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the boson @xmath0 discovered @xcite in 2012 by the cms and atlas collaborations was reported to have a mass about 125 gev and some decay modes predicted for the standard model ( sm ) higgs boson . since that time , the observed particle , called the higgs boson , has been intensively studied ( see , for example , papers  @xcite ) . a main goal of experiments on the higgs boson physics has been to prove or disprove the hypothesis that @xmath0 is the sm higgs boson . apart from the decay channels , the sm predicts that @xmath0 has @xmath13 . the followed thorough analysis has fine - tuned the mass of @xmath0 , which is @xmath14 gev according to ref . @xcite , and has yielded some information on its spin and its @xmath2 parity .    in particular , the observation of the @xmath15 and @xmath16 modes ( see , for example , ref . @xcite ) means that the higgs boson spin is 0 , 1 , or 2 while the fact that @xmath0 decays @xcite to @xmath17 and the landau - yang theorem exclude the spin - one variant . further , the analyses presented in @xcite rule out many spin - two hypotheses at a 99% confidence level ( cl ) or higher . therefore , we conclude that the spin of the higgs boson is 0 with a probability of about 99% .    to clarify the @xmath2 properties of @xmath0 , in ref . we study the decay of a spin - zero particle @xmath1 into two off - mass - shell @xmath3 bosons @xmath4 and @xmath5 . since @xmath1 is defined as an elementary neutral particle with zero spin , our study applies to the higgs boson . moreover , it can apply to the dilaton if this boson actually exists . the amplitude of the decay @xmath18 depends ( see eq . ( 4 ) in ) on 3 complex - valued functions of the invariant masses of @xmath4 and @xmath5 . these functions determine the @xmath2 properties of the boson @xmath1 and are called the @xmath19 couplings . using the cms and atlas experimental data on the decay @xmath8 ( where @xmath9 stands for @xmath10 , @xmath11 , or @xmath12 ) , these collaborations in refs . @xcite and we in ref . have obtained some constraints on the @xmath20 couplings . these constraints demonstrate that @xmath0 is not a @xmath2-odd state and it may be the sm higgs boson , another @xmath2-even state , or a boson with indefinite @xmath2 parity . besides , as shown in , a non - zero imaginary part of the @xmath20 couplings is not excluded , which can be related to small loop corrections and possibly to a non - hermiticity of the @xmath20 interaction .    thus , the @xmath2 parity of the higgs boson is not yet fully ascertained . moreover , in some supersymmetric extensions of the sm there are @xcite neutral bosons with negative or indefinite @xmath2 parity . that is why it is now important to establish the @xmath2 properties of the higgs boson . aiming at that , we consider the decay of the particle @xmath1 into @xmath4 and @xmath5 which then decay to fermion - antifermion pairs @xmath21 and @xmath22 respectively . while in ref . we study in detail the decays with the non - identical fermions , @xmath23 , in the present paper the case @xmath24 is under investigation . the masses of the fermions @xmath25 and @xmath26 is neglected in both papers . we are motivated to consider the decay into identical fermions by the following . in refs . @xcite the cms and atlas collaborations analyze 95 decays @xmath8 . 53 of them are the decays to identical leptons , namely to @xmath10 or @xmath11 . in spite of the fact that the decays to the identical leptons make up about 55% of the measured decays @xmath8 , the distributions of the former decays have not been properly analytically studied . the sm total widths of the decays into identical fermions are studied in refs . @xcite and are calculated in @xcite . some distributions of the decay @xmath27 are plotted in @xcite for the sm higgs boson and some spin - zero states beyond the sm . in the present paper we perform a more general study and consider the decay @xmath28 with allowance for all the possible @xmath2 properties of the particle @xmath1 .    in sec . [ section : the fully differential width ] we derive an analytical formula for the fully differential width of the decay to identical fermions . section  [ section : invariant mass and angular distributions ] shows a comparison of some distributions of the decay to identical leptons with those for the decay into non - identical ones . for this comparison we obtain an exact analytical formula for a certain differential width of the decay to non - identical fermions ( see appendix [ appendix : a formula for d gamma over d a of the decay into non - identical fermions ] ) . we analyze the usefulness of all the compared distributions for obtaining constraints on the @xmath20 couplings . in sec . [ section : comparison with experimental data ] we derive some sm histogram distributions of the decay @xmath8 by monte carlo ( mc ) integration and compare them with the corresponding simulations presented in ref . @xcite and with the experimental distributions from @xcite . we consider a neutral particle @xmath1 with zero spin and arbitrary @xmath2 parity . it can decay into two fermion - antifermion pairs , @xmath21 and @xmath22 , through the two off - mass - shell @xmath3 bosons ( @xmath4 and @xmath5 ) : @xmath29 if @xmath30 $ ] ( @xmath31 is the mass of the particle @xmath1 , @xmath32 is the mass of the @xmath33 quark , @xmath34 is the mass of the @xmath35 quark ) , which holds for @xmath36 , then @xmath37 . if @xmath38 , which is possible @xcite if @xmath1 is the dilaton , then @xmath39 can be the top quark as well .    in ref . we considered decays @xmath40 at the tree level . the present paper shows our analysis of decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 ] ) in the case of the identical fermions , @xmath41 : @xmath42 the matrix element of decay ( [ x- > z_1^ * z_2^ * - > f antif f antif ] ) is @xmath43 where the matrix elements @xmath44 and @xmath45 correspond to the diagrams ( a ) and ( b ) in fig . [ the feynman diagrams for the decay into identical fermions ] respectively . namely ,    @xmath46    where    * @xmath47 and @xmath48 ( @xmath49 and @xmath50 ) are the 4-momenta of the particles @xmath25 and @xmath51 ( @xmath26 and @xmath52 ) in the rest frame of @xmath1 ; * @xmath53 and @xmath54 are the 4-momenta of @xmath4 and @xmath5 respectively in the rest frame of @xmath1 in diagram fig . [ the feynman diagrams for the decay into identical fermions ]  ( a ) ; * @xmath55 ; * @xmath56 and @xmath57 are respectively the pole mass and the total width of the @xmath3 boson ; * @xmath58 is the amplitude of the decay @xmath18 where @xmath59 and @xmath60 are respectively the momentum and the helicity of the boson @xmath61 in the rest frame of @xmath1 ; * @xmath62 is the amplitude of the decay @xmath63 where @xmath64 and @xmath65 ( @xmath66 and @xmath67 ) are respectively the momentum and the polarization of @xmath68 ( @xmath69 ) in the rest frame of @xmath3 , @xmath70 is the helicity of decaying @xmath3 ; * @xmath71 and @xmath72 are the 4-momenta of @xmath4 and @xmath5 respectively in the rest frame of @xmath1 in diagram fig . [ the feynman diagrams for the decay into identical fermions ]  ( b ) ; * @xmath73 .    from the conservation of the energy - momentum 4-vectors we find all the possible values of @xmath74 and @xmath75 : @xmath76 where @xmath77 is the mass of the fermion @xmath39 . the amplitude @xmath58 is @xmath78 where @xmath79 , @xmath80 is the fermi constant , @xmath81 , @xmath82 , and @xmath83 are some complex - valued dimensionless functions of @xmath74 and @xmath75 , @xmath84 , @xmath85 is the polarization 4-vector of the @xmath3 boson with a momentum @xmath86 and a helicity @xmath70 , @xmath87 is the 4-momentum of the boson @xmath1 in its own rest frame , @xmath88 is the levi - civita symbol ( @xmath89 ) . the values of the couplings @xmath90 , @xmath91 , and @xmath92 reflect the @xmath2 properties of the particle @xmath1 . specifically , at the tree level the correspondence shown in table  [ the cp parity of x in the case of various sets of values of a_z , b_z , and c_z ] takes place . .the @xmath2 parity of the particle @xmath1 for various values of @xmath90 , @xmath91 , and @xmath92 . [ cols="^,^,^,^",options="header " , ] in this paper , we have considered the decay of a neutral particle @xmath1 with zero spin and arbitrary @xmath2 parity into two off - mass - shell @xmath3 bosons ( @xmath4 and @xmath5 ) each of which decays to identical fermion - antifermion pairs ( @xmath6 ) : @xmath7 . analytical formulas for the fully differential width of the decay in question and for the fully differential width of the decay @xmath8 are derived ( see eqs . ( [ the fully differential width ] ) and ( [ an explicit formula for the fully differential distribution of the decay h to z1 z2 to 4 ell ] ) ) . moreover , we present an exact formula for the differential width @xmath93 of a decay @xmath94 with @xmath23 ( see eq . ( [ an explicit formula for d gamma over d a for the decay into non - identical fermions ] ) ) .    integrating eq . ( [ the fully differential width ] ) with a mc method , we have obtained some non - histogram distributions for any decay @xmath95 ( @xmath96 ) with @xmath97 . these distributions are compared to those for the decay @xmath95 with @xmath98 ( see figs . [ plots of the distributions of a1 and a2 ] and [ plots of the distributions of a , theta , and phi ] ) . the comparison has revealed significant distinctions between the distributions for the case @xmath97 and the corresponding ones for @xmath98 . the dependences shown in fig . [ plots of the distributions of a , theta , and phi ] are calculated at 4 possible sets ( [ four sets of the possible values of a , b , c ] ) of values of the @xmath20 couplings @xmath90 , @xmath91 , and @xmath92 . at all the 4 sets these distributions almost coincide . therefore their measurement can yield notable constraints on @xmath90 , @xmath91 , and @xmath92 only if the distributions are measured at very high precision .    in order to determine the usefulness of eq . ( [ the fully differential width ] ) , we have computed some sm histogram distributions of the decay @xmath8 by means of integration of eq . ( [ an explicit formula for the fully differential distribution of the decay h to z1 z2 to 4 ell with the atlas or cms limitations ] ) . the distributions are calculated for atlas kinematical limitations ( [ the atlas limitations on m12 , m34 , eta_e , and eta_mu ] ) and for cms ones ( [ the cms limitations on m12 , m34 , eta_e , and eta_mu ] ) . we have compared our distributions with the atlas experimental ones and the atlas mc simulated ones ( see ref . the way our distributions are derived is almost purely analytical  its only numerical part is integration of eq . ( [ an explicit formula for the fully differential distribution of the decay h to z1 z2 to 4 ell with the atlas or cms limitations ] ) . besides , we have chosen the total number of decays more accurately than atlas during its simulations . however , our calculation does not allow for the fact that the invariant mass of @xmath9 may differ from @xmath99 while this fact is taken into account in the atlas simulations . the pros and cons of our technique and the atlas simulations make our distributions and the atlas simulated ones somewhat different but equally close to the atlas experimental data . we have also presented our distributions of @xmath100 , @xmath101 , @xmath102 , @xmath103 , and @xmath104 for the kinematic conditions specific for cms .    in summary , various distributions of the decays @xmath7 or @xmath8 have been obtained with a rather simple integration of eqs . ( [ the fully differential width ] ) and ( [ an explicit formula for the fully differential distribution of the decay h to z1 z2 to 4 ell ] ) respectively . this way of calculation gives an alternative to more traditional mc simulation . this research is partially supported by the national academy of sciences of ukraine ( project tso-1 - 4/2017 ) and the ministry of education and science of ukraine ( project 0115u000473 and project `` electromagnetic processes and processes involving short - lived particles in crystals at energies attainable at cern '' ) . the fully differential width of decay ( [ x- > z_1^ * z_2^ * - > f antif f antif ] ) is @xmath105 , & \end{aligned}\ ] ] where @xmath106 & \end{aligned}\ ] ] is the fully differential width of decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 , f_1 neq f_2 ] ) ( see eq . ( 5 ) in ref .  ) , @xmath107 is the weak isospin projection of the fermion @xmath68 , @xmath108 , @xmath109 is the electric charge of @xmath68 , @xmath110 is the electric charge of the positron , @xmath111 is the weak mixing angle , @xmath112 , @xmath113 @xmath114 @xmath115 @xmath116 , @xmath117 is the momentum of the fermion @xmath25 in the center - of - momentum frame of the particles @xmath25 and @xmath52 , @xmath118 , @xmath119 and @xmath120 are any unit and mutually orthogonal vectors such that @xmath121 , @xmath122 @xmath123 is the azimuthal angle of the @xmath25 momentum in the @xmath4 rest frame formed by the vectors ( @xmath119 , @xmath120 , @xmath124 ) , @xmath125 is the azimuthal angle of the @xmath26 momentum in the @xmath5 rest frame formed by the vectors ( @xmath119 , @xmath126 , @xmath127 ) , @xmath128 @xmath129 , @xmath130 is the momentum of the fermion @xmath26 in the center - of - momentum frame of the particles @xmath26 and @xmath51 , @xmath131 @xmath132 @xmath133 @xmath134 @xmath135 note that the dependence of expression ( [ the fully differential width ] ) on @xmath123 and @xmath125 reduces to a dependence on @xmath136 and in eq . ( [ the fully differential width ] ) the latter sum has to be substituted by @xmath104 . it follows from eq . ( [ an explicit formula for d gamma over d a for any decay x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 ] ) that for any decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 , f_1 neq f_2 ] ) @xmath137 where the differential width @xmath138 is determined by eq . ( 8) from ref .  .    if the functions @xmath139 , @xmath140 , @xmath141 , and @xmath142 are independent of @xmath74 and @xmath75 , integration of @xmath138 in eq ( [ a formula for d gamma over d a for the decay into non - identical fermions ] ) yields @xmath143 , & \end{aligned}\ ] ] where @xmath144 , @xmath145 , @xmath146 , in place of @xmath147 one may take either @xmath148 or @xmath149 ( this choice does not influence the dependence of @xmath93 on @xmath150 ) , @xmath151 we define the argument @xmath152 of a complex number @xmath153 as follows : @xmath154 where @xmath155 , @xmath156 , ~~\theta ( x ) \equiv 1 ~~ \forall x \in ( 0 , + \infty).\end{aligned}\ ] ] according to definition ( [ a definition of arg(z ) ] ) , @xmath157 . in this appendix we propose some general definitions of the differential widths @xmath158 and @xmath159 for any decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 ] ) , and show that the differential widths defined this way coincide with those defined in the standard fashion for decays ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 , f_1 neq f_2 ] ) and ( [ x- > z_1^ * z_2^ * - > f antif f antif ] ) separately . therefore , the distributions presented in fig .  [ plots of the distributions of a , theta , and phi]a are general distributions defined for any decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 ] ) which are calculated for the decay into non - identical leptons and the distributions in fig . [ plots of the distributions of a , theta , and phi]b are the same general distributions calculated for the decay into identical leptons . thus , comparison of fig . [ plots of the distributions of a , theta , and phi]a and fig . [ plots of the distributions of a , theta , and phi]b is sensible thanks to the existence of the general definitions of @xmath158 and @xmath159 . we define the function @xmath158 as @xmath160 where @xmath161 is the probability that in decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 ] ) there is a @xmath3 boson whose squared invariant mass lies in an interval @xmath162 $ ] . to derive an explicit formula for the distribution @xmath163 , we should recall that for decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 , f_1 neq f_2 ] ) @xmath164 where @xmath165 , @xmath166 is the number of the decays ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 , f_1 neq f_2 ] ) in which the squared invariant mass of @xmath4 ( @xmath5 ) is in an interval @xmath167 $ ] ( @xmath168 $ ] ) , the polar angle of @xmath25 ( @xmath26 ) lies in @xmath169 $ ] ( @xmath170 $ ] ) , and the azimuthal angle between the planes of the decays @xmath171 and @xmath172 is in an interval @xmath173 $ ] , among @xmath174 decays ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 , f_1 neq f_2 ] ) .    eq .  ( [ a relation between the fully differential width and the fully differential number of decays for the decay to non - identical fermions ] ) is consistent with the fact that for any decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 ] ) @xmath175 because @xmath176 using eqs . ( [ a definition of 1 over gamma d gamma over d a ] ) and ( [ a relation between the fully differential width and the fully differential number of decays for the decay to non - identical fermions ] ) , we obtain that for decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 , f_1 neq f_2 ] ) @xmath177 since if we neglect @xmath178 and @xmath179 , then @xmath180 ( see eq . ( 8) in ref .  ) .    for any decay ( [ x- > z_1^ * z_2^ * - > f antif f antif ] ) @xmath181 where @xmath182 is the number of the decays ( [ x- > z_1^ * z_2^ * - > f antif f antif ] ) in which there is a @xmath3 boson @xmath4 with a squared invariant mass lying in an interval @xmath167 $ ] and a @xmath3 boson @xmath5 whose squared invariant mass is in @xmath168 $ ] , the polar angle of @xmath25 ( @xmath26 ) lies in an interval @xmath169 $ ] ( @xmath170 $ ] ) , and the azimuthal angle between the planes of the decays @xmath171 and @xmath172 is in @xmath173 $ ] , among @xmath174 decays ( [ x- > z_1^ * z_2^ * - > f antif f antif ] ) . note that while for decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 , f_1 neq f_2 ] ) @xmath4 ( @xmath5 ) is defined as the @xmath3 boson decaying into @xmath21 ( @xmath22 ) , for decay ( [ x- > z_1^ * z_2^ * - > f antif f antif ] ) the choice of @xmath4 and @xmath5 is arbitrary , which leads to the difference between the definitions of @xmath166 and @xmath182 . ( [ a relation between the fully differential width and the fully differential number of decays for the decay to identical fermions ] ) accords with eq . ( [ a formula for gamma of any decay x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 ] ) due to the fact that @xmath183 the `` 2 '' in the right - hand side of eq . ( [ an integral of d^5 n_f_1 = f_2 over d^5 p ] ) emerges because of the double counting during the integration of @xmath184 on @xmath74 and @xmath75 . it follows from eqs . ( [ a definition of 1 over gamma d gamma over d a ] ) and ( [ a relation between the fully differential width and the fully differential number of decays for the decay to identical fermions ] ) that for decay ( [ x- > z_1^ * z_2^ * - > f antif f antif ] ) @xmath185 combining eqs . ( [ an explicit formula for 1 over gamma d gamma over d a for the decay into non - identical fermions ] ) and ( [ an explicit formula for 1 over gamma d gamma over d a for the decay into identical fermions ] ) , we infer that in the approximation @xmath186 for any decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 ] ) @xmath187      analogously , we define the differential width @xmath159 as @xmath188 where @xmath189 is the probability that in decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 ] ) there is a fermion whose polar angle lies in an interval @xmath190 $ ] . ( [ a definition of 1 over gamma d gamma over d theta ] ) and ( [ a relation between the fully differential width and the fully differential number of decays for the decay to non - identical fermions ] ) yield that for any decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 , f_1 neq f_2 ] ) @xmath191 according to eq . ( [ the fully differential width for the decays with f_1 different from f_2 ] ) , the differential width @xmath192 of decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 , f_1 neq f_2 ] ) is invariant under the substitution @xmath193 and @xmath194 if @xmath195 ( see eq . ( [ a definition of a_f ] ) for the definition of the quantity @xmath196 ) . that is why for decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 , f_1 neq f_2 ] ) in the case @xmath195 @xmath197 and therefore @xmath198 we find from eqs . ( [ a definition of 1 over gamma d gamma over d theta ] ) and ( [ a relation between the fully differential width and the fully differential number of decays for the decay to identical fermions ] ) that for decay ( [ x- > z_1^ * z_2^ * - > f antif f antif ] ) @xmath199 combination of eqs . ( [ an explicit formula for 1 over gamma d gamma over d theta for the decay into non - identical fermions ] ) and ( [ an explicit formula for 1 over gamma d gamma over d theta for the decay into identical fermions ] ) yields that for any decay ( [ x- > z_1^ * z_2^ * - > f_1 antif_1 f_2 antif_2 ] ) wherein @xmath195 @xmath200 it follows from eqs . ( [ a relation between the fully differential width and the fully differential number of decays for the decay to identical fermions ] ) , ( [ a relation between the fully differential width and the fully differential number of decays for the decay to non - identical fermions ] ) , ( [ the atlas limitations on m12 , m34 , eta_e , and eta_mu ] ) , and ( [ the cms limitations on m12 , m34 , eta_e , and eta_mu ] ) that @xmath201 where @xmath202 is the number of the decays @xmath203 selected by atlas or cms , among @xmath204 decays @xmath203 , @xmath205 @xmath206 , @xmath207 is the fully differential width of the decay @xmath203 ( see eqs . ( [ the fully differential width ] ) and ( [ the fully differential width for the decays with f_1 different from f_2 ] ) ) , @xmath208 @xmath209 @xmath210 moreover , the fully differential distribution of the decay @xmath8 is @xmath211 where    * @xmath212 is the number of the decays @xmath8 in which @xmath213 $ ] , @xmath214 $ ] , the polar angle of @xmath25 ( @xmath26 ) lies in an interval @xmath169 $ ] ( @xmath170 $ ] ) , and the azimuthal angle between the planes of the decays @xmath171 and @xmath172 is in @xmath173 $ ] , among @xmath215 decays @xmath8 ; * @xmath216 is the number of the decays @xmath217 in which @xmath213 $ ] , @xmath214 $ ] , the polar angle of @xmath25 ( @xmath26 ) lies in an interval @xmath169 $ ] ( @xmath170 $ ] ) , and the azimuthal angle between the planes of the decays @xmath171 and @xmath172 is in @xmath173 $ ] , among @xmath215 decays @xmath8 . it follows from eqs . 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in order to investigate various decay channels of the higgs boson @xmath0 or the hypothetical dilaton , we consider a neutral particle @xmath1 with zero spin and arbitrary @xmath2 parity . 
 this particle can decay into two off - mass - shell @xmath3 bosons ( @xmath4 and @xmath5 ) decaying to identical fermion - antifermion pairs ( @xmath6 ) : @xmath7 . 
 we derive analytical formulas for the fully differential width of this decay and for the fully differential width of @xmath8 ( @xmath9 stands for @xmath10 , @xmath11 , or @xmath12 ) . 
 integration of these formulas yields some standard model histogram distributions of the decay @xmath8 which are compared with corresponding monte carlo simulated distributions obtained by atlas and with atlas experimental data .