Time Delay Plots of Unflavoured Baryons N. G. Kelkar1,3, M. Nowakowski2,3, K. P. Khemchandani1 and S. R. Jain1 1Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085, India 4 2 Universita¨t Dortmund, Institut fu¨r Physik, D-44221 Dortmund, Germany 0 3 Departamento de Fisica, Universidad de los Andes, Cra. 1E No.18A-10, 0 2 Santafe de Bogota, Colombia n a Abstract J 0 We explore the usefulness of the existing relations between the 1 S-matrix and time delay in characterizing baryon resonances in pion- 2 nucleon scattering. We draw attention to the fact that the existence v of a positive maximum in time delay is a necessary criterion for the 7 existence of a resonance and should be used as a constraint in con- 9 1 ventional analyses which locate resonances from poles of the S-matrix 8 and Argand diagrams. The usefulness of the time delay plots of reso- 0 nances is demonstrated through a detailed analysis of the time delay 2 0 in several partial waves of πN elastic scattering. / h p PACS numbers: 14.20.Gk, 13.85.Dz, 03.65.Nk - p e h 1 Introduction : v Xi Ever since the discovery of the first excited nucleon state [1], the baryon r resonances have played a major role in particle and nuclear physics and have a contributed crucially to the search of the fundamental building blocks of nature. We perceive them now as the low energy manifestation of quan- tum chromodynamics (QCD) with three quark degrees of freedom. However, low energy QCD is still not well understood and very often one is left with models. The status of the resonances can differ from case to case as can their parameters extracted from different experiments. The N∗ programs at Jefferson Lab [2] and the forthcoming Japan Hadron Facility [3] have both revived the area. The various theoretical studies [4-7] and the hope to find exotic hadrons [8], make the area once again an exciting field of physics. It is then justified to look at the baryon resonances from a yet different per- spective and to analyze the existing data in a novel albeit well established 1 way, while waiting for new experimental results. Specifically, we refer to the time delay method which was introduced into scattering theory and es- pecially resonance physics by Eisenbud and Wigner [9, 10, 11]. In view of the considerable number of standard textbooks [12-17] and numerous papers [9-11,18-39] published since the seminal paper by Wigner [10], we provide only a short introduction. Time delay is a measure of the collision time in a scattering reaction which can be calculated directly from the phase shift or the T matrix. Obviously, such a concept has a close connection to the appearance of an unstable intermediate state (resonance), which, due to its finite lifetime, “delays” the reaction. Though the interest in time delay ever since the first papers was unabated, it is only recently that it has been used in practice in quantum scattering theory (in chaotic scattering [40], hadron resonances [33,37-39] and heavy ion collisions [35]) and tunneling phenom- ena [36], with success. The present work is simply a logical extension of this program carried over to baryon resonances, partly done already in [37]. To identify the baryon resonances, one performs a partial wave analy- sis of the meson baryon scattering data and obtains the energy dependent amplitude (or T-matrix) by fitting cross section data. Resonances are then determined by locating the poles of the T-matrix on the unphysical sheet and studying the Argand diagrams of the complex T-matrix. Due to model dependence in the analyses of the energy-dependent amplitudes, there are differences in the resonance parameters quoted by different groups [41]. The resonance receiving confirmation from several analyses is considered to be well established. Though we do not dispute the usefulness of the pole of the S-matrix, we note that there exist several views in literature, regarding the definition of a resonance. In a review article [42], Dalitz discussed various criteria for the existence of a resonance elaborately, with the conclusion that for the case of a pole in the S-matrix, S(E), in the unphysical E-plane lying sufficiently close to the physical E axis, there is no ambiguity in the conclu- sion of the existence of a resonance. However, the authors in [43] constructed examples in such a way that a sharp resonance was produced without an ac- companying pole in the unphysical sheet. They noted that even the inverse correspondence, namely, (pole of the S-matrix on the unphysical sheet) → (unstable particle) may be questioned. In [44], it was pointed out that a peak in the cross section cannot be conclusive evidence of a resonance. In [45], in addition to time delay, the exponential decay law was required as a signal of a genuine resonance (this may be in view of the existence of double 2 poles, which would lead to a non-exponential decay [46]). Cautious remarks on the use of Argand diagrams can be found in [19, 47]. The many different opinions reflect only the fact that the issue is not yet satisfactorily settled. Indeed, unstable particles remained to be problematic even until now [48]. We make use of the requirement stated in literature and text books [9-39], namely, the formation of a resonance should introduce a large positive time delay in the scattering of particles. We try to extract resonance parameters from the energy distribution of time delay by locating the position of the local maximum and reading off the width as advocated e.g. in [49]. Though the non-resonant background can deform the positive resonant structure in the vicinity of a resonance, we do expect some positive region around the resonance point, with perhaps a less dominant peak. This is confirmed by our study. Starting with the definition of time delay in terms of the S-matrix, we obtain its relation with the T-matrix and scattering phase shifts. We shall first demonstrate the usefulness of the method with examples of well-known N and ∆ resonances. Later on we proceed to the analysis of time delay in various partial waves of πN elastic scattering, using the available single energyvaluesaswellassomeenergy-dependent formsoftheT-matrix. Before we move on to the discussion of time delay, it is important to note that the time delay plots of the present work are not the same as speed plots [50] which have been sometimes referred to as time delay plots in literature. Speed plots are positive definite by definition. Time delay plots can also assume negative values and only a positive peak signals a resonance. In the elastic region, the speed is equal to time delay up to a constant factor, but once the inelastic channels open up, this is no longer true [37]. Considering the fact that the time delay method has so far not been applied to baryon resonances (but has been successfully applied to meson resonances [39]), our study is a practical test of time delay when applied directly to data. When applied to theoretical T matrix solutions, we could say that indeed the model is being tested, if we consider the resonances to be well established. In passing, we note that time delay is also related to the so called arrival time in quantum mechanics [51] and has also been used to obtain the density of resonances [52]. 3 2 Time delay in resonant scattering Weshallnowdiscuss theexpressions whichquantifytimedelayandcanhence be used to characterize resonances. 2.1 Relation to phase shifts In the early fifties, using a wave packet analysis, Bohm [15], Eisenbud [9] and Wigner [10], obtained an expression for the time delay ∆t in binary collisions. In the case of elastic scattering, they derived ∆t in terms of the energy derivative of the scattering phase shift as follows: dδ ∆t = 2h¯ . (1) dE The formation of a resonance in a scattering process, introduces a positive time delay between the arrival of the incident wave packet and its departure from the collision region. From the above relation, one expects the phase shift to increase rapidly in the vicinity of a resonance. The wave packet analysis of time delay was extended by Eisenbud to inelastic collisions [9]. He defined the delay time matrix ∆t, such that an element ∆t of this matrix, corresponded to the time interval between the ij outgoing wave in channel j and the ingoing wave in channel i. This time delay, ∆t , is related to the S-matrix as follows: ij dS ∆t = Re[−ih¯(S )−1 ij]. (2) ij ij dE Before we proceed further, we note that the phase shifts, in principle, depend on the orbital angular momenta, l, l′, of the initial and final states respectively and on the total angular momentum J. However, we have sup- pressed this dependence in the expressions whenever not relevant. In the present work, we consider πN elastic scattering, which is the scattering of a spin zero and spin one half particle. Since the total spin in the final and initial state is S = S′ = 1/2 and conservation of parity gives l = l′, the total angular momentum J takes the values l−1/2 and l+1/2. The S-matrix is diagonal in l and its elements are related to phase shifts as SJ = exp(2iδJ), ll for the elastic case in the absence of inelasticities. 4 We see that in the case of purely elastic scattering (j = i), and using a phase shift formulation for the S-matrix where S = e2iδ, we get, dδ ∆t = 2h¯ , (3) ii dE which is the same as Eq. (1). These ∆t are related to the lifetimes of ii metastable states or resonances in elastic scattering (see [18]). At high en- ergies, where apart from elastic scattering, the possibility of scattering into inelastic channels also opens up, the elastic S-matrix element is defined as S = ηe2iδ, where η is the inelasticity parameter defined such that 0 < η ≤ 1. Substituting the modified S (i.e. S = ηe2iδ) in Eq. (2), gives, dδ dη 1 dδ ∆t = Re −ih¯ 2i + = 2h¯ . (4) ii dE dE η dE (cid:20) (cid:18) (cid:19)(cid:21) The above equation is the same as Eqs (1) and (3). Thus it can be seen that the expression for the time delay, ∆t , for elastic scattering is the same, ii irrespective of the presence of inelastic channels. It is clear from the above expressions that time delay can also take nega- tive values resulting from phase shifts which decrease as a function of energy. However, the negative delay times cannot assume arbitrarily large values. In the case of elastic scattering (for the case of l = 0 and 1) it was shown by Wigner [10], that the causality condition puts a constraint on the lower value of the phase shift derivative (related in an obvious way to time delay), which in case of high momenta, i.e. for large k is given as, dδ /dk > −a. a can l be interpreted as the range of the interaction potential. We do observe some regions of large negative ∆t which will be discussed in Section 3. 2.2 Relation to T-matrix Instead of using the phase shift formulation of the S-matrix, we now start by defining the S-matrix in terms of the T-matrix, i.e., S = 1+2iT , (5) as is usually done in partial wave analyses of resonances [53, 54]. The matrix T contains theentire informationoftheresonant andnon-resonant scattering andiscomplex (T = TR+iTI). Substituting fromEq. (5)intotheexpression 5 for time delay in (2), the time delay ∆t , in terms of the real and imaginary ii parts of the amplitude T is given as, dTR dTI dTR S∗S ∆t = 2h¯ ii +2TR ii −2TI ii , (6) ii ii ii dE ii dE ii dE (cid:20) (cid:21) where S∗S can be evaluated using Eq. (5). In the present work, we have ii ii evaluated the time delay in πN elastic scattering and hence, i corresponds to πN in the above equation. Although a simple Breit-Wigner (BW) is not always a good choice to describe a broad hadronic resonance, it is instructive to see the results we get for time delay, starting from a BW matrix element. If we insert one such commonly used form of the T-matrix [54] in resonance regions, namely, Γ/2 T = , (7) E − E − iΓ/2 R in (6), we obtain, h¯Γ ∆t(E) = (8) BW (E −E)2 + 1Γ2 R 4 and the time delay at the resonance energy E (within the assumption that R the widths are not energy dependent) is, 4h¯ ∆t(E ) = . (9) R BW Γ A simple BW T-matrix, however, can be misleading, especially while dis- cussing time delay. The reason among others is that it lacks certain usually expected properties (threshold behaviour being one of them). We shall come to this point in greater detail in section 4. Before ending this section, we note the dependence of time delay on wave packets. It is well-known that the survival probability and lifetime of an un- stable quantum state depend on its preparation. Explicit formulae including wave packets can be found for unstable neutral kaons in [55]. We expect a similar dependence to be present in the expressions for time delay. Indeed, as given in [16], ∞ dδ ∆t(E) = 8π2h¯ dE′|A(E′,E)|22 , (10) 0 dE′ Z 6 whereA(E′)istheinitialwavepacketinmomentumspace. Ifthewavepacket is sharply centered around an energy E, we recover Eq. (1). In scattering processes where one measures the cross sections and distributions, the wave packets are indeed narrow, i.e., the energy spread ∆E ≪ Γ (see the second and last reference in [13] for a discussion of this issue). Hence we can use Eqs (1-4) to calculate time delay. In the next section, we shall evaluate the time delay in several partial waves of πN elastic scattering. We have checked that the values of time delay, ∆t , obtained either using the derivative of the real phase shifts as in ii Eq. (3) or the T-matrix as in Eq. (6) are the same. Since both the methods are equivalent, one can in fact use fits to the single energy values 1 of phase shifts to extract resonance parameters. 3 Time delay plots of resonances in πN elastic scattering We now analyze the existing πN scattering data using time delay plots. To demonstrate the usefulness of the method, we plot time delay in the energy regions where two well-known baryon resonances occur. In Fig. 1 are shown the real and imaginary parts of the complex T-matrices, the phase shifts and the corresponding time delay in the P and D partial waves in πN 33 13 scattering, evaluated using the T-matrices (solid lines) which fit the single energy values of T very well. The filled circles in Fig. 1 are the single energy values of phase shifts extracted from the cross section data on πN elastic scattering [56]. The widths of the P and D peaks at half maximum can be 33 13 readfromFig. 1 to bearound116 and50MeV respectively. The peaks inthe energy distributions occur at 1216 and 1512 MeV respectively. The average values of Breit-Wigner masses (widths) given in the Summary Table (ST) of the Particle Data Group [41] for these P and D resonances are 1232 (120) 33 13 and 1520 (120) MeV respectively. The ∆(1232) decays almost 100% to the 1 The values of phase shifts in different partial waves obtained by fitting the cross section data at the available energies are known as single energy (SE) values of phase shifts. Theerrorbarsonthesephaseshiftsnaturallydependontheerrorsinthemeasured cross sections. The elastic T-matrix element is related to the phase shift and inelasticity 2iδ parameter, ηl, as: Tl = (ηle l − 1)/2i. Thus, one can also obtain SE values of the T-matrix. 7 1 Re T Im T 0.5 0.5 T 0 0 Fit −0.5 150 150 g) e δ (d 100 100 P D 50 33 50 13 c) 3 1216 6 1512 e −2310 s 2 4 Γ ~ 50 MeV E) ( 1 Γ ~ 116 MeV 2 ∆t( 0 0 1.1 1.2 1.3 1.4 1.4 1.5 1.6 E (GeV) E (GeV) Figure 1: Single energy values of the real part of the T-matrix (filled tri- angles), imaginary part of T (open squares), phase shifts (filled circles) and the time delay ∆t evaluated in the P and D partial waves of πN elastic 33 13 scattering. The time delay is evaluated using the T-matrix given by the solid lines which fit the single energy values very well. πN channel and hence the time delay width seems to be in good agreement with the above value listed in the ST. The D has a branching ratio of 50 13 to 60% to the πN channel and the width of the time delay distribution is consistent with the partial width listed in the ST. Thus we see that in the case of purely elastic scattering as well as in the case of elastic scattering in the presence of inelastic channels, the method is quite useful. The peak position and width of the time delay distribution give the mass and elastic partial width of the resonance, respectively. Interestingly, the P phase shift of the only πN resonance (∆(1232)) 33 in the elastic region, remains positive and shows the characteristic resonant jump in this region. Hence, in this case, the speed defined in [50] is the same as time delay up to a constant factor. 8 3.1 New resonances from single energy values of phase shifts We shall now evaluate time delay from fits to single energy (SE) values of phase shifts. Since the results depend crucially on the quality of the data, we chose data sets with small error bars and made separate nth order polynomial fits to different energy regions of the phase shift. It would be more appro- priate to consider error bars and perform a χ2 fit, with a certain function. However, such a procedure would not be able to pick up the small structures and would amount to giving results similar to the energy dependent ones. We also chose to fit SE values of phase shifts rather than the SE values of real and imaginary parts of the T-matrix, simply as a matter of convenience. The time delay evaluated using fits to phase shifts or T-matrices is actually the same. The advantage of calculating time delay from such fits is that the results are directly related to data. The disadvantage is that they are sensi- tive to the quality of the data and hence to the fit. There also exists the well known continuum ambiguity problem with the SE values of phase shifts [58]. However, the present work does not aim at finding solutions to the problems related to the extraction of SE values. Hence, we use the values as available in literature and check if we still get some useful results for time delay. We perform this analysis for the I = 1/2 partial waves, P , P , D , 11 13 13 S and F in πN elastic scattering. We note that in spite of the above 11 15 mentioned problems, we get strikingly similar peak positions and widths as compared to the Summary Table resonance parameters. The results in Figs 2 and 3 reveal that time delay has the following main characteristics: (i) it locates well established resonances (ii) the positive peaks are more prominent than in the case where we calculated the same quantityfortheenergy-dependent solutions(seesection3.2below), (iii)there exist regions of negative time delay in addition to the positive peaks (iv) the new feature here is that we find additional resonant peaks. At present, given the quality of the data, it is not clear if these new structures are artifacts of the fit to the data or genuine indications of (new) resonances. For example, in the P and S cases, we have hints for new resonances in the higher 11 11 energy regions where the quality of data is worse. On the other hand, some one-star resonances like P (2100) and S (2090) are not excluded. We find 11 11 evidence for the 3-star resonances F (2000) and P (1900), with some indi- 15 13 cation that the latter consists actually of two nearby resonances. In the D 13 9 150 100 S11 g) e d δ( 50 ec) 2 1.65 s 1.508 −230 1.81 1.92 E) (1 0 ∆t( −2 1.4 1.6 1.8 2 E (GeV) Figure 2: Single energy values of phase shifts and the corresponding time delay in the S partial wave evaluated from a smooth fit to the phase shifts. 11 partial wave, we observe distinct peaks at 1512, 1695 and 1940 MeV, which could be associated with the 4-, 3- and 2-star resonances N(1520), N(1700) and N(2080) respectively. Note however that the existence of the two small peaks in this case depends crucially on two data points, and hence on the fit. These two points are sufficiently above continuum to justify the peaks (more so as they can be associated with known resonances). There seems to be more structure in the 1400−1700 MeV region of S . A fit made to 11 this detailed structure reveals the possibility of four resonances around 1650 MeV. Indeed, there is some support for this structure from recent works in literature [5, 59], where the existence of new resonances at 1.6 and 1.7 GeV is predicted within quark models. With the availability of more precise data on cross sections which would enable a better extraction of the SE values of phase shifts, one could locate theresonancesfromtimedelayplotsmoreaccurately. Welimitthediscussion in this section only to the 5 partial waves shown in Figs 2 and 3, since a detailed analysis with all partial waves would make sense only when the SE values of phase shifts would be better known. We list our findings in Table 1. 10