application/xmlComments on the σ and κD.V BuggPhysics Letters B 572 (2003) 1-7. doi:10.1016/j.physletb.2003.07.078journalPhysics Letters BCopyright © unknown. Published by Elsevier B.V.Elsevier B.V.0370-26935721-216 October 20032003-10-161-71710.1016/j.physletb.2003.07.078http://dx.doi.org/10.1016/j.physletb.2003.07.078doi:10.1016/j.physletb.2003.07.078http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/JournalsS300.3PLB20137S0370-2693(03)01211-510.1016/j.physletb.2003.07.078ExperimentsFig. 1(a) The denominator of the E791 Breit–Wigner amplitude vs. mass below threshold. (b) Phase shifts for ππ elastic scattering; dark points are phase shifts of Hyams et al. [13] above 450 MeV and from Ke4[10] data below 400 MeV; open circles shows π0π0 data from Refs. [15,16]; the curve shows the fit.Fig. 2(a) Error ellipses against mππ for the σ pole position. A: from elastic scattering and charge exchange; B: production data, Refs. [1,2]; the cross marks the pole position. (b) The intensity for the σ pole as a function of mass; the full curve shows the result for Eq. (7) and the dashed curve the parametrisation of E791. (c) The mass where the phase goes through 90° for complex s.Fig. 3(a) Kπ phase shifts. Points are from Lass [11]. The curve shows the fit to LASS data. (b) The fit to the intensity of Kπ elastic scattering from LASS data. (c) The intensity of the κ amplitude as a function of mass, normalised to 1 at its peak. (d) The kappa phase vs. Kπ mass.Fig. 4(a) F(s), (b) F(r) and (c) r2F2(r). The full curves show the optimum fit and dashed curves show results with a stronger exponential cut-off in s.Comments on the σ and κD.VBuggbugg@v2.rl.ac.ukQueen Mary, University of London, Mile End Rd, London E1 4NS, UKEditor: L. MontanetAbstractEvidence for the σ pole has been reported in production processes such as D+→π+π−π+; likewise evidence for the κ pole appears in D+→K−π+π+. Their effects in ππ and Kπ elastic scattering are much less conspicuous. However, consistent fits to both production data and elastic scattering may be obtained by including the Adler zero into an s-dependent width for each resonance. These zeros suppress strongly the effects of the σ and κ poles in elastic scattering; the zeros are absent from amplitudes for production data. With this prescription, data from ππ→ππ, Ke4 decays and CP violation in K0 decays give a σ pole position of (525±40)−i(247±25) MeV. A combined analysis with production data gives a better determination of (533±25)−i(249±25) MeV. The analysis of LASS data for Kπ elastic scattering, including the Adler zero, determines a κ pole at (722±60)−i(386±50) MeV.The Fourier transform of the matrix element for σ→ππ reveals a compact interaction region with RMS radius ∼0.4 fm.PACS13.25.Gv14.40.Gx13.40.HqThe σ and κ have a confused and controversial history. The E791 Collaboration observes a low mass π+π− peak in D+→π+π−π+ [1]; they fit it with a simple Breit–Wigner resonance which they interpret as the σ. A similar peak was observed in earlier DM2 data on J/Ψ→ωπ+π− [2]. The E791 Collaboration also reports a low mass Kπ enhancement in data on D+→K−π+π+ [3]; they again fit this enhancement with a simple Breit–Wigner resonance and claim it as the κ. Preliminary BES data on J/Ψ→K∗(892)κ presented at the Tokyo conference this year provide supporting evidence for a κ peak; there are two discrete solutions for the width if it is fitted as a resonance [4]. An essential question is why these peaks do not appear in ππ and Kπ elastic scattering.The answer has been known for some years to experts in the field; but it has not so far found its way into the formulae commonly used by experimentalists to fit data. A primary objective of this Letter is to introduce suitable formulae, consistent with the essential features of chiral perturbation theory. There is a problem in the over-simplified formulae presently used by some experimental groups in fitting the σ. A false pole appears below the ππ threshold. This Letter reports formulae which eliminate this false pole and provides examples of the application of these formulae. A re-analysis of the experimental data along these or similar lines is desirable, in order to achieve consistency with the data on elastic scattering.Consider first a production process such as D+→π+π−π+. The σ pole in π+π− may be fitted with a T-matrix element (1)TD+→π+σ=ΛσM2−s−iMΓ(s). As usual, s denotes mass squared for the ππ pair; M is the mass where the phase shift goes through 90° for real s. In fitting data, the conventional isobar model approach is to make Λσ a complex coupling constant: Λσ=gexp(iφ), where g is real and φ is a phase arising from rescattering in the 3-body final state; usually φ is taken to be constant, though it might vary slowly with s over a wide resonance such as the σ, as might g.It has been common practice to take (2)Γ(s)=Γ0ρ(s), where Γ0 is a constant and ρ(s) is Lorentz invariant phase space 2k/s; k is pion momentum in the σ rest frame. Zheng [5] points out that this formula suffers from a difficulty: the s-dependence of the Breit–Wigner denominator gives rise to a second pole below the ππ threshold. For the parameters of E791, it lies at s=198 MeV. This may easily be verified by evaluating the denominator of the Breit–Wigner amplitude. Below threshold this denominator is purely real if electromagnetic decays are neglected; Fig. 1(a) shows that the E791 denominator moves rapidly through zero at 198 MeV, producing a well defined pole. This is unphysical: a bound state decaying to γγ could not have escaped detection for 50 years. The formulae used by Ishida et al. [6,7] also contain bound state poles in the mass range 200–230 MeV; that problem is removed in Ref. [8] by a more complex energy dependence, but a second pole survives at 197−i245 MeV. Zheng points out that Eq. (2) is in contradiction with chiral perturbation theory, which is known to give an excellent account of low energy ππ and Kπ scattering.In order to understand how this problem may be avoided, consider next ππ elastic scattering. For present purposes, it is sufficient to consider the elastic region below the KK threshold. There, the amplitude must lie on the unitarity circle. It is conventionally written (3)T(ππ→ππ)=exp(2iδ)−12i=sinδexp(iδ). The corresponding Breit–Wigner amplitude is (4)T(ππ→ππ)=MΓ(s)M2−s−iMΓ(s). The phase factor exp(iδ) in Eq. (3) arises from the denominator in (4): tanδ(s)=MΓ(s)/(M2−s). The term sinδ of Eq. (3) is a consequence of elastic unitarity: (5)T−1=exp(−iδ)sinδ=cotδ−i=M2−sMΓ(s)−i. The denominator [M2−s−iMΓ(s)] in Eq. (4) is universal and should describe all channels where the σ appears; that is the foundation of the Particle Data Tables, where it is assumed that the mass and width of a resonance are the same for all channels where it appears. In particular, it should be the same for production data and elastic scattering.Note that Γ(s) appears in the numerator of Eq. (4) but not Eq. (1). For the production process, there is no corresponding unitarity constraint on the numerator. The process D+→π+π−π+ is only one of a large number of decay modes, so unitarity plays no essential role.In ππ elastic scattering, it is well known that there is an Adler–Weinberg zero just below the elastic threshold [9] at sA≃m2π/2. This zero is today a central feature of chiral perturbation theory; it may be confirmed experimentally by extrapolating phase shifts of Pislak et al. [10] below threshold. The Adler zero suppresses ππ elastic scattering strongly near threshold. To a first approximation, it cancels the pole in the denominator, with the result that δ(s) is rather featureless at low mass and shows little direct evidence for the pole. The same thing happens in Kπ elastic scattering, where there is an Adler zero at sA≃MK2−Mπ2/2. The LASS data for Kπ elastic scattering [11] display slowly rising phase shifts at low mass, again because the Adler zero suppresses the effect of the κ pole.In elementary quantum mechanics, the rate for a transition is (4π2/h)|F|2, where F is the matrix element for the transition. The width of a resonance is proportional to this transition rate. It is therefore logical to include the Adler zero explicitly into the width. The idea behind this is that the matrix element for ππ elastic scattering goes to zero for massless pions as k→0. In production reactions such as D+→π+σ, the pions are no longer ‘soft’ and no Adler zero is required.The Adler zero was introduced into the width in this way in an earlier publication [12], where Γ(s) was written: (6)Γ(s)=ρ(s)s−sAM2−sAexp−k2−k2rβ2, where kr is the pion momentum in the ππ rest frame at s=M2. The exponential cuts off the increasing width at large s. Today, that parametrisation is not quite accurate enough: it gives too large a scattering length. It is now replaced with (7)Γ(s)=ρ(s)s−sAM2−sAf(s)exps−M2A. Several simple forms for f(s) have been tried: f or f−1=b1+b2s or c1+c2k2. Empirically, the best form is f(s)=b1+b2s. A combined fit is made to CERN-Munich phase shifts for π−π+→π−π+[13], Ke4 data [10], CP violation in K0 decays [14] and data for π−π+→π0π0[15,16]. The contributions to ππ elastic scattering from f0(980), f0(1370) and f0(1500) have been added to the σ phase, following the Dalitz–Tuan prescription [17]. The σ pole lies at M=(525±40)−i(247±25) MeV, i.e., a full width ∼494 MeV.The fit is shown by the curve in Fig. 1(b). There are some systematic discrepancies above 740 MeV between phase shifts of Hyams et al. (filled circles) and π0π0 data (open circles). Kaminski et al. [18] have refitted the CERN-Munich moments. They find two slightly different solutions in this mass range. Using the Roy equations, they select one of these as the correct solution. Their result is close to ours, and does not reproduce the slight dip in the phases shifts of Hyams et al. above 800 MeV. This discrepancy is the largest source of error: the inclusion of the π0π0 data has the effect of increasing the mass of the σ pole by 22 MeV. Further systematic errors from the choice of f(s) are included in the quoted errors.The pole found here is in reasonable agreement with the result of Colangelo et al.: M=(490±30)−i(290±20) MeV [19]. Their determination has the virtue of fitting the left-hand cut consistently with the right-hand cut, using the Roy equations. They do not fit data above 800 MeV. Markushin and Locher [20] show a table of earlier determinations which cluster around the value obtained here, with quite a large scatter. Eq. (7) does give rise to other distant poles, which are ill-determined, but no others close to the region of interest.The mass and width are correlated as shown in Fig. 2(a) by the 1σ contour A. E791 and DM2 data, read from graphs, determine an optimum shown by ellipse B; the width is 50% larger than that determined by E791 because of inclusion of the Adler zero into the width. All data taken together determine a pole position of M=(533±25)−i(247±25) MeV. Errors are largely systematic and cover a variety of forms for f(s) in Eq. (7). From this combined fit, parameter values are M=0.9264 GeV, A=1.082 GeV2, b1=0.5843 GeV, b2=1.6663 GeV.The full curve on Fig. 2(b) shows the intensity of the σ in production data, Eq. (1), and normalised to 1 at the peak at s=460 MeV; incidentally, this curve is well approximated by Γ=const., without any phase space factor ρ(s). The intensity observed in a process such as D+→π+σ will be multiplied by the available 3-body phase space. The dotted curve on Fig. 2(b) shows the line-shape fitted by E791; it increases at low mass and is symptomatic of the second pole below threshold.A popular misconception is that the pole needs to be close to the mass M where the phase shift passes through 90° on the real axis. That is true only when Γ is constant. The Cauchy–Riemann relations for an analytic function state that d(Ref)/d(Res)=d(Imf)/d(Ims) and d(Imf)/d(Res)=−d(Ref)/d(Ims); because Γ varies with s, the phase of the amplitude varies as one goes off the real s axis to imaginary values. Fig. 2(c) shows the position on the complex plane where the phase goes through 90°; there is a rapid variation with Imm.Now we turn to the κ pole. The LASS data for Kπ elastic scattering are fitted with Eq. (7). However, there are no comparable data to the Ke4 data which determine the ππ scattering length and effective range, so one can set b2=0. As for ππ, it is assumed that phases of the κ and K0(1430) add. For K0(1430), a Flatté form is used: (8)Γ(s)=s−sAM2−sAg1ρKπ(s)+g2ρKη′(s). There is some freedom in fitting the K0(1430), because it rides on top of the κ ‘background’. It is therefore constrained to fit the line-shape of the strong K0(1430) peak in BES data presented by Wu [4]. As other authors have found, there is no improvement if decays to Kη are included. Fitted parameters for the κ are M=3.3 GeV, b1=24.49 GeV, A=2.5 GeV2. These parameters are highly correlated and one can obtain almost equally good fits over the range M=2.1 to 4 GeV by re-optimising b1 and A. The K0(1430) is fitted with M=1.513 GeV, g1=0.304 GeV, g2=0.380 GeV, giving a pole position of 1419−i158 MeV; that is close to the values fitted by Aston et al.: M=1412±6 MeV, Γ=294±23 MeV. (The full width Γ is twice the imaginary part of the pole position.)The fitted Kπ scattering length is 0.19mπ−1, very close to Weinberg's prediction from current algebra of 0.18mπ−1 [9]. Fits to Kπ phase shifts are shown in Fig. 3(a) and to the intensity of the elastic scattering amplitude in Fig. 3(b). The pole position of the κ is at M=722−i386 MeV. Realistic systematic errors are ±60 MeV on ReM and ±50 MeV on ImM from correlations between fitted values of M, b1 and A.The fitted value of M for the κ is very high. It is essential that it lies above the K0(1430) since the phase shift does not reach 90° before that resonance. As shown on Fig. 3(d), the fitted κ phase reaches ∼70° at 1600 MeV. It is possible that the phase for real s does not actually reach 90°. A similar case arises for the well-known NN 1S0 virtual state, where the phase shift rises to 60° and then falls again.The full curve in Fig. 3(c) shows the intensity of the κ amplitude as it will appear in production processes, i.e., according to Eq. (1) without Γ(s) in the numerator; the curve is normalised to 1 at the peak at 760 MeV. In production reactions such as D+→π+κ, the observed intensity will be multiplied by the available three-body phase space.We now return to the information contained in the matrix element F(s) for σ→ππ. This matrix element is proportional to the square root of Γ(s). It is interesting to take the Fourier transform of the matrix element in order to study its radial dependence: (9)F(r)∝∫0∞k2dkF(k)sinkrkr∝1r∫0∞dsF(s)sinkr. Here r is the relative separation r1−r2 between π+ and  π−. Fitted values of F(s), F(r) and r2F2(r) are shown in Fig. 4. The striking result is that r2F2(r) is quite compact, with an RMS radius of 0.39 fm.There is, of course, uncertainty about how to parametrise F(s) above 1 GeV. To examine the sensitivity of F(r) to this, a variety of exponentials have been used in Eq. (9) to parametrise the fall of Γ(s) at large s. What happens is familiar from the general properties of Fourier transforms. Changing F(s) at large s affects mostly F(r) at small r. An illustration is given by the dashed curves in Fig. 4. The cut-off in F(r) around r=0.55 fm is little affected, but F(r) changes considerably at small r. In this case, the RMS radius increases to 0.49 fm. Another remark concerns the small structure at r>0.7 fm. It seems likely that this is spurious, arising from small systematic errors in F(s): a well-known problem with Fourier transforms.Curves on Fig. 4 measure the matrix element for the ππ interaction. The wave function of the σ could however extend to larger radius if the interaction between the pions is of short range. This remark is also relevant to the f0(980) and a0(980), whose masses are so close to the KK̄ threshold that they inevitably have long-range KK̄ components in their wave functions (like the deuteron), even though their formation may depend on short range forces.The form of F(s) up to 1 GeV is not dependent on the precise form used for Γ(s) in Eq. (7). The phase shifts themselves determine Γ(s) via the relation tanδ=Γ(s)/(M2−s); the form for Γ(s) plays its role only in fitting production data to a formula consistent with elastic data.A further technicality is that a more general form for the Breit–Wigner denominator is [21]M2−s−m(s)−iMΓ(s), where (10)m(s)=M2−sπ∫MΓ(s′)ds′(s−s′)(M2−s′). Trials with this more general form produce only small changes. The reason is that, in the mass range of interest for the σ and κ poles, Γ(s) rises slowly and nearly linearly with s. Then m(s) is close to a constant; the change due to m(s) may be accomodated by shifting M2 a constant amount.We close with some general remarks on the nature of the σ and κ. The Adler zeros for both play key roles. It has been suggested [20,22,23] that meson exchanges, dominantly ρ exchange, can generate the σ. The range of this exchange is ∼0.25 fm, of the same order as the RMS radius of r2F2(r). Markushin and Locher illustrate in their Fig. 1 of Ref. [20] the variation of the σ pole position with increasing strength of attraction in three models. From this figure, it is apparent that the Adler zero plays the role of frustrating the formation of a bound state; it pushes the pole away into the complex s plane. The pole position in s for the σ is at 0.22−i0.27 GeV2 and for the κ at s=0.37−i0.56 GeV2.Many authors go on to suggest that the σ, κ, f0(980) and a0(980) may form a nonet [24–27]. For the a0(980) and f0(980), the dominant exchange diagram is K∗ exchange in ππ→KK̄. This may explain why f0(980) and a0(980) lie close to the KK threshold.11I am grateful to Prof. B.S. Zou for this remark. The recent discovery of the Ds(2317) and Ds(2460) by BaBar [28] and CLEO [29] suggests bound states of DK and D∗K; if so, it supports the possibility that f0(980) and a0(980) may be quasi-bound states of KK̄. In all these cases, there is no Adler zero to inhibit quasi-bound states. 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