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hep-ph/0211016 October 2002

Nambu-Goldstone Bosons in CP Violating theory with Majorana Masses

Darwin Changa,b, Wai-Yee Keungb and Chen-Pin Yeha

NCTS and Physics Departme

nt, National Tsinng Hua University, Hsinchu 30043, Taiwan, R.O.C. b Physics Department,University of Illinois at Chicago, IL 60607-7059, USA

a

arXiv:hep-ph/0211016v1 1 Nov 2002

ABSTRACT

We derive some properties of the Nambu-Goldstone boson coupling in theories that have CP violation and Majorana masses. We show explicitly that its diagonal coupling to a Majorana fermion is pseudoscalar not scalar. This clari?es some confusion in the literature. Some potentially useful o?-diagonal properties are also derived. We also show, in the process, that the Goldstone theorem often produces interesting and nontrivial identities in matrix theory which may be hard to prove otherwise. PACS numbers: 14.80.Mz, 11.30.Er, 14.60.Pq

Introduction

Theories with Majorana masses and CP violation have been taken more seriously recently because of the experimental evidence for neutrino masses[1, 2, 3]. They often contain some ingredients of spontaneous symmetry breaking of either gauge or global symmetries, and as a result, some accompanying unphysical Higgs or Nambu-Goldstone (NG) bosons. For example, the theory with spontaneously broken lepton number usually produces a NG boson, usually called majoron. It was well known that the coupling of the NG boson to the fermion should be always pseudoscalar in nature. This is because the degree of freedom of the NG boson usually can be identi?ed as the symmetry generator which is spontaneously broken[4]. Under the symmetry transformation, the associated NG boson is translated by a constant. For the theory to be invariant under this translation, the NG boson has to couple only through its derivative[5]. For the couplings that are diagonal in the fermion ?avor, such derivative couplings translate into pseudoscalar coupling using equation of motion. However, there are some recent doubt on the applicability of the above argument to the theory with Majorana masses and CP violation[6]. In this paper we wish to address this issue in detail and rea?rm that the above argument remains true. The conclusion is useful not only for the neutrino sector, but also for the chargino and the neutralino sectors of the supersymmetric theories[7]. For example, in Ref[8], it was

shown that the diagonal couplings of the unphysical NG Higgs boson to charginos are pseudoscalar even though the relevant sector of the theory is CP violating with Majorana gaugino masses. As a result of this analysis, we found that very interesting and nontrivial theorems in matrix algebra can be derived as a direct consequence of the Goldstone theorem. In the next section we will use the simplest example in SU (2)L × U (1) context with one ?avor of neutrino species (including a right handed neutrino) to illustrate the pseudoscalar nature of the NG boson coupling. We will prove that the NG boson coupling is pseudoscalar exactly. Then we will generalize it to N ?avors of neutrinos. In both cases, the fact the diagonal NG boson coupling has to be pseudoscalar gives rise to some nontrivial matrix theorems. To explore these matrix theorems further, we consider a very general theory with many NG bosons and then formulate a resulting matrix theorem which is simple but very nontrivial. In the appendix we include a general proof of the pseudoscalar nature of the diagonal NG boson couplings for completeness.

2 × 2 matrices of ν masses and couplings

The model is a combination of the singlet majoron model[9] and the triplet majoron model[10]. Details of the model can be found in Ref.[6]. In addition to the known fermions in the Standard Model (SM), there is an antineutrino (denoted as ν c ) which is a singlet under the gauge group with the lepton number L = ?1. The scalar bosons include one SU (2) doublet H (SM Higgs), one SU (2) triplet χ and one gauge singlet S . H has 1 hypercharge Y = ? 2 , lepton number L = 0, and χ has Y = 1, L = ?2, S has Y = 0, L = ?2. Let us start with one generation, the relevant quantum numbers are summarized below ν νc H χ S Y ?1 0 ?1 1 0 . 2 2 L +1 ?1 0 ?2 ?2

Rede?ning each scalar ?eld as deviation from its vacuum expectation value (vev), we write the Yukawa terms in the Hamiltonian after symmetry breaking as 2mD c 0? M m νν (H + vH ) + ν c ν c (S ? + vS ) + νν (χ0 + vχ ) + h.c. (1) vH vS vχ Bilinear form of Weyl fermions are understood to be connected by the Levi-Civita symbol. The neutrino mass terms are m mD ν ν νc + h.c. (2) mD M νc Two of the three complex masses can be made real by rotating the phases of the neutrino ?elds, leaving one CP violating mass parameter. The composition of the majoron can be derived from vacuum expectation values as √ 2 2 2 2 GM = ( 2/N )[2vχ vH Im(H 0 ) + vH vχ Im(χ0 ) + (vH + 4vχ )vS Im(S )] , (3)

4 v 2 + v 4 v 2 + (v 2 + 4v 2 )2 v 2 . The majoron-neutrinos coupling terms in where N = 4vχ H H χ H S χ the Hamiltonian are then mD m 2 2 vχ vH 2vH vχ ?v i ν vH c χ ν ν GM + h.c. (4) ?√ mD M 2 2 2 νc 2vH vχ vS vS (vH + 4vχ ) 2N vH

√ The above coupling 2 × 2 matrix has the simple form ?i( 2N )?1 C, where C is C≡

2 2 ?mvH 2mD vχ 2 2 2 2mD vχ M (vH + 4vχ )

.

(5)

The symmetric neutrinos mass matrix Mn can be diagonalized as U T Mn U = Mdiag , Mn = m mD mD M , (6)

where matrix U is unitary and Mdiag is a diagonal matrix with real non-negative elements. In the √ mass eigenstate basis the coupling matrix of the majoron to neutrinos becomes ?i( 2N )?1 U T CU . The scalar couplings of the majoron are proportional to imaginary parts of diagonal elements of U T CU . One can show directly that this diagonal scalar coupling vanishes by explicitly diagonalizing the mass matrix order by order in m/M (a procedure which was adopted in Ref[6]). We will give such an approximate calculation in the Appendix for comparison with Ref[6]. Here we are going to prove that the diagonal scalar couplings of the majoron are zero using the matrix property without assumption 2 2 about the size of m/M . De?ne r = 2vχ /vH , we have

2 C = vH

?m mD r mD r M (1 + 2r )

2 = vH (1 + r )

?m 0 0 M

2 + vH rMn .

(7)

To prove that the diagonal elements of U T CU are real, we only need to show that N′ ≡ U T has real diagonal elements.

2 2 N′ 11 = ?mU11 + MU21 2 2 N′ 22 = ?mU21 + MU22 .

?m 0 0 M

U

(8)

(9) (10)

Denote m1 , m2 as the neutrino mass eigenvalues. Then m mD mD M Express m and M as m1 and m2 ,

? 2 ? 2 m = (U11 ) m1 + (U12 ) m2 , ? 2 ? 2 M = (U21 ) m1 + (U22 ) m2 .

= U?

m1 0 0 m2

U? .

(11)

(12) (13)

Unitarity of U gives N′ 11 = = ′ N 22 = =

? 2 ? 2 2 ? 2 ? 2 2 ?((U11 ) m1 + (U12 ) m2 )U11 + ((U21 ) m1 + (U22 ) m2 )U21 ? 2 ? 2 ((U21 U21 ) ? (U11 U11 ) )m1 , ? 2 ? 2 2 ? 2 ? 2 2 ?((U11 ) m1 + (U12 ) m2 )U12 + ((U21 ) m1 + (U22 ) m2 )U22 ? 2 ? 2 ((U22 U22 ) ? (U12 U12 ) )m2 ,

(14) (15)

which are real. So the diagonal scalar couplings of the majoron to the neutrinos are zero even if the couplings are CP violating. Next consider the o?-diagonal terms of the majoron couplings. Note that U T CU and 2 vH (1 + r )N′ have identical o?-diagonal components. Using unitarity,

? ? N′ 12 = N′ 21 = (m1 + m2 )Re(U21 U22 ) + i(m1 ? m2 )Im(U21 U22 ) .

(16)

It means, if the neutrinos are degenerate, the o?-diagonal couplings of the majoron are also pseudoscalar. If (m1 + m2 ) = 0 then the o? diagonal couplings of the majoron are purely scalar. This is not surprising because if (m1 + m2 ) = 0 then one can rede?ne one of the neutrino by i, after that it is reduced to the degenerate case and o?-diagonal coupling picked up an i. These results are consequences of the Goldstone Theorem. If the global symmetry, say the lepton number in the current example, is an accidental symmetry only valid for renormalizable terms of ?elds in low energy physics, but broken by higher dimensional operators induced from very short distance physics at the scale MX . In general, the NG boson picks up a tiny mass[11] which is suppressed by the factor 1/MX . CP violation can probably give rise to a scalar diagonal coupling which is also suppressed by 1/MX , not by the less reducing factor 1/M from the neutrino sector. This mechanism[12] provides a possible source of a feeble ?fth force competing with gravity.

2N × 2N matrices of N generations

If there are N generations of neutrinos. The above proof can also be applied. We denote the 2N × 2N symmetry mass matrix as Mn = m ? m ?D T ? m ?D M , (17)

? are N ×N matrix. We can also ?nd a unitary matrix U to diagonalize where m ?, m ?D and M Mn such that ?11 U ?12 m ?1 0 U U T Mn U = , U= , (18) ?21 U ?22 0 m ?2 U ?11 , U ?12 , U ?21 , U ?22 are four N × N where m ? 1 and m ? 2 are real N × N diagonal matrices, and U matrix satisfying the unitary conditions,

? ? ? ? ?11 ?21 U U11 + U U21 = 1 , ? ? ?12 U ?12 + U ?22 U ?22 = 1 , U ? ? ? ? ?11 ?21 U U12 + U U22 = 0 .

(19) (20) (21)

The majoron coupling matrix in mass eigenstate basis is given by ?√ i UT 2N

2 ?mv ? H 2m ?D vχ2 T 2 2 2 ? 2m ?D vχ M (vH + 4vχ )

U .

(22)

Using the same trick as above, we only need to show that N′ = U T ?m ? 0 ? 0 M U (23)

? as m We express m ? and M ? 1 and m ? 2,

has real diagonal elements. First, we write down the 11 and 22 blocks, ?T m ? ?T ? ? ? ′11 = ?U N 11 ? U11 + U21 M U21 , ?T m ? ?T ? ? ? ′22 = ?U N 12 ? U12 + U22 M U22 .

? ?? m ?? ? ? ? 2U ?12 m ? = U , 11 ? 1 U11 + U12 m ? ? ? = U ?? m ? ? ? ? 2U ?22 . M 21 ? 1 U21 + U22 m

(24) (25) (26) (27)

Then, using the unitary conditions (19-21), we obtain ? ? ? ? ?T U ? ? ? 1U ?11 ?T U ? ? ? 1U ?21 ? ′11 = ?U N U11 + U U21 11 11 m 21 21 m ? ? T ?? ? ? = m ? 1 ? (m ? 1 U11 U11 + U11 U11 m ? 1) . Similarly,

? ? ?22 ?T U ?? ? 2 ? m ? ′22 = m N ? 2U U22 + U ?2 . 22 22 m

(28) (29)

In terms of the mass eigenvalues (m ? 1 )i and (m ? 2 )i , we have ? ? ? ? ?11 ?11 ? ′11 )ij = +(m ? 1 )i δij ? ((m ? 1 )i + (m ? 1 )j )Re(U U11 )ij ? i((m ? 1 )i ? (m ? 1 )j )Im(U U11 )ij , (N ? ? ?22 U ?22 )ij + i((m ?22 U ?22 )ij . ? ′22 )ij = ?(m (N ? 2 )i δij + ((m ? 2 )i + (m ? 2 )j )Re(U ? 2 )i ? (m ? 2 )j )Im(U (30) No dummy index summation occurs in above expression. It is obvious that the diagonal ? ′11 and N ? ′22 are real. The o?-diagonal block is components of both N ? ? ?21 ?T U ?? ? 2 ? ′12 = N ? ′21T = m N ? 1U U22 + U (31) 21 22 m ? ? ? ′12 )ij = ((m (N ? 1 )i + (m ? 2 )j )Re(U21 U22 )ij + i((m ? 1 )i ? (m ? 2 )j )Im(U21 U22 )ij . (32) These results all can be understood through arguments similar to that in the last section.

General N × N mass matrix

In this section, we demonstrate how the Goldstone theorem can be used to prove a matrix theorem. Theorem : Given any complex symmetric matrix of dimension N × N , M=

? ? ? ? ? ?

m11 m12 · · · m12 m22 . .. . . .

? ? ? ? ? ?

,

(33)

mN N

1 we decompose M as the sum of 2 Ci ,

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

..

. 0 ..

m1i . . .

0

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

Ci =

. mi?1,i 2mii mi,i+1 · · · miN .. . 0 .. .

(34)

where Ci is nonzero only along the i’th row and i’th column. Let U be the unitary matrix that diagonalizes the symmetric complex matrix M , that is, U T MU = MD where MD is a real diagonal matrix. Then the diagonal elements of the matrix U T CU are also real, for general real linear combination of Ci , that is, 2a1 m11 (a1 + a2 )m12 · · · (a1 + ai )m1i · · · (a1 + aN )m1N ? ? 2a2 m22 · · · (a2 + ai )m2i · · · (a2 + aN )m2N ? . ? .. N . ? . ··· ··· . C= ai Ci = ? ? 2 a m · · · ( a + a ? i ii i N )miN i=1 ? . . ? .. . . ? aN mN N

? ? ? ? ? ? ? ? ? ? ? ? ?

,

(35)

with arbitrary real coe?cients a1 , · · · , aN . We also know that if two of eigenvalues of MD are equal, say (MD )ii = (MD )jj , then the correspondent term (U T CU )ij is real. If (MD )ii = ?(MD )jj , then the correspondent term (U T CU )ij is pure imaginary. The proof of this seemingly very nontrivial theorem follows immediately from the Goldstone theorem. Consider a Lagrangian with U1 (1) × U2 (1) · · ·× UN (1) symmetry. For each Ui (1), there are one Weyl fermion ψi which carries a unit of its quantum number and one scalar Hi with ?2 units of its quantum number. In addition, for each pair of (Ui (1), Uj (1)) there is one scalar, hij carrying the quantum number (?1, ?1). There are 1 N (N ? 1) independent hij as hij ≡ hji and i = j . The Lagrangian of Yukawa couplings 2 can be written as

N N

L=

gi ψi ψi Hi +

i=1

i,j =1

fij ψi ψj hij + h.c.

i=j

(36)

where fij = fji . The couplings can be complex. After symmetry breaking, scalars develop vev’s, which are real when phases are absorbed into couplings with rede?nition of ?elds ( Hi = vi , hij = vij ). We de?ne mii = givi , mij = fij vij (no summation). The resulting complex symmetric Majorana mass matrix is exactly in the form of M above. The spectrum has N NG bosons √ N 2 Gi = ? vij Im(hij )) (37) (2vi Im(Hi ) + Ni j =1

j =i

where Ni =

2 4vi +

N j =i

j =1

√ 2 . The Yukawa coupling of Gi is ?i( 2N )?1 Ci . The concluvij

sion follows immediately from the Goldstone theorem. The explicit proof is very similar to those in the last two sections. Basically, we need to show Ci ? M after mass diagonalization has real diagonal entries. Note that the simple example of the one generation model described by Eq. (1) corresponds to the present case of N = 2 with U1 (1) generated by the quantum number Y , and U2 (1) generated by Y + L/2.

Conclusion

In this paper, we basically rea?rm that the diagonal Goldstone boson coupling should be pseudoscalar even when there are CP violation and Majorana masses in the system. Such

a property is probably known, however, it is still quite complicated to work it out explicitly considering that there exists confusion in the literature[6]. The pseudoscalar character of the unphysical Higgs couplings to chargino and neutralinos were not worked out explicitly until recently[8]. This nontrivial character implies that the Goldstone theorem often results in matrix theorems which are very nontrivial unless one uses the Goldstone theorem to motivate it.

Acknowledgment

WYK is partially supported by a grant from U.S. Department of Energy (Grant No. DEFG02-84ER40173). DC is supported by a grant from National Science Council(NSC) of Republic of China (Taiwan). We wish to thank X.G. He and H. Haber for discussions. DC wish to thank theory groups at SLAC and LBL for hospitality during his visit. WYK wish to thank NCTS of NSC for support during his sabbatical leave.

Appendix

Approximation for 2 × 2 Majoron Model

Below we work out the detailed calculation of the case studied in Ref.[6] for the neutrino mass matrix of size 2 × 2 of one generation. Physical CP phase φ appears in M = |M |eiφ of Eq. (6) in the basis that m and mD are real. We assume |M | ? mD ? m > 0 and m2 D /|M | the same order as m. First, we rotate phases of the two diagonal elements of Mn in opposite directions so that the resulting imaginary parts are equal. Then, leaving temporarily the unit matrix with imaginary coe?cient, we easily diagonalize the remaining real symmetric matrix. Putting back the imaginary part, we make the ?nal phase rotations to obtain the physical real diagonal mass matrix. These steps are summarized as U ??

?

e 0 ×

iφ 2

0 e

?i φ 2

? ?

1 ? im sin φ/|M 0 0 1 + im sin φ/|M | 0 e? 2 θ 0 1 ? im sin φ/|M |

φ i

with eiθ = m cos φ ? m2 D /|M | + im sin φ. We ?nd that

?

i

1 mD /|M | ?mD /|M | 1

,

Mdiag ?

Denote the two neutrino mass eigenstates as ψl and ψh . The majoron-neutrinos coupling terms become i ψl ψl ψh U T CU GM + h.c. (40) ?√ ψh 2N

m1 0 0 |M |

(1 ? im sin φ/|M |)e? 2 (θ?φ) mD /|M |ei 2 ? U ?? , φ i ?mD /|M |e? 2 (θ+φ) e?i 2 , with m1 =

2 m2 sin2 φ + (m cos φ ? m2 D /| M | ) .

?

(38) (39)

with

C=

2 2 ?mvH 2mD vχ 2 iφ 2 2 2mD vχ |M |e (vH + 4vχ )

.

(41)

Following the usual procedure, we pair up each left-handed Weyl ?eld ψ with its complex conjugated ?eld ?ψ ? to form a four-component Dirac ?eld Ψ. In term of the heavy and light Dirac ?elds Ψl , Ψh , the interaction is described by √ 1 ? (Ψ l 2N ? h )[Im(U T CU ) + Re(U T CU )iγ5 ] Ψ Ψl Ψh GM . (42)

In the leading order, we obtain

2 (U T CU )11 = ?vH m1 , 2 2 (U T CU )22 = (vH + 4vχ )| M | ,

(43)

with no imaginary parts to the accuracy of our expansion. This con?rms that the diagonal scalar couplings of the majoron are zero, contrary to the conclusion in Eq.(16) of Ref[6].

General Proof of Pseudoscalar Nature

The pseudoscalar nature of the Goldstone boson when coupled to the fermion is very generic, not limited to the neutrino models discussed above. It applies to the Goldstone boson coupling to chargino and neutralino as well. Below, we work out the detail in the basis of ψR and ψL . The fermion mass matrix M de?ned in the weak basis is diagonalized by bi-unitary transformation, U ′ MU ? = MD , ?R MψL = Ψ ? R MD ΨL . ?LM ? ψ (44)

Di?erent choices of vev’s correspond to various rotations eiT φ generated by the generator T of the spontaneously broken symmetry. With the overall physics unchanged, the physical diagonal mass matrix MD maintains basis independent, if we impose the following substitutions, U ? → eiφTL U ? , (45) U ′ → U ′ e?iφTR . The invariance δ (U ′ MU ? ) = 0 gives U ′ (δM )U ? = ?[(δU ′ )MU ? + U ′ M (δU ? )] = ?[(δU ′ )U ′ MD + MD (UδU ? )] . For an in?nitesimal φ, we have

d M , δM = iφ(T v )i dv i ?

(46)

δU ′ = ?iφU ′ TR ,

δU ? = iφTL U ? .

(47)

Therefore, Y ≡ [U ′ TR U ′ MD ? MD (UTL U ? )] ,

? d U ′ (T vi )( dv M )U ? = ?Y . i

(48)

As MD is diagonal and U ′ TR U ′ ? , UTL U ? are Hermitean, the diagonal entries of above expression Y is purely real.

The NG boson corresponding to the generator T is √ √ G = (T v )i Im 2Φi /N , or, 2ImΦi = (T v )iG/N + · · · ,

(49)

with the normalization N = |T v |. Here we adopt the convention that vi are all real. This can always be achieved by ?eld rede?ntion, sometimes inducing complex couplings. Its Yukawa terms in the Lagrangian become √ ? R U ′ (T v )i ( d M )U ? Ψ L , ? R U ′ (iImΦi d M )U ? ΨL = ?i( 2N )?1 GΨ (50) L ? ?Ψ dvi dvi √ √ ? R Y ΨL ? iΨ ? L Y ? ΨR ] = ?( 2N )?1 GΨ ? i Yii iγ5 Ψi + · · · . L ? ( 2N )?1 G[iΨ (51)

This proves the diagonal coupling of NG boson is pseudoscalar without assuming CP conservation. In the following, we provide another way to understand this result by the current conservation. The divergence of the T current is zero because of symmetry. ?L TL γ ? ψL + ψ ?R TR γ ? ψR + Φ? i ? ? T Φ , J T,? = ψ √ ? L γ ? UTL U ? ΨL + Ψ ? R γ ? U ′ TR U ′ ? ΨR ? 2(N/ 2)? ? G + · · · . . =Ψ √ ? L γ ? UTL U ? ΨL + Ψ ? R γ ? U ′ TR U ′ ? ΨR ] = ( 2N )?? ? ? G . ?? [Ψ Making use of the equation of motion, i ? ΨL,R = mΨR,L , we obtain √ ? R [U ′ TR U ′ ? MD ? MD (UTL U ? )]ΨL + H.c. . ( 2N )?? ? ? G = ?iΨ √ ? R Y ΨL ? iΨ ? L Y ? ΨR ] ( 2N )?? ? ? G = ?[iΨ

?

(52) (53)

J T,?

In the low energy limit, (54)

(55) (56)

Thus, this equation can be e?ectively generated by √ 1 ? i ? Ψ ? [Ψ ? R MD ΨL + H.c.] ? ( 2N )?1 G[iΨ ? R Y ΨL ? iΨ ? L Y ? ΨR ] . (57) (?G)2 + Ψ Le? . = 2 So we have consistent results from two di?erent approaches.

References

[1] Q. R. Ahmad et al., SNO Collaboration, Phys. Rev. Lett. 89, 011301, 011302 (2002). [2] S. Fukuda et al., Super-Kamiokande Collaboration, Phys. Rev. Lett. 86, 5656 (2001) and references therein; Q. R. Ahmad et al., SNO Collaboration, Phys. Rev. Lett. 87, 071301 (2001). [3] S. Fukuda et al., Super-Kamiokande Collaboration, Phys. Rev. Lett. 85, 3999 (2000) and references therein.

[4] J. Goldstone, Nuovo Cimento, 9, 154 (1961). See reviews by E. S. Abers and B. W. Lee, Phys. Rept. 9 1, (1973); H. Georgi, ‘Weak interactions and modern particle theory’ (Benjamin/Cummings Pub. Co., 1984); S. Coleman, ‘Secret Symmetry: an introduction to spontaneous symmetry breakdown and gauge ?elds,’ in Aspects of Symmetry: Selected Erice Lectures of Sidney Coleman (Cambridge University Press, Cambridge, 1985); S. Weinberg, ‘The Quantum Theory of Fields II’, (Cambridge University Press, Cambridge, 1996); and references therein. [5] An intrigue example of this derivative nature is given by D. Chang and W.-Y. Keung, Phys. Lett. B 217, 238 (1989). [6] X. G. He, B. H. McKellar and G. J. Stephenson, Phys. Lett. B 444, 75 (1998) [arXiv:hep-ph/9807338]. [7] H.P. Nilles, Phys. Rept. 110, 1 (1984); H. Haber and G. Kane, Phys. Rept. 117, 75 (1985); S. Dawson, arXiv:hep-ph/9612229; M. Drees, hep-ph/9611409; M. Peskin, hep-ph/9705479; etc.. [8] D. Chang, W. F. Chang and W.-Y. Keung, arXiv:hep-ph/0205084, Phys. Rev. D (in press). [9] Y. Chikashige, R. N. Mohapatra and R. D. Peccei, Phys. Lett. B 98, 265 (1981). [10] G. B. Gelmini and M. Roncadelli, Phys. Lett. B 99, 411 (1981). [11] E. Akhmedov, Z. Berezhiani, R. Mohapatra and G. Senjanovic, Phys. Lett. B299, 90(1993); I. Rothestein, K.S. Babu and D. Seckel, Phys. Lett. B300, 367(1993). [12] D. Chang, W.-Y. Keung and P. B. Pal, Phys. Rev. D 42, 630 (1990).

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