Q co *5% 0 E
CERN LIBRHRIES. GENE\/Fl ` DPTG931 \l\\\\l\\\\lll\\\lll\\\l\\\l\\\\\\\\\\\\\ l November,1993 PBQBEBBEA
BRST Formalism of the Quantum Field Theory of Charges and Monopoles
SHINJI MAEDAN
Department of Physics, Tokyo National College of Technology 1220-2 Kunugida-machi, Hachioji-shi, Tokyo 193, Japan
ABSTRACT
A local covariant operator formulation of the relativistic quantum iield theory of electric and magnetic charges is given. It is verified that the theory is unitary by the quartet mechanism which does not depend on perturbative calculations. Using the Maxwell equations, the renormalization constant of electric ( magnetic ) charge e ( g ) is obtained. The contribution of the string dependent terms in full gauge boson propagators to the renormalization constants is discussed. OCR Output 1. Introduction
The relativistic quantum field theory of electric and magnetic charges ( quan tum electro-magnetodynamics, QEMD ) has been studied by many authors m [2] Zwanziger presented a local lagrangian density including electrically and magneti cally charged Helds. ‘°’ That lagrangian density depends on a constant unit vector np , and Lorentz invariance of the theory is not so apparent. Using a functional rep resentation of Zwanziger’s formulation, Brandt, Neri and Zwanziger proved that the theory is Lorentz invariant when the charge quantization condition eg/41r : integer is satisfied. "' Because of the exsistence of the constant vector nl, and the charge quantization condition, the perturbative method will not be reliable. This makes it diflicult to prove the unitarity of the theory and to renormalize the coupling constants.
In this paper, without using the perturbative method, the proof of unitarity of the theory is given, and the renormalization constants of electric charge e and magnetic charge g are calculated based on the local covariant operator formalism. The quantum theoretical lagrangian with the gauge fixing term and the FP ghost term has the BRST symmetry. The physical states lphys > are defined according to the Kugo-Ojima condition
Qglphys > = 0, (1.1) where Q B is the generator of the BRST transformation. This subsidiary condi tion plays an essential role in proving the unitarity of the physical S-matrix‘°' and calculating the renormalization constants.
In Sec.2 , starting with Zwanziger’s formulation of the classical field theory of charges and monopoles, we give a quantum theoretical lagrangian with covariant gauge fixing and set canonical commutation relations. Besides the vector potential AH , Zwanziger introduced a dual vector potential Up . The classical lagrangian then has U (1),;;,. >< U (1),,,,,9 symmetry. The BRST charge and the ghost charge satisfy ordinary BRST algebra.
.. 2 _. OCR Output In Sec.3 , the unitarity of the physical S-matrix of the theory is proved on the assumption that the equations of motion and the canonical commutation relations of asymptotic fields are the same with those of free fields. In general, there exist some unphysical particles when the covariant gauge fixing is adopted. These par ticles have the possibility of breaking down the unitarity, but they decouple from the physical sector (1.1) by the quartet mechanism.
In Sec.4 , we calculate the renormalization constant of electric ( magnetic ) charge e ( g Panagiotakopoulos expressd the full gauge boson propagators by use of scalar functions C(p2) and D(p2) that don’t depend on n,,.m From these propagators two-point Green functions such as < 0|TF,,,,FpG|0 > are obtained. To see the relation between bare coupling constant and on-shell renormalized coupling constant, we make use of the Maxwell equations, which was first considered by Aoki ‘°' in the Weinberg—Salam model. The renormalization constants are written by the above scalar functions C(p2) and D(p2).
In Sec.5, some concluding remarks are presented.
2. BRST quantization
The classical local lagrangian of QEMD, elaborated by Zwanziger, is
Lz ; + L‘|TLCltiC7` 7
L-, 2 —@l¤· (6/\A)l · In- (6/\U)"l
+ (3/\U)] · (3/\A)"]
(2.2) OCR Output 27]/2 . Z - L . Z [rz (3 271/\ A)] 2 [n (3 A U ,
Cmatter Z l/J1 " ml `
.. 3 + (/@(16 · mz · 9(UW2
E 1/¤1(i16 · m1)¢1 —¤Jé’Ap
+ A/J2(i$6 · m2)¢2 · 9j5Up » (23) where U}, is a dual vector potential, nl, is an arbitrary but fixed four vector, and zpl ( rpg ) is an electrically (magnetically) charged spin é matter field with electric (magnetic) charge c ( g We have used Zwanziger’s notation : (8 /\ A),,,, 8,,,4,, — 8,,A,, ,(3 A A)Z,, : (1/2)e,,,,,,,,(6 A A)·°" and [n- (6 A A)],, = n”(8 A A),,,, The field strength Fw, and the dual field strength Fg, can be expressed locally in terms of the potentials alone
Fw Z (1/¤”)({¤ A l¤· (6 A A)l},»— {H A [¤· (6 A U)l}Z.))» (2-4)
FZ.) = (1/¤2)({¤ A ln- (6 A A)l}}l., + {H A l¤· (6 A U)l};») - (2-5)
The lagrangian (2.1) is invariant under the electric U(1),,le gauge transformation
An _’ An + 6pxe » $1 —* €Xp(_iCXe)¢1 » and also under the magnetic U (1),,wg gauge transformation
Up ·* Up + 6pXg »¤/¤2 —>€> To quantize the theory, we add a gauge fixing term LGF and a ghost term Lpp to (2.1) C I Eq + Cmmer + LGF + LFP , (2-6) OCR Output 47 = {§a.:1Z—a~i¤— w A w.>1-i~- w wbw>d1 - 4 OCR OutputTo set the canonical commutation relations, we will calculate the canonical moment H for each canonical variable ; (16,,, ,l1’12 ) ,(B,, ,11Ba ) ,(c,, ,Hc,_ ) , (Ea ,115,, The commutation relations of the matter fields $1 ,1pg are ordinary ones. Bisides ordinary potential A”(E 1/;**) , we have introduced the dual poten tial U ,,(E lé") which is not really independent variable, to express the lagrangian locally. The lagrangian (2.6) is then singular and we will use the Dirac method. Hereafter we shall choose the constant vector n" : (0,0,0,1). There are eight second class constraints qi, : -11** —n v#:" -11-3/\V¢1— M1 ¢2" B1+] 1 ¢3_1 16 323.·)( 2) 1 I ¢4-Ht 1— ; ¢7 - Ht +1 gw %14), ¢8 = HQ + gw/~1 mm Defining the matrix according to Dirac ; Cap E {cbc, ,¢>B} , we get non-zero elements with cv < B m] C12 =-6(><—y>. G’2.s=(n·6)6(><—y). Cu = —(¤· @)6(><— y) , 056 = —6(><— y) The Dirac brackets containing only canonical variables ( not time derivative ) be come 1G":1(>< J ). %":2(y J ) ln = (M-6)‘1(¤<—y), 1G":”(> {1£.°(>< J ). Bz»(y J )}D = 6ab6(><— y). and others are zero. From these Dirac brackets, the canonical commutation rela - 6 _ OCR Output tions are [3] 1/]”=1(> V{‘:“(>< J )» 1/§":‘(y J )\ = —¢ (H - 6)"(><— y) » (2-15) VQ{‘:°’(x ,t ), V},"="(y ,t )| : i6ab6(x— y) , (2.16) [V.f’(>< J )» Bb(y J )] = i 6¤.z»6(><—y) (2-17) The BRST charge QB is obtained from (2.8) and (2.9) BL i GL Q B 2 /3[dx(B8c—3Bc)—|—8‘———;—c—i¥ gt; a Da 0 aa K 1 Since the ghost fields ca are free fields, the last term would be neglected : 0.:1 The physical states lphys > are defined according to the Kugo-Ojima condition QBlphyS > : 0. (2.18) QB and the ghost charge Qc satisfy ordinary BRST algebra. The BRST transfor mation (2.8) and the physical states definition (2.18) lead to the following Ward Takahashi identity < 0|TBa(:z:)B;,(y)|0 > = < 0|{QB,T(Ba(x)6b(y))}|0 > = 0. (2.19) The Green function of 14,** and Bb can be obtained utilizing Eqs.(2.10), (2.17) and (2.19) FT- < 0lT(%"(=v)Bb(y))|0 > - 7 _ OCR Output E I d4¤¤¤*P(*·y’<¤nT<¤<.#<¤>>Bb Z -6ab 5-,; , (2.20) where F.T. represents the Fourie transformation. Multiplying (2.20) by ip" , we et [51 pupy - - a =v)% (y)I0 >——¤6az» FT <0|T*6"B( V - · (2-21) 3. Unitarity of the theory R.A.Brandt and F.Neri have argued perturbative unitarity of the QEMD. [u] They verified that in trivial order of perturbation theory the Feynman diagram leads to unitarity. In higher order , although , we are faced with the renormalization program which is very complicated due to the presence of the constant vector np , and they only suggested that the cross sections will be unitary in each order. Here we show that the physical S-matrix of the QEMD is unitary using the quartet mechanism. l°' Our only assumption is that the equations of motion and the canonical commutation relations of asymptotic fields are the same with those of free fields ( e : 0 : g). To verify the unitarity of the physical S—matrix in the covariant operator formalism , we need to prove that arbitrary physical states phys > (2.18) have positive semi-definite metric < phys | phys > 2 0 . (3.1) The free equations of motion of asymptotic field l/LBS M are <2’*Bi’” 2 (1/nllz2 (M ·<2)¥G“ " — (n ·<2)@"(n · WS ) —¤’*(¤· @(6 · WL`) + n’*|](n · WS ) — (n · 8)e”W/\n"8"V§‘S " ) , (3.2) - 8 ... OCR Output + n"l](n · KGS ) + (n · 8)e"WAn”8°l/Qas (3.3) Imposing (3.3) the boundary conditions to satisfy the following equation (1/¤2)(¤· GV? ”> I (1/¤2>{ @"(¤· Vé”> +¤"<6 · %°‘“’ ) —¤"<¤·6)”1¤(¤·%“> d*W,n"a¤iq=·S A ) + (n - 0)·*0#B;·‘* , (3.4) and substituting this to (3.2) , we get EWS ”— 8’*(6- 1Q°·") — 8**BfS = 0 (3.5) Choose the Feynman gauge oz,. : 1 , and we find mv? ” : 0 . (3.6) Similarly , for V;"" ", |j1IQ"’ " = 0 . The second class constraints 453, ¢4, 457 and 458 show that eight variables 1/T8 "(;4 = 1,2,3,4) and 1/;*8 "(;r : 1,2, 3,4) are not really independent variables. Two of them are written by the other six variables. Thus we can express the com mutation relations (2.14) and (2.15) only by l/TS1 e" :and*:2 w*with the help of (3.3). Differentiating ,u = 1 component of (3.3) by xg , ,u = 2 component of (3.3) by xl , and subtracting 33313%/;*8 #:2 : 33 {3233v;·‘* #:1 + 32(3 A v;¤)°2 4 31(3 A w~**)°1 } . (3.8) From (2.14) and (3.8) , i6(>< - y) : IWS ":l(>< J ). 63168 “:2(y J) - 9 OCR Output mr Wx .1 >, / 0%#0* .1:21 )dy +I1G”""‘:1(><,¢)»1£”(y,#)”":1 (3-9) Here we demand thats ifFand1 S {G*commute*:2 , then jr #:1, me ¤;1| Z mx - y) (3.10) Also, from (3.8) and (3.10) , we obtain F2, WS ”:2I = ¢6(><— y) 1 (3-11) where we demand the commutability of {QE2 **:and as ' w". :1 We finally obtain the commutation relations written by was " and it’s time derivative lQ°3"(x,i),Vf’S"(y,t)| :——ig’“’6(x—y), (3.12) where (2.10) and (2.17) (in Feynman gauge oz, : 1 ) are used. If we want to express the commutation relation in terms of 1/;** " , it also takes the similar form 1/;** "(x ,t) ,1};*8 "(y ,t )| = —ig’“’6(x — y) Since Kas " obeys the equation (3.6) or (3.7), and satisfies the commutation relation (3.12) or (3.13) , 1/},88 ” can be expanded as a Fourie integral and has four modes , i.e ; scalar mode a“(k, S) , longitudinal mode a“(k, L) and transverse modes a“(k, zh) , which have the metric a“(k, U) 1 ¤»°(q, T)'l = ¢7‘"6“°6(k— q) 1 +{ 1 0 0 0 :,0*1* "`I 0 1 0 0 L I 0 0 0 1 S \ 0 0 1 0 Because the ghost fields Ca and Ea are decoupled , the BRST transformations of - OCR Output asymptotic fields become 6BlQaS"=6”C2S, 6BC;S;0, 6BE;‘S = iB§”, 6BB§‘°’ : 0, 6B$2S : 0 , (a :1,2) . (3.15) These BRST transformations show that ( a“(k, L) , a“(k, S) , c“(k) , E“(k) ) (a. 1, 2 ) form the BRST quartet which are never observed by the quartet mechanism and that aa (k, :|:) and $f’;" are the BRST singlets. *°’ It looks as if two types of the transverse "photons" al (k, zi:) and a2 (k, :1:) were observed. But, as mentioned above, two degrees of freedom ( we shall choose a2 (k, zi;) ) are not really indepen dent variables. So, observable particles are the transverse photon a1(k, ;|:) , elec tron ( positron ) $1 and monopole ( antimonopole ) $2 , which all have the diagonal positive metric. We can thus show (3.1) and establish the unitarity of the theory without using the perturbative method. Free gauge boson propagators are obtained from the lagrangian (2.6) Gf,‘Z(p) E F·T- {< 0|Tl£."($)l6,"(y)|0 >};.... l6,, { —’“’ 1- ——;—,,p”p" "”"” 1 P2 b9(”‘P) ll OO€b€ nppa i — P2 3.16 ( ) Note that the commutation relations (3.12) and (3.13) are consistent with these propagators. - .. OCR Output To begin with , let’s find the form of full gauge boson propagators. Brandt et al. have proved that the full Green functions of the gauge-invariant operators , such as electric current Jp or magnetic current K ,, , are np-independent when the charge quantization condition cg/47r = integer is satisfied. W Then full Green functions < 0|T(J,,J,,)|0 > , < 0|T(K,,K,,)|0 > and < 0|T(J,)K,,)lO > don’t depend on np and have the f<>rm i(.<]puP2 — zap./)6"(z¤2) , i(gpvP2 — 1»,i1>»)D(1>2) and i(g),,,p2 - p,,p,,)F(p2) , respectively. C(p2), D(p2) and F(p2) are np-independent scalar functions. The full V] — V1 propagator becomes F.T. < 0|TVi”(a:)V]"(y)|0 > iff?) + Gif (P){i(9¤p1>2 — 1>¤ps)O (1¤)}Gl{(P)2'j’ + GTS (p){¢(y¤pz>2 — p¤pp)D(1>)}G€f2 (1)) + Gi'? (1>){i(y¤ra1>2 — p¤1>p)F(p)}G§f(p)23 + Gi'S'(P){i(yan1>2 — P¤PB)F (1¤2)}Gii’3 (P) 1 gpu { ¢— 1 —l— 2C —l-2 D —————-D,nl2p2 2 } 1 n"*n" 2 1 (p”n" +n”p") D(p2) + T ;TT-D T T ·TTTT——T——‘T i (M-P)2 (P) 1 (Trp) pz 1 {22p”p” } ————— (1 + -00—————;D C(1>> , 4.1) p2 M2 pz p2 F.T. < 0|Tl@"(x)V§’(y)|0 > 21 QW2 T{ T·— 1 + C + D ——-——-—C”2P2 2 } OCR Output - 1 n"n" 1 (p*‘n" -1- n"p") C'(pz) 2 + T *TG T T *‘TT—TTT p (p·p)z (P) p (pp) pz p”p" {(1- 4) D(pz) pz 2 } —— ;—— + —————————G' , 4.2 pz pz pz (p · p)z (P ) ( ) F.T. < 0|T%(x)”V2(y)"|0 >: e12 e*“""”n,,p.,m(1+C(p)+D(p))22 . (4.3) From (2.4), (4.1) and (4.3), it follows that F.T. < 0|T*6,,F’”’(x) V]'°(y)|0 > F.T. < 0|T*3.,{n/\ [n- (0 A l4)]}”"(x) V]’°(y)l0 > F-T < 0|T*@»{p» A In- (6 A %)l}d”"(w) W°(y)|0 > i (1+ C(p2))(pz gwz (4,4) Therefore the zero—mass pole (pz : 0) part of F.T. < O|T*3,,F*“’(:1:) Vip > is i (1 + C'(0))p”p"/pz . Comparing this fact with (2.21), we see that 6,,F** contains zero-mass part —(1 + C'(0))0”B1 . We repeat the similar calculation for F.T. < 0|T*6,,Fd*"’(:v) l@°(y)|0 > with the help of (2.5), (4.2) and (4.3) FT < 0|T*6 Fd”"( ’°—‘ 2 P. . ,, x) V5 (y)|0 >- -—z (1+D(p )) g" pup;)? — (). 4.5 It also results that 3,,Fd·‘“’ contains the zero—mass part —(1 —|— D(0))3"B2 . Now we use the Maxwell equation (2.12) in order to calculate the renormal ization constant of electric charge e .l°' There must exists the well-defined U (1),11e charge Q1 because U(1)e1e symmetry is not broken spontaneously. 8,,F’“’ containes zero-mass part —(1 + G'(0))6"B1 , and the U (1),11,., current jf in (2.12) thereby con tains zero—mass part e`*{(1 + C(O)) - 1}8*‘B1 . In order to have the well-defined charge Q1 , we set Q1 .-: / dgx (jfzo —w18kF°k) — 13 OCR Output E { 4% 3;%*: (4.6) where the constant wl is determined from the condition that contains no discrete massless spectrum"' ; 1 C(0) = ——-—— wl 4.7 ( ) . e (1 + C’(0)) In terms of , the Maxwell equation (2.12) becomes (1 + ew1)0,,F’“’ = —e§f — 6‘”B1 (4.8) Consider the matrix element (1 + wi) <<=(1>;)|@·»F""(=v)|€(1>¢) > ¢<¢(1>;)|5{‘(¤¤)|s(v4) >—<¤(v;)| B <<¤(1»;)|5i’ (¤>)|¤(1>i) > , (4-9) where |e(pi) > and |e(pf) > are the physical states of electron with momentum pi and pf , respectively , and these states are normalized as follows <€(1>;)I¢(;v¢) >= (2¢r)"(1>4 +p,·)o 6“(p1 — pi) (4-10) The term < e(pf)|6,,B1|e(p1) > in (4.9) vanishes due to the relation B1 = {QB, 61} and the definition of the physical states (2.18). We shall operate limq.,() f d4:vei to both sides of (4.9) where q = pf —— pi (1 + ew1)rd4me*"’”<}n>5/ e(pj)|6,,F’“’(:1;)|e(p;) > ard4w·¤=¤ In the limit ql, ——+ 0 , the contribution to the matrix element < e(pf)|3,,F’“’ |e(p4) > is dominated by the process where the zero-mass particle i/las') is contained , as the - OCR Output OCR Output(4.15) The field 1/Twp is a vector and the equation of motion (2.12) is invariant under the time reversal, so the vertex < e(pf)|lQaSp(x)|e(pi) > has the form —eR(pf+p,)" in the limit pj ——> pi. The normalized coupling constant eg is defined as on-shell coupling one. Using (4.12), (4.15) and the above eg , the p = 0 component of the 1.h.s of (4.11) becomes p_lm(1+ @w1)(2vr)46(pf — pi)(/1+ C(0) + D(0)(—eR) - 2(p,)O . (4.16) On the other hand, the ii : 0 component of (4.11) is _/¢ dw¤e‘q°”‘°<¤(p;)|Q1|¢(p¢) > (4-17) Z · if;p €(27F)46(P; — Pi) · 2(1>¢)¤» where the invariant normalization (4.10) has been used. From (4.16) and (4.17) , we finally get C].? I 1 + G(0) + D(0) with the help of (4.7). The renormalization problem of magnetic charge g can also be solved as in the case of electric charge renormalization, because the U (1),,,,,9 synnnetry is not broken spontaneously. The Maxwell equation (2.13) can be rewritten (1 + gw2)a,F*‘#" : -9}; - 0,,122 , (4.19) where well-defined U (1),,,,,9 charge Q2 is given by Q2 5 / d3:z: 5;:0 E / d3x(j§’=0 —w26kFd Ok) , and 1 D(0) Z *"' "‘‘”2 4.2 ( 0) OCR Output g 1+ D(o) We now consider the matrix element (1 + ywz) < m(p;)|6~F“””(=¤=>|m — 16 (4-21) where |m(p;) > and |m(pf) > are the normalized , as (4.10), physical states of monopoles. The same operation limq.,g f d4;reiq“‘ to both sides of (4.21) yields (1 + gwg)$3%/ 1d4xeiq“°< m(pf)]0,,Fd”"(x)\m(pi) > g %)/1111d‘*:re*q"’< q—} m(pf)|5§(x)|m(pi) > (4.22) In the limit qi, —> 0 , ¢“’“< m(1¤;)|3”F,i‘V(¤¤)lm(1>¢) > - ¤“’”< 0|8"F,‘ly(w)\V£’(q) 1>< (Q)m(1>,·)|1@°‘°’” (=¤)|m(1>¢) > ,. —\/l+O(O)+D(0) gH···<1»<1" 2q 1I ?? ><< m(1>,=)l1@aS”(=v)|m(p1) > (4-23) The magnetic current jg is taken to be a vector current and the equation of motion (2.13) is not invariant under both parity and time reversal. We can’t use the time reversal invariance to determine the form of the vertex < m(pf)]l@aS‘° (x)|m(pi) > However, we see that < m(p]=)|lQ3Sp(a:)|m(pi) > also has the form —gR(p_,< +pi)" in the limit pf —+ pi by use of the equation of motion (2.10) . The same step leads to the following relation between bare coupling constant g and on—shell renormalized coupling constant g R 1 + D 0 ( ) gR : 1 + C(0) + D(0) We thus get the renormalization constants of electric (4.18) and magnetic charge (4.24) . Note that the magnetic charge renormalization constant is obtained from the electric charge renormalization constant when we exchange C'(0) and D(0). — 17 OCR Output A local covariant operator formulation of the quantum field theory of electric and magnetic charges was given. The BRST charge has no explicit dependence on the constant vector ny , and the physical states are defined according to the Kugo—Ojima condition. Without using the perturbative method, the unitarity of the physical S-matrix of the theory was proved by the quartet mechanism. This proof was done on the assumption that the equations of motion and the canonical commutation relations of asymptotic fields are the same with those of free fields. The on—shell renormalized electric ( magnetic ) coupling constant CR ( g R ) is related to the bare one e (g) 1 C 0 + ( ) BR Z (5.1) 1 + C'(0) + D(0) 1 + D 0 ( ’ g . <5.Z> 1 + G(0) + D(0) where the scalar functions C'(p2) and D(p2) are defined in the expressions of the currenr-current full Green functions in Sec.4. We have derived the above eg and g R by use of the Maxwell equations (2.12) and (2.13) Since the theory contains both electric and magnetic charges, the gauge boson propagators (4.1) ,(4.2) and (4.3) have the string dependent terms which make it difficult to study the renormalization program."' We don’t know how these string dependent terms behave when the infinite summations of diagrams are carried out. However, we can see that the string dependent terms play an important role in deriving our results (5.1) and (5.2). The Green function (4.4) does not contain the scalar function D(p2) because of the existence of the string dependent terms in (4.1) 1 gpv ,nI2p2 T ——————D , Z p2 (TZ ·1Z)2 2(P ) 5.3) ( and 1 (1>“rZ” + Mp") D(1Z”> (5.4) OCR Output _ Z (TZ ·1>) P2 .. _. If it were not for these terms (5.3) and (5.4) , the Green function (4.4) would be . Z (1+ v The author would like to acknowledge the kind hospitality of High Energy Theory Group in Tokyo Metropolitan University. REFERENCES P.A.M.Dirac, Proc.Roy.Soc.A133 , 60 (1931) ; Phys.Rev.74 , 817 (1948). J.Schwinger, Phys.Rev.144 , 1087 (1966); Phys.Rev.151 , 1048 ; 1055 (1966). D.Zwanziger, Phys.Rev.D3 , 880 (1971). R.A.Brandt, F.Neri and D.Zwanziger, Phys.Rev.D19 , 1153 (1979). T.Kugo and I.Ojima, Prog.Theor.Phys.Supplement 66 , 1 (1979). K.A0ki, Soryushiron Kenkyu ( Kyoto )59 , 383 (1979). See also Ref.5 C.Panagiotakopou1os, J.Phys.A:Math.Gen. 16 , 133 (1983). T.Kugo and S.Ueha.ra, Nucl.Phys.B197 , 378 (1982). P.A.M.Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, Monograph Series ( New York, 1964 10. M.Blagojevié and P.Senjanovié, Nucl.Phys.B161 , 112 (1979). 11. R.A.Bra.ndt and F.Neri, Phys.Rev.D18 , 2080 (1978). 19 —