Electromagnetic effects in the K + π + π 0 π 0 decay

Transcription

1 Electromagnetic effects in the K + π + π 0 π 0 decay arxiv:hep-ph/ v3 10 Jan 2007 S.R.Gevorkyan, A.V.Tarasov, O.O.Voskresenskaya March 11, 2008 Joint Institute for Nuclear Research, Dubna, Russia Abstract The final state interactions of pions in decay K + π + π 0 π 0 is considered using the methods of quantum mechanics. We show how incorporate the electromagnetic effects in the amplitude of this decay and work out the relevant expressions valid both above and below the two charged pion production threshold M = 2m. The electromagnetic corrections are given as evaluated in a potential model. The ππ scattering at low energies provides a testing ground for strong interaction study [1]. As the free pion targets cannot be created the experimental evaluation of ππ scattering characteristics is restricted to the study of a dipion system in a final state of more complicated reactions. One of the most suitable reactions for such study are charged kaons decays to three pions K ± π ± π 0 π 0, (1) K ± π ± π + π. (2) The final state interaction of pions in these decays gives a valuable information on the S wave ππ scattering lengths a 0 and a 2 [2, 3]. From the other hand the Chiral Perturbation Theory (ChPT) (see, for instance, [4]) predicts their values with high precision. Recently, in the NA 48/2 experiment at CERN SPS the high quality data On leave of absence from Yerevan Physics Institute On leave of absence from Siberian Physical Technical Institute 1

2 on decay (1) have been obtained [5]. The dependence of decay rate on invariant mass of neutral pions M 2 = (p 1 + p 2 ) 2 reveals the anomaly ( cusp ) at the threshold of two charged pions M 2 = 4m 2. N. Cabibbo proposed the simple re-scattering model [6], in which the cusp is an effect due to charge exchange scattering process π + π π 0 π 0 in decay (2). According to the model [6] the amplitude of the decay (1) consists from two terms T = T 0 + 2ika x T + (3) where T 0, T + are unperturbated amplitudes for decays (1) and (2) respectively and k = 1 2 M2 4m 2 is momentum of charged pion. The second term in (3) is proportional to the difference of scattering lengths a x = (a 0 a 2 )/3 and flips from dispersive to absorptive at the charged pions threshold. As a result, the decay probability under threshold depends on the scattering lengths difference linearly, which allows one to extract the S wave scattering lengths from experimental data with high accuracy. The next important step in this direction has been done in [7], where the amplitude T was obtained including second order in scattering lengths terms using the unitarity of the S matrix. The results of [7] were supported by more rigorous approaches [8, 9], whose authors investigated the decay (1) in the framework of effective field theory and ChPT. The results from [7] were used in fitting procedure of experimental data [5], allowing to extract the difference a 0 a 2 with high accuracy. It is commonly believed that the tiny discrepancy between theoretical predictions and experimental data near threshold [5] is a result of disregard of electromagnetic effects in final state interaction. In view of importance of knowledge of scattering lengths with most possible accuracy the consideration of electromagnetic corrections in decay (1) becomes an actual issue. Later on we discuss the problem of Coulomb interaction among the charged pions using the methods of nonrelativistic quantum mechanics, which are completely suitable for considered case 1. We obtain the compact expressions for the amplitude 2 of the decay (1) with regard to the electromagnetic corrections which is valid both below and above the charged pions production threshold M = 2m. Leaving the strict derivation for separate publication, let us shortly discuss how to involve the electromagnetic effects in the considered problem and relevant modifications, which have to be done in the amplitude of decay (1). 1 The effects of radiation of real photons are out of the scope of present consideration and will be treated later on. 2 As the K decays are counterparts to the K + decays they are not treated separately. 2

3 Using the methods of nonrelativistic quantum mechanics it can be shown that the result of N. Cabibbo [6] can be generalized accounting the ππ scattering in the charge-exchange part of amplitude (second term in (3)) to all orders in scattering lengths T = T 0 + 2ikf x T +, f x = a x /D, D = (1 ik 1 R 11 )(1 ik 2 R 22 ) + k 1 k 2 R (4) Here k 1 = 1 2 M2 4m 2 0 ; k 2 = k = 1 2 M2 4m 2 are the neutral and charged pions momenta respectively. The elements of R matrix are real and in the isotopic symmetry limit can be expressed through the combinations of scattering lengths [7, 8] a x = (a 2 a 0 )/3; a 00 = (a 0 + 2a 2 )/3; a ± = (2a 0 + a 2 )/6 corresponding to inelastic and elastic pion-pion scattering as R 12 = 2a x ; R 11 = a 00 ; R 22 = 2a ±. (5) The replacement a x f x has small numerical impact on results of previous calculations done according [7, 8] in the dominant part of phase space, but as we will see is very crucial for inclusion of the electromagnetic interactions under threshold,were the formation of bound states (π + π atoms) can take place. The next step of our prescription is inclusion in expression (4) properly electromagnetic effects. The general receipt is known for many years (see for instance the textbook [10]) and implies the replacement of charged pion momenta k by logarithmic derivative of pion wave function in the Coulomb potential at the boundary of strong field r 0 i.e. ik τ = d log[g 0(kr) + if 0 (kr)] dr. (6) r=r0 Here F 0, G 0 are the regular and irregular solutions of the Coulomb problem. In the region kr 0 1 where the electromagnetic effects can be significant the above replacement gives [ ( τ = ik αm log( 2ikr 0 ) + 2C + ψ 1 imα )] = Re(τ) + iim(τ), [ Re(τ) = αm log(2kr 0 ) + 2C + Re ψ Im(τ) = ika 2, ( πξ A = exp 2 ) Γ(1 + iξ), 2k ( 1 imα 2k )], ξ = αm 2k where C = 0.577, α = 1/137 are Euler and fine structure constants, whereas ψ is digamma function. 3 (7)

4 To go under threshold it is enough to do the common replacement k iκ in the above expression [ ( τ = κ αm log(2κr 0 ) + 2C + ψ 1 mα )]. (8) 2κ At κ n = αm/(2n) where n is an integer, τ goes to infinity which correspond to Coulombic bound states in considered approach. On the other hand, the product κf x defining the amplitude behaviour under threshold remains finite due to the presence of dependence from τ in the denominator D in expression (4). This explains why the inclusion of electromagnetic effects can be done only after sum up all terms of infinite series in perturbation expansion. It is easy to see that the product κf x possess a resonance structure placed at the positions M n with relevant width = 2m κ2 n m, κ n = αm 2(n δ), δ = 1 [ π arctan, = αm a 22 k2 1 a ] 11a k1a Γ n = 4πk 1a 2 12 κ 3 n (10) m(1 + k1 2a2 11 ). The physical reason of resonance origin is transparent. Due to the process π + π ππ the Coulombic bound states of π + π system (A 2π atoms) becomes unstable 3. The considered effect of A 2π atoms creation in decay (1) is not the only contribution from electromagnetic interaction of pions. Outside of the resonance region the Coulomb interaction leads to the essential difference between the τ values calculated with electromagnetic corrections and without it. In particular, the nonzero contribution of the Coulomb corrections to the Re τ above the threshold leads to the interference term in decay rate provided by direct and charge-exchange contributions from (4). Thus above the threshold the interference is nonzero even at the lowest order in scattering lengths, unlike the original approach proposed by N. Cabibbo [6]. The further improvement of the theory consists in account of final state interactions in the direct term from (4). This can be done by simple substitution (9) T 0 T 0 (1 + ik 1 f 00 ) (11) 3 We don t discuss here the instability of exited states caused by there transition to ground state. Obviously this effect is very small and can safely be neglected. 4

5 where f 00 is the amplitude of π 0 π 0 scattering. It can be shown that 1 + ik 1 f 00 = 1 τa 22. (12) (1 ik 1 R 11 )(1 τa 22 ) ik 1 τa 2 12 To estimates the contribution from electromagnetic effects to decay rate of process (1) we introduce the ratio R(%) = ( T c 2 T 2 )/ T 2 where the amplitude T is given by expression (4),while the amplitude T c accounting for electromagnetic effects is given by expressions (4,11) with relevant modifications discussed above. The dotted line on the Fig. 1 gives the contribution of electromagnetic effects without bound states (pionium) corrections. The dashed line represent the same quantity, but with corresponding amplitude averaged using gaussian distribution [5] with expected mass resolution near threshold r.m.s. = 0.56 Mev. The solid line gives the contribution of all electromagnetic effects (bound states included) averaged as in the previous case with gaussian distribution. From this plot one concludes that after relevant averaging the essential contribution to the decay rate comes from the electromagnetic interactions don t leading to bound states. The developed approach allows to takes into account electromagnetic effects in the decay (1) and estimate their impact on decay rate of the process under consideration. We are grateful to V. Kekelidze and J. Manjavidze who draw our attention to the problem and advice during the work. We would like also to thank D. Madigozhin for many stimulating discussions. References [1] J. Gasser, H. Leutwyler, Ann. Phys. 158 (1984) 158. [2] V.N. Gribov, Nucl. Phys. 5 (1958) 653. [3] V.N. Gribov, ZhETF 41 (1961) [4] U.-G. Meissner, Rep. Prog. Phys. 56 (1993) 903. [5] J.R. Batley et al., Phys. Lett. B 633 (2006) 173. [6] N. Cabibbo, Phys. Rev. Lett. 93 (2004) [7] N. Cabibbo, G. Isidori, JHEP 03 (2005) 021. [8] G. Colangelo, J. Gasser, B. Kubis and A. Rusetsky, Phys. Lett. B 638 (2006)

6 [9] E. Gamiz, J. Prades, I. Sciemi, hep-ph/ [10] A.I. Baz, Ya.B. Zeldovich, A.M. Perelomov, Scattering, reactions and decays in nonrelativistic quantum mechanics (Nauka, Moscow, 1971). 6

7 R 2,6 2,4 2,2 2,0 1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0, M 2 [Gev 2 ] 0.08 Figure 1: The dependence of R on the square of invariant mass of neutral pions. 7