Annalen der Physik. 7. Folge, Band 27, Heft 2, 1971, S. 149-160 J. A. Barth, Leipzig CREMONA Problems Related to Static Pion-Nucleon Theory By H. J. KAISER With 3 Figures Abstract Certain mathematical aspects of the static pion-nucleon theory are investigated. We start from the fact that the theory in its uniformized form (cuts transformed away) leads to a system of functional equations for the 8-matrix. The nonlinear mapping involved in the functional equations is a second-order CREMONAtransformation. After a summary of the general properties of CRERIONAtransformations, the special transformations are studied which arise in the symmetric scalar and the CHEW-LOW theory respectively. The emphasis is on the possibility to separate, by means of a finite-order CREMONAtransformation, the functional equations into a set of uncoupled ones. For the symmetric scalar theory, the separation is trivial. For the CHEW-LOW theory, a proof of nonseparability is given. 1. Introduction Considerable interest has been centered in recent times on the static theory of pion-nucleon scattering as it provides a model to study the interplay of analyticity, unitarity, and crossing symmetry in 8-matrix theory. Unfortunately, in theory), no general the physically interesting case of three channels (CHEW-LOW solution is known. The uniformization technique [l, 2, 31 allows one t o construct the general solution for the two-channel case. I n the three-channel case, however, only a physically uninteresting class of solutions has been investigated so far [a, 31. It is the particular case where the S-matrix elements are restricted by cone”). The residue relation a t zero the condition Sis = X,,S,, (“ROTHLEITNER energy Sll(0) :#(, 0) :AY,J 0) = -4 : - 1: 2 demanded for a proper physical solution puts it outside the ROTHLEITNER cone. I n the present paper: an analysis of the static pion-nucleon theory is performed from the point of view of CREMONAtransformations [4, 51. I n section I1 a short summary about uniformization of the S-matrix is given. Section I11 provides an introduction t o the mathematics of CREMONAtransformations. I n section IV the CREMONAproperties of the symmetric scalar and the charged pseudoscalar theory are derived. Finally, in section V the problem of the existence of CREMONA transformations separating the coupled functional equations is discussed. It is proved that no “semiseparabel” CREMONAtransformation of finite order is equivalent t o the “coupled” CREMONAtransformation met in CHEW-LOW theory. 150 Annalen der Physik * 7. Folge * Band 27, Heft 2 * 1571 2. Uniformization of the S-matrix We consider the static pion-nucleon theory with 2 and 3 channels i.e. the symmetric scalar and the charged pseudoscalar theory respectively. The crossing matrices for these two cases are and The branch cuts of theX-matrix X,(z) in each of the channels formed away by means of the conformal mapping [2] (x can be trans- z = cash ( u ) . (3) Indeed, this mapping effects uniformization such that the only remaining singularities are poles. Expressed in both the x and the u plane, the conditions tjhe S-matrix must fulfil are : herm iticity S,(Z*) = X,*(Z), X,(-u*) = SZ(u), unitarit y To investigate the coupled system of functional equations (4)-(6) we will t r y t o separate them b y means of a (nonlinear) u-independent transformation replacing S , in a n one-to-onefashion by new functions V,. A necessary condition for the simultaneous separability of Eqs. (5) and (6) is that X;(u) = S,(u + in) = 2B C,#,3/Sp(u) (7) must be separabel. For our aim to solve the functional equations (4)-(6), complete separability V:, = FdV,) (8) is not necessary. The weaker demand of semiseparability V ; = Fl(V1) v:,= F,(V,, 7,) v::= F3(V1, v,, V3) (9) H. J. KAISER:CRE~WONA Problems Related to Static Pion-Pl’ucleon Theory 151 (written down for the 3-channel case) is sufficient. I n order t o be equivalent to (7), the equations (9) must have an unique reverse 3. CREMOXA transformations Let us consider the most general one-to-onetransformations in two variables. Whereas in the case of one variable we have only the broken linear transformat,ions 2’= ax cx ~~ +b +d as one-to-one mapping between 5 and X I , we are in the case of two variables not restricted t o the projective transformations (12) There exist one-to-one transformations of much more general kind. They are called CREMONAtransformations [ 3 , 51 and include the projective transformations (12) as CREMONAtransformations of first order. Consider two planes P and P’. A transformation mapping almost every straight line L of P into a curve L‘ of nth degree lying in P’ is a CREMONAtransformation C, of nth order provided any two of the curves L’ intersect in maximally one variable point. Two curves L‘, and L.1 will, of course, intersect generally in more than one point. To met the CREMONA condition, the additional intersection points must be “non-variable” i.e. common t o almost all curves L‘. Such a common point is called a fundamental point, in particular a fundamental point A ; of multiplicity s1 if the curves L’ have there a n sl-fold point. The behaviour in the plane P is, of course, completely analogous. Let us denote the multiplicities of the fundamental points A , in the plane P by r,. Table 1 Fundamental points (other than zeroth multiplicity) of the lowest-order CREMONAtransformations Number and multiplicities of fundamental points 0 3 single 1double 4 single 1triple 6 single or 3 double 3 single 1 quadruple 8 single or 1triple 3 double single or G double + + + + + + 152 Annalen der Phyeik * 7. Folge * Band 27, Heft 2 * 19il The multiplicities rk and s1 of the fundamental points belonging to an nthorder CREMONAtransformation fulfil the conditions 2 rk = k s1 = 3(n-l), 1 2 ri = 2 sf = n2 - 1, k 1 (13) where the sums run over all fundamental points of the respective plane. The simplest cases are summarized in Table 1. A fundamental point of multiplicity rk belonging to the plane P will be t,ransformed into a fundamental curve of degree rk belonging t o the plane P' and vice versa. All other (i.e. "normal") points will be transformed into points. I n this sense, every CREMONAtransformation is one-to-one. I n the plane P', a fundamental curve being the image of a fundamental point A , has in the fundamental point A ; of the plane P' an a,,-fold point. The aklfulfil the equations Cr,anl = ns, k Ca;l = s; k 2 k M,[LX,l' 1 xaEZ +1 = ri +1 1 (14) 2 akla,'l = r,rk* 1 = SPY/' = 38, - C & k l ~ s l a n l= nrk Cakl 1 k = 3r, - 1. 1 We may extend the scope of the equations (13) and (14) t o include normal points by associating with each normal point the multiplicity rk = 0 and putting ' 0 if either A , or A ; is a normal point, the other a true fundamental point a,, = -1 if both A , and A; are normal points, mapped into each other by C, 0 if both A , and A; are normal points but not mapped into each other. \ Consider two consecutive CREMONA transformations C: P -+P' and C' : P' + PI'.Their product C" = CC' : P -+ P" is always again a CREMONAtransformation. C and C' will in general not commute, CC' C'C. The order n" of C" depends on the orders and n' of C and C' as well as on details of the coincidence of fundamental points in the intermediate plane P'. All properties of C" can be expressed in compact form through the properties of C and C' if we arrange the order n , the multiplicities rt and sz, and the wkZin the square matrix + ir, ir, is, is, -al1 -al2 -a2, -a31 -0132 is, -0113 *-. -a23 -a33 ................................ Each row of this matrix corresponds to a particular point of the plane P,each column t o one of the plane P'. If we write t,he analogous matrix for the transformation C' and let their rows correspond to the same points of the plane P' as used for the columns of C then we obtain the matrix for C" as the product of the mat)rices C and C'. The number of points selected for representation as rows of 153 H. J. KAISER:CREMONAProblems Related t o Static Pion-Nucleon Theory the matrix C must include a t least all true fundamental points ( r k > 0), those for the columns all fundamental points (st > 0) of C in P' as well as all fundamental points (rk' >o) of in P'. Furthermore, if any one of the normal points in P is represented in a row of the matrix C then its image point in the plane P' must be included as a column of the same matrix. The same conditions must be met, of course, also with the roles of P and P' reversed. We take as a n example the product of the two CREMONAtransformations depicted in Fig.1. .'i' c' Fig.1. Product of two CREMONA transformations. 0, 0 , and denote fundamental points of multiplicities 0, 1, and 2 respectively. The numbers 1, 2, ..., l', ..., 6" serve only t o identify particular points of the planes P , P', and P". The akl cannot be read off the figure, they are specified i n the matrices (16) and (17) P P 'I I..0. I n accordance with Fig. 1, we choose the matrix for the second-order CREtransformation C as follows MONA c,= i -1 0 i -1 -1 i 0 -1 0 0 0 0 0 0 0 0 0 1 0 0-1 0 0 0-1 0 0 0 0 Here, the points of the planes P and P' 'corresponding to the rows and columns of the matrix are indicated explicitely. For example, we find a t the crossing of the 5t*1row with the 6th column the figure 1. It tells us that a45= -1, i.e. the CREMONAtransformation chosen transforms the (normal) point A, of the plane P into the (normal) point A ; of the plane P'.Likewise, we specify the thirdorder CREMONAtransformation occuring in Fig. 1 by the matrix 1" 2 ! 1 3" 4" 511 6,) p t ~ / p t ~ ~ 3 2i i -1 o i 2 i -1 i -1 0 -1 -1 0 0 i -1 i -1 1 0 0 0 0 0 i ~~~ i -1 -1 1 2' 0 0 0 0-1 0 0 0-1 5' ' G' I 154 Annalen der Physik * 7. Folge * Band 27, Heft 2 * 1971 The matrices (16) and (17) satisfy all the conditions (13) and (14) characteristic of CREMONAtransformations. The product C,Ci is the third-order CREMONA transformation 1" y , 3" 4" 5" 6" p"lp - 1 3 . 2i '%a-1 i - I i - 1 ,i-1 i _ _ ~ . ~ _ _ i 0 i 1 i ) II -1 -1 0 0 0-1 0 0-1 0 0 0 '4 0-$5 0 0 , G as shown in Fig. 1. It should be stressed again that (16) -.. (18) are only examples of CREMONAtransformations of second resp. third order, not the general ones. To every CREMONAtransformation, there exists an inverse. I t s matrix fulfils C-1 = CT j . e. it i 9 identical with the transposed matrix. 4. The CREMONA transformations occuring in nN theory We begin with the case of symmetric scalar theory. Inserting the crossing matrix (1)into Eq. (7) we get with the inverse This is a second-order CREMONAtransformation. Ignoring for the moment the fact that it can be easily separated by putting x = SJX,,y = S,,we will study the transformation in its present form (19), (20). It has (0, O), (00, c), (c, 00) as fundamental points in the S,- X, plane and (0, 0), (00, -am), ( 4 0 0 , ~ ) in the S;- 8.3 plane. Note that the points a t infinity are to be understood in the sense of projective geometry, i. e. for example (00, -200) = &lim I-.- ( M , - 2 M ) is regarded t o be distinct from (w, c ) = lim ( M ,c o n s t ) . Furthermore, a circle M-f- with infinite radius is regarded as limiting case of a straight line. Fig. 2 shows the fundamental points and fundamental curves of the CREMONA transformation (19), (20). I B Fig. 2. Fundamental points and fundamental curves of the CREMONAtransformation related t o symmetric scalar theory. The three J-;fundamental points i n the plane P and the corresponding fundamental curves in the plane P' are denoted by a, b, c. Analogously, A, B, C are fundamental points in P' and fundamental curves in P 155 H. J. KAISER: CREMONAProblems Related to Static Pion-Nucleon Theory The simplest possible matrix description of the transformation is 0 oo 400 s;l 0 -2.200 00 s.;1 s, s, _ _ _ ~ ~- i i 2 ~~ 0 -1 -1 0 c o o But to render possible the calculation of iterated transformations (C,)P, we extend the matrix t o include the successive images of normal points which become finally transformed into fundamental points (i. e. the predecessors of fundamental points). Furthermore, we let the jth column of the matrix denote the X,) = (w,ul).The fact that the same point (Si, 8;) = (w,w)as the jth row (IS,, transformation (19), (20) has in both planes fundamental points a t infinity compels us to distinguish carefully between points close t o the origin. Thus we find the infinite-dimensional matrix (22). I t s structure is simple enough to allow the calculation of its powers. 0 0 00 -26 c E 00 -200 ... ... 800 7~ 600 5~ 400 500 4 F 300 2E oo c E 00 48 200 3~ 400 ... 500 68 700 ... 9: 9; 4 & ... 0 0 0 0 i 0 0 0 0 o... 0 0 0 0 o... ... 0 0 0 0 - 1 ... 0 0 0 0 - 1 0 0 0 . .. 0 0 ... 0 0 0 0 0 1 0 0 0 0 o . . . ..................................................................................... 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0 . . . 0 0 0 0 0 ... 1 0 0 0 0 0 0 0 0 0 . . 0 0 0 0 0 ... 0 1 0 0 0 0 0 0 0 o... 0 0 0 0 0 1 . . . 0 0 1 0 0 0 0 0 0 o... 0 0 0 0 0 ... 0 0 0 1 0 1 0 0 0 0 o . . . i -1 0 0-1 I... 0 0 0 0 0 0 0 0 0 0 . .. 0 0 0 0 0 0 0 0 0 0 1 0 0 0 o... 0 0 0 i : : : o 0 0 0 0 0 0 ~ 10 0 0 0 . .. 0 0 0 0 0 .'.0 0 0 0 0 0 1 0 0 ... 0 0 0 0 0 ... 0 0 0 0 : i u 0 0 1 o... ................................................... 1 .......................... 2 i i 0 i -1 0 0 'I, _ I 0 0 0 1 0 i 0-1 0 0 0 0 I i 0 0 m c -28 E -200 00 .......... 554 800 7E 4F 600 300 5E 2E 400 00 c o 8 200 500 3E 6& 700 400 .......... (22) One may add to t,he description of the transformation (19), (20) that the points (1,1)and (-1, - 1)as well as the line X, = 8, remain invariant under the transformation, i. e. they are fixed points resp. a fixed curve. For the CHEW-LOW theory, we reduce the system (2), (7) by introducing the new variables w, = s, w2 = slIs2 w, = X,/S,. o 48 (23) 156 Annalen der Physik * 7. Folge * Band 27, Heft 2 * 1971 and obtain We will concentrate on the problem posed by the CREMONAtransformation between the W , - W , and the W i - Wg planes i. e. Eqs. (24b) and (24c). The fundamental points are located a t (-718, -1/2), (114, 1/16), and (-2, 4) in the W , - W , plane and a t (00, c ) , ( c , m),and (0, 0) in the W i - W ; plane. The iterable CREMONAmatrix given in Eq. (25) is again infinite dimensional. ... ... -6 5z 0 5 - 2 2 --. 0 -1 4 36 25 1 6 4 c 5 4 .... 2 25 00164 7 14 -7 ... -.. 5-2 i7 8 . . . 3_ _2 --1 22 17 2 - 8 1 77 14 17 c -2 2 00- ... .v; ... y', Wz 2 . . . 0 0 0 i o o i o o . . . ................................................ 0 ... 0 1 0 0 0 0 0 0 0 ... 0 ... 0 0 1 0 0 0 0 0 0 ... i ... 0 0 0 - 1 0 0 0 0 0 ... o * . . o o o 0 1 0 o o o . . . o . . * o o o 0 0 1 o o o . . . i ..- 0 0 0 0 0 0 -1 0 0 0 ... 0 0 0 0 0 0 0 1 0 ... o . . . o o o 0 0 0 O O l . . . 0 ... 0 0 0 0 0 0 0 0 0 ... ................................................ ................................................ o . . * o o o 0 0 0 O O O . . . o * * . o o o 0 0 0 o o o . . . i ... 0 0 0 -1 0 0 -1 0 0 ... o . . . o o o 0 0 0 o o o * * * o . . . o o o 0 0 0 o o o . . . o . . . o o o 0 0 0 0 0 0 - . * ................................................ . * a ~~ ~ ... 0 0 0 i 0 0 ... ................................ ... 0 0 0 0 0 0 ... *.. 0 0 0 * * a 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 *.* **. ... 0 0 0 ... ... 0 0 0 ... ... 0 0 0 ... *.. 0 0 0 ..0 0 0 ... 0 0 0 ... ................................ ................................ ... 0 1 0 0 0 0 ... -.* 0 0 1 0 0 0 ... 0 0 0 0 0 0 ... ... 0 0 0 0 1 0 ... . a * * a * * * * *-. 0 0 0 0 0 0 0 0 1 0 0 0 ................................ a * . **. **. w3 ........... 9/36 4/25 1/16 0 114 316 215 114 0 -112 -2 c 4 o o 4 16 512 2514 ........... ........... 7/22 3/22 14/17 2/17 -718 -112 0 0 c -2 17/14 1712 -a17 ............ (25) The transformation has the fixed pojnts (26) and the fixed curve W , = W ; (this parabola corresponds to the ROTHLEITNER cone). No other fixed curves of finite order do exist. I n Fig. 3, the main characteristics of the CREMONAtransformation CCL are shown. H. J. KAISER:CREMONAProblems Related to Static Pion-Nucleon Theory 157 Fig. 3. Fundamental points, fundamental curves, fixed points, and fixed curve of the CREMONAtransformation Coz related to CREW-LOW theory. 5. The separability problem We begin by asking under what circumstances an nth-order CREMONAtransformation C,: (z, y) -+ (%', y') is semiseparable in the sense that it may be put into the form x' = ax cx ~ +b + d' Two immediate consequences of (27) are that all parallels t o the y axis will p be transformed into parallels to the y' axis and that all straight lines y = Ax are transformed into one-valued functions y' = y'(z'). Now, a CREMONAtransformation has in general a lot of fundamental points of high multiplicity. Consequently, straight lines will be transformed by them into curves having multiple points. To secure, nevertheless, one-valuedness, all fundamental points of (27) with multiplicity > 1 must be situated a t (c, 00). For, the multiple points of y' = y'(z') then will lie a t ( c , 0 0 ) too, and this means these curves will have several asymptotes all parallel to the y' axis. Now, we know t h a t a CREMONAtransformation C, maps the straight lines of the plane P into curves of nth order of the plane P'. More precisely, this holds for straight lines avoiding all fundamental points of P. A straight line L passing through some fundamental points A , with multiplicities r, of the plane P will be transformed by C , into a curve L of order -+ m'=n- 2' 'k (28) (sum over all fundamental points on L).L' has in the point A ; of P' a &-fold point where q; = SL - 2' OLkl. (29) Now we concentrate on the semiseparable case i. e. we demand that all straight lines passing through the fundamental point A, = (6,~) and no other fundamental point should be mapped exclusively into straight lines having a t the same point Ad = (6,~) a fundamental point but passing through no other fundamental point. Accordingly, we have t o put = 0. m' = 1 q; = 1 q' = q: = (30) ... 158 Annalen der Physik * 7. Folge * Band 27, Heft 2 * 1971 From (28) and (29) we conclude ro = so = n - 1. (31) According t o the arguments given above we introduce the notion of CREMONA transformations of type S . D e f i n i t i o n : An nth-order CREMONAtransformation belongs t o type X if it possesses in both planes P and P' a t the same place ( E , q ) a fundamental point of multiplicity n - 1. (32) The properties (13) valid for all CREMONAtransformations are sufficient t o secure that the only other fundamental points (with multiplicity > 0) of a type X CREMONAtransformation are 2n - 2 fundamental points of multiplicity 1.CREMONAtransformations of type S attain the semiseparated form (27) as soon as (E, 7) = (c, 0 0 ) . From (13) and (14) we recognize that a CREMOWA transformation of type 8 exists in every order n. Its matrix reads up to permutations among the last 2n - 2 columns and/or the last 2n - 2 rows and up to the possibility to introduce any number of normal points n (n - 1 ) i i i i (n - 1)i -n 2 -1 -1 - 1 i -1 -1 0 0 i -1 0 -1 0 = i -1 0 0 -1 + c: i ... . a * * a - ................................................. i -1 0 0 0 -1 0 0 0 ... -1 6 7 x1 Y1 x2 Yz x2 Y3 ........... X2n-2 Y2n-2 (33) It is easy t o derive from (33) that the product of any two CREMONA transformations of type S belongs again to type S and that the square of a transformation C," of type S has its order within the limits 1 order ((C2)2) 5 2n - 1. (34) Consequently, the pth power of C," fulfils 1 5 order ((C,S)P)< pn. (35) Within these limits, the order depends on the relations among the fundamental points of multiplicities 0 and 1 i.e. on the akl. Let us now apply this formalism t o the CREMONAtransformations (22) and (25) of symmetric scalar and CHEW-LOWtheory respectively. The CREMONAmatrix (22) is of type S because the fundamental point (0, 0) is common t o both planes and a t the same time of order n - 1 with an 01 = n - 2. We shift the point (0, 0) to (c, 0 0 ) by the substitution x = X,/Sz Y = 8, (36) H. J. KAISER: CREXOXA Problems Related to Static Pion-Nucleon Theory 159 and arrive a t the semiseparat,ed transformation 42 - 1 x’ __ x 22‘ 1 4-x‘ ’ x+2’ + y =--- (22’ (37) 3 + 1) y’. This is a CREMONAtransformation of third order having fundamental points of multiplicity 1 a t (x,y) (x‘,y’) = (-2 + - E , 01, (-2 E, (0, -1, 01, (-1 c) (38) O), (-V, a), (4, 0) and, of course, the common fundamental point of multiplicity 2 a t (c, 00). On the other hand, the CREMOXA transformation of the CHEW-LOWtheory (25) is not of type S. This does not exclude, a priori, that it could be transformed into type S. Our problem is therefore: Is the matrix CCL given in E q . (25) equivalent t o a semiseparated one! I n other words, we ask whether there exists a (finite order) CREMONAtransformation T such that = (00. 0). (00, SI TCCLT-’ = CA(39) is an Nth-order CREMONAtransformation of type S. Suppose (39) holds, then the s same T will transform the pthpower of CCLinto the pth power of Cn- T ( C ~ ~ T) -’ l = (c$).. (40) We know already from Eq. ( 3 5 ) that (CL$)’ is necessarily of less than order. Furthermore it is not too difficult t o derive from the explicit form (25) of Ccz the structure of its powers. For our present purposes, it is sufficient t o notice that the recursion formula M , = 2M,-1 - M D . 4 (41) holds for the order of the pt” power of CCJ,.The first M , are given in (42). p l ~- M, 1 2 3 4 3 G 7 8 9 1 0 - 1 2 4 8 15 28 52 ‘36 1 7 7 32G 11 ~~ GOO 1104 (42) The asymptotic behaviour of the 31, for large p is easy to derive from ( 4 1 ) . Denoting by I , the root with largest absolute value of the equation ~4 - we find M, + 1= 0 21~3 - A17 w 1.35 * 1.8393’. (43) (44) If we insert the (Cf-)”of a t most linearily increasing order (35) and the ( C C L ) ~ of exponentially increasing order (44) into (40) and let p tend to infinity then we recognize that no finite-order CREMONAtransformation T can bring CCL t o type S . And, consequently, CCLis not equivalent to a finite-order semiseparated CREMONAtransformation. This excludes the possibility to solve the functional equations of CHEW-LOWtheory by the methods developed for static theories with two channels. 160 Annalen der Physik * 7. Folge * Band 27, Heft 2 * 1971 References [I] [2] [3] [4] L5] WANDERS, G., Nuovo Cim. 23 (1962) 817. ROTHLEITNER, J.,Z. Phys. 177 (1964) 287. MESCHERYAEOV, V. A., Dubna preprint P-2369 (1965). CREMONA, L., Opere Matematiche, vol. 2, Milano 1915. MLODZEYEVSKY, B. K., Matem. Sbornik 31 (1922) 7. Z e u t h e n , Institut fur Hochenergiephysik der Deutschen Akademie der Wissenschaften zu Berlin. Bei der Redaktion eingegangen am 4. November 19iO. Anschr. d. Verf.: Dr. H. J. KAISER,Institut fur Hochenergiephysik der DAW DDR-1615 Zeuthen, Platanen-Allee 6:

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