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LUTP13-02 January2013 h p 3 and quark masses 3 1 → 0 2 n a J 0 3 Stefan Lanz∗ ] AlbertEinsteinCenterforFundamentalPhysics,InstituteforTheoreticalPhysics, h UniversityofBern,Sidlerstrasse5,CH-3012Bern,Switzerland p - DepartmentofAstronomyandTheoreticalPhysics,LundUniversity, p e Sölvegatan14A,S-22362Lund,Sweden h E-mail:[email protected] [ 1 v In recent years, the decay h 3p has received considerable attention from experimental and 2 → theoreticalside. Itis ofparticulartheoreticalinterest becauseit allowsforthe determinationof 8 2 (m m ). In addition, formany yearsnow theory has had difficultiesto understandthe slope u d 7 − oftheneutralchannelDalitzplotdistribution,whichincontrastisverywellmeasured. Idiscuss . 1 heretherelationofthedecaytothemassesofthelightquarksandreviewanumberoftheoretical 0 3 andexperimentalworksthatareconcernedwiththesequestions. 1 : v i X r a Tobepublishedintheproceedingsof: The7thInternationalWorkshoponChiralDynamics, August6-10,2012 JeffersonLab,NewportNews,Virginia,USA Speaker. ∗ (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ h 3p andquarkmasses StefanLanz → 1. Introduction The masses of the light quarks, i.e., the up-, down-, and strange-quark, are not directly ac- cessible toexperimental determination duetoconfinement, whichprevents quarks fromappearing as free particles. Since these masses are much smaller than the typical scale of QCD at around 1 GeV, their contribution to hadronic quantities as, e.g., the nucleon mass is rather small. There is, however, a prominent exception to this rule, formulated in the famous Gell-Mann–Oakes–Renner relation[1]. Itstatesthatthemassesofthelightestmesonsaredeterminedbythecombinedeffects of spontaneous and explicit chiral symmetry breaking, that is by the chiral quark condensate and the light quark masses. At leading order in the quark mass expansion and including h -p 0 mixing andfirstorderelectromagnetic corrections, onefinds 2e m2 =B (m +m )+ B (m m )+..., m2 =B (m +m )+D p +..., p 0 0 u d √3 0 u− d p + 0 u d em m2 =B (m +m )+..., m2 =B (m +m )+D K +..., (1.1) K0 0 d s K+ 0 u s em m +m +4m 2e mh2 =B0 u d s B0(mu md)+.... 3 −√3 − The parameter e = √3/4(m m )/(m mˆ) 0.015, mˆ = (m +m )/2, is the h -p 0 mixing d u s u d − − ≈ angle. According to Dashen’s theorem [2], the electromagnetic corrections to the pion and kaon masscoincideatleadingorderwithD p /K (35MeV)2. BecauseB isnotaprioriknown,onecan em 0 ∼ only extract ratios of quark masses from Eq. (1.1). Assuming Dashen’s theorem to be true, one findsthefamousrelations firstderivedbyWeinbergusingcurrentalgebra[3]: m m2 m2 +m2 m m2 +m2 m2 d K0− K+ p + 1.79, s K+ K0− p + 20.2. (1.2) m ≈ m2 m2 m2 +2m2 ≈ m ≈ m2 m2 +m2 ≈ u K+− K0− p + p 0 d K0− K+ p + Accesstom m ismadedifficultbythefactthatthissmallquantityentersEq.(1.1)onlyquadrat- u d − icallyandbythepossibility ofDashenviolating contributions tothecharged kaonmass. Lattice calculations are able to relate the quark masses to measurable meson masses, thus leading to reliable predictions for m and mˆ. I will not discuss these methods further and instead s refertothemanycontributions inthese proceedings (e.g.[4–6])aswellastotheextensive report byFLAG[7]. Thedetermination of(m m )ishoweverdifficultalsoinLatticecalculations, due u d − tothe reasons discussed above: itenters quadratically orishidden behind sizable electromagnetic corrections, whichareonlybeginning tobestudiedontheLattice[8,9]. 2. h 3p → The main focus of this article is the decay process h 3p . It can appear in two variants: → either the decay goes into three neutral pions, h 3p 0, or into a pair of charged pions together → with a neutral one, h p +p p 0. I will denote the amplitude by A (s,t,u) for the neutral and − n → by Ac(s,t,u) for the charged channel. The Mandelstam variables are defined as s=(pp ++pp ), − t =(pp 0+pp ), and u=(pp 0+pp +) in the charged channel, where they satisfy the relation s+ − t+u=mh2 +2mp2++mp20. Duetochargeconjugation symmetry,theamplitudeissymmetricunder 2 h 3p andquarkmasses StefanLanz → t u. Theadaptation ofthesedefinitions totheneutralchannelisobvious. Theneutral amplitude ↔ iseventotallysymmetricins,t,andu. The particular importance of the decay h 3p for the determination of (m m ) is due u d → − to the fact that it is forbidden by isospin symmetry. The reason is that three pions can not at the sametimecoupletovanishingangularmomentumandzeroisospin. TheonlyoperatorintheQCD Lagrangian thatcanproduce suchatransition is m m L = u− d(u¯u d¯d). (2.1) IB − 2 − As a consequence of being generated by this D I =1 operator, the decay amplitude must be pro- portional to (m m ) and can be used to extract this quantity. The decay width can be seen as a u d − measureforthesizeofisospin breakinginQCD. Duetothedifferentelectricchargesoftheup-andthedown-quark,alsotheelectromagneticin- teractionisisospinviolatingandcancontributetothedecaywidthofh 3p . Thesecontributions → have been predicted to be small [10, 11], which has been confirmed by explicit one-loop calculations within Chiral Perturbation Theory (ChPT) [12, 13]. Let me stress, however, that Ref. [13] indicates thattheymightnotbeentirely negligible. Still,h 3p providesarathercleanaccessto → isospin breaking withinQCDandhenceto(m m ). u d − InChPT,quarkmassesarealwaysmultipliedbyB . Toavoidtheappearanceofthisparameter, 0 itisconvenienttorewritetheprefactortotheamplitudeintermsofquarkmassratios. Twodifferent conventions areinuse: Ah →3p (cid:181) B0(mu−md)=−Q12m2K(mm2K2p−mp2)+O(M3)=−R1(m2K−m2p )+O(M2), (2.2) with m2 mˆ2 m mˆ Q2= s− , R= s− . (2.3) m2 m2 m m d− u d− u Note that in Eq. (2.2) the term containing Q is accurate up to O(M3), while the other one is corrected already at O(M2). Fromthesedefinitions, onefindsthatthetworatiosarerelated by m 1 R=2Q2 1+ s − . (2.4) mˆ (cid:16) (cid:17) The ratio m /mˆ can be determined on the lattice. The current lattice average from FLAG is s m /mˆ =27.4 0.4 [7]. Since the decay width is basically given by the phase space integral over s ± theamplitude squared, 1 1 G h →3p (cid:181) Z |Ah →3p (s,t,u)|2 (cid:181) Q4 (cid:181) R2, (2.5) an accurate theoretical description of the decay amplitude can be used to extract either Q or R by comparison with an experimental value for G h 3p . Note that this also means that finding a value → forQorRisequivalent tofindingthecorrectnormalisation ofthedecayamplitude. Of course the aforementioned procedure can be reversed: given a reliable value for Q (or R) thedecaywidthcanbepredicted. Thetheoreticalchallenge indoingthisstaysthesame: oneneeds 3 h 3p andquarkmasses StefanLanz → anaccuratedescription ofthedecayamplitude. Theobvious choicetotreattheprocessinquestion is of course ChPT. But this is not entirely successful, as one sees immediately by comparing the tree-level (or current algebra) [14, 15], one-loop [16] and two-loop prediction [17] for the decay width1 withthecurrentPDGvalue, G h p +p p 0 =296 16eV[18]: → − ± G h p +p p 0 =66eV+94eV+138eV+...=298eV+.... (2.6) → − Even though the result at two-loops happens to coincide almost perfectly with experiment, the chiral series does not exhibit good convergence behaviour. Since the decay width is based on the square of the amplitude, Eq. (2.6) is somewhat misleading: the numbers are enhanced by interference such that, e.g., the two-loop number also contains contributions that are of O(p8) andO(p10). Theamplitudeisconverging morequickly. IthasbeenshownindependentlyofChPTthatthewidthisenhancedbylargefinalstaterescat- teringeffects[19]. Thisismirroredinthechiralseriesandisamotivationtotreattheprocess with dispersive methods, whichallowtosumupthecontributions fromfinalstaterescattering. Furthermore, the theoretical understanding of the energy distribution in the neutral channel is not complete. Due to symmetry reasons, the square of the amplitude can in this situation be parametrised by a single parameter a , which has been measured by many experiments with good mutualagreement. Whileexperimentsallfindthisparameter tobenegative,ChPTatone-andtwo- looporderpredictsapositivevalue. Alsoothercalculationshavefailedtoreproducethisparameter satisfactorily ascanbeseeninthecompilation ofresultsinFig.4. But difficulties appear also on the experimental side. There are hints of a tension between the manymeasurements of the neutral channel and the only available high-statistics measurement of the charged channel by KLOE.Such a statement can be made because the charged and neutral channel decayamplitudesarerelatedby A (s,t,u)=A (s,t,u)+A (t,u,s)+A (u,s,t), (2.7) n c c c if only first order isospin breaking, i.e. D I = 1, is considered. This relation allows for certain consistency checksamongexperiments, whichwillbediscussed inmoredetaillater. Toconclude this section, I want to discuss an important property of the decay amplitude, the so-called Adler zero [20]. This soft-pion theorem states that in the SU(2) chiral limit, the decay amplitude hastwozerosat pp + 0 s=u=0, t =mh2 , and pp 0 s=t =0, u=mh2 . (2.8) → ⇔ − → ⇔ Movingawayfromthechirallimit(butkeeping mp + =mp 0 mp ),thepositionsoftheAdlerzeros ≡ areshiftedbyacontribution oftheorderof m2 to p 4 m2 4 m2 s=u= m2p , t =m2h + p , and s=t = m2p , u=m2h + p . (2.9) 3 3 3 3 1Thetree-levelandone-loopvalueshavebeentakenfrom[16],whereQ=24.15isused. Thisvaluefollowsfrom Dashen’stheoremwiththePDGmesonmassesfromthetimethearticlewaspublished.Notwo-loopresultforthedecay width is quoted in [17] since there, R has been calculated from the experimental value for G . But fromtheir results togetherwithQ=24.15andms/mˆ =27.4,onefindsG =298eV. 4 h 3p andquarkmasses StefanLanz → d2N 10000 dXdY 10000 5000 1 0 1 0.6 1 0.6 0.2 0.2 -0.2 Y -0.2 -0.6 -1-1 -0.6 X 0 Figure1: TheDalitzplotdistributionmeasuredbytheKLOEcollaboration(figurefrom[21]). As one expects from a SU(2) soft-pion theorem, the corrections to the position of the Adler zero areoforder m2p ,sincethesymmetryforbidslarge O(ms)contributions. Atone-looporder, thereal part has Adler zeros at s=u=1.35m2p and s=t =1.35m2p , where also the imaginary part of the amplitude issmall. 3. Dalitzplotmeasurements Themomentumdistribution ofathree-particle decayistypicallydisplayedinformofaDalitz plot, where it is plotted against two independent kinematic variables. A common choice are the so-called Dalitzplotvariables, whichinthecharged channelaredefinedby √3 3 X = (u t), Y = (mh mp 0)2 s 1, (3.1) 2mh Qc − 2mh Qc − − − (cid:0) (cid:1) with Qc =mh 2mp + mp 0. In the neutral channel it is for symmetry reasons convenient to use − − the variable Z =X2+Y2. The physical region of the decay process lies for both channels inside thecirclewithX2+Y2=1butdoesnotentirelycoverit. Thepoint X =Y =0isreferredtoasthe centreoftheDalitzplot. Anexampleofathree-dimensional DalitzplotisshowninFig.1. It is common to parametrise the Dalitz plot distribution as a polynomial in X andY. For the charged channel anduptocubicorder, theDalitzplotparametrisation reads G (X,Y)= A (s,t,u) (cid:181) 1+aY+bY2+cX+dX2+eXY+ fY3+gX3+hX2Y+lXY2, (3.2) c c | | where the coefficients a,b,... are called Dalitz plot parameters. Charge conjugation symmetry requires termsoddinXtovanishsuchthat c=e=g=l=0. Several experiments have measured at least some of the Dalitz plot parameters in Eq. (3.2) [22–24,21],butonlythemeasurement byKLOEhascollected enoughstatistics tobereliable for more than one or two of them. Their result for the Dalitz plot distribution from about 1.3 106 · h p +p p 0 events is shown in Fig. 1. They find Dalitz plot parameters consistent with charge − → conjugation. Alsotheparameter hisconsistent withzero. Fortheothers,theyfind a= 1.090+0.009, b=0.124 0.012, d=0.057+0.009, f =0.14 0.02. (3.3) − 0.020 ± 0.017 ± − − Thisisthefirsttimethat f hasbeenmeasured. 5 h 3p andquarkmasses StefanLanz → The Dalitz plot measurement of the charged channel by the WASA-at-COSY collaboration, which has been announced at this conference [25], has since been published in a PhD thesis [26]. The statistics is considerably smaller than for KLOE. The new data confirms the low value for b, but deviates from KLOE in d. Also the CLAS collaboration has collected large statistics on the charged channel, buttheanalysis hasnotyetbeencompleted [27]. Furthermore, anewanalysis of amuchlargerdatasetfromKLOEisunderwaythatalsointends toimproveoncertainlimitations ofthepreviousanalysis (seethecontribution byL.BalkeståhlinRef.[28]). The neutral channel has been measured much more often in recent years [29–37] such that thepictureisconsiderablyclearer. Duetothesymmetryin s,t,andu,thenumberofpossibleterms intheDalitzplotparametrisation ismuchsmallerinthiscase. Uptofourthorder,itreads Y2 G (X,Y)= A (s,t,u)2 (cid:181) 1+2a Z+6b Y X2 +2g Z2. (3.4) n n | | (cid:18) − 3 (cid:19) So far, experiments have not reached the accuracy needed to determine b and g , but a has been measured bymanyexperiments ingoodagreement, leading totheaverage a = 0.0315 0.0015 − ± [18]. Allthemeasurementsentering thisnumberarecompiledinFig.4. 4. Theoretical work Duringthelastfewyears,h 3p hasalsoreceivedconsiderableattentionfromthetheoretical → side. I will in the following briefly discuss a few of the relevant works. An exhaustive discussion oftheliterature availableonthissubjectishowevernotwithinthescopeofthisshortarticle. The strong contribution in the isospin limit has been calculated up to the two-loop level in • ChPT[17]. Thecorrectionstotheone-loopresultarefoundtobesizable(seealsoEq.(2.6)). Unfortunately, a large number of low-energy constants enter, some of which are not well known. Thesizeofthe p6 contribution cantherefore notbeestimated reliablyatthe present stage. The ChPT amplitude is used to extract a value for the quark mass ratio R = 41.3. Using Eq.(2.4)togetherwiththeFLAGaverageform /mˆ,thisleadstoQ=24.2. Furthermore,the s result forthe neutral channel slope parameter is positive, a =0.013 0.032, but due to the ± largeuncertainty alsoencompasses negativevalues. Based on Eq. (2.7), anupper limit for a in termsof charged channel Dalitz plot parameters isderived: 1 1 a b+d a2 . (4.1) ≤ 4(cid:18) −4 (cid:19) Itisthisrelationthatcanbeusedtochecktheconsistencyofchargedandneutralchannelex- periments. TheDalitzplotparametersfromKLOEandWASA-at-COSYbothleadtoanup- perlimitthatisnegativebutlargerthanthePDGaveragefor a . WhileforKLOEtherelation becomesalmostanequality(a < 0.029),theupperlimitislargerfortheWASA-at-COSY − result(a < 0.0036). Thereasonforthisismainlythelargervaluefor d inthelattercase. − 6 h 3p andquarkmasses StefanLanz → The complete electromagnetic corrections up to order p4 and e2m have been calculated q • withinChPTinRef.[13]. Comparedtoanearliersimilarcalculation[12],thetermsoforder e2(m m ) have been added. Contrary to the assumption made in the older publication, u d − these terms are of comparable size to other e2m contributions and can not simply be ne- q glected. However, the total electromagnetic contribution is still shown to be small; of the orderofafewpercent intheamplitude, lessthanonepercent inthevalueofQ. The decay has been analysed within the framework of non-relativistic effective field the- • ory (NREFT) up to two loops [38]. The method has before been successfully applied to K 3p [39–41]andconsistsbasicallyofanexpansioninsmallpionthree-momentaaround → thecentreoftheDalitzplot. Thecalculation requires twomaininputs. Thelow-energy con- stantsappearinginthetree-levelNREFTdecayamplitudearedeterminedbymatchingtothe one-loop amplitude from ChPT at the centre of the Dalitz plot. Final state pp rescattering is included in the NREFT calculation. The additional low-energy constants that appear are fixedfromtwoRoyequationanalyses([42,43]and[44]). Inthisway,arepresentationofthe shape of the Dalitz plot distribution in both channels is constructed. Since the overall nor- malisation of the amplitude can not be reliably determined within the NREFT framework, novalueforthequarkmassratio Qisgiven. Aparticularadvantageofthecalculationisthat isospin-breaking effectshavebeenincluded. It is found that rescattering effects lead to sizable corrections to the Dalitz plot parameters. One- and two-loop contributions are in general of similar size, thus emphasising the partic- ularimportance ofrescattering effectsinthisprocess. Especially inthecaseofa ,wherethe alreadypositivetree-levelvalueisfurthershiftedinthepositivedirectionbytheoneloopcor- rection, the two-loop contribution is large and negative. This leads to a = 0.025 0.005, − ± inmarginalagreementwithexperiment. Regardingisospinbreaking,itisfoundthatthemost sizable corrections totheDalitzplotparameters arekinematiceffectsduetothefactthatthe charged andneutralpionmassesarenotthesame. Using theNREFTmethod, itispossible toturn theupperlimit inEq.(4.1) intoanequality. This is achieved by expressing the required correction in terms of the Dalitz plot parameter a. Since information on h 3p enters the NREFT amplitude through the Dalitz plot → parameters, no input from one-loop ChPT is needed in this case. The Dalitz plot parameters from KLOE lead to a = 0.059 0.007 in clear disagreement with the PDG average − ± as well as KLOE’s own value [37]. As the possible source of the problem, the parameter b is identified, since b = 0.308 > b = 0.124. However, the new measurement NREFT KLOE by WASA-at-COSY indicates that d rather than b might be responsible. Indeed, from their Dalitz plot parameters onefinds a = 0.033 0.03, which encompasses the PDGaverage. − ± Clearly, moredataisneededinordertofinallysettlethismatter. The process has been treated with a mostly analytical dispersive approach [45], where two • rescattering processes are taken into account. It has to be noted that chiral power counting isstrictly followed suchthatthenumberofsubtraction constants coincides withthenumber of low-energy constants at two-loop order which is six. Also, the dispersive result can be matchedexactlytothetwoloop-result byanappropriate choiceofsubtraction constants. 7 h 3p andquarkmasses StefanLanz → 2.5 one-loopChPT KKNZdispersive ) 2.0 s s, 2 1.5 − 0 3s 1.0 s, ( M 0.5 0.0 0 2 4 6 8 10 s=uinm2p Figure2: Theone-loopamplitudefromChPTtogetherwiththedispersiveamplitudefromRef.[45]along the line s=u. Dashed linesrepresentrealparts, dottedlinesimaginary parts. Itis clearly visible thatthe dispersiveamplitudehasnoAdlerzero.IthankK.Kampfforprovidingtheprogramthatproducedthisdata. The main result of this work is a dispersive representation where the subtraction constants are fitted to the charged channel data from the KLOE collaboration. Since the overall nor- malisationdependsonQ,itcannotbeobtainedfromthedata. Instead,theimaginary partof thedispersiveamplitudealongthelinet=ufromzerouptothresholdisrequiredtobeclose tothetwo-loopresult. ThemotivationforthisprocedureisthattheO(p6)correctionshappen toberathersmallalongthisline. TheresultingamplitudereproducestheexperimentalDalitz plotdistributioninthephysicalregionandleadstoQ=23.1 0.7anda = 0.0044 0.004. ± − ± There ishowever asevere problem with the dispersive amplitude. Ascan be seen in Fig. 2, ithasnoAdlerzero, whichmeans thatitisnotconsistent withSU(2)chiral symmetry. Itis hardtojustify anyuseofChPTinacalculation thatsoblatantly violatesthesymmetryatits foundation. 5. Ourdispersiveanalysis I want to conclude the discussion of the theoretical literature on h 3p by a somewhat → more detailed description of an ongoing numerical dispersive calculation in collaboration with G. Colangelo, H. Leutwyler, and E. Passemar. Intermediate status reports of this work have been presented atconferences before[28,46,47]andhavebeenpublished informofaPhDthesis[48]. Amoredetaileddiscussion oftheformalismcanbefoundinthesereferences andIwillfocushere onnewerdevelopments thatarenotdescribed there. The method we apply has been described in detail in Refs. [49, 50]. Because the calculation is numerical, it is possible to include an arbitrary number of rescattering processes. Also, chiral powercounting isnotfollowed: wealwaysusethebestavailable input, e.g.,wedonotexpandthe pp scattering inputtotheappropriate orderineachiteration. Themethodinvolves twomainsteps thatcanbetreated entirely independently: Adispersive representation mustbederivedandsolved 8 h 3p andquarkmasses StefanLanz → numerically. The result is an amplitude that is a function of a number of unknown subtraction constants. Thesecondstepisthentodetermine theseconstants inagoodway. The dispersion relations are based on the decomposition of the normalised decay amplitude as[51,49] 2 M(s,t,u)=M (s)+(s u)M (t)+(s t)M (u)+M (t)+M (u) M (s), (5.1) 0 1 1 2 2 2 − − −3 where the subscript stands for the isospin of the scattered pion pair. This decomposition has the advantage that one deals with functions of a single variable only. Using analyticity and unitarity, onearrivesatasetofdispersion relations forthethreefunctions M (s)thatarealloftheform I M (s)=W (s) P(s)+snI ¥ ds′ sind I(s′)MÎ(s′) , (5.2) I I (cid:26) I p Z4m2p s′nI |W I(s′)|(s′−s−ie )(cid:27) whereW (s)aretheso-called Omnèsfunctions [52]givenby I ¥ s d (s) W (s)=exp ds I ′ . (5.3) I p Z ′s′(s′ s ie ) 4mp2 − −   The functions Mˆ (s) are angular averagesover all the M (s), suchthat the dispersion relations are I I recursiveandcoupledwitheachother. Twokindsofinputsareneeded. Ontheonehand,oneneeds toknowthepp scattering phaseshifts d (s),whichwetakefromRef.[43]. Ontheotherhand,the I polynomials P(s) contain the subtraction constants, which must be determined with information I fromoutsidethedispersive machinery. Itisonthisissuethatprogresshastakenplacerecently. Thedispersive representations inEq.(5.2)canbeTaylorexpanded: M (s)=a +b s+c s2+d s3+.... (5.4) I I I I I TheM (s) areuniquely determined once allthesubtraction constants arefixed. Theyhave unique I Taylorexpansions, suchthatonecandefineaunique relation amongasetofsubtraction constants andanappropriateequallysizedsetofTaylorcoefficients. Asolutionofthedispersionrelationscan therefore bespecified by giving values forthe subtraction constants or forthe Taylorcoefficients. The advantage of working with the latter isthat they pick up imaginary parts only at O(p6), such thattheycansafelybeapproximatedasreal. Thisthenautomaticallyleadstosubtractionconstants, wheretheimaginarypartisnon-zero, butsuppressed comparedtotherealpart. Thesplitting ofthe amplitude into the M (s) is not unique because of the relation s+t+u= I mh2+2mp2++m2p 0: onecanaddpolynomialstotheMI(s)insuchawaythatM(s,t,u)isnotchanged. This gauge freedom allows to choose some Taylor coefficients arbitrarily and the number of free parameters is therefore smaller than the number of Taylor coefficients (or subtraction constants) thatareused. We have tried to work with different numbers of subtraction constants, but comparison with data has shown that satisfactory agreement is only achieved, if eleven constants are used. The corresponding Taylorexpansions read M (s)=a +b s+c s2+d s3+..., 0 0 0 0 0 M (s)=a +b s+c s2+..., (5.5) 1 1 1 1 M (s)=a +b s+c s2+d s3+.... 2 2 2 2 2 9 h 3p andquarkmasses StefanLanz → The reason for ending the series one order lower for M (s) is that this function is multiplied by 1 anotherpowerofmomentainEq.(5.1),suchthatallthreefunctions leadtocontributions ofO(s3) inthetotalamplitude. We determined the subtraction constants with two different methods. The first one relies en- tirelyonone-loop ChPTasinputandmakesnouseofdataatall: Sinceone-loop ChPTcannotbetrusted fortermsbeyond O(s2),thecoefficients d ,c ,and 0 1 • d aresettozero. Thedispersion relations thuscontaineight subtraction constants. 2 The parameters a , b , a , and a are set to their tree-level value using the gauge freedom. 0 0 1 2 • Thismeansthatthereareactually onlyfourfreeparameters inthedispersion relations. Finally, theremaining parameters c ,b ,b ,andc aresettotheirone-loop value. 0 1 2 2 • In this way, the dispersive solution is entirely fixed. It will later be referred to as “dispersive, one loop”. I stress again that for this solution, only ChPT at low energy is used in order to fix four subtraction constants. This low-energy information is then extrapolated to the physical region by meansofthedispersion relations. Thesecondmethodusesthefullsetofelevensubtraction constants: Theparameters a ,b ,a ,a ,c ,b ,b ,andc aredetermined exactlyasbefore. 0 0 1 2 0 1 2 2 • The presence of more parameters also enlarges the gauge freedom since polynomials of • higherordercannowbeaddedtothe M (s). Theparameter d ischosensuchthatd ,which I 2 2 isthecoefficienttos3inP (s),iszero. Thenumberoffreeparametersisthussixinthiscase. 2 Finally, the remaining two parameters d and c are determined by fitting the square of the 0 1 • amplitude inthephysicalregiontothecharged channel data fromtheKLOEcollaboration. This leads to another solution of the dispersion relations which will be referred to as “dispersive, fittoKLOE”.Wehavealsofittedotheravailable datasets,butforsimplicity, onlyoneofthesefits ispresented hereasanexample. The Dalitz plot distributions from both solutions are depicted in Fig. 3 together with the corresponding one-loop results. The left panel shows the Y distribution along the line with X =0, which is equivalent to the s distribution for t =u. Note that small s correspond to large Y due to the minus sign in the definition ofY in Eq. (3.1). Clearly, ChPT is successful at low energy, but fails towards the upper end of the physical region. It is noteworthy that the extrapolation of the low-energy ChPTamplitude through the dispersion relation already leads to aconsiderably better agreement with experiment, which is then further improved by the fit. It seems not, at first sight, remarkable that the fit does agree with the data it is fitted to. However, the fit must at the same timebeconsistent withthechiralconstraintsatlowenergy anditisimportanttoshowthatthiscan actually beachieved. ThroughEq.(2.7),theneutralchannel amplitudecanbecalculated fromthecharged channel. TherightpanelofFig.3showstheZ distributions thatonefindsinthiswayfromthethreecharged channel amplitudes. Accordingly, the blue curve involves no experimental information on the neutral channel. But the influence of the charged channel data is exactly what is needed to bring 10