Topcolor assisted technicolor

23 February

1995

PHYSICS

Physics

Letters

B 345 (1995)

LETTERS

B

483-189

Topcolor assistedtechnicolor Christopher T. Hi!1 Fermi

National

Received

Accelerator

30 November

L.ubnratmy,

f?O. Box MO,

1994; revised Editor:

manuscript

Barah,

received

IL 6OJlO. 5 Deccmbcr

USA ’ 1994

H. Georgi

Abstract A condensate, 71, arising from O(TeV) scale “topcolor”, in addition to technicolor (and ETC) may naturally explain the gauge hierarchy, the large top quark mass, and contains a rich system of testable consequences. A triplet of :tro,rgly coupled pseudo-Nambu-Goldstone bosom, “top-pions”, near !he top mass scale is a generic prediction of the models. A new class of technicolor schemes and associated phenomenology is suggested in this approach.

1. Introduction

and synopsis

The large top quark mass is suggestive of new dynamics associated with electroweak symmetry breaking (ESB j. Top quark condensation models try to identify all of the ESB with the formation of a dynamical top quark mass. In the fermion-loop approximation one can write a simple Pagels-Stokar formula which connects the Nambu-Goldstone boson (longitudinal W and 2) decay constant, frr. to the dynamical mass, mc [ I] (we fix the normalization of f,, in Jzq. (7) below):

Here rn,: is the dynamical mass, k a constant of O( I ), and A the cut-off scale at which the dynamical mass is rapidly going to zero. If electroweak symmetries are broken dynamically by the top quark mass, then = 175 GeV, and taking the .fn = hvk = (2&&‘/2 cut-off A N I .5 TeV, and k x I, we would predict !oo r Electronic

address:

0370-2693/95/$U9.50 SsDIO370-2693(94)01660-7

(intemet) @

[email protected].

1995 Elsevier

Science

B.V. All rights

reserved

large a top mass, m, - 900 GeV. Ergo, top condensation &modelsmust either allow A/m, >> I with drastic fine-tuning, or invoke new dynamical mechanisms to try to obtain a natural scheme. 2 In this letter we wish to sketch another possibility, which seems to carry some intriguing implications. We consider the possibility that: (i) electroweak in!eractions are indeed broken by technico!or (TC) [ 31 witin an extended technicolor (ETC) (yet, one could replace these elements of our discussion with Higgs scalars,either as an approximation to theTC/ETC dynamics, or as a fundamental structure as in SUSY); (ii) the top quark mass is large because it is a combination of a dynamical condensate com.vonent, ( I - E) mr, generated by a new strong dynamics, together with a small fundamental component, em, (i.e. l < I, generated by the extended technicolor (ETC) or Higgs); (iii)

the

new

strong

dynamics

is assumed

to be chiral-

2 In theories, such as SUSY schemes, in which the scale of new physics may be large, A N IOr5 GeV, the top quark mass surprisingly saturates the Pagels-Stokar formula. In this case rn, is precisely determined by the infra-red quasi-fixed point 12 1, which subsumes all corrections !o Eq. ( 1).

484

C. T. Hill /Physics

L.t-trm

cri:ically strong but spontaneously broken by TC at the scale - 1 TeV, and it is coupled preferentially to the third generation. The now strong dynamics thcrcfore occurs primarily in interactions that involve N71, frbi~, and %$/I, while the ETC interactions of the form T/aQ. where Q is a techniquark, arc relatively feehlc. Our basic assumptions, (i)-(iii). leave littlc frcedom oi choice in the new dynamics: We require a new class of tcchnicolor models incorporating “topcolor” (TopC) 141. 1 In TopC the dynamics at the - I TeV scale involves the following structure (or it generaliration thereof): W(3),

x W(3),

-+ SU(3)urr,

x U( I)Y, x U( I )yz x SU(2)[. x U( 1 )I:M

(2)

where SU(3)1 X U(;)Yl (SU(3)2 X U(l)Yz) gencrally couples preferentially to the third (first and second) generations. The U( 1 )s are just strongly rcscaled versions of clcctroweak U( 1) Y. Hence we are advocating a kind of gauge group “replication” which is generation sensitive. ,cL’(3)i x U( I)Yl is assumed strong enough to form chiral condensates which will naturally be tilted in the top quark direction by the (I( 1)Yl couplings. This strong interaction is non-confining, since the theory spontaneously breaks down to ordinary QCDxU( I )EM at the TEV scale by the technicolor gauge group G-rC. U( l)y~ and U( 1 )Yz are stronger than the usual U( l)Y, and there need occur no significant fine-tuning to arrange a (tt) condensate, but not a (zb) condensate, by the simultaneous effects of SU(3)1 and U(I),‘, in the gap equation. The b-quark mass is then an intcrcsting issue, involving a combination of ETC effects and instantons in SO(3) 1. The &term in SU( 3) 1 may ultimately be the origin of CKM CP-violation in these schemes. Above all, the new spectroscopy of such a system should begin to materialize indirectly in the third generation (e.g., in Z + j;b) or perhaps at the Tevatron in top and bottom quark production. A triplet of strongly coupled pseudo-Nambu-Goldstone bosonr. (PNGBs), 3, we dub “top-pions”, near the top mass scale is a generic prediction of the models. The top-pions will have a decay constant of f,, x 50 3 Else, we could lry to use. the W(2) degrees of freedom of the third generation. a possibility which we will not consider presently.

E 345 (I 995) 483489

GeV. and a strong coupling given by a GoldbegerTrieman relation, $,I,~ z ~n:/&f~ x 2~5, potentially observnhlc in ii’ -7 f $ 7; if r)r+ :. rn, -t Nt,, ’

2. Topcolor dynamics We arc relaxing the rcquircment that a top condensate account for the full ESB and we are gcneralizing the structure in the interest in naturalness. ESB can be primarily driven by a technicolor group GTC. and/or TC can also provide condensates which generatc the breaking of topcolor to QCD and V( I ) y. The coupling constants (gauge fields) of SU( 3) I x SU(3)2 are, respectively, 81 and 112 (A& and A&> while for UC 1 )YI x C’( l)Yz they are, respectively, 41 and qz (BI,, Bz@). ‘i’he U( 1 )u; couplings arc then (Ii;+ where Y is usual clcctroweak hyperchargc. A (3,3) x (41, i&) tcchni-condensate breaks SU( 3) I x sU(3)2xU(l)YlxU(l)Y~ -+SU(3i~[,xU(I),at a scale Agtrsim240 GeV. or it fully breaks .I%( 3) I x

SU(3)2xU(

l)y, xU( l)Y2XSU(2)L

-+ SU(3)qcr,x

U( I)EM at the scale A,rc = 240 GcV. This typically leaves a residual global spmetv. SiJ( 3)’ x U( I )‘, implying a degenerate. massive color octet of “colorons”, Bi, and a singlet heavy ZL. The gluon A$ and coloron Bi (the SM lJ( 1 )Y field B, and the (I ( 1)’ field ZL), are then defined by orthogonal rotations with mixing angle 6 [e’] :

I =7+---. I hj ij

I !I;’

(3) 2nd g:3 (gl) is the QCD (U(t )Y) coupling constant at I\TC. We ultimately demand cot 8 z$ I and cot@’ > I to select the top quark direction for condensation. The ma$ses of the degenerate octet of colorons and Z’ are given by MB x g3A/ sin 0 cos 0, The usual QCD gluonic Mi? x gtA/ sint?‘cose’. ( I!/( I)Y electroweak) interactions are obtained for 4 Or the top quark may disappear into a dominant t - h + (F -+ c + z) if M, > W+ + UIJ, in which IIOI been detected at the Tevatmn.

decay mode case top has

CT. Hill/

Phytiss

Leers

any quarks that carry either SL1(3) 1 or XI(?): triplet quantum numbers (or rcppropriatcly scaled I!I( I ); couplings). Integrating out II and Z’ we obtain au cffcctivc low cncrgy four-fermion interaction:

(4) where &.R = i ( 1 f y’)$. K = gi cot’ 6/4a and uyl = gf cot2 9’/47r,with cut-offs of MB and .Mz,. The symmetry breaking leading to the top mass is triggered by the interactions of Eq. (4) and can be estimated in the Nambu-Jona-Lasinio (NJL) approximation. For sufficiently large K the attractive fourfermion TopC interaction would alone trigger formation of a condensate, (7r+&), which is globally custodially SL1(2) symmetric. However, the V( 1)yt force is attractive in the Tt channel and repulsive in the gb channel. Thus, one obtains the pair of gap equations for tn, and ml, (Mp x MB for simplicity here) : w

=

$(K

x

+

l-

=

jnh

$- In( Mi/mf)

( fn/,

$W

B

$(K

-

&Ky,)W,

- 2 f In(M:l$))

I

(5)

Demanding nonvanishing m, and vanishing rnb, we require critical and subcritical combinations: K+

&KY1 (Ken,

=

>

Kcrit;

Kc,+

$r in NJL).

>

K -

$Ky,

;

(6)

We can readily satisfy Eqs. (6) without fine-tuning. Note that in the color singlet channels the V( I )rt effects are actually l/N,. If M.p << MB then we should treat the Z.J(I)rt as a radiative enhancement (suppression) of the Tf (Ebb) channel. Moreover, an analysis of the full effective Lagrangian reveals that one obtains a composite 2 Higgs-doublet model. One doublet, ‘Ht. couples to tR and develops the VEV;

B 345 (1995)

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485

the other, Hz, couples to I!R and remains a massive (non-tachyonic) bound state. In the limit of switching off KY]. HI and Hz form a (custodial) SU(2),. doublet and the effective Lagrangian is X’(2), invariant. The techniquarks (Q;), which have condensed by the confining TC interactions, have acquired constituent massesof order 500 GeV and can be neglected on the scales,X N n, as well. Thus, (Qr) condensates, which would break technicolor, do not form. Of course, the NJL approximation is crude. but as long as the associated phase transitions of the full strongly coupled theory are approximately second order, then analogous rough-tuning in the full theory should be possible. Arranging that the couplings are simultaneously large at N I TeV is a further issue having to do with a GUT scale boundary condition. It suggests that low energy couplings are small because of the familiar imbedding relations of Eq. (3), and GUT scale couplings are larger than usually assumed, perhaps 0( i ). Further strong dynamics probably occurs in the “desert” (e.g. imbedding involving S(I( 2) ,,, etc.). Of course, without knowing the ETC theory N lo” GeV, we cannot imagine reliable extrapolations to the GUT scale. In a theory like this we are clearly a priori abandoning the few “successful predictions” of perturbative (SUSY) unification. ETC interactions (or fundamental Higgs) gene; ate the light fermion masses, and give small contributions to the t and b quark masses as well. The ETC masses are potentially subject to resonant enha;.czments in the full theory, [ 51, and without significant fine-tuning we expect that the largest fermion mass scale that ETC need provide is O( m,) N 1.O GeV to 0( tnc) N 0. I GeV. As described below the b quark receives instanton contributions in the gauge group SU( 3) t . Since ETC is required to generate 0( 1.O) to O(O.l) GeV masses, it may need to be a walking ETC [6]. Since the top condensation is a spectator to the TC (or Higgs) driven ESB, there must occur a multiplet of top-pions. A chiral Lagrangian can be written:

(7) When and q = (1.6). I: = exp(i+?/\/Zf,,). E = 0, Eq. (7) is invariant under $L -+ eiP’“12#~, iiu ---) iiu + Pfw/fi. Hence, the relevant currents

486

are left-handed.

jj

= (jlLyp$@~,

C. T. Hill / Physk.s

Lertus

and (ii.“jj~]O)

=

B 345 ( 1995) 483-#89

Z'h.f7p,/d?. The Pagels-Stoknr relation. Eq. ( I ), then follows by demanding that the +?‘I kinetic tcnn is gencratcd by integrating out the fcrmiots. The toppion decay constant rstimated from Eq. (1) using A = MN and tn, = 175 GcV is .f7 z SOGeV.The couplings of the top-pions to I and b take the form:

to the t and 0 masses. It can lead to induced scalar couplings of the neutral top-pion, as in Ref. [ 71, and an indllccd CKM CP-phase, however, WC will prcscntly neglect the effects of 01 (these cft’ccts wit1 bc small, of order e 171). We gcncrally cxpcct k - I to !‘)-I as in QCD. Bosonizing in fcrmion bubble approxitnation ~‘IN &lnlM;q) where 2’., =cxp( i#'r"/&.fn)~ yields:

and the coupling

This implies

strength

is governed

by the rckuion

an instanton

indticcd

6-quark

mass:

g/a?= w/Jzf7r. The sm:tll ETC mass component ol the top quark implies that the masses of the top-pions will depend upon E and A. Estimating the induced top-pion nass from the fermion loop yields [7] :

where the Pagels-Stokar formula is used for f$ (with k = 0) in the last expression. For e = (0.03, 0. I ), MB z ( I .5, I .O) TcV, and in, = I80 GeV this predicts in+ = ( 180, 240) GcV. The bare value of r, ea. generated at the ETC scale AurC, is subject to very large radiative enhancements by topcc!or and U( 1) rt by factors of order (AETc/MB)Y N 10’. Thus, we expect that a bare value of EO - 0.005 can produce sizeable tn* > m,. Note that ii will generally receive gauge contributions to its mass; these are at most clectroweak in strength, and therefore of order - IO GeV. The b quark receives mass contributions from ETC of 0( 0. I ) to O( I .O) GeV, but also an induced mass from instantons in SU(3)t which may be dominant. The instanton effective Lagrangian may be approximated by the ‘t Hooft flavor determinant (we place the cut-off at MB) :

This is not an unreasonable estimate of the obscrvcd 0 quark mass as WC might have fcnred it would bc too large. Expanding Xi, thcrc also occur induced top-pion couplings to 6~ proportional lo rrri:

(13)

3. Some observables The t and b quarks appearing in, e.g., Eq. (8), are current-basis quarks. The combination of TopC masses and ETC masses yields a general fermion mass matrix, Diagonalization leads to the CKM matrix. For the up-type (down-type) quarks we take the ticld redefinition LO be given by unitary matnces CJL.R and DL.R. where the CKM matrix is V = ULD,,. The leading flavor changing interactions involve then mixing to the second generation:

+ i&+(&&D,.

where 8: is the SU( 3) t strong CP-violation phase. 131 cannot be eliminated because of the ETC contribution

,,$ + F&U&.)

+ h.c.

1

(14)

Exchange of top-pions (as well as topgluons, Z’, and the deeply bound Hz) generates flavor changing effects. By and large we find that these can be tolerably

CT. Hill/

Phvsics

Letters

small in the low lying states, up to the B mesons. hut may show up in processes like Z ---t 30. (i) h -b s -t y : The top-pion interactions lcad in principle to contributions to the process b - s+y. WC estimate the ratio to the SM result (WC expect QCD corrections to largely cancel) :

(15) [In lowest order we have the standard model contribution plus the top-pion contribution CT = -$A(mf/M$) - (c)2A(mf/m?,)/6 where c = DL h.&.~/V~,vfv. comparing Eq. (14) to Eqs. (2.3, 2.29b) in Grinstcin et al. [8]; c is essentially cot p in model I, and there is no iI(y) in the present case.] For us, A(m~/m~)/3A(mf/&) M 0.15. The SM result with QCD saturates the observed branching ratio. However, the QCD corrections are very large, and one cannot assume the NNLO QCD effects are not also significant. Conservatively, we might require, w 5 0.1, hence, DL h.$/Vhz5 0.3 using il,,.k/fn N 3.5. Since DL ,,.,.is not measured (only the CKM element is) this constraint is not strictly binding. Identifying, however, DL~~ with the corresponding clement in the square root of the CKM matrix would fa\L)f &h;,‘vh,, - $, the constraint becomes slightly binding. fit note th?t the situation is not completely settled [ 81. There are, of course, other apparent!y smaller effects due to Z’, b-coupled top-pions from instantons, and the deeply bound Higgs. J-‘I and HZ. (ii) AS = 2 and AC = 2 effects: There occljr FCNC effects induced by the CKM mixing in the mass basis to the current basis third generation. In the current basis, we have the neutral top-plon coupled to the t and b quarks as i(m,i$t + m$wb)?rO/&f,,. Exchange of these neutrals will induce AC = 2 and AS = 2 effective interactions when we rotate the t and 0 quarks to their mass eigenbases, I -+ t + 0( A2)c + 0( p)u and b + b + O(A2)s + O(A”)d. Thus, we obtain effective AC = 2 and AS = 2 interactions: miO( A”) _ 5 m@(A’“)5 _ 5 sy f&d cy ucy u + 2mifi 24fi

+ . .. (16)

B 34.5 (199s)

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487

With A N 0( IO-‘), rrz~gtr,siram,, thcsc arc of an acccptahlc strength, c.g., in comparison to ($.A’/ I287r’& )?yfidsy,n. Charged top-pions give box diagrams of a similar strength. (iii) t ----)ii’ + IX The mode t - 5+ + 6, if kinematically allowed, is ruled out if the top is seen IO have the convmtiona! rate r -+ W“ + 6. because the ??coupling is very strong. Small tni, is disfavored by h -+ s + y in any case. From our perspective the observation of a strongly colrpled ii’ + t + 5 is a natural consequence of new strong dynamics associated with the generation of the top quark mass. The ii+ is expected to be a broad state and may be difficult to detect; the i?’ may be narrow if 11le < 31, and would decay through anomalies to gg and yy, (and to 51, through Eq. (13)) and imitates some effects of states in two-scale tcchnicolor (in contrast to [ 91 we do not expect color octet PNGBs associated with the f,, N 50 GeV scale). ai,: It is particularly intriguing that, (iv) Rhr q+ while ETC interactions generally lead to a suppression [ 10 ] , TopC schemes catt contain signifccmt ettkmcenlents of Rh = T(Z --i i&)/T(Z -+ hadrons) [II]. In the models we have described both the topgluons and the Z’ will enhance Rb. This is a desirable fcature, because when the observed LEP central value for f?h is !it topgluons idone give too much elhiUlcCmerit to top production at the Tevatron [ 12,111. On the other hand, Z’ can enhance Rt, with smaller impact upon ai,. In our present schemes’ we might expect Mz,, MB N 500-1000 GeV to accommoda!c acceptable observable effects in top production and &,. The Z’ may then be observable in as,, at the Tevatron. These potentially important effects, as well as S, T and U, will be discussed in grentcr detail elsewhere.

4. An example of a new model We note that a number of new models is suggested by this approach. In model building we have several optincs: (I) TC breaks both the EW interactions and the TopC interactions; (II) TC breaks EW, and something else breaks TopC; (III) TC breaks only TopC and something else drives ESB (e.g., a fourth generation condensate driven by TopC). We presently show an example of a very skeletal model in category (I) in Table I.

488

CT.

hill/Physics

Lerters B 345 (1995)

483489

Table I Gauge charge assignments of quarks in a schematic model SU(3)mt x SU(3)r~z x SU(3)t x SC’(3)z x SU(2); x rr( I )yt x U( I )n. Additional fieldc (such a~ Icptonst required for anomaly cancellation and are not shown. @Q) breaks X/(3) I x Z/(3)2 x I/( i )rt x U(i),-+ W(3) Y /I( 1)~. and (TT) hrcaks Sf/(?)~. x U(l)r --) U( l)u~. fir) forms via SU(3)t x O(l),-1 --ticld

.SU(3~~~l-1

su(3)~!r~2

sir(3),

SU(3)?

SCi(2)I.

L’( lh!

UC I))?

01.

3

I

C)R

3

I

3 I

I 3

1 I

I 0

0 I

7-L = (T. B)L

I I I I I I I

3 3 I I I I I

I I 3 3 I I I

I I I I 3 3 3

2 1 2 I 2 I 2

0 0 I if,-+) 4‘ c$.-f, 0

f ($-f, 0 0 0 0 !.

1

I

I

3

I

0

rR = (T,B)R

II. = tR = CL = CR = UL =

(1. b)l. (t,b)R (C.S)L (c, c?)R (u. cl)r.

UR = (ti,d)R

---.

For simplicity we choose GrC = SU(3jxl x Su(3)rCz and we have indicated !hc U( l)i hypercharge assignments. The leptons and other technifields that are required to cancel anomalies are not shown. The techniquark condensate (QQ). breaks SU(3)t x SU(3)2 x V(l)yt x CJ(l)yz + ‘%s notbreak SL1(2)t x V( 1)~. SU(3) XU~l)Y,’ SU(2)~xU(l)r --J(l)n~occutsthroughthecondensate of techniquarks TL,R which feel the weaker Su( 3)rc2 x u( 1)~ interactions, thus (m) is approximately custodially SL1(2) invariant. The third generation develops the tilted condensate through the SU( 3) t x CJ(1) t interaction with rough-tuning of the tilting. We have also assigned the second generation (c, s) to the stronger V( 1) I perhaps permitting a resonant enhancement of the ETC mass scale for charm and strange, so we assume that the lJ( 1)t coupling is subcritical by itself. Tb pattern suggests a further “SfJ( 3)~” replication for the first generation. We believe these models offer new insights into the dynamical origin of fetmion masses and electroweak symmetry breaking, and merit further study. Further model studies and phenomenological applications will be presented elsewhere. Our key result is that, if the top mass arises by a dynamical chiraf symmetry breaking, together with some additional mechanism leading to the light quark masses and electroweak breaking (TC/ETC or Higgs), then there will necessarily occur a triplet of top-pions. This result is generic to all such models and possibly testable in the foreseeable future.

Acknowledgement

I thank Bill Bardeen, Estia Eichten and Ken Lane for many insightful discussions. This work was performed at the Fermi National Accelerator Laboratory, which is operated b:t Universities Research Association, Inc., under contract DE-AC02-76CH03000 with the U.S. Department of Energy. References ] I ] W.A. Bat&en, CT. Hill and M. Lindner Phys. Rev. D 41 (1990) 1647. ] 2 ] C.T. Hill, Phys. Rev. D 24 ( 1990) 69 I ; CT. Hill. C.N. Zeung and S. Rao. Nucl. Phys. B 262 ( 1985) 517; J. Bagger, S. Dimopoulous and E. Maxso. Phys. Rev. Lett. 55 ( 1985) 920. [3] S. Weinberg, Fhys. Rev. D I3 ( 1976) 974; L. Susskind, Phys. Rev. D 20 (1979) 2619; S. Dimopoulos and L. Susskind, Nucl. Phys. B 155 237 (1979); E. Eichten and K. lane. Phys. Lett. B 90 (1980) 125. [4] C.T. Hill, Phys. J&t. B 266 (1991) 419; S.P. Martin. Phys. Rev. D 45 (1992) 4283; D 46 (1992) 2197; Nucl. Phys. B 398 (1993) 359; M. Lindner and D. Ross, Nucl. Phvs. B 370 ( 1992) 30; R. Btinisch. Phys. Lett. B 268 ( 1991) 394; C.T. Hill, D. Kennedy, T. Onogi and H.L. Yu, Phys. Rev. D 47 (1993) 2940. [ 51 T. Appelquist. M. Binhom. T. Takeuchi and L.C.R. Wijewardhane, Phys. Lett. B 220 (1989) 223; T.W. Appelquist. D. Karabali and L.C.R. Wijewardhana,

489 Phys. Rev. Lett. 57 ( 1966) c,S7; K.R. Mendcl and V. Miransky, Phys. Lett. B ‘:6@ ; 1991) 384: V. Miran,;ky. Phys. Kev. lctt. 69 (1992) 1022: N. Evans, Phys. Lett. B 331 (1994; 77R. 16 1 B. Holdom. Phys. Rev. Lett. 60 ( 1988) 1223: see also 15 1. 17 1 See C.T. Hill and G.G. Ross. Nuci. Phys. B 3 I I ( 1988) 25.3; Phys. lxtt. B 203 ( 1988) 125; for ;r discussion of analogous chin1 Lagrangians and the effects of CP-violation. 181 See e.g., N: Deshpande, in: B-Decays. ed. S. Stone, World Scientific ( 1992): J.L. Hewett, Top Ten Models Constrained By /I + sy, SLAC-PUB-6521

(1994):

Phys.

Rev. lxtt.

70 t 1993)

1045:

V. Barper. M. Hcrgcr and R.J.N. Phillips. I’hys. Kcv. lxtt. 70 ( 199.1) IMU: B. Grinskin. K. Springrr ;md M. Wise. Nucl. I’hyn. B 3.19 ( 1990) 269. 191 E. Eichten and K. Lane, Phys. Lett. B 222 ( IO:?,) 274; K. Lane and M. Kamana, Phys. Rev. D 44 [ 1991) 257X. S. Chivukula, S. Selipsky and E. Simmons, Phys. Rev. Lrtt. 69 (1992) 575. C.T. Hill and Xinmin Zheng. % -+ %/> versus Dynamicill Elcctrowcak Symmetty Breaking Involving the Top Quark. Fermilab-Poh-94i231, hep-ph/9409315, to appear in Phyr. Rev. D. Hill

1121C.T.

and S.J. Parke,

Phys.

Rev. D 49 ( 1994)

4454.