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recently , ultracold atomic gases with strong p - wave interaction have attracted broad interest both experimentally expk40 , expk40b , expk40mola , expk40dipo , expk40molb , expli6a , expli6b , expli6c , expli6molsca , expli6r87 , explik , expli6mol , expkrb and theoretically pyou,2dpsfa , psfi,2dpsfb , psf1certa , psf1certb , psferta , psfa , p2dsfb , psff,2dpsfc , psfh , p2dsfa , psfc , 1dpsfb , psfe , pssfa , psfertc , gpra , 1dpsfa , psfb , psfertb , psf1certc , pfrvortex , psfola , pbosonert , psfbosefermi , twofermionerta , maqidajpsj1 , maqidajpsj2,zinnerepjd , maqidaprl1,pra2010a , maqidapra1,maqidapra2,maqidapra3,xxxa , stoofprl , p3bert , ppseudopotentialerta , p2bfr , ppseudopotentialertb , ppseudopotentialc , ppseudopotentiald , ppseudopotentiale , prlblume , nf1,nf2,nf3,nf4 . the p - wave magnetic feshbach resonances , which can generate tunable p - wave ineratomic interactions , have been observed in the cold gases of @xmath11k @xcite , @xmath12li expli6a , expli6b , @xmath12li-@xmath13rb mixture expli6r87 , @xmath12li-@xmath14k mixture @xcite , and @xmath15k-@xmath13rb mixture @xcite . the p - wave feshbach molecules have also been created and studied in the gases of @xmath16k @xcite and @xmath12li expli6a , expli6c , expli6mol , expli6molsca . these experimental achievements stimulate theoretical researches on the quantum superfluid in ultracold atomic gases with strong p - wave interactions pyou,2dpsfa , psfi,2dpsfb , psf1certa , psf1certb , psferta , psfa , p2dsfb , psff,2dpsfc , psfh , p2dsfa , psfc , 1dpsfb , psfe , pssfa , psfertc , gpra , 1dpsfa , psfb , psfertb , psf1certc , pfrvortex , psfola , pbosonert , psfbosefermi , maqidajpsj1,maqidajpsj2 , zinnerepjd , maqidaprl1,pra2010a , maqidapra1,maqidapra2,maqidapra3,xxxa , stoofprl , as well as the relevant few - body problems @xcite . until now , most theories of ultracold atomic gases with strong p - wave interactions psf1certa , psf1certb , psferta , psfertc , psfertb , psf1certc , pbosonert , stoofprl , twofermionerta , p3bert , ppseudopotentialerta , p2bfr , ppseudopotentialertb are based on the low energy expansion of the p - wave scattering amplitude @xmath17 given by the effective - range theory @xcite @xmath18here @xmath19 is the relative momentum of the two atoms ; @xmath20 is the scattering volume and @xmath4 is the effective - range . the effective - range theory for the scattering amplitude is used in both theories of p - wave atomic superfluids psf1certa , psf1certb , psferta , psfertc , psfertb , psf1certc , pbosonert , stoofprl and related few - body problems @xcite . in particular , the separable two - body potential @xcite @xmath21 used in the many - body hamiltonian for the p - wave atomic superfluid is derived directly from low energy expansion ( 1a ) of the scattering amplitude .    however , the effective - range theory is correct only for the _ short range _ potentials ( _ e.g. _ , yukawa potential ) @xcite which decays faster than any power function @xmath22 in the large interatomic distance limit @xmath23 . here @xmath24 is the relative coordinate between two atoms . for a realistic interaction between two cold atoms , which is described by a _ long - range _ potential dominated by the van der waals term @xmath25 in the limit @xmath26 . here @xmath27 is the single - atom mass , @xmath28 is the plank constant and @xmath29 is the van der waals length . due to the long - range van der waals potential , the effective - range theory and the low - energy expansion ( [ 1a ] ) of the p - wave scattering amplitude is not applicable any longer @xcite .    in the presence of a p - wave magnetic feshbach resonance ( pmfr ) in the ultracold gases of polarized fermionic atoms , the p - wave scattering amplitude of the atomic collision is contributed by both the background potential in the open channel and the bound state in the closed channel . the long - range nature of the background potential makes the final scattering amplitude to be inconsistent with the effective - range theory . therefore , it is essential to investigate the condition under which the effective - range theory ( [ 1a ] ) can be used as an approximation of the exact p - wave scattering amplitude under a pmfr . if the effective - range theory provides a good approximation of the scattering amplitude , then the previous theories on p - wave superfluid would be applicable ; nevertheless , if the exact scattering amplitude is found to be significantly different with the one in eq . ( [ 1a ] ) , then the previous theories should be modified . especially , the separable two - body potential can not be used any more . the low - energy p - wave scattering amplitude near pmfr has been investigated in refs . @xcite . however , these studies are based on simplified models of the atomic interaction , e.g. , zero background potential p2bfr or a separable background potential that decays exponentially in the momentum space @xcite . the long - range van der waals potential is not taken into account in either case . due to these simplifications , the scattering amplitudes given in refs . @xcite automatically have the form of eq . ( [ 1a ] ) , and can not be used to judge the applicability of the effective - range theory .    in this paper , based on the realistic long - range inter - atomic potential , we provide an explicit calculation for the low - energy p - wave scattering amplitude of two spin polarized fermonic atoms near a pmfr , and then discuss the condition under which the effective - range theory can be used as a good approximation . we get sufficient conditions for the effective - range theory , and show that for the ultracold gases of of @xmath9li and @xmath10k with the fermi temperature of the order 1@xmath30k , the effective - range theory can be used as a good approximation in the resonance regime where the scattering volume is enhanced      the main results of this paper are summarized as follows .    in this work we first calculate the exact expression of the low energy p - wave scattering amplitude with a pmfr . we prove that the scattering amplitude can be expressed as @xmath31with @xmath32 the strength of the magnetic field applied along the @xmath33 axis . here @xmath34 is the @xmath33-component of the angular momentum ; @xmath35 @xmath36 and @xmath8 are @xmath2-dependent scattering parameters . we obtain the general expressions for them . it is pointed out that , the denominator of @xmath37 can not be expressed as a laurent series with @xmath2-independent coefficients because @xmath38 @xmath36 and @xmath39 are not analytical functions of @xmath2 . equation ( [ 1c ] ) shows that , in the presence of a pmfr the inconsistency of the scattering amplitude with the effective - range theory is displayed in a more complicated manner . the low - energy p - wave scattering amplitude in eq . ( [ 1c ] ) is different from the one obtained from the standard effective - range theory in the following two senses :    \1 . the scattering parameters @xmath40 depend on the incident momentum @xmath41 . the term @xmath42 $ ] can not be included in the effective - range theory .    after obtaining the p - wave scattering amplitude under a pmfr , we discuss the applicability of the effective rang theory as an approximation of the scattering amplitude ( [ 1c ] ) . we find that , in the bec side of the pmfr where @xmath43 and @xmath8 have the same sign , sufficient conditions for the validity of effective - range theory are @xmath44 . in the bcs side of the resonance , the sufficient conditions become @xmath45 . here @xmath46 , @xmath47 and @xmath48 are defined in eqs . ( [ r1new ] ) , ( [ r2 ] ) and ( [ r3 ] ) . if these conditions are satisfied , the scattering amplitude of our system can be approximated as @xmath49 which has the same form as eq . ( [ 1a ] ) derived by the effective - range theory . qualitatively speaking , the above sufficient conditions means that we can use the effective - range theory if the fermonic momentum of the cold gas is low enough , the magnetic field is tuned close enough to the resonance point and the background scattering potential in the open channel is far away from the zero - energy shape resonance point . for the realistic cold gases of fermi atoms , if the background scattering is far away from the shape resonance , the effective range theory can usually be used in the total region where the @xmath50-wave interaction is negligible . the paper is organized as follows . in sec . ii , we calculate the p - wave scattering amplitude near a pmfr , and obtain the low energy expansion in ( [ 1c ] ) . the parameters @xmath51 are expressed in terms of the background scattering parameters and the magnetic field . in sec . iii we discuss the range of applicability of the effective - range theory . we show that , under some simple sufficient conditions @xmath52 or @xmath53 , the scattering amplitude given by the effective - range theory is a good approximation for the exact one obtained in sec . we further show that these conditions are well satisfied in the cold gases of @xmath14k and @xmath12li when the scattering volume is enhanced by a pmfr , and then the previous results based on the effective - range theory are applicable for these systems . in sec . iv there are some conclusion and discussions . we describe some detail of our calculations in the appendixes . in this section we calculate the p - wave scattering amplitude in presence of a pmfr induced by a magnetic field along the @xmath33 direction . we begin with the two - channel hamiltonian for the relative motion of two atoms ( fig . 1 ) : @xmath54where @xmath55 is the kinetic energy of relative motion , @xmath56 is the background scattering potential in the open channel , @xmath57 is the coupling between the open and closed channel , and @xmath58 is the interaction potential in the close channel , that has a @xmath32-dependent positive threshold @xmath59 . in this paper , for simplicity , we assume that the background potential @xmath60 is independent on the direction of @xmath24 and invariant under the so(3 ) rotation . we further assume that , in the closed channel there are only three bare p - wave bound states @xmath61 which are near resonance with the threshold of the open channel . here @xmath62 is the angular momentum along the @xmath33 axis . the self energy @xmath63 of @xmath64 with @xmath62 depends on the atomic magnetic dipole . for the atoms with small magnetic dipole , @xmath9li , the values of @xmath65 and @xmath66 are close with each other p2bfr , while for the the atoms with large magnetic dipole , _ @xmath67k , the difference between @xmath68 and @xmath69 is quite large @xcite . the p - wave scattering amplitude in the open channel can be defined with the standard scattering theory @xcite . to this end we firstly introduce the two - component stationary scattering state@xmath70where @xmath71 is the eigenstate of the atomic relative momentum with eigen value @xmath72 , and @xmath73 is the mller operator defined as@xmath74 in the large interatomic distance limit @xmath26 , the asymptotic behavior of the state @xmath75 can be expressed as@xmath76with @xmath77 and @xmath78 the scattering amplitude which can be further expanded in terms of different partial waves:@xmath79here @xmath80 is the spherical harmonic function . for a scattering potential with so(3 ) symmetry , the partial wave scattering amplitude only depends on the quantum number of the angular momentum @xmath81 . in our case , the so(3 ) symmetry is broken by the interaction between the atomic magnetic dipole and the magnetic field . then we have a @xmath82-dependent scattering amplitude @xmath83 .    in the case of low - energy scattering between two spin polarized fermonic atoms , one can neglect all the high partial wave scattering amplitudes @xmath84 with @xmath85 , and only consider the p - wave amplitudes @xmath86 , which can be further expressed in terms of the p - wave phase shifts @xmath87:@xmath88    during the scattering process , the bare bound state @xmath89 is coupled with the p - wave background scattering states in the open channel and significantly change the p - wave scattering amplitude @xmath37 . this effect can be directly treated via the feshbach resonance theory , _ e.g. _ , the methods in ref . @xcite and ref . @xcite .    after a straightforward calculation in appendix a , we find that the final phase shift @xmath87 is the sum of the background phase shift @xmath90 for the background potential @xmath91  and a correction @xmath92 given by the closed channel : @xmath93 here @xmath92 satisfies@xmath94 with @xmath95 and @xmath96 given by@xmath97 in the above we have used the background green s function @xmath98 @xmath99 and the background scattering state @xmath100 defined as@xmath101    in the following subsections , we evaluate the low - energy expression of the scattering amplitude @xmath37 by expanding the term @xmath102 in eq . ( [ fdelta ] ) in the limit @xmath103 . as shown in eq . ( [ delta1a ] ) , the phase shift @xmath104 is the sum of @xmath105 and @xmath92 . the low energy behavior of background phase shift @xmath106 is already known to be @xcite @xmath107 therefore , if we can further obtain the low - energy expansion of term @xmath108 , then the expressions of @xmath109 and @xmath37 can be calculated straightforwardly .      in this subsection we investigate the expression of @xmath111 in the limit @xmath103 . to this end , we need to expand both the numerator and the denominator of ( [ delta1 ] ) in the low - energy limit .    in this paper we assume the background scattering volume in the open channel is finite . it can be proved that ( appendix c ) , in this case the function @xmath112 can be expanded as@xmath113with @xmath114 .    on the other hand , due to the long - range nature of the van der waals potential , the partial wave scattering state @xmath115 is not an analytical function of the incident momentum @xmath2 in the neighborhood of @xmath116 @xcite . to investigate the low - energy behavior of @xmath95 and then the denominator of ( [ delta1 ] ) , we separate the non - analytical part of @xmath117 by introducing the background jost function @xmath118 @xcite defined as@xmath119 here @xmath120 is the canonical solution of the radial equation@xmath121 with boundary condition@xmath122 where @xmath123 is the first - order regular riccati - bessel function @xcite . according to the standard scattering theory @xcite , @xmath124 is an analytical function of @xmath2 , and can be expanded as a talyor series of @xmath2 : @xmath125 it is pointed out that , all the odd order terms of the above taylor series are exactly zero @xcite . thus the non - analytical part of @xmath95 is included in the term with the jost function @xmath118 . substituting eqs . ( [ ftuter ] ) , ( [ jost1 ] ) and ( [ gk2 ] ) into eq . ( [ delta1 ] ) , we find that in the low - energy limit the factor @xmath126 takes the form @xmath127here we have the @xmath2-dependent parameters:@xmath128 ^{-1}.   \nonumber \\ & &   \label{rdet}\end{aligned}\]]with the parameters @xmath129 and @xmath130 defined by@xmath131 _ { k=0 }   \nonumber \\ w_{m_{z}}^{\prime } & = & \frac{1}{120}\frac{d^{5}}{dk^{5}}\left [ |\langle \phi _ { \mathrm{res}}^{(m_{z})}|w|\psi _ { k1m_{z}}^{(\mathrm{bg}+)}\rangle    & & \end{aligned}\ ] ]      in the above subsection we get the expansion ( [ kcot2 ] ) of the factor @xmath108 . substituting eq . ( [ kcot2 ] ) , ( [ delta1bg ] ) and ( [ delta1a ] ) into ( [ fdelta ] ) , we finally get the low - energy behavior of the p - wave scattering amplitude @xmath37 in presence of a pmfr : @xmath132where the @xmath2-dependent scattering parameters are given by@xmath133here the parameters @xmath134 and @xmath135 are defined as@xmath136    so far we have obtained the low - energy expression of the p - wave scattering amplitude @xmath37 in the case of pmfr . with the help of the scattering theory , we obtain the general expressions ( [ f1]-[v2effa ] ) for the scattering amplitude @xmath37 as well as the scattering parameters @xmath137 , which are formulated in terms of the background scattering parameters @xmath138 and the magnetic field . it is pointed out that , although due to the long - range nature of the van der waals potential we can not express the denominator of @xmath37 as an laurent series with @xmath2-independent coefficients , we successfully include all the @xmath2-dependence of the parameters @xmath40 into the jost function @xmath139 . ( [ v0eff ] ) clearly show the effect of pmfa that the scattering volume @xmath140 diverges under the magnetic field @xmath141 . in the realistic cold atom gases , for the creation of an observable effects with @xmath50-wave interaction , the scattering volume @xmath142 should be large enough . particularly , @xmath143 should be much larger than the van der waals length so that in the bcs region , the transition temperature @xmath144 $ ] @xcite of superfluid is realizable and in the bec region the binding energy of the @xmath50-wave feshbach molecule be roubst with respect to the detail of the atom - atom interaction potential .    in the end of this section , we consider the dependence of the effective range @xmath145 on the magnetic field @xmath32 . according to eq . ( [ v2effa ] ) , @xmath145 depends on @xmath32 through the ratio @xmath135 between @xmath146 and @xmath140 , and the quantatity @xmath147 . in the cold gases of @xmath9li and @xmath10k , the background scattering volumes @xmath146 are of the order of @xmath148 . according to our above discussion , they are too small for the creation of @xmath50-wave superfluids @xcite . therefore in these systems the strong enough @xmath50-wave interactions can only be obtained in the resonance region with @xmath149 or @xmath150 which implies @xmath151 . on the other hand , according to eq . ( [ rdet ] ) , the dependence of @xmath152 on @xmath32 is significant when the magnetic field is far away enough from from the resonant point @xmath129 so that the factor @xmath153 is comparable or larger than @xmath154 . the values of @xmath155 and @xmath130 are not available for @xmath9li and @xmath10k . nevertheless , the binding energies of the @xmath50-wave feshbach molecules are measured to be linear functions of the magentic field @xcite in the region with large enough @xmath50-wave scattering volumes ( @xmath156 ) . this observation shows that in these regions the term @xmath157 is negligible and the effective range @xmath145 can be approximated as a @xmath32-independent constant @xmath158 . in the above section , we have obtained the expression ( [ f1 ] ) of the p - wave scattering amplitude @xmath37 in the region near the point of pmfr . it is apparent that , this expression is different from the one ( 1a ) given by the effective - range theory in the following two senses :    \1 . in the standard effective - range theory , the scattering volume @xmath159 and effective - range @xmath4 are independent on the incident momentum @xmath2 . nevertheless , in the expression ( [ f1 ] ) the scattering parameters @xmath160 depend on @xmath2 through the jost function @xmath118 . the term @xmath42 $ ] can not be included in the effective - range theory . it is apparent that , if under some condition the scattering amplitude ( f1 ) can be approximated as@xmath161i.e . , both the @xmath2-dependence of @xmath162 and the term @xmath163 $ ] can be neglected , the behavior of the system would be approximately described by the effective - range theory . in this section , we investigate the conditions for the approximation ( f1er ) or ( [ 1 g ] ) , or the applicability of effective - range theory . we will consider the importance of the term @xmath164 $ ] and the @xmath2-dependence of the jost function respectively . in this subsection we search the sufficient condition for the ignorance of the @xmath2-dependence of the scattering parameters @xmath165 . as we have discussed above , the @xmath2-dependence of the scattering parameters comes from the mode square of the jost function @xmath118 . in the ultracold gases of the fermionic atoms , the maximum value of the relative momentum of two atoms is on the order of the fermi momentum @xmath166 . therefore , the importance of the @xmath2-dependence of the parameters parameters @xmath167 can be described by the factor @xmath168 obviously , when @xmath169 , we can replace @xmath170 with @xmath171 and neglect the @xmath172 dependence of the parameters @xmath160 . ( blue dash - dotted line ) , @xmath173 ( open circle ) , @xmath174 ( black solid line ) and @xmath175 ( red dashed line ) in eq . ( [ jost]).,width=283 ]    to investigate the behavior of the the ratio @xmath176 , we first calculate the jost function @xmath118 . by means of the quantum defect theory gao3 , we can obtain the expression of @xmath170 ( see appendix d ) : @xmath177^{-1 }   \nonumber \\\end{aligned}\ ] ] where @xmath178 is a @xmath2-independent coefficient and @xmath179 @xmath180 with @xmath181 defined in @xcite . in fig . 2 we plot the functions @xmath182 in the low - energy case .    the parameter @xmath183 is denoted as @xmath183 is related to the background scattering parameters @xcite . expanding the p - wave phase shift in eq.(7 ) of ref . @xcite , we can express @xmath184 in terms of @xmath183 : @xmath185^{2}}\beta _ { 6}^{3 } ;   \label{v0bg } \\ \mathcal{s}^{(\mathrm{bg)}}&=&-\frac{35(1+k_{l=1}^0)^{2}\pi } { % 324(k_{l=1}^{0})^2\gamma \lbrack 3/4]^{4}}\beta _ { 6}^{2}.   \label{v1bg}\end{aligned}\ ] ]    the above expression shows that , when @xmath186 ^ 2)\sim0.1 $ ] , we have @xmath187 and the background scattering potential is far away from the shape resonance ; when @xmath188 is much smaller than @xmath189 , the background potential is in the shape resonance region which gives @xmath190 . now we consider the features of the ratio @xmath176 , which is determined by the parameter @xmath188 . 2 shows that in the low energy case with @xmath191 , the function @xmath174 is almost a @xmath2-independent constant and much larger than the other three d - functions . therefore , if the parameter @xmath188 is large or the background scattering potential in the open channel is far away from the shape resonance , then according to eq . ( [ jost ] ) , the jost function @xmath192 is dominated by the term with @xmath193 . in this case the variation of @xmath170 with respect to @xmath2 is negligible and we have @xmath169 . on the other hand , if @xmath188 is close to zero and the background scattering potential is close to the shape resonance , then @xmath192 becomes a rapid changing function of @xmath2 and the ratio @xmath176 would be significant . the above argument is quantitatively verified by fig . 2 , where the ratio @xmath46 is plotted as functions of @xmath2 with respect different values of @xmath183 or @xmath194 . it is clearly shown that , if the fermonic momentum @xmath195 , then the ratio @xmath46 and the @xmath2-dependence of the scattering parameters can be neglected when the background potential is far enough from the shape resonance so that @xmath196 . if @xmath197 , this restriction can be further relaxed to @xmath198 . defined in eq . ( [ r2 ] ) as a function of the fermi momentum @xmath199 with @xmath200 ( @xmath201 , blue solid line ) , @xmath202 ( @xmath203 , green empty square ) , @xmath204 ( @xmath205 , black empty circle ) , @xmath206 ( @xmath207 , blue dash - dotted line ) , @xmath208 ( @xmath209 , green filled square ) , @xmath210 ( @xmath211 , red dashed line).,width=283 ]      now we discuss the importance of the term @xmath164 $ ] . since the purpose of this paper is to obtain the sufficient condition for the effective - range theory , or the approximation ( [ f1er ] ) , for simplicity , in this subsection we assume the condition @xmath169 obtained in the above section is already met , and the @xmath2-dependence of the coefficients @xmath212 can be neglected .    in the bec side of the pmfr where @xmath213 , since @xmath214 , the parameters @xmath8 and @xmath43 have the same sign . in that case , if the absolute value of @xmath215 is much smaller than the one of @xmath216 , then it would be also much smaller than @xmath217 , and can be neglected . it is obvious that , in the limit @xmath103 , the term @xmath215 would be much smaller than @xmath218 . therefore the importance of the term @xmath219 is actually determined by the ratio @xmath47 between the two terms for the upper limit of the relative momentum @xmath199 : @xmath220}{1/[% \mathcal{v}^{\mathrm{eff}}(0;b;m_{z})k_{f}^{2}]}\right\vert .\end{aligned}\ ] ] when @xmath47 is much smaller than unity , we can neglect the term @xmath221 . if @xmath47 is comparable or larger than unity , the term @xmath222 would be necessary for the theory .    the straightforward calculation with eqs . ( [ v0eff ] ) , ( [ v1eff ] ) , ( v0bg ) and ( [ v1bg ] ) yields @xmath223 in the practical cold atom systems we have @xmath224 . therefore , in the resonance region with @xmath225 , the ratio @xmath47 in eq . ( [ r1new ] ) is much smaller than unity , and then the term @xmath226 is negligible .    in the bec side of the resonance with @xmath227 , the parameters @xmath228 and @xmath8 have different signs . in that case , there is a speical momentum@xmath229which makes the terms  @xmath230 $ ] and @xmath231 $ ] cancel with each other or@xmath232    therefore , if the atomic relative momentum @xmath2 is far away from @xmath233 , the absolute value of @xmath234 would be quite different with the one of @xmath235 . in that case we can still neglect the term @xmath222 under the condition @xmath236 or @xmath237 . if the atomic relative momentum @xmath2 is in the neighborhood of @xmath238 and the terms @xmath239 is canceled with @xmath235 , the scattering amplitude ( [ f1 ] ) can be expressed as@xmath240 in that case , if the absolute value of @xmath241 $ ] is much smaller than @xmath238 , we an also neglect the term with @xmath36 , even in the neighborhood of @xmath238 . we define a parameter @xmath48 as @xmath242then the term with @xmath36 can be neglected when @xmath243 . a further calculation with eqs . ( [ kstar ] ) , ( [ v0eff ] ) , ( [ v1eff ] ) , ( [ v0bg ] ) and ( [ v1bg ] ) implies that @xmath244    as shown above , the condition @xmath245 is obtained for the momentum region @xmath246 . since the realistic momentum of the atomic relative motion takes the value between zero and @xmath199 , in the cases with @xmath247 , we can disregard the restriction of the ratio @xmath48 , and use effective - range theory under the condition @xmath248 in both the bec and the bcs sides of the resonance .      in summary , the general sufficient conditions for the effective - range theory in the bcs side of the resonance can be summarized as @xmath249while the ones for the bec side are @xmath250from the definition of the ratios @xmath251 and @xmath48 , we notice that in the realistatic cold gases of fermi atoms , the crucial factors for the usage of effective range theory is the background @xmath50-wave scattering volume @xmath252 and the @xmath32-dependence of the factor @xmath253 . if the the background @xmath50-wave scattering is far away from the shape resonance so that @xmath254 then according to our previous discussions and eqs . ( [ r1new ] ) , the conditions @xmath44 can be satisfied in the region @xmath255 where the @xmath50-wave interaction is strong enough for the creation of @xmath50-wave superfluids . in that region we also have @xmath256 . if @xmath257 can be further approximated as a @xmath32-independent constant which is of the order @xmath258 , then the condition @xmath245 can also be satisfied , and the effective range theory can be used a good approximation for the realistic scattering amplitude . in the following subsection we show that the pmfrs in the cold gases of @xmath10k and @xmath9li are just of this case .      in the above subsections we obtained the sufficient conditions ( [ sc1 ] ) , ( [ sc2 ] ) of the effective - range theory for the p - wave scattering amplitudes of polarized fermonic atoms near a pmfr . in this subsection , with the help of the conditions , we perform a discussion on the usage of effective - range theory in the ultracold gases of @xmath10k and @xmath9li .    for the ultracold gas with @xmath10k atoms in the state @xmath259 , we have @xmath260(a.u . ) @xcite and @xmath261 @xcite . these parameters leads @xmath262 and @xmath263 . if the fermi temperature @xmath264k , then we have @xmath265 . the straightforward calculation shows that @xmath266 . therefore @xmath2-dependence of the scattering parameters can be safely neglected . the p - wave feshbach resonance for the states with @xmath267 occurs at @xmath268 g with width @xmath269 g and effective - range @xmath270 . the resonance for the states with @xmath271 occurs at @xmath272 g with width @xmath273 g and effective - range @xmath274 expk40dipo . according to these data we have @xmath275 when @xmath276 . then effect from the ratio @xmath48 is also negligible . therefore the sufficient condition for the usage of effective - range simply becomes @xmath277 . further calculation shows that @xmath278 when @xmath279 . then the effective - range approximation ( [ f1er ] ) is applicable for @xmath10k atoms in the state @xmath259 in the whole region of pmfr with @xmath280 . the condition for the effective - range approximation is broken only in the small region @xmath281 g ( @xmath271 ) or @xmath282 g ( @xmath267 ) where we have @xmath283 or @xmath284 . now we consider the gas with @xmath9li atoms in the ground hyperfine state @xmath285(a.u . ) li6parameter , xxxgao and @xmath286 . these parameters leads @xmath287 and @xmath288 . if the fermi temperature @xmath289k , we have @xmath290 which implies @xmath291 . then similar as above , the effective range approximation ( [ f1er ] ) is also applicable for @xmath9li atoms in the whole region of pmfr with @xmath292 . in this work we obtain the explicit expression of the p - wave scattering amplitude of two ultracold spin polarized fermonic atoms near the p - wave feshbach resonance . we show that due to the long rang nature of the van der waals potential , the scattering amplitude is explicitly described by eq . ( [ 1c ] ) in the low - energy case . with the help of the quantum defect theory , we formulate all the scattering parameters @xmath293 in terms of the background parameters and the inter - channel coupling . based on this result , we discussed the applicability of the effective - range theory , or the eq . ( [ 1d ] ) as an approximation of the exact scattering amplitude . we show that , in the bec side of the resonance , the sufficient conditions of the effective - range theory can be quantitatively described as @xmath294 while in the bcs side the conditions become @xmath295 , where @xmath296 are defined in eqs . ( r1new ) , ( [ r2 ] ) , and ( [ r3 ] ) . the applicability of the effective - range theory for the ultracold gases of @xmath10k and @xmath9li are examined with our results . the effective - range theory is shown to be a good approximation in both of the two cases in the resonance regime where the absolute value of the scattering volume is equal to or larger then the background one . in this appendix we post the derivation of the p - wave phase shift @xmath297 in eqs . ( [ delta1a ] ) and ( [ delta1 ] ) . our calculation is based on the method in ref . we begin from the scattering state @xmath75 in eq . ( [ phik ] ) . according to the scattering theory @xcite , the open channel component @xmath298 and closed channel component @xmath299 of @xmath75 satisfy the two channel lippmman - schwinger equation    @xmath300    with the free green s functions@xmath301we further define the background green s function @xmath302 and the closed channel green s function @xmath303 as@xmath304then we have the relationships @xmath305substituting eqs . ( [ g01 ] ) and ( [ g02 ] ) into the last terms of the right hand side ( r.h.s . ) of eqs . ( [ lp1a ] ) and ( [ lp1b ] ) , and using the lippmman schwinger equation ( [ psibg ] ) for the background scattering state @xmath306 , we get the equation which relates @xmath75 with the background scattering state @xmath306 @xcite : @xmath307    to calculate the p - wave phase shifts @xmath87 , we operate the projection operator @xmath308 for the manifold @xmath309on both of the two sides of eqs . ( [ m ] ) and ( [ n ] ) . then we have@xmath310here we have used@xmath311which are guaranteed by the rotational symmetry along the @xmath33-axis of the system . we also made the approximation @xmath312 that is , we only take into account the contribution from the near - resonance bound state @xmath313 in the closed channel .    substituting eqs . ( [ nn ] ) into eq . ( [ mm ] ) , we get@xmath314with @xmath315replacing the @xmath316 in the r.h.s . of eq . ( [ p ] ) with the r.h.s . of ( [ o ] ) , we get@xmath317 substituting eq . ( [ q ] ) into eq . ( [ o ] ) , and using the asymptotic expression ( [ asypsi ] ) of the scattering state and the definition ( parf ) of the partial wave scattering amplitude , we can obtain the p - wave scattering amplitude    @xmath318    here @xmath95 is defined in eq . ( psik1mz ) . @xmath319 is defined as @xmath320with@xmath321the background state with in - going boundary condition . in the above calculation we also used the asymptotic behavior of the background green s function:@xmath322with @xmath77 .    with straightforward calculation , we can further rewrite the scattering amplitude @xmath37 in eq . ( [ fcomplex ] ) as @xmath323with@xmath324 in the derivation of eq . ( [ finalf ] ) we have used the relationship ( appendix b)@xmath325and @xmath326here @xmath327 refers to the principle value of the integral .    considering the relationship ( [ fdelta ] ) between the scattering amplitude @xmath37 and the phase shift @xmath87 , it is easy to prove that the phase shift @xmath87 corresponding to the scattering amplitude ( [ finalf ] ) is the one given in eqs . ( [ delta1a ] ) and ( [ delta1 ] ) . it is pointed out that , this result can also be proved with the method in ref . ( @xcite ) . in this appendix we prove the formula ( [ nppsi ] ) in appendix a. we begin from the relationship @xcite between the three dimensional scattering states with ingoing and outgoing boundary conditions : @xmath328considering the definitions ( [ psik1mz ] ) and ( [ psik1mzn ] ) of @xmath329 , we can obtain @xmath330we further define the one dimensional functions @xmath331 as taylor @xmath332using the relationships @xmath333we get@xmath334    on the other hand , we know that @xmath335 and @xmath336 satisfy the same differential equation @xmath337with the same boundary condition @xmath338 . then @xmath339 is proportional to @xmath340 . to calculate the ratio between @xmath341 , we consider their asymptotic behaviors in the limit @xmath26 : @xmath342 ;   \nonumber \\ \\ f_{k1}^{(\mathrm{bg}-)}(r ) & = & \hat{j}_{1}(kr)+kf_{k1}^{(\mathrm{bg}% ) } ( k)^{\ast } [ \hat{n}_{1}(kr)-i\hat{j}_{1}(kr ) ]   \nonumber \\\end{aligned}\]]with @xmath343defined in ( [ j1 ] ) and @xmath344 the irregular first order riccati - bessel function . comparing the coefficients of @xmath345 , we have@xmath346here we have used@xmath347substituting eq . ( [ c10 ] ) into eq . ( [ c3 ] ) , we get eq . ( [ nppsi ] ) . in this appendix we prove the eq . ( [ gk2 ] ) . we firstly rewrite the factor @xmath96 as    @xmath348    here we have used the identity@xmath349the first term in the r.h.s . of ( [ expgk ] ) is independent on @xmath2 . it contributes the constant term @xmath350 in eq . ( [ gk2 ] ) .    on the other hand , the second term in eq . ( [ expgk ] ) can be re - written as @xmath351 in the limit @xmath116 , we have@xmath352 . on the other hand , as we have shown in sec . iii.b , the factor @xmath353 tends to a non - zero constant in the low energy limit if the background scattering volume in the open channel is finite . using the relationship ( [ jost1 ] ) between @xmath118 , @xmath354 and @xmath355 , and the low - energy behavior ( [ ftuter ] ) of @xmath356 , it is easy to prove that the factor @xmath357 is proportional to @xmath358 in the limit @xmath359 . therefore , the above integration in eq . ( [ c44 ] ) converges to a finite constant in the limit @xmath360 . then the expansion in eq . ( [ gk2 ] ) is proved and we have @xmath361 in this appendix we calculate the jost function @xmath118 of the background scattering state . to this end , we introduce a function @xmath362 where @xmath363 is defined in eq . ( [ jost1 ] ) . it is apparent that @xmath364 is a solution of the radial equation @xmath365with a @xmath2-independent boundary condition @xmath366    following the spirit of quantum defect theory @xcite , we assume the scattering potential @xmath367 can be approximated as the van der waals potential @xmath25when @xmath368 is larger than a critical distance @xmath369 which is much smaller than @xmath258 . in the region with @xmath370 , @xmath367 is assumed to be so large that the atomic kinetic energy @xmath371 is negligible in comparing with @xmath91 , and then @xmath372 is independent on @xmath2 .    in the region @xmath373 , @xmath372 is the superposition of the two independent solutions @xmath374 and @xmath375 of eq . ( [ radical ] ) @xcite ( in ref . @xcite , @xmath374 and @xmath375 are denoted as @xmath376 and @xmath377):@xmath378 in the short distance region with @xmath379 , @xmath380 and @xmath381 are almost independent on @xmath2 @xcite . the wave function @xmath372 in the two regions are connected at point @xmath382 , where @xmath372 , @xmath383 and @xmath375 are approximately independent on @xmath2 . then we know that @xmath384 and @xmath385 are independent on @xmath2 and we have @xmath386with @xmath387 . the above equation yields@xmath388    in the region @xmath26 , the asymptotic behaviors of @xmath389 and @xmath390 can be expressed in terms of @xmath181 ( @xmath391 ) defined in @xcite @xmath392 ;   \nonumber \\ & &   \label{ff1 } \\ \kappa _ { \epsilon_k1}^{(0)}(r ) & \rightarrow & \sqrt{\frac{2}{\pi k}}\left [ z_{gf}\sin \left ( kr-\frac{\pi } { 2}\right ) -z_{gg}\cos \left ( kr-\frac{\pi } { % 2}\right ) \right ] . \nonumber \\ & &   \label{g1}\end{aligned}\]]on the other hand we know that in the same limit we have @xcite @xmath393 .   \nonumber   \label{sc } \\ & & \end{aligned}\]]together with eqs . ( [ ss ] ) , ( [ ff1 ] ) , ( [ g1 ] ) as well as ( [ sc ] ) , we obtain the expression of ( [ jost ] ) of @xmath162 and the background phase shift : @xmath394where @xmath179 and @xmath395 is the one in sec . the result ( [ tgdeltabg ] ) is also given in @xcite . e. wille , f. m. spiegelhalder , g. kerner , d. naik , a. trenkwalder , g. hendl , f. schreck , r. grimm , t. g. tiecke , j. t. m. walraven , s. j. j. m. f. kokkelmans , e. tiesinga , and p. s. julienne , phys . . lett . * 100 * , 053201 ( 2008 ) .
most of the current theories on the p - wave superfluid in cold atomic gases are based on the effective - range theory for the two - body scattering , where the low energy p - wave scattering amplitude @xmath0 is given by @xmath1 $ ] , where @xmath2 is the incident momentum , and @xmath3 and @xmath4 are the @xmath2-independent scattering volume and effective - range , respectively . however , due to the long - range nature of the van der waals interaction between two colliding ultracold atoms , the p - wave scattering amplitude of the two atoms is not described by the effective - range theory @xcite . in this paper 
 we provide an explicit calculation for the p - wave scattering of two ultracold atoms near the p - wave magnetic feshbach resonance ( pmfr ) . 
 we show that the low energy p - wave scattering amplitude in the presence of pmfr takes the form @xmath5 $ ] where @xmath6 @xmath7 and @xmath8 are @xmath2-dependent parameters . 
 based on this result , we show sufficient conditions for the effective range theory to be a good approximation of the exact scattering amplitude . using these conditions 
 we show that the effective - range theory is a good approximation for the p - wave scattering in the ultracold gases of @xmath9li and @xmath10k when the scattering volume is enhanced by the resonance .