Contribution of sigma meson pole to K_L-K_S mass difference

a rXiv:h ep-ph/8225v122Aug2YITP-00-49Contribution of sigma meson pole to K L -K S mass difference ∗K.Terasaki Yukawa Institute for Theoretical Physics,Kyoto University,Kyoto 606-8502,Japan Abstract The hypothesis of σmeson pole dominance in the |∆I |=12rule in these decays [1],and recently revived in connection with the direct CP violation in the K →ππdecays [2].If it is the case,however,the matrix elements, σ|H w |K ,should survive and give a significant contribution to the K L -K S mass difference,∆m K ,where H w is the strangeness changing (|∆S |=1)effective weak Hamiltonian.Dynamical contributions of various hadron states to hadronic processes in which pion(s)take part can be estimated by using a hard pion technique (with PCAC)in the infinite momentum frame (IMF)[3].For later convenience,we review briefly it below.As an example,we consider a decay,B (p )→π1(q )π2(p ′),in the IMF,i.e.,p →∞,and assume that its amplitude M (B →π1π2)can be approximately evaluated at a slightly unphysical point,q →0,i.e.,q 2→0but (p ·q )is finite:M (B →π1π2)≃lim p →∞,q →0M (B →π1π2).(1)In this approximation,the σ→π+π−amplitude is described in terms of the asymptotic matrix element, π−|A π−|σ ,(matrix element of A π−taken between π−and σwith infinite momentum)as M (σ→π+π−)≃√f π π−|A π−|σ ,(2)which has been symmetrized with respect to exchange of π+and π−in the final state since isospin symmetry is always assumed in this note.The asymptotic matrix element, π−|A π−|σ ,is given bylimp→∞,q→0 π−(p′)|Aπ−|σ(p)=(2π)3δ(3)(p−p′) π−|Aπ−|σ√√√m2n−m2K π2|A¯π1|n n|H w|K +(π1↔π2),(9)M(l)S(K→π1π2)=i2fπm2π−m2Kfπm2π−m2K|M (σ)|≫|M ETC |,|M (n =σ)S |,|M (l )S |,|M fact |,(12)unless the amplitudes in the right-hand-side cancel accidentally each other.However,if the σpole contribution dominates K S →ππamplitudes,it may be worried about that its strange partner,κ,also plays a role in the same amplitudes.The κpole amplitude can be obtained in the same way as M (σ)and its ratio to M (σ)is approximately given byM (κ)(K S →π+π−)m 2κ π|H w |κ4πf 2πm 2σ(m 2σ−m 2π)2| π−|A π−|σ |2,(14)andΓ(σ)(K S →π+π−)≃q Km 2K −m 2σ 2| π−|A π−|σ σ|H w |K 0 |2,(15)respectively,where q σand q K are the center-of-mass momenta of the final pions in the corresponding decays.Since the K S →ππmode dominates the decays of K S ,its total width,ΓK S ,is approximately given by ΓK S ≃32Γ(σ)(K S →π+π−)under the σpole dominance hypothesis.The σmeson pole dominance in the K S →ππmeans that the matrix element, σ|H w |K ,exists and its magnitude should be sizable.Therefore,under this hypothesis,the σmeson pole may give a substantial contribution to ∆m K .The formula describing dynamical con-tributions of hadrons to ∆m K has been given in the IMF long time ago [5].Using it,we obtain the following pole contribution of σmeson,∆m (σ)K =−| K L |H w |σ |2ΓK S ≃1q K (m 2σ−m 2K )m 2K −m 2π 2m K2Γ(σ→π+π−)for m σless than the K ¯Kthreshold (≃1GeV).Now we study whether the above σpole dominance in the K S →ππdecays can be realized in consistency with ∆m K .σ(GeV)mσ(GeV) 1.0- 0.130.50Γσ(GeV)Γm 1.0- 0.130.25RR(σ)(σ)0.50.30.40.50.30.4σ(GeV)(a)(b)Fig.I.R (σ)=∆m (σ)/ΓK S for 0.4<m σ<1.0GeV and 0.3<Γσ<0.5GeV.R (σ)is cut at 0.25in (a)and at 0.50in (b)in order not to exceed the esti-mated ππcontinuum contribution R (ππ)and the measured R exp ,respectively,as discussed in the text.It has been known that contribution of S -wave ππintermediate states to ∆m K can occupy about a half [6]of the observed value [7],i.e.,R (ππ)≡∆m (ππ)K ΓK S exp =0.477±0.022.(18)The above ∆m (ππ)K was estimated by using the Muskhelishvili-Omn`e s equation and the mea-sured ππphase shifts,etc.,in which any indication of σmeson was not obviously seen.Therefore,if σexists,its contribution should be included in the above ∆m (ππ)K ,so that we may put loosely the upper limit of the σpole contribution to ∆m K around the above esti-mate of ∆m (ππ)K ,i.e.,∆m (σ)K /ΓK S <0.25,and look for values of m σand Γσto satisfy it since σmeson is still hypothetical,i.e.,its mass and width are still not confirmed.At energieslower than 900MeV,the ππphase shift analyses have excluded any narrow I =0scalar state but a broad one (Γσ∼500MeV)may have a room in the region [7],0.4<m σ<1.2GeV.In fact,various broad candidates of σmeson with different masses (∼500−700MeV),different widths (∼300−600MeV)and different structures have been studied at this workshop [8].R (σ)in Eq.(17)increases rapidly as m σincreases.It is beyond not only the estimated R (ππ)for m σ>0.55GeV but also the measured R exp in Eq.(18)for m σ>0.57GeV andis much larger than the above cuts in the region m 2σ≫m 2K .Therefore,even if κpolecontribution to the K →ππdecays is taken into account,the result,R (σ)≫R exp form 2σ≫m 2K ,is not changed as discussed before.In this way,it is seen that the σmeson pole dominance in the K →ππamplitudes is not compatible with ∆m K if m σ>0.57GeV and 0.3<Γσ<0.5GeV,unless any other contribution cancels ∆m (σ)K .However,the above does not necessarily imply that the σmeson pole dominance is compatible with the K L -K S mass difference if m σ<0.55GeV,since we have so far con-sidered only the long distance effects on the K L -K S mass difference.The short distance contribution from the box diagram [9]which is estimated by using the factorization may saturate the observed (∆m K )exp although it is still ambiguous because of uncertainty of the so-called B K parameter.If it is the case,however,we need some other contribution to cancel the ππcontinuum contribution (including σmeson pole).Possible candidates are pseudo-scalar(PS)-meson poles since the other contributions of multi hadron intermediate states will be small because of their small phase space volumes.The above implies that the matrix elements, P |H w |K ,P =π0,η,η′,···,survive and their sizes are large enough tocancel ∆m (ππ)K .In this case,however, π|H w |K ’s can give large effects on the K →ππamplitudes [10]through Eq.(4)with Eq.(5)and break the σmeson pole dominance.For the K L →γγdecay,it is known that short distance contribution is small [9].To reproduce the observed rate for this decay,we again need contributions of PS-meson poles given by the matrix elements, P |H w |K ’s,with sufficient magnitude,although their con-tributions are sensitive to the η-η′mixing and are not always sufficient.In fact,the above PS-meson matrix elements can approximately reproduce Γ(K L →γγ)exp ,Γ(K →ππ)exp ’s and (∆m K )exp ,simultaneously,with the help of some other contributions (non-factorizable amplitudes with PS-and K ∗-meson poles for the K L →γγdecay,factorized ones for the K →ππdecays and the short distance contribution to the K 0-¯K0mixing,etc.)but without any contribution of σpole [10].Namely,we do not necessarily need the σpole contribution in the K S →ππdecays.As was seen above,it is unlikely that the σmeson pole amplitude dominates the K S →ππ.It will be seen directly by comparing M (σ)(K S →π+π−)with M ETC (K S →π+π−).If the asymptotic matrix elements, π|H w |K ’s,with sufficient magnitude exist and satisfy the |∆I |=1M ETC (K 0S →π+π−)≃2 m 2K −m 2π π+|H w |K + .(19)The mass dependent factor |(m 2K −m 2π)/(m 2σ−m 2K )|from M(σ)can be enhanced only if m σis very close to m K and σis narrow.However,if Γσwere small,| π−|A π−|σ |also would be small.When we smear out the singularity at m σ=m K using the Breit-Wigner form,the size of |(m 2K −m 2π)/(m 2σ−m 2K ) π−|A π−|σ |is at most ≃2for 0.4<m σ<1.0GeV and0.3<Γσ<0.5GeV.However,any narrowσstate around m K is not allowed[7]as mentioned before.Moreover,σdoes not belong to the same ground state asπand K(for example, 3P0of{q¯q}state in the quark model,etc.),so that the matrix elements,| σ|H w|K |,will be much smaller than| π|H w|K |since wave function overlapping betweenσand K mesonstates will be much smaller than that betweenπand K which belong to the same1S0state of{q¯q}.Therefore,it is unlikely that theσmeson pole amplitude dominates the K→ππamplitudes.An amplitude for dynamical hadronic process can be decomposed into(continuum con-tribution)+(Born term).Since M S has been given by a sum of pole amplitudes,M ETC corresponds to the continuum contribution[11].In the present case,M ETC(K S→ππ)will be dominated by contributions of isoscalar S-waveππintermediate states and develop a phase(≃isoscalar S-waveππphase shift at m K)relative to the Born term which is usually taken to be real in the narrow width limit.The estimated phase difference between|∆I|=12amplitudes for the K→ππdecays is close to the measured isoscalar S-waveππphaseshift at m K[12].It suggests that the isoscalar S-waveππcontinuum contribution will be dominant in the K S→ππamplitudes.In summary,we have studied contribution of theσmeson pole to∆m K under the hypothesis thatσmeson pole dominates the K S→ππamplitudes,and have seen that it provides too large contributions to∆m K and that,to cancel out such effects,contributions of pseudo-scalar-meson poles will be needed.We also have discussed,comparing theσmeson pole amplitude with M ETC in the K S→ππamplitudes,that enhancement of theσmeson pole contribution is not sufficient if it is broad.Additionally,a recent analysis in the K→ππdecays within the theoretical framework of non-linearσmodel suggests that theσmeson pole contribution can occupy,at most,about a half of the|∆I|=12amplitude for the K→ππdecays is very unlikely.The author thanks Dr.D.-X.Zhang and Dr.Y.Y.Keum for discussions and comments. 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