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description of nuclear pairing correlation has been a major subject in recent nuclear physics . for instance , in self - consistent meanfield ( scmf ) description of atomic nuclei , there have been various approaches in order to take the nuclear pairing correlation into account @xcite . these approaches based on the scmf or the nuclear density functional theory ( dft ) have provided considerable agreements with the measured binding energy as well as its odd - even staggering for bound nuclei . even with these efforts , however , the complete character of the nuclear pairing correlation has not been revealed . for example , whether the phenomenological pairing interaction should have the volume or surface type of density dependence is still an open question @xcite . at present , one can find several candidates for the sophisticated nuclear pairing model @xcite . in order to validate these models , one may need reference observables to fit not only for bound nuclei . also , it should be emphasized that the pairing correlation near and beyond the neutron and proton driplines could play a fundamental role to determine the limit of existence on the nuclear chart @xcite . recently , it has been expected that the two - proton ( ) radioactivity may provide novel information on the nuclear pairing correlation . in the true  emission @xcite , a pair of protons is emitted simultaneously from the parent nucleus , whereas the single - proton emission is forbidden or strongly suppressed due to the pairing correlation energy . the proton - proton pairing plays an essential role to determine the observables , especially the released energy ( q - value ) as well as the -decay width or lifetime @xcite . the released q - value can be related with the proton - proton pairing strength in bound nuclei . on the other hand , the -decay width has no correspondence in bound systems , whose lifetime is trivially infinite . thus ,  decays may provide another lodestar with new experimental input to optimize and validate the pairing models . thanks to the experimental developments , there has been a considerable accumulation of data for the -emitting nuclei @xcite . on the other side , however , theoretical studies have not been sufficient to clarify the relation between the  radioactivity and the pairing correlation @xcite . because  emission is a typical many - body problem , its elucidation could also provide an universal knowledge on the multi - particle quantum phenomena in various domains . those include , _ e.g. _ the quantum entanglement @xcite , bcs - bec crossover @xcite , and efimov physics @xcite .    in this article , we discuss how the theoretical characters of pairing models are reflected on -decay properties , connected to a specific interest in sophisticating those models . for this purpose , we employ the time - dependent three - body model @xcite , whose simplicity enables us to phenomenologically understand the pairing model - dependence of  radioactivity . we focus on the  emission from the ground state of the @xmath3be nucleus , assuming the configuration of two valence protons and a rigid @xmath4 core . formalism of our model is given in sec . [ sec : form ] . in sec . [ sec : calc ] , we present numerical calculations and discussions . finally sec . [ sec : sum ] is devoted to summarize the article . details of the time - dependent three - body calculations are present in ref.@xcite . in this article , we employ the same method with some expansions . the  decay from @xmath0be is described as a time - evolution of the two protons in the spherical mean - field generated by the @xmath4 core . the three - body hamiltonian is given as @xcite , @xmath5 where @xmath6 ( @xmath7 ) is the single particle ( s.p . ) hamiltonian between the core and the @xmath8-th proton . @xmath9 is the reduced s.p . mass with @xmath10 . the core - proton potential is given as @xmath11 . woods - saxon potential is expressed as v_ws(r ) = v_0 f(r ) + u_ls ( @xmath12 @xmath13 ) , [ eq : cp - ws ] with a function , @xmath14 $ ] , which can be related to the density - distribution of the core . @xmath15 describes coulomb potential . for @xmath16 and @xmath17 , we employ the same parameters as in ref.@xcite , in order to reproduce the resonance energy and width in the @xmath18-channel of @xmath19 scattering , @xmath20 mev and @xmath21 mev , respectively @xcite . note that this resonance is attributable to the centrifugal potential barrier @xcite . the @xmath2 pairing potential is introduced as @xmath22 . here , we employ a schematic density - dependent contact ( sddc ) potential , v_p - p^(n)(_1 , _ 2 ) & = & w ( @xmath23 ) ( _ 1-_2 ) , + w(r ) & = & w_0 , [ eq : w_sddc ] for nuclear pairing interaction . here two protons have an contact pairing correlation , whose density - dependence is schematically approximated as the @xmath24 term . notice also that @xmath25 . for intrinsic two - nucleon structures , including the dinucleon correlation , similar three - body model calculations with sddc pairing models have been utilized @xcite , with a consistency between other theoretical results @xcite .    with the contact interaction , it is generally known that one should introduce the energy cutoff , @xmath26 , in order to avoid the ultra - violet divergence @xcite . in present case , the bare strength , @xmath27 , can be determined so as to reproduce the empirical scattering length of nucleons in vacuum , @xmath28 fm , consistently to the energy - cutoff @xcite . that is , w_0 = , with @xmath29 . the cutoff energy is set as @xmath30 mev similarly in ref.@xcite , yielding @xmath31 mev@xmath32@xmath33 . total expectation value of @xmath34 , which is conserved during the time - evolution , corresponds to the released q - value of the three - body decay . that is , q_2p & = & , + & = & _ ( t ) + _ pair(t ) , + _ pair(t ) & = & , where @xmath35 is the pairing correlation energy ( pce ) . in order to reproduce the empirical q - value , @xmath36 kev @xcite , we also employ a density - dependence parameter , @xmath37 in eq.([eq : w_sddc ] ) , corresponding to the surface type of the sddc pairing interaction . indeed , it schematically describes the surface type of the density - dependent pairing interaction in meanfield calculations @xcite . note also that pce approximately corresponds to the pairing gap when the system is bound . we assume the @xmath38 configuration for two protons and the @xmath4 core , consistently to the total spin - parity which is also @xmath38 for the ground state of @xmath0be . thus , the eigen - states of the three - body hamiltonian , @xmath39 , can be expanded on @xmath38 basis : & & = _ m u_nm , [ eq : expansion ] + & & _ m ( _ 1 , _ 2 ) = @xmath40 [ _ n_a ljm(_1 ) _ n_b lj(-m)(_2)]^0^+ , [ eq : basis_0p ] with the simplified notation , @xmath41 . @xmath42 is the anti - symmetrization operator . the expansion coefficients , @xmath43 , are determined by diagonalizing the hamiltonian matrix for @xmath34 . state satisfies @xmath44 , with the radial quantum number @xmath45 , the orbital angular momentum @xmath46 , the spin - coupled angular momentum @xmath47 and the magnetic quantum number @xmath48 . we employ the s.p . states up to the @xmath49-channel . in order to take into account the pauli principle , we exclude the first @xmath50 state , which is occupied by the protons in the core nucleus . the continuum s.p . states ( @xmath51 ) of @xmath52 are discretized in the radial box of @xmath53 fm . thus , continuum eigenstates of @xmath34 are also discretized . as we present in sec.[subsec : width ] , this radial box is sufficiently large to provide a good convergence in terms of the decay width . it is worthwhile to mention that , if one neglects pce , -resonance state locates at @xmath54 , where @xmath55 is the @xmath19 resonance energy . in this case , where -wave function keeps the pure @xmath56 configuration , it was confirmed that the decay process is well described as a sequential  emission @xcite . taking pce into account , the resonance energy is decreased mainly due to the attractive pairing interaction . figure [ fig : levels ] schematically describes this situation . in ref.@xcite , the finite - range and density - independent minnesota potential was employed to describe the pairing force @xcite , and then the strongly correlated  emission was suggested . '' should be corrected as `` @xmath57 '' . ] to occur the spatial correlation in this process , mixture of other configurations from @xmath56 plays an essential role @xcite . in order to fix the initial state for time - evolution , we employ the same confining potential , @xmath58 , as in ref.@xcite . this confining potential method has provided good approximations for quantum resonance phenomena @xcite , together with an intuitive way to understand their dynamics . the initial state solved within the confining hamiltonian can be expanded on the eigen - basis of the original hamiltonian : @xmath59 . thus , the time - evolution is represented as = _ n f_n ( t ) , [ eq : ex_e ] where @xmath60 . it is worthwhile to note that the time - invariant discrete energy spectrum can be given as d(e_n)=@xmath61 ^ 2=@xmath62 ^ 2 . if one takes the continuous energy limit , @xmath63 resembles the breit - wigner ( bw ) spectrum , which characterizes the quantum resonance properties of concerning radioactive process @xcite .    in fig . [ fig:1 ] , we plotted the normalized density distribution for the initial state . that is , ( t;r_1,r_2,_12 ) = 8 ^ 2 r_1 ^ 2 r_2 ^ 2 _ 12 @xmath64 ^ 2 , at @xmath65 fm . for convenience , @xmath66 is translated to a function of the relative distance between the two protons , @xmath67 , and that between the core and the center of mass of two protons , @xmath68 . from fig . [ fig:1 ] , we find the similar result in ref.@xcite , where a finite - range pairing interaction was used instead : the higher peak at @xmath69 fm and @xmath70 fm , as well as at @xmath71 , indicates a strong localization of two protons . notice also that this localization is attributable to the spin - singlet configuration , suggesting a diproton correlation @xcite .      in fig . [ fig : dens_whole_tide ] , we plotted the time - development of  state , in terms of the probability - density distribution . it is well described that the confined two protons at @xmath65 are released during the time - development . the probability - density outside the core - proton barrier gradually increases , indicating an evacuation of two protons . in order to monitor their decay dynamics more preciously , it is helpful to focus on the projected decay state @xcite . that is , - ( t ) , with @xmath72 . because the initial state is well confined , this projected decay state mainly represents the outgoing components released from around the core . in fig . [ fig : dens_tide ] , we plot the density distribution of the projected decay state normalized at each point of time . that is , _ d(t;r_1,r_2,_12 ) = , [ eq : decay_dens ] where @xmath73 is the decay probability . in fig . [ fig : dens_tide ] , the strongly correlated -emission is suggested with our surface sddc pairing model . the diproton correlation , which can be detected as a peak at @xmath74 fm and @xmath75 fm , as well as at @xmath71 , is dominant during the time - evolution . notice also that the sequential  emission , which is graphically indicated as a ridge along the @xmath76 line @xcite , is strongly suppressed . this dynamical behavior of protons is similar to that suggested from the finite - range minnesota pairing model @xcite . ? = ?     @ ccccc ccc & @xmath27 & @xmath77 & @xmath78 & @xmath79 & @xmath80 & @xmath81 & + & ( mev@xmath32@xmath33 ) & & ( kev ) & ( kev ) & ( kev ) & ( kev ) & +  sddc ( surface ) & @xmath82 & @xmath83 & @xmath84 & @xmath85 & @xmath86 & @xmath87 & +  sddc ( mixed ) & ? @xmath88 & @xmath89 & @xmath90 & @xmath91 & @xmath92 & @xmath87 & +  sddc ( volume ) & ? @xmath93 & @xmath94 & @xmath95 & @xmath96 & @xmath97 & @xmath98 & +  sddc ( emitter ) & @xmath99 & @xmath100 & @xmath101 & @xmath102 & @xmath103 & @xmath104 & +  minnesota @xcite & & & @xmath84 & @xmath105 & @xmath106 & @xmath107 & +  experiment @xcite & & & @xmath108 & @xmath109 & @xmath110 & @xmath110 & +     ? = 12 we next investigate the decay width , which is one of the directly measurable quantities of the  emission . from the decay probability , @xmath111 , the -decay width is calculated as ( t ) = - = n_d(t ) . [ eq : width ] it is worthwhile to note that , if the time - evolution follows the exponential decay rule , which is a fundamental property of radioactive processes , the decay probability is given as @xmath112 . here @xmath113 is the constant decay width . in this case , @xmath114 becomes identical to @xmath115 , which determines the mean lifetime of this system , @xmath116 .    in fig . [ fig : plpg ] , we plotted the actually calculated decay width as a function of time . with the sddc pairing interaction , we finely obtain the convergence of result after a sufficient time - evolution . from krylov - fock theorem @xcite , this coincides with that the energy distribution , @xmath117 , well approximates the bw spectrum , and thus reproduces the exponential decay rule . @xmath114 at @xmath118 fm is @xmath85 kev , as tabulated in table [ table : cp ] . in fig . [ fig : plpg ] , for a comparison , we fetch the result with the finite - range minnesota pairing force @xcite .    from these results , we find that the sddc pairing interaction underestimates the experimental -decay width , whereas the finite - range minnesota pairing showed a good agreement with it . remember that , as present in table [ table : cp ] , both pairing models are adjusted so as to reproduce the same q - value , corresponding to an equivalent condition on the pce . also , the core - proton potential and the cutoff parameters are common in both cases . thus , the difference in -decay width should be purely attributed to the pairing properties . in fig . [ fig : plpg ] , the partial decay widths for the spin - singlet and spin - triplet channels are also plotted @xcite : @xmath119 . one finds again the dominance of the spin - singlet configuration in  emission consistently to the density distribution in fig . [ fig : dens_tide ] . the exact values of @xmath120 at @xmath118 fm are also summarized in table [ table : cp ] . with the sddc pairing , the spin - singlet -decay width is remarkably small compared with the minnesota pairing case , whereas the spin - triplet width shows the similar values in both cases . +    the smaller width in the sddc pairing case might be attributable to the numerical sensitivity . for radioactive processes governed by the quantum tunneling effect , even a small difference in q - value can yield a large change of the resultant decay width . in order to check this sensitivity , we run the same calculations but changing the absolute strength of the pairing force . that is , @xmath121 with @xmath122 , and @xmath123 , whereas @xmath124 exactly corresponds to the empirical q - value . because the pairing interaction is attractive , a reduction of it ( @xmath125 ) yields the enhanced q - value , meaning that the resonance energy is shifted to the higher region . the @xmath78-@xmath126 relation obtained with the different pairing strengths is present in fig . [ fig:5 ] . obviously , the enhanced ( reduced ) q - value leads to the increase ( decrease ) of the decay width . this can be kinetically understood as that the enhanced q - value increases the penetrability of the potential barrier and then yields the larger width , which is equivalent to the shorter lifetime . similarly in table [ table : cp ] , the spin - singlet width is universally dominant , whereas the spin triplet width , indicated by the gap of the total and the spin - singlet widths , shows the similar values in both cases . in our calculations throughout several @xmath78 values , the sddc pairing commonly yields the smaller width than the minnesota pairing . consequently , the difference in @xmath126 is not attributable to the q - value sensitivity , but concluding the pairing model - dependence . note also that , with the sddc pairing , we could not reproduce the empirical @xmath78 and @xmath126 simultaneously , even within the experimental uncertainties .    in fig . [ fig:6 ] , the discrete energy spectra , @xmath117 , and their continuous distributions smeared by a cauchy - lorentz function are plotted . we see that the spectrum width obtained with the sddc pairing model is not sensitive to the peak position , and it is narrower than that with the minnesota pairing model . this result coincides with the @xmath126 values shown in fig . [ fig:5 ] .      in the remaining of this section , we investigate what in the sddc pairing model essentially controls the -decay width . in scmf or dft calculations , so - called volume and mixed types of density - dependent pairing interaction have been employed as often as the surface type @xcite . these characters of density - dependence can be approximated by @xmath127 and @xmath128 in eq.([eq : w_sddc ] ) for volume , mixed , and surface types , respectively . in order to investigate the role of density - dependence , we perform the similar calculations by changing @xmath77 , in addition to the original , surface sddc case ( @xmath37 ) .    for @xmath129 and @xmath130 , we change the pairing strength , @xmath27 , so as to reproduce the same reference q - value . those parameters are summarized in table [ table : cp ] . notice that changing @xmath27 corresponds to the different value of vacuum - scattering length .    in fig . [ fig : plpg2 ] , we plot the decay widths obtained with different sddc parameterizations . it is found that the volume sddc pairing yields the smallest @xmath126 value , whereas the mixed pairing reproduces the medium result between the surface and volume cases . considering the experimental reference quantities , the surface sddc pairing somehow provides the best improvement among three cases . here we also employ an _ emitter _ sddc potential with @xmath131 to help our discussion . its parameters are also tabulated in table [ table : cp ] . with this model , we indeed obtain a good agreement with experiment : as shown in fig . [ fig : plpg2 ] and table [ table : cp ] , the emitter sddc pairing can reproduce the experimental q - value and decay width simultaneously . consequently , fixing the same reference q - value , -decay width is connected to the density - dependence parameter , @xmath77 .    in the top panel of fig . [ fig : rho_1p ] , we plot the contact pairing strength , @xmath132 in eq.([eq : w_sddc ] ) , for several sddc parameterizations . it is worthwhile to emphasize that , with the emitter sddc model , the pairing interaction is repulsive and thus the pairing correlation becomes unstable deeply inside the core . also , due to the deeper @xmath27 value , two protons in vacuum earn a larger pce than that with other sddc models . the sensitivity of -decay width is now understood from a synergy of pairing correlations in two regions . for this purpose , in fig . [ fig : rho_1p ] , we present the one - proton probability - density distribution of the initial and time - developed states . that is , _ 1(t;r ) & = & 8 ^ 2 _ 0^r_box dr_2 r_2 ^ 2 _ -1 ^ 1 d(_12 ) + & & @xmath133 ^ 2 . because our -basis functions are anti - symmetrized , @xmath134 is symmetric for the exchange of @xmath135 and @xmath136 . thus , @xmath137 represents the mean radial distribution of  probability .    from fig . [ fig : rho_1p ] , in the surface and emitter sddc cases , the probability density shows a dispersive shape . this is a product of ( i ) weak or repulsive pairing inside the core , as well as ( ii ) strong @xmath2 interaction in vacuum . the first effect drives out the two - proton wave function tunneling the potential barrier , whereas the second favors a large probability density outside the core . this mechanism then yields a loose stability with shorter mean lifetime . on the other hand , in the volume sddc case , two - proton density hardly diverges with the isotropically attractive pairing interaction . note that this mechanism commonly exists during the time - evolution , because the total hamiltonian itself is time - independent .    with the present simple sddc interaction , @xmath77 and @xmath27 are not independent parameters to fit : when one is fixed , another should be determined consistently to @xmath138 . thus , model - dependence of @xmath126 contains two effects from ( i ) the density - dependent pairing inside the core , and ( ii ) @xmath2 scattering properties in vacuum @xcite . at this moment , however , separation of these effects is not trivial . one possible approach is to employ a non - trivial parameterization of the density - dependence . this approach could also help to resolve _ tri - lemma _ problem detailed in the next subsection . lastly , we have confirmed that our conclusions do not change even if we employ a smaller value of @xmath26 . we recalculated the decay width with @xmath139 mev , with sddc pairing strength renormalized to reproduce @xmath140 mev . the obtained results reproduced the same width @xmath114 , as with the higher cutoff . we mention an unnegligible problem raised from our results . namely , there is a _ tri - lemma _ of @xmath141 , and the two - nucleon scattering length in vacuum , @xmath142 .    in three - body model calculations for bound systems , all the two - body interactions are prepared to reproduce the partial two - body subsystems . additionally , in order to reproduce the three - body binding energy , one often takes the dynamical effect into account . density - dependent pairing model is a phenomenological procedure for this purpose . for bound systems , this method works if the three - body picture is adequate . the situation changes for unbound @xmath0be . with sddc pairings combined with the well - adjusted core - proton potential for @xmath19 , ( i ) in the case of @xmath143 ( surface ) , @xmath144 is not reproduced , where we keep @xmath142 and @xmath78 consistently to experiments . ( ii ) with @xmath145 ( emitter ) , we reproduce the experimental @xmath78 and @xmath144 , whereas the pairing strength , @xmath27 , is inevitably modified inconsistently to the experimental scattering length . this is the tri - lemma of @xmath142 , @xmath78 , and @xmath126 . note that this problem could be also found in ref.@xcite , where the softened minnesota pairing reproduces only @xmath78 and @xmath144 .    in this work , certain sddc parameterization to resolve this problem has not been found . one possible solution is to employ a non - trivial parameterization of the density - dependence , @xmath132 , in the pairing force . the new @xmath132 should differ from our surface sddc function inside the core , whereas they should have the same @xmath27 value in vacuum . another possible solution is phenomenological three - body force @xcite . it may provide both the three - body dynamical effect and the asymptotic two - body properties consistently to experiments . because these considerable solutions inevitably harm the simplicity of the present model , we leave these developments as a subject of future study . although this tri - lemma remains , our conclusion about the pairing model - dependence in 2p - decay width is not changed . in order to improve theoretical pairing models , which can be connected to the effective pairing force in dft calculations , modern experimental data of -emitters may provide adequate reference quantities .      before closing our discussion , we mention another sensitivity to the range of pairing force . the @xmath2 minnesota potential used in previous calculations was given as v_p - p^(n ) = v_min & = & v_r e^-d^2/2q^2 + & + & v_s e^-d^2/2_s^2 q^2 _ s=0 + & + & v_t e^-d^2/2_t^2 q^2 _ s=1 , [ eq : pp_min ] where @xmath146 , @xmath147 mev , @xmath148 mev , @xmath149 mev , @xmath150 fm , @xmath151 , and @xmath152 . @xmath153 is the projection to the spin - singlet ( triplet ) channel . remember that @xmath154 was softened from the original value , @xmath155 mev @xcite , in order to reproduce the reference q - value . here the first term describes a soft repulsive core . decomposing @xmath156 , this term reads as v_r e^-d^2/2q^2 = w_r , where @xmath157 mev@xmath32@xmath33 , and similarly as expected for the following attraction terms . utilizing a well known formula , _ q0 = ( x/ ) = @xmath158 ( x ) , then at zero - range limit , we obtain _ q0 v_min(_1,_2 ) = w_0 ( _ 1-_2 ) , [ eq : minne_zr ] where @xmath159 mev@xmath32@xmath33 . notice also that , for two - proton basis in @xmath38 configuration , matrix elements of the spin - triplet contact potential become zero automatically from the angular - momentum algebra @xcite . indeed , this zero - range form is identical to the volume sddc pairing given in table [ table : cp ] , but with the deeper bare strength .    employing the volume sddc pairing given in eq . ( [ eq : minne_zr ] ) , however , we confirmed that the @xmath160 three - body system fictionally becomes bound with @xmath161 mev , where the decay width is trivially zero . in order to reproduce the experimental q - value , @xmath140 mev , we needed to use the shallower bare strength as in table [ table : cp ] . as a result , we obtained @xmath162 and @xmath96 kev with the finite - range and zero - range minnesota potentials , respectively . this shows a sensitivity of -decay width to the interaction range .    to reinforce our conclusion , we repeat the same calculation but with the shorter range , @xmath163 fm in eq.([eq : pp_min ] ) . in this case , we also need to employ the enhancement factor , @xmath164 , to reproduce the reference q - value : @xmath165 . resultant -decay widths are displayed in fig.[fig:10 ] . as expected , the short - range minnesota yields the medium value of the decay width between the original and zero - range minnesota cases . notice that , because minnesota forces are density - independent , this range - sensitivity should be attributed only to the vacuum scattering properties . here we simply emphasize that the finite range could be an alternative option instead of the density - dependence , in order to reproduce the concerning -decay width and q - value simultaneously . we have discussed the dependence of  radioactivity on nuclear pairing models within the time - dependent three - body model calculations . comparing the zero - range sddc and the finite - range minnesota pairing forces , the  dynamics is interpreted as the correlated  emission similarly in both cases . evaluating the absolute decay probabilities , we found that the two - proton decay width is remarkably sensitive to the pairing models , even with the equivalent condition on the emitted q - value .    within the sddc pairing model , the -decay width significantly reflects the synergy of pairing correlations inside and outside the core . it controls the penetrability of  density , confined in the core - proton potential barrier at the initial state reproducing the reference q - value . the surface or emitter sddc pairing provided the looser stability and thus the larger decay width . because our three - body hamiltonian itself is not time - dependent nor self - consistent , it is not completely clear whether the similar density - dependence effect can be reflected on the scmf or dft calculations . thus , it is suggestive to expand our procedure to these frameworks @xcite , which enables us to perform the systematic investigation along the -drip line . the work toward this direction could be helpful to sophisticate the theoretical pairing models , utilizing the -decay data as the reference quantities . an experimental survey widely for the -emitter candidates could be profitable for this purpose @xcite .    with the simple sddc pairing force , there remains a tri - lemma of @xmath141 , and the two - nucleon scattering length in vacuum , @xmath142 . in order to reproduce the whole two - body and three - body properties consistently to experiments , further model development is necessary . we also found another kind of sensitivity to the range of pairing force . considering the experimental q - value , with the finite - range minnesota potential , we finely reproduced the experimental -decay width @xcite . on the other side , at the zero - range limit of that potential , theoretical -decay width is significantly narrow . because the density - dependence is absent in these cases , there should be another logic to explain the different width . for this clarification , a further systematic analysis with respect of the interaction range may be helpful . even it demands computational efforts , the work toward this direction is in progress now . also , it might help to resolve the tri - lemma problem of 2p - decay . comparison with other kind of experimental data , _ e.g. _ momentum distributions in refs . @xcite , is an important topic for future work . for this purpose , however , the present model space should be expanded sufficiently to reliable handle the long - range coulomb effects . although the computational cost is highly increased , it may provides another procedure to validate effective pairing models . t. oishi sincerely thank kouichi hagino and hiroyuki sagawa for fruitful discussions . this work was supported by academy of finland and university of jyvskyl within the fidipro programme . we acknowledge the csc - it center for science ltd . , finland , for the allocation of computational resources . 51ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12  & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty http://books.google.co.jp/books?id=k2haur-cq4ac[__ ] , cambridge monographs on particle physics , nuclear physics and cosmology ( ,  ,  )  and ,  eds . , http://www.worldscientific.com/worldscibooks/10.1142/8526#t=aboutbook[__ ]  ( ,  ) link:\doibase 10.1103/revmodphys.75.607 [ * * ,   ( ) ] link:\doibase 10.1103/revmodphys.75.121 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.53.2809 [ * * ,   ( ) ] link:\doibase 10.1140/epja / i2009 - 10780-y [ * * ,   ( ) ] ,   link:\doibase 10.1103/physrevc.71.054303 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.78.024315 [ * * ,   ( ) ] link:\doibase    10.1103/physrevc.88.034314 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.87.064308 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.91.024305 [ * * ,   ( ) ] link:\doibase    10.1103/physrevc.81.034307 [ * * ,   ( ) ] \doibase    http://dx.doi.org/10.1038/nature11188 [ * * ,   ( ) ] link:\doibase    10.1103/physrevlett.110.222501 [ * * ,   ( ) ] @noop * * ,   ( ) @noop * * ,   ( ) \doibase    http://dx.doi.org/10.1016/j.physletb.2009.04.085 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.87.034328 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.90.034303 [ * * ,   ( ) ] link:\doibase 10.1134/s1063779609050049 [ * * ,   ( ) ] http://stacks.iop.org/0034-4885/71/i=4/a=046301 [ * * ,   ( ) ] link:\doibase 10.1103/revmodphys.84.567 [ * * ,   ( ) ] link:\doibase    10.1103/physrevc.86.044301 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.89.014331 [ * * ,   ( ) ] \doibase http://dx.doi.org/10.1016/j.physletb.2008.06.062 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.76.064316 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.99.022506 [ * * ,   ( ) ] \doibase http://dx.doi.org/10.1016/0370-2693(70)90349-7 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.82.463 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.90.052514 [ * * ,   ( ) ] \doibase    http://dx.doi.org/10.1016/0003-4916(91)90033-5 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.56.3054 [ * * , ( ) ] link:\doibase 10.1103/physrevc.72.044321 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.82.024315 [ * * ,   ( ) ] ,   \doibase    http://dx.doi.org/10.1016/0375-9474(88)90124-8 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.71.064326 [ * * ,   ( ) ] http://stacks.iop.org/1402-4896/2012/i=t150/a=014024 [ * * , ( ) ] link:\doibase 10.1103/physrevc.75.021301 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.81.044308 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.88.042504 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0375-9474(77)90007-0 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.59.262 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.38.1747 [ * * , ( ) ] link:\doibase 10.1103/physreva.69.042705 [ * * ,   ( ) ] @noop _ _ ,  reidel texts in the mathematical sciences , vol . 3  ( ,  ,  ) @noop * * ,   ( ) link:\doibase 10.1103/physrevc.86.044331 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.87.014605 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.90.024303 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.63.054313 [ * * ,   ( ) ] link:\doibase 10.1103/physrevc.80.034602 [ * * ,   ( ) ]
the sensitivity of two - proton emitting decays to the nuclear pairing correlation is discussed within a time - dependent three - body model . 
 we focus on the @xmath0be nucleus assuming @xmath1 configuration , and its decay process is described as a time - evolution of the three - body resonance state . 
 a noticeable model - dependence of two - proton decay width is found by utilizing schematic density - dependent contact ( sddc ) and the finite - range minnesota pairing models . 
 the model - dependence with the sddc pairing interaction can be understood by monitoring the time - dependent density distribution , which reflects ( i ) density dependence of pairing correlation inside the core , and ( ii ) @xmath2 scattering properties in vacuum . 
 our result suggests that two - proton decay width can be a suitable reference quantity to sophisticate the nuclear pairing model beyond the nucleon driplines .