LPT-ORSAY/15-08 CPT-P001-2015 On some aspects of isospin breaking in the decay K± π0π0e± (ν−) e → V. Bernard ∗ Groupe de Physique Th´eorique, Institut de Physique Nucl´eaire Bˆat. 100, CNRS/IN2P3/Univ. Paris-Sud 11 (UMR 8608), 91405 Orsay Cedex, France S. Descotes-Genon † Laboratoire de Physique Th´eorique, CNRS/Univ. Paris-Sud 11 (UMR 8627), 91405 Orsay Cedex, France M. Knecht ‡ Centre de Physique Th´eorique, CNRS/Aix-Marseille Univ./Univ. du Sud Toulon-Var (UMR 7332) CNRS-Luminy Case 907, 13288 Marseille Cedex 9, France 5 (Dated: January 29, 2015) 1 0 Two aspects of isospin breaking in the decay K± → π0π0e± (ν−e) are studied and discussed. The 2 first addresses the possible influence of the phenomenological description of the unitarity cusp on n the extraction of the normalization of the form factor from data. Using the scalar form factor a of the pion as a theoretical laboratory, we find that this determination is robust under variations J of the phenomenological parameterizations of the form factor. The second aspect concerns the 8 issue of radiative corrections. We compute the radiative corrections to the total decay rate for 2 K± → π0π0e± (ν−e) in a setting that allows comparison with the way radiative corrections were ] handled in the channel K± → π+π−e± (ν−e). We find that once radiative corrections are included, h the normalizations of the form factor as determined experimentally from data in the two decay p channels come to a better agreement. The remaining discrepancy can easily be accounted for by - otherisospin-breaking corrections, mainly thoseduetothedifferencebetween themasses of theup p and down quarks. e h [ 1 v I. INTRODUCTION 2 0 The program of analysing K decays of the charged kaon conducted by the NA48/2 collaboration at 1 ℓ4 the CERN SPS has so far been very successful. In the π+π channel of the electron mode, ℓ = e [the decay 7 − .0 tKh±e c→orrπe+spπo−ned±in(gν−e)bwrainllchhienngcerfaotritohabnedrhefaedrreodntico faosrmKe+f4a−c]t,oirtsh[1a]s,lteod,abveesridyesacacumraotreedperteecrismeindaettieornmionfatthioenπoπf 1 0 S-wave scattering lengths a00 and a20 [2, 3], that constitutes a stringent test of the QCD prediction obtained 5 within the framework of chiral perturbation theory [4–7]. 1 Morerecently,the resultsconcerningananalysisofthe dataobtainedinthe π0π0 channelofthe electron v: mode [the decay K π0π0e (ν−) will henceforth be referred to as K00] have also become available [8]. Xi Although the numbe±r o→f events±is loewer [∼ 6.5·104 events in the Ke040 mode4e vs. ∼ 106 events for Ke+4−], this allows for some cross checks at the level of the structure of one of the form factors, that is identical for the two r a channels in the isospin limit. The normalisation of this common form factor, as measured in the two channels, reads [1, 8] |Vus|fs[Ke+4−] = 1.285±0.001stat±0.004syst±0.005ext, (1+δ )V f [K00] = 1.369 0.003 0.006 0.009 . (I.1) EM | us| s e4 ± stat± syst± ext Ignoringforthetimebeingthecorrectionfactorδ (wewilldiscussradiativecorrectionsbelow),thedifference EM of the two values, as compared to the value measured in the Ke+4− channel, amounts to 6.5% in relative terms. This might be considered as a small difference, but given the uncertainties, it is, in statistical terms, quite significant. Adding all errors in quadrature [38] this gives f [K00] (1+δ ) s e4 =1.065 0.010. (I.2) EM fs[Ke+4−] ± ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] ‡Electronicaddress: [email protected] 2 It seems difficult to ascribe a variation of 6.5% to the radiative correction factor δ alone. While in EM some regions of phase space radiative corrections can reach the 10% level, they usually sum up to 1% in the decay rate. The radiative corrections to the Ke+4− decay mode±have been discussed in several place±s [9–11] at one-loop precision in the low-energy expansion. But no comparable study has been done for the K00 decay e4 mode. There exists an older, less systematic, analysis [12] that covers the corrections due to virtual photon exchanges and real photon emission, which could provide the relevant contributions at a first stage, but its practical use is somewhat limited, since the expressions given there are not always very explicit, and moreover need to be checked. Furthermore, not all radiative corrections occurring in the charged Ke+4− channel [9–11] have been taken into account in the analysis of the experimental data. These additional radiative corrections could affect fs[Ke040] and fs[Ke+4−] in different ways, and make up for another part of the discrepancy. If δ alone does not explainthe discrepancy(I.2), one has to look for other sources of isospin-breaking EM effects. These can be due to the difference between the up and down quark masses m and m , conveniently u d described by the parameterR, with 1/R=(m m )/(m m ), where m is the mass ofthe strangequark, d u s ud s − − whereasm denotesthe averagemassofthe upanddown quarks,m =(m +m )/2. Forinstance,atlowest ud ud u d order in the chiral expansion, one has [9, 13] f [K00] 3 s e4 = 1+ . (I.3) fs[Ke+4−] (cid:18) 2R(cid:19) Barring contributions of higher-order corrections, values of R as small as [14] R = 35.8(1.9)(1.8) can account for about two thirds of the effect in Eq. (I.2). Finally,therearealsoisospinbreakingeffectsinducedbythemassdifferencebetweenchargedandneutral pions. Most notable from this point of view is the presence of a unitarity cusp [8] in the form factor describing the amplitudeoftheK00 mode. Theinterpretationofthiscuspisbynowwellunderstood,andasinthecaseof e4 the K π0π0π decay [15–19], it arises from the contribution of a π+π intermediate state in the unitarity ± ± − → sum [for a generaldiscussionof the properties of the K formfactors fromthe pointof view ofanalyticity and e4 unitarity, see Ref. [20] and references therein]. This cusp contains information on the combination a0 a2 that describes the amplitude for the process 0− 0 π+π π0π0 at threshold. Although this information probably cannot be extracted from the K00 data in a wa−y →as statistically significant as the determination from the cusp in K π0π0π [21], it is neev4ertheless ± ± → importanttoincludeacorrectdescriptionofthiscuspintheparameterisationoftheformfactorusedtoanalyse the data. This necessity has been demonstrated in full details in the case of the K π0π0π decay, and it ± ± → is to be expected that the same attention to these matters should be paid also in the analysis of the K00 data. e4 Failure to do so may introduce a systematic bias which would make the comparison with the information on the form factor extracted from the Ke+4− data spurious to some extent. It is the purpose of the present note to address some of these issues. In a first step, we investigate the possible influence that variousparameterisationsof the formfactors couldhave onthe outcome ofthe analysis. In order to control inputs and ouputs fully, we choose to work with a simplified model, where the exact form factors are known from a theoretical point of view, and where one can assess the effects of various choices of parameterisations for the form factors used in order to analyse the numerically generated data [that we will henceforth refer to as pseudo-data]. This frameworkis providedby the scalarform factorsof the pions, defined as π0(p )π0(p )m(uu+dd)(0)Ω = +Fπ0(s) [s (p +p )2], h 1 2 | | i S ≡ 1 2 π+(p )π (p )m(uu+dd)(0)Ω = Fπ(s) [s (p +p )2]. (I.4) h + − − | | i − S ≡ + − b Expressions of these form factors, with isospin-breaking contributions due to the difference of masses between b charged and neutral pions included, have been recently obtained in [22] up to and including two loops in the low-energyexpansion. Wewillusetheseexpressionsinordertogeneratepseudo-data,whichwecanthensubmit to analysis,usingvariousparameterisationsfor the formfactors,inspiredby those inuse forthe analysesofthe Ke+4− andKe040 experimentaldata. The reasonforworkingwiththe scalarformfactorsisatleasttwofold. First, the form factors, with isospin-breaking effects included, are known at two loops in both channels, whereas in the K case, only the form factors in the channel with two charged pions have been studied at the same level e4 of accuracy as far as isospin-breakingcorrectionsare concerned[20] [see Ref. [23] for a systematic study at one loop]. Second, the K form factors depend on two more kinematical variables, besides the di-pion invariant e4 mass. The scalar form factors depend only on the latter, and offer therefore a simple kinematical environment, so that the issues we wish to focus on can be addressed without unnecessary additional complications. In a second step, we address the issue of radiative corrections to the total decay rate of the decay K π0π0e (ν−). Our intent here is not to develop a full one-loop calculation, at the same level of precision ∓ ∓ e → as those that exist for the decay channel into two charged pions [9–11]. We rather aim at providing a simple estimate for the radiative corrections to the total decay rate, much in the spirit of Refs. [12] and [24] or, on 3 a more general level, of Ref. [25]. This will allow us to assess how much of the discrepancy (I.2) has to be ascribed to other isospin-breaking effects in the form factors, such as discussed above. Theremainderofthisstudyisthenorganisedinthefollowingway. First,wegive(SectionII)atheoretical discussionof the structure ofthe scalarformfactors of the pions using the explicit expressionsobtained in Ref. [22]. We will thus adapt the discussion of Ref. [8] to the case at hand. Working on this analogy will allow us to give an assessment of some additional assumptions regarding the structure of the form factors implicitly madeinRef. [8]. Next,wegeneratepseudo-data(SectionIII)usingtheknowntwo-loopexpressionsofthe form factors,thatwethenanalyzeusingvariousphenomenologicalparameterisations,thatdonotnecessarilycomply with the outcome of Section II. The purpose here is to discuss in a quantitative way the possible systematic biases that can be induced by these different choices. The last part of Section III addresses the determination of the combination a0 a2 of S-wave scattering lengths. Radiative corrections, aiming at an estimate of the 0− 0 correction factor δ in Eq. (I.1), are discussed in Section IV. We first compute radiative corrections to the EM Ke040 decayrateinasimilarset-uptotheoneusedforthetreatmentofthedataintheKe+4− channel,inorderto obtaina meaningfulcomparisonbetweenthe twochannels. Thenwe compute the effects ofadditionalphotonic corrections, not included in this treatment. Finally, we end our study with a summary and conclusions. Two Appendices contain technical details relevant for the discussions in Section II and in Section IV, respectively. II. DESCRIBING THE CUSP: THEORY According to the generalanalysis of Ref. [17], the occurence of both π0π0 and π+π intermediate states − at differentthresholds leadsto the following structurefor the scalarformfactorofthe neutralpionFπ0(s) [M S π stands for the charged-pionmass, whereas M is the mass of the neutral pion]: π0 π0(s) i π0(s) [s 4M2] e iδ(s)Fπ0(s) = F0 − F1 ≥ π , (II.1) − S   F0π0(s)+F1π0(s) [4Mπ20 ≤s≤4Mπ2] where π0(s)isafunctionofsthatissmoothaslongasnootherthreshold,correspondingtohigherintermediate F0 states, is reached. Here δ(s) represents a phase. It can be chosen arbitrarily, as long as it is also a smooth function of s. The cusp at s=4M2 observed in the differential decay rates corresponding to this simplified [as π compared to K ] situation then results from this decomposition, since e4 2 2 π0(s) + π0(s) 2Im π0(s) π0 (s) [s 4M2] 2 F0 F1 − F0 F1 ∗ ≥ π Fπ0(s) =  (cid:12) (cid:12) (cid:12) (cid:12) h i . (II.2) (cid:12) S (cid:12)  (cid:12)(cid:12) π0(s)(cid:12)(cid:12)2+(cid:12)(cid:12) π0(s)(cid:12)(cid:12)2+2Re π0(s) π0 (s) [4M2 s 4M2] (cid:12) (cid:12) F0 F1 F0 F1 ∗ π0 ≤ ≤ π (cid:12) (cid:12) Apart from the dependence(cid:12)(cid:12)with res(cid:12)(cid:12)pect(cid:12)(cid:12)to the(cid:12)(cid:12)second kihnematical variaible s , the empirical parameterisation (cid:12) (cid:12) (cid:12) (cid:12) e used for the fit to the K00 data, Eq. (9.1) in Ref. [8], complies with this general representation provided [the e4 variable S used in this reference corresponds to the variable s used here]: π 1) π0(s) is parameterised as a real constant times σˆ(s), with F1 4M2 q2 σˆ(s)= 1 π = , (II.3) s − s s 1+q2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) for 4M2 s 4M2, or q2 0, with s=4M2((cid:12)1+q2). (cid:12) (cid:12) (cid:12) π0 ≤ ≤ π ≤ π 2) π0(s) is set to zero (its value for s=4M2) for s 4M2 (q2 0) F1 π ≥ π ≥ 3) For 4M2 s 4M2 (q2 0), π0(s) is replaced by a constant, equal to π0(4M2). A more theoreticπa0lly≤bas≤edpaπramet≤erizatFio0n,adaptedfromthe simple discussionofFth0e cuspπin K π0π0π ± ± → given in Ref. [16], is considered in Sec. 9.4 of Ref. [8], though not used for the data analysis. As compared to Eq. (II.2), its validity also rests on additional assumptions, which, once transposed to the present situation, read: 1’) The phase δ(s) can be chosen such as to make the two functions π0(s) and π0(s) simultaneously F0 F1 real, so that Eq. (II.2) takes the simpler form π0(s)2+ π0(s)2 [s 4M2] Fπ0(s)2 = |F0 | |F1 | ≥ π . (II.4) | S |   [F0π0(s)+F1π0(s)]2 [4Mπ20 ≤s≤4Mπ2]  4 2’) π0(s) is related to the scalar form factor Fπ(s) of the charged pion, multiplied by a combination of the two S-Fw1ave ππ scattering lengths a0 and a2 in theSisospin limit, 0 0 2 π0(s)= a0 a2 π(s)σˆ(s). (II.5) F1 −3 0− 0 FS (cid:0) (cid:1) In view of the discussion in Ref. [8], π(s) should be identified with the phase-removed scalar form factor of the charged pion. The latter is giveFnSby e−iδ0π(s)FSπ(s), where the phase δ0π(s) is defined as FSπ(s+i0) = e2iδ0π(s)FSπ(s−i0). Our purpose in this Section is twofold. First, we will rewrite the two-loop representation of the form factor Fπ0(s) obtained in Ref. [22] in the form (II.2), that makes the cusp structure explicit. Second, we will S assessto whichextenttheadditionalfeaturesmentionedaboveandassumedinRef. [8]areactuallyreproduced by the structure of the form factors at two loops in the low-energy expansion. In particular, we will establish the precise relation between π(s) and Fπ(s) in Eq. (II.5) at this order. In what follows,and unless otherwise stated, it will always be undFeSrstood thatS s 4M2 . Furthermore, in practice s 4M2 will actually mean ≥ π0 ≥ π 4M2 s M2, where M is the mass of the chargedkaon, so that we need not worry about thresholds other π ≤ ≤ K K than those produced by two-pion intermediate states. A. The cusp in the one-loop form factor We start with the study of the cusp using the one-loop expression of the form factor Fπ0(s), S ϕ00(s) Fπ0(s)=Fπ0(0) 1+aπ0s+16π 0 J¯(s) 16πFπ(0)ϕx(s)J¯(s). (II.6) S S " S 2 0 # − S 0 In this expression,aπ0 denotes a subtractionconstant,thatwe neednotspecify further forthe time being. The S loop functions J¯(s) and J¯(s) are given by 0 J¯(s) = s ∞ dx 1 σ (x) 0 16π2 x x s i0 0 Z4Mπ20 − − J¯(s) = s ∞ dx 1 σ(x), (II.7) 16π2 x x s i0 Z4Mπ2 − − with 4M2 4M2 σ (s) = 1 π0 , σ(s) = 1 π. (II.8) 0 − s − s r r The functions ϕ00(s) and ϕx(s) denote the lowest-order real parts of the S-wave projections of the amplitudes 0 0 of the processes π0π0 π0π0 and π0π0 π+π , respectively. Their expressions read − → → s 4M2 ϕ00(s)=a , ϕx(s)=a +b − π, (II.9) 0 00 0 x x F2 π with [20, 22, 26] 2 ∆ 2 ∆ 1 F2 a = a0+2a2 1 π , a = a0 a2 +a2 π , b = 2a0 5a2 π , (II.10) 00 3 0 0 − M2 x −3 0− 0 0M2 x −12 0− 0 M2 (cid:18) π(cid:19) π π (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) and ∆ M2 M2 . πIn≡theπra−ngeπo0f s under consideration, the function J¯(s) is complex, but both its real and imaginary 0 parts are smooth, 1 1 σ (s) J¯(s)= [2+σ (s)L (s)+iπσ (s)], L (s) ln − 0 [s 4M2 ], (II.11) 0 16π2 0 0 0 0 ≡ 1+σ (s) ≥ π0 (cid:18) 0 (cid:19) whereas J¯(s) may be rewritten as iσˆ(s) [s 4M2] J¯(s) = J¯[0](s)+J¯[1](s) − ≥ π . (II.12) × +σˆ(s) [4M2 s 4M2]  π0 ≤ ≤ π  5 The two functions J¯[0](s) and J¯[1](s) are smooth, and read 1 1 J¯[0](s)= 2+σ(s)Lˆ(s) , J¯[1](s)= , (II.13) 16π2 −16π h i where 1 σ(s) Lˆ(s)=ln − [s 4M2 ]. (II.14) 1+σ(s) ≥ π0 (cid:18) (cid:19) In these expressions, the definition of σ(s) has been extended below s=4M2 by[39] π 4M2 1 π =σˆ(s) [s 4M2] − s ≥ π  r σ(s) = . (II.15)  i 4Mπ2 1=iσˆ(s) [4M2 s 4M2] s − π0 ≤ ≤ π According to Eqs. (II.1)and (II.2), J¯(s)rexhibits a cuspstructure ats=4Mπ2. One thus obtains, atthis order, the decomposition of the form (II.1) for Fπ0(s), with S ϕ00(s) ϕx(s) π0(s) = Fπ0(0) 1+aπ0s+ 0 [2+σ (s)L (s)] Fπ(0) 0 2+σ(s)Lˆ(s) + (E6), F0 S ( S 2π 0 0 )− S π O h i π0(s) = Fπ(0)ϕx(s)σˆ(s) + (E6), (II.16) F1 S 0 O provided one factorises the global phase 1 δ(s)= σ (s)ϕ00(s) + (E4). (II.17) 2 0 0 O Therefore, up to so far unspecified higher order corrections, the one-loop expression of the form factor can be brought into the form (II.1). Both functions π0(s) and π0(s)/σˆ(s) are real and smooth for s 4M2 at F0 F1 ≥ π0 this stage. However, the expression for π0(s) in (II.16) does not quite comply with Eq. (II.5). Whereas at F1 this stage π(s) is equal to the constant Fπ(0), which, at this order, can be identified with the phase-removed form factoFr,Sthe combination of scatteringSlengths that occurs in Eq. (II.5) corresponds to ϕox(4M2), where 0 π ϕox(s) is the expression of ϕx(s) in the isospin limit. Thus, at this order, the expression (II.5) misses both the 0 0 dependence with respect to s in ϕx(s), and the isospin-breaking corrections in the scattering lengths. 0 For laterreference, we briefly extend the discussionto the scalarformfactorof the chargedpion. At one loop, it is given by 1 Fπ(s)=Fπ(0) 1+aπs+16πϕ+−(s)J¯(s) 16πFπ0(0) ϕx(s)J¯(s). (II.18) S S " S 0 # − S 2 0 0 Besides the subtraction constant aπ, that differs from aπ0 (they become identical in the isospin limit), this S S expression involves the lowest-order real part of the S-wave projection of the amplitude for the scattering process π+π π+π , − − → s 4M2 ϕ+−(s)=a +b − π, (II.19) 0 +− +− F2 π where [20, 22, 26] 1 ∆ 1 F2 a = 2a0+a2 2a2 π, b = 2a0 5a2 π . (II.20) +− 3 0 0 − 0M2 +− 24 0− 0 M2 π π (cid:0) (cid:1) (cid:0) (cid:1) After having factorised the global phase 1 Fπ0(0) δ˜(s)= σ (s)ϕx(s) S + (E4), (II.21) −2 0 0 Fπ(0) O S one can decompose Fπ(s) according to Eq. (II.1), with S ϕ+−(s) ϕx(s) π(s) = Fπ(0) 1+aπs+ 0 2+σ(s)Lˆ(s) Fπ0(0) 0 [2+σ (s)L (s)] + (E6), F0 S S π − S 2π 0 0 O (cid:26) h i(cid:27) π(s) = Fπ(0)ϕ+−(s)σˆ(s) + (E6). (II.22) F1 − S 0 O Both functions π(s) and π(s)/σˆ(s) are real and smooth for s 4M2 at this stage. F0 F1 ≥ π0 6 B. The cusp in the two-loop form factor Fπ0(s) of the neutral pion S Let us now go through the same analysis, but with the two-loop expression of the form factor. The expressions of the pion scalar form factors at two loops and in presence of isospin breaking have been worked out in Ref. [22] using a recursive construction based on general properties like relativistic invariance, unitarity, analyticity, and chiral counting. The scalar form factor of the neutral pion can be written as[40] Fπ0(s) = Fπ0(0) 1+aπ0s+bπ0s2 S S S S (cid:16) (cid:17) 1 +8πFπ0(0)ϕ00(s) 1+aπ0s+ ϕ00(s) J¯(s) S 0 S π 0 0 (cid:20) (cid:21) 2 16πFπ(0)ϕx(s) 1+aπs+ ϕ+−(s) J¯(s) − S 0 S π 0 (cid:20) (cid:21) M4 + π Fπ0(0) ξ(0)(s)J¯(s)+ξ(1;0)(s)K¯0(s)+2ξ(2;0)(s)K¯0(s)+ξ(3;0)(s)K¯0(s) F4 S 00 0 00 1 00 2 00 3 π h +ξ(1;∇)(s)K¯∇(s)+ξ(3;∇)(s)K¯∇(s)+2ξ(2;±)(s) 16π2J¯(s) 2 J¯(s) 00 1 00 3 00 − 0 M4 (cid:2) (cid:3) i 2 π Fπ(0) ξ(0)(s)J¯(s)+2ξ(2;±)(s)K¯ (s)+ξ(1)(s)Kx(s)+ξ(3)(s)Kx(s) − F4 S x x 2 x 1 x 3 π h +2ξ(2;0)(s) 16π2J¯(s) 2 J¯(s)+∆ ξ (s)¯x(s) + (E8). (II.23) x 0 − 1 x K O (cid:2) (cid:3) i In this formula, the functions ξ(0)(s),...,ξ(0)(s),... are polynomials of at most second order in the variable s. 00 x Their expressions can be found in Ref. [22], except for ∆ ξ (s), that reads 1 x ∆ s s F2 M2 ∆ ξ (s)=8 π b b a π +2b 1+ π0 . (II.24) 1 x M2 +0M2 9M2 +0− +0M2 +0 M2 π π (cid:20) π π (cid:18) π (cid:19)(cid:21) It is also useful to be aware of the relations F4 F4 F4 F4 ξ(2;0)(s)=2 π [ϕ00(s)]2, ξ(2;0)(s)=2 π ϕ00(s)ϕx(s), ξ(2;±)(s)=4 π [ϕx(s)]2, ξ(2;±)(s)=4 π ϕx(s)ϕ+−(s). 00 M4 0 x M4 0 0 00 M4 0 x M4 0 π π π π (II.25) In order to achieve the decomposition (II.1), we need to extend the decomposition of the function J¯(s) in Eqs. (II.12) and (II.13) to the other functions, denoted generically by K¯α(s), that appear in the expression n (II.23). This may be done as follows. First, we may observe that, like J¯(s) or J¯(s), these functions can also 0 be defined by a dispersive representation of the form [for J¯(s) K¯0(s), one has k0(s) = σ (s)/16π, whereas for J¯(s) K¯ (s), k (s)=σ(s)/16π, see Eq. (II.7)] 0 ≡ 0 0 0 0 0 ≡ K¯α(s)= s ∞ dx 1 kα(x). (II.26) n π x x s i0 n Zsthr − − Explicitexpressionsforthefunctions kα(s)aregiveninAppendix A.Forthe setoffunctionsK0(s)andK¯∇(s), n n n one has s = 4M2 . These functions will therefore each develop an imaginary part for s s = 4M , ImK¯α(s) t=hrkα(s)θ(πs0 4M2 ), while the real part displays a cusp at s = 4M , but is smoot≥h fotrhrs 4Mπ0. The snituationnis differ−ent foπr0the remaining functions, K¯ (s), K¯x(s), and ¯x(sπ),0for which s =4M2,≥so thπa0t, n n K thr π in a generic way, they have the following structure ikα(s) [s 4M2] K¯α(s) = ReK¯α(s)+ n ≥ π n n  0 [4M2 s 4M2]  π0 ≤ ≤ π iσˆ(s) [s 4M2] = ReK¯α(s)+knα(s) ≥ π . (II.27) n σ(s) × 0 [4M2 s 4M2]  π0 ≤ ≤ π In general, the function kα(s)/σ(s), although real, is not smooth for the whole range s 4M2 , but only for n ≥ π0 s 4M2. Suppose one can find a function kˆα(s) such that it coincides with kα(s) for s 4M2, and such that ≥ π n n ≥ π kˆα(s)/σ(s) is real and smooth for all s 4M2 . Then one can perform the decomposition n ≥ π0 iσˆ(s) [s 4M2] K¯(s) = K¯[0](s)+K¯[1](s) − ≥ π , (II.28) × +σˆ(s) [4M2 s 4M2]  π0 ≤ ≤ π  7 in terms of two real and smooth functions K¯[0](s) and K¯[1](s), given by ReK¯α(s) [s 4M2] K¯α[0](s) = n ≥ π , K¯α[1](s) = kˆnα(s). (II.29) n ReK¯α(s) ikˆα(s) [4M2 s 4M2] n − σ(s)  n − n π0 ≤ ≤ π Such a decomposition canindeed be achieved for the various functions considered here, as discussed in detail in App. A. The decomposition (II.1) of the form factor now follows immediately, with eiδ(s) π0(s) = Fπ0(0) 1+aπ0s+bπ0s2 F0 S S S (cid:16) (cid:17) 1 +8πFπ0(0)ϕ00(s) 1+aπ0s+ ϕ00(s) J¯(s) S 0 S π 0 0 (cid:20) (cid:21) 2 16πFπ(0)ϕx(s) 1+aπs+ ϕ+−(s) J¯[0](s) − S 0 S π 0 (cid:20) (cid:21) M4 + π Fπ0(0) ξ(0)(s)J¯(s)+ξ(1;0)(s)K¯0(s)+ξ(1;∇)(s)K¯∇(s)+2ξ(2;0)(s)K¯0(s) F4 S 00 0 00 1 00 1 00 2 π h +ξ(3;0)(s)K¯0(s)+ξ(3;∇)(s)K¯∇(s)+2ξ(2;±)(s) 16π2J¯[0](s) 2 J¯(s) 00 3 00 3 00 − 0 M4 h i i 2 π Fπ(0) ξ(0)(s)J¯[0](s)+ξ(1)(s)K¯x[0](s)+2ξ(2;±)(s)K¯[0](s)+ξ(3)(s)K¯x[0](s) − F4 S x x 1 x 2 x 3 π h +2ξ(2;0)(s) 16π2J¯(s) 2 J¯[0](s)+∆ ξ (s)¯x[0](s) + (E8), (II.30) x 0 − 1 x K O i (cid:2) (cid:3) and 2 eiδ(s) π0(s) = σˆ(s) 16πFπ(0)ϕx(s) 1+aπs+ ϕ+−(s) J¯[1](s) F1 − S 0 S π 0 (cid:26) (cid:20) (cid:21) M4 2 π Fπ0(0)ξ(2;±)(s) 16π2J¯(s)J¯[1](s) − F4 S 00 × 0 π M4 +2 π Fπ(0) ξ(0)(s)J¯[1](s)+ξ(1)(s)K¯x[1](s)+2ξ(2;±)(s)K¯[1](s)+ξ(3)(s)K¯x[1](s) F4 S x x 1 x 2 x 3 π h +2ξ(2;0)(s) 16π2J¯(s) 2 J¯[1](s)+∆ ξ (s)¯x[1](s) + (E8). (II.31) x 0 − 1 x K O i(cid:27) (cid:2) (cid:3) Bothfunctionseiδ(s) π0(s)andeiδ(s) π0(s)/σˆ(s)aresmoothfors 4M2 ,butcomplex. Itremainsto discuss F0 F1 ≥ π0 the phase δ(s). If we want to make the function π0(s) real,while keeping it smooth, then its choice is unique, F0 1 δ(s) σ (s) ϕ00(s)+ψˆ00(s) , (II.32) ≡ 2 0 0 0 h i with ψˆ00(s) given by 0 1 M4 σ (s)ψˆ00(s) = π ξ(0)(s)k0(s)+ξ(1;0)(s)k0(s)+ξ(1;∇)(s)k∇(s)+ξ(2;0)(s)k0(s) 2 0 0 F4 00 0 00 1 00 1 00 2 π (cid:20) 1 +ξ0(30;0)(s)k30(s)+ξ0(30;∇)(s)k3∇(s)+ξ0(20;±)(s)8πσ0(s)σ(s)Lˆ(s) . (II.33) (cid:21) Notethatψˆ00(s)differsfromthequantityψ00(s)definedinEq. (4.6)ofRef. [22]bythepresenceofthefunction 0 0 Lˆ(s) insteadof L(s) inthe last termbetween squarebrackets,see Eq. (A.2). This makes σ (s)ψˆ00(s) a smooth 0 0 function for s 4M2 , whereas σ(s)L(s), and hence ψ00(s), displays a cusp at s = 4M2. Making use of Eq. ≥ π0 0 π (II.27), the removal of the phase δ(s) indeed leads to a real and smooth expression for the function π0(s): F0 π0(s) = Fπ0(0) 1+aπ0s+bπ0s2 F0 S S S +8πF(cid:16)π0(0)ϕ00(s) 1+a(cid:17)π0s ReJ¯(s) S 0 S 0 h i 2 16πFπ(0)ϕx(s) 1+aπs+ ϕ+−(s) J¯[0](s) − S 0 S π 0 (cid:20) (cid:21) 8 M4 + π Fπ0(0) ξ(0)(s)ReJ¯(s)+ξ(1;0)(s)ReK¯0(s)+ξ(1;∇)(s)ReK¯∇(s)+ F4 S 00 0 00 1 00 1 π h +ξ(2;0)(s)ReK¯0(s)+ξ(3;0)(s)ReK¯0(s)+ξ(3;∇)(s)ReK¯∇(s) 00 2 00 3 00 3 M4 i 2 π Fπ(0) ξ(0)(s)J¯[0](s)+ξ(1)(s)K¯x[0](s)+2ξ(2;±)(s)K¯[0](s) − F4 S x x 1 x 2 π h +ξ(3)(s)K¯x[0](s)+∆ ξ (s)¯x[0](s) x 3 1 x K +8Fπ0(0)[ϕx(s)]2 16π2J¯[0](s) 2 ReJ¯(s) i S 0 − 0 8Fπ(0)ϕx(s)ϕ00((cid:16)s) 16π2ReJ¯(s)(cid:17) 2 J¯[0](s) − S 0 0 0 − +Fπ0(0) ϕ00(s) 2 2R(cid:2)eK¯0(s)+8ReJ¯(cid:3)(s)+ 1 1 4Mπ20 + (E8). (II.34) S 0 2 0 8 − s O (cid:20) (cid:18) (cid:19)(cid:21) (cid:2) (cid:3) As far as π0(s) is concerned, we may even proceed in a more direct way by noticing that, up to higher order F1 corrections, Eq. (II.31) rewrites as eiδ(s)F1π0(s) = e2iσ0(s)ϕ000(s)σˆ(s) ϕx0(s)+ψˆ0x(s) h i Fπ0(0) Fπ(0) 1+aπs+16πϕ+−(s)J¯[0](s) 8π S ϕx(s)J¯(s) + (E8), (II.35) × S S 0 − Fπ(0) 0 0 O (cid:26) S (cid:27) with [for the notation, see Appendix A] M4 1 ψˆx(s) = 2 π ξ(0)(s)k (s)+ξ(2;±)(s)kˆ (s)+ξ(1)(s)kx(s)+ξ(3)(s)kx(s) 0 F4 σ(s) x 0 x 2 x 1 x 3 π ( +ξ(2;0)(s)kx(s)+∆ ξ (s)kx(s) . (II.36) x 2 1 x ) Now, the phase that appears factored out on the right-hand side of this equation can be identified with the phase δ(s) onthe left-hand side, since the difference generates contributionsof order (E8), that areneglected O anyway. Taking into account Eqs. (II.21) and (II.22), one finally obtains π0(s)=σˆ(s) ϕx(s)+ψˆx(s) (s) + (E8), (II.37) F1 0 0 Fπ O h i with (s) eiδ˜(s) π(s) + (E6). (II.38) Fπ ≡ F0 O It is possible to give a more precise interpretation of the combination ϕx(s)+ψˆx(s) that occurs in (II.37). To 0 0 this end, let us recall from Ref. [22] that the ℓ = 0 partial-wave projection fx(s) for the scattering amplitude 0 of the process π0π0 π+π is given, at order one loop and for s 4M2 , by → − ≥ π0 1 fx(s)=ϕx(s)+ψx(s)+iϕx(s) σ (s)ϕ00(s)+σ(s)ϕ+−(s)θ(s 4M2) + (E6), (II.39) 0 0 0 0 2 0 0 0 − π O (cid:20) (cid:21) where ψx(s) is defined inEq. (4.15)ofRef. [22][the contribution∆ ψx(s), ofsecondorderin isospinbreaking, 0 2 0 is numerically quite small, and is omitted for simplicity]. It differs from ψˆx(s) by the replacement of kˆ (s) by 0 2 k (s) in Eq. (II.36). Then applying the decomposition (II.28) to fx(s), one finds 2 0 iσˆ(s) [s 4M2] e iδ(s)fx(s) = fx[0](s)+fx[1](s) − ≥ π , (II.40) − 0 0 0 × +σˆ(s) [4M2 s 4M2]  π0 ≤ ≤ π with fx[0](s)=ϕx(s)+ψˆx(s)+ (E6), and fx[1](s)= ϕx(s)ϕ+−(s)+ (E6). 0 0 0 O 0 − 0 0 O We may summarise this theoretical study of the cusp in the scalar form factor of the neutral pion with a couple of remarks: Itis,ingeneral,notpossibletochosethephaseδ(s)inEq. (II.1)suchastomakeboth π0(s)and π0(s) • F0 F1 real simultaneously. A relative phase remains, see Eqs. (II.37) and (II.38). At lowest order, this phase is given by the S-wave projection of the inelastic rescattering of a pair of neutral pions through a pair of charged pions, cf. Eq. (II.21). 9 The structure of π0(s)is morecomplicatedthanjust the productofthe scatteringlengthcorresponding • F1 tothisrescatteringamplitudetimesthephaseremovedscalarformfactorofthechargedpion. Attheorder we have been working, it involves the decomposition (II.40) of the S-wave projection of this amplitude times the part π(s) of the decomposition (II.1) of Fπ(s). This is different fromthe phase-removedform F0 S factor, as already seen at order one loop: 0 [s 4M2] e−iδ0π(s)FSπ(s)−F0π(s)=F1π(s)×1 [4M≥2 πs 4M2] + O(E6). (II.41)  π0 ≤ ≤ π Note howeverthatthis difference onlyconcernstheregion4M2 s 4M2,whichcontributesverylittle π0 ≤ ≤ π to the total decay rate as defined by Eqs. (III.1) and (III.2) below. C. Description of the two-loop form factor Fπ(s) of the charged pion S We now briefly address the scalar form factor of the charged pion. The issue here is not to describe the cusp, that occurs below the physical threshold at s=4M2, but to provide the expressions that will be used in π the sequel. Again, we will rely on the results obtained in Ref. [22], and rewrite the form factor Fπ(s) at two S loops in a way that is adapted to our purposes. In particular, we will consider the phase-removed form factor, which reads, in the relevant domain s 4M2, ≥ π e−iδ0π(s)FSπ(s) = FSπ(0) 1+aπSs+bπSs2 1 8πFπ(cid:0)0(0)ϕx(s) 1+a(cid:1)π0s+ ϕ00(s) ReJ¯(s) − S 0 S π 0 0 (cid:20) (cid:21) 2 +16πFπ(0)ϕ+−(s) 1+aπs+ ϕ+−(s) ReJ¯(s) S 0 S π 0 (cid:20) (cid:21) M4 π Fπ0(0) ξ(0)(s)ReJ¯(s)+ξ(1)(s)ReK¯x0(s)+2ξ(2;0)(s)ReK¯0(s)+ξ(3)(s)ReK¯x0(s) − F4 S x 0 x 1 x 2 x 3 π n +∆ ξ (s)Re ¯x0(s)+2ξ(2;±)(s) 16π2ReJ¯(s) 2 ReJ¯(s) 1 x K x − 0 M4 (cid:2) (cid:3) o 1 4M2 +2 π Fπ(0) ξ(0) (s)ReJ¯(s)+ξ(1;±)(s)ReK¯ (s)+2ξ(2;±)(s) ReK¯ (s)+ 1 π F4 S +−;S +−;S 1 +−;S 2 32 − s π (cid:26) (cid:20) (cid:18) (cid:19)(cid:21) +ξ(3;±)(s)ReK¯ (s)+ξ(1;∆)(s)ReK¯∆(s)+ξ(3;∆)(s)ReK¯∆(s)+ +−;S 3 +−;S 1 +−;S 3 1 4M2 +2ξ(2;0)(s) 16π2ReJ¯(s) 2 ReJ¯(s)+ 1 π0 + (E8). (II.42) +−;S 0 − 64 − s O (cid:20) (cid:18) (cid:19)(cid:21)(cid:27) (cid:0) (cid:1) The functionsK¯α(s)thatappearinthis expressionhaveagainadispersiverepresentationofthe formdisplayed n in Eq. (II.26). The absorptive parts are in part given in Appendix A. For the remaining one, we have σ (s) σ (s) kx0(s)= 0 kx(s), kx0(s)= 0 kx(s), (II.43) n σ(s) n σ(s) for the functions corresponding to s =4M2 , and, for those whose dispersive integrals start at s =4M2, thr π0 thr π 1 σ(s) 3 M2 k∆(s) = L (s 4∆ ), k∆(s) = π0 L2(s 4∆ ), (II.44) 1 8π σ (s 4∆ ) 0 − π 3 16π sσ(s) 0 − π 0 π − withthedefinitionsofthefunctionsσ(s), σ (s)andL (s)fors 4M2 giveninEqs. (II.8),(II.11),and(II.15). 0 0 ≥ π0 III. GENERATION AND ANALYSIS OF THE PSEUDO-DATA In order to study the effect that particular choices of phenomenological parameterisations of the form factorscanhaveontheoutput,wewillfirstgeneratenumericaldatasetsforthescalarformfactorsoftheneutral and charged pions. The pseudo-data in question consist of the (unnormalized) decay distribution defined by d2Γπ0(s,s ) 1 4M2 ℓ 1 π0 Fπ0 (s)2λ3/2(M2,s,s ), (III.1) dsds ≡ 2 − s | S;data | K ℓ ℓ r 10 withλ(x,y,z)=x2+y2+z2 2xy 2xz 2yz. Thetotaldecayrateisobtainedbyintegratingthedistributions − − − (III.1), convolutedwiththe K phase space,overthe whole physicalrange[we now considerthe electronmode ℓ4 only, and set m =0]: e Γπ0 = MK2 ds (MK2−√sd)s2 d2Γπ0(s,se). (III.2) e N dsds Z4Mπ20 Z0 e Since we consider the scalar form factor instead of K form factors, the integrationwith respect to s involves ℓ4 ℓ the phase space only. The overallnormalisation factor has been chosen to be the one of the K decay, e4 G2 V 2 1 = F| us| , (III.3) N 3 212π5M5 Fπ(0)2 · K | S | up to the factor 1/Fπ(0)2, introduced for convenience. | S | A. Form factors and input parameters used for the generation of pseudo-data The form factors involved in the preceding expressions are considered as known exactly and are con- structed as follows. For Fπ0 (s), we will basically use the decomposition of Eq. (II.1), with π0(s) given by S;data F0 Eq. (II.34), and π0(s) given by Eqs. (II.37) and (II.38). Unfortunately, for some of the functions involved in Eq. (II.34),likeRFe1K¯∇(s)orReK¯x(s),explicitanalyticalexpressionsarenotknown. Foranumericalapproach, n n wecouldusetheirdispersiverepresentation,asgivenbyEq. (II.26). Wehavehoweverfounditmoreconvenient to start from expressions upon which we have full analytical control. For that purpose, one may replace the functionsReK¯∇(s)bythecorrespondingfunctionsReK¯0(s),andlikewiseReK¯x(s)byReK¯ (s). Wealsodrop n n n n thecontributionproportionalto∆ ξ (s). Intherangeofsweareinterestedin,4M2 s M2,thedifference 1 x π0 ≤ ≤ K induced in π0(s) by these changes is numerically very small. For the scalar form factor of the neutral pion, F0 the resulting expression then reads π0 (s) i π0 (s) [s 4M2] Fπ0 (s) = F0;data − F1;data ≥ π , (III.4) S;data   F0π;0data(s)+F1π;0data(s) [4Mπ20 ≤s≤4Mπ2] with  π0 (s) = Fπ0(0) 1+aπ0s+bπ0s2 F0;data S S S +8πF(cid:16)π0(0)ϕ00(s) 1+a(cid:17)π0s ReJ¯(s) S 0 S 0 h i 2 16πFπ(0)ϕx(s) 1+aπs+ ϕ+−(s) J¯[0](s) − S 0 S π 0 (cid:20) (cid:21) M4 + π Fπ0(0) ξ(0)(s)ReJ¯(s)+ ξ(1;0)(s)+ξ(1;∇)(s) ReK¯0(s)+ F4 S 00 0 00 00 1 π n h i +ξ(2;0)(s)ReK¯0(s)+ ξ(3;0)(s)+ξ(3;∇)(s) ReK¯0(s) 00 2 00 00 3 M4 h i o 2 π Fπ(0) ξ(0)(s)J¯[0](s)+ξ(1)(s)K¯[0](s)+2ξ(2;±)(s)K¯[0](s)+ξ(3)(s)K¯[0](s) − F4 S x x 1 x 2 x 3 π h i +8Fπ0(0)[ϕx(s)]2 16π2J¯[0](s) 2 ReJ¯(s) S 0 − 0 8Fπ(0)ϕx(s)ϕ00((cid:16)s) 16π2ReJ¯(s)(cid:17) 2 J¯[0](s) − S 0 0 0 − +Fπ0(0) ϕ00(s) 2 2R(cid:2)eK¯0(s)+8ReJ¯(cid:3)(s)+ 1 1 4Mπ20 , (III.5) S 0 2 0 8 − s (cid:20) (cid:18) (cid:19)(cid:21) (cid:2) (cid:3) and F1π;0data(s) = e−2iσ0(s)ϕx0(s)FFSπSπ0((00))σˆ(s)FSπ(0) ϕ+−(s) Fπ0(0)ϕx(s) 1+aπs+ 0 2+σ(s)Lˆ(s) S 0 [2+σ (s)L (s)] (III.6) × S π − Fπ(0) 2π 0 0 (cid:26) h i S (cid:27) M4 1 Lˆ(s) M2 ϕx(s)+ π ξ(0)(s)+2ξ(1)(s) +2ξ(2;±)(s)σ(s)Lˆ(s)+3ξ(3)(s) π Lˆ2(s) . ×( 0 Fπ4 8π " x x σ(s) x x s−4Mπ2 #)