Phase Motion in the Scalar Low-Mass pi+pi- Amplitude in D+ ->pi-pi+pi+ Decay

Applying the Amplitude Difference method to Fermilab experiment E791 D+ ->pi-pi+pi+ data, we measure the low mass pi+pi- phase motion. Our results suggest a significant phase variation, compatible with the existence of an isoscalar sigma(500) meson, as previously reported using an isobar model fit to the full Dalitz-plot density.

Recently we proposed the isobar-based Amplitude Difference (AD) method to extract the phase motion of a complex amplitude in three-body heavy-meson decays [1]. With this method, the phase variation of a generic complex amplitude can be directly revealed through interference in the Dalitz-plot region where it crosses a well established resonant state, used as a probe. As a test, this method was successfully applied to data [2] to extract the well known phase motion of the scalar amplitude f 0 (980) observed in D + s → π − π + π + 1 decay. In the present paper we use the same method to study the low π + π − mass region of the D + → π − π + π + decay where Fermilab experiment E791 showed evidence for the existence of a light and broad scalar resonance [3].
To obtain good fit quality in a full Dalitz-plot analysis, E791 found it necessary to include an extra scalar particle, in addition to the well-established di-pion resonances [4]. For this new scalar state, parameterized as an S-wave Breit-Wigner resonance, they measured a mass and a width of 478 +24 −23 ± 17 MeV/c 2 and 324 +42 −40 ± 21 MeV/c 2 , respectively. These parameters are compatible with those expected for the isoscalar meson σ(500). The D + → σ(500)π + [3] decay appeared as the dominant contribution, accounting for approximately half the D + → π − π + π + decays.
The E791 result has been widely discussed [5]- [10] and new data has become available [11]. However it is desirable to be able to confirm the result through a direct observation of the phase motion expected for a resonance [6,7,8]. In this context we apply the AD method to the low π + π − mass region of the D + → π − π + π + decay. We also compare the phase variation of the Breit-Wigner function found in the isobar Dalitz-plot analysis [3] to the model-independent method of this paper.
The present study is a reanalysis of the Fermilab experiment E791 data. Here we investigate a subset of the total phase space used by the experiment in their full Dalitz-plot analysis. A description of the experiment, data selection criteria, background parametrization and detector acceptance are found in references [3,12]. The final π − π + π + invariant mass distribution is shown in Fig. 1. There are 1686 events with invariant mass between 1.85 and 1.89 GeV/c 2 shown in the shaded region of Fig. 1. The integrated signal-to-background ratio in this range is about 2:1. Fig. 2 shows the folded Dalitz-plot. The horizontal and vertical axes are the squares of the π + π − invariant mass high (s 12 ) and low (s 13 ) combinations. The analysis presented here uses the hatched area of Fig. 2. We estimate 60 background events in a total of 197 candidate events. The background does not show any dependence on the s 12 variable [3].
The detector acceptance in this region is almost constant. There is a very mild slope in the s 13 acceptance, but no variation with s 12 . Nevertheless the acceptance is taken into account to correct the event distributions shown later.
There are two conditions necessary to extract the phase motion of a generic am-1 Charge conjugate states are implied throughout the paper. Figure 1: The π − π + π + invariant mass spectrum. The dashed line represents the total background. Events used for the D + isobar Dalitz-plot analysis [3] and in this AD method analysis are in the hatched area.
plitude with the AD method: • A crossing region between the amplitude under study and a probe resonance has to be dominated by these two contributions.
• The integrated amplitude of the probe resonance must be symmetric with respect to an effective mass squared (m 2 ef f ). 2 To study the low mass region in s 13 , three well known resonances could serve as a probe in s 12 in the D + → π − π + π + decay: ρ(770), f 0 (980) and f 2 (1270). However the broad ρ(770) and f 0 (980) are too close to each other to pass the isolation criteria mentioned above, as can be seen in Fig. 2. On the other hand, the tensor f 2 (1270), m 2 0 = 1.61 GeV 2 /c 4 , is located where the ρ(770) reaches a minimum due to its decay angular distribution in the crossed (s 13 ) channel, (see Fig. 3). In the D + → π − π + π + decay, the f 2 (1270) contribution satisfies the necessary conditions of having a substantial contribution crossing the low mass region, this being the regions where all other amplitudes can be considered negligible. In particular, we estimate a contamination of 5% of ρ(770)π + events in the region of interest.
We assume that the only contributions in this region are the f 2 (1270) amplitude in s 12 and the ππ complex amplitude under study in s 13 . We write: where γ is the overall relative phase-difference between the two amplitudes (assumed to be constant and arising from both production and final-state-interaction (FSI) between the di-pion system and the bachelor pion); sinδ(s 13 )e iδ(s 13 ) represents the most general amplitude for a two-body elastic scattering, p * / √ s 13 is a phase space integration factor to make this description compatible with ππ scattering and p * is the pion momentum measured in the resonance rest frame; J=2 M f 2 (1270) (s 12 , s 13 ) is the angular function for the f 2 (1270) tensor resonance given by 4 3 (| p 3 || p 2 |) 2 (3cos 2 θ − 1), θ being the angle between the pions 2 and 3, and J the angular momentum of the resonance; and a R and a s are the production strengths. The AD method assumes that the π + π − production is constant over the effective mass range considered and a R and a s are energy independent. Finally the Breit-Wigner distribution is given by: The width is given by Γ , where the central barrier factor is J=2 F = 1/ 9 + 3(rp * ) 2 + (rp * ) 4 . The parameter r is the radius of the resonance (∼ 3f m) [13] and p * = p * (m) is the momentum of decay particles at mass m, measured in the resonance rest frame, p * 0 = p * (m 0 ), where m 0 is the resonance mass.  For events in s 12 between m 2 ef f and m 2 ef f + ǫ in s 12 , J=2 M f 2 (1270) (s 13 ) is shown in Fig. 5a and for those between m 2 ef f and m 2 ef f − ǫ in Fig. 5b 4 . We can see that these two plots are just slightly different. In our analysis we consider the approximation J=2 M + f 2 (1270) (s 13 ) ≈ J=2 M − f 2 (1270) (s 13 ) and take the average function J=2M f 2 (1270) (s 13 ). An important effect that we have to take into account is the zero of this function at s 13 ∼0.48 GeV 2 /c 4 . Below we discuss the consequences of that for this AD method application. To be brief, from here on we use J=2M f 2 (1270) (s 13 ) =M and p * / √ s 13 = p ′ . The main equation of the AD method applied to the integrated amplitude-square difference is [1,2]: where C is an overall constant. From Eq. 2, it follows that the function ∆ A 2 p ′ /M directly reflects the behavior of δ(s 13 ). A constant ∆ A 2 p ′ /M implies a constant δ(s 13 ) which is the case for a non-resonant contribution. In the same way, a slow phase motion will produce a slowly varying ∆ A 2 p ′ /M distribution, and a full resonance phase motion produces a clear signature in ∆ A 2 p ′ /M with the presence of zero, maximum and minimum values.
As mentioned previously, the background has no dependence on s 12 and its contribution vanishes from the ∆ A 2 distribution. The A 2 in s 13 for events integrated in s 12 m 2 ef f and m 2 ef f + ǫ and m 2 ef f and m 2 ef f − ǫ, corrected by the acceptance shape, are presented in Figs. 6a and 6b, respectively; these events correspond to the hatched area of Fig. 2. We obtain ∆ A 2 shown in histogram 6c, by subtracting the 6b histogram from that in 6a. To extract the phase motion, we divide ∆ A 2 (Fig. 6c) byM (average of the distributions in Figs. 5a and b), and multiply by p ′ , both known functions of s 13 . Then the only s 13 dependence of the right hand side of Eq. 2 is in the phase motion δ(s 13 ). From Fig. 5, the zero at s 13 ∼ 0.48 GeV 2 /c 4 in the angular function, produces a singularity around this value in ∆ A 2 p ′ /M. In Fig. 7 we show the ∆ A 2 p ′ / M distribution. To treat the effect of the singularity, we have used a binning such that the singularity is placed in the middle of one bin. Doing this, we isolate the singularity in a single bin (bin 6) and discard it in the analysis. We point out that the location of this singularity can only affect the exact position of the minimum of ∆ A 2 p ′ /M. It does not change the general features observed, in Fig. 7, that this quantity starts at zero, has maximum and minimum values, and comes back to zero, the signature for a strong phase variation. The confidence level for a straight line fit to the data in Fig. 7 is 4.6%, while the separation between the maximum and minimum values has a significance level of 2.6 r.m.s.
We can see that the 6 th bin has a huge error, which corresponds to the bin due to the presence of the singularity. Assuming that δ(s 13 ) is an analytical function of s 13 , Eq. 2 allows us to set the two following conditions at the maximum and minimum values of ∆ A 2 p ′ /M, respectively. and With these two conditions, we obtain γ and C, and with these values we can extract directly δ(s 13 ) from Fig. 7 by inverting Eq. 2.
To propagate the statistical errors from Fig. 6 to the values of the γ and δ(s 13 ), we "produce" statistically compatible "experiments" by allowing each bin of Fig. 6a and Fig. 6b to fluctuate randomly following a Poisson law. We then solve the problem for each set. The statistical error in each bin for δ(s 13 ) is the r.m.s. of the δ(s 13 ) distributions from the Monte Carlo experiments. For the systematic errors, we change the ǫ parameter (ǫ = 0.22 GeV 2 /c 4 and ǫ = 0.30 GeV 2 /c 4 ); we examine the possible influence of other neglected amplitudes contributing in this region of the phase space (based on the E791 amplitude measurements for non-resonant, f 0 (1370)π + and ρ 0 (1450)π + [3] contributions); and, to study the effect of averaging J=2M f 2 (1270) (s 13 ) distributions, we use each distribution of Fig. 5a and Fig. 5b separately. The three systematic errors, while treated separately bin-by-bin, are found to be of an average size, relative to the statistical uncertainty, of 1, 0.6, and 0.4, respectively. They are added in quadrature.
We measure γ = 3.31 ± 0.33 ± 0.49 (with the first error statistical and the second systematic) 5 . The value is somewhat larger than the E791 full Dalitz-plot analysis value (γ Dalitz = 2.59 ± 0.19) [3]. The asymmetry of the distribution in Fig. 4 and the consequent use of an effective mass-squared for the f 2 (1270) = 1.535 GeV 2 /c 4 instead of the nominal mass is responsible for the observed shift. To evaluate the magnitude of this effect, we generated 1000 fast MC samples with only two amplitudes, f 2 (1270) and σ(500). For both, we used Breit-Wigner functions with the E791 parameters, including the phase-difference of 2.59 rad. We extract γ from these 1000 samples with the method presented here. The result has a mean value of 3.07 ± 0.10 rad, instead of the input value 2.59 rad. We estimate an offset of -0.48 (2.59 -3.07) for γ from the difference between the generated and measured values in this Monte Carlo test. This yields to a corrected γ corr = 2.83 ± 0.38 ± 0.49. The production phasedifference between the f 2 (1270)π + and the σ(500)π + decays of D + measured in the isobar Dalitz analysis is in good agreement with γ corr from the AD method.
With our γ and C values we solve Eq. 2 for δ(s 13 ) for each s 13 bin. However, there are ambiguities that arise due to the sin −1 operations. Table 1 shows the four possible solutions for δ(s 13 ) 6 . To resolve the ambiguities we use the assumption that the phase is zero at threshold and is an increasing, monotonic, and smooth function of s 13 .
The solution, after the above criteria for δ(s 13 ), including systematic and statistical errors, is shown in Fig. 8 and in bold values in Table 1. We see a strong phase variation of about 180 0 , starting at threshold and saturating around s 13 = 0.6 GeV 2 /c 4 . The limited sample size does not allow us to perform an accurate measurement of the mass and width parameters of a BW resonance fitted to this plot. We can only say that, at least for the preferred solution, the phase variation is compatible with the complete phase motion through 180 0 expected for a resonance. This result is in  Fig.7; note the arbitrary normalization. Columns 3 to 6 are four possible solutions for δ(s 13 ): δ 0 (s 13 ) for (2δ(s 13 ) + γ), δ 1 (s 13 ) for (2δ(s 13 ) + γ) + 2π, δ 2 (s 13 ) for π − (2δ(s 13 ) + γ)) and δ 3 (s 13 ) for π − (2δ(s 13 ) + γ)) + 2π). The systematic errors are described in the text.
agreement with the evidence for a broad and low mass scalar resonance suggested by the previous E791 result using the full Dalitz-plot analysis [3]. The Breit-Wigner phase motion for the E791 mass and width parameters, combined with other scalar contributions obtained from that fit [3], is shown as the continuous line of Fig. 8. We can see a two standard deviation difference at lowest and highest π + π − invariant masses squared, in opposite directions, but in overall agreement with the mass region where a scalar meson σ must have its strong phase variation. Both results, the E791 with a Breit-Wigner phase variation, and the one presented in this paper, show a stronger phase variation than that obtained with theoretical constraints in ππ → ππ elastic scattering data in the scalar-isoscalar channel below 1 GeV [9,14,15]. The discrepancy between these results could be an indication that applying Watson's theorem [16] is not straight-forward when comparing the phase motion of a two-body elastic interaction to the three-body decays.
We have presented an extraction of the phase motion of the low mass π + π − scalar amplitude using the well known f 2 (1270) tensor meson in the crossing channel acting as an interferometer. The result is obtained with an event counting procedure in a region of the phase space which is dominated by a D-wave f 2 (1270) interfering with the S-wave. The derivation of the phase motion relies heavily on the assumption that the maximum and minimum bins in Fig. 7 correspond to the quantity S = sin(2δ + γ) = +1 and −1. The clear presence of a maximum and a minimum, separated from each other by 2.6 standard deviations, supports this assumption. Given this caveat, the solution for δ(s 13 ) has a variation of about 180 0 , consistent with a resonant σ(500) contribution. We also obtain good agreement between the FSI γ corr determined with the AD method and the γ observed in the full Dalitz-plot analysis using an isobar model [3]. These results support the previous evidence for an important contribution Figure 8: The phase values δ(s 13 ) from our preferred solution versus the invariant π + π − mass squared with statistical and systematic errors added in quadrature. The continuous line is the Breit-Wigner phase motion with the E791 parameters for the σ(500) [3].