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Searching for the standard model Higgs particle in Bs decays
Volume 215, number l
PHYSICS LETTERS B
8 December 1988
S E A R C H I N G FOR T H E S T A N D A R D M O D E L H I G G S PARTICLE IN B~ DECAYS G. EILAM ~and A. SONI
Department QfPhysics. Universityof California, Los Angeles, CA 90024. USA Received 6 September 1988
Purely leptonic B~decay, B ~ + z and B~-~la+p- are shown to be very sensitive to a standard model (SM) Higgs particle, for a t quark of mass ~>100 GeV, and for a Higgs mass of up to about 7 GeV, the latter being exchanged in the s-channel. Bd-o~+~decays are also discussed. Theoretical cleanliness (i.e. absence of long-distanceeffects) of these modes also suggests that they should be very useful as precision tests of the SM and indicators of new physics.
The mass of the standard model (SM) Higgs particle is practically u n c o n s t r a i n e d a~, but its couplings are well-defined [2 ]. A search strategy for the Higgs particle is then determined by the mass range scanned, the accelerator and by the experimental setup. B decays have been suggested as a source of Higgs particles if M H < M s , through the flavor-changing loop process b-~ s + H [ 3-8 ]. The relevant vertex is given by [3] A (bsH) = ~
3
change, the other SM processes contributing to b g - ~ + ~ - are [4,11,12] s-channel Z exchange (with a flavor-violating loop bsZ vertex) and a box diagram. Thus: A ( b g ~ + ~ - ) = [C~ ( r h { ) g R b . ~
+ C2 (rhZ)~7~,Lb'~7~'~ ] vt,
(2)
where R, L = ( 1 _+75 )/2, and
3g4rhjh~x g3rhbrhZvtg( 1 + 7 5 ) b ,
( 1)
where rh-m~/Mw, vt- Vtt,V'~, g2/8Mw=Gv/21/2 and external masses and MH are assumed to be smaller than m~ and Mw. Although doubts have been raised about the validity of the above result (see ref. [ 2 ] for reviews and further references) a recent calculation using different methods [ 8 ] arrives at the same effective coupling as in eq. (1). Furthermore, the bsH vertex suffers only m i n o r q u a n t u m c h r o m o d y n a m i c s ( Q C D ) corrections [ 6, 9,10 ]. In the present letter we propose to use the bsH vertex for the related processes Bs--,~+~ - where ~ =~t or z. In addition to the s-channel Higgs particle ex1 Permanent address: Department of Physics, Technion, Haifa 32000, Israel. *~ The theoretical lower limit ofMu> 7 GeV that is often quoted [ 1] holds only if the top quark mass is below 80 GeV andwith three generations of quarks and leptons. Furthermore, Mn < 1 TeV from perturbative unitarily. See ref. [2] for reviews and further references. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
C, (x) = 2567r2(m~ _ M 2 ) ,
(3)
g4x C2(x) = 3 2 ~ w V ( X ) ,
(4)
with
1
3
x
3
1
F ( x ) = ~ + 4 ( l - x ) 21°gx+ 4 1----~'
(5)
and m~ is the lepton mass. Thus, placing the b and g in Bs we find (note that there is no interference t e r m ) F(B~-~+~
T M
)
G 4r2 . 3 / -
F J Bs rrl Bs ~ 1 - -
32n 5
\"
4m~'~ '/2 2 | 2 4 mB~J m~rn~]vt{ 2
(9m~(1-4m2/rn2~) X
64(m2~-M2) 2
1 F(rh()2) ' + m~
(6)
Note also that although the Higgs particle is off-shell, the on-shell vertex in eq. (1) is valid as long as 171
Volume 2 l 5, number 1
PHYSICS LETTERS B
[ 13,14 ] rnH << Mw, which applies to the present discussion. We should also add that the part of our result corresponding to Z exchange plus the box contribution is in agreement with ref. [ 12 ] ~2 The branching ratio is deduced by writing
--- 0.27
F(B~--*£+ £ - )
iv, i ~/~o
,
l
~
I
I
;
I
~
.....
7"
I cff
F(B, ~£+£ - ) B R ( B - , e v + X) = F(B--,ev+X)
BR(B~£+£-)
1 0 -4 ~
8 December 1988
10"6
(7) 2
where
6
I0
14
18
Mu (GeV) 5
G~M~
Fo = 192rc3 ,
(8)
andwe used ] Vb~l2_~ t V~I2, B R ( B ~ e v + X ) =0.12 and
r ( ~ - ~ e v + X ) =£ol r ~ 120.44,
(9)
where QCD and phase space tilctors are included in the last equation. Results are presented with the following values of the parameters:f~ = 0.16 GeV [ t 5 ] (we use a normalization such that f~ = 0.132 GeV) and mB~=5.38 GeV (based on m ~ , - m ~ = m D , rnr~ ). The precise value o f f ~ is unknown at present, and it is therefore the largest source of uncertainty for the branching ratio once the values o f MH and rn~ are fixed. WithfB~ =0.2 GeV (which is just about the maximum allowed by the current lattice computations [ 15 ]) our results are increased by approximately 60%. In fig. I the branching ratio for B~--,x+x- is plotted versus MH for rrh = 200 GeV (full line). The prediction of the SM without the s-channel Higgs particle exchange is given by the dashed line. Only for ?fin > 7 GeV can the Higgs particle exchange be completely ignored. With 106 B , the mass range 3 ~
172
Fig. 1. Full line: Branching ratio for B ~ + z ~ as a function of the SM Higgs boson mass, for m , = 2 0 0 GeV. Dashed line: SM without the Higgs particle exchange in the s-channel.
in constraining the Higgs particle an accurate measurement of B~-~x+x - should therefore be very useful. In fig. 2 we present the same branching ratio, for m, = 100 GeV; results are typically" about an order of magnitude lower than for m,=200 GeV. Clearly (in the three-generation SM) with m,= 50 GeV the mode becomes uninterestingly small. On the other hand, the large Bd-f~0 mixing that has been observed [ 16 ] and confirmed [17 ] ensures that SM (with three families) requires a very heavy t quark (i.e. m~>~100 GeV). Since the z+x- final state is difficult to identify, the Bs~g+g - decay mode is obviously preferred. However, one loses a factor of r n J m , from the H££ coupling. The loss in the branching ratio is slightly smaller than ( m J m , ) 2 as can be seen in fig. 3, where the 10"5
i 10-T
_? 2
6
mt:IOOGeV
10
14
Mn{GeV) Fig. 2, Same as fig. t with m r = 100 GeV.
18
Volume 215, number 1 t ........
PHYSICS LETTERS B
t
m t = 200GeV
7" 10"7
t
10"~
I
2
6
I
I
~.__L~
,.~-t---£-
10
MH(GeV)
14
18
Fig. 3. Same as fig, 1, for B~/,t +~a-. branching ratio for B ~ rt + I.t- for m, = 200 GeV is depicted as a function of Mr~, With 10 s B , which is perhaps not an unreasonable number for planned B factories, the mass range 4~
.B~
<0.2
(10)
Then using [ 15 ] j ~ J J ~ = 0. 7
elusive modes suffers from hadronic uncertainties. All these problems are absent from the purely leptonic B-,~+~ - decays discussed here. It was shown [21,22 ] that CPviolation will lead to lz polarization in K 1,2-* ~t+ ~t- decays, where K L,2 are CP eigenstates. It was subsequently shown [ 13 ] that a light Higgs boson ( M . < 10 GeV) will cause ~t polarization in excess of 10 -3 for KL--,~t+~t TM" A nonvanishing polarization for a CP eigenstate decay, is then equivalent - for B--.~+~ - decays - to a non-zero polarization asymmetry, i.e. a difference between the polarization in B and in 13 decays [23]. Unfortunately the SM, even with a light Higgs particle gives a small (at most 1%) polarization asymmetry. Nevertheless, it will be interesting to look for such an effect in B-~z+z - through angular distributions of the decay products and thus to search for deviations from the SM, So far we have discussed the sensitivity o f Ba,s-,~+~ - for search of light Higgs particles. We also want to use the opportunity to stress that these decay modes are remarkably clean i.e. free o f "long-distance" effects which tend to make the analogous decay KL--, ~t+~t- difficult for theoretical interpretation. Pure leptonic decays o f Bd,~ do not have that problem, the relevant flavor-changing transition being characterized predominantly by heavy virtual quarks only. Furthermore, there are very good reasons to believe that lattice techniques will continue to yield improved determination of the relevant decay constants. These decays should therefore serve as precision tests of the SM and indicators o f exotic physics.
(11)
the ratio is given by B R ( B ~ - ~ + ~ - ) ~0.1. 0.03 ~< B R ( B _ ~ + ~ _ )
8 December 1988
(12)
Therefore, for m~=200 GeV, with 10 6, l0 s Bd's (via ~c+ z - , ~t+ ~t- decays respectively ) a mass range o f approximately 1 GeV width around the Bd mass can be explored in search of the Higgs particle. The Wilczek mode [ 19 ] "t'~TH has been recently used to set limits on MH [20]. However, there are serious questions regarding Q C D and relativistic corrections for that decay [2 ]. Furthermore, while predictions for the inclusive quark process b ~ s H are relatively free from ambiguities, the transition to ex-
The work o f G.E. has been supported in part by the V P R Research Fund, and by the Fund for Promotion o f Research at the Technion. The work o f A.S. was supported in part by the Department o f Energy.
References [ l ] S. Weinberg, Phys. Rev. Lett. 36 294 ( 1976); A. Linde, JETP Lett. 23 (1976) 64. [ 2 ] M.S. Chanowitz, LBL report LBL-24878 (February 1988), in: Annu, Rev. Nucl. Part. Sci. 38, 1988, to be published; J. Ellis and F. Pauss, CERN preprint CERN-TH4992/88 (March 1988), in: Proton-antiproton collider physics (World Scientific, Singapore, 1988), to be published. 173
Volume 215, number l
PHYSICS LETTERS B
[3] R.S. Willey and H.E. Yu, Phys. Rev. D 26 (1982) 3086, 3287. [ 4 ] B. Grzadkowski and P. Krawczyk, Z. Phys. C 18 ( 1983 ) 43. [ 5] R.M. Godbole, U. Turke and M. Wirbel, Phys. Lett. B 194 (1987) 302. [6] H.E. Haber, A.S. Schwarz and A.E. Synder, Nucl. Phys. B 294 (1987) 301. [ 7 ] R.S. Chivukula and A.V. Manohar, Phys. Lett. B 207 ( 1988 ) 86. [8] B. Grinstein, L. Hall and L. Randall, preprint LBL-25095, UCB-PTH-88/6 (April 1988 ), unpublished. [9] L.J. Hall and M.B. Wise, Nucl. Phys. B 187 ( 1981 ) 397. [ 10] J.-M. Fr6re, M.B. Gavela and J.A.M. Vermaseren, Phys. Lett. B 125 (1983) 275. [ 11 ] T. lnami and C.S. Lim, Prog. Theor. Phys. 65 ( 1981 ) 297; see also W.-S. Hou, R.S. Willey and A. Soni, Phys. Rev. Lett. 58 (1987) 1608. [ 12 ] B.A. Campbell and P.J. O'Donnell, Phys. Rev. D 25 (1982) 1989.
174
8 December 1988
[ 13 ] F.J. Botella and C.S. Lira, Phys. Rev. Lett. 56 (1986) 1651. [14] F,J. Botella and C.S. Lira, Phys. Rev. D 34 (1986) 301. [ 15 ] C. Bernard, T. Draper, G. Hockney and A. Soni, preprint IUHET-141, U C L A / 8 7 / T E P / 2 8 (May 1988), Phys. Rev. D., to be published. [ 16 ] ARGUS Collab., H. Albrecht et al., Phys. Lett. B 192 ( 1987 ) 245. [ 17 ] CLEO Collab., A. Jawahery, talk XXIV Intern. Conf. on High energy physics (Munich, August 1988). [18]S. Stone, preprint CLNS-87/103 (October 1987), unpublished. [ 19 ] F. Wilczek, Phys. Rev. Lett. 39 (1977) 1304. [20] M. Narain et al., 1987 Intern. Syrup. on Lepton and photon interactions at high energies (July 1987), eds. W. Bartel and R. Riickl (North-Holland, Amsterdam, 1988). [21 ] L.A. Sehgal, Phys. Rev. 181 (1969) 215. [22] P. Herczeg, Phys. Rev. D 2 7 (1983) 1512. [23] G. Eilam and A. Soni, preprint U C L A / 8 8 / T E P / 1 9 ( J u l y 1988).
PHYSICS LETTERS B
8 December 1988
S E A R C H I N G FOR T H E S T A N D A R D M O D E L H I G G S PARTICLE IN B~ DECAYS G. EILAM ~and A. SONI
Department QfPhysics. Universityof California, Los Angeles, CA 90024. USA Received 6 September 1988
Purely leptonic B~decay, B ~ + z and B~-~la+p- are shown to be very sensitive to a standard model (SM) Higgs particle, for a t quark of mass ~>100 GeV, and for a Higgs mass of up to about 7 GeV, the latter being exchanged in the s-channel. Bd-o~+~decays are also discussed. Theoretical cleanliness (i.e. absence of long-distanceeffects) of these modes also suggests that they should be very useful as precision tests of the SM and indicators of new physics.
The mass of the standard model (SM) Higgs particle is practically u n c o n s t r a i n e d a~, but its couplings are well-defined [2 ]. A search strategy for the Higgs particle is then determined by the mass range scanned, the accelerator and by the experimental setup. B decays have been suggested as a source of Higgs particles if M H < M s , through the flavor-changing loop process b-~ s + H [ 3-8 ]. The relevant vertex is given by [3] A (bsH) = ~
3
change, the other SM processes contributing to b g - ~ + ~ - are [4,11,12] s-channel Z exchange (with a flavor-violating loop bsZ vertex) and a box diagram. Thus: A ( b g ~ + ~ - ) = [C~ ( r h { ) g R b . ~
+ C2 (rhZ)~7~,Lb'~7~'~ ] vt,
(2)
where R, L = ( 1 _+75 )/2, and
3g4rhjh~x g3rhbrhZvtg( 1 + 7 5 ) b ,
( 1)
where rh-m~/Mw, vt- Vtt,V'~, g2/8Mw=Gv/21/2 and external masses and MH are assumed to be smaller than m~ and Mw. Although doubts have been raised about the validity of the above result (see ref. [ 2 ] for reviews and further references) a recent calculation using different methods [ 8 ] arrives at the same effective coupling as in eq. (1). Furthermore, the bsH vertex suffers only m i n o r q u a n t u m c h r o m o d y n a m i c s ( Q C D ) corrections [ 6, 9,10 ]. In the present letter we propose to use the bsH vertex for the related processes Bs--,~+~ - where ~ =~t or z. In addition to the s-channel Higgs particle ex1 Permanent address: Department of Physics, Technion, Haifa 32000, Israel. *~ The theoretical lower limit ofMu> 7 GeV that is often quoted [ 1] holds only if the top quark mass is below 80 GeV andwith three generations of quarks and leptons. Furthermore, Mn < 1 TeV from perturbative unitarily. See ref. [2] for reviews and further references. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
C, (x) = 2567r2(m~ _ M 2 ) ,
(3)
g4x C2(x) = 3 2 ~ w V ( X ) ,
(4)
with
1
3
x
3
1
F ( x ) = ~ + 4 ( l - x ) 21°gx+ 4 1----~'
(5)
and m~ is the lepton mass. Thus, placing the b and g in Bs we find (note that there is no interference t e r m ) F(B~-~+~
T M
)
G 4r2 . 3 / -
F J Bs rrl Bs ~ 1 - -
32n 5
\"
4m~'~ '/2 2 | 2 4 mB~J m~rn~]vt{ 2
(9m~(1-4m2/rn2~) X
64(m2~-M2) 2
1 F(rh()2) ' + m~
(6)
Note also that although the Higgs particle is off-shell, the on-shell vertex in eq. (1) is valid as long as 171
Volume 2 l 5, number 1
PHYSICS LETTERS B
[ 13,14 ] rnH << Mw, which applies to the present discussion. We should also add that the part of our result corresponding to Z exchange plus the box contribution is in agreement with ref. [ 12 ] ~2 The branching ratio is deduced by writing
--- 0.27
F(B~--*£+ £ - )
iv, i ~/~o
,
l
~
I
I
;
I
~
.....
7"
I cff
F(B, ~£+£ - ) B R ( B - , e v + X) = F(B--,ev+X)
BR(B~£+£-)
1 0 -4 ~
8 December 1988
10"6
(7) 2
where
6
I0
14
18
Mu (GeV) 5
G~M~
Fo = 192rc3 ,
(8)
andwe used ] Vb~l2_~ t V~I2, B R ( B ~ e v + X ) =0.12 and
r ( ~ - ~ e v + X ) =£ol r ~ 120.44,
(9)
where QCD and phase space tilctors are included in the last equation. Results are presented with the following values of the parameters:f~ = 0.16 GeV [ t 5 ] (we use a normalization such that f~ = 0.132 GeV) and mB~=5.38 GeV (based on m ~ , - m ~ = m D , rnr~ ). The precise value o f f ~ is unknown at present, and it is therefore the largest source of uncertainty for the branching ratio once the values o f MH and rn~ are fixed. WithfB~ =0.2 GeV (which is just about the maximum allowed by the current lattice computations [ 15 ]) our results are increased by approximately 60%. In fig. I the branching ratio for B~--,x+x- is plotted versus MH for rrh = 200 GeV (full line). The prediction of the SM without the s-channel Higgs particle exchange is given by the dashed line. Only for ?fin > 7 GeV can the Higgs particle exchange be completely ignored. With 106 B , the mass range 3 ~
172
Fig. 1. Full line: Branching ratio for B ~ + z ~ as a function of the SM Higgs boson mass, for m , = 2 0 0 GeV. Dashed line: SM without the Higgs particle exchange in the s-channel.
in constraining the Higgs particle an accurate measurement of B~-~x+x - should therefore be very useful. In fig. 2 we present the same branching ratio, for m, = 100 GeV; results are typically" about an order of magnitude lower than for m,=200 GeV. Clearly (in the three-generation SM) with m,= 50 GeV the mode becomes uninterestingly small. On the other hand, the large Bd-f~0 mixing that has been observed [ 16 ] and confirmed [17 ] ensures that SM (with three families) requires a very heavy t quark (i.e. m~>~100 GeV). Since the z+x- final state is difficult to identify, the Bs~g+g - decay mode is obviously preferred. However, one loses a factor of r n J m , from the H££ coupling. The loss in the branching ratio is slightly smaller than ( m J m , ) 2 as can be seen in fig. 3, where the 10"5
i 10-T
_? 2
6
mt:IOOGeV
10
14
Mn{GeV) Fig. 2, Same as fig. t with m r = 100 GeV.
18
Volume 215, number 1 t ........
PHYSICS LETTERS B
t
m t = 200GeV
7" 10"7
t
10"~
I
2
6
I
I
~.__L~
,.~-t---£-
10
MH(GeV)
14
18
Fig. 3. Same as fig, 1, for B~/,t +~a-. branching ratio for B ~ rt + I.t- for m, = 200 GeV is depicted as a function of Mr~, With 10 s B , which is perhaps not an unreasonable number for planned B factories, the mass range 4~
.B~
<0.2
(10)
Then using [ 15 ] j ~ J J ~ = 0. 7
elusive modes suffers from hadronic uncertainties. All these problems are absent from the purely leptonic B-,~+~ - decays discussed here. It was shown [21,22 ] that CPviolation will lead to lz polarization in K 1,2-* ~t+ ~t- decays, where K L,2 are CP eigenstates. It was subsequently shown [ 13 ] that a light Higgs boson ( M . < 10 GeV) will cause ~t polarization in excess of 10 -3 for KL--,~t+~t TM" A nonvanishing polarization for a CP eigenstate decay, is then equivalent - for B--.~+~ - decays - to a non-zero polarization asymmetry, i.e. a difference between the polarization in B and in 13 decays [23]. Unfortunately the SM, even with a light Higgs particle gives a small (at most 1%) polarization asymmetry. Nevertheless, it will be interesting to look for such an effect in B-~z+z - through angular distributions of the decay products and thus to search for deviations from the SM, So far we have discussed the sensitivity o f Ba,s-,~+~ - for search of light Higgs particles. We also want to use the opportunity to stress that these decay modes are remarkably clean i.e. free o f "long-distance" effects which tend to make the analogous decay KL--, ~t+~t- difficult for theoretical interpretation. Pure leptonic decays o f Bd,~ do not have that problem, the relevant flavor-changing transition being characterized predominantly by heavy virtual quarks only. Furthermore, there are very good reasons to believe that lattice techniques will continue to yield improved determination of the relevant decay constants. These decays should therefore serve as precision tests of the SM and indicators o f exotic physics.
(11)
the ratio is given by B R ( B ~ - ~ + ~ - ) ~0.1. 0.03 ~< B R ( B _ ~ + ~ _ )
8 December 1988
(12)
Therefore, for m~=200 GeV, with 10 6, l0 s Bd's (via ~c+ z - , ~t+ ~t- decays respectively ) a mass range o f approximately 1 GeV width around the Bd mass can be explored in search of the Higgs particle. The Wilczek mode [ 19 ] "t'~TH has been recently used to set limits on MH [20]. However, there are serious questions regarding Q C D and relativistic corrections for that decay [2 ]. Furthermore, while predictions for the inclusive quark process b ~ s H are relatively free from ambiguities, the transition to ex-
The work o f G.E. has been supported in part by the V P R Research Fund, and by the Fund for Promotion o f Research at the Technion. The work o f A.S. was supported in part by the Department o f Energy.
References [ l ] S. Weinberg, Phys. Rev. Lett. 36 294 ( 1976); A. Linde, JETP Lett. 23 (1976) 64. [ 2 ] M.S. Chanowitz, LBL report LBL-24878 (February 1988), in: Annu, Rev. Nucl. Part. Sci. 38, 1988, to be published; J. Ellis and F. Pauss, CERN preprint CERN-TH4992/88 (March 1988), in: Proton-antiproton collider physics (World Scientific, Singapore, 1988), to be published. 173
Volume 215, number l
PHYSICS LETTERS B
[3] R.S. Willey and H.E. Yu, Phys. Rev. D 26 (1982) 3086, 3287. [ 4 ] B. Grzadkowski and P. Krawczyk, Z. Phys. C 18 ( 1983 ) 43. [ 5] R.M. Godbole, U. Turke and M. Wirbel, Phys. Lett. B 194 (1987) 302. [6] H.E. Haber, A.S. Schwarz and A.E. Synder, Nucl. Phys. B 294 (1987) 301. [ 7 ] R.S. Chivukula and A.V. Manohar, Phys. Lett. B 207 ( 1988 ) 86. [8] B. Grinstein, L. Hall and L. Randall, preprint LBL-25095, UCB-PTH-88/6 (April 1988 ), unpublished. [9] L.J. Hall and M.B. Wise, Nucl. Phys. B 187 ( 1981 ) 397. [ 10] J.-M. Fr6re, M.B. Gavela and J.A.M. Vermaseren, Phys. Lett. B 125 (1983) 275. [ 11 ] T. lnami and C.S. Lim, Prog. Theor. Phys. 65 ( 1981 ) 297; see also W.-S. Hou, R.S. Willey and A. Soni, Phys. Rev. Lett. 58 (1987) 1608. [ 12 ] B.A. Campbell and P.J. O'Donnell, Phys. Rev. D 25 (1982) 1989.
174
8 December 1988
[ 13 ] F.J. Botella and C.S. Lira, Phys. Rev. Lett. 56 (1986) 1651. [14] F,J. Botella and C.S. Lira, Phys. Rev. D 34 (1986) 301. [ 15 ] C. Bernard, T. Draper, G. Hockney and A. Soni, preprint IUHET-141, U C L A / 8 7 / T E P / 2 8 (May 1988), Phys. Rev. D., to be published. [ 16 ] ARGUS Collab., H. Albrecht et al., Phys. Lett. B 192 ( 1987 ) 245. [ 17 ] CLEO Collab., A. Jawahery, talk XXIV Intern. Conf. on High energy physics (Munich, August 1988). [18]S. Stone, preprint CLNS-87/103 (October 1987), unpublished. [ 19 ] F. Wilczek, Phys. Rev. Lett. 39 (1977) 1304. [20] M. Narain et al., 1987 Intern. Syrup. on Lepton and photon interactions at high energies (July 1987), eds. W. Bartel and R. Riickl (North-Holland, Amsterdam, 1988). [21 ] L.A. Sehgal, Phys. Rev. 181 (1969) 215. [22] P. Herczeg, Phys. Rev. D 2 7 (1983) 1512. [23] G. Eilam and A. Soni, preprint U C L A / 8 8 / T E P / 1 9 ( J u l y 1988).