Zero-contours in low-energy K-π scattering

SummaryThe paths of zeros of the low-energy K-π scattering amplitudes

$$A^{I_s = \tfrac{3}{2}} $$

 and

$$A^{I_t = 1} $$

 are examined in a simple K* and ρ dominance model, where it is found that the roles of PCAC zero, K* and ρ Legendre zero, and double-pole-killing zero are played by the same zero-contour in the (s, t, u)-plane, an ellipse for the first amplitude, an ellipse plus the lines=u for the second one. The behaviour of these zero-contours in theu-channel physical region is studied in comparison with the experimental zeros determined from the K+π−→K+π− and K+π0→K0π− scattering results, and it is found that up to the K* mass region the elliptic contour of

$$A^{I_s = \tfrac{3}{2}} $$

 is essentially unaffected by unitarity, whilst the elliptic contour of

$$A^{I_t = 1} $$

 is slightly more sensitive to such constraints. A comparison with the behaviour of the zeros of the corresponding π-π amplitudes

$$A_{\pi \pi }^{I_s = 2} $$

 and

$$A_{\pi \pi }^{I_t = 1} $$

 is also performed.RiassuntoSi esaminano i percorsi di zero delle ampiezze di scattering a bassa energia

$$A^{I_s = \tfrac{3}{2}} $$

 e

$$A^{I_t = 1} $$

 in un semplice modello di dominanza di K* e ρ, in cui si trova che i ruoli degli zeri di PCAC, di Legendre K* e ρ, e di eliminazione dei poli doppi sono sostenuti dagli stessi contorni di zero nel pianos,t,u: un ellisse per la prima ampiezza e un ellisse più la lineas=u per la seconda. Si studia l’andamento di questi contorni di zero nella regione di interesse fisico del canaleu, a confronto degli zeri sperimentali determinati a partire dai risultati degli scattering K+π−→K+π− e K+π0→K0π− e si trova che fino alla regione della massa del K* il contorno ellittico di

$$A^{I_s = \tfrac{3}{2}} $$

 essenzialmente non risente dell’unitarietà, mentre il contorno ellittico di

$$A^{I_t = 1} $$

 è leggermente più sensibile a queste condizioni. Si fa anche un confronto con l’andamento degli zeri delle ampiezze ππ corrispondenti

$$A_{\pi \pi }^{I_s = 2} $$

 e

$$A_{\pi \pi }^{I_t = 1} $$

.РеэюмеИследуются траектории нулей амплитуд

$$A^{I_s = \tfrac{3}{2}} $$

 и

$$A^{I_t = 1} $$

 рассеяния K-π при ниэких знергиях в простой модели с доминантностью K* и ρ, где обнаружено, что роли нуля PCAC, нуля Лежандра для K* и ρ и нуля, подавляюшего двойной полюс, исполняют нулевой контур в (s, t, u)-плоскости, зллипс для первой амплитуды, зллипс плюс линияs=m для второй амплитуды. Поведение зтих нулевых контуров в фиэической области μ-канала сравнивается с зкспериментальными нулями, определенными иэ реэультатов рассеяния K+π−→K+π− и K+π0→K0π−. Получено, что вплоть до массовой области K* зллиптический контур

$$A^{I_s = \tfrac{3}{2}} $$

, по сушеству, не эависит от унитарности, тогда как зллиптический контур для

$$A^{I_t = 1} $$

 является чувствительным к таким ограничениям. Также проводится сравнение поведения нулей соответствуюших π-π амплитуд

$$A_{\pi \pi }^{I_s = 2} $$

 и

$$A_{\pi \pi }^{I_t = 1} $$

.


-Introduction.
A few years ago ODORICO has provided a possible explanation of mesonmeson scattering data in termt~ of a hypothesis of a global structure for the straight-line propagation of nearby zeros of the scattering amplitude, whereby lineR passing through the inter~;ectionH of resonances in different channels also pass through the Legendre zeros of resonances in the physical regions of the  1 Since then numerous studies of zero-contours of the 7t-7t scattering amplitude have been performed using the experimental phase shifts { 4 -7 ). Several of them disagree with certain predictions made by ODORICO, such as for example the analyses of PENNINGTON and PROTOPOPESCU { 4 ) and of EGUCHI et al. ( 5 ) which contest the suggestion that the anomalous behaviour of the experimental moment (Y~) at 0.98 GeV may be seen as a consequence of the entry in the physical region of a zero at fixed s = 4f1 2 -2m~. Nevertheless on the whole these studies find this straight-line zero hypothesis as a reasonable approximation to reality when viewing the Mandelstam plane from afar. However in any local region zero-contours are far from straight, and this is especially evident in the low-energy region as is discussed by PENNINGTON and SCHMID in ref. ( 8 ). The theoretical analysis of ARNEODO, GUERIN and DoNOHUE ( 9 ) is also very significant of this feature since it emphasizes that, in a simple nonunitary p dominance model, the nearby zero-contours of low-energy 7t-7t scattering amplitudes A= (s, t, u) and A~;; 2 (s, t, u) form a closed curve: a circle in the (s, t, u)plane. This circle plays the respective roles of POAO zero, p Legendre zero and double-pole-killing zero, and is in large part rather stable when unitarity is enforced.
Such experimental and theoretical studies have not been performed in K-7t scattering. Thus we propose in this article to examine the behaviour of the zero-contours of the low-energy K-7t amplitudes; first in a pure K* and p dominance model, which is a model for the whole amplitude and consequently permits us to define the zero-contours in the unphysical as well as the physical regions; then as determined from the experimental phase shift analyses of the reactions K+n-~ K+7t-and K+1t 0 ~ K 0 1t+ ( 10 -13 ), i.e. only in particular physical ( 4 ) M. R. PENNINGTON       The main re~ult of our investigation i:-; that, in the narrow-wiclth K* and p dominance approximaJ.ion, t.he zero-eontours of the K-rr amplitude:-; AI,~! an<l AI,~t are re,;peetively an ellip1-1e and an ellipse plm the line s = u in the (s, t, u)-plane. Tlwse contour;; are related to the different zero:-; arising at low energy: the Adler-Weinberg zero, the Legendre zero of K* and p mesons and the double-pole-killing zeros. Moreover the behaviour of the experimental complex zeros imggeHts that in the low-energy u-ehannel physieal region (u ~ ~ mi:.) the elliptic zero-contour of the amplitu<Le AI,~! i:-; e;;sentially unmodified by the unitarity com;traintK, while the elliptic zero-contour of the amplitude A I,~t iH more sensitive to unitarization and is drawn into a somewhat straight shape lying at t ,__,-0.:! (GeV)2. This ,;imple K* and p dominance model is found to be unreliable for energies above 0.95 GeV, since it predicts nearby zero-contours for the both K-rr seattering amplitudes while experimentally the zeros appear to be far away in the complex (s, u)-space, and consequently the term zero-contour is ambiguom; at wch energies.
We organize the paper as follow~>. In Sect. 2 we examine the zero-contours of the simple K* and p dominance model. In Sect. 3 we discuss how they are influenced in the K* mass region by the unitarity comtraint~. In Seet. 4 we compare them with the zero-contours ealculated from the experimental phase shifts. In Seet. 5 we perform a compariRon of zero-contours in rr-rr and K-rr scattering. Kinematie~ and amplitudes in K-rr scattering are defined in the Appendix.
The contribution of both p and K* resonances to the K-rr scattering amplitudes A(s, t, u) and B(s, t, u), which respeetively represent in the s-channel physical region the reactions K+rr+--+ K+n+ and K+rr---+ K"n", may be written in the Breit-Wigner approximation ail and where it is explicitly assumed that the p couples in the same way to the 1t1t and KK channels. The respective kaon and pion mass are m and ,u, and the details of the K-7t kinematics are defined in the Appendix.
This simple model for the low-energy region has the analytic structure needed to incorporate the cuts due to elastic unitarity, but it has no cuts corresponding to inelastic unitarity. One can justify this absence of double-spectral function in so far as the inelastic effects are unimportant, i.e. up to 1 GeV ("). Moreover this model gives the correct threshold behaviour for the real parts of all K-7t partial-wave amplitudes, and whilst one cannot determine the scattering amplitudes everywhere, this nonunitary approximation may be considered as reasonable in the low-energy region. (Some problems arising from the incorrect behaviour of the imaginary part of the K-7t partial-wave amplitudes will be discussed later.) In the limit of narrow-width resonances: rK. and rp small, both amplitudes A(s, t, u) and B(s, t, u) are real except at t =m~, u = mi• and s = mi• (only for B(s, t, u)), and the respective zero-surfaces intersect the real (s, t, u)-plane along the curves defined by  Before developing this study of complex zeros we must resolve a problem inherent in this simple Breit-Wigner model which consists in an incorrect behaviour of the imaginary part of the I=! K-n S-wave amplitude. If, on the one hand, the expressions (2.1) and (2.2) do not give a correct threshold behaviour for the imaginary parts of the partial-wave amplitudes, on the other hand they  imply more dramatieally that the imaginary part of the I=·~ 8-wave is negative, though quite small (< ().06), up to the K* mass. In order to eliminate this difficulty we added linear terms in q" q 1 and q,. to the simple Breit-\Vigner approximation to the K-7t amplitudes where ABw(s, t, u) and BBw (8, t, u) are given by the expressiom (2.1) and (2.2), and ex, (J, y are parameters. vVe fixed these parameters by imposing the unitarity contraints at threshold in the 8-and t-channels. In the 8-channel we have where g(s) are the 8-wave amplitudes (I=·~ or il defined from (2.6) and (2.7) (see the Appendix), and a~ the corresponding seattering lengths. In the t-channel we have where ~(t) is defined in the Appendix and IIg is the I= 0 8-wave scattering length of the 7t-7t amplitude. In the following we Hhall consider IIg to be equal to 0.17p-1 , and we shall diHcuss the importance of thiH choice at the eml of this Section. We then find that ex=-0.575 (GeV}-1 , (J = 0.078 (GeV)-', Similarly eq. (2.8) makes the I= l S-wave unitary only at threshold. The other partial waves are not unitary even at threshold. Now, having resolved the important problem of the positivity of the imaginary part of the I= t S-wave (but not the general problem of unitarity), we are allowed to define the complex zeroR of the K-11: scattering amplitudes.
(1 5 ) J. P. ADER, C. MEYERS which defines a two-dimensional surface in the complex (s, u)-space. The following points lie on this surface: These points may be associated, in this order, with the zero of the Legendre polynomial at the p mass in the t-channel and at the K* mass in the u-channel physical region, and the double-pole-killing zero at the intersection of the p and the K*. The projections of these points on the real (s, t, u)-plane are eSSentially independent Of rp and FKo l provided that rp « mp and FKo « mKo, Their respective distances from this plane are characterized by Im t of order mPFP and Im u of order mK.rK., hence one expects the zeros of A(s, t, u) to be nearby zeros. But this generalization is not evident in view of the presence of linear terms in q,, qt and qu in the expression of A(s, t, u). So we traced in Fig. 1 the modulus contours of this amplitude in the real (s, t, u)-plane. It is clear from this Figure that there exists a curve along which A(s, t, u) shows a pronounced sharp minimum, and it is reasonable to associate this with a nearby zero-contour as is discussed in ref.
( 9 ). Moreover this curve of minimal JA J is not very different from the elliptic zero-contour of the narrow-width meson resonance approximation. The unitarization procedure at the threshold, defined by (2.8) and (2.9), tends somewhat to draw the original ellipse of zeros into a triangular shape with the horizontal base passing through the K*-p intersection, but this effect is indeed not very pronounced. As we have previously remarked, this quasi-elliptic contour of nearby zeros plays the roles of the p Legendre zero, the K* Legendre zero and the K*-p crossing zero. In addition it also playR the roh~ of the Adler-Weinberg zero, since it passes through the real (s, t, u)-plane region: {t.;;;4,u 2 , s and u,;;; (m+ ,u) 2 } not very far from the line s = m 2 + ,u 2 , which correspondH to a smooth extrapolation of the <·.urrent. algebra z1;ro to the on-maHH-Hhell amplitude.
2·2. Complex zeros of R(s, t, u). -From (:J.7) the zeroH of the amplitude B(s, t, u) are given by which defineH a t.wo-climensional wrface in the eomplex (s, u)-spaee. The following points lie on thiH ~mrfaee: These pointH emTeHpond respeetively to the K* Legendre zero in the sand u-channel physical regimu;, the p J,egendre zero in the t-ehannel and to the double-pole-killing zero at the K*-p and K*-K* interHectiom. They exhibit the ~mme features as tlw previously des<•ribed points of the zero-surfaee of am.
plitude A(s, t, u). In the same way the modulus contours of the amplitude B(s, t, u) attest to the existence of a nearby zero-contour, which is characterized by a curve (plus a line) of minimal \B\lying very close to the ellipse (the line s = u) of the original narrow-width meson resonance model (Fig. 2). Consequently the unitarization procedure at the threshold defined by (2.8) and (2.9) does not greatly disturb the behaviour of the zero-contour of B(s, t, u). This result is easily explained by the small value taken by the {J-parameter To conclude this description we want to discuss the sensitivity of the paths of zeros of the amplitudes A(s, t, u) and B(s, t, u) to the value of the I= 0 1t1t S-wave scattering length IIg injected in the unitarity constraint at the threshold in the t-channel (2.9). Previously this scattering length took the value given by the current algebra theory and the zero-contours lay very close to the elliptic forms given by the narrow-width resonance approximation. . Thus the original elliptic shape of the zero-contours is distorted, in particular for the amplitude A(s, t, u). Moreover the zero-surfaces go away from the real (s, t, u)-plane in certain regions such as the u-channel physical region of the two amplitudes A(s, t, u) and B(s, t, u), where they respectively represent the experimentally well-known reactions K +7t----* K +7t-and K +no---* ---* K 0 7t+, and the definition of the zero-contours becomes ambiguous. Therefore it appears clear that in a low-energy Breit-Wigner picture of the K-7t am- plitudes the zero-contours are stable features against the unitarization procedure at the threshold (in the three channels) only so long the I= 0 ' TC-' TC S-wave scattering length is compatible with the current algebra predictions.
3. -Zero-contours in the K * region and unitarity constraints.
In order to investigate the consequences of unitarity for the nearby zerocontours, we transpose the analysis which PENNINGTON and ScHMID have performed for ' TC-' TC scattering ( 8 ) to the case of K-'TC scattering.
In the u-ehannel physical region of the K-'TC amplitude A(s, t, u), only the S and P waves are important near the K* resonance. Hence the equation of zeros may be written as Moreover in the K* region we can reasonably neglect the I= ! P-wave and represent the I=-! P-wave by a unitary Breit-Wigner form. Using a phase shift representation for the S-waves, we obtain for the projection of the nearby complex zeros onto the real (s, t, u)-plane Re (cosO*(u)) = On the line u = mi:. the zero occurs for which is proportional to the value taken at the K* mass by the imaginary part of t,he S-wave amplitude in the u-channel physical region. This expression implies that the unitarity constraints cannot shift the intercept beyond the following limits:

-Zero-contours and low-energy K-7t scattering data.
There exist two different methods of describing zero-contours of an amplitude A(s, t, u): the contours of minimal modulus and the intersection of the hyperplane Im u = 0 with the surface A(s, t, u) = 0. One method, which we employed in our definition of the theoretical paths of zeros of the K-7t amplitudes in Sect. 2, corresponds experimentally to finding the minimum of the differential cross-sections. The other consists in determining the zero-contours for A(s, t, u) in the u-channel using the phase shifts of the dominant waves, which are experimentally accessible; one solves for the complex values of cos O*(u) such that A(u, cos O*(u)) = 0, and then projects onto the real (s, t, u)plane. The respective properties of these two methods are discussed in detail in ref. ( 9 • 23 ). The main result of these analyses is that both methods lead to the same definition of zero-contours if the amplitude possesses nearby zeros, which are illustrated by the existence of a curve (or curves) along which the amplitude shows a pronounced sharp minimum. On the other hand, if the zeros acquire a nonnegligible imaginary part, the modulus contours no longer present such strongly marked features and the definition of zero-contours becomes so ambiguous that the two methods lead to different conclusions. Consequently if one uses the second method of describing the experimental paths of zeros, as we propose to do in this Section, we must take into account not only the projection of the complex zeros onto the real (s, t, u)-plane but also their distance from this plane, i.e. the imaginary part of these zeros. Moreover, as it is always interesting to look at what happens in the differential crosssection, we shall do so and compare the eventual minima with the real parts of the complex zeros.
We determine the experimental zeros by using the three following phase ~>hift analyses: i) The analysis of the S and P K-n scattering wave phase shifts from threshold to mKrc = 1.  the D-wave starts to contribute and a <<second>> zero begins to approach the physical region. Thus, in this region of higher energies we have to refer to the only D-wave phase shift analysis, i.e. to the Firestone et al. experiment (Fig. 4 and 5). As is well known, we cannot determine the zero-contours far outside the physical region since a finite number of partial waves provides a poor rep-resentation of the amplitude there. Nevertheless we see that this <{second)} zero enters the physical region through the forward direction at approximately the place predicted by the simple Breit-Wigner model of Sect. 2, and with Jlm cos O*(u)J ~ 0.9. It is not a nearby zero and its entrance in the u-channel physical region causes the real part of the <{first )} zero to recoil somewhat towards the backward direction. Then the real part of the <{first )} zero lines up, ready to become one of the Legendre zeros of the K**(1420)-resonance, while the real part of the <{ second )} zero moves, ready to become the other K** Legendre zero. .At the same time that they approach the K** region, these complex zeros approach the real (s, t, u)-plane and they produce a more and more marked double-dip structure in the differential cross-section. Hence at u ~ mi-• both zeros are respectively characterized by  it represents in the physical region the reaction K+7t 0 ~ K 0 7t+) obtained from the analyses i) and ii) (*). One can see from these Figures that the deviation from the elliptic contour of the narrow-width K* and p dominance model seems to be more important for this amplitude than for the amplitude A(s, t, u).
Indeed the zero enters the physical region through the backward direction before the elliptic contour and with a small imaginary part, e.g. Jim cos{}*( u) I "' the B-and P-waves do not yield a good approximation to the amplitude B (s,t,u) at such energies. Moreover this lack of D-wave analysis does not allow us to study the entrance and the behaviour of the double-pole-killing zero in the u-channel physical region. We remark in addition that the experimental results suggest a possible contour of nearby zeros at t ~-0.2 (GeV) 2 for .Az•= 1 (s, t, u). Finally for completeness we have tried to define the paths of zeros of the K-7t scattering amplitude .A(s, t, u) in the t-channel from the experimental 7t-7t scattering results. This hope was illusory for the following reasons: where a finite number of partial waves provides a good representation of the amplitude, only the phase of the complex zeros is determined, i.e. Im cos O*(t)jRe cos O*(t) and not the separate real and imaginary parts of cos O*(t). For example, considering that this phase is nothing but the difference of the phases of I= 0 B-wave and P-wave of the 7t-7t scattering amplitude, we know that the complex zero of .A(s, t, u) passes through the real (s, t, u)plane for t ~ 0.5 (GeV) 2 , but we are unable to say where it does this on the straight line t = 0.5 (GeV) 2 • 5. -Comparison of zero-contours in 7t-7t and K-7t scattering.
It is interesting to compare the zero-contours predicted by the simple Breit-Wigner model for 7t-7t and K-7t scattering. In both cases the nearby zerocontours are found to be simple closed curves which act as Adler-Weinberg zeros, zeros of P-wave resonance Legendre polynomials and double-pole-killing zeros. For the 7t-7t amplitudes .A:t; 2 and .A~;; 1 the curves are circles ( 9 ), whereas for the corresponding K-7t amplitudes .A(s, t, u) and B(s, t, u) the curves are ellipses n.
These zero-contour predictions, in spite of the very simple model from which they are drawn, appear to be in good agreement with the experimentally obtained contours of nearby zeros, at least for energies up to ~ 0.95 GeV. However, the prediction that the double-pole-killing zero and the upper branch (*) The l't"-l't" amplitude Af[;; 1 and the K-1t amplitude B(s, t, u) are antisymmetric under the exchange 8~ u, and they also possess a zero line for 8 = u. of the contours form a single smooth curve fails for both n-n and K-n scattering, but for different reasons. In n-n scattering, due in part to the presence of the S*-resonance situated between the p and f, the experimental contours of zeros remain well defined, with Jim cos O*(u) J ~ 0.2, and it is clear that the <<first >~ and << second >> zero branches do not form a simple closed curve. On the other hand, in K-n scattering there are experimentally no nearby zeros in the energy range (0.95--;-1.30) GeV, hence the concept of zero-contour is not well defined. However it is again clear that the model prediction of a closed curve of nearby zeros is not verified. In order to illustrate this difference we show in Fig. 8 the experimental differential cross-sections for n+n-(a)) and K+n-(b)) elastic scattering for energies approximately half-way between the p and f(a)), and between the K*(890) and K**(1420). In case a) a pronounced minimum is visible, whereas the K-n cross-section has only a broad relatively poorly defined minimum. We have shown that in the narrow-width K* and p dominance model the zero-contour of the K-n amplitude .A(s, t, u) = .A 1 .-i(s, t, u) is a closed curve, an ellipse in the real (s, t, u)-plane. This curve fills three distinct roles: Adler-Weinberg zero inside the region below threshold in all channels, p and K* Legendre zeros respectively in the t-channel near the p mass and in the u-channel physical region near the K* mass, and double-pole-killing zero at the intersection of the lines t = m~ and u = m~.. The experimental zeros determined from the data of ref. ( 10 " 13 ) show clearly that the zero-contour is essentially that predicted by this simple model in the low-energy u-channel physical region (u >(m~.). This result lends support to the analysis of .ARNEODO, GUERIN .and DoNOHUE { 9 ), which emphasizes how, in the low-energy region (u ~m~), unitarization gives rise to a very small distortion of the circular zero-contour given by the narrow-width p dominance approximation of the corresponding n-n scattering amplitude A:t; 2 (8, t, u).
For the antisymmetric (under the exchange 8~u) K-n amplitude B(8, .t, u) = -Az'"" 1 (8, t, u)jv'2 the narrow-width K* and p dominance model predicts the zero-contour to consist of the line 8 = u, plus an ellipse passing through the Legendre zero of the K* in the 8-and u-channel physical regions and through both K*-p double poles. But this elliptic zero-contour is slightly more sensitive to unitarization at low energy (u ~m~.) than the elliptic zero-contour of the previous K-n amplitude A(8, t, u), and the experimental results show evidence for a possible contour of nearby zeros at t c::::-0.2 (GeV) 2 • This model fails for u > m~., since it predicts a closed contour of nearby zeros for the two K-n amplitudes A(8, t, u) and B(8, t, u), while experimentally the zeros have a nonnegligible imaginary part and that consequently the notion ·of zero-contour is not well defined at such energies. We must remark that the €xperimental zeros of the corresponding n-n amplitudes (respectively A~2 and -A;o;; 1 /v2) also disagree with the predictions of the narrow-width p 1iominance model, but not for the same reason, since they are well defined nearby zeros up to the f mass which do not propagate along the predicted cir-·cular curves for u > m!.
It remains a puzzle why the simple Breit-Wigner model, which strongly violates unitarity, still predicts zero-contours in good agreement with ex-Jleriment at low energy. The success of the analysis for n-n scattering of PEN-NINGTON and ScHMm ( 8 ), which is based on the idea of zero-contours being relatively stable against unitarization, suggests that this idea may be of some phenomenological importance.
Finally we discussed how it was illusory to hope for the experimental de- In the t-channel the incoming momentum Q(t) and the scattering angle 6(t) are given by v't-4f-t 2 Vt-4m 2 We use explicitly in this paper the convenient variables In our definition of zero-contours in Sect. 2, the limit one takes to approach the real (8, t, u)-plane corresponds to a particular prescription for these variables. For example, the (8 + iO) limit corresponds to the prescription q,= -ilq.l for 8 real >(m+ ft) 2 , the (8-iO) limit to q,= ilq.l. In the same way the (t + iO) limit corresponds to qt= -ilqtl for t real > 4ft 2 , the (t-iO) limit to qt= ilqtl· The K-1t isospin amplitudes (I,= l and j) can be expressed in terms of two invariant amplitudes A ±(8, t, u) which exhibit definite crossing properties under 8 ~ u exchange The physical partial waves Tf(t) are defined by +1 T{(t) = l J A 1 (t, 8, u)P,( cosO(t)) d( cosO(t)) -1