EPJ Web of Conferences 3, 01008 (2010) DOI:10.1051/epjconf/20100301008 © Owned by the authors, published by EDP Sciences, 2010

Cusp effects in decays

Bastian Kubis,a Helmholtz-Institut f¨ur Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical , Universit¨at Bonn, D-53115 Bonn, Germany

Abstract. The 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 . 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

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low-energy 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 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 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]. It had even been pointed out very early by Bu- ysis. The resulting scattering length determination will be dini and Fonda [26] that these cusps may be used to inves- shown in the comparison in Sect. 4. tigate ππ scattering: as suggested in Fig. 2, the strength of

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π+ π+ π0 π+ K+ K+ π0 + π− π0 π0 (a) (b) Fig. 2. “Direct” and “rescattering” contribution to the decay c K+ π0π0π+. The black dot marks the charge-exchange ππ scat- 1 → tering vertex proportional to the scattering lengths at threshold. b d a 2 the cusp is proportional to the charge-exchange pion–pion c d scattering amplitude at threshold, hence a combination of ( ) 3 ( ) scattering lengths might be extracted from a precision anal- Fig. 3. Topologies for ππ rescattering graphs at one and two ysis of the cusp effect. loops. The double line denotes the decaying , while single The challenge for theory is to provide a framework that lines stand for either charged or neutral pions. matches the tremendous accuracy of the experimental data: the partial data sample analyzed in Ref. [22] was based on 7 0 0 3.1 Power counting, Lagrangians 2.3 10 K± π π π± decays, subsequently expanded to more× than 6.0→ 107 decays [27]. Different theoretical ap- × First, we need to specify our power counting scheme. We proaches have been suggested to this end: a combination of introduce a formal non-relativistic parameter ǫ and count analyticity and unitarity with an expansion of the rescatter- 3-momenta of the pions in the final state according to ing effects in powers of the ππ threshold parameters [28], p /Mπ = (ǫ). Consequently, the pions’ kinetic energies and chiral perturbation theory beyond one-loop order [29]. are| | O In the following, we advocate the use of non-relativistic ef- fective field theory [30] as the appropriate systematic tool 2 2 2 Ti = ωi(pi) Mi = (ǫ ) , where ωi(pi) = M + p , to analyze these decays. − O i i q (13) with i = 1, 2, 3, M1 = M2 = Mπ0 , M3 = Mπ+ , and the Q-value of the reaction has to be counted as (ǫ2), too, as O 3 Non-relativistic effective field theory 2 MK Mi = Ti = (ǫ ) . (14) − i i O Consider as a starting point a generic ππ partial wave am- X X plitude T. Close to threshold, its real part can be written in In addition, we adopt the suggestion of Ref. [28]: as the ππ the effective range expansion according to scattering lengths are small due to the Goldstone nature of the pions, their final-state rescattering can be taken into ac- Re T = a + b q2 + c q4 + ..., (12) count perturbatively, in contrast to what one has to do e.g. in the treatment of three-nucleon systems. Hence in this with the scattering length a, the effective range b, a shape case, we can make use of a two-fold expansion in ǫ and a, parameter c etc. Chiral perturbation theory allows to cal- by which we generically denote all ππ threshold parame- culate the parameters a, b, c to a certain accuracy in the ters. This scheme allows for a consistent power counting quark-mass (or pion-mass) expansion, see Eq. (4) for an in the sense that at any given order in a and ǫ, only a finite example, but in principle, each of these parameters re- number of graphs contributes. ceives contributions from each loop order. On the other The polynomial terms contributing at tree level are or- hand, one can set up a non-relativistic effective field the- ganized in even powers of momenta, hence there are terms ory (NREFT) in such a way that the scattering length a of order ǫ0, ǫ2, ǫ4, .... Slightly more complicated is the is entirely given in terms of tree graphs, without any fur- power counting of the loop graphs, of which the typical ther loop corrections; similarly, the effective range b can topologies at one- and two-loop order are shown in Fig. 3. be calculated from tree and two-loop graphs only, but then Generically, the non-relativistic pion propagators are of a no further contributions. In other words, NREFT allows form 1 2 to parameterize T directly in terms of threshold parame- = (ǫ− ) (15) ∝ ω(p) p0 O ters. Note that this is exactly what we want: the aim here is − not to predict the scattering lengths, but to provide a repre- (note however the discussion below on its precise form), sentation of the (scattering or decay) amplitude in terms while a loop integration is counted according to of the ππ threshold parameters that allows for an accu- d4 p = dp0d3p = (ǫ5) . (16) rate extraction of the latter from experimental data. This O is similar in spirit to the use of NREFT in the analysis of Consequently, we find that each additional loop induced by hadronic atoms in order to extract scattering lengths of dif- two-body rescattering is suppressed by a factor of ferent systems from their life times or energy level shifts 2 2 5 (see Ref. [18] and references therein). (ǫ− ) ǫ = ǫ , (17)

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such that the one-loop diagram (a) in Fig. 3 is of (a1ǫ1), subsequently resummed. In particular the presence of the while the two two-loop graphs (b) and (c) are of O(a2ǫ2). square roots ω(p) in the propagator Eq. (20) leads to sig- We therefore find a correlated expansion in a andOǫ: loops nificant technical complications in the calculation of the are not only suppressed by powers of the ππ threshold pa- loops. rameters, but in addition by powers of ǫ. We need two types of interaction terms in the effective Finally, a rescattering graph due to three-body interac- Lagrangian, generating the ππ interaction as well as the tions that first appears at two-loop order, see diagram (d) K 3π tree level amplitudes, in Fig. 3, is suppressed by → 2 2 = Φ† Φ† Φ + . . + ... + ǫ , 2 3 5 2 4 ππ Cx +( 0) h c ( ) (23) (ǫ− ) (ǫ ) = ǫ . (18) L − O G0 2 H0  2 2 K3π = K+†Φ+(Φ0) + K+†Φ (Φ+) + h.c. + (ǫ ) , This diagram is not proportional to any ππ threshold pa- L 2 2 − O rameters; however, the graph is a constant, and its main effect apart from coupling constant renormalization is to where we have only displayed the leading, energy-in- dependent couplings, and the ellipsis in ππ denotes sim- give the K 3π vertex a small imaginary part. Specif- L ically, denoting→ the leading (ǫ0) K+ π0π0π+ vertex ilar interaction terms for the other possible ππ scattering O 0 0 + → channels. The current accuracy of the calculation of the by G0 (see Eq. (23) below), the π π π intermediate state, K 3π decay amplitude includes all terms up to and in- with elastic three-particle rescattering approximated by the → 0 4 1 5 2 4 cluding (a ǫ , a ǫ , a ǫ ); for this purpose, K3π as well leading-order vertex derived from chiral perturbation the- O L 4 ory, leads to an imaginary part of as S-wave interaction terms in ππ are needed up to (ǫ ), while only P-wave scattering lengthsL and no D-waveO con- 2 Im G (M 3M )2 M tributions are necessary in ππ. 0 = K π π + 3 L − 2 (MK 3Mπ) The whole framework briefly sketched here is a com- Re G0 256π 24 √3F4 O − π pletely Lagrangian-based quantum field theory, hence all 5   1.5 10− , (19) constraints from analyticity and unitarity are automatically ≃ · obeyed. which therefore indeed turns out to be formally of (ǫ4), but numerically entirely negligible. O As the cusp effect depends essentially on the analytic 3.2 Matching properties of the amplitude, it is clearly desirable to pre- serve the latter exactly, i.e. to correctly reproduce the sin- The coupling constants of the non-relativistic Lagrangians gularity structure of the relativistic decay amplitude in Eq. (23) have to be related to physical observables in the the low-energy region p Mπ; only far-away singu- underlying relativistic field theory. In the case of the cou- larities associated e.g. with| | ≪ the creation and annihilation plings of ππ, they can be matched using the effective of particle–antiparticle pairs (inelastic channels) should be range expansionL of the ππ scattering amplitude. For ex- subsumed in effective coupling constants. To this end, we ample, Cx as defined in Eq. (23) is related to the charge- use a pion propagator of the form exchange amplitude T = T(π+π π0π0) by x − → 1 1 Re T = 2C + (ǫ2) , 0 . (20) x x 2ω(p) ω(p) p O 2 2 − 32π M + M 0 = 0 2 + π − π + 2 2 This corresponds to the complete particle-pole piece of the 2Cx a0 a0 1 2 (e p ) − 3 − 3Mπ O full relativistic propagator,   ( ) 32π 0 2 2 = a a 1 + (0.61 0.16) 10− 1 1 1 1 1 − 3 0 − 0 ± × = + , (21) M2 p2 2ω(p) ω(p) p0 2ω(p) ω(p) + p0 + (e2 p4) .  n o (24) π − − O and therefore reproduces the correct relativistic dispersion The isospin-breaking corrections in relating the charge- law. The propagator Eq. (20) can be generated by a non- exchange amplitude at threshold to the scattering lengths local kinetic-energy Lagrangian of definite isospin are calculated in chiral perturbation the- ory, where the second line in Eq. (24) shows the analytic 2 2 kin = Φ†(2W)(i∂t W)Φ, W = Mπ ∆ , (22) correction at (e ), while the numerical estimate in the L − − third line includesO the higher order of (e2 p2) [32,17]. q O + 0 0 + where Φ represents the pion field operator and ∆ is the The polynomial terms G0, G1, ... for K π π π + + + → Laplacian. kin generates all relativistic corrections in the and H0, H1, ... for K π π π− are not strictly matched, propagatorL and leads to a manifestly Lorentz-invariant and but used as a parameterization→ of the amplitudes in ques- frame-independent amplitude. tion. They replace the more traditional Dalitz plot param- In order to restore the naive power counting rules for eters used for that purpose in experimental fits neglecting loop graphs in the presence of explicit heavy (pion) mass non-trivial final-state rescattering effects. The strategy is to scales, one has to apply the threshold expansion [31,17, fit (in principle) all parameters of the non-relativistic rep- 18]: all loop integrals are expanded in powers of the in- resentation to data of both decay channels, and then deter- verse pion mass, integrated order by order, and the results mine the scattering length combination a0 a2 via Eq. (24). 0 − 0

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In practice, one may decide to use some of the parameters, π0 π+ π0 π0 for instance higher-order ππ threshold parameters such as π− π0 π0 π− effective ranges or P-waves, as input, employing their the- K+ K+ oretically predicted values [6] instead. π+ π0 π+ π+ π+ π0 3.3 Analytic structure of the non-trivial (a) (b) two-loop graph Fig. 4. Two-loop diagrams for K+ π0π0π+ displaying anoma- lous thresholds. Double, single,→ and dashed lines stand for The function F(s) describing the non-trivial, genuine two- charged , charged, and neutral pions, respectively. loop graph (c) in Fig. 3 can be expressed analytically in terms of logarithms (see Ref. [33] for an explicit closed representation). Close to threshold, it can be approximated rim. Diagram (b), on the other hand, has complex anoma- according to lous thresholds s3±, which lead to special complications in a dispersive representation of this loop graph, necessitating 2 2 v (s) MK 9Mπ a deformation of the integration path over the discontinu- F(s) ± − (25) ≃ 256π2 s M2 M2 ity. Again, on the upper rim of the cut, the complete loop K − π function F(s) is analytic also in that case. (for all pion masses running in the loop equal), which is We wish to emphasize once more that the singularity manifestly of (ǫ2) as required by the power counting set structure of the explicit non-relativistic representation of up in Sect. 3.1.O However, a decomposition of the full two- F(s) [33] in the low-energy region, i.e. in the decay region loop function according to and slightly beyond, is identical to the one of the fully rel- ativistic amplitude. It therefore includes all the anomalous F(s) = A(s) + B(s) v (s) , (26) thresholds discussed above, in precisely the right kinemat- ± ical positions. with both A(s), B(s) analytic functions of s as suggested in Ref. [28],turnsout not to hold.In fact, it can be shown[34] that if one enforces such a decomposition, both A(s) and 3.4 Two-loop representation B(s) diverge at the border of phase space for maximal s 2 like 1/ √sp s , sp = (MK M3) , in such a way that the − − The full representation of the K 3π decay amplitudes up sum A(s) + B(s) v (s) is finite. 0 4 1 5 →2 4 ± to and including (a ǫ , a ǫ , a ǫ ) comprises tree, one- What is more, at least for certain pion mass assign- loop, and two-loopO graphs of the topologies shown in ments within the loop, the decomposition Eq. (26) even Fig. 3, with all possible charge combinations of interme- fails as a representation of the analytic structure of F(s) diate pions. The only loop function at one loop, see graph within the decay region. With the pions in the loop labelled (a) in Fig. 3, is given by as indicated in Fig. 3, the solutions of the Landau equa- tions [35,36] show that anomalous thresholds exist for i v(s) J(s) = , (28) 16π 1 2 2 2 2 2 s± = M + M + M + M (Ma + Mb) 2 K 3 c d − hence produces precisely the analytic structure discussed ( 2 2 2 2 in Sect. 2. The two-loop graph (b) in Fig. 3 is given as a (M M )(M M ) √λ1λ2 + K − c d − 3 ± , product of two functions J(s), hence it is real abovethresh- 2 (Ma + Mb) 2 ) old and, if it contains singular behavior at s3 = 4Mπ+ (in a 2 2 2 λ1 = λ (Ma + Mb) , M3, Md , product of one “charged” and one “neutral” loop), the real square root that interferes with the dominant tree graphs is 2 2 2 λ2 = λ M , (Ma + Mb) , M  , (27) 2 K c also seen for s3 > 4Mπ+ . The same is true for the threshold   behavior of the genuine two-loop function discussed in the with the standard K¨all´en function λ(a, b, c) = a2 +b2 +c2 last section, see Eq. (25). Finally, the two-loop graph with − 2(ab+ac+bc). According to Eq. (27), the analytic structure three-body rescattering, diagram (d) in Fig. 3, is a constant of F(s) is particularly intricate for Ma + Mb , M3 + Md. and can essentially be absorbedin a redefinitionof the tree- The two relevant graphs in K+ π0π0π+ fulfilling this level couplings, hence it does not affect the analytic struc- → condition are shown in Fig. 4. In the case of diagram (a), ture in a non-trivial way. the s1± are real and yield branch points in the amplitude The cusp up to two loops is therefore of the follow- + at s1− = 308 MeV and s1 = 356MeV, compared to ing generic structure: while the one-loop diagrams gener- the phase space limits in this variable, given by threshold ate a structure i a v (s3) which interferes with the (dom- p p ∝ ± + √st = 275MeV and pseudothreshold √sp = 359 MeV. inant) tree amplitude below the π π− threshold (where the The functions A(s) and B(s) in the decomposition Eq. (26) square root turns real), the two-loop graphs include terms 2 display singular behavior at s1±, while the complete ampli- a v (s3), hence a singular structure (in interference with tude is analytic at the upper rim of the cut (that starts at ∝the tree± parts) above that point. This is illustrated schemat- 2 st = (Mπ+ + Mπ0 ) ); it becomes singular only at its lower ically in Fig. 5. The “two-loop cusp” above threshold is a

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"two-loop cusp" (a) (b) (c) Fig. 6. Examples for diagrams with “external” radiative correc- "one-loop cusp" tions. Double, single, and dashed lines stand for charged kaons, charged, and neutral pions, respectively. Wiggly and curly lines denote Coulomb and transverse , respectively.

0.270 0.275 0.280 0.285 1/2 s ables arrives at the percent level, the effects of electromag- 3 netic or radiative corrections have to be taken into account. Such corrections in K 3π decays have already been considered earlier in the→ framework of chiral perturbation "two-loop cusp" theory [37–39], or in a quantum-mechanicalapproach [40, 41]. As we employ a Lagrangian framework, the inclusion "one-loop cusp" of photons via minimal substitution is completely straight- forward:

∂µΦ (∂µ ieAµ)Φ , (30) ± → ∓ ± and similarly for the kaons. Furthermore, all possible non- minimal gauge invariant terms can be added. In the con- text of a non-relativistic theory, it is useful to work in the 0.276 0.278 0.280 0.282 Coulomb gauge and differentiate between Coulomb and 1/2 transverse photons, which feature differently in the gener- s 3 alized power counting scheme. In addition, for transverse Fig. 5. Sketch of the cusps in the decay spectrum at (a) be- photons, one has to differentiate between “soft” and “ultra- 2 + O low, and at (a ) above the π π− threshold, denoted by the ver- soft” modes: while both have zero components that have to O 2 tical dotted line. In the lower panel, focussing even closer on the be counted according to l0 = (ǫ ), the three components O 2 threshold region, the tree-level spectrum (dashed) is subtracted are either l = (ǫ) for soft, or l = (ǫ ) for ultrasoft pho- for better illustration of the small “two-loop cusp”. tons. The summaryO of the countingO rules for diagrams with virtual photons is then as follows [42]: much smaller effect, yet given the precision of the data in 1. Adding a Coulomb to a hadronic “skeleton” di- the NA48/2 analysis, it has turned out to be a vital ingre- agram modifies its counting by a factor of e2/ǫ. An ex- dient in the theoretical representation in order to achieve ample of this is diagram (a) in Fig. 6: with a constant a statistically adequate description. As the cusp strength K 3π vertex of (ǫ0), this diagram will scale as ff →2 1 O at two loops also incorporates ππ rescattering e ects other (e ǫ− ), the negative power in ǫ indicating the pres- than the charge exchange channel, there is in principle also Oence of the Coulomb pole in that graph. (reduced) sensitivity to another linear combination of S- 2. Transverse photons couple to with vertices of 2 wave scattering lengths, e.g. a0 alone. A fit of this form (ǫ), hence soft transverse photons are suppressed rel- yields [27] ativeO to Coulomb photon exchange by two orders in the ǫ-expansion. As an example, diagram (a) in Fig. 6, a0 a2 = 0.2815 0.0043 ..., 0 − 0 ± stat ± with the Coulomb photon replaced by a transverse one, 2 2 a0 = 0.0693 0.0136stat .... (29) contributes at (e ǫ). − ± ± 3. Ultrasoft transverseO photons added to a hadronic A comparison to the theoretical prediction Eq. (6) shows “skeleton” diagram finally change its power counting that a0 a2 comes out uncomfortably large (by more than by a factor of e2ǫ2. As an example, the transverse pho- 0 − 0 3.5σ). This turns out not to be a statistical accident, but ton in diagram (b) of Fig. 6 can be shown to be ultra- there is a theoretical explanation for this discrepancy that soft, hence with a constant K 3π vertex, the graph we will discuss in the following section. scales as (e2ǫ2). → O As is well known, the inclusion of virtual photon ef- 4 Radiative corrections fects requires the simultaneous consideration of radiation of additional real photons in order to obtain well-defined, Once the theoretical and experimental precision in the de- infrared-finite quantities. The observable that can be cal- termination of hadronic, strong-interaction physics observ- culated including effects of (α) (where α = e2/4π is the O

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Fig. 7. Examples for diagrams with “internal” radiative correc- tions. The wiggly lines denote Coulomb photons; otherwise, see Fig. 8. Multi-photon exchange inside the charged-pion loop, re- the line style in Fig. 6. sponsible for pionium formation.

fine structure constant) is [42] leads to an infinite number of bound-state poles close to threshold. The analytic solution to the resummation of an dΓ dΓ(K 3π) dΓ(K 3πγ) infinite number of exchanged Coulomb photons is known = → + → + (α2) as the Schwinger Green’s function [46] replacing the sim- ds3 E

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2 2. Isospin-breaking corrections in the matching relations. a0 These are small, the uncertainty is estimated to be . 0.01 1%, see Eq. (24). DIRAC 3. The effects of higher-order derivative interactions in 0.00 the two-loop contributions, i.e. terms of (a2ǫ4) and higher. Their impact is still under investigationO [34], al- -0.01 though there are indications that the scattering lengths are very stable under such modifications of the ampli- -0.02 NA48/2 tude. cusp 4. Higher loop contributions, starting at three loops NA48/2 3 3 -0.03 (a ǫ ). Ke4 O For this last point, we wish to discuss the so-called thresh- -0.04 old theorem [26,23,30,42] (valid in the absence of pho- tons). It states that the coefficient of the leading cusp be- -0.05 havior (or v (s3)) is proportional to the product ± + + + + 0 0 T(K π π π−) T(π π− π π ) , (36) -0.06 → thr × → thr where the second factor is just the combination of scatter- 0.25 0.26 0.27 0.28 0.29 0.30 a0 - a2 ing lengths given in Eq. (24), and the “threshold” at which 0 2 2 the first factor is to be evaluated refers to s1 = 4Mπ+ , s2 = Fig. 9. Combined experimental results on the S-wave ππ scatter- 2 2 s3 = (MK Mπ+ )/2. In other words, knowing the decay ing lengths from the pionium width as obtained by the DIRAC − + + + amplitude for the charged final state T(K π π π−) to 0 0 collaboration [12], K [14], and the cusp in K± π π π± [27], n → e4 → (a ) allows one to determine the dominant cusp strength the latter two as measured by the NA48/2 collaboration. For de- Oof T(K+ π0π0π+) at (an+1). As we have the full two- tails, see text. Data obtained from Ref. [27]. loop representation→ availableO for all K 3π channels, we can estimate the size of the cusp at three→ loops by the ex- 0 0 pansion the cusp analysis in K± π π π± described here [27], → the NA48/2 results from Ke4 decays [14] (dashed) using + π+π+π . . + . , T(K −) thr 1 0tree 0 13 i1 loop 0 0142 loop theoretical input on isospin-breaking corrections [21], and → ∝ − − − − (37) the constraint from the pionium lifetime obtained by the which suggests that the three-loop cusp will modify the DIRAC experiment [12] as the vertical dash-dotted band. leading (one-loop) effect by about 1.5%. This estimate is 0 The narrow dotted band shows the correlation between a0 no substitute for a complete three-loop calculation, as it 2 ¯ 3 and a0 dictated by the relation of ℓ4 to the scalar radius does not yield a representation of (a ) elsewhere in the of the pion [6], the smaller ellipses are fits to the two decay region except near the cusp, andO neither does it con- NA48/2 experiments using the chiral perturbation theory tain information about subleading non-analytic behavior 3 constraint. Altogether, a most impressive agreement be- near threshold (e.g. v (s3)). Still, we regard Eq. (37) tween the different experiments as well as experiment and as a good indication∝ for± the rate of convergence in the theory has been achieved. K+ π0π0π+ amplitude. →

5 On the accuracy of the extraction 6 Cusps in other decays 0 2 of a − a 0 0 0 0 6.1 KL → 3π , η → 3π

0 0 An important ingredient yet missing to finally assess the The mechanism generating the cusp in the π π invariant accuracy of the extracted values for the ππ scattering mass distribution is rather generic and only due to the final- lengths is a reliable estimate of the theoretical uncertainty state interactions between the pions. We may therefore an- inherent in the representation of the amplitude. In the ticipate that other decays into two neutral pions plus a third first publication of an experimental cusp analysis [22], a particle will show a very similar cusp effect, the most ob- vious examples being K 3π0 and η 3π0. The effect generic theoretical error of 5% was assumed, following L → → a suggestion made in Ref. [28], thus the theoretical input of the cusp in these channels has been investigated theo- dominated the final uncertainty. The following main points retically [28,29,33,47,48], and first efforts to see it exper- 0 may be responsible for the theoretical error. imentally have been reported both for KL 3π [49] and η 3π0 [50–52] decays. The main diff→erence between 1. Radiative corrections. These are now taken care these→ and K+ π0π0π+, however, is the following. As of [42], we regard the remaining uncertainty from indicated in Fig.→ 10, the extent to which the decay spec- higher-order radiative corrections as entirely negligi- trum with respect to the invariant mass squared of the π0π0 ble. pair is perturbed by the cusp effect does not only depend

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π0 π+ ably no competitive scattering length determination from these channels seems feasible in the near future, the cusp π0 + effect should at least be taken into account in ongoing or 0 G H Cx future precision determinations of the η 3π Dalitz plot π− slope parameter α [50–52]; compare also→ Ref. [47]. π0 π0

Fig. 10. Mechanism for cusp effects in generic decays into three final-state , including two π0. 6.2 η′ → ηπ0π0

At least from a theoretical perspective, much more promis- ing in this respect is the decay η ηπ0π0 [54]. With the ′ → η′ and the η both being particles of isospin 0, it is obvi- ous that in the approximation of isospin conservation, the ππ pair is produced with total isospin 0, which immedi- + ately shows that the amplitude for η′ ηπ π− is en- hanced compared to the η ηπ0π0 one→ by a factor of ′ → √2 (the sign is according to the Condon–Shortley phase conven− tion).We thereforeexpecta cusp much more promi- 0 nent than in KL, η 3π , if not quite as pronounced as in K+ π0π0π+.→ From the experimental perspective, the upcoming→ high-statistics η experiments at ELSA [55], 0.276 0.278 0.280 0.282 ′ 1/2 MAMI-C [56–58], WASA-at-COSY [59,60], KLOE-at- Φ s3 [GeV] DA NE [61,62], or BES-III [63] are expected to increase the data basis on η′ decays by orders of magnitude, so an Fig. 11. Sketches of the leading (one-loop) cusp effects on the ff + 0 0 + 0 investigation of the cusp e ect in this channel seems very decay spectra for K π π π (dashed line) and KL 3π → + → promising. (dash-dotted line) in the vicinity of the π π− threshold, marked What makes this channel somewhat different from by the dotted vertical line. The full line denotes the unperturbed those investigated so far is the presence of the η in the final spectrum without ππ rescattering. state, and hence of πη rescattering as a new ingredient to final-state interactions. There is no experimental informa- on the charge-exchangescattering length as encoded in the tion on πη threshold parameters, and it turns out that chiral coupling constant C , but strictly speaking, it is rather pro- symmetry constrains these quantities only very badly [64, x 65]: the (p4) corrections to the current algebra prediction portional to O H of the S-wave scattering length, for example, can easily be C , (38) as big as or bigger than the leading order. The one thing G x × that chiral perturbation theory does seem to predict reli- where G and H generically denote the coupling strengths ably is the fact that πη threshold parameters are systemati- to the “neutral” and “charged” final state, i.e. π0π0 and cally smaller than the ππ ones. In conventions comparable + π π− plus a third meson, respectively. In other words, to those chosen in ππ scattering (see Ref. [54] for details), the strength of the cusp depends crucially on the rela- the S-wave πη scattering length is given at leading order tive branching fractions into the charged and neutral final by states: the more the decay into charged pions is preferred, 2 Mπ 4 ff a¯0 = + (Mπ) , (39) the better the magnification of the e ect in the spectrum. 96πF2 O It turns out that the ratio H/G is very different for the π | | 0 different decays with three-pion final states. While for the which compared to a0, see Eq. (4), is smaller by a factor of K+ decays considered so far, it is approximately2, both for 21. We therefore expect the effect of the πη final-state in- KL and η it is closer to 1/3,in otherwordsthe KL and η pre- teractions to be significantly smaller than that of ππ rescat- fer to decayinto 3π0. To illustrate the difference, we sketch tering. the leading (one-loop, (a)) cusps for K+ π0π0π+ and For an investigation of the impact of the πη threshold 0 O 0 → 0 0 KL 3π (the picture for η 3π is very similar to parameters on the cusp effect in η′ ηπ π , we vary them the latter→ case) in Fig. 11. While→ the square-root-like struc- in a sensible range, suggested by→ various sets of next-to- ture is clearly visible to the naked eye for the K+ decay, leading order low-energy constants. In Fig. 12, we show 0 0 it is much harder to discern in the case of the KL. For this the decay spectrum for η′ ηπ π with respect to the reason, it is also much harder experimentally to achieve a invariant mass of the π0π0 pair,→ comparing the spectrum precision determination of ππ scattering lengths from an calculated from the tree-level amplitude to that given by 0 0 investigation of the cusps in KL 3π or η 3π . More the full two-loop result. The tree-level couplings are fixed quantitatively, while the cusp effect→ reduces the→ number of by (the central values of) the Dalitz plot parameters deter- + + 0 0 + events below the π π− threshold in the K π π π spec- mined in Ref. [66], and we assume isospin symmetry be- 0 → 0 0 + trum by about 13% [53], for e.g. η 3π , the correspond- tween these couplings for the η′ ηπ π and η′ ηπ π− ing reduction amounts to only 1 2%→ [48]. So while prob- channels. There is a very clear→ signal of the cusp→ effect −

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more. On the other hand, the prediction of the cusp seems extremely stable. A remarkable feature of Figs. 12, 13 is the fact that there seems to be hardly any trace left of what we dis- cussed as the “two-loop cusp” in K+ π0π0π+ decays, → + i.e. a square-root-like behavior above the π π− threshold. This observation bears up against closer scrutiny: numeri- 0 0 cally we find that the cusp above threshold in η′ ηπ π is suppressed by about a factor of 250 compared→ to the leading, (a) cusp. The explanation for this suppression can be foundO with the help of the threshold theorem again, see Sect. 5, and it turns out to be the result of residual ap- proximate isospin symmetry between the amplitudes for 0 0 + η′ ηπ π and η′ ηπ π− [54]. So even if we al- lowed→ for small isospin→ breaking in the tree-level couplings (which was neglected here), a strong relative suppression 0 0 Fig. 12. Decay spectrum for η′ ηπ π , divided by pure phase would persist. → + space. The insert focuses on the cusp region around the π π− Finally, we can estimate the size of a potential three- threshold. The dashed line corresponds to the tree result, the gray loop cusp in analogy to Sect. 5, which turns out not to be band shows the full result to two loops, with the uncertainty due suppressed by similar arguments, although, naturally, by to the variation of the πη threshold parameters. Figure taken from the high power of scattering lengths involved. In this case, Ref. [54]. we find that the cusp of (a3) should reduce the leading (a) cusp by about 0.5%O [54]. So in contrast to K+ Oπ0π0π+ decays, for the description of which the (a2) cusp→ O turned out to be absolutely necessary, in the case of η′ ηπ0π0 the singularity is entirely dominated by the leading,→ one-loop rescattering term.

6.3 The role of πη interactions in η′ → ηπ0π0

The cusp effect in the π0π0 invariant mass spectrum of the 0 0 decay η′ ηπ π is dominantly (and, as we have seen above, practically→ completely) due to ππ final-state rescat- tering. An even more interesting question, however, may be whether there is access to information on the πη thresh- old parameters, too, in this decay. The observation in the last section that at least a large part of their effect can be absorbed in a redefinition of the tree-level couplings shows 0 0 Fig. 13. Decay spectrum for η′ ηπ π as in Fig. 12, after renor- that this is not a trivial endeavor,and at least not in the cen- malization of tree couplings in→ order to reproduce the Dalitz plot ter of the Dalitz plot. parameters from Ref. [66] with the full amplitude. Figure taken The obviousquestion to ask is whether there is an inter- from Ref. [54]. esting non-analytic behavior near the πη thresholds, i.e. for 2 s1 / s2 close to (Mη + Mπ) . In contrast to what makes the + cusp at the π π− threshold so special, we obviously can- + below the π π− threshold, plus a significant deviation be- not go below threshold and really see a cusp as a change tween tree and two-loop spectrum mainly due to ππ final in slope below vs. above a certain kinematical point; how- = 2 state interactions at large s3 Mπ0π0 . The width of the ever, we still may look for square-root-like behavior at the band gives an indication of the size of πη rescattering ef- border of the Dalitz plot, difficult as it would be to in- fects, which however hardly affect the cusp region. vestigate such a phenomenon experimentally. From what Obviously, the Dalitz plot parameters of the distribu- we have learnt so far, such a cusp above threshold would tion including loop corrections in Fig. 12 are not identical have to be a two-loop effect and therefore is expected to to the input parametersanymore. In Fig. 13, we have there- be small. However, things turn our to be even worse: with fore renormalized the tree-level couplings in such a way the use of the threshold theorem again, applied now to the 2 that the full amplitude reproduces the Dalitz plot expan- threshold s1 = (Mη + Mπ) , one can show that the interfer- sion as measured in Ref. [66]. The result is very striking, ence of genuine two-loop graphs with the tree-level ampli- as the by far largest part of the final-state interactions above tude (both real) is always exactly cancelled by the product + the π π− threshold can be absorbed into such a redefinition of two corresponding one-loop graphs (both imaginary in of the tree-level parameters. In particular, hardly any effect our formalism), such that no square-root behavior survives of the variation of πη threshold parameters is visible any in the squared amplitude [67]. This cancellation is shown

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∗

0 0 2 Fig. 14. Visualization of the cancellation of threshold cusps in η′ ηπ π at s1 = (Mη + Mπ0 ) . Double lines denote both η′ and η fields, dashed lines stand for neutral pions. Figure taken from Ref. [67].→ schematically in Fig. 14 for a specific set of graphs, and it References can be shown to persist for all diagrams up to two loops. We therefore have to conclude that, with the methods de- 1. S. Weinberg, Physica A 96 (1979) 327. scribed here, we cannot identify a method to extract πη 2. J. Gasser and H. Leutwyler, Annals Phys. 158 (1984) scattering lengths in a similar fashion as the cusp effect al- 142. lows for the ππ ones. 3. M. Gell-Mann, R. J. Oakes and B. Renner, Phys. Rev. 175 (1968) 2195. 4. M. Knecht, B. Moussallam, J. Stern and N. H. Fuchs, Nucl. Phys. B 457 (1995) 513 [arXiv:hep- 7 Summary and conclusions ph/9507319]. 5. J. F. Donoghue, J. Gasser and H. Leutwyler, Nucl. Non-relativistic effective field theory provides a system- Phys. B 343 (1990) 341. atic framework for an analysis of the cusp phenomenon 6. G. Colangelo,J. Gasser and H. Leutwyler,Nucl. Phys. and pion–pion scattering lengths in K+ π0π0π+ decays. B 603 (2001) 125 [arXiv:hep-ph/0103088]. The representation of the decay amplitude→ is calculated 7. G. Colangelo, J. Gasser and H. Leutwyler, Phys. Lett. in a combined expansion in a non-relativistic parameter ǫ B 488 (2000) 261 [arXiv:hep-ph/0007112]. and ππ threshold parameters a, which is currently available 8. B. Ananthanarayan, G. Colangelo, J. Gasser and up to (ǫ4, aǫ5, a2ǫ4). In order to match the enormous ex- H. Leutwyler, Phys. Rept. 353 (2001) 207 [arXiv:hep- perimentalO accuracy achieved by the NA48/2 collaboration ph/0005297]. theoretically, radiative corrections have to be included. The 9. S. Descotes-Genon, N. H. Fuchs, L. Girlanda and effect of the latter on the ππ scattering lengths is surpris- J. Stern, Eur. Phys. J. C 24 (2002) 469 [arXiv:hep- ingly large, as photon effects modify the analytic structure ph/0112088]. + of the decay amplitude near the π π− threshold. Similar 10. F. J. Yndur´ain, R. Garc´ıa-Mart´ın and J. R. Pel´aez, cusp phenomena are also present in other decays such as Phys. Rev. D 76 (2007) 074034 [arXiv:hep- 0 0 KL 3π or η 3π , where they are however far less ph/0701025]. prominent→ and much→ harder to use for a precision determi- 11. R. Kami´nski, J. R. Pel´aez and F. J. Yndur´ain, Phys. nation of scattering lengths. More promising in that respect Rev. D 77 (2008) 054015 [arXiv:0710.1150[hep-ph]]. 0 0 is the decay η′ ηπ π , which, on the other hand, seems 12. B. Adeva et al. [DIRAC Collaboration], Phys. Lett. B → not to offer easy access to πη threshold parameters. 619 (2005) 50 [arXiv:hep-ex/0504044]. 13. S. Pislak et al., Phys. Rev. D 67 (2003) 072004 [arXiv:hep-ex/0301040]. 14. J. R. Batley et al. [NA48/2 Collaboration], Eur. Phys. Acknowledgements J. C 54 (2008) 411. 15. S. Deser, M. L. Goldberger, K. Baumann and I am very grateful to my coworkers Moritz Bissegger, W. E. Thirring, Phys. Rev. 96 (1954) 774. Gilberto Colangelo, Andreas Fuhrer, J¨urg Gasser, Akaki 16. A. Gall, J. Gasser, V. E. Lyubovitskij and A. Rusetsky, Rusetsky, and Sebastian Schneider for the most fruitful Phys. Lett. B 462 (1999) 335 [arXiv:hep-ph/9905309]. collaborations leading to the results presented here, and 17. J. Gasser, V. E. Lyubovitskij, A. Rusetsky and to J¨urg Gasser for useful comments on this manuscript. A. Gall, Phys. Rev. D 64 (2001) 016008 [arXiv:hep- I furthermore wish to thank Luigi Di Lella and Dmitri ph/0103157]. Madigozhin for providing me with the data for Figs. 1 18. J. Gasser, V. E. Lyubovitskij and A. Rusetsky, Phys. and 9. We acknowledge the support of the European Com- Rept. 456 (2008) 167 [arXiv:0711.3522 [hep-ph]]. munity Research Infrastructure Integrating Activity “Study 19. K. M. Watson, Phys. Rev. 88 (1952) 1163. of Strongly Interacting Matter” (acronym HadronPhysics2, 20. A. Pais and S. B. Treiman, Phys. Rev. 168 (1968) grant agreement No. 227431) under the Seventh Frame- 1858. work Programme of the EU. Work supported in part by 21. G. Colangelo, J. Gasser and A. Rusetsky, Eur. Phys. J. DFG (SFB/TR 16, “Subnuclear Structure of Matter”) and C 59 (2009) 777 [arXiv:0811.0775 [hep-ph]]. by the Helmholtz Association through funds provided to 22. J. R. Batley et al. [NA48/2 Collaboration], Phys. Lett. the virtual institute “ and strong QCD” (VH-VI-231). B 633 (2006) 173 [arXiv:hep-ex/0511056].

01008-p.11 EPJ Web of Conferences

23. N. Cabibbo, Phys. Rev. Lett. 93 (2004) 121801 47. C. Ditsche, B. Kubis and U.-G. Meißner, Eur. Phys. J. [arXiv:hep-ph/0405001]. C 60 (2009) 83 [arXiv:0812.0344 [hep-ph]]. 24. E. P. Wigner, Phys. Rev. 73 (1948) 1002. 48. C.-O. Gullstr¨om, A. Kup´s´cand A. Rusetsky, Phys. 25. U.-G. Meißner, G. M¨uller and S. Steininger, Phys. Rev. C 79 (2009) 028201 [arXiv:0812.2371 [hep-ph]]. Lett. B 406 (1997) 154 [Erratum-ibid. B 407 (1997) 49. E. Abouzaid et al. [KTeV Collaboration], Phys. Rev. 454] [arXiv:hep-ph/9704377]. D 78 (2008) 032009 [arXiv:0806.3535 [hep-ex]]. 26. P. Budini and L. Fonda, Phys. Rev. Lett. 6 (1961) 419. 50. C. Adolph et al. [WASA-at-COSY Collaboration], 27. J. R. Batley et al. [NA48/2 Collaboration], Eur. Phys. Phys. Lett. B 677 (2009) 24 [arXiv:0811.2763 [nucl- J. C 64 (2009) 589 [arXiv:0912.2165 [hep-ex]]. ex]]. 28. N. Cabibbo and G. Isidori, JHEP 0503 (2005) 021 51. M. Unverzagt et al. [Crystal-Ball-at-MAMI, TAPS, [arXiv:hep-ph/0502130]. and A2 Collaborations], Eur. Phys. J. A 39 (2009) 169 29. E. G´amiz, J. Prades and I. Scimemi, Eur. Phys. J. C 50 [arXiv:0812.3324 [hep-ex]]. (2007) 405 [arXiv:hep-ph/0602023]. 52. S. Prakhov et al. [Crystal-Ball-at-MAMI and A2 30. G. Colangelo, J. Gasser, B. Kubis and A. Rusetsky, Collaborations], Phys. Rev. C 79 (2009) 035204 Phys. Lett. B 638 (2006) 187 [arXiv:hep-ph/0604084]. [arXiv:0812.1999 [hep-ex]]. 31. M. Beneke and V. A. Smirnov, Nucl. Phys. B 522 53. D. Madigozhin, Pion scattering lengths from the cusp (1998) 321 [arXiv:hep-ph/9711391]. effect analysis, talk given at: FlaviAnet Kaon Work- 32. M. Knecht and R. Urech, Nucl. Phys. B 519 (1998) shop, Anacapri, , June 12–14, 2008. 329 [arXiv:hep-ph/9709348]. 54. B. Kubis and S. P. Schneider,Eur. Phys. J. C 62 (2009) 33. M. Bissegger, A. Fuhrer, J. Gasser, B. Kubis 511 [arXiv:0904.1320 [hep-ph]]. and A. Rusetsky, Phys. Lett. B 659 (2008) 576 55. Project C.5 of the SFB/TR 16, “Subnuclear Structure [arXiv:0710.4456 [hep-ph]]. of Matter”, U. Wiedner, R. Beck et al.. 34. J. Gasser, B. Kubis and A. Rusetsky, in preparation. 56. A. Starostin, in Proc. 2nd Int. Workshop on Eta 35. L. D. Landau, Nucl. Phys. 13 (1959) 181. Meson Physics, Pe˜niscola, Spain, 2007, edited by 36. R. Eden, P. V. Landshoff, D. I. Olive and J. C. Polk- M. Jacewicz and B. H¨oistad, arXiv:0710.1809 [nucl- inghorne, The analytic S-matrix, Cambridge Univer- ex]. sity Press (1966). 57. P. Achenbach, arXiv:0802.2870 [nucl-ex]. 37. A. Nehme, Phys. Rev. D 70 (2004) 094025 58. M. Unverzagt [for the Crystal-Ball-at-MAMI Collab- [arXiv:hep-ph/0406209]. oration], arXiv:0910.1331 [hep-ex]. 38. J. Bijnens and F. Borg, Eur. Phys. J. C 39 (2005) 347 59. H. H. Adam et al. [WASA-at-COSY Collaboration], [arXiv:hep-ph/0410333]. arXiv:nucl-ex/0411038. 39. J. Bijnens and F. Borg, Eur. Phys. J. C 40 (2005) 383 60. B. R. Jany and D. Duniec [WASA-at-COSY Col- [arXiv:hep-ph/0501163]. laboration], Proc. 11th Int. Conf. on Meson–Nucleon 40. S. R. Gevorkyan, A. V. Tarasov and O. O. Voskre- Physics and the Structure of the Nucleon (MENU senskaya, Phys. Lett. B 649 (2007) 159 [arXiv:hep- 2007), J¨ulich. ph/0612129]. 61. KLOE-2, Letter of Intent [http://www.lnf.infn.it/lnf- 41. S. R. Gevorkyan,D. T. Madigozhin, A. V. Tarasov and admin/direzione/roadmap/LoIKLOE.pdf]. O. O. Voskresenskaya, Phys. Part. Nucl. Lett. 5 (2008) 62. C. Bloise, AIP Conf. Proc. 950 (2007) 192. 85 [arXiv:hep-ph/0702154]. 63.H. B. Li, J. Phys. G 36 (2009) 085009 42. M. Bissegger, A. Fuhrer, J. Gasser, B. Kubis [arXiv:0902.3032 [hep-ex]]. and A. Rusetsky, Nucl. Phys. B 806 (2009) 178 64. V. Bernard, N. Kaiser and U.-G. Meißner, Phys. Rev. [arXiv:0807.0515 [hep-ph]]. D 44 (1991) 3698. 43. G. Isidori, Eur. Phys. J. C 53 (2008) 567 65. M. Koles´ar and J. Novotn´y, Eur. Phys. J. C 56 (2008) [arXiv:0709.2439 [hep-ph]]. 231 [arXiv:0802.1289 [hep-ph]]. 44. G. Gamow, Z. Phys. 51 (1928) 204. 66. V. Dorofeev et al., Phys. Lett. B 651 (2007) 22 45. A. Sommerfeld, Atombau und Spektrallinien, Volume [arXiv:hep-ph/0607044]. 2, Vieweg, Braunschweig (1939). 67. S. P. Schneider and B. Kubis, PoS CD09 (2009) 120 46. J. Schwinger, J. Math. Phys. 5 (1964) 1606. [arXiv:0910.0200 [hep-ph]].

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