Radiative resonance couplings in γπ→ππ
aa r X i v : . [ h e p - ph ] D ec INT-PUB-17-041
Radiative resonance couplings in γπ → ππ Martin Hoferichter, Bastian Kubis,
2, 3 and Marvin Zanke Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA Helmholtz-Institut f¨ur Strahlen- und Kernphysik (Theorie), Universit¨at Bonn, D–53115 Bonn, Germany Bethe Center for Theoretical Physics, Universit¨at Bonn, D–53115 Bonn, Germany
Studies of the reaction γπ → ππ , in the context of the ongoing Primakoff program of the COM-PASS experiment at CERN, give access to the radiative couplings of the ρ (770) and ρ (1690) reso-nances. We provide a vector-meson-dominance estimate of the respective radiative width of the ρ ,Γ ρ → πγ = 48(18) keV, as well as its impact on the F -wave in γπ → ππ . For the ρ (770), we establishthe formalism necessary to extract its radiative coupling directly from the residue of the resonancepole by analytic continuation of the γπ → ππ amplitude to the second Riemann sheet, without anyreference to the vector-meson-dominance hypothesis. PACS numbers: 11.55.Fv, 13.75.Lb, 11.30.Rd, 13.60.LeKeywords: Dispersion relations, Meson–meson interactions, Chiral Symmetries, Meson production
I. INTRODUCTION
Apart from the two-photon decay of the neutral pion,the process γπ → ππ is the simplest manifestation of theWess–Zumino–Witten anomaly [1, 2]. The leading orderin the chiral expansion [3–5], F π = eN c π F π = 9 . − , (1)is determined by the number of colors N c , the elec-tric charge e = √ πα , and the pion decay constant F π = 92 . F π = 12 . .
0) GeV − [7], sug-gested some tension with the low-energy theorem, cor-rections beyond the leading order (1) have been workedout [8–13], with the net result that higher-order andelectromagnetic corrections reduce the value to F π =10 . .
2) GeV − . Together with a similar extraction from π − e − → π − e − π [14], leading to F π = 9 . .
1) GeV − ,the low-energy theorem is now tested at the 10% level,far behind the 1 . π → γγ [15, 16]. Meanwhile, a first lattice calculationof γ ∗ π → ππ has been reported in [17, 18].In contrast to earlier measurements, the Primakoffstudies at COMPASS cover not only the threshold regionof γπ → ππ , but extend to much higher center-of-massenergies. As pointed out in [19], this allows one to usethe ρ resonance as a lever to vastly increase the statisticsof the anomaly extraction, combining constraints fromanalyticity, unitarity, and crossing symmetry into a two-parameter description of the amplitude whose normal-ization coincides with F π . More recently, interest inthe γπ → ππ reaction has been triggered by its relationto the hadronic-light-by-light contribution to the anoma-lous magnetic moment of the muon, where it appears asa crucial input quantity for a data-driven determinationof the π → γ ∗ γ ∗ transition form factor [20], which inturn determines the strength of the pion-pole contribu-tion in a dispersive approach to hadronic light-by-lightscattering [21–25]. In fact, the kinematic reach of the COMPASS exper-iment extends up to and including the ρ (1690), thefirst resonance in the F -wave. In this paper, we esti-mate its impact on the γπ → ππ cross section basedon vector-meson-dominance (VMD) assumptions, whichcorresponds to an estimate of the radiative width Γ ρ → πγ ,see Sect. II. For the ρ (770) such a simplified approach isnot adequate anymore, precisely due to the amount ofstatistics available at the ρ peak that should allow oneto significantly sharpen the test of the chiral low-energytheorem in the future [26]. Instead, the analytic con-tinuation of the γπ → ππ amplitudes that underlie thisextraction, in combination with the known ρ -pole param-eters and residues from ππ scattering [27, 28], determinesthe ρπγ coupling constant, g ρπγ , once the free parametersof the representation have been fit to the cross section.The precise prescription for how to extract the radiativecoupling of the ρ , defined through the residue of the polein a model-independent way, is spelled out in Sect. III.Combining all currently available information, prior tothe direct COMPASS measurement, we predict the lineshape of the cross section in Sect. IV. A short summaryis provided in Sect. V. II. VECTOR MESON DOMINANCE
Throughout, we follow the conventions of [19]. Theamplitude for the process γ ( q ) π − ( p ) → π − ( p ) π ( p ) (2)is decomposed according to M γπ → ππ ( s, t, u ) = iǫ µναβ ǫ µ p ν p α p β F ( s, t, u ) , (3)in terms of the scalar function F ( s, t, u ), the photon po-larization vector ǫ µ , and Mandelstam variables chosen as s = ( q + p ) , t = ( p − p ) , and u = ( p − p ) , with s + t + u = 3 M π , particle masses defined by the chargedstates, and a relation to the center-of-mass scattering an-gle z = cos θ according to t = a ( s ) + b ( s ) z, u = a ( s ) − b ( s ) z,a ( s ) = 3 M π − s , b ( s ) = s − M π σ π ( s ) ,σ π ( s ) = r − M π s . (4)Crossing symmetry implies that the scalar function F ( s, t, u ) is fully symmetric in s, t, u . In the conventionsof (3) the cross section becomes σ ( s ) = ( s − M π ) / ( s − M π )1024 π √ s Z − d z (cid:0) − z (cid:1) |F ( s, t, u ) | . (5)Later, we also need the partial-wave decomposition [29] F ( s, t, u ) = X odd l f l ( s ) P ′ l ( z ) , (6)where P ′ l ( z ) denotes the derivative of the Legendre poly-nomials, and the inversion is given by f l ( s ) = 12 Z − d z (cid:0) P l − ( z ) − P l +1 ( z ) (cid:1) F ( s, t, u ) . (7)Elastic unitarity relates these partial waves to the isospin I = 1 ππ phase shifts δ l ( s ), M I =1 ππ ( s, t ) = 32 π X odd l (2 l + 1) t l ( s ) P l (cid:18) ts − M π (cid:19) ,t l ( s ) = e iδ l ( s ) − iσ π ( s ) , (8)by means ofIm f l ( s ) = σ π ( s ) (cid:0) t l ( s ) (cid:1) ∗ f l ( s ) θ (cid:0) s − M π (cid:1) . (9)The fact that the phase of f l ( s ) coincides with δ l ( s )is a manifestation of Watson’s final-state theorem [30].Finally, the dominant electromagnetic correction [12]amounts to F ( s, t, u ) → F ( s, t, u ) − e F π t F π . (10) A. ρ (770) The VMD amplitude for γπ → ππ can be constructedby combining the ρ → ππ amplitude from L ρππ = g ρππ ǫ abc π a ∂ µ π b ρ cµ , (11)with isospin indices a, b, c , together with M ρπγ = eg ρπγ ǫ µναβ ǫ µρ ǫ νγ p α p β , (12) γ ωπ ππρ FIG. 1: VMD mechanism for the ρ contribution to γπ → ππ . where p and p refer to the momenta of the pion and thephoton, and ǫ µρ , ǫ νγ to the ρ and γ polarization vectors.The result reads f VMD1 ( s ) = 2 eg ρπγ g ρππ M ρ − iM ρ Γ ρ − s , (13)where the finite width of the ρ has been taken intoaccount by means of a Breit–Wigner propagator. InSect. III we will reinterpret both couplings, g ρππ and g ρπγ , as residues of the respective poles, but for the mo-ment we first collect the phenomenological informationavailable when treating the ρ as a narrow resonance. Inthis approximation, the width becomesΓ ρ → ππ = | g ρππ | πM ρ (cid:0) M ρ − M π (cid:1) / , (14)i.e. | g ρππ | ∼ . ρ → πγ = e | g ρπγ | πM ρ (cid:0) M ρ − M π (cid:1) . (15)Within the narrow-width approximation, one could thenextract | g ρπγ | from the measured cross section for γπ → ππ and thereby determine Γ ρ → πγ . At a similar levelof accuracy, SU (3) symmetry (see e.g. [31]) suggestsΓ ρ → πγ = Γ ω → π γ / ρ → π γ = 69(9) keV, Γ ρ ± → π ± γ = 68(7) keV [6]. Amodel-independent extraction of the radiative couplingof the ρ from γπ → ππ will be discussed in Sect. III. B. ρ (1690) For the generalization to the ρ (1690) contribution to γπ → ππ and the determination of its radiative width, wefollow [32, 33]. To this end, we first remark that G -paritydictates the photon in this process to have isoscalar quan-tum numbers. In the VMD picture, it therefore couplesto the ρ and a pion predominantly via the ω meson (as-suming the φ to be negligible due to Okubo–Zweig–Iizukasuppression). As we aim for a prediction of the radiativedecay of the ρ , we in particular need to assume strict VMD without a direct ρ πγ coupling. Figure 1 thereforesuggests that we need to determine the coupling con-stants g ρ ππ , g ρ πω , as well as g ωγ .Starting from [33] L ρ = g ρ ππ F π h ρ µνλ (cid:2) ∂ µ π , ∂ ν ∂ λ π (cid:3) i + g ρ πω F π ǫ λαβγ h ρ µνλ ∂ µ ∂ α π i ∂ ν ∂ β ω γ , L ωγ = − eM ω g ωγ A µ ω µ , (16)with spin-3 fields ρ µνλ = ρ aµνλ τ a , pion isotriplet π = π a τ a , the isoscalar vector field ω µ , and the electromag-netic field A µ , one finds the partial decay widthsΓ ρ → ππ = | g ρ ππ | πF π M ρ (cid:0) M ρ − M π (cid:1) / , Γ ρ → πω = | g ρ πω | πF π M ρ λ (cid:0) M ρ , M ω , M π (cid:1) / , (17)where λ ( x, y, z ) = x + y + z − xy + xz + yz ). To-gether with M ρ = 1688 . .
1) MeV, Γ ρ = 161(10) MeV,BR( ρ → ππ ) = 23 . . ρ → πω ) =16(6)% [6] this fixes the parameters according to | g ρ ππ | = 0 . , | g ρ πω | = 1 . − . (18)Similarly, one then finds for the radiative widthΓ ρ → πγ = e | g ρ πω | πF π | g ωγ | M ρ (cid:0) M ρ − M π (cid:1) , (19)and with | g ωγ | = 16 . ω → e + e − = e ( M ω − m e ) / π | g ωγ | (cid:18) m e M ω (cid:19) , (20)we obtain the predictionΓ ρ → πγ = 48(18) keV . (21)This result lies slightly higher than the quark-model es-timate Γ ρ → πγ = 21 keV [34]. Finally, the resonant con-tribution to the γπ → ππ F -wave becomes f VMD3 ( s ) = eg ρ ππ g ρ πω ( s − M π )( s − M π ) F π g ωγ s ( M ρ − iM ρ Γ ρ − s ) . (22) III. RADIATIVE COUPLING OF THE ρ (770)A. ππ scattering In a model-independent way, the properties of the ρ (770) are encoded in the pole position and residues ofthe S -matrix on the second Riemann sheet. The primeprocess to determine the parameters is I = 1 ππ scatter-ing, whose partial-wave amplitude in the vicinity of thepole can be written as t , II ( s ) = g ρππ ( s − M π )48 π ( s ρ − s ) , s ρ = (cid:18) M ρ − i Γ ρ (cid:19) , (23) Ref. M ρ [MeV] Γ ρ [MeV] | g ρππ | arg( g ρππ ) [40][28], GKPY 763 . +1 . − . . +2 . − . . +0 . − . (cid:0) − . +1 . − . (cid:1) ◦ [28], Roy 761 +4 − . +3 . − . . +0 . − . (cid:0) − . +1 . − . (cid:1) ◦ [27], Roy 762 . .
8) 145 . . ρ (770) from dispersion re-lations. The phase arg( g ρππ ) in the last column is only deter-mined modulo 180 ◦ . where the conventions have been chosen in such a waythat in the narrow-width limit the coupling g ρππ matchesonto the Lagrangian definition (11). Elastic unitarityfor ππ scattering relates the amplitudes on the first andsecond Riemann sheets according to t , I ( s ) − t , II ( s ) = − σ π ( s ) t , I ( s ) t , II ( s ) , (24)where we have introduced [35] σ π ( s ) = r M π s − , σ π ( s ± iǫ ) = ∓ iσ π ( s ) , (25)so that the pole parameters can be determined from thecondition that t , I ( s ρ ) = 1 / (2 σ π ( s ρ )) = − i/ (2 σ π ( s ρ ))(since Im s ρ < t ( s )on the first sheet is available. Such a representation isprovided by dispersion relations, in the form of Roy equa-tions [36–38] or variants thereof, the so-called GKPYequations [39]. The latter produce the pole parametersgiven in the first line of Table I, in good agreement withthe determination from Roy equations, but with smalleruncertainties. In the following, we use the GKPY pa-rameters from [28] together with the I = 1 phase shiftsfrom [39]. Within uncertainties, this covers similar de-terminations listed in the table. Note that g ρππ is a com-plex coupling, with a phase that is observable (modulo180 ◦ ), although Table I shows that this phase is rathersmall [40]. B. Pion form factor
The simplest quantity that probes the electromagneticinteractions of the pion is its form factor F Vπ ( s ). Giventhe wealth of experimental data, it provides an ideal test-ing ground to study how well VMD predictions fare whenconfronted with real data, in this case for g ργ instead of g ρπγ . In analogy to (24), the elastic unitarity relation,Im F Vπ ( s ) = σ π ( s ) (cid:0) t l ( s ) (cid:1) ∗ F Vπ ( s ) θ (cid:0) s − M π (cid:1) , (26)defines the analytic continuation of the form factor ontothe second sheet F Vπ, I ( s ) − F Vπ, II ( s ) = − σ π ( s ) F Vπ, I ( s ) t , II ( s ) . (27)In the vicinity of the pole we may write F Vπ, II ( s ) = g ρππ g ργ s ρ s ρ − s , (28)where the conventions are chosen in such a way that inthe narrow-width and SU (3) limit g ργ = g ωγ /
3, cf. (16).Altogether one finds1 g ργ g ρππ = i σ π ( s ρ )24 π F Vπ, I ( s ρ ) , (29)which allows one to extract g ργ from the form factorevaluated at s ρ on the first sheet and the previouslydetermined g ρππ . The dispersive formalism for the an-alytic continuation to s ρ has been studied in detail inthe literature, see [25, 41–49], and data abound, mostlymotivated by the ππ contribution to hadronic vacuumpolarization in the anomalous magnetic moment of themuon. In this way, the dominant uncertainties actuallyarise from the error in g ρππ as well as the systematics ofthe fit, e.g. whether ρ – ω mixing (as present in the fit to e + e − data [50–55], but not in τ → ππν [56]) is includedin the definition of the form factor.In the end, the results of the fits fall within the range | g ργ | = 4 . | g VMD ργ | = | g ωγ | / . ρ → e + e − in analogy to (20), without SU (3) assump-tions, even produces | g VMD ργ | = 5 . g ρππ g ργ ) ∼ − ◦ , so that g ργ is almostreal (with the same sign as the one chosen for g ρππ ). C. γπ → ππ The derivation for γπ → ππ proceeds in close analogyto the pion form factor. From the unitarity relation (9)we find the analytic continuation f , I ( s ) − f , II ( s ) = − σ π ( s ) f , I ( s ) t , II ( s ) , (30)and writing f , II ( s ) = 2 eg ρπγ g ρππ s ρ − s (31)in the vicinity of the pole (to match onto (13) in theVMD limit), the analog of (29) becomes eg ρπγ g ρππ = i s ρ σ π ( s ρ )48 π f , I ( s ρ ) . (32)However, the analytic continuation is less straightfor-ward than for F Vπ ( s ), due to the fact that, in contrastto the form factor, the scattering process γπ → ππ pro-duces a left-hand cut, which, in addition, needs to be con-structed in such a way that crossing symmetry is main-tained. The corresponding formalism has been derivedin [19]. Starting from the decomposition F ( s, t, u ) = F ( s ) + F ( t ) + F ( u ) , (33)which holds if imaginary parts from partial waves with l ≥ -1.5 -1 -0.5 0 0.5 1 1.5-0.8-0.6-0.4-0.200.20.40.6 PSfrag replacements s (cid:2) GeV (cid:3) F ( ) -1.5 -1 -0.5 0 0.5 1 1.5-0.4-0.200.20.40.60.811.2 PSfrag replacements s (cid:2) GeV (cid:3) F ( ) (cid:2) G e V (cid:3) FIG. 2: Basis functions F ( i )2 for γπ → ππ . The black solid(red dashed) lines refer to the real (imaginary) parts. the dispersion relation for F ( s ) can be represented in theform F ( s ) = C (1)2 F (1)2 ( s ) + C (2)2 F (2)2 ( s )= 13 (cid:0) C (1)2 + C (2)2 s (cid:1) + 1 π Z ∞ M π d s ′ s ′ s s ′ − s × (cid:0) C (1)2 Im F (1)2 ( s ′ ) + C (2)2 Im F (2)2 ( s ′ ) (cid:1) , (34)where C ( i )2 refer to the subtraction constants in the twice-subtracted dispersion relation. These are the free param-eters of the fit. In contrast, the basis functions F ( i )2 ( s )can be calculated once and for all, for a given input ofthe ππ phase shift δ ( s ) (the results for the phase shiftfrom [39] are depicted in Fig. 2). The partial wave f ( s )follows from f ( s ) = 34 Z − d z (cid:0) − z (cid:1)(cid:0) F ( s ) + F ( t ) + F ( u ) (cid:1) = C (1)2 + C (2)2 M π + 1 π Z ∞ M π d s ′ K ( s, s ′ ) × (cid:0) C (1)2 Im F (1)2 ( s ′ ) + C (2)2 Im F (2)2 ( s ′ ) (cid:1) , (35)with integration kernel K ( s, s ′ ) = s s ′ ( s ′ − s ) + 3 b ( s ) n(cid:0) − x s (cid:1) Q ( x s ) + x s o − s ′ + s − M π s ′ , x s = s ′ − a ( s ) b ( s ) , (36)and the lowest Legendre function of the second kind Q ( z ) = 12 Z − d xz − x ,Q ( z ± iǫ ) = 12 log (cid:12)(cid:12)(cid:12)(cid:12) z − z (cid:12)(cid:12)(cid:12)(cid:12) ∓ i π θ (cid:0) − z (cid:1) . (37)For the GKPY ρ parameters from [28] we obtain f , I ( s ρ ) = C (1)2 (cid:0) . . i (cid:1) − C (2)2 (cid:0) . . i (cid:1) GeV , (38)where the uncertainties reflect the propagated errors onthe pole parameters only (when experiment reaches few-percent accuracy, also the uncertainties in the ππ phaseshift will have to be included). Once the C ( i )2 are fitto cross-section data, this relation determines f , I ( s ρ ),and thus, by means of (32), the radiative coupling of the ρ (770) (including its phase). The current knowledge ofthese couplings, see (40) below, indicates that, similar to g ργ , g ρπγ is almost real with the same sign as g ρππ . IV. LINE SHAPE OF γπ → ππ Currently available information on the radiative cou-pling of the ρ (770) [6] largely derives from the high-momentum Primakoff experiments [57–59], while no ex-perimental result is available for the ρ (1690) at all.Thanks to its high-statistics data, COMPASS has theunique opportunity to determine these couplings eitherfor the first time or with unprecedented accuracy; com-pare their results for the radiative widths of the a (1320)and the π (1670) as extracted from the similar Primakoffreaction γπ → π [60]. For the ρ (770) such a measure-ment is intimately related to the determination of the chi-ral anomaly, and, building upon [19], the previous sectionestablishes the formalism to extract both simultaneouslyin a consistent, model-independent way.In this section, we reverse the argument and collect thecurrently available information to predict the line shapeto be expected in the γπ → ππ cross section. First ofall, the combination C (1)2 + C (2)2 M π is related to the chi-ral anomaly, but only up to an additional quark-massrenormalization C (1)2 + C (2)2 M π = ¯ F π ≡ F π (cid:0) M π ¯ C (cid:1) , (39)estimated from resonance saturation to 3 M π ¯ C =6 .
6% [8]. We use the corresponding central value, but, given that we wish to extract ¯ F π from the data, as-sign a 10% uncertainty, ¯ F π = 10 . .
0) GeV − , to re-flect the level of accuracy that previous measurementshave established. The second combination of couplingconstants corresponds to the radiative coupling of the ρ (770), for which we take the SU (3) VMD result | g ρπγ | =0 . − , but, in view of the results for the pionform factor in Sect. III B, attach a 10% uncertainty aswell. These constraints translate into C (1)2 = 9 . .
0) GeV − , C (2)2 = 24 . .
5) GeV − , (40)where the uncertainty in C (1)2 and C (2)2 is entirely dom-inated by ¯ F π and | g ρπγ | , respectively. For the ρ (1690)we use the parameters as given in Sect. II B.The twice-subtracted dispersion relation (34) is per-fectly suited to extract the coefficients C ( i )2 from cross-section data up to and including the ρ resonance, butdisplays a pathological high-energy behavior. To obtaina description that remains valid in the whole region below2 GeV, we implement a version of the basis functions F ( i )2 with a relatively low cut-off parameter Λ = 1 . f ( s ) ∼ /s , in agreement with gen-eral arguments based on the Froissart bound [61]. In ad-dition to the P -wave, the symmetrized version (33) pro-duces non-vanishing contributions to f l ( s ) for all (odd) l in the partial-wave projection. However, we checkedthat the corresponding F - and higher partial waves canbe ignored, and similarly effects from excited ρ states, ρ ′ and ρ ′′ , are likely negligible [32, 62] (assuming that theseresonances couple with a comparable relative strength asin the pion form factor [56]). While inelastic correctionsincluded directly in the dispersive description by meansof the inelasticity parameter are typically small [33], suchexcited ρ states provide an indicator for the size of thedominant inelasticities from 4 π intermediate states; seealso [45]. Finally, the narrow-resonance approximationfor the ρ is strictly meaningful only at the resonancemass, while the additional momentum dependence in (22)distorts the resonance shape. To obtain a more realisticline shape we follow [63, 64] and introduce centrifugal-barrier factors, which amounts to the replacement f VMD3 ( s ) → f VMD3 ( s ) B ( q f ( s ) R ) B ( q i ( s ) R ) B ( q f ( M ρ ) R ) B ( q i ( M ρ ) R ) , (41)with B ( x ) = 15 / √
225 + 45 x + 6 x + x , initial- and We wish to emphasize that our dispersive representation of the P -wave is very reliable mostly below 1 GeV, and the model forthe F -wave around the ρ resonance. Despite the indicationsfor comparably smaller ρ ′ , ρ ′′ contributions, we do not claim tomake a high-precision prediction of the line shape between 1 and1 . PSfrag replacements √ s (cid:2) GeV (cid:3) σ [ µ b ] FIG. 3: Total cross section for γπ → ππ . The dashed linerefers to our central solution, using (18) and (40). final-state momenta q i ( s ) = s − M π √ s , q f ( s ) = r s − M π , (42)and a scale R ∼ F -wavecross section, σ F (cid:0) M ρ (cid:1) = 3 (cid:0) M ρ − M π (cid:1) / (cid:0) M ρ − M π (cid:1) πM ρ (cid:12)(cid:12) f (cid:0) M ρ (cid:1)(cid:12)(cid:12) = 56 πM ρ (cid:0) M ρ − M π (cid:1) Γ ρ → ππ Γ ρ → πγ Γ ρ = 1 . µ b , (43)amounts to roughly 6% of the dominant ρ (770) peak.While currently the uncertainties are large, an improvedmeasurement of the energy dependence would immedi-ately translate to better constraints on the underlyingQCD parameters, most notably the chiral anomaly F π ,but also, as we have shown in this paper, the radiativecouplings of the ρ (770) and ρ (1690) resonances. V. SUMMARY
Extending the dispersive formalism for γπ → ππ de-veloped in [19], we have worked out the analytic contin-uation necessary to extract the radiative coupling of the ρ (770), as defined by the residue at its resonance pole.Throughout, we have indicated the correspondence to theparameters that would occur within a narrow-resonancedescription, and collected the current phenomenologicalinformation. Combined with a VMD estimate for the ρ (1690), we have obtained a prediction for the cross sec-tion of γπ → ππ up to 2 GeV, with uncertainties domi-nated by the current knowledge of the underlying param-eters: the chiral anomaly F π and the radiative couplingsof the ρ (770) and ρ (1690) resonances.This prediction can be considered a benchmark forthe ongoing Primakoff program at COMPASS. Measur-ing the cross section with reduced uncertainties comparedto Fig. 3 would allow one to test the narrow-width esti-mate of the ρ → πγ decay rate, Γ ρ → πγ = 48(18) keV,and to improve the determination of the chiral anomalyand the ρ → πγ coupling, both without relying on modelassumptions while still profiting from the full statisticsof the ρ resonance. Such improved experimental infor-mation on γπ → ππ is particularly timely given its rela-tion to hadronic light-by-light scattering in the anoma-lous magnetic moment of the muon as well as recent lat-tice calculations. We look forward to the results of theongoing analysis of the π − π channel at COMPASS thatwill provide an important step in this direction. Acknowledgments
We would like to thank Jan Friedrich, Boris Grube,Misha Mikhasenko, Stephan Paul, Jacobo Ruiz de Elvira,and Julian Seyfried for helpful discussions. Financialsupport by the DOE (Grant No. DE-FG02-00ER41132)and the DFG (CRC 110 “Symmetries and the Emergenceof Structure in QCD”) is gratefully acknowledged. [1] J. Wess and B. Zumino, Phys. Lett. , 95 (1971).[2] E. Witten, Nucl. Phys. B , 422 (1983).[3] S. L. Adler, B. W. Lee, S. B. Treiman and A. Zee, Phys.Rev. D , 3497 (1971).[4] M. V. Terent’ev, Phys. Lett. , 419 (1972).[5] R. Aviv and A. Zee, Phys. Rev. D , 2372 (1972).[6] C. Patrignani et al. [Particle Data Group], Chin. Phys.C , 100001 (2016).[7] Y. M. Antipov et al. , Phys. Rev. D , 21 (1987).[8] J. Bijnens, A. Bramon and F. Cornet, Phys. Lett. B ,488 (1990).[9] B. R. Holstein, Phys. Rev. D , 4099 (1996) [hep-ph/9512338].[10] T. Hannah, Nucl. Phys. B , 577 (2001)[hep-ph/0102213].[11] T. N. Truong, Phys. Rev. D , 056004 (2002)[hep-ph/0105123].[12] L. Ametller, M. Knecht and P. Talavera, Phys. Rev. D , 094009 (2001) [hep-ph/0107127].[13] J. Bijnens, K. Kampf and S. Lanz, Nucl. Phys. B ,245 (2012) [arXiv:1201.2608 [hep-ph]].[14] I. Giller, A. Ocherashvili, T. Ebertshauser,M. A. Moinester and S. Scherer, Eur. Phys. J. A , 229 (2005) [hep-ph/0503207]. [15] I. Larin et al. [PrimEx Collaboration], Phys. Rev. Lett. (2011) 162303 [arXiv:1009.1681 [nucl-ex]].[16] A. M. Bernstein and B. R. Holstein, Rev. Mod. Phys. ,49 (2013) [arXiv:1112.4809 [hep-ph]].[17] R. A. Brice˜no, J. J. Dudek, R. G. Edwards, C. J. Shultz,C. E. Thomas and D. J. Wilson, Phys. Rev. Lett. ,242001 (2015) [arXiv:1507.06622 [hep-ph]].[18] R. A. Brice˜no, J. J. Dudek, R. G. Edwards, C. J. Shultz,C. E. Thomas and D. J. Wilson, Phys. Rev. D , 114508(2016) [arXiv:1604.03530 [hep-ph]].[19] M. Hoferichter, B. Kubis and D. Sakkas, Phys. Rev. D , 116009 (2012) [arXiv:1210.6793 [hep-ph]].[20] M. Hoferichter, B. Kubis, S. Leupold, F. Niecknig andS. P. Schneider, Eur. Phys. J. C , 3180 (2014)[arXiv:1410.4691 [hep-ph]].[21] G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer,JHEP , 091 (2014) [arXiv:1402.7081 [hep-ph]].[22] G. Colangelo, M. Hoferichter, B. Kubis, M. Procura andP. Stoffer, Phys. Lett. B , 6 (2014) [arXiv:1408.2517[hep-ph]].[23] G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer,JHEP , 074 (2015) [arXiv:1506.01386 [hep-ph]].[24] G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer,Phys. Rev. Lett. , 232001 (2017) [arXiv:1701.06554[hep-ph]].[25] G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer,JHEP , 161 (2017) [arXiv:1702.07347 [hep-ph]].[26] J. Seyfried, Master’s thesis, TU M¨unchen, 2017.[27] G. Colangelo, J. Gasser and H. Leutwyler, Nucl. Phys. B , 125 (2001) [hep-ph/0103088].[28] R. Garc´ıa-Mart´ın, R. Kami´nski, J. R. Pel´aez andJ. Ruiz de Elvira, Phys. Rev. Lett. , 072001 (2011)[arXiv:1107.1635 [hep-ph]].[29] M. Jacob and G. C. Wick, Annals Phys. , 404 (1959)[Annals Phys. , 774 (2000)].[30] K. M. Watson, Phys. Rev. , 228 (1954).[31] F. Klingl, N. Kaiser and W. Weise, Z. Phys. A , 193(1996) [hep-ph/9607431].[32] M. Zanke, Bachelor’s thesis, University of Bonn, 2017.[33] F. Niecknig, B. Kubis and S. P. Schneider, Eur. Phys. J.C , 2014 (2012) [arXiv:1203.2501 [hep-ph]].[34] T. Maeda, K. Yamada, M. Oda and S. Ishida,Int. J. Mod. Phys. Conf. Ser. , 1460454 (2014)[arXiv:1310.7507 [hep-ph]].[35] B. Moussallam, Eur. Phys. J. C , 1814 (2011)[arXiv:1110.6074 [hep-ph]].[36] S. M. Roy, Phys. Lett. , 353 (1971).[37] B. Ananthanarayan, G. Colangelo, J. Gasserand H. Leutwyler, Phys. Rept. , 207 (2001)[hep-ph/0005297].[38] I. Caprini, G. Colangelo and H. Leutwyler, Eur. Phys. J.C , 1860 (2012) [arXiv:1111.7160 [hep-ph]]. [39] R. Garc´ıa-Mart´ın, R. Kami´nski, J. R. Pel´aez, J. Ruizde Elvira and F. J. Yndur´ain, Phys. Rev. D , 074004(2011) [arXiv:1102.2183 [hep-ph]].[40] J. Ruiz de Elvira, private communication.[41] J. F. de Troc´oniz and F. J. Yndur´ain, Phys. Rev. D ,093001 (2002) [hep-ph/0106025].[42] H. Leutwyler, hep-ph/0212324.[43] G. Colangelo, Nucl. Phys. Proc. Suppl. , 185 (2004)[hep-ph/0312017].[44] J. F. de Troc´oniz and F. J. Yndur´ain, Phys. Rev. D ,073008 (2005) [hep-ph/0402285].[45] C. Hanhart, Phys. Lett. B , 170 (2012)[arXiv:1203.6839 [hep-ph]].[46] B. Ananthanarayan, I. Caprini, D. Das and I. Sen-titemsu Imsong, Phys. Rev. D , 036007 (2014)[arXiv:1312.5849 [hep-ph]].[47] B. Ananthanarayan, I. Caprini, D. Das and I. Sen-titemsu Imsong, Phys. Rev. D , 116007 (2016)[arXiv:1605.00202 [hep-ph]].[48] M. Hoferichter, B. Kubis, J. Ruiz de Elvira, H.-W. Ham-mer and U.-G. Meißner, Eur. Phys. J. A , 331 (2016)[arXiv:1609.06722 [hep-ph]].[49] C. Hanhart, S. Holz, B. Kubis, A. Kup´s´c, A. Wirzbaand C. W. Xiao, Eur. Phys. J. C , 98 (2017)[arXiv:1611.09359 [hep-ph]].[50] M. N. Achasov et al. , J. Exp. Theor. Phys. ,380 (2006) [Zh. Eksp. Teor. Fiz. , 437 (2006)][hep-ex/0605013].[51] R. R. Akhmetshin et al. [CMD-2 Collaboration], Phys.Lett. B , 28 (2007) [hep-ex/0610021].[52] B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. , 231801 (2009) [arXiv:0908.3589 [hep-ex]].[53] F. Ambrosino et al. [KLOE Collaboration], Phys. Lett.B , 102 (2011) [arXiv:1006.5313 [hep-ex]].[54] D. Babusci et al. [KLOE Collaboration], Phys. Lett. B , 336 (2013) [arXiv:1212.4524 [hep-ex]].[55] M. Ablikim et al. [BESIII Collaboration], Phys. Lett. B , 629 (2016) [arXiv:1507.08188 [hep-ex]].[56] M. Fujikawa et al. [Belle Collaboration], Phys. Rev. D , 072006 (2008) [arXiv:0805.3773 [hep-ex]].[57] T. Jensen et al. , Phys. Rev. D , 26 (1983).[58] J. Huston et al. , Phys. Rev. D , 3199 (1986).[59] L. Capraro et al. , Nucl. Phys. B , 659 (1987).[60] C. Adolph et al. [COMPASS Collaboration], Eur. Phys.J. A , 79 (2014) [arXiv:1403.2644 [hep-ex]].[61] M. Froissart, Phys. Rev. , 1053 (1961).[62] S. P. Schneider, B. Kubis and F. Niecknig, Phys. Rev. D , 054013 (2012) [arXiv:1206.3098 [hep-ph]].[63] F. Von Hippel and C. Quigg, Phys. Rev. D , 624 (1972).[64] C. Adolph et al. [COMPASS Collaboration], Phys. Rev.D95