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Addenda
Volume 213, number 4
PHYSICS LETTERS B
3 November 1988
ADDENDA
I. Antoniadis, J. Ellis, J. Hagelin and D.V. Nanopoulos, An improved flipped SU (5) X U ( 1 ) model from the four-dimensional string, Phys. Lett. B 208 (1988) 209. Note the following two features of the fermion mass matrices (5), (6) and (7) on page 212: (a) Before the reduction of physical states due to the projection induced by the boundary condition vector a ofeq. (2g) on page 210, the model has an SO (10) symmetry. The couplings of the states which survive the a projection are still related by SO(10). This implies that the top left-hand side entries in the matrices M-l~3 (5) and M_~ (7) are equal, and hence that mb = ms [ 17 ] at the superunification scale. Renormalization effects [ 16 ] then yield mb ~ 5-6 GeV for the physical bottom quark mass. (b) Although the F~t'2fi ~2 and F2 ~l fil 2 couplings are equal at the superunification scale because of the same SO(10) symmetry, this does not mean that me= mr. One reason is that the FI, 2 fields have as yet undetermined mixings with the H~.2 fields. Since the (F~, H~ ) and (F2, H2) diagonalizations are independent, the couplings of the physical t and c quarks may be different. Moreover, the renormalization group equations for the couplings of F~ and F2 are different because of the large 21Ft F Ih I coupling, implying that mc # m, after renormalization.
A. Masiero and G. Ridolfi, b--,s+7 and b ~ s + g in N = 1 supergravity theories, Phys. Lett. B 212 (1988) 171. In ref. [ 9 ] we studied in detail the enhancement of the decay for the process B R ( b ~ s + 7 ) due to the exchange of supersymmetric particles in the loop. In the final part of the same paper, we also considered supersymmetric contributions to the process BR(b--,s+g), stressing the fact that, in this case, the 562
external gluon line can be attached also to the gluino internal leg. In view of the experimental results that will be available in the next future, we think it is useful to give a complete calculation also for the process BR(b--,s+g). We find F ( b - - , s + g ) = 9a3 m~, 4 (lEVI2+ IF~'I2), 32;¢ 2 m~
(7)
where F v'A = (F/_2,FL,3 +/-~2,FR,3 ) [H(x,) + }P(x,) ]
+ __m~(1-'*L2iFRi3+V~2il'ci3) [K(xi) -~Q(' xi) ] • mb
(8) The same notations of ref. [ 9 ] are used. This expression represents the total contribution to B R ( b ~ s + g ) coming from the exchange of the supersymmetric partners of down squarks and gluons at the one-loop level. It is interesting to compare it with the decay rate in eq. (6a) of ref. [ 9 ], which refers solely to the contribution of the diagram with the gluon emitted by the gluino internal line (We take this opportunity to notice that, in ref. [ 9 ], the factor a 2 in eq. (6a) should be replaced by a3, and that the function Q(x) and K(x) should be taken with the opposite sign). The new result is shown in fig. 5, which is to be compared with fig. 4 of ref. [ 9 ]. We see that the area in the m B - m~ plane which is forbidden by the limit B R ( b ~ s + g ) <20%
(9)
remains essentially unaltered, with the only exception of the narrow diagonal strip, which is now allowed. The limit (9) implies bounds on m~ and m~ which are slightly more severe than those imposed by B R ( b - , s + y). We take the opportunity of this addendum to notice that in refs. [ 9,10 ], when we compute the supersymmetric contributions to b ~ s + 7 and b ~ s + g , we
Volume 213, number 4
PHYSICS LETTERS B
200
3 November 1988
physical charged Higgs scalar is present, a n d its ex, change can also give a sizeable contribution, d e p e n d ing on its mass a n d on the ratio o f the v a c u u m expectation values o f the neutral Higgs scalars. We thank W.-S. H o u for discussing this point with us, a n d we refer to his p a p e r in ref. [ 11 ] for a quantitative analysis.
150
10(]
We t h a n k S. Bertolini a n d F. B o r z u m a t i for discussions which p r o m p t e d the m o r e complete analysis presented in this a d d e n d u m .
E
50
0
50
100 m~ (GeV)
150
200
Fig. 5. Region of the me -m~ plane where BR(b-,s+g) > 0.2, for rn~= 100 GeV and VE/V~=2 (shaded area). The region allowed by SU(2) ×U( 1) breaking is also shown (dotted area).
References [9] A. Masiero and G. Ridolfi, preprint CERN-TH.5081/88 (1988). [ 10] S. Bertolini, F. Borzumati and A. Masiero, Phys. Lett. B 192 (1987) 437; Nucl. Phys. B 294 (1987) 321. [ 11 ] W.-S. Hou and R.S. WiUey, preprint MPI-PAE/Pth 81/87 (1987).
consider only the exchange o f squarks a n d gluinos in the loops. In addition, in s u p e r s y m m e t r i c m o d e l s a
563
PHYSICS LETTERS B
3 November 1988
ADDENDA
I. Antoniadis, J. Ellis, J. Hagelin and D.V. Nanopoulos, An improved flipped SU (5) X U ( 1 ) model from the four-dimensional string, Phys. Lett. B 208 (1988) 209. Note the following two features of the fermion mass matrices (5), (6) and (7) on page 212: (a) Before the reduction of physical states due to the projection induced by the boundary condition vector a ofeq. (2g) on page 210, the model has an SO (10) symmetry. The couplings of the states which survive the a projection are still related by SO(10). This implies that the top left-hand side entries in the matrices M-l~3 (5) and M_~ (7) are equal, and hence that mb = ms [ 17 ] at the superunification scale. Renormalization effects [ 16 ] then yield mb ~ 5-6 GeV for the physical bottom quark mass. (b) Although the F~t'2fi ~2 and F2 ~l fil 2 couplings are equal at the superunification scale because of the same SO(10) symmetry, this does not mean that me= mr. One reason is that the FI, 2 fields have as yet undetermined mixings with the H~.2 fields. Since the (F~, H~ ) and (F2, H2) diagonalizations are independent, the couplings of the physical t and c quarks may be different. Moreover, the renormalization group equations for the couplings of F~ and F2 are different because of the large 21Ft F Ih I coupling, implying that mc # m, after renormalization.
A. Masiero and G. Ridolfi, b--,s+7 and b ~ s + g in N = 1 supergravity theories, Phys. Lett. B 212 (1988) 171. In ref. [ 9 ] we studied in detail the enhancement of the decay for the process B R ( b ~ s + 7 ) due to the exchange of supersymmetric particles in the loop. In the final part of the same paper, we also considered supersymmetric contributions to the process BR(b--,s+g), stressing the fact that, in this case, the 562
external gluon line can be attached also to the gluino internal leg. In view of the experimental results that will be available in the next future, we think it is useful to give a complete calculation also for the process BR(b--,s+g). We find F ( b - - , s + g ) = 9a3 m~, 4 (lEVI2+ IF~'I2), 32;¢ 2 m~
(7)
where F v'A = (F/_2,FL,3 +/-~2,FR,3 ) [H(x,) + }P(x,) ]
+ __m~(1-'*L2iFRi3+V~2il'ci3) [K(xi) -~Q(' xi) ] • mb
(8) The same notations of ref. [ 9 ] are used. This expression represents the total contribution to B R ( b ~ s + g ) coming from the exchange of the supersymmetric partners of down squarks and gluons at the one-loop level. It is interesting to compare it with the decay rate in eq. (6a) of ref. [ 9 ], which refers solely to the contribution of the diagram with the gluon emitted by the gluino internal line (We take this opportunity to notice that, in ref. [ 9 ], the factor a 2 in eq. (6a) should be replaced by a3, and that the function Q(x) and K(x) should be taken with the opposite sign). The new result is shown in fig. 5, which is to be compared with fig. 4 of ref. [ 9 ]. We see that the area in the m B - m~ plane which is forbidden by the limit B R ( b ~ s + g ) <20%
(9)
remains essentially unaltered, with the only exception of the narrow diagonal strip, which is now allowed. The limit (9) implies bounds on m~ and m~ which are slightly more severe than those imposed by B R ( b - , s + y). We take the opportunity of this addendum to notice that in refs. [ 9,10 ], when we compute the supersymmetric contributions to b ~ s + 7 and b ~ s + g , we
Volume 213, number 4
PHYSICS LETTERS B
200
3 November 1988
physical charged Higgs scalar is present, a n d its ex, change can also give a sizeable contribution, d e p e n d ing on its mass a n d on the ratio o f the v a c u u m expectation values o f the neutral Higgs scalars. We thank W.-S. H o u for discussing this point with us, a n d we refer to his p a p e r in ref. [ 11 ] for a quantitative analysis.
150
10(]
We t h a n k S. Bertolini a n d F. B o r z u m a t i for discussions which p r o m p t e d the m o r e complete analysis presented in this a d d e n d u m .
E
50
0
50
100 m~ (GeV)
150
200
Fig. 5. Region of the me -m~ plane where BR(b-,s+g) > 0.2, for rn~= 100 GeV and VE/V~=2 (shaded area). The region allowed by SU(2) ×U( 1) breaking is also shown (dotted area).
References [9] A. Masiero and G. Ridolfi, preprint CERN-TH.5081/88 (1988). [ 10] S. Bertolini, F. Borzumati and A. Masiero, Phys. Lett. B 192 (1987) 437; Nucl. Phys. B 294 (1987) 321. [ 11 ] W.-S. Hou and R.S. WiUey, preprint MPI-PAE/Pth 81/87 (1987).
consider only the exchange o f squarks a n d gluinos in the loops. In addition, in s u p e r s y m m e t r i c m o d e l s a
563