Weak isospin violations in charged and neutral Higgs couplings from supersymmetric loop corrections

Transcription

1 PHYSICAL REVIEW D 69, 751 Weak isospin violations in charged and neutral Higgs couplings from supersymmetric loop corrections Tarek Ibrahim Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt* and Department of Physics, Northeastern University, Boston, Massachusetts 115-5, USA Pran Nath Department of Physics, Northeastern University, Boston, Massachusetts 115-5, USA Received 19 November 3; published April The supersymmetric QCD and the supersymmetric electroweak loop corrections to the violations of weak isospin to Yukawa couplings are investigated. Specifically it involves an analysis of the supersymmetric SUSY loop corrections to the Higgs couplings to the third generation quarks and leptons. Here we analyze the SUSY loop corrections to the charged Higgs couplings which are then compared with the supersymmetric loop corrections to the neutral Higgs couplings previously computed. It is found that the weak isospin violations can be quite significant, i.e., as much as 5% or more of the total loop correction to the Yukawa coupling. The effects of CP phases are also studied and it is found that these effects can either enhance or suppress the weak isospin violations. We also investigate the weak isospin violation effects on the branching ratio BR(H t b)/br(h ) and show that the effects are sensitive to CP phases. Thus an accurate measurement of this branching ratio along with the branching ratio of the neutral Higgs boson decays can provide a measure of weak isospin violation along with providing a clue to the presence of supersymmetry. DOI: 1.113/PhysRevD PACS numbers: 1.6.Jv, 1.8.Cp I. INTRODUCTION In this paper we investigate the effects of supersymmetric QCD and the supersymmetric electroweak corrections to the violations of weak isospin in the couplings of Higgs bosons to quarks and leptons 1. Specifically, we compute in this paper the supersymmetric loop effects to the couplings of the charged Higgs boson with quarks and leptons. These are then compared with the supersymmetric corrections to the couplings of the neutral Higgs boson. We also study the effects of CP phases on the charged Higgs couplings. The CP phases that appear in the soft parameters, however, are subect to strong constraints from the electric dipole moments EDMs of the electron, and of the neutron 3 and of the Hg 199 atom. The mechanisms available for the suppression of the EDMs associated with large phases consist of the cancellation mechanism 5,6, the mass suppression 7 and phases ust in the third generation 8 among others 9,1. Large CP phases affect a variety of low energy phenomena including the Higgs sector. One such phenomenon is the CPeven CP-odd mixing among the neutral Higgs bosons which has been studied in great detail Here we study the effects of CP phases on the charged Higgs couplings to quarks and leptons and also the effect of CP phases on the breakdown of the weak isospin invariance. The outline of the rest of the paper is as follows: In Sec. II we give the basic formalism. In Sec. III we compute the SUSY QCD and SUSY electroweak loop correction to the charged Higgs couplings to third generation quarks and leptons. In Sec. IV we analyze the loop corrections to the charged Higgs decays *Permanent address. H t b and H. In Sec. V we give a numerical analysis of the loop effects and estimate the sizes of the loop corrections to violations of weak isospin. We also study the effects of CP phases on the charged Higgs couplings and on the ratio of the decay branching ratios of the charged Higgs boson to t b and. Finally, the conclusions are given in Sec. VI. II. THE BASIC FORMALISM We will use the framework of the minimal supersymmetric standard model MSSM which contains two doublets of Higgs bosons and for the soft breaking sector we will use the extended SUGRA framework 1 with nonuniversalities and with CP phases. Thus for the Higgs sector we have H 1 H 1 1 H 1, H H 1 H. The components of H 1 and H interact with the quarks and the leptons at the tree level through 15 L f b, h f f Rf L H 1 1 h t t Rt L H h b b Rt L H 1 h t t Rb L H 1 h R L H 1 H.c. The loop corrections produce shifts in these couplings and generate new ones as follows: //697/75111/$ The American Physical Society

2 TAREK IBRAHIM AND PRAN NATH PHYSICAL REVIEW D 69, 751 L eff f b, h f h f f Rf L H 1 1 h f f Rf L H *h t h t t Rt L H h t t Rt L H 1 1 *h b h b b Rt L H 1 h b b Rt L H 1 * h t h t t Rb L H 1 h t t Rb L H 1 *h h R L H 1 h R L H 1 *H.c. 3 * has been used to get a gauge invariant L eff. We rewrite Eq. 3 in a form which is illustrative, L eff i h b h b b RH 1 i Q L h h RH 1 i l L h t h t t RQ L i H h b b RQ L i H i *h Rl L i H i *h t t RQ L i H 1 i * L violation H.c. L violation h b h b b Rt L H 1 h b h b b Rt L H 1 *h h R L H 1 h h R L H 1 * h t h t t Rb L H 1 h t h t t Rb L H 1 *. 5 The corrections h b, h, h t, h b, h, and h t have been calculated in Refs with the inclusion of CP phases. Their effects on the decay of neutral Higgs couplings have been studied in Ref. 18. In this paper we analyze the SUSY QCD and the SUSY electroweak loop corrections to the couplings of the charged Higgs bosons with quarks and leptons in the MSSM. Specifically we calculate the corrections h b, h t, h, h b, h t, and h from exchange of sparticles at one loop. We then study their effects on the decay of the charged Higgs boson into quarks and leptons for the third family. We note that in the approximation h b,t, h b,t,, h b,t, h b,t, 6 one finds that the right hand side of Eq. 5 vanishes and SUSY loop correction preserves weak isospin. This is the approximation that is often used in the literature 1. However, in general, the equalities of Eq. 6 will not hold and there will be violations of weak isospin given by Eq. 5. In this paper we will investigate the size of these violations and their implications on phenomena. down quark and the lepton. We discuss now the various contributions in detail. For h b we need b th interaction which is given by i and L b th H 1 b *t i i H 1 b it * i H.c. gm b m W cos m gm t A b D* b D t1i m W sin D * b1 D ti gm b gm bm t m W cos D * b D ti m W cos D * b1 D t1i 7 g m Wcos D* b1 D t1i 8 III. SUSY QCD AND SUSY ELECTROWEAK CORRECTIONS TO THE CHARGED HIGGS COUPLINGS Figure 1 gives the SUSY QCD loop correction through gluino exchange and SUSY electroweak correction via neutralino and chargino exchanges. In the analysis of these graphs we use the zero external momentum approximation 19,1,18,17 rather than the alternate technique of Refs.,1. We will work within the framework of MSSM and SUGRA models allowing for nonuniversalities. We will characterize the parameter space of these models by using the parameters m A, m, m im 1/ e i i (,,3), A t, A b, A and tan. Here m A is the mass parameter in the Higgs sector, m is the universal scalar mass, m i(i,,3) are the gaugino masses corresponding to the gauge groups U(1), SU() and SU(3) C, A t,b, are the trilinear couplings in top quarks, sbottoms, and staus, and tan H /H 1 H gives mass to the up quark and H 1 gives mass to the i gm t m W sin m gm b A t D b1i D* t m W cos D bid* t1 gm t gm bm t m W sin D bid* t m W sin D b1id* t1 g m Wsin D b1i D* t1. 9 Using the above one finds from Fig. 1 h b g 1 f (m,m i,m ) is defined so that s 3 ei 3D b D t1i * *m i g f m g,m,mb t i

3 WEAK ISOSPIN VIOLATIONS IN CHARGED AND... PHYSICAL REVIEW D 69, 751 U i1 * D* b1 b U i * D* b fm b,mi,m k 15 FIG. 1. Exhibition of the supersymmetric loop contribution to the charged Higgs couplings with third generation quarks. All particles in the loop are heavy supersymmetric partners with t i(b ) being heavy top quarks sbottoms, g the gluino, and k ( i ) neutralinos charginos. f m,m i,m 1 m m i m m m m i m m ln m m m m i ln m m m i m ln m i i for the case i and f m,m i,m 1 i m i m m ln m i m m m i. m 11 1 Further in Eq. 1 D bi is the matrix that diagonalizes the b squark mass matrix so that b L D b1i b i, b R D bi b i 13 b i are the b squark mass eigenstates. Similarly D ti is the matrix that diagonalizes the t squark mass matrix so that t L D t1i t i, t R D ti t i 1 t i are the t squark mass eigenstates. The SUSY electroweak correction h EW b arising from the neutralino exchange and from the chargino exchange see Fig. 1 is given by the following: ki gx k V i1 * g X kv i * g X 1kV i *. 16 Here U and V diagonalize the chargino mass matrix, X diagonalizes the neutralino mass matrix, and k b is defined by m b(t) k b(t) m W cos sin. 17 Finally, bk, bk and bk for the b quark are defined so that and bk gm bx 3k m W cos, bk eq b X 1k * g X cos k *T 3b Q b sin W W bk eq b X 1k gq bsin W cos W tk gm tx k m W sin, X k tk eq t X 1k * g X cos k *T 3t Q t sin W W tk eq t X 1k gq tsin W cos W X k Q b(t) 1 3 ( 3 ) and T 3b(t) 1 ( 1 ) and h b EW 1 k1 tk *D t1i * tk D ti * i * bk D b1 bk D b m k 16 f m k,m,mb t i X 1k X 1k cos W X k sin W X k X 1k sin W X k cos W. Thus the total correction h b is given by m k m i g ki f m t,mi 16 b U i * D t1 tk *D* t1 tk D* t,m k bk D b1 bk D b h b h b g h b EW. 1 Similarly the SUSY QCD and SUSY electroweak correction h b is computed to be 751-3

4 TAREK IBRAHIM AND PRAN NATH PHYSICAL REVIEW D 69, 751 h b 1 s 3 ei 3D b D t1i * i m g f m g,m,mb t i h 1 k1 * k D 1 k D 1 k1 i bk D b1 bk D b k * m k 16 f m k,m,m tk *D t1i * tk D ti * m k 16 f m k,m,mb. t i m k m g k k 16 U* k * f m,m,m k An analysis similar to the above gives for h t and for h t the following: 1 k1 g ki m k m i 16 h t 1 s 3 ei 3D b1i * D t i *m g f m g,m b,m i t U i1 * D* 1 k U i * D* k D 1 k D fm,mi,m k 6 1 k1 bk * D b1i * bk D bi * 1 k1 i * tk D t1 tk D t m k 16 f m k g ki * m m i k 16,mb,m i t k t V i * D b1 bk * D* b1 bk D* b fm,mi b,mk D i, k, k, k, k, k are defined similar to D bi, k b etc. Finally, for h we have h 1 k * k1 m k k D 1 k D 16 f m,m,m k 7 tk D t1 tk D t V i1 * D* t1 k t V i * D* t f m t,mi,m k 3 gm m W cos D g m Wsin D 1 ki gx 3k * U i1 g U ix k * g U ix 1k *. Similarly for h t one has the following: h t 1 s 3 ei 3D b1i * D t i m g f m g,m b,m i t 1 k1 bk * D b1i * bk D bi * i tk D t1 tk D t m k 16 f m k,mb,m i t. 5 The analysis of h and of h is free of the SUSY QCD correction while the SUSY electroweak correction gives gm m W cos m gm A D* m W cos D * 1 g m Wcos D* 1. 8 One measure of the size of the violation of the weak isospin is the deviation of the barred quantities from the unbarred quantities. Thus as a measure of violations of weak isospin in b quark couplings we define the quantity r b hb h b r b. hb h b 9 The deviation of this quantity from unity is an indication of the violation of weak isospin in the Higgs couplings. Similarly we can define r t and r by replacing b with t and in Eq

5 WEAK ISOSPIN VIOLATIONS IN CHARGED AND... PHYSICAL REVIEW D 69, 751 IV. SUSY LOOP CORRECTION TO CHARGED HIGGS DECAYS: H À \ t b AND H À \ À In this section we study the branching ratio involving the decays H t b and H. One may recall that in the neutral Higgs sector, the ratio R h BR(h bb )/BR(h ) is found to be sensitive to the supersymmetric loop corrections and to CP phases. In an analogous fashion in this paper we define the ratio R H BR(H t b)/br(h ) and show that this ratio is a sensitive function of the supersymmetric loop corrections, a sensitive function of the CP phases and in addition sensitive to the violations of weak isospin. To this end it is convenient to display the charged Higgs interaction L int b Bbt s B p bt 5 th B s B p 5 H H.c., 3 B s bt 1 h bh b e i btsin 1 h be i btcos 1 h th t *e i btcos 1 h t *ei btsin B p bt 1 h th t *e i btcos 1 h t *ei btsin 1 h bh b e i btsin 1 h be i btcos B s p B 1 h h e i / sin, e i / 1 h cos, 31 bt ( b t )/ and b, and t are defined by the following: Im h b h b tan h b h b tan b 1Re h b h 3 b tan h b h b FIG.. Plot of r b,r t,r as a function of for the following inputs: solid curves m A GeV, tan, m 35 GeV, m g 3 GeV, 1.1,., 3.3, A t 3, At, A b 7, Ab. The curves in descending order at correspond to r b, r and r t ; dashed curves same input as for solid curves except that m 375 GeV and A b 8. h b m b v 1 1Re h b Re h b tan h b h b Im h b Im h 1/ b tan h b h b 33 and similar relations hold for h. For h t a similar relation holds but with v 1 replaced by v and tan replaced by cot. Notice that h b and h b in Eqs. 3, 33 are not barred quantities. Quantities of interest for the purpose of illustration of loop effects are R tb an R defined by and R tb H t b H t b 3 and the same holds for tan with b replaced by on the right hand side of Eq. 3. For tan t an expression similar to Eq. 3 holds with b replaced by t and tan replaced by cot. The coupling h b is related to the b quark mass by the relation R H H,

6 TAREK IBRAHIM AND PRAN NATH PHYSICAL REVIEW D 69, 751 FIG. 3. Plot of R tb (H t b) loop /(H t b) tree as a function of the phase. The input parameters are m A GeV, m GeV, m g GeV, 1,, 3, At Ab, and A t A b. The curves in ascending order at the point correspond to tan 5, 1, 15,, 3. H t b 3 3 m t m b m H m t m b 1/ M H 1 B bt s B p bt m H m t m b 1 B bt s B p bt m t m b Here (1) is the QCD enhancement factor and is given by s 9.1 s so that (1)1.5 for S.1. Similarly we have H 1 8M H 3 m H m B s B p The quantities (H t b) and (H ) correspond to the tree value, i.e., when the loop corrections in FIG.. Plot of R tb as a function of the phase. The input parameters are m A GeV, m GeV, At Ab, A t A b, 1, 3, and tan 1. The curves in ascending order at correspond to values of m g 3,, 6, 8, 1 GeV. (H t b) and (H ) are set to zero. Finally, to quantify the breakdown of the weak isospin arising from SUSY QCD and SUSY electroweak loop effects we define the following quantities: R tb/ RH H R R H, 39 the first term in the numerator includes the full loop correction including the effects of weak isospin violation and the quantities with subscripts are evaluated at the tree level. Further, we define r tb/ R H H nobarr H R, r tb/ is defined identical to R bt/ except that no barred quantities are used, i.e., we set h b,t, h b,t, and h b,t, h b,t, in Eq. 31. A comparison of r tb/ and R bt/ exhibits the amount of weak isospin violation induced by SUSY loop effects

7 WEAK ISOSPIN VIOLATIONS IN CHARGED AND... PHYSICAL REVIEW D 69, 751 TABLE I. EDMs at tan 5 for Figs. 5 and 8. Case d e ecm d n ecm C Hg cm a b c isospin is a symmetry of the tree level Lagrangian but is violated at the loop level. The size of the weak isospin violation arising from loop corrections can be quantified by the r b defined by Eq. 9 and by r t,r similarly defined. In Fig. we give a plot of r b,r t,r as a function of. Recalling that deviations of r b,t, from unity register the violations of weak isospin, we find that indeed such deviations can be as much as 5% or more depending on the region of the parameter space one is in. Thus in general the violations of weak isospin arising from Eq. 5 would be significant. Next we investigate the question of how large the loop corrections themselves are relative to the tree values. In Fig. 3 we give a plot of R tb defined by Eq. 3 as a function of for values of tan ranging from 5 to 3 for the specific set of inputs given in the caption of Fig. 3. The analysis of the figure shows that the loop correction varies strongly with the phase with the correction changing sign as varies FIG. 5. An exhibition of R tb as a function of tan with and without phases. The three lower curves are for the cases a, b and c with parameters given by a m A GeV, m m 1/ 3 GeV, A, A 1, 1.5,.659, 3.633,.5 solid; b m A GeV, m m 1/ 555 GeV, A, A, 1.6,.653, 3.67,.5 long-dashed; c m A GeV, m m 1/ 8 GeV, A 3, A.8, 1.,.668, 3.6,.5 dot-dashed. A t A b A, At Ab A in all cases. The EDM constraints including the H 199 g are satisfied for the above curves at tan 5 as shown in Table I. A plot of the above three cases but with the phases set to zero is given by the similar upper curves. V. NUMERICAL ANALYSIS The analytical analysis of Secs. II IV is valid for MSSM. However, the parameter space of MSSM is rather large and for this reason for the numerical analysis we use the framework of SUGRA models but allowing for nonuniversalities. Specifically in the numerical analysis we use the parameters m A, m, A t, A b A A s are in general complex, tan and i (,,3) i are the phases of the gaugino masses, i.e., m im 1/ e i i. In addition one has the Higgs mixing parameter which appears in the superpotential as H 1 H ) which is also in general complex, i.e., exp(i ), is determined by radiative breaking of the electroweak symmetry while is arbitrary. We note, however, that not all the phases are independent since the phases appear only in certain combinations in physical quantities. We then evolve them through renormalization group equations to the low energy scales see e.g., Ref. 5. Now as seen from the discussion in Secs. I and II, the weak FIG. 6. Plot of R (H ) loop /(H ) tree as a function of the phase. The input parameters are m A GeV, m GeV, m g GeV, 1,, 3, At Ab, and A t A b. The curves in descending order at the point correspond to tan 5, 1, 15,,

8 TAREK IBRAHIM AND PRAN NATH PHYSICAL REVIEW D 69, 751 FIG. 7. Plot of R as a function of the phase. The input parameters are m A GeV, m GeV, At Ab, A t A b, 1, 3, and tan 1. The curves in ascending order at correspond to values of m g 3,, 6, 8, 1 GeV. from to. Further, the analysis shows that the loop correction can be as large as about 5% of the tree contribution in this case. In Fig. we give a plot of R tb as a function of for the specific set of inputs given in the caption of Fig.. The analysis of Fig. shows that the loop corrections are substantial and further that they have some sensitivity to though the sensitivity is significantly smaller when compared to the sensitivity to variations in seen in Fig. 3. The reason for this difference is that is the phase that appears only in the electroweak loops whose contributions are relatively smaller than those arising from SUSY QCD while the variations in arise from both QCD and electroweak contributions. In Fig. 5 we give a comparison of the loop correction to R tb with and without phases as a function of tan. The lower curves are for three cases a, b and c whose inputs are given in the figure caption. These cases at tan 5 satisfy the EDM constraints including the Hg 199 constraint as shown in Table I taken from Ref. 18. The upper three similar curves have all the same inputs as the lower three curves except that the phases are all set to zero. Figure 5 shows that the loop corrections to R tb depend sensitively on the phases and the correction can change sign from its tree value in the presence of phases. Further, the ratio R tb is sensitive to tan with or without the inclusion of phases. The analysis of Figs. 6 8 is identical to the analysis of Figs. 3 5 except that the analysis of Figs. 6 8 is for the FIG. 8. An exhibition of R as a function of tan with and without phases. The three upper curves are for the cases a, b and c with parameters given by a m A GeV, m m 1/ 3 GeV, A, A 1, 1.5,.659, 3.633,.5 solid; b m A GeV, m m 1/ 555 GeV, A, A, 1.6,.653, 3.67,.5 long-dashed; c m A GeV, m m 1/ 8 GeV, A 3, A.8, 1.,.668, 3.6,.5 dot-dashed. A t A b A, At Ab A in all cases. The EDM constraints including the H 199 g are satisfied for the above curves at tan 5 as shown in Table I. A plot of the above three cases but with the phases set to zero is given by the similar lower curves. ratio R. The main difference here from the R tb case is that for the case of R there are no SUSY QCD corrections. Thus the sensitivity to the variations in the electroweak parameters is larger. Thus a comparison of Fig. and Fig. 7 shows that R is more sensitive to variations in than R tb because the electroweak corrections are not masked by QCD as in the case of R tb. Finally, in Fig. 9 we plot R tb/ and r tb/ defined in Eq. 39 and Eq. as a function of. A comparison of the two shows that the effect of weak isospin violation on the branching ratios can be in the neighborhood of 5%. As mentioned in the Introduction, the technique used in this paper is different from the one in the analysis of Ref. 1. While the main focus of the present work is to calculate the loop effects on the decay widths (H t b) and (H ), the main focus of the work of Ref. 1 is the calculation of the CP asymmetry parameter CP (H tb )(H t b)/(h tb )(H t b)

9 WEAK ISOSPIN VIOLATIONS IN CHARGED AND... PHYSICAL REVIEW D 69, 751 VI. CONCLUSIONS FIG. 9. Plot of R tb/ and r tb/ as a function of for the following inputs: solid curves m A GeV, m 3 GeV, m g 3 GeV, tan, 1.1,., 3.3, A t 3, At, A b 7, Ab. The curves in descending order at correspond to r tb/ and R tb/ ; dotted curves same input as for solid curves except that m 35 GeV; dashed curves same input as solid curves except that m 375 GeV and A b 8. The analysis of Ref. 1 also gives an estimate of the branching ratio for the mode H t b. However, a comparison with this estimate requires that one compute all the decay modes of H including the mode. In this analysis we did not compute this decay mode but propose to do so in a future work which would allow a direct comparison with the analysis of Ref. 1. In this paper we have investigated the effects of supersymmetric loop corrections on the violations of weak isospin in Yukawa couplings of the Higgs bosons to quarks and leptons. Specifically we have computed the gluino, chargino and neutralino loop corrections to the charged Higgs couplings to the third generation quarks and leptons. We find that the loop corrections to the charged Higgs couplings can be as much as 5% of the tree level contribution. We also compared the supersymmetric loop corrections to the charged Higgs couplings with the supersymmetric loop corrections to the neutral Higgs couplings. The disparity between the charged Higgs and the neutral Higgs couplings is a measure of the violations of weak isospin in the effective low energy Lagrangian. The analysis shows that the effects of violations of weak isospin on the Yukawa couplings can be as much as 5% or more. It is also found that such violations are in fact also sensitive to CP phases. Using these results we have investigated the charged Higgs decays H t b and H. It is shown that the branching ratios for these decays are sensitive to weak isospin violation effects, and the effects of the violations of weak isospin on the branching ratio can be as much as 5%, and thus accurate measurement of the branching ratios of the charged and neutral Higgs decays can provide a measure of such violations. The new results of this paper are contained in Secs. III V. Specifically in Sec. III we have given computations of h b,t, and h b,t, not previously computed in the literature in the current framework. The analysis of this paper will also be useful in the more complete computations of decays of the top quarks and sbottoms 6 and in the more accurate computation of charged Higgs decays 7. ACKNOWLEDGMENT This research was also supported in part by NSF grant PHY APPENDIX A: For the convenience of comparison of the barred and the unbarred quantities, we exhibit below the unbarred quantities for the bottom quark 18. First we exhibit h b. We have h b 1 s 3 ei 3m g G *D i b1i * D b f m g,m b,mb i 1 1 k1 k1 g E*V i k1 * D t1i * k t V k * D ti * k b U k * D t1 m k 16 f m k m i m g C i V i1 * D t1k * k t V i * D tk * k b U* D t1k f m 16 t k,m,m t i t,mi,m 751-9

10 TAREK IBRAHIM AND PRAN NATH PHYSICAL REVIEW D 69, k1 G i * bk D b1 b D b bk * D b1i * bk D bi * m k 16 f m k,mb,mb i 1 k1 m i m i b D b1k b D bk bi * D b1k * bi D bk * f m,mi 16 b,m A1 k E i gm Z cos W 1 3 sin W D t1i * D t1 3 sin W D ti * D t sin gm t M W sin D t1i * D t1 D ti * D t gm tm A t M W sin D ti * D t1 A and C i g m i sin M i Q*cos i R i W * A3 and G i gm Z cos W sin W D b1i * D b1 1 3 sin W D bi * D b sin gm b M W cos D b1i * D b A Q i 1 U iv 1 R i 1 M W m *U i1 V 1 *U i V A5 i appearing in Eq. A1 is defined by i g m i sin M i Q*cos i R i W * A6 gq i 1 X 3i *gx* gx* 1 i R i 1 m M 1*X 1i * X* 1 m *X i * X* *X 3i * X* X i * X* 3. A7 W Next we exhibit h b 18. We have h b 1 s 3 ei 3m g H i D b1i * D b f m g,m b,mb i 1 1 k1 k1 g F i V k1 * D t1i * k t V k * D ti * k b U k * D t1 m k 16 f m k H i bk D b1 bk D b bk * D b1i * bk D bi * m k,m,m t i t 16 f m k,mb,mb A8 i 751-1

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