb)mn. Hence we find a further gauge invariant coupling Umim(x 1, X, Ox) ('.Ta)mn U,.„(X, a;,, C2) Vfjj,x, x* 0 . ) .

(2.18)

Starting from (2.17) and using appropriate combinations of 3mn and traces one can couple also a larger number of phase factors in adjoint and fundamental representation. This includes e.g. the coupling of three adjoint factors by means of fabc= —i/Ttr[Ta,Tb]Tc). Finally, (2.15), (2.18) and the following remark remain valid if the end of a phase factor is replaced by a local field with the corresponding colour transformation property. T™n,

2.3.

Wilson functional W(C)

and constraints on

W(C)

For applications in connection with W I L S O N ' S confinement criterion [ 5 ] and the loop space formulation of QCD [10, 11] one needs the trace of phase factors for closed curves W(G) = trU(x,x,G).

(2.19)

This is a gauge invariant quantity. B y taking the trace the dependence on the reference point x drops out. The values of W(G) for different contours are not independent from one another. They fulfill the so called Mandelstam constraints [3]. Since W{G) are traces of NxN matrices (in the adjoint case N iVa — 1) one finds for arbitrary

16

H. DORIC, Renormalization of P a t h Ordered Phase Factors

contours [19] • C%) = W(C2 • CJ 0 =

Z

s

"CSIV+I

(2.20)

§ n ^ • ^»(Ci. • • •>

(2-21)

where Wx(Clt ..., = JF(W(C • C) W{C) + 2W{C • C • C),

N = 3

etc.

From (2.10), (2.16), (2.22) we get a relation between W and W'idj W^(C)

= \W{C)\2 - 1.

(2.24)

Already, this algebraic constraint indicates the different behaviour of matter particles in the fundamental or adjoint colour representation with respect to the confinement property [70].

2'4.

Functional derivatives of path ordered phase factors

The attempt to describe the dynamics of gauge fields by path dependent phase factors necessarily involves the study of U(x2, xlt C) in the course of variation of the contour G = {x{&) | 0 a ^ 1}. We understand the, functional derivative dF/dx(a) of a general functional F f a c ) } in a weak sense according to the following construction [21]. The first variation dF(x(a) + ey(

1

eXp

/

x2/2e) 1 1

[11].

Finally, in view of applications to phase factors a superregularization has been construc2*

20

H. DORN, Renormalization of Path Ordered Phase Factors

ted, which yields finite results in removing the regularization 1/x2 —> l/x2 • l/2[(c/|x|)"+i£ + (c/|z|)^],/?-->0, e ->0[26]. In the following we use for technical simplicity dimensional regularization. We get in this way enough information on anomal dimensions to study via the renormalization group the small distance behaviour of renormalized phase factors and related hadron operators. The renormalization problems for phase factors are solved [12, 16, 18, 27—30]. We present the results in the language of an auxiliary field formalism [31, 32, 16, 18]. In this formalism the nonlocal object U(x2, xlt C) is reduced to a local object in a modified theory. 3.2.

Auxiliary z-field formalism

Writing the path ordering in eq. (3.1) with the help of the step function 8(a) one gets {ig)k £ J PiA^xJ k c

... AJxk))

dx*> ••• dap

= (ig)k f'e(ak - fft_0 - 8(a2 - ffj) Tak ... TaiA%{x(ak)) o - AfSxfa)) x»*(ak) ••• iax ••• dak.

(3.6)

This can be described in the following language: A field z(a) is defined in one dimensional parameter space interacting by means of the classical field a^(cr) with the four dimensional gluon field A^x). The building blocks of this effective theory are z-propagator:

dmnd(a2 — oj,

dab8(a2 — a^

2 . - A vertex: igx^a) (Ta)mu,

^ ^

gU^^o)

in the fundamental or adjoint representation. These graphical rules correspond to the action [31, 16, 18] + iJ Z — iga>"A^ z da + iz(0) z(0) (3.8) o describing the gauge invariant minimal coupling of z and A^. The action (3.8) is invariant with respect to changes of the curve parameter, too. Now there is an obvious operator identity U(x2, xu G) = (z(a2) z(aj)z (3.9) s

=

SQCD

with {•••% denoting the expectation value with respect to the z-field only. This reduces the nonlocal (with respect to its gluon content) phase factor to an usual two point function. The endpoint term izz(0) in (3.8) is a consequence of the finite er-interval. Due to our choice c = 0 in (d/d —gA'(x) x(a)

x(ae

(transposed)

— a)

(3.11)

z(a)

—> z(ae — a),

z(a) —> z(ae — a) if z Bose

z(a)

—> iz(ae

z(a) —> iz(ae

— a),

— a) if z F e r m i ,

if quarks are present, they have to undergo a charge conjugation. As far as renormalization is concerned,' the problem of restricting the gluon propagator to an one dimensional contour now appears as the problem of defining the product of the z-propagator and the gluon propagator. The restriction of the gluon propagator itself is relevant for diagrams with more than one through-going z-lin^, only [16]. Our aim is the handling of path ordered phase factors, hence we can confine ourselves to the sector with one z-line. Finally, the formalism can be extended to the case of more than one contour. To each contour C{i> one introduces a z{i\ The different z(i) have no correlation among themselves and each by its own couples via x {i \a) to the gauge field. 3.3.

General discussion of the renormalization of path ordered phase factors for smooth simple contours

In this section we review the renormalization of the z — A system [16] based on the use of the BRS transformation [34] and functional techniques in full analogy to the elegant discussion of renormalization of the usual quark-gluon system [35]. The formalism we use later on also in the context of composite operators built out of z, A, tp. All manipulations require a gauge invariant regularization. Our field theory is invariant with respect to the BRS transformation (c, c FaddeevPopov-ghosts, e infinitesimal Grassmann variable, quarks dropped to shorten the discussion) ÛApa — Dn,abcb ' ÔCa=