Journal of Modern Physics, 2013, 4, 904910 http://dx.doi.org/10.4236/jmp.2013.47122 Published Online July 2013 (http://www.scirp.org/journal/jmp) The KhuriJones Threshold Factor as an Automorphic Function B. H. Lavenda Università Degli Studi, Camerino, Italy Email: info@bernardhlavenda.com Received February 8, 2013; revised March 10, 2013; accepted April 6, 2013 Copyright © 2013 B. H. Lavenda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT The KhuriJones correction to the partial wave scattering amplitude at threshold is an automorphic function for a di hedron. An expression for the partial wave amplitude is obtained at the pole which the upper halfplane maps on to the interior of semiinfinite strip. The Lehmann ellipse exists below threshold for bound states. As the system goes from below to above threshold, the discrete dihedral (elliptic) group of Type 1 transforms into a Type 3 group, whose loxodromic elements leave the fixed points 0 and ∞ invariant. The transformation of the indifferent fixed points from −1 and +1 to the sourcesink fixed points 0 and ∞ is the result of a finite resonance width in the imaginary component of the angular momentum. The change in symmetry of the groups, and consequently their tessellations, can be used to dis tinguish bound states from resonances. Keywords: Threshold Factor; Automorphic Function; Elliptic and Loxodromic Elements 1. Introduction and Summary This paper suggests that the origin of strong interaction symmetries may be found in functions that are auto morphic with respect to a group generated by a fractional linear transformation. Examples of automorphic func tions are the circular functions which are automorphic with respect to , and elliptic functions which are periodic with respect to a group generated by two translations. Elliptic functions live on tilings that are parallelograms which fill up the entire plane. By cutting and pasting, the tessellations can be made into a torus of genus 1. 2nn 1z Poincaré and Klein were interested in looking for automorphic functions of higher genus; that is, complex valued functions invariant under specific groups. Klein had already examples in the shape of regular solids, of which there are only a finite number. We will show that, for real angular momenta, the partial wave amplitude is such an automorphic function corresponding to the dihe dral group. According to Khuri [1] and Jones [2], the partial wave amplitude derives its form from the asymp totic large angular momentum limit of the Legendre function of the second kind which has three singular points. These singular points are homologues of vertices of triangles in a conformal mapping, and the group we will be dealing with is the triangular group. In the unphysical region the angular momentum be comes complex, and, consequently, the vertex angles of the triangles in the division of the sphere, which would otherwise have the form of a double pyramid, also be come complex.The real part corresponds to a rotation, while the imaginary part represents a stretching. What were indifferent fixed points of an elliptic transformation below threshold, cosz, where 0,z , become sourcesink fixed points, , of a loxodromic transformation above threshold. What mathematically can be obtained by conjugation has a completely diffe rent physical explanation: the transition from the physical, 1, 1z 1, 1z 3SO 0, 2 3 , to the unphysical, , region as the system passes through the threshold. This can also be viewed as a transformation from the discrete, elementary group of Type 1, consisting of elliptic elements whose group is conjugate to a subgroup of , to a discrete, elementary group of Type 3, with loxodromic elements that leave invariant the fixed points . In this way, it may be feasible to study the strong interaction sym metries through the tessellations of the hyperbolic plane, , and ball, , that Poincaré used to study the sym metries of kleinian groups, or discrete groups of Möbius transformations with complex coefficients. C opyright © 2013 SciRes. JMP
B. H. LAVENDA 905 2. Modified Scattering Amplitude It is wellknown that the scattering amplitude can be cast into what is now known as the WatsonSommerfeld representation [3], , ik k , sin i ii kz k P z k 1Pz integer k (1) although its history goes back much further to Poincaré, who used it to study the bending of electromagnetic waves by a sphere [4]. The poles occur at integer values of the angular momentum, , and the residue is written as the product of the Legendre function of the first kind, , , which contains all the angular dependencies, and a factor , which is a function of the linear momentum, . Pz k Partial waves can be projected out of (1) by using 1 1sin d, 21 1Pz Pz z (2) for integer and complex . The contribution from one Regge pole to the partial wave scattering amplitude is . 1 k kk 2.kk Ak The residue has been found to vary as [5,6] This gives incorrect threshold behavior by predicting that 20 . 001 0 k A In order to correct the result, Khuri [1] and Jones [2] modified the residue, 1 cosh ,z expkk (3) where 2 2 1 1, 2 m k 20km 0 cosz (4) in the unphysical region where , and is the particle’s mass. In the limit as , they obtained the correct threshold behavior 2 k 1 cosh 2 zkm 22 2 e. kk m The added factor (3) supplies the partial wave with a branch point at 22 4km because the square root in 2 22 22 1 22 mm kk 1 coshln 11z vanishes there, but it does not give the additional branch points at 22 2jm k2,3,j for . 3. The Dihedral Group Unwittingly, Khuri and Jones transformed the partial wave scattering amplitude into an automorphic function. It is the form that the Legendre function of the second kind takes in the limit as [7], viz., 1 1cosh 2 1e .Qz (5) Said differently, the automorphic function, 21wz z (6) 1 2 2 2 1 , 1 zz zz (7) is the ratio of two independent solutions of the Fuchsian differential equation, 2 2 d0, d yIzy z where 2 22 2 2, 111 11 44 2 . 21 21 21 Iz wz z zz wz (8) The curly brackets denote the Schwarzian derivative of with respect to , 2 3 ,. 2 ww wz ww It is a projective invariant that was first discovered by Lagrange in his studies of conformal mapping of a sphere onto a plane, although the name was coined by Cayley in favor of Schwarz. The Schwarzian (8) clearly shows that Qz 1, 1,z has three singular points at w , which are homo logues of the vertices of a triangle in the plane. Since the sum of angles, 11 , 22 z1, 1, the triangles are spherical, and there is no real orthogonal disc which encloses the triangular tessellations. Hence, the need to project them onto a sphere. The spherical triangles are conformally represented in the half plane by the angular points, . The plane is then projected onto a sphere stereographically Copyright © 2013 SciRes. JMP
B. H. LAVENDA 906 [8]. The triangle on the sphere is bounded by arcs of great circles that cut the equator orthogonally. They form a double pyramid with the summits at the poles, each triangle having a vertex angle , and two right angles at the base on the equator, as shown in Figure 1. At the pole, , the conformal mapping (7) be comes 2 2 21 1 1 21 ln1cosh . w z coshwuiv in , ln zz zz zz Since , cosh coshzw cosh cosandsinh s uv yuv 0u 0v the upper half of the plane is conformally mapped onto the interior of a strip in the plane bounded by the line on the right, and the lines on the bottom and zwv on the top. The partial wave amplitude, (2), then reduces to 121 at the pole, so that the contribution from one Regge pole at 20k to the partial wave projection, as , is 2 4ln 1. 21 kmk 1 A Inverting the conformal map (7) gives the dihedral equation [9], cosh ln,nw 1.n 11 2 n n zw w (9) where1 (10) Introducing homogeneous coordinates (9) can be written as 22 11 2 12 1, 2 nn nn w ww 2 zw z (11) which is the arithmeticgeometric mean inequality. The vanishing of the functional determinant of the numerator and denominator of (11) [9], 2 2 1 2 40 , n n w 1n 21 21 212 12 12 11 12 12 22 22 nnnn nn nn nwnw nw ww nwwnww indicates that there are zeros at Figure 1. The sphere for the dihedron where the sphe rical triangles consist of hatched and unhatched parts with bases on the equator. The critical points −1, 0, 1, ∞, corre spond to the branch points, −1, +1, and ∞ on the sphere. w 10w and , and another zeros at 20w2n 1 2 1. n w w Therefore, we can write the ratio of homogeneous coordinates as either, 11 22 ,or0,1, ,1, rr ww gggrn ww (12) where the first term 2 e, e, in in gg w is the nth root of unity. Since the values of the homogenous coordinates have absolute value of unity, they lie on the equator, and (12) correspond to rotations of the sphere either through angles 2rn , or rn 0w. According to (9), the critical points w and z of the wsphere correspond to z r wg of the sphere. In addition, the points correspond to the point on the sphere, while those of 1zz r wgg 1z all correspond to 1, 1, on the same sphere. Hence, there will be three branch points, z on the sphere—the same three singular po ints of the Legendre function of the second kind. 4. Complex Angular Momentum For close to we can perform a Taylor series ex pansion in terms of the energy difference EEE , , and retain only linear terms [10], I Ei (13) Re where R , I Im , and the prime denotes differentiation with respect to the energy, evaluated at the energy . It is clear from the dihedral equation, (9), that E when both are real. This also follows from the condition that the fixed energy dispersion relation, 1The most interesting cases for angular momenta are 0 0 1d, z zQzAzz (14) converges. For large z, zz 2n [10, p. 148], while , and 1 n, although we can choose some multiple of h can be made arbitrarily close to an integer whose sole effect is to change the number of iterates on the elliptic element, , whic g. Copyright © 2013 SciRes. JMP
B. H. LAVENDA 907 1 Qz z Pz for all values of , and not like which requires 1 Re 2 Re Re , so that (14) will converge only when ei wr . Introducing (13) into (9), together with , give 22 ln 2 cosh1 4 R Er i z E 11 ln 2 . E r (15) The resonance width, 1, 2 vanishes below threshold, but zxiy is always positive definite. Introducing into (15) it becomes clear that only below threshold do we have a conformal mapping [11] of circles, rconst , in the plane, onto ellipses, w 22 2 1, RR y rE z 2 cosh lnsinhln x rE (16) in the plane, whose semimajor axis and eccentricity are coshln rE R and sech lnR rE , respec tively. In other words, ln h , R ri E cosz (17) represents an ellipse with semimajor axis ln rE cosh R, and eccentric angle, RE . In high energy physics, (17) would be referred to as a Leh mann ellipse [12]. Moreover, the straight lines const w., in the  plane, are mapped onto hyperbolas, 22 2 1, RR y EE z 2 cos sin x (18) in the plane, with semiaxes cos E R and sin E 1z ln r R. The foci of the hyperbolas are the same as those of the ellipses (16), viz., . The families of ellipses, (16) and hyperbolas, (18), that constitute a system of confocal conics is destroyed above threshold by a mixing of and in the real and imaginary parts of the argument in (15) due to the presence of a finite resonance width, . 5. Discontinuous Elementary Groups Below threshold, the resonance width, , vanishes iden tically, and the group of rigid motions are pure rotations, n. R RR E E 1 cos si sin cos R Ei iE V (19) The rotation matrix (19) ensures that the fixed points are at , instead of the usual rotation matrix which has indifferent fixed points, i . The motion on a sphere is shown in the upper sphere in Figure 2. We are in the physical region of bound states where cos 1,1 . Dynamical resonances for decaying unstable particles give a finite width above threshold. This introduces de formations which are hyperbolic elements, 11 cosh sinh 22 11 sinh cosh 22 RR RR . U (20) The fixed points of (20) are the same as those of (19), and, consequently the trace of their commutator is ,2tr UV . The combination of stretching and rotation give rise to loxodromic elements, 11 cosh sinh 22 11 sinh cosh 22 RR RR iE iE iE iE TUV (21) which can be thought of as a rotation through a complex angle, 1 2 REi . The map, 1, 1 z zz 1 moves to 0 and 1 to . Mathematically, this is mere conjugation, but, physically, it represents the trans Figure 2. Elliptic (above) and loxodromic (below) motion on a sphere. Copyright © 2013 SciRes. JMP
B. H. LAVENDA 908 formation of real (bound states) into complex (dynamical resonances) angular momenta. The loxodromic motion is shown in the lower part of Figure 2, which is described by the scaling map, ,zaz 2 eRiE a 1 1 0 (22) where the modulus of the multiplier, , is greater than 1. The fixed point at (or ) is an attracting fixed point, or sink, while the fixed point at 0 (or ) is a repelling fixed point, or source. Therefore, points will spiral out of the source, 0, and spiral into the sink, , as shown in lower figure in Figure 2. Below threshold, , and the fixed points are indifferent, giving rise to the pattern shown in the upper figure in Figure 2. For a positive integer, the contour 1 in the de finition of the Legendre function of the first kind [13], 1 11 22 Pz i 2 1 1d, z z z 11 1 (23) is a closed curve, oriented in the positive direction, that encloses both points . The integrand is real when is on the real axis between and . This is the phy sical region. For noninteger , (23) is a hypergeometric function with a branch cut going from to in order to keep it singlevalued. In contrast, Legendre function of the second kind, 2 11 4s in2 Qz i 2 1 1d, z 1 (24) where 2 is a figure 8 encircling the points and . The Legendre function of the second kind, (24), is regular and singlevalued in the plane which has been cut along the real axis from 1 to . According to (24), cannot be an integer, and it has logarithmic singulari ties at and because the path must cut the lines joining to and . 1z 1 zz U1 U 1 1zz 1 The isometric circles of and are 1 2 :e0and : R IzI z 1 2 e 0, R T T respectively, which are exterior to one another. Isometric circles define a complete locus of points in the neigh borhood of which lengths and areas remain invariant under substitutions of the form [14]. It can be shown that the fundamental region for the group of motions generated by consists of that part of the plane ex terior to and [14, p. 54]. A point belongs to the fundamental region if a circle can be drawn about the point as its center that does not contain an interior of the isometric circles. For both real and complex 3 , the groups which are formed are discrete and elementary. By elementary it is meant that there exist finite orbits in [15]. In the case that of real angular momenta, they constitute the group of finite, cyclic rotations. More specifically, if is number of vertices of the polygon, where each vertex has n elements, then the relation to the size of the group, G, and the number of elements of the vertices is [15, p. 85] 1 11 21 1 s jj Gn 2n . (25) Moreover, since j it also follows that the right hand side of (25) is bounded by 1 11 1. 2 s jj s n 2s Now, for we have a crescent for by two non concentric intersecting circles with two vertices, and (25) reduces to 12 1111 , 2Gnn which identifies G as the harmonic mean. It can be satisfied by 12 Gnn G 2 3s . By conjugation, the fixed points, or vertices, can be brought to 0 and . Then is a finite, cyclic group of rotations in [15]. The case applies to a dihedron, for which (25) becomes 1 12 1. s jj nG n (26) Suppose the number of elements at the vertices are ordered as 123 nn 3n 12n . The choice 1 is incom patible with (26) since the sum on the left will be inferior to 1. We, therefore, choose , and (26) reduces to 23 1112 . 2nn G 4nn n Since 2 will lead to a contradiction, 2 can be either 2 or 3. In the former case 3 is free to take on any value from 2 to so there will be 2Gn 2n sides of the various triangles in the division of the sphere, as shown in Figure 1. Now another elementary group has , and has every element of the group leaving 0 and invariant [15, p. 84]. The group cannot have the Poincaré metric, 2 1dz z 2 2 2 , because loxdromic motion does not pre serve the unit disc, . Rather, what is required is Poincaré’s extension which adds on an additional dimen sion so that points on the plane are stereographi cally projected onto the sphere . Rotations, like loxo Copyright © 2013 SciRes. JMP
B. H. LAVENDA 909 dromic transformations, also do not have an invariant real disc, , so that both elliptic and loxodromic generators live on the sphere, . 2 2 1 What appears mathematically as conjugation, by trans ferring the fixed points and to 0 and 1 , is physically quite different for it takes the system from real to complex angular momenta. In other words, merely by conjugation, the vertices, , transform the physical region , or in (4), into 0 and 1, 1 z20 11,k , of the unphysical region, , or . The two types of transformations will give a different tessellation of hyperbolic space, and so can be studied in much the same way that Poincaré characterized his kleinian groups, which are much less wellknown than their fuchsian counterparts [15, Ch. 8]. 20k z 1,1z 1, 20k 1, 1z 6. Strengths and Weaknesses of the Analysis Different definitions of the Legendre function may be adopted according to whether is considered to be an unrestricted complex variable, or a real variable confined to the open interval . However, there is no relation between complex values of the degree of a Legendre function and the unphysical region of . 1z According to (4), bound states, , lie in the phy sical region. But, what is missing is a conformal map ping, 2,k (27) between the plane and the plane. If we go back to the WatsonSommerfeld representation, (1), we can write it, in the vicinity of , as [3, p. 46] k 2 222 cos sin cos regular function, cos k kP 2 k kP kkk (28) where is considered a real function of , the energy. Thus, if we consider the conformal mapping 2 k 2 sin,wk then a semiinfinite strip of width 1 in the plane is mapped onto the upper half plane where the boun daries of the strip are homologues of the points w 1 and on the real axis. The poles of (28) will then be at 1 1 2 20k . It is assumed that for , the Regge trajectories (27) are real, but once , the Regge trajectories become complex. This being a bound state would lead to the conclusion that Regge trajectories are the desired interpolating functions between bound states [16]. 20k 20k For , 2 k Re is complex, and this describes resonances. It is argued [16] that can coincide with integer 0 for k, while Im remains small. Then appealing to continuity, is considered complex where 20 2 k 2 k becomes exactly equal to 0. Con sequently, we can take physical with unphysical , which is a resonance, or unphysical 2 k with physical , which would correspond to a metastable state known as a shadow state [3, p. 47]. However, we are not at liberty to consider k2 complex as we originally con sidered complex. 2 k k Rather, discrete, elementary groups can be employed to distinguish between bound and resonance states. Only for maps of the form 2 ,1,1, s zaz as Re will the angular momentum become complex, and so transform the indifferent fixed points of Type 1 group into a Type 3 group whose orbits leave invariant the sourcesink fixed points at 0 and [15, p. 84]. What is still missing is the form of the Regge trajectory (27) which would connect with k, and so lead to physical values in (4), and complex values of 20 with which result in unphysical values of (4). 20k REFERENCES [1] N. N. Khuri, Physical Review, Vol. 130, 1963, pp. 429 435. doi:10.1103/PhysRev.130.429 [2] E. Jones, Lawrence Radiation Laboratory Report, UCRL 10700, 1963. [3] R. Omnès and M. Froissart, Mandelstam Theory and Reg ge Poles, Benjamin, New York, 1963. [4] H. Poincaré, Rendiconti del Circolo Matematico di Pal ermo, Vol. 29, 1910, pp. 169259. doi:10.1007/BF03014067 [5] A. O. Barut and D. E. Zwanziger, Physical Review, Vol. 127, 1962, pp. 974977. doi:10.1103/PhysRev.127.974 [6] R. G. Newton, Journal of Mathematical Physics, Vol. 3, 1962, pp. 867883. doi:10.1063/1.1724301 [7] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tri comi, Eds., Higher Transcendental Functions, Vol. 1, Mc GrawHill, New York, 1953, p. 162. [8] A. R. Forsyth, “Theory of Functions of a Complex Vari able,” Vol. 2, 3rd Edition, Cambridge U.P., Cambridge, 1918, p. 692. [9] F. Klein, “Elementary Mathematics from an Advanced Standpoint,” Macmillan, New York, 1932, pp. 115117. [10] S. Frautschi, “Regge Poles and SMatrix Theory,” Ben jamin, New York, 1963, p. 114. [11] L. Bieberbach, “Conformal Mapping,” Chelsea, New York, 1953. [12] H. Lehmann, Nuovo Cimento, Vol. 10, 1958, pp. 579589. Copyright © 2013 SciRes. JMP
B. H. LAVENDA Copyright © 2013 SciRes. JMP 910 doi:10.1007/BF02859794 [13] D. Hilbert and R. Courant, “Methods of Mathematical Physics,” Vol. 1, Interscience, New York, 1953, pp. 501 508. [14] L. R. Ford, “Automorphic Functions,” 2nd Edition, Chel sea, New York, 1929, p. 25. [15] A. F. Beardon, “The Geometry of Discrete Groups,” Sprin ger, New York, 1983, p. 82. doi:10.1007/9781461211464 [16] R. Omnès, “Introduction to Particle Physics,” WileyIn terscience, New York, 1971, p. 347.
