# Effective Flavor-changing Weak Neutral Current in the Standard Theory and Z-Boson Decay.

код для вставкиСкачатьAnnalen der Physik. 7. Folge, Band 40, Heft 6, 1983, S. 334-346 J. A. Bsrth, Leipzig Effective Flavor-changing Weak Neutral Current in the Standard Theory and 2-Boson Decay G. MA” Zentralinstitut fur solar-terrestrische Physik der Akademie der Wissenschaften der DDR, Berlin T. RIEMANN Institut fur Hochenergiephysik der Akademie der Wissenschaften der DDR, Berlin-Zeuthen’) Abstract. We have studied the effective Zfi f,-vertex in the Glashow-Salam-Weinbergtheory arising from 1-loop corrections, allowing for arbitrary fermion masses miin the loop. We use our results to study the branching ratio of the 2-boson decay into two different fermions. This is of phenomenological interest if there exist yet unknown fermions with large mass which could show up at present and designed accelerators only through radiative corrections. Effektiver flavor-andernder schwacher neutraler Strom in der Standard-Theorio und Z-Boson-Zerfall - Inhaltsubersicht. Wir untersuchen den durch 1-loop-Korrekturen entstehenden Zf,f.-Vertex in der Glashow-Salam-Weinberg-Theorie. Im Loop sind beliebige Fermionmassen zugelassen. Die Resultate werden benutzt, um das Verzweigungsverhiiltnisfiir den 2-Boson-Zerfall in zwei unterschiedliche Fermionen zu untersuchen. Diese Zerfallsmode ist von phanomenologischem Interesse bei dcr Suche nach unbekannten Fermionen sehr grol3er Masse, die an derzeitigen und geplanten Beschleunigern nur durch Strahlungskorrekturen nachweisbar sein werden. 1. Introduction I n recent years one distinguished theory of electroweak particle interactions was obtained -the GLASHOW-SALAM-WEINBERG theory [11. One of the interestingfeatureswithin this theory is that of the non-diagonal weak neutral currents. Being flavor-diagonal in tree approximation, the weak neutral current obtains a small flavor-violating component from loop corrections in the presence of symmetry breaking. Earlier studies of the corresponding phenomenological consequences were limited t o the low energy region. I n refs. [2, 31, e.g., the effective Zd-coupling has been studied in the limit of vanishing external momenta. Here we allow for a non-vanishing momentum of the weak neutral current, qi 0. Furthermore, in the effective Zf,f,-coupling we do not specify whether f, are up- or down-type leptons or quarks in the left handed weak isospin dublets. Already a rough estimate shows the dependence of the form factor V on the mass splitting between the fermions f i with mass mi in the loop. With N families of fermions, + N V - 93 1) 2 PziPR V(m?/M&). i=l Now: Joint Institute for Nuclear Research, Dubna, USSR. (1.1) 335 Effective Flavor-changing Weak Neutral Current in the Standard Theory Unitarity of the KOBAYASHI-MASKAWA mixing matrix Pii [4] forces V to vanish if not the masses mi are different2). This reflects the possibility to define a separate, conserved fermion number for each family if all the up- or all the down-type fermion masses are equal thus preventing family mixing and, in turn, flavor-violation by the neutral current ,). Eq. (1.1) implies rather small predictions for all presently known fermions (mi Hw). But it opens the possibility to observe effects of rather heavy fermions (mi 3 M,) which cannot be produced as free particles at present or designed accelerators. These rather massive fermions could, however, be virtually exchanged in the loop diagrams describing the Z&fz-vertexenhancing the couplings of this vertex so strongly that the vertex and, in this indirect way, the massive fermions may be observable. The article is organized as follows: I n Sect. 2 we shortly discuss renormalization of the vertex. In Sect. 3 we present the vertex for arbitrary q i and mi in terms of the 3-point functions C,, Cii of 'T HOOFT and VELTMAN[9, 101 as functions of I, and Q of the virtual fermion. Sect. 4 contains an application of the results of Sect. 3-the decay Z -+ f2. After t h e Discussion we define in App. A the lagrangian and present in App. B some results on the 3-point functions C,, Cii which may be useful in other contexts too. We also relate them to the functions I, introduced in ref. [5] for the study of muon number violation. < &+ 2. Renormalization Although there is no independent 2-factor of the Zf,f,-vertex, it gets an infinite contribution from fermion wave function renormalization.In the on mass shell renormalization program [ll, 121 g and 4 = sinz Ow are fixed as follows: a, = M ~ / W W&,, = 1 - ozl,g2& = ez = 4 ~ 0 1 . (2.1) Diagrams with self-energy insertions on externaI legs give a vanishing contribution to the matrix element under consideration. Nevertheless, th6 self-energy diagrams of Fig. 1 must be calculated since they define the non-diagonal part of the lepton wave function renormalization; and it is this procedure which induces and completely fixes a counter term for the ZKf2-vertex. 'I Fig. 1. Self-energy diagrams defining non-diagonal fermion wave function renormalization + + a) The same phenomenon occurs in the processes ,u+ e y, vl +. v2 y and related ones as studied, e.g., in refs. [5, 61. 3) VELTMAN'B observation [7] of the influence of mass splitting within a fermion doublet on the charged to neutral current ratio in neutrino scattering is another example of large massea showing up in low energy physics, but there it is due to breaking of weak isospin. All these phenomena demonstrated the limited validity of the decoupling theorem [S] in theories with spontaneously broken gauge invariance. G . NANN and T.RIEMANN 336 For the left-handed fermion fields fz,8we introduce a matrix of 8-factors: -. = (zL):p fL,t ( 8 , t = 1, . 9 N ) . (2.2) The same must be done for the right-handed fields. Since mass eigenstates are already introduced, mass renormalization may be treated by mt = (2L);1’2 m:Zm,8(ZR);1’z (no sum over 8 . ) The following conditions on the regularized self-energies (including counter term contributions) define the 8-factors introduced: z;,(p) u, = 0 I p s , -m’t f%,8 9 ( 8 , t = 1, . ..,N ) (2.3) a, &(p) = 0 -m; (u and a are on mass shell wave functions of fermions.) The lengthy procedure of solving (2.3) [13] simplifies considerably in the approximation ma = 0 which will be assumed to be valid throughout the paper. Then, ZR,sl- dat and with no loss of generality one may choose ZL,st= 2z,ra. Corresponding to the diagrams of Pig. 1 we get for the nondiagonal elements of the 2-factor matrix Ips= where A i= m:/M& and the 2-point function B, is defined in eqs. (Bl-2) of App. B. I f inserted into the originally diagonal lagrangian, among other contributions the matrix 8, induces the following counter term of order g3: Y,, = BZ,,, f,’y”(l + 7’) f:Z‘P + h.c., (2.5) % (v a ) (zL)y~ BZ,,, = 2% with Z, from eq. (2.4) and v, a being the vector and axial vector couplings of the weak neutral current to f, and f t . The expression eq. (2.5) is the counter term for the flavorchanging weak neutral current we searched for. We stress once again that it is completely determined by fermion wave function renormalization. + 3. The Amplitude Neglecting masses of external (on mass shell) fermions the Z&f,-vertex may be described by two form factors. With the conventions and Feynman rules of refs. [13,10] and App. A the vertex is p = ix2 $3 -y P ( 17 4C @ + (3.1) Fig. 2. Dingrams contributing to the effective Zf,f,-vertes Effective Flavor-changingWeak Sciitr,tl Current in the Standard Theory 337 I n the (:lashow-Salam-Weinberg theory the vector and axial vector form factors are equal. Contributions to them come from t h e diagrams of Fig. 2. Using dimensional regularization, one gets in the 't Hooft-Fepnman gauge (3.2) a ) 2ACo] + 1" ,(?;+a)The first curly bracket is calculated from the first diagram in Fig. 2, and so on. Frirthermore, c g = oil, c = -yi/Mf,,, :7 = Z(13 - 2 Q 4 ) , u = 213. A i= m:/M&, (3.4) (33) (I,, Q are weak isospin and electric charge of ferinion f i ) . The dimensionless 3-point functions C = C( l , O, d ) and d = C(d, I, 1)and B, are defined following refs. [9, 101 and discussed in App. B. The functions C2,, 6, and B, are divergent. While V ( A i ,a) is free of divergencies proportional to A;, i t has a constant divergent contribution. This constant divergenc.v vanishes as a result of the (nondiagonal) unitary sum when V(di,.a)is used to calcnlatc the form factors (3.2)*). We have compared our numerical results in detail with those of MA and PRAMUI)ITA [3] for a = 0. This is a good check of (3.3) which expresses the vertex in terms of loop functions C, 6, Bl.I n numerical computer calculations we have used the analytic expressions (B. 1-7) of App. B. The reduction of 3-point functions to C,, 6, and complex logarithms has the advantage of avoiding roughly cancelling large contributions in intermediate steps which plague other calculational schemes. Furthermore, we made an independent computer numeric integration of the 3-point functions and the form factor (3.4) and got complete agreement with the analytic results. The dependence of V(d ,a) on /I is shown in Fig. 3 for the special choice a = a, = M i / M & = 1.25 ( M , = 94 GeV, M , % 84 GeV). As long as A < a/4, the vertex has an imaginary part. For applications the large A behaviour is of special interest. Using 4 ) Concerning renormalization one further comment may be useful. In Sect. 2 we have renormalized using Z-factors. Alternatively, one could add diagrams containing the square root of full propagators at external fermion legs. This rule leads to the addition of half the non-diagonal self-energydiagrams to t,he vertex as ha8 been done, e.g., in ref. 131 for the case of zero momentum transfrr u = 0. G . MANN and T. RIEMANN 338 eqs. (B. 10-16) of App. B one gets the following approximation: + [ - - - -I3-1b + 1 18 8 + [ - -+ 3 -0.- ) 4] -+-u ( l : 8: 2 3 A(y) = ?J arctan (1/2y), !J = (1/0 - 1/4)’’!!. (3.7) Eq. (3.6) implies a dependence of observable quantities on the fourth power of the large ferinion masses in the loop 5). On the 2-boson mass shell the numerical value is V(O 1, 1.25) 21 13[d + 2.84,5 l n d - 4.4!12] + &[-0.04861. (3.8) The large A hehaviours of the four vertices (rip- and down-type leptons and quarks) are rather siniilar to each other; up to an overall sign (of relevance in interference effects) it is only the numerically unimportant dependence on the charge Q which makes a difference. So we have drawn in Fig. 3 only two of the vertices. The situation changes for small A where the vertices split up. For later use we need Y ( 0 ,a) which may be calculated from (3.3), (B. 17-26). Nunierically, V ( 0 ,1.25) = (2.498 + i 2.037) + (-0.234 13 - i 0.411) Q . (3.9) The (T has only a non-leading influence on I’ in (3.t;). One may get a further impression of the smooth dependence of the vertices on (J from the coinparison of (3.8) with i’(4 I, a = 0 ) . The general expression (3.3) yields in this limit: < V ( A , 0) = 13[d(0- 10) I l + 8L + 61. (3.10) This function is also shown in Fig. 3. 1 For I3 = Tit is in accordance with ref. [3]. The functions I,, have been introduced for the study of (low energy) muon number et al. [ b ]and are discussed together with L in -4pp. B. The form violation by ALTARELLI factor V ( A , 0) has no dependence on sin2 @,, or Q as may he seen also from the Ward identity [el. Especially, ?‘(A 9 I , 0) rr, 13(0+ 3 I n d - 3) (3.11) (which should be compared to (3.8)) and l*(d < 1,O) rr, 13[2A In A + 641. (3.12) The dependence on /lIn A for small A is absent if CT =+= 0. In the next section we will apply our results to the nondiagonal decay of the %boson into fermions. 5) Three diagrams contain terms of order A In A which canccl each other. The remaining leading term in (3.6) comes from the Higgs diagrams including the counter term. 339 Effective Flavor-changing Weak Neutral Current in the Standard Theory 100 10 1 0.1 10 Fig. 3. Form factor V(d, u), (3.3), of the effective Zf,f,-vertex as function of the fermion mass mi. External fermions are charged leptons (dashed curve) and downtype quarks (solid curve) 100 1000 mi [ GeV 1 Fig. 4. Dependence of the branching ratio B(f,, f2), (4.4), on the fermion mass mi. External fermions are charged leptons (r = 1; dashed curve) and down-type quarks ( r = 3; solid curve) 4. Non-Diagonal 2-Boson Decay The next generation of accelerators will allow the production of a large number of 2-bosons. This raises interest in all its decay modes, including rather rare ones. The non-diagonal 2-boson decay is possible through radiative corrections containing virtual ferinions of necessarily different mass. This allows the search for these fermions even if they are too heavy to be produced directly. Of course they must have lighter partners f, with m, < M , and Kobayashi-Maskawa mixing. Then a decay of Z into f, and f,, is a signal of the existence of the f i . Here we will study this option in more detail using the results of Chapter 3. G. M ~ N and N T.RIEMANN 340 I n order to define a branching ratio from the width (2+ f2 a norinalization : + f l ) we have to choose The sum over the two charge configurations makes B(f,, f z ) insensitive to CP violation; neglecting this, the two contributions to the numerator of (4.1) are equal. With (3.1), the branching ratio becomes (4.8) ( r = l ( 3 ) for leptons (quarks)). Using sin20rV2: 0.8 and (3.2), one inay estimate the order of ma.gnittide : I?, I N (4.3) Another factor of about 2/(11.3 N ) must be taken into account if one wants to relate (Z-t fl to the anticipated total decay rate [14]. The resulting number has to be confronted with the proposed possibility to produce millions of 2-boson decays per experiment a t a future accelerator like LEl'or SLC [14]. It is evident that there remains a chance to observe non-diagonal 2-boson decay only if the form factor V(Ai,ax) is larger (better: much larger) than 1. But this is not enough. What really is necessary is the splitting between the form factors V ( A i ,az) of different generations. Constant contributions are killed by the (non-diagonal) unitary sum. The main features of the problem inay be demonstrated in the following example choosing N = 3, fl = b-quark, f 2 = q u a r k [lj]. The following compact expression arises for the branching ratio (4.2): + 7,) Here we have used that V ( A u ,a) = V(0,a) + const - A , 'v V(0,o) (4.5) (and so for Ac). The expression for V ( 0 ,a) is given in (3.9). I n Fig. 4 we demonstrate the dependence of B(b, s) of (4.4) on the mass of the tquark. We also show the corresponding function for external charged leptons. The other two cases are rather similar as is evident from the preceding discussion. The range of interest is between m(t)N 20 GeV (experimental lower limit. [16] and m(t) O(SO0 GeV). The upper limit stems from the analysis of partial wave unitary bounds and applicability of perturbation theory [17]. I n Fig. 4 we have divided out froin B(b, s) the influence of the unknown Kobayashi-Maskawa matrix elements [18]. Furthermore, another factor of roughly 2/(11.3N) must be included to get the total rate. For comparison we have also drawn (4.4)for a = 0 in which case there is no threshold behaviour. Apart from this the curve approaches that for a = az already at moderate A . This behaviour is an indication of the relative unimportance of external masses for the physical effect we are interested in. Of similar restricted numerical relevance should be the inclusion of external fernlion masses. The spirit of the approach has been demonstrated. To go one step further in the analysis of possible Z-boson decay one had t o speculate on relevant parameter sets and possible further fernlion generations, so we will stop here [19]. N 341 Effective Flavor-changingWeak Neutral Current in the Standard Theory Conclusions We have generalized earlier work in two directions. We allowed for 0 0 which made possible the investigation of non-diagonal 2-boson decays. Our results demonstrate that the search for flavor-changing 2-boson decays could give information on the existence of rather massive fermions. Of course, the sensitivity of 2-boson decays with a t best millions of events is much siiialler than that of, e.g., the rare muon decay p +e y and corresponding light quark physics, but i t is sensitive also t o other particle mixings and with this in mind of its own value. Secondly, we generalized earlier work in that we allowed for non-diagonal couplings of 2-boson and neutrinos, charged leptons, up- and down-type quarks6). Of course, massive fermions may show up in the loop effect studied here ifthey have couplings to particles with mass smaller than M,. I n view of this our technically motivated restriction to vanishing external fermion masses (m, < M,) is rather restrictive even if the results are not too strongly affected numerically. . Nuinerical calculations of the 2-boson decay (0 =az) along the line of our Chapter 4 but allowing for arbitrary quark masses have been presented in preprints we got knowledge of in preparing the present article. CAPDEQUI-PEYRANERE and TALON[20] coinputed the cross section of e+e--annihilation into two different massive quarks; they investigated several parameter sets but did not analyze the large A limit. AXELROD [all mainly investigated the decay Z+ t + ij (generally, into charge 213 quarks) and got for this choice small rates. Finally, GANAPATHIet al. [ 2 2 ] and CLEMENTSe t al. [ 2 3 ] studied the general case of 2 + + Q allowing rather large ferniion masses in the loop arriving a t similar conclusions as those contained in our Sect. 4. A c k n o w l e d g e m e n t s . We thank D. Yu. BARDW,C.-J. BIEBL, S. M. BILENKY, P. CH.CHRISTOVA,D. EBERT,R . YECCEI and E. WIECZOREK for useful comments and discussions. One of us (T. R.) thanks S. SCHLENSTEDT for introducing him to the use of a computer. + + a -4ppendix A : Charged Current Interactions We use the lagrangian of ref. [lo]. Fermion masses are introduced by nondiagonal Yukawa couplings of SU(2)x U(1) eigenstates t o the Higgs dublet. These couplings imply the following charged Higgs boson-fermion interactions some of them are given in [lo] : y;P=- 2 1/2 + y5) P,ifyw”- y@(l + %z 2)/2 [-Mw (1 + y s ) md mzL - -(1- f ) ] 1 Here j d ( f u ) is a ferniion with I3 = 2 Maskawa mixing matrix. Mw F8ifp- + h.c. (+ T1 ) ,P,; are elements of the Kobayashi- Appendix B: The Vertex Integrals Our definition of one-loop functions agrees with that of ’T HOOFTand VELTMANN [9] (see also [lo]). The only exception is the normalization-our functions are diniensionless. They depend on the dimensionless variables ni(Mwni = pi) and cri(M&cri = mf). 6) We should mention here that it is not totally excluded by experiment (including cosmology) t,o have rather heavy (unstable)neutral leptons so that we decided to handle qnarks and leptons with equal right here. 342 %point function B1(d;al,az) J dnq q"/N2 - i ~ [)( q p1l2+ rn; - i ~ ] . in2piB1(n: ;ul) ~ N2 = (a2 + m4 2= ) ) + We need t,his function in the limit $ = 0: 1 + B;, B l ( O ; d ,1) = - 7 8 2 1 A Bi(O;A,l)=-2 1-A S = 2/(4 - n ) - Y E 1 - I n n - In (M&/p2). %point functions C(z!, ng;al, as, a3) in2(Co;CY;C"v)= M $ J d n q { l ; q"; ql"q"}/N,, N , = (q2 + mf - [ ( q + p1)' + mz - i ~ []( q + p2)' + mg - i ~ ] (B.3) ) +p v c "C + P;P;c22 + (PYP; + ZGP3 c2, + JG6/"'C2*. it) c" = p",c,, C"" = 2 12) 1Pl 21 The dimensionless functions C,, Cii depend on the arguiiients {-(nl- n2)2,n;,n;; bl, G2' GZ}. Here they are t o be calculated for arbitrary -(PI - P ~ ) ~ / M =$u but vanishing masses of external fermions. The remaining two sets of scalar 3-point functions depend on two different sets of parameters : { b , O , O ; l , d , d ) and { a , O , O ; A , 1). I n short: ( l , A , d ) and ( A , 1,l). We may drop the actual calculation of them. It has been shown in ref. [ 9 ] how to express Co in terms of Spence functions @ ( Z ) : + co, + c o 3 ; ac,i co = -co1 = @(A) (b), - @ x -1 x 2. x -x. d The definition of variable9 X~ has been taken over from ref. [ 9 ] too. Their values in the limit of two vanishing external monenta squared are listed in Table 1. Table 1. Variables being used to calculate the two types of 3-point)functions 1 - (1- A ) / o 1/(1- A ) -iE Effective Fliivor-changing Weak Neutral Current in the Standard Theory 343 All the other 3-point functions Cii inay be expressed by C, and complex logarithms also depending on the xi. Of course, performing the integrations one carefully must do the necessary subtractions as demonstrated for C, in [I)]. The result may be expressed using 3 auxiliary functions 1 - xi -2. Gi = (1- xi)ln-x;ln U = -1 - G, G2 G,; x, - 2i z, - zi 1 1-xi 1-xi H=--H,+H,+H,: H a. -- - - - In 1 - x, x, - xi x, - xi 1 -xi -xi In xo zo - zi x, - xi + + 3-point functions tor u = 0 Much simpler results map be obtained if CT = 0 as, e.g., in [3] or in the study of the rare muon dacey u , +e y [5]. I n this case the Feynman integrals allow one trivial integration leaving the integrals I, [ 5 ] , L : + 1 L(u,~)= jdxxIn(az+b). 0 (B-8) We found it convenient to calculate the I, using I, and the recursion relation UI,, + bInp1 = -.n1 The 3-point functions are then (c = 0, arbitrary A ) : c, = I,, c11 - C 12 --- + , C,, c j 24 = C,, = 1 2C2, = ---b--L, 1 1 - 4 2 (B.9) 1 I,, G. M ~ N N and T.RIEMANN 344 whereli=u(d,1,1)= -u(l,d,A)= l - J , b = h ( d , Furthermore, to arrive at (3.10) we made use of - 2dZ, - 41 - I - 1; - I, - AIT l , l ) = d , b ( l , d , A ) = 1(B.lO) = 0, 1 +-= 2 (B.11) 0. Vertex functions tor A < 1 Using the general expressions (B.4- 7) it is rather simple to study further specialized situations. Here we write down vertex funct,ions in the limit of a large fermion inass d in the loop. We neglect all terms and omit all vertex functions which are not larger than O(l/d) when d 1 in the context of this paper (coinpare eq. (3.3)): (B.12) 1 C23(1,A, A ) = 1SA’ (B.13) (B.14) 1 C o ( A , l , l ) = -1n3 A 1 +[1 - 4A(y)]-, (B.13) ._1 1 c~~(A,I,I)=--~~A 4 + [- 1-2 + 3 1 2 B;(O;d,1)=--nil - 3 4 (B.1G) - 1: 1 3 (4 - a) A(!)) a --d 72 +- 1 + -1n :I - 9 1 1 -2A ’ (B.17) A(y) and y are defined in (3.7). Vertex functions for A =0 1 -d(Col,0, 0) = z 2 / 6 - @,[l/(a I)] - _In2 (a 2 -+ + 11+ i z In (a + I ) , (B.18) aC,,(l, 0,O) = oc,, = c, - 1 -d2C2,(1, 0, 0) = + 111 a 1 (a + 2 ) c, - -a a - - 7.3, (€3.19) 2 -c 2(ln a - i z ) , (B.20) + I) C, + 30 + 2 - (a -t 2) (111 0 - i n ) kC;,(l,O, 0) = - 2 ( ~ 1 B;(O, 0 , l ) = - -, 4 7 (B.21) (B.22) Effc ctive Flavor-changing Weak Neutral Current in the Standard Theory b(C00, 1, 1)= n2/G - @(l- a) 345 + 2 Re @ (B.23) d , , ( O , 1, 1) = -[Go - B + I], (B.24) a2C2,(0, I, I) = 2 (B.25) 20c;,(O, 1, 1) = - C, B = ByCarctan (2y) - I3 + 1 - 3 2 1 + nay - 2ayarctan(2y) , (B.26) + arctan [‘Ly(a - I)/(Y- 41). (B.27) As long as 0 < CT < 4, only the functions C(1, A , A ) develop an imaginary part (for A < Atl,, = --d/4).C,, Cll, 5, contain terms of order d In d if 0 < d < 1,but these cancel when they come together in the form factor eq. (3.3) [12]. Also, the combined limit d + 0, a+ 0 is well-defined for the form factor although this is not the case for all the single vertex integrals. References [l] GLASHOW, 8. L.: Nucl. Phys. 22 (1961) 679; WEINBERG, S.: Pliys. Rev. Lett. 19 (1967) 1264; SALAM, A.: Proc. Sth Nobel Symposium, ed. SVARTHOLM, N., Stockholm 1968. [2] GAILLARD, hf. K.; LEE,B. W.: Phys. Rev. D 10 (1974) 897. A.: Phys. Rev. D 22 (1980) 214; INAMI,T.; LIM, C. 8.: Progr. Theor. [3] MA, E.; PRAMUUITA, Phys. 66 (1981) 297. [4] KOBAYASHI, M.; MASKAWA, K.: Progr. Theor. Phys. 49 (1973) GG2. [5] ALTARELLI, G. et al.: Nucl. Phys. B 125 (1977) 285 and ref. therein. [GI CHENG,T. P. ;LI, L.-F.: Phys. Rev. Lett. 46 (1980) 1908; PAL,P. B.; WOLFENSTEIN, L.: Phys. Rev. D 26 (1982) 766. [7] VELTMAN,M. : Nucl. Phys. B 193 (1977) 89. J.: Phys. Rev. D 11 (1976) 2856. [8] APPELQUIST,T.; CARRAZONE, M.: Nucl. Phys. B 163 (1979) 365. [9] ’T HOOFT,G.; VELTMAN, G.; VELTMAN, M.: Nucl. Phys. B 160 (1979) 151. [lo] PASSARINO, [ll]AOKI,K., et al.: Progr. Theor. Phys. 64 (1980) 707; 66 (1981) 1001; SIRLIN, A.: Phys. Rev. D 22 (1980) 971; BARDIN,D. Yu.; CHRISTOVA,P. CH.; FEDORENKO, 0. M.: Nucl. Phys. B 175 (1980) 435; B 197 (1982) 1. [l?] MANN,G.; RIEMANN, T.: Berlin-Zeuthen prepr. PHE 82-5 (1982); Proc. of Int. Conf. v’82, A., Budapest 1982. June 1982, Balatonfiired ed. FRENKEL, N. G. ;EILAM,G.: Phys. Rev. [13] With slightly changed notation, this may be found in: DESHPANDE, D 26 (1982) 2463. [14] ELLIS,J.: Pror. of the LEP Summer Study, CERN-prepr. 79-01 (1979), vol. 2; Proceedings of the SLC workshop on experimental use of the SLAC linear collider: SLAC-Report-246 (1982). 1151 Ma”, G . ; RIEMANN, T.: Berlin-Zeuthen prepr. PHE 82-10 (1982). [lG] HEINzELnImN, G.: Proc. XXIst Int. Cod. on High Energy Physics. ed. PETIAU, P. e t al., Paris 1982. [17] CHANOWJTZ, M. S.; FURMAN, M. A.; HI-NCHLIFFE, I.: Phys. Lett. B 78 (1978) 285; Nucl. Phys. B 163 (1979) 402. S.: Proc. XXIst Int. Conf. on High Energy Physics. ed. PETIAU,P. et al., Paris 1982. [18] PAKVASA, 346 G. MANXand T. RIEYAXN [19] XIYAMA, S.; SATO,K.: Progr. Theor. Phys. 66 (1Ui8) 1703; STRAUJIANN, N.: GIFT lectures on weak interactions and astrophysics, Peniscola 1980, SIN-prepr. PR-80-013 (1980) and ref. therein. [20] CAPDEQUI-PEYRANERE,hr. ; TALON, h1. : &fOlltpelher prepr. PM/86/J (1982). [21] AXELROD,A.: LBL-prepr. LBL-14GG3 (1982). [ 2 2 ] GaNaP-aTHI, v., et a].: SLAC-prepr. SLAC-Pub-2955(198'2). M.. et al.: Cornell-prepr. CLNS-82/544(1986). [23] CLEJIENTS, Bei der Redaktion eingegangen am 12. September 1983. Anschr. d. Vcrf.: Dr. G. MANN Zentralinstitut fiir solnr-tcrrestrischc Physik der AdW der DDR Observatorium fiir Radioastronomie DDR-1,501Tremsdorf Dr. T. RIEMANN Institut fur Hochenergiephysik der AdW der DDR DDR-1615 Zeuthen Platanenallee G

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