# spectrum and decay properties with coupled channel effects

###### Abstract

The mass spectrum of states has been obtained using the phenomenological relativistic quark model (RQM) with coupled channel effects. The Hamiltonian used in the investigation has confinement potential and confined one gluon exchange potential (COGEP). In the frame work of RQM a study of M1 and E1 radiative decays of states has been made. The weak decay widths in the spectator quark approximation have been estimated. An overall agreement is obtained with the experimental masses and decay widths.

###### keywords:

relativistic quark model (RQM); radiative decay; confined one gluon exchange potential (COGEP); meson states^{†}

^{†}journal: Journal of LaTeX Templates\geometry

top=0.75in,left=0.80in,bottom=0.75in,right=0.80in

## 1 Introduction

The meson is a double heavy quark-antiquark bound state and carries flavours explicitly and provides a good platform for a systematic study of heavy quark dynamics. mesons are predicted by the quark model to be members of the pseudo scalar ground state multiplet EC94 (). The first successful observation of meson was made by CDF collaboration in 1998 from run I at TEVATRON through the semileptonic decay channel FA98 (). They measured the mass of to be MeV and the life time ps. The more precise measurement of mass of i.e., was done by the CDF collaboration through the exclusive non-leptonic decay AA06 (); AA706 (); TAA07 (). The results of the CDF collaboration was confirmed by the observations made by the D0 collaboration VM08 (); VM09 () at TEVATRON. The LHCb has reported several new observations on decays recently. More experimental data on meson are expected in near future from LHCb and TEVATRON.

A suitable theoretical model is required to explain the properties such as mass spectrum, decays, reaction mechanism and bound state behaviour of mesons which involve heavy quarks. The properties of the light and heavy mesons were studied using the phenomenological models. A De Rujula et al DG75 () proposed first QCD based model for the study of hadron spectroscopy. The model had a reasonable success and predicted the masses of charmed mesons and baryons. Several non-relativistic phenomenological potentials with radial dependencies for the confinement along with one gluon exchange potential (OGEP) were examined by Bhaduri et al BC81 (). The ground state heavy meson spectrum has been studied by Vijaya Kumar et al KB13 (). Radiative decay properties of light vector mesons have been studied by Monteiro et al AP14 (). Bottomonium spectrum and its decay properties have been studied in a non relativistic model using OGEP by Monteiro et al APM11 (). Bhagyesh et al BG12 () in their non relativistic model used Hulthen potential to study the orbitally excited quarkonium states. In these models the relativistic effects were completely ignored.

There have been many calculations of baryon properties using relativistic models, like MIT bag models AC74 (); AR74 (), cloudy bag modelsAW84 (); GE79 (), chiral bag models GM79 (); AG81 () etc. Relativistic calculations, where constituent quarks are confined in a potential, have also been performed SB83 (); NB86 (); NS90 (). There are other bag models in literature too. In Budapest bag model the volume energy term is replaced by a surface energy term PJ78 (). Another model which effectively contains a surface tension term is the ’SLAC’ bag, developed by Bardeen et al WB74 () which begins from a local field theory in which heavy quarks interact through a neutral scalar field. Ferreria et al PL77 (); PL80 () used relativistic quark model to study several properties of low lying hadrons. In this model both, the linear and quadratic confinement schemes were used. Bander et al BA84 () used a relativistic bound state formalism to make simultaneous study of all meson systems. Isgur et al SN85 (); SN86 () in their relativized quark model used a parametrized potential and incorporated relativistic kinematics to describe all mesons in the same frame work.

In NRQM formalism though the mass spectra of ground state meson has been produced successfully, the radiative decay rates, particularly hindered decay rates are significantly influenced by relativistic effects. Therefore, it is necessary to include these effects for the correct description of the decays. Radiative decays are the most sensitive to relativistic effects. Hindered radiative decays are forbidden in the non relativistic limit due to the orthogonality of initial and final meson wave functions. They have decay rates of the same order as the allowed ones. In the relativistic description of mesons an important role is played by the confining quark-antiquark interaction, particularly its Lorentz structure. Thus comparison of theoretical predictions with experimental data can provide valuable information on the form of the confining potential. Hence we use relativistic quark model formalism to study the properties of meson states.

The paper is organized in 4 sections. In sec. 2 we briefly review the theoretical background for relativistic model, the framework of the coupled-channel analysis and the relativistic description of radiative decay widths. In sec. 3 we discuss the results and the conclusions are drawn in sec. 4 with a comparison to other models.

## 2 Theoretical Background

### 2.1 The Relativistic Harmonic Model

We investigate properties of states using confined one gluon exchange potential in the frame work of relativistic harmonic model (RHM) SB83 (). The Hamiltonian used has the confinement potential and a two body confined one gluon exchange potential(COGEP) PC92 (); SB91 (); KB93 (); KB97 ().

### 2.2 Confined One Gluon Exchange Potential

In the present existing models for low energy nuclear phenomena the gluon degrees of freedom have been eliminated from the theoretical frame work and it is assumed that the gluon exchange can be incorporated into the theory through OGEP. But in deriving the OGEPDG75 () the gluon propagators used are similar to the free photon propagators used in obtaining Fermi-Breit interaction in QED. Since the confinement of color means the confinement of quarks as well as gluons, the confined dynamics of gluons should play a decisive role in determining the hadron spectrum and in the hadron-hadron interaction. The confinement schemes for quarks and gluons have to be more closely connected to each other in QCD and the confinement of gluons has to be taken into account. The COGEP is obtained from the scattering amplitude using confined gluon propagatorsPC92 (); SB91 (); KB93 (); KB91 (). Here are the gluon propagators in the momentum representation in current confinement model (CCM)SB87 (). The CCM was developed for the confinement of gluons in the spirit of RHM and aims at a unified confinement theory for the study of quark-gluon bond system in the spirit of RHM for the confinement of gluons. In the CCM the coupled non-linear terms in the Yang-Mill tensor is treated as a color gluon super current in analog with Ginzburg-Landuâs theory of superconductivity. The coupled non-linear terms in the equation of motion of a gluon are simulated by a self induced color current (=) or equivalently an effective mass term for all the gluons with . The equation of motion is solved using harmonic oscillator modes in the general Lorentz gauge imposing a secondary gauge condition termed âoscillator gaugeâ. The two confined gluon propagators are then obtained in this gauge using the property of the harmonic oscillator wave functions.
The RHM with COGEP has been quite successful in obtaining the N-N phase shifts and in hadron spectroscopyKB93 ().

The COGEP is obtained from the scattering amplitude PC92 (); SB91 (); KB93 (); KB97 ()

(5) |

where , are the wave functions of the quarks in RHM. The are the zero energy CCM gluon propagators in momentum representation, where is the four momentum transfer. is the quark gluon coupling constant. In CCM, propagators in the momentum representation are given by,

(6) |

The are given by,

(7) |

where and are the creation and destruction operators in the momentum space.

The scattering amplitude (5) is written as

(8) |

We express the 4-component RHM wave function in terms of 2-component wave function by a similarity transformation.

i.e.

(9) | |||||

(10) |

where

(11) |

and

(12) |

The above expression can be simplified to

(13) |

We have,

(14) |

i.e.

(15) |

Similarly we can write,

(16) | |||

(17) |

Substituting (13), (15), (16) and (17) in (8), the scattering amplitude now expressed in terms of the two component spinor and the momentum dependent operator can be written as,

(18) |

The function is the particle interaction operator in the momentum representation and by taking the Fourier transform of each term in the scattering amplitude we get the potential operator in the co-ordinate space. We drop all the higher order momentum dependent terms in to obtain the scattering amplitude which is given by

(19) |

The terms which contribute to the central part of COGEP are,

, and

In CCM the propagator satisfies the differential equation

(20) |

The term , has angular dependence. But the tensor operator is constructed in such a way that the average value of tensor operator over the angular variables vanishes. The averaging over the direction of r gives

(21) |

Substituting for , the central part of the COGEP becomes

(22) |

where and are the propagators given by

(23) | |||

(24) |

where and are color matrices, , W’s are Whittaker functions and (fm) is a constant parameter which gives the range of propagation of gluons and is fitted in the CCM to obtain the glue-ball spectra and r is the distance from the confinement center.

The terms which contribute to the spin orbit part of the COGEP are

(25) |

Operating on and and defining

The spin orbit part of COGEP is

(26) |

The spin orbit term has been split into the symmetric and anti symmetric terms.

The terms which contribute to the tensor part of the COGEP are,

(27) |

The tensor part of the COGEP is,

(28) |

where

(29) |

### 2.3 Coupled Channel Effects

In this section we briefly review coupled channel models. For detailed discussions on coupled channel effects see TO79 (); ON84 (); TO95 (); TO96 (); BE83 (); KH84 (); BE80 (); MI69 (); Li12 (); TB97 (); TS08 (); ES96 (); TO85 (); PG91 (); HZ91 ().

Current QCD inspired potential models generally neglect the hadron loop effects (continuum couplings). These couplings lead to two body strong decays of the meson above threshold and below threshold they give rise to mass shifts of the bare meson states.

(30) |

where we have considered open flavour strong decay . Here A, B, C denote mesons.

The wave function obeys the equation

(31) |

The Hamiltonian H for this combined system consists of a valence Hamiltonian and an interaction Hamiltonian which couples the valence and continuum sectors.

(32) |

where

(33) |

(34) |

where is the decay amplitude.

### 2.4 Radiative Decays

Radiative decays are a powerful tool for the study of the quark structure of mesons, and the calculation of corresponding amplitudes is a subject of the increasing interest. We consider two types of radiative transitions of the meson:

a) Electric dipole (E1) transitions are those transitions in which the orbital quantum number is changed (, ). E1 transitions do not change quark spin. Examples of such transitions are and . The partial widths for electric dipole (E1) transitions between states and are given by

(39) |

where is the energy of the emitted photon,

in relativistic model.

is the fine structure constant. is the charge of the c quark and is the charge of the quark in units of , is reduced mass, and are the masses of b quark and c quar respectively, and are the masses of initial and final mesons.

and

(40) |

is the radial overlap integral which has the dimension of length, with being the normalized radial wave functions for the corresponding states.

b) Magnetic dipole (M1) transitions are those transitions in which the spin of the quarks is changed () and thus the initial and final states belong to the same orbital excitation but have different spins. Examples of such transitions are vector to pseudo scalar (, ) and pseudo scalar to vector (, ) meson decays.

The magnetic dipole amplitudes between -wave states are independent of the potential model.

The M1 partial decay width between S wave states is WJ88 (); NL78 (); CE05 (); WE86 (); BB95 (); NB99 (); BA96 ()

(41) |

where is the overlap integral for unit operator between the coordinate wave functions of the initial and the final meson states, is the spherical Bessel function. , , , and are the spin quantum number, orbital angular momentum and total angular momentum of initial and final meson states respectively.

### 2.5 Weak Decays

The weak decays of mesons provide information about the underlying quark dynamics within the system. The weak decays of bound state of a quark and an anti-quark, which carries heavy flavour c and b - enable us to probe the validity of the standard model of elementary particle interactions and determine several parameters of this model. A rough estimate of the weak decay widths can be done by treating the -quark and -quark decay independently so that decays can be divided into three classes AA99 (); GS91 () (i)the -quark decay with spectator -quark, (ii) the -quark decay with spectator -quark, and (iii) the annihilation (), where .

## 3 Results and Discussion

### 3.1 Mass Spectrum of states with coupled-channel effects

The quark-antiquark wave functions in terms of oscillator wave functions corresponding to the relative and center of mass coordinates have been expressed here, which are of the form,

(42) |

where N is the normalising constant given by

(43) |

are the associated Laguerre polynomials.

The six parameters are the mass of charm quark , the mass of beauty quark , the harmonic oscillator size parameter , the confinement strength , the CCM parameter and the quark-gluon coupling constant . The parameters are obtained by a square fit to the available experimental data of charmonium, bottomonium and meson mass spectra. The CCM parameter is taken from ref (PC92 (); SB85 (); SB87 ()) which was obtained by fitting the iota (1440 MeV) as a digluon glue ball.There are several papers in literature where the size parameter is defined SN86 (); IM92 (). We obtain the value ’b’ by minimizing the expectation value of the Hamiltonian i.e, . We then tune the parameter to reproduce the experimental mass value. In literature we find different sets of values for and , which are listed in Table 1.

Parameter | Ref.AM80 () | Ref. EJ78 () | Ref. WS81 () | Ref. CJ77 () | Ref.DR03 () |
---|---|---|---|---|---|

1800 | 1480 | 1480 | 1480 | 1550 | |

5174 | 5180 | 4880 | 4880 | 4880 |

The values of strong coupling constant in literature are listed in Table 2. The value of strong coupling constant (=0.3) used in our model is compatible with the perturbative treatment.

Parameter | Ref. SN85 () | Ref. DR03 () | Ref.AA05 () | Ref. EC94 () | Ref. SA95 () |
---|---|---|---|---|---|

0.21 | 0.265 | 0.357 | 0.361 | 0.391 |

We use the following set of parameter values.

(44) |

Bare State | BD | Total | ||||||
---|---|---|---|---|---|---|---|---|

0 | 0 | 0 | -5.661 | -5.033 | -10.434 | -9.328 | -30.456 | |

-2.046 | -1.805 | -2.052 | -3.955 | -3.496 | -7.293 | -6.488 | -27.135 | |

-57.922 | -57.406 | -57.946 | 0 | 0 | -19.088 | -18.932 | -211.294 | |

0 | 0 | 0 | -18.49 | -18.393 | -37.603 | -37.901 | -112.387 | |

0 | 0 | 0 | -38.390 | -38.049 | 0 | 0 | -76.439 | |

-40.618 | -40.314 | -40.632 | 0 | 0 | 0 | 0 | -121.557 | |

0 | 0 | 0 | -1.547 | -1.361 | -2.837 | -2.523 | -8.268 | |

-0.546 | -0.476 | -0.548 | -1.929 | -1.711 | -1.049 | -0.920 | -7.179 | |

-30.675 | -30.312 | -30.682 | -15.326 | -15.146 | -3.077 | -3.044 | -128.262 | |

0 | 0 | 0 | -3.147 | -3.111 | -27.643 | -27.49 | -61.391 | |

0 | 0 | 0 | -27.214 | -27.552 | -69.486 | -68.957 | -193.209 | |

-40.753 | -40.359 | -40.772 | -54.308 | -53.783 | -20.835 | -20.606 | -230.663 | |

-148.72 | -146.395 | -148.828 | 0 | 0 | -48.589 | -47.903 | -540.435 | |

0 | 0 | 0 | -25.081 | -24.744 | -49.343 | -48.741 | -147.909 | |

0 | 0 | 0 | -98.623 | -97.088 | 0 | 0 | -195.711 | |

-79.114 | -77.890 | -79.171 | 0 | 0 | 0 | 0 | -236.175 |

State | |||||||||
---|---|---|---|---|---|---|---|---|---|

This work | Ref.SJ96 () | Ref. VA95 () | Ref. ZV95 () | Ref. EC94 () | Ref.DR03 () | Ref.SN85 () | Ref.CT96 () | Ref.FL () | |

6275 | 6247 | 6253 | 6260 | 6264 | 6270 | 6271 | 6280 | 6286 | |

6314 | 6308 | 6317 | 6340 | 6337 | 6332 | 6338 | 6321 | 6341 | |

6672 | 6689 | 6683 | 6680 | 6700 | 6699 | 6706 | 6727 | 6701 | |

6766 | 6738 | 6717 | 6730 | 6730 | 6734 | 6741 | 6743 | 6737 | |

6828 | 6757 | 6729 | 6740 | 6736 | 6749 | 6750 | 6765 | 6760 | |

6776 | 6773 | 6743 | 6760 | 6747 | 6762 | 6768 | 6783 | 6772 | |

6838 | 6853 | 6867 | 6850 | 6856 | 6835 | 6855 | 6960 | 6882 | |

6850 | 6886 | 6902 | 6900 | 6899 | 6881 | 6887 | 6990 | 6914 | |

7078 | 7008 | 7010 | 7012 | 7072 | 7028 | 7019 | |||

7009 | 7001 | 7020 | 7012 | 7077 | 7041 | 7028 | |||

7154 | 7016 | 7030 | 7009 | 7079 | 7036 | 7028 | |||

6980 | 7007 | 7040 | 7005 | 7081 | 7045 | 7032 | |||

6914 | 7088 | 7100 | 7108 | 7091 | 7122 | ||||

7259 | 7113 | 7140 | 7135 | 7126 | 7145 | ||||

7322 | 7124 | 7150 | 7142 | 7145 | 7150 | ||||

7232 | 7134 | 7160 | 7153 | 7156 | 7164 |

We evaluate the bare state masses and shifts due to ,, , , , and loops (with , , , , , , , , and ).

For the case of a bound system of quark and antiquark of unequal mass, charge conjugation parity is no longer a good quantum number so that states with different total spins but with the same total angular momentum, such as the and pairs, can mix via the spin orbit interaction or some other mechanism. The meson states with are linear combination of spin triplet and spin singlet states which we describe by the following mixing

(45) | |||

(46) |

where is a mixing angle, and the primed state has the heavier mass. For we have mixing of P states

(47) | |||

(48) |

The values of the mixing angles for P states are and

Similarly for we have mixing of D states,

(49) | |||

(50) |

The value of mixing angle for D states is

The calculated masses of the states are listed in Table 4. Our calculated mass value for is 6275.851 MeV and for (1S) is 6314.161 MeV. (1S) is heavier than (1S) by 38.193 MeV. This difference is justified by calculating the splitting of the ground state which is given by

(51) |

The mass of first radial excitation (2S) is 6838.232 MeV which is heavier than (1S) by 562.381 MeV. This value agrees with the experimental value of (2S) 684245 MeVAG14 (). The difference between the (2S) and (1S) masses turns out to be 536.412 MeV. Our prediction for masses of orbitally excited states are in good agreement with the other model calculations.

### 3.2 Radiative Decays

The calculation of radiative (EM) transitions between the meson states can be performed from first principles in lattice QCD, but these calculation techniques are still in their developmental stage. At present, the potential model approaches provide the detailed predictions that can be compared to experimental results.

The possible decay modes have been listed in Table 5 and the predictions for E1 decay widths are given. Also our predictions have been compared with other theoretical models. Most of the predictions for transitions are in qualitative agreement. However, there are some differences in the predictions due to differences in phase space arising from different mass predictions and also from the wave function effects. For the transitions involving and states which are mixtures of the spin singlet and spin triplet states, there exists huge difference between the different theoretical predictions. These may be due to the different mixing angles predicted by the different models. Wave function effects also appear in decays from radially excited states to ground state mesons such as . The overlap integral for these transitions in our model vanishes and hence we get decay width for these transitions zero.

The M1 transitions contribute little to the total widths of the 2S levels. Because it cannot decay by annihilation.
Allowed M1 transitions correspond to triplet-singlet transitions between S-wave states of the same n quantum number, while hindered M1 transitions are either triplet-singlet or singlet-triplet transitions between S-wave states of different n quantum numbers.

The possible radiative M1 transition modes are as follows, (i) 2 , (ii) 2 , (iii) 2 , (iv)1 .
In the above (ii) and (iii) represent hindered transitions and (i) and (iv) represent allowed transitions.
In order to calculate decay rates of hindered transitions we need to include relativistic corrections.There are three main types of corrections: relativistic modification of the non relativistic wave functions, relativistic modification of the electromagnetic transition operator, and finite-size corrections. In addition to these there are additional corrections arising from the quark
anomalous magnetic moment. Corrections to the wave function that give contributions to the transition amplitude are of two categories:

1) higher order potential corrections, which are distinguished as a) Zero recoil effect and b) recoil effects of the final state meson and 2)Colour octet effects. The colour octet effects are not included in potential model formulation and are not considered so far in radiative transitions.

Transition | k | This work | Ref. DR03 () | Ref. EC94 () | Ref. VA95 () | Ref.FL () |
---|---|---|---|---|---|---|

MeV | keV | keV | keV | keV | keV | |

348.527 | 42.384 | 75.5 | 79.2 | 65.3 | 74.2 | |

437.575 | 83.879 | 87.1 | 99.5 | 77.8 | 75.8 | |

494.615 | 121.143 | 13.7 | 0.1 | 8.1 | 26.2 | |

446.371 | 89.04 | 122 | 112.6 | 102.9 | 126 | |

473.213 | 106.088 | 18.4 | 0 | 11.6 | 32.5 | |

529.934 | 148.992 | 147 | 56.4 | 131.1 | 128 | |

175.953 | 3.635 | 5.53 | 7.8 | 7.7 | 9.6 | |

83.181 | 0.384 | 7.65 | 14.5 | 12.8 | 13.3 | |

22.416 | 0.00751 | 0.74 | 0 | 1.0 | 2.5 | |

73.879 | 0.269 | 7.59 | 17.7 | 14.8 | 14.5 | |

70.979 | 0.238 | 1.05 | 0 | 1.9 | 6.4 | |

10.104 | 0.00068 | 4.40 | 5.2 | 15.9 | 13.1 | |

573.927 | 0 | 21.9 | 16.1 | |||

88.815 | 0 | 22.1 | 15.3 | |||

938.854 | 0 | 2.1 | 2.5 | |||

859.837 | 0 | 25.8 | 19.2 | |||

917.035 | 0 | 3.1 | ||||

971.788 | 0 | 20.1 | ||||

63.253 | 0.422 | 34.0 | 41.2 | 25.5 | ||

397.439 | 104.751 | 45.3 | 54.3 | 32.1 | ||

456.65 | 158.896 | 10.4 | 5.4 | 5.9 | ||

371.628 | 85.639 | 75.3 | 73.8 | 49.4 | ||

409.075 | 114.223 | 13.8 | 8.1 | |||

468.191 | 171.244 | 90.5 | 58.0 |

The spherical Bessel function introduced in equation (41) takes into account the so called finite-size effect (equivalently, re summing multipole-expanded magnetic amplitude to all orders). For small , , so that transitions with have favoured matrix elements, though the corresponding partial decay widths are suppressed by smaller factors. For large value of photon energy () transitions with have favoured the matrix element, since becomes very small. transition rates are very sensitive to hyperfine splitting of the levels due to the factor in equation (41).

There have been many models which study the radiative decays of meson using non relativistic and relativistic quark models. Eichten and Quigg EC94 () calculated the radiative M1 transition rates for the allowed and hindered transitions. They used the equation (41) in their potential model approach to determine the M1 transition rates of meson. Allowed transition rates for processes (i) and (iv) were found to be 0.0040 keV and 0.130 keV respectively. Hindered transition rates for the processes (ii) and (iii) were 0.253 keV and 0.223 keV respectively. Abd El-Hady et al AA05 () have investigated the radiative decay properties of meson in a Bethe-Salpeter model. The allowed transition rates for processes (i) and (iii) were found to be 0.0037 keV and 0.0189 keV respectively. The hindered transition rates for the processes (ii) and (iv) were found to be 0.135 keV and 0.1638 respectively. Ebert et al DR03 () have studied these M1 transitions including full relativistic corrections in their relativistic model. They depend explicitly on the Lorentz structure of the non relativistic potential. Several sources of uncertainty make M1 transitions particularly difficult to calculate. The leading-order results depend explicitly on the constituent quark masses, and corrections depend on the Lorentz structure of the potential. They estimated the allowed transition rates to be 0.033 keV and 0.017 keV respectively. For the hindered transition, decay rates were found to be 0.428 keV and 0.488 keV. Also it is clear from their calculations that the predicted decay rates for hindered transitions which are increased by relativistic effects almost by a factor of 3 and they are larger than the rates of allowed M1 transitions by an order of magnitude.

We have calculated the M1 transition rates for meson states using equation (41). The resulting M1 radiative transition rates of these states are presented in table 6. In this table we give calculated values for decay rates of M1 radiative transition in comparison with the other relativistic and non relativistic quark models. We see from these results that the relativistic effects play a very important role in determining the meson M1 transition rates. The relativistic effects reduce the decay rates of allowed transitions and increase the rates of hindered transitions. The M1 transition rates calculated in our model agree well with the values predicted by other theoretical models.

Transition | ||||||||
---|---|---|---|---|---|---|---|---|

This work | Ref.AA05 () | Ref.DR03 () | Ref.FL () | Ref.SA95 () | Ref.DR03 () | Ref.EC94 () | ||

38.193 | 0.0185 | 0.0189 | 0.033 | 0.059 | 0.060 | 0.073 | 0.135 | |

12.329 | 0.0018 | 0.0037 | 0.017 | 0.012 | 0.010 | 0.030 | 0.029 | |

550.614 | 0.193 | 0.1357 | 0.428 | 0.122 | 0.098 | 0.141 | 0.123 | |

515.410 | 0.123 | 0.1638 | 0.488 | 0.139 | 0.096 | 0.160 | 0.093 |

### 3.3 Weak Decays and Life Time of meson

In accordance with the classification given in section 2.5, the total decay width can be written as the sum over partial widths

(52) |

In the spectator approximation:

(53) |

Calculated value of is