Rare B → K∗ℓ+ℓ− decay, two Higgs doublet model, and light cone QCD sum rules

2 October 1997

PHYSICS

EUEVIER

LETTERS

6

Physics Letters B 410 (1997) 216-228

Rare B -+ K * p/decay, two Higgs doublet model, and light cone QCD sum rules T.M. Aliev a,‘, M. Savcl a,2,A. 6zpineci a, H. Koru b ’ Physics Department. Middle East Technical Vniuersity, 06531 Ankara, Turkey b Physics Department, Gazi University. 06460 Ankara. Turkey Received 22 April 1997; revised 25 July 1997 Editor: R. Gatto

Abstract The decay width, forward-backward asymmetry and lepton longitudinal and transversal polarization for the exclusive B + K * FEdecay in a two Higgs doublet model are computed. It is shown that all these quantities are very effective tools

for establishing new physics. 0 1997 Elsevier Science B.V.

1. Introduction Experimental discovery of the inclusive and exclusive B + X,y and B + K *y [I] decays stimulated the study of rare B meson decays in a new manner. These decays take place via flavor-changing neutral current (FCNC) b + s transitions which are absent in the Standard Model (SM) at tree level and appear only at the loop level. Therefore the study of these decays can provide sensitive tests for investigating the structure of the SM at the loop level, searching for new physics beyond the SM [2]. Currently, the main interest is focused on the rare B meson decays for which the SM predicts large branching ratios and that can be potentially measurable in the near future. The rare B + K * pF(f= e, p, 7) decays are such decays. For these decays the experimental situation is very promising [3] with e+e- and hadron colliders focusing only on the observation of exclusive modes with I= e, p and T as the final states. At the quark level, the process B -+ K * FF is described by the b -+ sZ+I- transition. In literature [4-171 this transition has been investigated extensively in both SM and two Higgs doublet model (2HDM). It is well known that in the 2HDM, the up type quarks acquire their masses from Yukawa couplings to the Higgs doublet Hz (with the vacuum expectation value u,) and down type quarks and leptons acquire their masses from Yukawa couplings to the other Higgs doublet H, (with the expectation value u, ). In 2HDM there exist five physical Higgs fields: neutral scalar Ho, ho, neutr a 1 pseudoscalar A0 and charged Higgs bosons H *. Such a model

I

E-mail: [email protected]. ’ E-mail: [email protected].

0370.2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00968-4

TM. Aliev et al. / Physics ktters B 410 (1997) 216-228

217

occurs as a natural feature of the supersymmetric models 1181. In these models the interaction vertex of the Higgs boson and fermions depends on the ratio tanp = 2 which is a free parameter in the model. The constraints on tan/3 are usually obtained from B - B, K - K mixing, b + sy decay width, semileptonic decay b -+ cry7 and is given by [19,20]:

(1) (the lower bound m,+z 200 GeV is obtained in [ZO]). In 2HDM the charged and neutral Higgs interactions with fermions induce new contributions to the FCNC CL’= e, pu) decay the contributions from neutral Higgs boson exchange are usually processes. For b + SF/ neglected, due to the fact that the interaction of the neutral Higgs and leptons is proportional to the lepton mass. But for the b + sr+ T- decay the mass of the 7 lepton is not too small compared to the mass of the b-quark, therefore the neutral Higgs boson exchange diagrams can give considerable contributions to the process. Recently it has been emphasized by Hewett [21] that T lepton longitudinal polarization is an important observable and may be accessible in the B + X,7+ T- decay. In [22] it is shown that in addition to longitudinal component PL of polarization of the T lepton two other orthogonal components P,, which lies in the decay plane, and PN which is perpendicular to the decay plane, are also very significant for the r+~- channel, since they are proportional to the mass of the 7 lepton. The polarization components PL, P, and PN involve different combinations of the Wilson coefficients C,, CG” and C,, (see below) and hence contain independent information. Therefore, polarization effects can play a central role for the investigation of the structure of the SM and for establishing new physics beyond it. In calculating the branching ratios and other observables at hadronic level, e.g., for B + K * ptdecay, we face the problem of computing the matrix element of the effective Hamiltonian responsible for this decay, between B and K * states. This problem is related to the non-perturbative sector of QCD and it can be solved only by means of a non-perturbative approach. These matrix elements have been investigated in the framework of different approaches such as chiral theory [23], three point QCD sum rules method 1241, relativistic quark model by the light-front formalism [25], effective heavy quark theory [26] and light cone QCD sum rules [27] (see also [28]). The aim of the present work is to calculate these matrix elements using the light cone QCD sum rules formalism in the framework of the 2HDM, taking into account the newly appearing operators due to the neutral Higgs exchange diagrams, and to study the forward-backward asymmetry and final lepton polarization for the exclusive B + K * /‘+/decay. Taking into account the additional neutral Higgs boson exchange diagrams, the effective Hamiltonian is calculated in [29] as

where the first set of operators in the curly brackets describe the effective Hamiltonian responsible for the b + sl+Z- decay in the SM. The value of the corresponding Wilson coefficients includes the contribution of the diagrams with H * running in the loop (see [7]), i.e. C,( M,) =

C7”“(M,) + cy-( M,),

T.M. Alieu et al. /Ph.vsics

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Letters B 410 (19971 216-228

where

C,“-(G)

-

- 3(y-1)3

6(y-Q2 1 - 4sin28, = sit&

&

(Y-

1)’

1 (y-l)‘lny 3y”-6y+4

47y2 - 79y + 108

GWW) =

&tg’/? y I

lny I

W

- ctg’P Y

12

8y3+5y-

3y-2

5y - 3

108(y-

1)3

-

18( y-

- -

i

W

y-l

1.

1

1 ctg$

1)4

+

(Y_q'lny

lny

1 ,

The explicit forms of Cs”(Mw >, C,$“(MW), and C,“,“
*

+

(sin2cu+ hC0s2a)ft( X,Y)

;

m2,o

+

i Z)

(sin2a + hcos2a!)( 1 -

I mW

sin22 (Y + ___ 2m2,k

1

f2( X,Y)

1i

(m$ +rn2,0)~ 2m2,o

L

(3)

f,(Y) '

rn$ -

(4) (5) cQ,(mw)=

s

[C,,(mw) - cu:bdl

e

(6)

’

(7)

i=5,...,10,

C,im,)=O,

where 2

2 .X=2,

_Y=-

>

rnfj*

m,

xlnx fI(X,Y) = x_l f3(Y) =

X

mt

m,

-

1 -y+ylny (y_ 1)2

‘=G

ylny y-l ’

2 +!?I,

Y'

m2,0

xlnY fdx3y)=

(z-x)(x-

lnz 1) + (z-1)(x-

1) ’

219

TM. Alieo et al. /Physics Letters B 410 (1997) 216-228

The QCD correction to the Wilson coefficients Ci(mw) and Cp,(mw)canbe calculated using the renormalization group equations, In [29] it was shown that the operators 0, and O,, do not mix with Qi (i = 1,. . . ,lO), so that the Wilson coefficients C, and C,, remain unchanged and their values are the same as in the SM. Their explicit forms can be found in [29], where it is also shown that 0, can mix with Qi. But additional terms due to this mixing can safely be neglected since the corrections to the SM value of C, arising from these terms are less than 5% when tan@ I 50. 10) do not mix with Q, and Q2 and also there is Moreovertheoperators Oi(i= l,...,lO)and Qi(i=3,..., no mixing between Q, and Q2. For this reason the evolutions of the coefficients Co, and CpI are controlled by the anomalous dimensions of Q, and Q2 respectively: Ce(mb) = 7j-Ye/PoCQXmw),

i= 1,2,

where 7; = - 4 is the anomalous dimension of the operator S, b, [30]. Neglecting the strange quark mass, the matrix element for b + se’F

-2C,2SioP,,q’(1

+ ys)beypf+

C,,3(1 + ys)bek’+

decay is [29]:

CQzS(l + ys)bey5f

4

, 1

(8)

where q2 is the invariant dileptonic mass. Here the coefficient CG“( p,q2) E C,( p) + Y( p, p2), where the function Y contains the contributions from the one loop matrix element of the four-quark operators [7,31,32]. It is clear that the last two terms in Eq. (8) can give considerable contribution only for the B --f K *T+r-decay mode, since C,, and CQl are proportional to the lepton mass m/. In addition to the short distance contributions, it is possible to take into account the long distance effects associated with real CZin the intermediate states 7 i .e -9 the cascade process B + K *J/J/(+‘) -+ K *PT. These contributions are taken into account by introducing a Breit-Wigner form of the resonance propagator and this procedure leads to an additional contribution to CG“ of the form [I 1,331 --

37T ff2 V=J/$.$‘,... =

mvT(V+/+T) (q2-m2y)

-im,T,

’

From Eq. (8) it follows that, in order to calculate the branching ratio for the exclusive B + K *T+~-decay, the matrix elements (K’ISy,(l - y,)blB), (K*lGa,,q”(l + y,)blB), and (K*lS(l + y,)blB) have to be calculated. These matrix elements can be written in terms of the form factors in the following way: (K*(p.~)lsy,(l-y,)blB(p+q))=

-~p,pg~*YpPqum

2V( q2) +m B

-i$(ms+mK.)A,(q2)

K’ A21q2)

+i(2P+qM~*dm

+m B

+

iq,(E*q)

K’

F [A,(q2)- ‘%]

I

+2i(E*q) 4,-(2P+q), where E is the polarization vector of K * meson.

(9)

1

q2 qq”),

rni-m2,.

I

(10)

220

T.M. Aliev et d/Physics

Letters B 410 (1997) 216-228

To calculate the matrix element ( K * 1S( I + +ys)blB > we multiply both sides of Eq. (9) by qP and use the equation of motion. Neglecting the mass of the strange quark, we get: (K*(P&(~

+ y$(B(p+q))

= ~j’(.y)(m,+m,,)A,(q2) +i(E*q)(mB-mK*)Az(q’) +G*q)2m,&%(q’) Using the equation

-A&‘)]}.

(11) can be written as a linear combination of the formfactors

A3(q2)

of motion, the formfactor

A, and A2 (see [241): m,+??l,.

A3(q2)=

“B,mmK’ A*( 9’).

A,( q2) -

zrn

(12)

KS

K’

Making use of this relation we obtain:

(K*(p,+(l

+rx)@(~+q))=

%{-+*q)Ao(q2)}.

(13)

From Eqs. (8), (91, (10) and (13) we obtain for the matrix element of the B + K * p/decay: Ga A=

-I/,bV,:(ey’e[2AE,,,,,~*YpPqu+iBIEL;

-iB,(e*q)(2p+q).-iB,(e*q)q,]

2fi7r

+ey’*Ys~[2C~~,p,~*vpPqu+iD,~~~

-iD2(e*q)(2p+q),-iDde*q)qp]

(14

+iF,(E*q)Pf+iF2(E*q)?~s~), where A = C;” m

+4C,qT,.

,“,

4

K’

B

B,=C;ff(mB+mK.)A,+4C7+m;-m;+‘2,

llZb

4 B, = C;ff m

+4C,q

,“2, B

B, = - C;“-

T2+ 4

K’

2m,. (A,-A,)+4C+, q2

A2 D2

=

Cl0

m,+m,.

2m,. D, = C,, -(A3 q2

’ -A,)

1

i

q2 mi--m,.

2

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TM. Aliev et al. / Physics Letters B 410 (1997) 216-228

2m,. F, = C,, -A

0'

mb

2m,. -A F, = CQ2

0'

mb

The formfactors V, A,, A,, A,, T,, T2 and T3 are calculated in the framework of light-cone QCD sum rules in [27] and their q2 dependence, to a good accuracy, can be represented in the following pole forrn: 0.55

0.26

v(q2) =

Ao(q2)

0.18 TI(q2)

0.40 A2(q2)=

0.18 2 3 T,(q2)=

=

0.36 4(q2)=

=

0.14 2t

T,(q’)

=

2'

(16)

which we will use in further numerical analysis (for A, we use Eq. (12)). Note that in the analysis of the q2 dependence of the formfactors, we have also used the linear pole form as an alternative fit. No significant difference has been observed in the numerical calculations resulting from Eq. (16) and the linear pole form fit. Using Eq. (14) and performing summation over final lepton polarization, we get for the double differential decay rate: dT -= dq’dz

G2a21V,,v,,* j2h1’2u 2hmi[4m:(

2’27r 5m B

(A(’ - ICI’) + m;,s( 1 + u2z2)(( Al2 + Icl’)]

+

+2Re(D,D;)

+ 4zp’[Re(F’2D;)

-2(1-r)miRe(D,D;)]

-(l-r)miRe(F2D;)

+misRe(F,D;)]

b

-miA’/2z

4m,- ~~(m~Re(B,F,‘)-(1-r-s)Re(B,FT))

[ +8misv{Re(

B,C*)

+Re(

AD;))

,

(17)

T.M. Aliev et d/Physics

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Letters B 410 (19971 216-228

where z = cos0, 8 is the angle between the momentum of the E’ lepton and that of the B meson in the center of mass frame of the lepton pair, A= 1 +r’+s2-2r-2s-2rs, r=m~~/rn~, s=q2/mi and u q---=1 - 4m,/q- , m, are the lepton velocity and mass respectively. If we put F, = 0 and F, = 0 (which corresponds to the case where the contributions of the neutral Higgs boson exchange diagrams are neglected) we get the results of [27]. As we noted earlier, the forward-backward asymmetry and final lepton polarization asymmetry are very useful tools for extracting more precise information on the Wilson coefficients C,, CGff and C,, and also for searching new physics. Therefore in this work we shall study these quantities as well. The forward-backward asymmetry A, is defined in the following way:

I

A&q?)=

‘dz-

dl-

’

d;T

/0

‘dz -+ dq2dz

-

dT ’ dz1 -I d&z dr ’ ’ dz/ -1 dq2dz

Let us now discuss the final lepton polarization.

eL=

eN=

fi,

PK’

‘Pi xp,I

IPK,

’

We define the following

three orthogonal

unit vectors:

eT=eNXeL’

where p, and pK * are the three momenta of the F and the K * meson, respectively, in the center of mass of the fir system. The differential decay rate for any given spin direction n of the F lepton, where II is a unit vector in the f lepton rest frame, can be written as

(18) where the subscript “0” corresponds to the unpolarized case and it can be obtained from Eq. (17) by integration over Z. From definition of e, it is obvious that P, lies in the decay plane whose orientation is organized by the vectors p, and pK., and PN is perpendicular to this plane. The polarization components Pi (i = L, T, N) are defined as:

$(n =e,) pj(q2)=

dr

dyz(n =ei) After standard calculations pL= f

phm6,Re(

i

$(n =-e,) + $(n =-ei) ’ -

for P,(i = L, T) we get

AC*)

h2m6 + t LRe(B,D; r

--:+(I--r-s)[Re(B*D;)+Re(B,D;)1

-8~ zp[Re(,F,D;) b

( 19)

-(I

) +4y

4 [

3 +4rs

1

Re(B,D;)

-8FRe(F,I;)

-r)Re(F,D;)

+m:sRe(F,D;)ii,

(20)

T.M. Aliev et al./Physics

Re(B,D;) fi - Amim/--Re

-

fi +mim,(l-r-s)TRe(B,D;)

\rRe(B,F;) r

6

(1

-Re(B,F,*)-mimr(l-r-s)mb Tr r-d

r d--s -r-s)-

+m”,&(l

fi mb

+m4,(1 -r--s) +mBm/(l

+8mimfiRe(AB;)

( B, 0; ) - hm6,-

r

223

Letters B 410 (1997) 216-228

-

r)

Re( B,Dl)

i-v7 u2

Re( B,D;)

- hm6,fiTRe(DzF1*) mb

u2

Re( D,F,*)

’

-

(21)

mb&

The denominator A in Eqs. (20) and (21) can be obtained from Eq. (17) by integration over z of the terms within the curly bracket. We also calculate the Phi component of the lepton polarization, but the numerical results show that its value is quite small and because of that we do not present its explicit form here.

2. Numerical analysis The values of the main input parameters, which appear in the expression for the decay width are: mb = 4.8 GeV, m, = 1.35 GeV, m7 = 1.78 GeV, mP = 0.105 GeV, AocD = 225 MeV, mB = 5.28 GeV, and mK. = 0.892 GeV. We use the pole form of the formfactors given in Eq. (16). For B meson lifetime we take T( Bd) = 1.56 X lo-i’s [34]. The values of the Wilson coefficients Cf”(mb) and Cs,M(mb) to the leading logarithmic approximation are [35,36]: C, = -0.315, C,, = -4.642. The expression CGff for the b -+ s transition in the next to leading order approximation is given as (see for example [35]) Cgff(mb)=C9SM(mb)+C9H-(mb)+0.124W(8)+g(mc^,S^)(3C,+C2+3C3+Cq+3CS+Cg) - +g(m@,?)( C, + 3C,) - +g( mb,s^)(4C,

+ 4C, + 3C, + C,)

+ $(3C, + c, + 3c, + C,) )

(22)

with C, = -0.249,

C,= 1.108,

c, = 7.4 x 1o-3 )

C, = 1.112x 10-2,

C,= -2.569x

10-2,

C,““( mb) = 4.227,

C, = -3.144 X 10v2,

where mq^= %, s^= $. In the above expression, w(s^) represents the one gluon correction to the matrix element 0, and its explicit form can be found in [l31, while the function g(mB,s^) arises from the one loop contributions of the four quark operators 0, - 0, (see for example [35,36]), i.e. g( mg,i)

= - $lnmg + j$ + $y, - $(2 + y,)/E

+

0(1 -y4) i

where y,= y, and s’= $$

I In

1+/K I-\li--y,

-

irr

+ 6( y, - l)arctanI

, &

1

(23)

T.M. Alieo et al./Physics

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12.3

Letters B 410 (19971 216-228

IS.0

17.5

q2

.I1

(GeV2)

Fig. 1. Invariant mass squared (q*> distribution of the 7 lepton pair for the decay B --f K *T+ T-. Lines 1, 3 and 4 correspond to the short distance contributions for the values of tanP = 50, tanP = 30 and tanfl = 1, respectively; while lines 2, 5 and 6 correspond to the sum of the short and long distance contributions for the values of tanp = 50, tanp = 30 and tanP = 1. respectively.

In all numerical calculations we use the following values for the masses of the Higgs particles: mH * = 200 GeV, rnp = 80 GeV, mHo = 150 GeV, and mAo = 100 GeV. We also take sina = a/2 and for tanP we choose the following set of values: tan /3 = 1, tan/3 = 30 and tanp = 50. In Fig. 1 we present the q* dependence of the differential branching ratio for the B, -+ K *T+T- decay with and without the long distance effects. It follows from this figure that the differential branching ratio is sensitive to the value of tanp if the I,//’ mass region is excluded. For example at tan/3 = 50 the differential branching ratio is approximately 2 times larger than the one at tan/3 = 1. In Fig. 2 we plot the dependence of the forward-backward asymmetry A,, on q2 with and without the long-distance effects at different values of tanp. From this figure we see that A, is positive for all values of q2 except in the I,!J’resonance region and it is sensitive to the value of tanp.

I

12.:

I

I

I

13.0

17.5

2n.n

q2 Fig. 2. The dependence lines 1 to 6.

of the forward-backward

asymmetry

A,,

(GeV2)

on y2 for the decay B + K *~+r-.

See Fig. 1 for the interpretation

of

225

T.M. Alieu et al./ Physics Letters B 410 (1997) 216-228

q2 Fig. 3. The dependence

of the longitudinal

polarization

(GeV*)

PI, on q2 for the B 4 K ‘T+ T-

See Fig. 1 for the interpretation

of lines 1 to 6.

In Fig. 3 we depicted the q2 dependence of the longitudinal polarization of the final lepton PL with and without the long distance effects at different values of tan& As we see from this figure if we exclude the resonance mass region of I//, PL is negative for all values of q2 at tanp = 1 as well as at tanp = 30 and tanp = 50. In Fig. 4 we present the q2 dependence of the transversal polarization P, of the T lepton which lies in the decay plane, without long distance effects at tanp = 1, tanjl = 30 and at tanp = 50. From this figure it follows that at tanp = 1 P, is positive at all values of q*. For tan/3 = 30 and tan j3 = 50, P, is positive near the threshold region but far from the threshold it becomes negative. Therefore the determination of the sign of P, in future experiments is a very important issue and can provide a direct information for the establishment of new physics. In Fig. 5 we present the differential branching ratio versus q2 for the B + K */l+pL-decay. From this figure and Fig. 1 it follows that the differential branching ratio for the B + K *pL+p-decay is approximately one order

I

0.13

I

I

I

-

T-

+L

0.00

& t

-

-

-0.13

Q

-‘.

-Ti

a,

-0.30 -

‘h

! %.

%_-L_

__.__

,!’

tan3 = 30

-.-___---._._._,___._____.---’ -0.43

f

_.I.

I

,

I

1

12.5

15.0

li.5

20.0

q2

(Gel’*)

Fig. 4. The same as in Fig. 3 but for the transversal

polarization

P,.

226

734. Alieu et d/Physics

Letters B 410 (1997) 216-228

I

I

I

I

1

(1.0

3.0

10.0

l%Xl

2o.n

4’ (GeV*) Fig. 5. The same as in Fig. 1. but for the B + K *lLfpL

decay

of magnitude greater compared to that for the B + K *r+ r-decay. Note that for tan p = 30 and tan/? = 50, the results for the differential branching ratio are practically the same. on q’ at tanp = 1 and tan/3 = 50. From a In Fig. 6, we depict the dependence of A,(B -+ K *p’p-) comparison of this figure with Fig. 2 we observe that the behaviour of A, for the B + K *p+~-decay is totally different from that for the B + K *~+r-decay. In the B + K *r+r-case A, is positive for all values of q2 (without the long distance effects), while it changes its sign here in the B + K *p+/Lcase. This indicates that for the B -+ K *r+r-case, the neutral Higgs boson exchange diagrams give considerable contribution. It follows from these two figures then that the determination of the sign of A, in different kinematical regions is also a very important tool for establishing new physics. The behaviour of the longitudinal polarization with changing q2 is presented in Fig. 7. In the B -+ K *p+ p-decay, PL is firstly positive and then it changes sign (without the long distance effects), which is absolutely different from that of the B + K *r+r-case, for which PL is negative for all q*.

-0.6

q* (GeV*) Fig. 6. The same as in Fig. 2, but for the B + K *~+II.- decay. Lines 1 and 3 correspond to the short distance contributions for the values of tanp = 50 and tanp = 1, respectively; while lines 2 and 4 correspond to the sum of the short and long distance contributions for the values of tan p = 50 and tan p = 1, respectively.

221

TM. Alieu et al. / Physics L.etters E 410 (1997) 216-228

q2(GeV*) Fig. 7. The same as in Fig. 3, but for the B + K *p+p-

decay. See Fig. 6 for the interpretation

of lines 1 to 4.

PL, and PT contain independent information, their investigation in Since, as we have already noted, A,, experiments will be a very efficient tool in establishing new physics. At the end of this section we present the values of the branching ratios for the B, + K *T+T- decay. After integrating over q2 we get for the branching ratios for the B, --) K *r+ T- decay, without the long distance contributions,

B(B/K*T+T-)=

1.05 x lo-’

(tat@ = 1)

0.76~10-~

(tanp = 30)

I 0.92 x 1O-7

(24)

(tat@ = 50).

The ratio of the exclusive and inclusive channels is defined as B(B~K*T+T-) R=

B(b + ST’T-)

’

In the SM this ratio is given as R = 0.27 + 0.07 when B(b -+ ST+ 7-j = (2.6 + 0.5) X lo-’ [36]. In our case we have

R=

0.40

(tan j3 = 1)

0.29

(tanp=

I 0.35

30)

(25)

(tan/3 = 50))

where we use the SM value for the inclusive B( b -+ ST+ 7-j. In conclusion, we calculate the rare B + K * FF decay in 2HDM. It is observed that A,, PL and the transversal polarization P, of the charged lepton are very sensitive to the value of tan p. Therefore, in search of new physics, their experimental investigation can serve as a crucial test.

References [I] MS. Alam et al., CLEO Collaboration, R. Ammar et al., CLEO Collaboration,

Phys. Rev. Lett. 74 (1995) 2885; Phys. Rev. Lett. 71 (1993) 674.

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T.M. Alieu et d/Physics

Letters B 410 (1997) 216-228

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Physic;