We review our evaluation of forward dispersion relations for direct fits to
the different, and often conflicting, pion-pion scattering experimental
analyses. We find that some of the most commonly used data sets do not satisfy
these constraints by several standard deviations. We also provide a consistent
pion-pion amplitude by improving a global fit to data with these dispersion
relations.Comment: To appear in the Proceedings of the XI. International Conference on
  Hadron Spectroscopy- HADRON05. Rio de Janeiro. 21st-25th August. 200

Pelaez, J. R.

English

arXiv

José R. Peláez

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Consistency checks of different ππ scattering data sets using forward dispersion relations

©Amer Inst Physics.
ISSN: 0094-243X
International Conference on Hadron Spectroscopy (11. 2005. Rio de Janeiro, Brasil).We review our evaluation of forward dispersion relations for direct fits to the different, and often conflicting, ππ scattering experimental analyses. We find that some of the most commonly used data sets do not satisfy these constraints by several standard deviations. We also provide a consistent ππ amplitude by improving a global fit to data with these dispersion relations.Depto. de Física TeóricaFac. de Ciencias FísicasTRUEpu

Peláez Sagredo, José Ramón

Docta Complutense

Consistency checks of different ππ scatteringdata sets using forward dispersion relationsJosé R. PeláezDept. Física Teórica II. Universidad Complutense, 28040-Madrid, SpainAbstract. We review our evaluation of forward dispersion relations for direct fits to the different,and often conflicting, ππ scattering experimental analyses. We find that some of the most commonlyused data sets do not satisfy these constraints by several standard deviations. We also provide aconsistent ππ amplitude by improving a global fit to data with these dispersion relations.Keywords: mesons, pions, scattering, dispersion relationsPACS: 12.39.Fe, 13.75.Lb, 11.55.Jy, 11.80.EtA precise knowledge of the ππ scattering amplitude provides crucial tests for ChiralPerturbation Theory (ChPT), as well as information on light meson spectroscopy, pionicatom decays and CP violation in kaons. However, there are several ππ scattering datasets in the literature, in conflict among themselves, even within the same experiment.The reason is that the data are extracted from other reactions and thus with largetheoretical and systematic uncertainties. Here we review our recent works [1, 2, 3]where we checked dispersion relations on the different sets of data and provided simpleparameterizations of ππ scattering amplitudes consistent with such requirements.Fits to different sets of dataWe first consider fits to data [3] for the S0, S2, P waves, below s1 2    1 GeV. Weparameterize the phase shifts, δ  s, taking into account the analytic properties, zerosand poles of the amplitude, in a conformal expansion of cotδ  s.The P wave, up to  1GeV, comes from the following fit to the pion form factor [4]cotδ1 s s1 22k3 M2ρ s B0B1ss0 sss0 s; s1 20  105 GeV  (1)B0  10690011 B1  013005 Mρ  773609(s0 is the point where inelasticity begins to be nonnegligible). The fit is seen in Fig.1a.For the S2 wave at low energies, we first fix the Adler zero at z2  Mπ and fit only thelow energy data, s1 2   10 GeV; later on we allow z2 to vary. We havecotδ  20  s s1 22kM2πs2z22 B0B1ss0 sss0 s s01 2  105 GeV (2)B0  80428 B1 736126; (3)706300 400 500 600 700 800 900 1000 √s  (MeV)050100150 δ1(s)Pion form factor (PY)Protopopescu et al. Estabrooks & Martin300 400 500 600 700 800 900 1000 1100 1200 1300 1400s1/2 (MeV)0306090120150180210240270300330 δ00(s)Solution B Solution ASolution C Solution DSolution EProtopopescu et al.  (Table VI)Estabrooks & Martin s-channelEstabrooks & Martin t-channelKaminsky et al.PY from dataGrayer et al.400 600 800 1000 1200 1400s1/2  (MeV)-7-6-5-4-3-2-10 δ2(2)(s)Losty et al.Hoogland et al.Cohen et al.PY from dataPY improvedDurusoy et al. (OPE)Durusoy et al (OPE-DP)FIGURE 1. a) P wave ππ phase shifts [5, 6], versus Eq.(1) (solid line below 1 GeV), which is not a fit tothese data, but to the pion form factor [4]. The uncertainty is the line thickness. Above 1 GeV, the dottedline and error are as in [3]. b) S0 phase shifts and error band as given by Eq.(4) below 1 GeV, and from[3] above. The Kl4 and K2π decay data are not shown. (see our [3] for details). c) Continuous line: The fitto I   2, D-wave phase shift data. Broken line: fit improved with dispersion relations. The experimentalpoints are from [8].Except for the very reliable Kl4 and K  2π decay experiments [7], that we alwaysfit, the S0 data is very confusing and thus we have adopted two approaches. In the firstmethod, the “global fit”, we fit averaged phase shift data between 081 GeV  s1 2 097 GeV, (where the experiments agree within 15σ ), composing their errors carefully.The fit, shown in Fig.1.b, is valid for s1 2  095 GeV, and with the Adler zero fixed atz0  Mπ , corresponds tocotδ  00  s s1 22kM2πs 12z20M2µ  sM2µ B0B1ss0 sss0 s (4)B0  y x; B1  662259x; y  2104070 x  026The second method is to fit only Kl4 and K 2π data, or to add to them the data fromthe various experimental analyses separately. The results can be found in Table 1.The D2 inelasticity is negligible below 4M2ρ . A pole term is needed since the data [8]gives a small negative phase above   500 MeV, but the Froissart–Gribov representa-tion, yields [9] a positive scattering length, a 22   272 036 10 4 M 5π , which isincluded in the fit. Also, the inflection seen in data around 1 GeV asks for a third orderconformal expansion. So we writecotδ  22  s s1 22k5B0B1w sB2w s2 Mπ 4s4 Mπ2∆2 s w s ss0 sss0 sAnd we find B0   24 03 103, B1   78 08 103, B2   237 38 103,∆  19620 MeV. The fit is shown in Fig.1.c.For brevity, we do not discuss here the D0 and F waves as they do not present specialfeatures, nor the intermediate energy region 1 GeV  s1 2  142 GeV, but a detailedaccount of their parameterizations can be found in [3].We also need the imaginary part of the scattering amplitude at s1 2  142 GeV, thatwe take from a Regge fit to experimental data ( see [2] for details and the slightlyimproved rho residue of [3]), shown in Fig.2. As discussed in [2, 3], standard Reggefactorization describes experiment [10, 2] and is consistent with crossing sum rules.7071.4 1.6 1.8 2 2.2 2.4s GeV1020304050TotalΣΠ0Πmb Regge PYPYData: Biswas et al.TotalΣΠ0Πmb1.5 2 5 10 15s GeV1020304050TotalΣΠΠmbRegge: PYData: Robertson et al.Biswas et al.Abramowicz et al.PYZakharov et al.Hoogland et al.TotalΣΠΠmb2 4 6 8 10s GeV1020304050TotalΣΠΠmbData: Robertson et al.Biswas et al.Hanlon et al.PYZakharov et al.Regge PY Hyams et al.TotalΣΠΠmbFIGURE 2 The ππ cross sections. Experimental points from [10]. The stars at 1.38 and 1.42 GeV (PY)are from the phase shift analysis of experimental data given in [3]. Continuous lines, from 1.42 GeV(PY): Regge formula, with parameters as in [2] (the gray bands cover the uncertainties). Below 2 GeV,the dotted line corresponds to the π π  cross section from the Cern–Munich analysis of Hyams et al.[6]B0 B1 Mµ (MeV)It1χ2 dofπ0π0χ2 do f PY, Eq.(4) 21 04 6 62 782 24 0 3 3 5K decay only 18 5 1 7  0 766 95 0 2 1 8K decay + Grayer B [6] 22 7 1 6 12 3 3 7 858 15 1 0 2 7K decay +Grayer C [6] 16 8 0 85 0 34 2 34 787 9 0 4 1 0K decay + Grayer E [6] 21 5 3 6 12 5 7 6 1084 110 2 1 0 5K decay + Kamisnki [6] 27 5 3 0 21 5 7 4 789 18 0 3 5 0K decay + Grayer A [6] 28 1 1 1 26 4 2 8 866 6 2 0 7 9K decay+ EM, s-channel [6] 29 8 1 3 25 1 3 3 811 7 1 0 9 1K decay+ EM, t-channel [6] 29 3 1 4 26 9 3 4 829 6 1 2 10 1K decay+Protopopescu VI [5] 27 0 1 7 22 0 4 1 855 10 1 2 5 8K decay+Protopopescu XII [5] 25 5 1 7 18 5 4 1 866 14 1 2 6 3K decay+ Protopopescu 3 [5] 27 1 2 3 23 8 5 0 913 18 1 8 4 2TABLE 1 Fits to K decays and different data sets of S0 phase shifts. PY is our global fit, Eq.(4), and itsB0 and B1 uncorrelated uncertainties can be obtained from Eq.(4). Note that many of these sets have verylarge χ2d o  f   for the It   1 and π0π0 dispersion relations below 950MeV. Let us also remark that someof them (like that of Sol. E) have a relatively small χ2d o  f   just because it has huge uncertainties in itsparameters. Ideally one would require a low χ2d o  f   with small uncertainties in the parameters.Checking and improving amplitudes with forward dispersion relationsLet us study how well the previous fits to data satisfy three independent scatteringamplitudes. We choose the t-symmetric or antisymmetric combinations, that form acomplete set: F00 	 F π0π0  π0π0, F0 	 F π0π  π0π, and the t channelisospin one amplitude, F  It1. Final uncertainties are small for the two first, since theydepend only on two isospin states, and their imaginary parts are sums of positive terms.Thus, we find two dispersion relations by choosing either F  F00 or F  F0 inReF sF 4M2π s s4M2ππPP  ∞4M2πds 2s4M2π ImF ss s s s4M2π s s4M2π (5)708By setting s  2M2π , and F  F00, we find a sum rule important to fix the Adler zeros.F00 4M2π  F00 2M2π8M4ππ  ∞4M2πdsImF00 ss s2M2π s4M2π (6)Finally, for isospin unit exchange, which does not require subtractions,ReF It1 s0 2s4M2ππPP  ∞4M2πdsImF  It1 s0 s s s s4M2π (7)at threshold this is known as the Olsson sum rule.Depending on the method we use to fit the S0 wave we find the results in Table 1,where, we have separated on top those fits to data with a total χ2do f    6 for theπ0π0 and It  1 dispersion relations up to 0.925 GeV, a fairly reasonable χ2do f since these fits were obtained independently of the dispersive approach.However, in Table 1 we also list the very frequently used t and s-channel solutionsof Estabrooks and Martin [6], those of Protopopescu et al.[5], from Table VI, VIII andtable XII, as well as the solution A of Grayer et al. [6]. Their It  1 plus π0π0 dispersionrelation total χ2do f  is surprisingly poor: 11.3, 10.1, 7, 6, 7.5, 9.9, respectively.Therefore, any result that relies heavily on these sets should be taken very cautiously.We have also improved the previous low energy fits parameters by fitting also thedispersion relations up to 0.925 GeV, thus obtaining parameterizations more compatiblewith analyticity and s  u crossing. Improving from Eq.(4), we find, in Mπ units,S0; s1 2  2mK : B0  17405; B1  4314;Mµ  79021 MeV; z0  195 MeV [Fixed];S2; s1 2  10 : B0 80817; B1 775; z2  147 MeV [Fixed];P; s1 2  105 : B0  1064011; B1  01700040; Mρ  773609 MeV;D2; s1 2  142 : B0   2902103; B1   7308103;B2   25436103; ∆  21219 (8)The D0 and F waves do not change appreciably and we just refer to [3] for details. InFig.3 we show the improved curves for S0 and S2, and that of D2 in Fig.1.c.Concerning the improved fits to individual data sets, we get somewhat different resultsfor S0, listed in Table 2 with the χ2do f  of each forward dispersion relation and thestandard deviations for the sum rule in Eq.(6) (which are more than four for K decay plusthe Grayer B or E or Kaminski improved solutions). For other waves, no matter what S0fit is used, we find very similar values to those in Eq.(8). This can be checked in Fig.3.b,where we show the improved “K decay + Grayer Sol. B” S2 wave. Even though it is theone for which we obtained the most different central values for the S0 wave comparedwith those given in Eq.(8), it falls within the uncertainty of our improved solution.In summary, with forward dispersion relations, we have checked the consistency ofdifferent phase shift analyses available in the literature. Surprisingly, some of the mostfrequently used phase shift sets do not satisfy these dispersive constraints and sum rules,and should therefore be used cautiously. We have provided a simple parameterizationof ππ scattering consistent simultaneously with some data sets and all three forwarddispersion relations, that we hope could be of use for future studies of ππ scattering.709300 400 500 600 700 800 900s1/2 (MeV)020406080100 δ00(s)Solution B Solution C Kaminsky et al.K decay dataK decay improved fitK decay + Sol. C improved fitPY from dataPY improvedGrayer et al.400 600 800 1000 1200 1400s1/2 (MeV)-30-25-20-15-10-50 δ0(2)(s)Cohen et al.Losty et al.Hoogland et al. AHoogland et al. BDurusoy et al. (OPE)Durusoy et al.(OPE+DP)PY improvedK decay+ Sol. B improvedFIGURE 3. a) The improved S0 phase shift (PY improved, Eq.8), the global fit (PY from data, the S0in Eq. (4)), and the improved solutions “K decay only" and “Grayer C" of Table 2 (almost on top of PYimproved). b) S2 improved Phase shift (PY improved, Eq. (8)); global fit (PY from data, Eq. (4)) and theimproved parameterization with K decays and So. B of Grayer et al.[6].TABLE 2. Improved fits (only those already with χ2do f  6 in Table 1). Although errors are givenfor the Adler zero, we fix it when evaluating other errors, to break the otherwise very large correlations.Improved Improved K decay only K decay+ K decay+ K decay+ K decay+fits: PY, Eq.8 Grayer C Grayer B Grayer E KamińskiB0 17 4 0 5 16 4 0 9 16 2 0 7 20 7 1 0 20 2 2 2 20 8 1 4B1 4 3 1 4  0 0 5 1 8 11 6 2 6 8 4 5 2 13 6 43 7Mµ (MeV) 790 30 809 53 788 9 861 14 982 95 798 17z0 (MeV) 195 30 182 34 182 39 233 30 272 50 245 39It   1 χ2d o  f   0.40 0.30 0.37 0.37 0.60 0.43π0π0 χ2d o  f   0.66 0.29 0.32 0.83 0.09 1.08π π  χ2d o  f   1.62 1.77 1.74 1.60 1.40 1.36Eq.(6) 1.6σ 1.5σ 1.5σ 4.0σ 6.0σ 4.5σREFERENCES1. Peláez, J. 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Consistency checks of different pion-pion scattering data sets using forward dispersion relations

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