aa r X i v : . [ h e p - ph ] D ec Final state interaction in K → π decay E.P. Shabalin Institute for Theoretical and Experimental Physics, B.Cheremushkinskaya,25, 117218, Moscow, Russia
Abstract
Contrary to wide-spread opinion that the final state interaction (FSI) enhancesthe amplitude < π ; I = 0 | K > , we argue that FSI does not increase the absolutevalue of this amplitude. PACS: I = 1 / K → π decays was achieved in the paper [1], where the authors hadfound a considerable increase of contribution of the operators containing aproduct of the left-handed and right-handed quark currents generated bythe diagrams called later the penguin ones. But for a quantitative agreementwith the experimental data, a search for some additional enhancement of the < π ; I = 0 | K > amplitude produced by long-distance effects was utterlydesirable. A necessity of additional enhancement of this amplitude due tolong-distance strong interactions was also noted later in [2].The attempts to take into account the long-distance effects were under-taken in [3] - [15].In [3], the necessary increase of the amplitude < π ; I = 0 | K > wasassociated with 1/N corrections calculated within the large-N approach (Nbeing the number of colours).In [4], [5], the strengthening of the < π ; I = 0 | K > amplitude arised dueto a small mass of the intermediate scalar σ meson.One more mechanism of enhancement of the < π ; I = 0 | K > amplitudewas ascribed to the final state interaction of the pions [6] - [14]. But as itwill be shown in present paper, unitarization of the K → π amplitude inpresence of FSI leads to the opposite effect: a decrease of the < π ; I = E-mail address: [email protected] (E.Shabalin) | K > amplitude. The similar conclusion had been obtained formerly in[15] where a different from our approach to investigation of FSI effect wasused. We exploit the technique based on the effective ∆ S = 1 non-leptonicLagrangian [1] L weak = √ G F sin θ C cos θ C X i c i O i . (1)Here O i are the four-quark operators and c i are the Wilson coefficients calcu-lated taking into account renormalization effect produced by strong quark-gluon interaction at short distances. Using also the recipe for bosonizationof the diquark compositions proposed in [2], one obtains the following result: < π + ( p + ) , π − ( p − ); I = 0 | K ( q ) > = κ (0) ( q − p − ) , (2)where κ (0) is a function of G F , F π , θ C and some combination of c i . Thenumerical values of κ (0) obtained in [1] and [2] turned out to be insufficientfor a reproduction of the observed magnitude of the < π ; I = 0 | K > amplitude.Could a rescattering the final pions occuring at long distances changethe situation? To answer this question, we consider at first the elastic ππ scattering itself. The elastic ππ scattering. The general form of the amplitude of elastic ππ scattering is T = < π k ( p ′ ) π l ( p ′ ) | π i ( p ) π j ( p ) > = Aδ ij δ kl + Bδ ik δ jl + Cδ il δ jk , (3)where k, l, i, j are the isotopical indices and A, B, C are the functions of s = ( p + p ) , t = ( p − p ′ ) , u = ( p − p ′ ) .The amplitudes with the fixed isospin I = 0 , , T (0) = 3 A + B + C, T (1) = B − C, T (2) = B + C. (4)To understand the problems arising in description of ππ scattering in theframework of field theory, let’s consider the simplest chiral σ model, where A tree = g σππ m σ − s − g σππ m σ − m π = g σππ m σ − m π · s − m π m σ − s (5)and B and C are obtained from A by replacement s → t and s → u , respec-tively. 2t follows from Eqs.(4) and (5), that the isosinglet amplitude T (0)tree is asum of the resonance part A treeRes = 3 A tree (6)and the potential part A treePot = B tree + C tree . (7)The resonance part must be unitarized summing up the chains of pion loops,that is , taking into account the repeated rescattering of the final pions.At the one loop order A one − loopRes = A treeRes (1+ ℜ Π R + i ℑ Π) = A treeRes (1+ ℜ Π R + i A treeRes q − m π /s π ) , (8)where ℜ Π R is the renormalized real part of the closed pion loop [16] ℜ Π R ( s ) = ℜ Π( s ) − ℜ Π( m σ ) − ∂ ℜ Π( s ) ∂s | s = m σ ( s − m σ ) . (9)The last two terms in r.h.s. of this equation are absorbed in renormaliza-tion of the resonance mass and coupling constant g σππ . In leading order ofperturbation theory in g σππ one has (see also [17]) ℜ Π R ( s ) = A treeres ( s )16 π q − m π /s ln 1 − q − m π/s q − m π /s . (10)But in view of very big value of this constant such a calculation does notgive a proper estimate of ℜ Π R ( s ). It will be explained below, how to get areliable magnitude of ℜ Π R ( s ).The unitarized expression for A Res is A unitarRes = A treeRes ( s )1 − ℜ Π R ( s ) − i ℑ Π Res = A treeRes ( s )1 − ℜ Π R ( s ) · − i tan δ Res , (11)where tan δ Res = A treeRes ( s ) q − m π /s π (1 − ℜ Π R ( s )) . (12) Strictly speaking, the 4 π intermediate state brings a correction in Eq.(11). But its con-tribution to ℑ Π( s ) is equal to zero because 4 m π > m K . As for ℜ Π R ( s ), in our approach,all separate contributions to it will be taken into account phenomenologically introducinga form factor, see below. A unitarRes = 16 π sin δ Res e iδ Res q − m π /s , (13)leading to the cross section σ Res = 4 π sin δ Res k , k = √ s · q − m π /s. (14)Of course, the amplitude T (0) must be unitarized including the potentialpart B + C too. But if this potential part is considerably smaller than theresonance one, the effect of FSI can be estimated roughly from A unitarRes . Tounderstand what gives the unitarization of A treePot , we use the form of the S matrix of elastic scattering with the total phase shift as a sum of thephase shifts produced by separate mechanisms of scattering [18]. In otherwords, if there is a number of resonances and if, in addition, there is potentialscattering, the matrix S looks as S = e iδ Res1 e iδ Res2 ...e iδ Pot . (15)Then, in terms of δ Res = X j δ Resj and δ tot = δ Res + δ Pot , (16) A unitar = 16 π q − m π /s sin δ tot e iδ tot (17)or A unitar = 16 π q − m π /s (sin δ Res cos δ Pot + sin δ Pot cos δ Res ) e iδ tot . (18)The phase shifts δ Res and δ Pot can be taken from [19], where the ResonanceChiral Theory of ππ Scattering (RChT) was elaborated. This model incor-porates two σ mesons, f (980), ρ (750) and f (1270). In addition, some phe-nomenological form factors were introduced in the vertices σππ, ρππ, f ππ .Their appearence follows in the field theory from the result (11), accordingto which the effect of ℜ Π R ( s ) may be incorporated in g σππ ( s ), where g σππ ( s ) = g σππ − ℜ Π R ( s ) = g σππ F ( s ) . (19)4he RChT gives a quite satisfactory description of the observed behavior ofthe phase shifts δ ( s ), δ ( s ), δ ( s ) in the range 4 m π ≤ s ≤ . For δ and δ at s = m K we have δ = 36 . ◦ , δ = − . ◦ (20)in agreement with the experimental values δ = 37 ◦ ± ◦ , δ = − ◦ ± ◦ (see [14]). The phase shifts δ ( s ) and δ ( s ) turn out to be consistent withthe results obtained using the Roy’s dispersion relations.A good fit to experimental data on δ and δ was obtained in [19] with F ( q ) = exp (cid:16) − . q − m π ) / (GeV) (cid:17) . (21)Then ℜ Π R ( s = m K ) = − .
12 (22)and has the same sign as in Eq.(10), but turns out to be considerably smallerin absolute value. For √ s = m K , the phase shifts obtained in [19] are δ Res = 46 . ◦ , δ Pot = − . ◦ . (23)Then A unitarPot A unitarRes = sin δ Pot cos δ Res sin δ Res cos δ Pot = − . . (24)Therefore, the amplitude A unitarPot is small and may be neglected in a roughestimate of FSI effect. FSI in K → π decay. Basing on the result (24), we shall estimate effects of FSI in the K → π amplitude, taking into account only the resonance rescattering effect. Then,in one loop approximation, the amplitude (2) is < π + ( p + ) π − ( p − ); I = 0 | K ( q ) > one − loopRes = κ (0) (cid:20) ( q − p − ) + A treeRes ( q )(2 π ) i R ( q − p ) d n p [( p − q ) − m π ][ p − m π ] + i A treeRes ( q )16 π ( q − p − ) q − m π /q (cid:21) . (25) A difference 1 . ◦ between δ P ot and δ is caused by intermediate ρ meson giving thecontributions to these phase shifts which are different in value and sign [19].
5n the t’Hooft-Veltman scheme of dimensional regularization [20] π ) i R d n p [( p − q ) − m ][ p − m ] = π (cid:18) ln M m + 2 + q − m /q ln − √ − m /q √ − m /q (cid:19) ; π ) R p d n p [( p − q ) − m ][ p − m ] = m π (cid:18) M m + 3 + q − m /q ln − √ − m /q √ − m /q (cid:19) M → ∞ . (26)After renormalization excluding the parts of these integrals independent ofthe external momentum, we come to < π + π − | K ( q ) > one − loopon − mass − shell == κ ( m K − m π ) (cid:20) A treeRes ( s )16 π q − m π /s ln − √ − m π /s √ − m π /s + i A treeRes ( s )16 π q − m π /s (cid:21) . (27)This result agrees with the Cabibbo–Gell-Mann theorem [21], according towhich the K → π amplitude vanishes in the limit of exact SU (3) symmetry.The unitarization of the amplitude (27) done in accordance with theprescription (11) leads to the result | < ππ ; I = 0 | K ( q = m K ) > | Res = κ (0) ( m K − m π ) cos δ Res − ℜ Π R ( m K ) , (28)where ℜ Π R ( m K ) is determined by Eq.(22). In accordance with Eq.(12), thefactor characterising FSI effect can be expressed through δ Res and A treeRes : χ (0) = cos δ Res ( s = m K )1 − ℜ Π R ( m K ) = 16 π sin δ Res ( s = m K ) A treeRes ( s = m K ) q − m π /m K . (29)The phase shift δ Res obtained in the framework of the theory [19] and given byEq.(23) is compatible with the data giving δ ≈ ( δ − δ ) exp = 44 . ◦ ± . ◦ .A value of A treeRes evidently depends on the used σ model which, besidesa good description of data on the phase shifts, must reproduce the otherexperimental observations. In particular, the result for a mass of the lightest σ meson: m σ ≤
700 MeV [22]. An existance of σ particle with mass m σ =600 ÷
700 MeV explains also why the observed value of the form factor F ( Q = 4 m π ) of K e decay is by 50% bigger than a value predicted in thenon-physical point Q = m π by the theory based on algebra of currents andsoft-pion technics [23] . In the region m π ≥ Q ≤ m π the amplitude is purely real σ mesons in U (3) L ⊗ U (3) R symmetric mesontheory, one can obtain that at m σ ≤
700 MeV the normalized amplitude A treeRes = 32 ( s − m π ) " g σ ππ ( m σ − m π )( m σ − s ) + g σ ππ ( m σ − m π )( m σ − s ) ≥ . (30)In the theory [19] m σ = 697 . A treeRes = 72 .
4. Therefore, using thedata on the phase shifts, one obtains χ (0) = 0 . +0 . − . . (31)This value is compatible with χ (0) = 0 .
61 given by Eq.(28) where ℜ Π R ( s = m K ) = − .
12. The part connected with the potential rescattering, beingsmall, can not change the conclusion that FSI diminishes the < π ; I = O | K > amplitude.In the case of the < π ; I = 2 | K > amplitude, FSI adds to the initialexpression κ (2) ( m K − m π ) the factor χ (2) = 16 π sin δ e iδ / ( A tree2 q − m π /m K ) (32)At the used in [19] parameters, the normalized A tree2 = − . , δ = − . ◦ and as a result χ (2) = 1 . K → π decay was studied in the frameworkof σ model in [25]. In this paper, the authors, however, put ℜ Π = 0. Then A unitar = A tree / (1 − i ℑ Π)and this formula was used by them to estimate the FSI effects in the K → π decay. But earlier the same authors had found that ℜ Π = 0 [17]. In thiscase, the unitarization leads to A unitar for the elastic ππ scattering given inEq.(11) and to A unitarRes ( K → π ; I = 0) in Eq.(28). As it is seen from Eq.(28),FSI could increase or diminish the K → π amplitude depending on relativemagnitudes of cos δ and (1 −ℜ Π R ) . We have shown that cos δ/ (1 −ℜ Π R ) < K → π amplitude. Conclusion.
Carring out our investigation in the framework of field theory with the phe-nomenological mesonic Lagrangian we have not found an enhancement of the7mplitude < ππ ; I = 0 | K > due to final state interaction of pions. On thecontrary, our analysis has shown that FSI diminishes this amplitude. Hence,FSI is not at all the mechanism bringing us nearer to explanation of the∆ I = 1 / K → π decay. Acknowledgements
I am very grateful to Yu.A.Simonov for discussions and comments con-cerning this letter.
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