CP violation and electric-dipole-moment at low energy τ production with polarized electrons

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1 Nuclar Physics B 763 (2007) CP violation and lctric-dipol-momnt at low nrgy τ production with polarizd lctrons J. Brnabéu a,b, G.A. Gonzálz-Sprinbrg c,j.vidal a,b, a Dpartamnt d Física Tòrica Univrsitat d València, E Burjassot,València, Spain b IFIC, Cntr Mixt Univrsitat d València-CSIC, València, Spain c Instituto d Física, Facultad d Cincias, Univrsidad d la Rpública, Iguá 4225, Montvido, Uruguay Rcivd 5 Octobr 2006; accptd 22 Novmbr 2006 Availabl onlin 1 Dcmbr 2006 Abstract Th nw proposals for high luminosity B/Flavor factoris, nar and on top of th Υ rsonancs, allow for a dtaild invstigation of CP-violation in th τ-pair production. In particular, bounds on th tau lctric dipol momnt can b obtaind of gnuin CP-odd obsrvabls rlatd to th τ-pair production. W prform an indpndnt analysis from low nrgy (10 GV) data by mans of linar spin obsrvabls. W show that for a longitudinally polarizd lctron bam a CP-odd asymmtry, associatd to th normal polarization trm, can b masurd at ths low nrgy facilitis both at rsonant and non-rsonant nrgis. In this way stringnt and indpndnt bounds to th tau lctric dipol momnt, which ar ordrs of magnitud blow othr high or low nrgy prvious bounds, can b obtaind Elsvir B.V. All rights rsrvd. 1. Introduction Th standard modl dscribs with high accuracy most of th physics found in prsnt xprimnts [1]. Nowadays, howvr, nutrino physics offrs a first clu to physics byond this low nrgy modl [2]. Othr signals of nw phnomna may also appar in CP violation physics. Whil CP violation in th standard modl can b asily introducd by quark mixing, as in th CKM mchanism, th discovry of CP violation in th lpton sctor would stablish nw sourcs of CP violation and th apparanc of nw physics. Th tim rvrsal odd lctric dipol momnt * Corrsponding author. addrss: jorg.vidal@uv.s (J. Vidal) /$ s front mattr 2006 Elsvir B.V. All rights rsrvd. doi: /j.nuclphysb

2 284 J. Brnabéu t al. / Nuclar Physics B 763 (2007) (EDM)ofthτ is th sourc of CP violation in th τ -pair production vrtx. In th framwork of local quantum fild thoris th CPT thorm stats that CP violation is quivalnt to T violation. Whil th T-odd lctric dipol momnts (EDM) of th lctron and muon hav bn xtnsivly invstigatd both in xprimnt and thory, th cas of th tau is somwhat diffrnt. Th dipol momnt ffctiv oprators flip chirality and ar thrfor rlatd to th mass mchanism of th thory. Th tau lpton has a rlativly high mass: this mans that tau lpton physics is xpctd to b mor snsitiv to contributions to chirality-flip trms coming from high nrgy scals and nw physics. Furthrmor, th tau has a vry short liftim and can dcay into hadrons, so diffrnt tchniqus to thos for th (stabl) lctron or muon cas ar ndd in ordr to masur th dipol momnts. Thr ar vry prcis bounds on th EDM magnitud of nuclons and lptons; th most prcis on is th lctron EDM, d γ = (0.07 ± 0.07) cm, whil th loosr on is th τ EDM [2] 0.22 cm < R(d τ γ ) 1016 < 0.45 cm. From th thortical point of viw th CP violation mchanisms in many modls provid a kind of accidntal protction in ordr to gnrat an EDM for quarks and lptons. This is th cas in th CKM mchanism, whr EDM and wak-edm ar gnratd only at vry high ordr in th coupling constant. This opns a way to tst many modls: CP-odd obsrvabls rlatd to EDM would giv no apprciabl ffct from th standard modl and any xprimntal signal should b idntifid with byond th standard modl physics.following th idas of [3] and [4],th tau wak-edm has bn studid in CP-odd obsrvabls [5,6] at high nrgis through linar polarizations and spin spin corrlations. EDM bounds for th tau, from CP-vn obsrvabls such as total cross sctions or dcay widths, hav also bn considrd in [7 9]. InRf.[10] th snsitivity to th WEDM in spin spin corrlation obsrvabls was studid for tau-charm-factoris with polarizd lctrons. Whil most of th statistics for th tau pair production was dominatd in th past by high nrgy physics data, mainly at LEP, nowadays th situation has volvd. High luminosity B factoris and thir upgrads at rsonant nrgis (Υ thrsholds) hav th largst τ pair sampls. In th futur, th possibility for a Supr-B/Flavor factory with a τ -pair production rats many ordrs of magnitud highr than prsnt sampls is bing intnsivly analyzd [11]. Ths facilitis may also hav th possibility of polarizd bams. This calls for a ddicatd study of th obsrvabls rlatd to CP violation and th EDM of th τ lpton at low nrgis. In this papr w study a st of diffrnt obsrvabls for th tau systm that may lad to comptitiv rsults with th prsnt bounds for th EDM. 2. Effctiv Lagrangian Dviations from th standard modl, at low nrgis, can b paramtrizd by an ffctiv Lagrangian built with th standard modl particl spctrum, having as zro ordr trm just th standard modl Lagrangian, and containing highr dimnsion gaug invariant oprators supprssd by th scal of nw physics, Λ [12]. Th lading non-standard ffcts com from th oprators with th lowst dimnsion. For CP violation thos ar dimnsion six oprators and thr ar only two oprators of this typ that contribut [12,13] to th tau EDM and wak-edm: O B = g (1) 2Λ L 2 L ϕσ μν τ R B μν, O W = g 2Λ L 2 L τϕσ μν τ R W μν. Hr L L = (ν L,τ L ) is th tau lptonic doublt, ϕ is th Higgs doublt, B νν and W μν ar th U(1) Y and SU(2) L fild strngth tnsors, and g and g ar th gaug couplings. Othr possibl oprators that on could imagin rduc to th abov ons of Eq. (1) aftr using th standard modl quations of motion. In so doing, th couplings will b proportional to th tau-lpton Yukawa couplings.

3 J. Brnabéu t al. / Nuclar Physics B 763 (2007) Fig. 1. Diagrams (a) dirct γ xchang, (b) Υ production, (c) EDM in γ xchang, (d) EDM at th Υ -pak. Thus, th ffctiv Lagrangian for th EDM is: L ff = iα B O B + iα W O W + h.c., whr th couplings α B and α W ar ral. Not that complx couplings do not brak CP consrvation and lad to magntic dipol momnts which ar not considrd in this papr whr w ar mainly intrstd on CP-odd obsrvabls. Aftr spontanous symmtry braking th nutral scalar gts a vacuum xpctation valu and th intractions in Eq. (2) can b writtn in trms of th gaug boson mass ignstats A μ and Z μ as: L γ,z ff = idγ τ τσ μνγ 5 τf μν idz τ τσ μνγ 5 τz μν, (3) whr F μν = μ A ν ν A μ and Z μν = μ Z ν ν Z μ ar th Ablian fild strngth tnsors of th photon and Z gaug boson and dγ τ and dτ Z ar th lctric and wak-lctric dipol momnts, rspctivly. W hav not writtn in Eq. (3) som of th trms coming from Eq. (2) bcaus thy do not contribut at lading ordr to th obsrvabls w ar intrstd in. Ths trms ar th non-ablian couplings involving mor than on gaug boson and th trm rlatd to th CP-odd ν τ τ W ± couplings. In th ffctiv Lagrangian approach th sam nw physics couplings that contribut to th EDM form factor, F nw (q 2 ), also contribut to th EDM which is dfind at q 2 = 0. Only highr dimnsion oprators contribut to th diffrnc F nw (q 2 ) F nw (0) and, if q 2 Λ 2, as rquird for th consistnc of th ffctiv Lagrangian approach, thir ffcts will b supprssd by powrs of q 2 /Λ 2. This allows us to mak no distinction btwn th lctric dipol momnt and th lctric form factor in this papr. Th + τ + τ cross sction has contributions coming from th standard modl and th ffctiv Lagrangian Eq. (3). At low nrgis th tr lvl contributions com from γ xchang or Υ xchang in th s-channl. Th intrfrnc with th Z-xchang (γ Z, Υ Z at th Υ pak) and th Z Z diagrams ar supprssd by powrs of (q 2 /MZ 2 ).Th tr lvl contributing diagrams ar shown in Fig. 1 whr diagrams (a) and (b) ar standard modl contributions, and (c) and (d) com from byond th standard modl trms in th Lagrangian. Notic that standard modl radiativ corrctions that may contribut to CP-odd (2)

4 286 J. Brnabéu t al. / Nuclar Physics B 763 (2007) obsrvabls (for xampl, th ons that gnrat th standard modl lctric dipol momnt for th τ ) com in highr ordr in th coupling constant, and at prsnt lvl of xprimntal snsitivity thy ar not masurabl. On ths grounds th bounds on th EDM that on may gt ar just th ons coming from byond th standard modl physics. Taking into account that our obsrvabls ar gnuin CP-odd, highr ordr CP-vn amplituds can b nglctd. 3. Low nrgy polarizd bams and th EDM As w will show, th EDM can b studid at lading ordr in th angular distribution of th + τ + (s + )τ (s ) diffrntial cross sction for longitudinally polarizd lctrons. Th polarization of th final frmions is dtrmind through th study of th angular distribution of thir dcay products. In our analysis w only kp linar trms in th EDM, nglcting trms proportional to th mass of th lctron. Whn considring th masurmnt of th polarization of just on of th taus, th normal to th scattring plan polarization (P N ) of ach tau is th only componnt which is T-odd. For CP-consrving intractions, th CP-vn trm (s + + s ) N of th normal polarization only gts contribution through th combind ffct of both an hlicity-flip transition and th prsnc of absorptiv parts, which ar both supprssd in th standard modl. For a CP-violating intraction, such as an EDM, th (s + s ) N CP-odd trm gts a non-vanishing valu without th nd of absorptiv parts. As P N is vn undr parity (P ) symmtry, any obsrvabl snsitiv to th EDM will nd, in addition to dγ τ, an additional P -odd contribution coming from longitudinally polarizd lctrons. A standard axial coupling, coming from a Z-xchang in th s-channl, could also b considrd as an altrnativ but in that cas th contribution is supprssd by powrs of (q 2 /MZ 2 ). Following th notation of rfrncs [13] and [14], w now show how to masur th EDM using low nrgy CP-odd obsrvabls. Our aim is to idntify gnuin CP-odd obsrvabls that ar linar in th EDM and not (additionally) supprssd by ithr (q 2 /MZ 2 ) or unitarity corrctions. In th cntr of mass (CM) rfrnc fram w choos th coordinats as in Fig. 2. Ths ± ar th τ ± spin vctors in th τ ± rst systm, s ± = (0,s± x,sy ±,sz ± ). With this stting, polarization along th dirctions x,y,z corrspond to what is calld transvrs (T), normal (N) and longitudinal (L) polarizations, rspctivly. W first considr th τ -pair production in + collisions though dirct γ xchang (diagrams (a) and (c) in Fig. 1). Nxt, w will show that th basic rsults of this sction still hold for rsonant Υ production (diagrams (b) and (d), Fig. 1). Lt us assum from now on that th tau production plan and dirction of flight can b fully rconstructd. This can b don [15] if both τ s dcay smilptonically. Th diffrntial cross sction for τ pair production with polarizd lctrons with hlicity λ is: dσ dω = dσ0 τ λ dω + dσs τ λ dω +. τ λ Th trms to b considrd includ th lading ordr standard modl contributions and th ffctiv oprator EDM. Th dots tak account for highr ordr trms in th ffctiv Lagrangian that ar byond xprimntal snsitivity and which ar not considrd in this papr. (4)

5 J. Brnabéu t al. / Nuclar Physics B 763 (2007) Fig. 2. Coordinat systm for h ± production from th τ ±. Th first trm of Eq. (4) rprsnts th τ spin-indpndnt diffrntial cross sction. Th scond trm dω dσs τ λ includs th linar trms in th spin of th τ s and has snsitivity to th EDM in thir normal polarization: whr dσ S dω = α2 τ λ 16s β{ λ [ ] (s + s + ) x X + + (s + s + ) z Z + + (s s + ) y Y + (s s + ) x X + (s s + ) z Z }, (5) X + = 1 γ sin θ τ, X = 1 2 sin(2θ)2m τ Im { dτ γ }, Z + = cos θ τ, Z = 1 γ sin2 θ 2m τ Y = γβ 2 2m τ cos θ τ sin θ τ R { d γ } τ Im { dτ γ }, and α is th fin structur constant, s = q 2 is th squard CM nrgy and γ = (6) s 2m τ, β = 1 1 γ 2 ar th dilation factor and τ vlocity, rspctivly. As can b sn in Eq. (6), th EDM is th lading contribution to th normal polarization of th singl tau.

6 288 J. Brnabéu t al. / Nuclar Physics B 763 (2007) Singl tau normal polarization obsrvabl W now show how to gt an obsrvabl proportional to th EDM trm from th normal polarization of a singl tau in th complt procss + (pol) γ τ + τ h + ν τ h ν τ. Th cross sction can b writtn as a function of th kinmatical variabls of th hadrons into which ach tau dcays [16] as: dσ ( + γ τ + τ h + ν τ h ν τ ) λ with = 4 dσ ( + τ +( n +) τ ( n )) λ Br ( τ + h + ) ( ν τ Br τ h )dω h + dω h ν τ 4π 4π (7) n ± = α q ± ± q ± = α ( ± sin θ ± cos φ ±, sin θ± sin φ ±, cos θ± ). Th linar trms in th spin of th taus dpnd on svral kinmatic variabls that w hav to tak into account: th CM polar angl θ τ of production of th τ with rspct to th lctron, th azimuthal φ ± and polar θ± angls of th producd hadrons h± ( ˆq ± )inthτ ± rst fram (th mans that th quantity is givn in th τ rst fram; s Fig. 2). Ths angls appar in a diffrnt way on ach trm: th θ τ angl ntrs in th cross sction as cofficints (of th (s + s + ) x trm, for xampl) whil th hadron s angls appar in th cross sction through th polarization paramtrs n ±. Th whol angular dpndnc of ach contribution is uniqu and it is this dpndnc that allows to slct on of th trms in th cross sction. Indd, it is by an intgration on th θ τ angl, followd by a ddicatd intgration on th hadronic angls, that on can slct a polarization trm and thr, th contribution of th EDM. For th normal polarization trm this works as follows. Th intgration ovr th τ variabls dω τ rass all th information on th Z + and Y + trm of th cross sction. Thn, th cross sction can b writtn only in trms of th surviving trms as: d 4 σ S λ = π 2 α 2 β Br ( τ + h + ) ( ν τ Br τ h )dω h + dω h ν τ 4s 4π 4π { λ [( ) n γ x + ( n ) ] [( ) + x + λγβ n y ( n ) ]2m τ + R { d γ } y τ + 4 [( ) n 3γ z ( n ) ]2m τ + Im { d γ } } z τ. As can b sn up to this point, hlicity indpndnt normal (longitudinal) polarization trms du to absorptiv (EDM imaginary) parts may also surviv. On may gt rid of thm by subtracting th cross sctions for diffrnt hlicitis. d 2 σ S Pol( ) d4 σ S λ=1 d 4 σ S λ= 1. Thn, in ordr to nhanc and slct th corrsponding P N obsrvabl, on has intgrat as much as kinmatic variabls as possibl without rasing th signal of th EDM (R{dτ γ } dτ γ from (8) (9) (10)

7 J. Brnabéu t al. / Nuclar Physics B 763 (2007) now on). Kping only azimuthal angls and intgrating all othr variabls on gts: d 2 σ S dφ dφ = πα2 β Br ( τ + h + ) ( ν τ Br τ h ) ν τ + Pol( ) 32s { 1 [ ] (α ) cos φ (α + ) cos φ + γ + βγ [ (α ) cos φ (α + ) cos φ + ]2m τ Now, to b snsitiv only to th EDM w can dfin th azimuthal asymmtry as: whr A N = σ L σ R σ 3πγβ 2m τ = α 8(3 β 2 ) dγ τ, 2π [ π σl = d 2 σ S dφ ± dφ dφ dφ Pol( ) = Br ( τ + h + ) ( ν τ Br τ h ) (παβ) 2 γ ν τ α 8s 2π [ 2π σr = d 2 σ S ] dφ ± dφ dφ dφ + 0 π Pol( ) ] dγ τ 2m τ dγ τ, = Br ( τ + h + ) ( ν τ Br τ h ) (παβ) 2 γ 2m τ ν τ α (14) 8s dγ τ. It is asy to vrify that all othr trms in th considrd cross sction ar liminatd whn w intgrat in this way. Notic that this intgration procdur dos not ras contributions coming from th CP-vn trm of th normal polarization as it will b shown in th nxt sction. To gt rid of CP-vn trms on has to dfin a tru CP violation obsrvabl by summing up th dfind asymmtry (12) for τ + and for τ A CP N = 1 A + (15) 2( N + A ) 3πγβ 2m τ N = αh 8(3 β 2 ) dγ τ. This obsrvabl is fr of th CP-vn contributions dscribd in what follows and it is a gnuin CP-odd obsrvabl γ Z intrfrnc }. (11) (12) (13) Contributions to this obsrvabl can also com from th standard Z γ intrfrnc: dσ 0 Zγ dω = α2 P Z 2 τ λ 16 (2s w c w ) 2 β( s MZ 2 ) Zγ M 0, dσ S Zγ dω = α2 P Z 2 τ λ 16(2s w c w ) 2 β{ [ Γ Z M Z (s + s + ) y Y Zγ ] + + ( s MZ 2 )[ (s + s + ) x X Zγ + + (s + s + ) z Z Zγ ]} +, (16) (17)

8 290 J. Brnabéu t al. / Nuclar Physics B 763 (2007) whr P Z 2 1 = (s MZ 2 )2 + ΓZ 2, a = 1 M2 2, v= s2 w, Z M Zγ 0 = λav ( 2β cos θ + 2 β 2 sin θ ) a 2 β cos θ v 2( 2 β 2 sin 2 θ ), X Zγ + = 1 [ ( λ 2v 2 + a 2 β cos θ ) av(2 β cos θ) ], γ Z Zγ + = [ a 2 β ( 1 + cos 2 θ ) + 2v 2 cos θ ] av [ β ( 1 + cos 2 θ ) + 2 cos θ ], Y Zγ + = 1 aβ(λv a)sin θ. γ (18) Subtracting th cross sctions for diffrnt hlicitis and intgrating, as in th prvious sction, on gts Zγ σl = πα2 β 2(2s w c w ) 2 s P Z 2 Br ( τ + h + ) ( ν τ Br τ h ) ν τ av { 4 ( s MZ 2 ) ( ) } 1 β2 πβ α Γ Z M Z, 3 4γ Zγ σr = πα2 β 2(2s w c w ) 2 s P Z 2 Br ( τ + h + ) ( ν τ Br τ h ) ν τ av { 4 ( s MZ 2 ) ( ) } 1 β2 πβ ± α Γ Z M Z 3 4γ so that th Z contribution to th asymmtry (12) is {}}{ Zγ A N = α 3πβγ av sγ Z M Z 8(3 β 2 ) γ 2 (2s w c w ) 2 (s MZ 2 )2 + (Γ Z M Z ) 2. ε (19) (20) (21) At 10 GV, th valu of th ε factor is of th ordr , which maks this contribution to th asymmtry two ordrs of magnitud blow th xpctd snsitivity for th EDM. Anyway, this asymmtry dos not contribut to th CP-odd A CP N of Eq. (15) Obsrvabls at th Υ rsonancs All ths idas can b applid for + collisions at th Υ pak whr th τ pair production is mdiatd by th rsonanc: + Υ τ τ.atthυ production nrgis w hav an important tau pair production rat. W ar intrstd in τ pairs producd by th dcays of th Υ rsonancs, thrfor w can us Υ(1S), Υ(2S) and Υ(3S) whr th dcay rats into tau pairs hav bn masurd. At th Υ(4S) pak, although it dcays dominantly into B B, high luminosity B-factoris hav an important dirct tau pair production. Excpt for this last cas, that can b studid with th rsults of th prcding sctions, w assum that th rsonant diagrams (b) and (d) of Fig. 1. dominat th procss on th Υ paks. This has bn xtnsivly discussd in Rf. [17]. Th main rsult is that th tau pair production at th Υ pak introducs th sam tau polarization matrix trms as th dirct production with γ xchang (diagrams (a) and (c)). Th only

9 J. Brnabéu t al. / Nuclar Physics B 763 (2007) diffrnc is an ovrall factor H(s) 2 in th cross sction which is rsponsibl for th nhancmnt at th rsonant nrgis, th pur rsonant (imaginary) amplitud bing H ( M 2 ) 3 Υ = i (22) α Br( Υ + ). Bsids, it is asy to show that, at th Υ pak, th intrfrnc of diagrams (a) and (d) plus th intrfrnc of diagrams (b) and (c) is xactly zro and so it is th intrfrnc of diagrams (a) and (b). Finally, th only contributions proportional to th EDM com with th intrfrnc of diagrams (b) and (d), whil diagram (b) squard givs th lading contribution to th cross sction. Th computations w did bfor can b rpatd hr, and finally w obtain no changs in th asymmtris: th only diffrnc is in th valu of th rsonant production cross sction at th Υ pak that is multiplid by th ovrall factor H(MΥ 2 ) 2. In fact, on can tak th four diagrams (a) (d) togthr and still gt th sam rsults w hav alrady shown. Enrgis off or on th rsonanc will automatically slct th significant diagrams. 4. Bounds on th EDM W can now stimat th bounds on th EDM that can b achivd using this obsrvabl. For numrical rsults w assum a st of intgratd luminositis for high statistics B/Flavor factoris. W also considr th π ± ν τ or ρ ± ν τ (i.. h 1,h 2 = π,ρ ) dcay channls for th tracd τ ±, whil w sum up ovr π ν τ and ρ ν τ hadronic dcay channls for th non-tracd τ. For comparison with othr rfrncs, w show th bounds for th τ -EDM that can b st in diffrnt scnarios: dτ γ cm BaBar + Bll at 2ab 1, dτ γ cm Supr-B/Flavor factory, 1 yr running, 15ab 1, dτ γ cm Supr-B/Flavor factory, 5 yr running, 75ab 1. Ths bounds improv prsnt ons by up to 3 ordrs of magnitud. W can also dfin othr obsrvabls that ar snsitiv to th imaginary part of th EDM. Th analysis is similar to th on w hav don hr and in Rf. [17] and will b publishd lswr. To conclud, w hav shown that low nrgy data maks possibl a clar sparation of th ffcts coming from th lctromagntic-edm, th wak-edm and intrfrnc ffcts. Polarizd lctron bams opn th possibility to put bounds on th τ EDM looking at singl tau polarization obsrvabls with low nrgy data. Ths obsrvabls allow for an indpndnt analysis of th EDM bounds from what has bn don with othr high and low nrgy data. Acknowldgmnts This work has bn supportd by CONICYT-PDT-094-Uruguay, by MEC and FEDER, undr th grants FPA and FPA , and by Gnralitat Valnciana undr th grant GV05/264. Rfrncs [1] W.-M. Yao, t al., J. Phys. G 33 (2006) 1. [2] K. Inami, t al., Bll Collaboration, Phys. Ltt. B 551 (2003) 16. (23)

10 292 J. Brnabéu t al. / Nuclar Physics B 763 (2007) [3] J. Brnabéu, G.A. Gonzálz-Sprinbrg, J. Vidal, Phys. Ltt. B 326 (1994) 168. [4] W. Brnruthr, U. Low, J.P. Ma, O. Nachtmann, Z. Phys. C 43 (1989) 117. [5] M. Acciarri, t al., L3 Collaboration, Phys. Ltt. B 434 (1998) 169. [6] D. Buskulic, t al., ALEPH Collaboration, Phys. Ltt. B 346 (1995) 371; K. Ackrstaff, t al., OPAL Collaboration, Z. Phys. C 74 (1997) 403; H. Albrcht, t al., ARGUS Collaboration, Phys. Ltt. B 485 (2000) 37. [7] F. dl Aguila, M. Shr, Phys. Ltt. B 252 (1990) 116. [8] J.A. Grifols, A. Mndz, Phys. Ltt. B 255 (1991) 611; J.A. Grifols, A. Mndz, Phys. Ltt. B 259 (1991) 512, Erratum. [9] R. Escribano, E. Masso, Phys. Ltt. B 395 (1997) 369. [10] B. Ananthanarayan, S.D. Rindani, Phys. Rv. D 51 (1995) [11] S for xampl [12] W. Buchmullr, D. Wylr, Nucl. Phys. B 268 (1986) 621; C.N. Lung, S.T. Lov, S. Rao, Z. Phys. C 31 (1986) 433; M. Bilnky, A. Santamaria, Nucl. Phys. B 420 (1994) 47. [13] G.A. Gonzálz-Sprinbrg, A. Santamaria, J. Vidal, Nucl. Phys. B 582 (2000) 3. [14] J. Brnabéu, G.A. Gonzálz-Sprinbrg, M. Tung, J. Vidal, Nucl. Phys. B 436 (1995) 474. [15] J.H. Kuhn, Phys. Ltt. B 313 (1993) 458. [16] Y.S. Tsai, Phys. Rv. D 4 (1971) [17] J. Brnabéu, G.A. Gonzálz-Sprinbrg, J. Vidal, Nucl. Phys. B 701 (2004) 87.