EPJ Web of Conferences 3, 01008 (2010) DOI:10.1051/epjconf/20100301008 © Owned by the authors, published by EDP Sciences, 2010 Cusp effects in meson decays Bastian Kubis,a Helmholtz-Institut f¨ur Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, Universit¨at Bonn, D-53115 Bonn, Germany Abstract. The pion mass difference generates a pronounced cusp in the π0π0 invariant mass distribution of K+ π0π0π+ decays. As originally pointed out by Cabibbo, an accurate measurement of the cusp may allow one to pin→ down the S-wave pion–pion scattering lengths to high precision. We present the non-relativistic effective field theory framework that permits to determine the structure of this cusp in a straightforward manner, including the effects of radiative corrections. Applications of the same formalism to other decay channels, in particular η and η′ decays, are also discussed. 1 The pion mass and pion–pion scattering alternative scenario of chiral symmetry breaking under the name of generalized chiral perturbation theory [4]. The approximate chiral symmetry of the strong interac- Fortunately, chiral low-energy constants tend to appear tions severely constrains the properties and interactions in more than one observable,and indeed, ℓ¯3 also features in of the lightest hadronic degrees of freedom, the would-be the next-to-leading-order corrections to the isospin I = 0 0 Goldstone bosons (in the chiral limit of vanishing quark S-wave pion–pion scattering length a0 [2], masses) of spontaneous chiral symmetry breaking that can be identified with the pions. The effective field theory that 7M2 0 = π + ǫ + 4 , a0 2 1 (Mπ) systematically exploits all the consequencesthat can be de- 32πFπ O rived from symmetries is chiral perturbation theory [1,2], 5M2 n 3 o 21 21 which provides an expansion of low-energy observables in = π ¯ + ¯ ¯ + ¯ + ǫ 2 2 ℓ1 2ℓ2 ℓ3 ℓ4 . (4) terms of small quark masses and small momenta. 84π Fπ − 8 10 8 One of the most elementary consequences of chiral ℓ¯ and ℓ¯ are known from ππ D-waves a0, a2, while ℓ¯ can symmetry is the well-known Gell-Mann–Oakes–Renner 1 2 2 2 4 be determined from a dispersive analysis of the scalar ra- relation [3] for the pion mass M in terms of the light quark dius of the pion r2 S [5,6], such that the correction term ǫ masses (at leading order), π in Eq. (4) can beh rewritteni as 2 0 uu¯ 0 2 S ¯ M = B(m + m ) , B = h | | i . (1) r π 200π 15ℓ3 353 u d − F2 ǫ = M2 h i + F2 a0 + 2a2 − . π 3 7 π 2 2 − 672π2F2 ( π ) A non-vanishing order parameter B, related to the light (5) 0 quark condensate via the pion decay constant F (in the Consequently, a measurement of a0 can lead to a determi- chiral limit), is a sufficient (but not necessary) condition nation of ℓ¯3, and hence to a clarification of the role of the for chiral symmetry breaking. Chiral perturbation theory various order parameters of chiral symmetry breaking in allows to calculate corrections to this relation [2], nature. We wish to point out that Eq. (5) only rewrites the 0 4 ¯ dependence of a0 on the (p ) low-energy constants ℓ1 4 M4 in the form of a low-energyO theorem. The theoretical pre-− M2 = M2 ℓ¯ + (M6) , (2) π − 32π2F2 3 O dictions of the two S-wave ππ scattering lengths of isospin 0 and 2 from a combination of two-loop chiral perturbation ¯ with the a priori unknown low-energy constant ℓ3. Another theory [7,6] and a Roy equation analysis [8] (for QCD in way to write Eq. (2) is therefore the isospin limit), 2 2 3 0 M = B(mu + md) + A(mu + md) + (m ) , (3) a = 0.220 0.005 , π O u,d 0 ± a2 = 0.0444 0.0010 , and the natural question arises: how do we know that the 0 − ± leading term in the quark-mass expansion of M2 really a0 a2 = 0.265 0.004 , (6) π 0 − 0 ± dominates the series? ℓ¯3 could actually be anomalously large, the consequence of which has been explored as an do not depend on the D-wave scattering lengths as input, but rather yield values for all ππ threshold parameters as re- a e-mail: [email protected] sults. The predictions Eq. (6) are among the most precise in This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited. Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20100301008 EPJ Web of Conferences low-energy hadron physics and present a formidable chal- 50 lenge for experimental verification. For other recent phe- nomenological determinations of the scattering lengths, 150 see Refs. [9–11]. 40 Traditionally, information on pion–pion scattering has been extracted from reactions on nucleons, which is diffi- 2 cult to achieve in a model-independent way, and data are usually not available very close to threshold kinematics. 30 The latest precision determinations therefore mainly con- 100 cern three different methods: the lifetime measurement of 0.077 0.078 0.079 pionium [12], Ke4 decays [13,14], and, most recently, the so-called cusp effect in K 3π decays. Let us very briefly discuss→ the first two modern ex- perimental approaches. Pionium is the electromagnetically + 50 bound state of a π π− pair, with an ionization energy of about 1.86 keV and a ground state width of about 0.2 eV. 1000 events / 0.00015 GeV Its energy levels as given by purely electromagnetic bind- ing are perturbed by the short-ranged strong interactions: + they are shifted by elastic strong rescattering π π−, but in particular, even the ground state is not stable, it decays dominantly into π0π0. The decay width is given by the fol- 0 0.08 0.09 0.10 0.11 0.12 0.13 lowing (improved) Deser formula [15,16] 2 s3 [GeV ] 2 3 0 2 2 Γ = α p a0 a0 (1 + δ) , (7) Fig. 1. Cusp in the decay spectrum dΓ/ds3 of the decay K± 9 − 0 0 → π π π± as seen by the NA48/2 collaboration. The dotted vertical + where α is the fine structure constant, p the momentum of line marks the position of the π π− threshold, the insert focuses a final-state π0 in the center-of-mass frame, and δ is a nu- on the cusp region. Data taken from Ref. [22]. merical correction factor accounting for isospin violation beyond leading order, δ = 0.058 0.012 [17]. Given the 2 The cusp effect in K± → π0π0π± decays theoretical values for the ππ scattering± lengths of Eq. (6), 0 0 the pionium lifetime can be predicted to be In an investigation of the decay K± π π π±, the NA48/2 collaboration at CERN has observed→ a cusp, i.e. a sud- 15 τ = (2.9 0.1) 10− s , (8) den, discontinuous change in slope, in the decay spectrum ± × with respect to the invariant mass squared of the π0π0 pair while ultimately, the argument should be reversed, and a = 2 dΓ/ds3, s3 Mπ0π0 [22]; see Fig. 1. A first qualitative ex- measurement of the lifetime is to be used for a determina- planation was subsequently given by Cabibbo [23], who tion of a0 a2. The current value from the DIRAC experi- pointed out that a K+ can, simplistically speaking, either 0 − 0 ment [12], decay “directly” into the π0π0π+ final state, or alternatively +0.49 15 decay into three charged pions π+π+π , with a π+π pair τ = 2.91 0.62 10− s , (9) − − − × rescattering via the charge-exchange process into two neu- agrees with Eq. (8), but is not yet comparably precise. For tral pions, compare Fig. 2. The loop (rescattering) diagram a comprehensive review of the theory of hadronic atoms, has a non-analytic piece proportional to see Ref. [18]. + + + 4M2 The decay K π π−e νe (Ke4) can be described in π+ 2 → i 1 , s3 > 4M + , terms of hadronic form factors, which, in the isospin limit, − s3 π i v (s3) = r (11) share the phases of ππ scattering due to Watson’s final state ± 4M2 π+ 2 theorem [19]. What can be extracted unambiguously from 1 , s3 < 4Mπ+ , − s3 − the decay, using the so-called Pais–Treiman method [20], r is the difference of ππ I = 0 S-wave and I = 1 P-wave and as the charged pion is heavier than the neutral one by phase shifts nearly 4.6 MeV, the (then real) loop diagram can interfere + δ0(s ) δ1(s ) , (10) with the “direct” decay below the π π− threshold and pro- 0 ππ 1 ππ 2 − duce a square-root-like singularity at s3 = 4Mπ+ , the cusp and as the energy of the two pions is kinematically re- visible in the experimentally measured spectrum Fig. 1. stricted to √sππ < MK, these phases are accessible close Such threshold singularities have of course been known for to threshold. It has been pointed out [21] that, given the a long time [24] and have been re-discovered for the scat- precision of the latest NA48/2 data [14], it is necessary to tering of neutral pions in the context of chiral perturbation include an isospin-breaking correction phase in the anal- theory [25].