Charmless decays and the new physics effects in the minimal supergravity model
Abstract
By employing the QCD factorization approach, we calculate the new physics contributions to the branching radios of the twobody charmless decays in the framework of the minimal supergravity (mSUGRA) model. Within the considered parameter space, we find that (a) the supersymmetric (SUSY) corrections to the Wilson coefficients () are very small and can be neglected safely, but the leading order SUSY contributions to and can be rather large and even change the sign of the corresponding coefficients in the standard model; (b) the possible SUSY contributions to those penguindominated decays in mSUGRA model can be as large as ; (c) for the well measured decays, the significant SUSY contributions play an important rule to improve the consistency of the theoretical predictions with the data; (d) for decays, the theoretical predictions of the corresponding branching ratios become consistent with the data within one standard deviation after the inclusion of the large SUSY contributions in the mSUGRA model.
pacs:
13.25.Hw, 14.40.Nd,12.60.Jv, 12.15.JiI Introduction
As is well known, the precision measurements of the B meson system can provide an insight into very high energy scales via the indirect loop effects of the new physics beyond the standard model (SM) slac504 ; hurth03 . Although currently available data agree well with the SM predictions, we generally believe that the Bfactories can at least detect the first signals of new physics if it is there.
Among the (P stands for the pseudoscalar light mesons) decay channels considered in this paper, twelve of them have been measured with good accuracy. And the data indeed show some deviations from the SM expectations:

The puzzle, the observed branching ratios cleo03 ; babar03 ; belle03 are much larger than the corresponding SM predictions, appeared several years ago, and there is still no convincing theoretical interpretation for this puzzle after intensive studies in the framework of SM as97 and the new physics modelsetapnp .
Although not convincing, these discrepancies together with the socalled anomaly phiks may be the first hints of new physics beyond the SM in B experiments london04 ; sil04 .
Up to now, the possible new physics contributions to rare B meson decays have been studied extensively, for example, in the Technicolor modelstc2 , the twoHiggsdoublet modelsetapm3 ; 2hdm1 and the supersymmetric models hmdx98 ; tyy96 ; tyy97 ; mssm . Among the various new physics models, the supersymmetric models are indeed the most frequently studied models in searching for new physics in B meson system. The minimal supersymmetric standard model (MSSM)as99 is the general and most economical lowenergy supersymmetric extension of the SM. But it is hardly to make definite predictions for the physical observables in B meson decays since there are more than one hundred free parameters appeared in the MSSM. In order to find the possible signals or hints of new physics beyond the SM from the data, various scenarios of the MSSM are proposed by imposing different constraints on it as99 . The minimal supergravity (mSUGRA) model msugra seems to be a very simple constrained MSSM model, since it have only five free parameters , and at the high energy scale.
The previous works in the framework of mSUGRA model focused on the semileptonic, leptonic and radiative rare B decays. In Refs.hmdx98 ; tyy96 ; tyy97 ; huang03 , for example, the authors studied the rare decays , , and the mixing in the mSUGRA model, and found some constraints on the parameter space of this model.
For decays, they have been studied in the SM ali9804 ; chen99 ; bbns99 ; mgmc01 ; mm03 ; du02 ; pqcd , the Technicolor modelstc2 and the twoHiggsdoublet modelsetapm3 . In Ref.sa04 , Mishima and Sanda calculated the supersymmetric effects on decays in the PQCD approachpqcd and predicted the values of CP asymmetries with the inclusion of the supersymmetric contribution. In this paper, we calculate the supersymmetric contributions to the branching radios of the twenty one decay modes in the mSUGRA model by employing the QCD factorization approach (QCD FA) bbns99 ; mgmc01 ; mm03 . The contributions from chirally enhanced power corrections and weak annihilations are also taken into account. We find that the branching ratios of some decay modes can be enhanced significantly, and these new contributions can help us to give a new physics interpretation for the socalled puzzle.
This paper is organized as follows. In section II, we give a brief review for the minimal supergravity model. In section III, we calculate the new penguin diagrams induced by new particles and extract out the new physics parts of the Wilson coefficients in the mSUGRA model. The calculation of decays in QCD factorization approach is also discussed in this section. In section IV, we present the numerical results of the branching ratios for the twenty one decay modes in the SM and the mSUGRA model, and make phenomenological analysis for those well measured decay modes. The final section is the summary.
Ii Outline of the mSUGRA model
In the MSSM, the most general superpotential compatible with gauge invariance, renormalizability and Rparity conserving is written as as99 :
(3) 
where , and are Yukawa coupling constants for downtype, uptype quarks, and leptons, respectively. The suffixes are SU(2) indices and i,j=1,2,3 are generation indices, is the antisymmetric tensor with . In addition to the SUSY invariant terms, a set of terms which explicitly but softly break SUSY should be added to the supersymmetric Lagrangian. A general form of the soft SUSYbreaking terms is given as as99 :
(4)  
where , , , , , and and are scalar components of chiral superfields , , , , , , and respectively, and , , and are , , and gauge fermions. And the terms appeared in Eq.(4) are the mass terms for the scalar fermions, mass and bilinear terms for the Higgs bosons, trilinear coupling terms between sfermions and Higgs bosons, and mass terms for the gluinos, Winos and binos, respectively.
In the mSUGRA model, a set of assumptions are added to the MSSM. One underlying assumpsion is that SUSYbreaking occurs in a hidden sector which communicates with the visible sector only through gravitational interactions. The free parameters in the MSSM are assumed to obey a set of boundary conditions at the Plank or GUT scale:
(5) 
where , and (i=1,2,3) denotes the coupling constant of the , , gauge group, respectively. The unification of them is verified according to the experimental results from LEP1pdg02 and can be fixed at the Grand Unification Scale . Besides the three parameters , and , the supersymmetric sector is described at GUT scale by the bilinear coupling B and the supersymmetric Higgs(ino) mass parameter . However, one has to require the radiative electroweak symmetrybreaking (EWSB) takes place at the low energy scale. The effective potential of neutral Higgs fields at the treelevel is given by (to be precise, oneloop corrections to the scalar potential have been included in the program we used later)
(6)  
where we have used the usual shorthand notation: , , . The radiative EWSB condition is
(7) 
where the value , denotes the vacuum expectation values of the two neutral Higgs fields as , with . From Eq.(7), we can determine the values of and :
(8) 
Through Eq.(II) we can see the sign of is not determined. Therefore only four continuous free parameters, and an unknown sign is left in the mSUGRA model. They are:
(9) 
In the mSUGRA model, all other parameters at the electroweak scale are then determined through the five free parameters by the GUT universality and the renormalization group equation (RGE) evolution. In this paper, we calculate the SUSY and Higgs particle spectrum through a Fortran code: SUSPECT version 2.1 ajg02 . The important features of this code include (a) the renormalization group evolution between low and high energy scales; (b) consistent implementation of radiative electroweak symmetry breaking; and (c) calculation of the physical particle masses with radiative corrections. Using this code, we obtain the SUSY and Higgs particle masses, and the mixing angles of squarks at the electroweak scale. From these Lowenergy supersymmetric parameters, the mixing matrices , for the uptype and the downtype squarks, the mixing matrices for charginos and neutralinos are determined. The explicit expressions of the two mixing matrices and , two matrices and , and a matrix can be found in Refs.ajg02 ; ctfj ; pm96 .
Iii The basic theoretical framework for
In this section, we present the theoretical framework and the relevant formulas for calculating the exclusive nonleptonic decays of the and mesons into two light pseudoscalar mesons.
iii.1 Effective Hamiltonian and relevant Wilson coefficients in SM
In the SM, if we take into account only the operators up to dimensions , and assume , the effective Hamiltonian for the quark level threebody decay at the scale reads ajb98
(10)  
where is the products of elements of the CabbiboKabayashiMaskawa quark mixing matrixckm . And the currentcurrent (), QCD penguin (), electroweak penguin (), electromagnetic and chromomagnetic dipole operators ( and ) can be written as gam96
(11) 
where stands for generators, and are the color indices, and by definition. The sum over runs over the quark fields that are active at the scale , i.e., .
To calculate the nonleptonic B meson decays at nexttoleading order in and to leading power in , we should determinate the Wilson coefficient through matching of the full theory onto the fivequark low energy effective theory where the gauge boson, top quark and all SUSY particles heavier than are integrated out, and run the Wilson coefficients down to the low energy scale by using the QCD renormalization group equations. In table 1, we simply present the numerical results of the LO and NLO Wilson coefficient in the NDR scheme in different scales. More detailed analytical expressions can be found for example in Refs.ajb98 ; gam96 .
LO  NLO  LO  NLO  LO  NLO  

1.179  1.134  1.115  1.080  1.072  1.043  
0.019  0.020  0.012  0.013  0.008  0.008  
0.010  0.012  0.008  0.010  0.006  0.007  
0.018  0.028  0.007  0.046  0.030  
0.055  0.055  0.035  0.035  0.023  0.023  
0.415  0.395  0.286  0.273  0.191  0.183  
iii.2 Wilson coefficients in the mSUGRA model
In the mSUGRA model, the new physics contributions to the rare decays will manifest themselves through two channels. One is the new contributions to the Wilson coefficients of the same operators involved in the SM calculation, the other is to the Wilson coefficients of the new operators such as operators with opposite chiralities. In the SM, the latter is absent because they are suppressed by the ratio . In the mSUGRA model, they can also be neglected, as shown in Ref.ctf02 . Therefore we here use the same operator base as in the SM.
It is well known that there is no SUSY contributions to the Wilson coefficients at the tree level. There are five kinds of contributions to the quark level decay process at oneloop level, depending on specific particles propagated in the loops:

the gauge boson and uptype quarks , which leads to the contributions in the SM;

the charged Higgs boson and uptype quarks ;

the charginos and the scalar uptype quarks ;

the neutralinos and the downtype squarks ;

the gauginos and the downtype squarks .
The new physics contributions from those superparticle loops may induce too large flavor changing neutral currents (FCNCs). To escape from the socalled SUSY flavor problem, degeneracy of masses of squarks and sleptons among different generations has been assumed in the minimal SUGRA model.
In order to determine the new physics contributions to Wilson coefficients , , and (we ignore the new physics contributions to because they are suppressed by a factor of ) at the scale, we need to calculate the Feynman diagrams appeared in Fig.1. First, by employing conservation of the gluonic current, we can define the effective vertex of the penguin processes as in Ref.swi98 :
(12) 
with
(13)  
where and are the electric and magnetic form factors, is the gluon momentum, and are the chirality projection operators.
By calculating the Feynman diagrams as shown in Figs.1(b)(f), we find (in the naive dimensional regularization (NDR) scheme,) the new physics parts of the Wilson coefficients at the scale
(14)  
(15) 
where for , and . In addition, since where is the mass of the heavy scalar fermions, we set for the form factors as an approximation ^{1}^{1}1Eqs.(14) and (15) differ from those appeared in Ref.whi99 , but our final analytic expressions for are the same as that in Ref.tyy97 except for the definition for Wilson coefficients.. The explicit expressions of the form factors and induced by supersymmetric particles are the following
(16)  
(17)  
(18)  
(19)  
(20)  
(21)  
(22)  
(23)  
where for and for decay, respectively. and is the mass of the particle i. In our calculations, we have set for since and . The oneloop integration functions and the coupling constants which appear in and are listed in Appendix A. Using the form factors in Eqs.(16)(23), we obtain the analytic expressions for and . The Wilson coefficient as given in Eq.(15) is the same as that in Ref.tyy97 except for some differences in expression. In Ref.tyy97 , the CKM factor has not been extracted from Wilson coefficients, and the CKM matrix elements have been absorbed into the definition of the coupling constant . See Appendix A for more details.
For the effective vertex of the supersymmetric penguin processes, we only consider it’s contributions to . The explicit analytical expressions of the SUSY contribution to induced by new particles have been given in Ref. tyy97
(24)  
(25)  
(26)  
(27)  
Now, we found all the supersymmetric contributions to the relevant Wilson coefficients. We should remember that, the only source of flavor violation in the mSUGRA model is the usual CKM matrix in the SM. The flavor violation in the sfermion sector at the electroweak scale is generated radiatively in the mSUGRA model and consequently small. Therefore, If we take the mixing matrices and as given in the Appendix of Ref.ctfj
(28) 
the gluino and neutralinomediated diagrams will not contribute to the decay processes considered here. The new physics contributions will come from the chargedHiggs and chargino diagrams only.
iii.3 decays in QCD factorization
To calculate the decay amplitude of the processes , the last but most important step is to calculate hadronic matrix elements for the hadronization of the finalstate quarks into particular final states. At the present time, many approaches have been put forward to settle the intractable problem. Such as the native factorization bsw87 , the generalized factorization ali9804 ; chen99 , the QCD FA bbns99 ; mgmc01 ; mm03 and the PQCD approach pqcd . In this paper, we employ the QCD FA to calculate the branching ratios of decays.
In QCD FA, the contribution of the nonperturbative sector is dominated in the form factors of transition and the nonfactorizable impact in the hadronic matrix elements is controlled by hard gluon exchange. In the heavy quark limit and to leading power in , the hadronic matrix elements of the exclusive nonleptonic decays of the B meson into two light pseudoscalar mesons ( absorbs the spectator quark coming from the B meson) can be written as bbns99
(29)  
where is the form factor describing decays. and denote the perturbative shortdistance interactions and can be calculated by the perturbation approach. are the universal and nonperturbative lightcone distribution amplitudes (LCDA) for B and meson respectively mgmc01 . Weak annihilation effects are not included in Eq.(29).
Consider the low energy effective Hamiltonian Eq.(10) and the unitary relation of the CKM matrix, the decay amplitude can be written as
(30) 
the effective hadronic matrix elements can be calculated by employing the QCD factorization formula Eq.(29). When considering order corrections to the hard scattering kernels and from nonfactorizable single gluon exchange vertex correction diagrams, penguin diagrams and hard spectator scattering diagrams and the contributions from the chirally enhanced power corrections ^{2}^{2}2For more details of various contributions and the corresponding Feynman loops, see for example Refs. mgmc01 ; mm03 and references therein., Eq.(30) can be rewritten asdu02
(31) 
Here