O(2,1) decomposition of the equal-mass multipheripheral equation at t=0

We extend the results of a group-theoretical analysis of the t=0 multiperipheral equation to the case t<0 for pairwise equal masses. Using variables discussed in a previous paper, we diagonalize the equation in the Bali-Chew-Pignotti (BCP) model with respect to the 0(2, 1) group and relate the solutions to the equation so obtained with the solutions obtained after diagonalization with respect to the 0(3, 1) group. Poles in the 0(3, 1) partial-wave amplitude give rise to the expected sequence of daughter poles in the 0(2, 1) partial-wave amplitude. At general momentum transfer, we establish factorization at the 0(1, 1) poles in the decomposition of the BCP amplitude, and present further simplifications to the diagonalized equations based upon this model.

P H Y S I C A L R E V I E W D VOLUME 1, N UMB E R 10 15 MAY 1970 0(2,1) Decomposition of the Equal-Mass Multipheripheral Equation at f=0f Marcello Ciafaloni* Department of Physics, University oj California, Berkeley, California 94720 and Carleton DeTar Lawrence Radiation Laboratory, University of California, Berkeley, California 94720 (Received 15 December 1969) We extend the results of a group-theoretical analysis of the t<0 multiperipheral equation to the case ( = 0 for pairwise equal masses. Using variables discussed in a previous paper, we diagonalize the equation in the Bali-Chew-Pignotti (BCP) model with respect to the 0(2, 1) group and relate the solutions to the equation so obtained with the solutions obtained after diagonalization with respect to the 0(3, 1) group. Poles in the 0(3, 1) partial-wave amplitude give rise to the expected sequence of daughter poles in the 0(2, 1) partial-wave amplitude. At general momentum transfer, we establish factorization at the 0(1, 1) poles in the decomposition of the BCP amplitude, and present further simplifications to the diagonalized equations based upon this model. I. INTRODUCTION HE recent group-theoretical analysis1-3 of the multiperipheral equation4-6 with respect to the 0(3, 1) and 0(2, 1) groups has provided a natural framework in which to investigate the constraints that unitarity imposes upon the residues and trajectories of the Regge-daughter family near t = 0. In this paper, we shall examine some preliminary problems in this direc­tion. Since different sets of variables have been used to write the <=06,1 and t<02,3 equations, it is important to study first how they match in the limit t=0. More­over, if we take the Bali-Chew-Pignotti7 (BCP) model for the production amplitudes at t= 0 as CD did, it is essential to translate this model in the t<Q variables by keeping the nonleading powers in the asymptotic expansion. The BCP variables, used by CD and MM1 at t= 0, are essentially the parameters of the 0(2, 1) groups which preserve the momentum transfers in the multi­peripheral chain. The 2<0 variables,2,3 which we shall call "three-dimensional BCP variables," are instead the t Work supported in part by the U.S. Atomic Energy Com­mission and in part by the Air Force Office of Scientific Research, Office of Aerospace Research, U.S. Air Force, under Grant No. AF-AFOSR-68-1471. * On leave of absence from Scuola Normale Superiore, Pisa, Italy, and Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Italy. 1 A. H. Mueller and I. J. Muzinich, Ann. Phys. (N.Y.) (to be published); hereafter referred to as MM1. 2 A. H. Mueller and I. J. Muzinich, Ann. Phys. (N.Y.) (to be published); hereafter referred to as MM2. s M. Ciafaloni, C. DeTar, and M. N. Misheloff, Phys. Rev. 188, 2522 (1969); hereafter referred to as CDM. 4 G. F. Chew, M. L. Goldberger, and F. Low, Phys. Rev. Letters 22, 208 (1969). 61. G. Halliday and L. M. Saunders, Nuovo Cimento 60A, 177 (1969). 6 G. F. Chew and C. DeTar, Phys. Rev. 180, 1577 (1969); hereafter referred to as CD. 7 N. F. Bali, G. F. Chew, and A. Pignotti, Phys. Rev. 163, 1572 (1967); hereafter referred to as BCP. 1 parameters of the little groups of the Lorentz three- vectors k, associated with each upper and lower momen­tum transfer Qu,i by the formula Qu,i=\_k, w±|(-/)1/2], valid in a Breit frame of the over-all momentum trans­fer Q. Since the most important contribution to the phase space comes, for t small, from spacelike k's,3 we shall often refer to the three-dimensional BCP variables as "0(1, 1) variables" and to the poles in the respective Fourier transforms as "0(1, 1) poles." In this language, the purpose of this paper is (a) to establish the factorization at the 0(1, 1) poles in the 0(1, 1) decomposition of the BCP model at general momentum transfer, and (b) to use the three-dimen­sional BCP variables at < = 0, giving a relation between the 0(2, 1) and 0(3, 1) decompositions of the incom­plete absorptive part of the scattering amplitude. The latter relation, which is model dependent, gives, so to speak, the eigenfunctions of the Regge daughter poles in terms of the ones of the Lorentz poles. It is therefore similar to the off-shell relation found8 for the Bethe-Salpeter equation. As we mentioned before, that would be the natural starting point for the dynamical study of derivatives and residues of the daughter sequence near t- 0. However, we have not extended our analysis further in this direction. The 0(1, 1) expansion of the BCP model for the production amplitudes has been given in MM2. We derive a simplified form of this expression and of the resulting multiperipheral equation in Sec. If, and we show that to each Regge pole in the BCP expansion there corresponds an infinity of integrally spaced 0(1, 1) poles with factorizable residues. In Sec. Ill, we take the <=0 limit of this equation for pairwise equal masses and relate the incomplete absorp- 8 D. Z. Freedman and J. M. Wang, Phys. Rev. 153, 1596 (1967). 2917 2918 C I A F A L O N I A ND C. D e T A R 1 Fig. 1. Connection between t<0 and t = 0 frames in the middle of the chain. Only the lower t = 0 frames are shown. The i<0 frames are shown halfway between upper and lower momenta. The notation js defined in the text. tive part in this limit with that of the t= 0 equation of CD. This relationship then implies a connection between the 0(3, 1) and 0(2, 1) decompositions of the respec­tive incomplete absorptive parts, from which we can derive the eigenfunctions of the Regge poles in the daughter sequence from that of a given Lorentz pole. In Appendix B we also simplify the diagonalized t = 0 equation of MM1, using a technique similar to that developed by CDM for the /<0 equation. In Appendix D a model of the Amati-Stanghellini-Fubini (AFS) type is treated as an example. II. t<0 EQUATION FOR BCP MODEL We begin with a review of the three-dimensional and four-dimensional BCP variables, which we have indi­cated schematically for an internal segment of the multiperipheral ladder in Fig. 1 and for the end of the ladder in Fig. 2. The three-dimensional BCP variables (cf. CDM and MM2), consisting of the x boosts gs- and y boosts fi, build up the 0(2, 1) transformation a defined recursively,9 , &i-\- X i-t-ly (2.1) while the four-dimensional BCP variables, for the lower amplitude, consisting of the z boosts qu and 0(2, 1) transformation gu=rz(jxu)bx(lu)rz(vu), build up the 0(3, 1) transformations au, defined recursively, ai,i+i=a-uqiigi,i+i- (2.2) An analogous set of four-dimensional BCP variables is defined for the upper part of the ladder, which we dis­tinguish with the label u: qUi, gUi, etc. The initial trans­formations «o and ait, are defined, respectively, in terms of the initial z rotation 4>a and initial rotation10 rta= ^z(4)a)'l'y(l3la) : Oq-<j)a, aio=?ia. (2.3) 9 For the sake of economy, we use the same label for a one- parameter transformation as for the parameter itself. 10 We have set equal to unity the arbitrary initial Lorentz transformations, mentioned in previous approaches. A similar set of variables defines transformations at the other end of the ladder, and we obtain the transforma­tions h and bm defined in CD, CDM, and MM1 and MM2: bb an+iqn+i<j>b, bib ai ifn-y\yih, (2.4) where r»= ry (fim) rz (<f>b). MM2 have given the Lorentz transformation, which relates the three-dimensional BCP frames (i,r), in which Q=Qui-Qu=LO, 0, 0, (-*)1'2], Qii=Z0, ki, o, w - %( - 01/2], (2.5) Qui=£0, ki, 0, W;+i(-01/2]> and Qi,i+1 and Qu,i+i lie in the xzt plane, to the four­dimensional BCP frame (li, r) in which e„=no, o,o,(-*K)i/2] (2.6) and Qi,i+1 lies in the tz plane. The transformation con­sists in a y rotation 6i,i, which brings Qi,i in (2.5) to the form (2.6), followed by an x boost11 hi,i, which removes the x component of Qi,i+1.12 Similarly, we can transform from the frames (i+1,1) to (li+1,1) by a y rotation di,i+1 followed by an x boost11 fi,i+i~l. The parameters of these Lorentz transformations may be calculated in terms of ki, Wi, ki+1, w,-+i, Mi, and t, or equivalently in terms of tu, tui, tili+1, ta,i+i, Mi, and t. The formula for du is simply sin du=ki/ ( - tu)112, cos 9u=[wi-h( - t)my ( - tu)112, (2.7) ^ua Qui Fig. 2. Connection between <<0 and <=0 frames at the left end of the chain. 11 Our notation differs from MM2. Our hu is their m i+i{ + and our fu is their u, ;+i'+1 +. 12 Note that in this way we specify the frame (li, r) completely with no arbitrary z rotation left, as in CD.1 0 ( 2 , 1 ) D E C O M P O S I T I O N OF T H E E Q U A L - M A S S 2919 while fi,i+i and hi,i depend upon all of these variables.18 Analogous variables are defined for the upper half of the ladder. The fact that the fs and h's adjacent to one rung of the ladder depend only upon the Lorentz scalars associated with the rung is crucial to the factor­ization condition. At the ends of the ladder, the above approach must be modified with 0 being replaced by the z boost Uu and h by the y rotation /3z„, as indicated in Fig. 2. From these two figures, one can now read off the important identities relating the three- and four-dimensional BCP variables14: gli fifoli i^li^ HVli, (2.8 a) @n 'qidi.i+i hiityii(2.8b) (tli- tila (2.8c) The Toller angle o)u = vu-{-ixi,i+1 is fixed by formula (2.8a) in terms of fs-, fi+i, and four sets of ki and »,■; however, to leading order in exp| ft | and exp| f,-+1 | the dependence is reduced to the variables, sgnf,-, sgnf<+i, ki, Wi, and ki+1, Wi+i.3 In the same approximation, it is proper to consider the dependence upon the Toller angle as residing in the multi-Regge vertex function, and the reduced kinematical dependence then forms the basis for a simple factorization of the residues as functions of the k's and w's. In general, however, the Toller angles are not convenient kinematical variables for t< 0. They have, in effect, been replaced by the extra set of momentum-transfer variables. The procedure for the 0(2, 1) diagonalization of the /<0 equation given by MM2 and CDM begins with a decomposition of the unitarity integrand with respect to the 0(1, 1) group parameters fi. We concentrate upon the 0(1, 1) decomposition of the lower BCP amplitude and later combine lower and upper ampli­tudes to form the unitarity integrand. We begin with the BCP amplitude for the production of N particles: ■L*J-mamQ'‘'mN+xmb / - J-^ma to v aJ '-7£0»H)iU v^l/ 7i'li.Pi Xapllrail(tl)-Kgi)GiimWiy»(th fe) • • • GlN+imN+lPN+2yN+lbDpN+2mbSh{rb), (2.9) 13 We have, in terms of the lower variables, cosh/*ii = [£,-+i sinhgJ/[(-sinhg;,] and cosh/i„+i = [£,• sinhg;]/[(-tu)m sinhgjj]. 14 It would appear that the first two identities, viewed as equations relating the various boost parameters, do not always have a solution. Indeed, when nu^O, f;» = 0, vu = 0, the first equation cannot be solved. There are two reasons for these apparent difficulties. The first has to do with the assumptions about the sign of fa and ki2. For k? <0 (timelike three-momentum) and tu<0 we would replace with a z rotation <£; and 9u with a z boost. Equation (2.8a) would then read fu4>ihu=imUivu, which spans the necessary remaining portion of the 0(2, 1) group. However, as discussed in CDM, spacelike three-momentum transfers span the most important part of the phase space for small t, and the whole phase space in the limit <->0 if one adheres to the definition (2.5) in this limit. The second reason for the apparent inadequacy of (2.8a) is that our prescription for going from the three-dimensional to the four-dimensional BCP frames does not leave room for an arbitrary z rotation in the four­dimensional BCP frames. This restricts the choice of the BCP 0(2, 1) transformation. where <7 ,~l~1=TJ ln ,~l-1TJ ~l~l Ujmm - ^ m 'J/mm t-/ m j UJ=T{l+m+\)/T(-l+m), (2.10) and a is Toller's16 0(2, 1) representation function of the second kind. For the lower amplitude, is the z com­ponent of the spin of particle i in the frame (li, r) for i~ 1, ..., 2V+1, and sama, sbmb describe the spins of the initial particles. Conservation of helicity requires that G imp - dm, l-p Gmp. (2.11) If we use the formula a,r~-\g)~ £ Dpf{f)ajk-«~WDkf{h) (2.12) jk for g=f£h, which is valid term by term in an asymptotic expansion of both sides in exp| f |, and the 0(1, 1) decomposition of the a function, given by (A51),16 Z! H Vj,nTa exp[rf (a- w)]PFTCr,J:"0(Tf), n=0 T=± _ (2.13) we may write «i>r"_I(g)~ Z Dp,n"{f) exp [r^((x-n)2d(T^)DnT,f{h), TIT (2.14) where A^"(/)=£A>/(/m.»T", 3 Dnr,f(h)^ZWnT,k-Dkf(h). (2.15) k Equation (2.14) expresses the decomposition of an 0(2, 1) contribution in terms of a series of factorized 0(1, 1) contributions, and is valid as an asymptotic relation in exp| f |. If we substitute Eq. (2.14) into (2.9), we obtain the 0(1, 1) decomposition of the BCP amplitude, a simplification of an expression already given by MM2: MW™ Z exp( - ima4>a) Umaini:T1a'm(Ea, wa; k\, Wi) ni,Ti X exp[nfi(ai-«i)]0(Tifi) X UnitTun2,T2mi(ki, Wi; k2, w2) exp[r2f2(a2-«2)]0(r2f2) • • * Unii+i,rit+y,mi,m^+i'b (^n+1, Eb, Wb) exp ( ifflb(j>b) , (2.16) where we have omitted the sum over y for the sake of clarity. We have defined Un, ,.v-= l5„T/(«GkA,.v"'(/). (2.17) l,pf 16 M. Toller, Nuovo Cimento 37, 631 (1965). 16 Note that, although formulas given in the text do not depend formally on the choice of the basis for the representation functions, the actual expression for o(f) of course does. Formulas (A3), (A51), and (A52) are written in Toller's conventions. Since the o(f) is evahaated for a y boost, it differs from the expression given by Toller by a factor2920 M. C I A F A L O N I A ND C. D e TAR 1 D*th11)-G*-D*(f'u) x I \ ~C D(X) c/- N D(h^,) - G - D(f^ ) X Fig. 3. Index summation scheme for the expression of the residue in Eq. (2.22). We shall now apply the above results for the decom­position of the production amplitude to the decomposi­tion of the unitarity integrand. In writing the unitarity integrand with the BCP form (2.9) for the production amplitude, one must use care in summing over the intermediate particle helicities wii. With the convention adopted above, which gives a simple form (2.10) for the conservation of helicity at the vertex, the helicity of particle i is measured with respect to different axes for the lower and upper amplitudes (see Fig. 1). For the lower amplitude, it is measured along the z axis in the frame (pi, I), a rest frame of particle i, which is related to (li, r) by a z boost The corresponding frame for the upper amplitude (pi, u) differs from the frame (pi, I) by a y rotation,17 which we designate by Xi- (That only a y rotation is required is most easily seen by observing that the sequence of transformations ViT'-htT'du^dihiVi does not affect the y component.) Naturally, this rotation is zero when <=0, since in this limit the frames (li, r) and (ui, r) are equivalent. The rotation x» depends upon the variables ki, w», ki+h w,-+i, nii2, and /,18 and therefore introduces no new complica­tions for the factorization condition. To sum over the intermediate helicities, we must therefore insert for each intermediate particle the function Dmimus(x), and sum over mi and mu, where s is the spin of the intermediate particle. If we now apply the decomposition (2.16) to the lower and upper amplitudes alike and combine the intermediate particle helicities as prescribed above, we obtain the 0(1, 1) decomposition of the unitarity inte­grand. To each pair of Regge trajectories au and a.ui, there corresponds an infinite sequence of 0(1, 1) con­tributions, the first of which factorizes directly, the second of which is a sum of two factorizable terms, and so on. The degeneracy comes from the "cross terms" in the product of two series of the form (2.14). The meaning of this degeneracy becomes clear when it is understood that the product of two a functions may be represented asymptotically as a sum of a functions, "" 1(f)]*C®ii*i al 1(f)D'^£C (au, Oil, v, ju, ji, j) V Xajlr(-ot'+al-,'>-1(i)C(au, at, v; ku, ki, k), (2.18) 17 We are idebted to Michael Misheloff for assistance on this point. 18 cos x«: _ [2M j2(t- tlj - tui) -h (tl,i+l~ tli-M?) (tuti+l - tui - Mf) ] where ctu, ai are real, and O(ctu, &l, V, ku, kl, k) $]e,ki-huC((Xu, Oil, V, ku, kl) , (2.19) and similarly for C7.19 Each a function in the series contributes in turn a single series of factorizable 0(1, 1) contributions via Eq. (2.13), beginning with the term exp^l f |(au+a(- »<)]. Rather than working with Eq. (2.16) directly in the unitarity integrand, we adopt the following strategy, which makes the connection with the <=0 formalism more transparent. We substitute Eq. (2.12) in Eq. (2.9), expressing the upper and lower BCP amplitude in terms of the a(f) 's. Then we combine the upper and lower amplitudes to form the unitarity integrand. If we then apply formula (2.18) to the product of upper and lower a functions at each link, the result is the following unitarity integrand: Mm)M'£ exp ( - ima4>a) ja,jb,yi,3iM y^lUmaji'a''Yl(E'a, ka'y ki)(Xjxjtl "T1 1(^*) XUhlj2^(w1, k\) w2, k2)ahkfay^(^) • • • Ut!N+1,mi''N+lib(wN+1, kN+1; Eb, h) exp( - imyfrb), (2.20) where we have lumped together into y the sums over yu, yi, and v and have written at each link a.y=a.yU-\-oiyi-v. (2.21) The vertex functions are (see Fig. 3) Ukj^'(w,k-,w',h')= £ C(au,cti,v,ku,khk) 3uf ,3l/,ku,kl,mu,mi XCL Dhui^(K)GKmt,Pa'Dpjj^a'‘,(fu')JtDm,m‘(x) lu<Put xCL ii.vi' XC'(au',ai',v'-,ju',ji',j'), (2.22) with similar expressions for Uia'~<x and we have put ma=mia-mua and mb- mn,- and the sum over ja and jb includes the usual channel spins at the ends of the ladder. If we apply the 0(1, 1) decomposition (2.13) of the a function to Eq. (2.20), we obtain the form of the unitarity integrand required by CDM for the 0(2, 1) diagonalization of the multiperipheral equation. We define, accordingly, the incomplete absorptive part Bm„,„ry(a) and its partial-wave projections bmanTly. For a discussion of the diagonalization of the integral equa­tion, see CDM and MM2. After diagonalization the 19 The coefficients C and C' are related to the vector addition coefficients for the representations of 0(2, 1). |jSee Kuo-hsiang Wang, UCRL Report No. UCRL-19306, 1969 (unpublished)]. For practical applications involving a few leading terms, they may be obtained directly by comparing asymptotic expressions for1 0 ( 2 , 1 ) D E C O M P O S I T I O N OF THE E Q U A L -MA S S 2921 equation reads bma,n'7,^f {k , W ) = (0)frmo,ji'T'^7 t W ) + E vSdkdwbm„nrly(k,w)UnT,rl'T^'{k,w}k',w') y,n,r Xd\ ay-n) ,tt'(a (r1), (2-23) where the index n refers to the 0(1, 1) contributions resulting from a single ay. The function d is described in CDM. The vertex function U is defined through Eqs. (2.13) and (2.22): UnT.n^yy'= E Wm,^Ukj>yy'Vr,n^'. (2.24) h,jf The functions bma,nTly are related to the functions bm.,nr+ly appearing in the modified 0(2, 1) expansion (3.8b) and (A45) by A ly Vma ,W7+ _ r[/+1+r{ay-n) ]rp+1 - r(ay- n) ] r(2/+2) ""'"T ' (2.25) which follows from Eq. (4.14) of CDM. III. 0(2, 1) AMPLITUDES AT *=0 We have shown in Sec. II that the 0(2, 1) and 0(1, 1) expansions of the production amplitudes are equivalent as asymptotic series in the parameters exp| f |, con­nected with the subenergies. At t< 0 we have also defined, through the unitarity integral, the incomplete absorptive part Bm,a,nry(a; k,w), a function of the over­all 0(2, 1) transformation a, for the wth 0(1, 1) "daughter" of a given angular momentum ay=ayu-j- otyi- Vj resulting from the addition of the upper and lower Regge-pole contributions. At t=0, the incomplete absorptive part can be defined either as a function of the 0(2, 1) transformation a, or in terms of the 0(3, 1) transformations au = ai=a. They are not the same func­tion in different variables because they are constructed by splitting off different factors from the complete absorptive parts, depending upon whether they are derived from a factorized 0(1,1) or 0(2, 1) expansion of the unitarity integrand. By using the explicit form of these expansions, we shall now derive a relation between the two incomplete absorptive parts, which eventually will give the relation between 0(2, 1) and 0(3, 1) partial-wave amplitudes. Since gi=gu=g at t = 0, the Clebsch-Gordan combina­tion of upper and lower amplitudes is simple. Following CD, we assume I M CJV) (2,-, V n ayl (/,) mi,yi Xa„!0™1_"Tl~1 (&)-Rmi7178(h, k) • • • RmK+1yN+lb(tN+i) DmN+imJb(rb), (3.1) and we define the incomplete absorptive part Bmamy (a, t) as in CD, by removing the last factors RD in the unitar- ^ „ D*(h)- -D(h)-Cv = -C " D (h )- Fig. 4. The property of the Clebsch-Gordan coefficients used in the text. ity integral. If we compare the above expansion of the unitarity integrand with that obtained directly from (2.9) using (2.18), we see, by matching terms in the asymptotic expansion, that 8ip'Rp'yy - 0(oj«, <xi, v \ Xu, li, 0 lu,ll,Pur >Plf W X (Giump„')*Gi,mpi'C'(au', a/, /; pJ, p{, p'), (3.2) where the factor Sip> follows from (2.11) and (2.19) (helicity conservation), au is short for ayu, etc., and 7 is short for {yu, yi,v}. The 0(1, 1) expansion of (3.1) can be obtained from (2.12) and (2.13), and we get I |2~ Z exp(-ima<j>a)Uma,nir1ayl(khWi) yi,ni,Ti X exp[nfi(aTl~%)]0(rifi) X Uniri,n2T2*^Z(.felj ^1? ^2? ^2) CXp[jT2^*2 (<^ 72 * * * ^nN+VTN+lt'frtb yN+ib(kN+iWif+i) exp(-imb<f>b), (3.3) where Unr,n'r'yy,= E A,r ,pay (h) Rpyy'Dp,n,*,<*-<'( f) . (3.4) p Equation (3.4) after substitution of (3.2) is to be compared with the expression (2.24) after substitution of (2.22), in the limit t= 0. With the present procedure we first combine the upper and lower a~""1( /f/z) in the Clebsch-Gordan sequence and then factor the functions Da{h) for asymptotic f's, whereas in Sec. II we per­formed the same operations in opposite order. The equivalence of the two procedures and the equality of respective i=0 residue functions and absorptive parts follow from the property of the coefficients C and C' schematically shown in Fig. 4. Hence we conclude that Unr ,n/r' Ujit tnr7f • Looking at the expansions in Eqs. (3.1) and (3.3) with (3.4) in mind, we see that their equivalence implies that20 (sind)Bmamy{adh)^ E y(a)Dnr,ma?(h), (3.5) nr ao Bmam{a6h) is the same function of a = a8h as the CD in­complete absorptive part, but satisfies a different integral equation in which the variables q, 8, f replace q, Ji, f, and v. The reason is that in the CD integral equation the integration over jj. replaces the summation over intermediate helicities, whereas here it has been performed explicitly. The equivalence of the two equations can be proved by noting that, owing to helicity conservation, the equation satisfied by B(aBh) in invariant under the substitution /(->/«/3, h'-^h'p', /'-where /3 and fi' are z rotations, and that an extra integration over fl can therefore be added. This invariance permitted an arbitrariness with respect to s rotations in the definition of the CD frames.where the factor sin0,= ki/(-ti)112 comes from the phase space,21 and use has been made of the relation [Eq. (2.8c)] a=u<rla!dh (3.6) and of the fact that ua=I at t=0 for pairwise equal masses. Note that, if we parametrize a=rrjg, ?G0(3), Tj=Bz(rj), g£0(2,l), a=<t>y£, <l>=Rz((t>), y-Bx(ri), t=By{Q, (3.7) Eq. (3.5) is valid as a relation between asymptotic series in el{l. This follows from Eqs. (2.12) and (2.13), on which (3.3) is based, which are valid in the same asymptotic sense. Having derived the relation between incomplete absorptive parts, we now proceed to relate the partial- wave expansions. Consistently with the asymptotic meaning of (3.5) we shall perform some manipulations on the 0(2, 1) and 0(3, 1) decompositions in order (a) to express the left- and right-hand sides of (3.5) in terms of the residue functions bc/M and ba,nrl which are the meaningful quantities in the asymptotic sense and which can be directly deduced from the diagonalized equations (B15) and (2.23), and (b) to extract the h dependence of the left-hand side consistently with the right-hand side.22 Problem (a) is solved in Appendix A, in which we prove that, for asymptotic g and £,23 we have 24 5».»(«)~ E f d[X\ba^MDjama,asmM(a), (3.8a) M ,s J o bma>nTr^&ma,anTr^{$'Q) r X exp(3.8b) where we have dropped the index y, we have defined anT= -r(a-n), and n. ™=yn. , ^Jama.asm - Z-i ■LyJaina', ls,m' V,''// mr X Um'aam'm~a~l(g) Vm~a-\ (3.9) and j=± is the label of the two 0(2, l)'s which occur in the reduction of the 0(3, 1) group.26 21 The volume elements dt sinhg dsg and dk dw d£ are appropriate for the integral equations after the factors ( - f)1/2 and k are removed from the respective incomplete absorptive parts that come directly from the BCP expansion. For a given a, B and B are therefore normalized in a different way; hence, the factor sinO. 22 Since h depends on the variables k, w, k', and w', this is needed in order to have a relation involving only one set of variables, k and w. 23 Equations (3.8a) and (3.8b) are valid as asymptotic expres­sions in coshf and elfl, respectively, where g = R,(/i)Bx(f)R,(p). 24 The notation is as follows (see Refs. 1-3): The 0(3, 1) re­presentation functions are labeled by (X, M), which specifies the unitary representation, by (j, m) for the 0(3) basis, by (Is, m) or (Is, nr) for the 0(2, 1) basis. In the last case, I specifies the 0(2, 1) representation for each (s=±) of the two classes of 0(2, 1) cosets which occur in the 0(3, 1) group, and (,u±) refers to the two classes of 0(1, 1) cosets in 0(2, 1), with a given eigenvalue ( - i/j.) of Ky. d{f\ and d[X] are the relevant measures in the I and X planes.^ 26 A. Sciarrino and M. Toller, J. Math. Phys. 8, 1252 (1967). 2922 M. C I A F A L O N I Problem (b) is solved by noting that with the parametrization (3.7) in (3.6), when | £ | is asymptotic so is S = hh, (3.10) where /G0(2, I).23 Then substituting Eqs. (2.12) and (2.13) into Eq. (3.9), we obtain the right h dependence in the form Dj„ma,aSmM(a)~ 5Z Djama.aSjmXM{4yqQ) n,T X exp{- &nT)Dmma(h)6(T£), (3.11) where f). XM/'A\=y' T) tM(K\V , « ■L^ jm; as ,nr \V j - \U / V m' ,nr = 2ir lim (-t)[(m- a„r)Z?yTO;„s^+XAf(i)], (3.12) and the last equality follows from the definition of Vm',nTa in Eq. (A50). By making use of the group multiplication properties and noting that 6, Ky,2i and £ commute, we obtain Djama-,ae,iJ.+*M{a8) = £ MX] W ,m?M{I) 8f ,r (3-13) and going to the residues at the poles /j. = a„T, we get Djama-as,nrXM{4>v6)= £ $d[f\ \_Kma(ls',ja)y- sf ,r X Dma ,a„Trl (<fa)Dl*>,anTr, «*,»" (6), (3.14) where the K function is written in Toller's26 notation. We can now substitute (3.14) into (3.11), and then (3.11) and (3.8a), (3.8b) into (3.5) to get the final result bma,nrl(k,w) = (smd) X) M>] M.s.sl XbJM(.t)Dis,,anr+.,as,m™(0), (3.15) where bma,nTl=bma,nT+l, and the ( -) amplitude can be obtained by the use of the conjugation properties of CDM [see Eq. (Al)]. An expression forZ)XM(0) can be obtained from (3.12) and (C6). A N D C. D e TAR 1 Fig. 5. Location of the poles in the X plane for the integration of Eq. (3.15).1 0 ( 2 , 1 ) D E C O M P O S I T I O N OF THE E Q U A L -MA S S 2923 Equation (3.15) solves the problem of connecting the solutions of the t=Q diagonalized equations with respect to 0(2, 1) [Eq. (2.23)] and to 0(3, 1) [Eq. (B15)]. When ba*M has Lorentz poles at X= ±Xo,26 the singular­ities of bm^nr1 come from the pinchings of the A contour, and it is evident from Fig. 5 that they may occur at l = \0-n- 1 and at the symmetric positions I= - Xo+». Actually, only the sequence 1-\q- n- 1 can occur in bin*,nr1, because this amplitude is, according to CDM, analytic in the right-half I plane. It is possible to show that this is true for Eq. (3.15) by the methods of Appendix A, which are briefly summarized here, for the convenience of the reader. Whenever we have a summation of the form E (3-16) s,M where/;s,3-xm transforms contravariantly under conjuga­tion with respect to 0(3, 1) (poles at X=- l+n and X = /+l+w) and gj,i^M covariantly [poles at \ = l-n and X = - (J+1+m)U, we can replace the s summation above by (3.17) where a^M, /3fM, and U/'M are defined in Appendix A and has zeros in X at the position of the poles of It is clear, therefore, that the first term of (3.17) has no /-dependent poles in X, while the second has poles only at \=l- n and X= - (H-l+w). We now apply this result to the s' summation of Eq. (3.15), with g^-^K^n and>bXMDXM. We note that, in this case,/y,;+_x has poles at X= ±Xo coming from ba^M, and does not have the poles X = l-n in DXM because of the (+) character of the function Dm,*+1 occurring in its induction construction [see Eq. (C6)]. The first term of Eq. (3.17), which has no /-dependent poles in X, cannot give rise to poles in I in the left-hand side of (3.15). The singularities in the X plane of the Fig. 6. Location of the poles in the X plane of Eq. (3.15) after substitution of Eq. (3.17). We have shown with small circles the poles coming from basl, and with crosses all other poles. 26 We assume Xo<0, so that the completeness relation is properly convergent [[cf. (A33) ]. Note that, because of the conventions of Toller (Ref. 22) and MM1 for the X plane, ba+XM has a pole at X= - Xo, and is well behaved in the left-half X plane. second term of (3.17) are illustrated in Fig. 6. It is clear that the only possible pinchings are between X= - (H~l+») and X=-Xo, and this implies the above­stated result. The Regge-pole eigenfunctions, which can be calcu­lated by this method in terms of the Lorentz-pole eigenfunction, are useful, in principle, to obtain dynam­ical quantities such as derivatives of the Regge family at £=0. Note finally that the relation between the total absorptive parts which follows from (3.15) is model independent and is, of course, the same as the one obtained from the general group-theoretical analysis.27 ACKNOWLEDGMENTS We thank Dale Snider for reading the manuscript. We are particularly grateful to Michael Misheloff for his collaboration in the early stage of this work. APPENDIX A: MODIFIED COMPLETENESS RELATIONS 1. Conjugation Properties in Noncompact Basis It has been shown by CDM that the (r= -) partial- wave projection of Bo,,,1 is a linear combination of the (f=+) function and the function with I replaced by - /-l, through the relation28 B0,»J=V.B0,,*+(- 0/ r(H-1), (Al) where OLpl=- cosx/i/cos7r I, j3M*=[ir/r(-/I-0r(/x-0 costt/J. (A2) Equation (Al) is a consequence of the equivalence of the representations Dl and Dwhich, in the mixed basis, may be expressed as follows: Dm,^l~1(g)= L (Uml)*Dm,„r'l(g)yr',hrl, (A3) rf where, as in Sciarrino and Toller,26 Uml=T(l+l+m)/T(-l+m) (A4) and 7 is a unitary matrix in the r basis, 7i^ y2nl\ ) • (A5) 72m' 7l/x / By inspection, one finds "mj=(-7i//72^)*, 0/=(1/72mO*- (A6) With g-I in (A3), the expression reads Km,^r~l~1= Yi {Uml)*Km^r'lyr',v.rl, (A7) r' 27 M. Toller, Nuovo Cimento 53A, 671 (1968). 28 The restriction in CDM can be removed by choosing the appropriate sheet in continuing to tj<0. However, (A2) and (A5) need slight modifications for fermion representations.2924 M. C I A F A L O N I AND C. D e TAR 1 where (£*■,»')*=Dm,»rl{I) ■ (A8) Analogous relations hold for the 0(3, 1) group in the noncompact 0(2, 1) basis, where the equivalence is between the representations (X, M) and ( - X, - M). We shall show that this equivalence leads to a simplifi­cation of the 0(3, 1) completeness relation in this basis. Instead of (A3) we have Djn.,ism~^~M{a)=Y. (U^yDjn.Mm^(a)Ts^M, (A9) s' where Toller has defined v>*= n (aio) i=\M\ *ta and T is unitary in the s basis.29 Sciarrino and Toller25 have discussed the function (All) and have obtained the identities s;j) = s;j), (A12a) K™(1, - -J) = (A12b) 1, s-j) = U„ri_1 USMlKm*M(l, s;j), (A12c) which follow directly from the integral representation KJM(l, ±;j)=U2sa+iy2 J" (coshr)*-1 XrMJ(e±) (f) d coshf, (A13) where tan|0+= tanh|f, cotj0-= tanh|f. (A14) In addition the K function has orthogonality prop­erties, implied by the identification KJM(l,s;j)<r^{l,s,m\jm), (A15) which follow from group multiplication properties of the D functions. We shall make use of the property E K™(1, s;j)lKm^(l', s'-,j)J = S8Ml, I'). (A16) j Combining (A9), (All), and (A16), we obtain the following expression for T: 8(1, E s;j)J* j s';j) UjXM. (A17) From (A12a) and (A12b) it then follows that /ruw (_)2«ivn IY"=( , (A18) \iVM (-)2<ivV where, as in Sciarrino and Toller, (- )2e= (_^ 2Af 29 That must be independent of m may be verified by putting a->ag in (A4) and using the irreducibility of D‘. Furthermore, from (A12), ra*x'-M= (-)2Tazxilf for a= 1, 2, r1,_;_1XM=(-)2fIVM, (A19) r2,_z_1XM= ivM. If we define ai™=-(Tu*M/Tn*M)*, /SiXJ*= (l/lV^)*, (A20) so that LDjn, ]* = i+,m™(a) ]* +^M[.Djn., ]WM) *, (A21) then from (A19) we conclude that ap-M=afM^ a_;_iXM= ( - )2ea^M, p^M=l3?M. (A22) In order to continue the foregoing relations to values of I and X corresponding to nonunitary representations, we make the replacement 1, X->- X in expressions involving the complex conjugation of the functions K, T, a, and j3. Thus (a^)* = a_w-^, (/?^)*=|3_;_r^. (A23) To conclude our summation of general properties of the functions a and /? we observe that the unitarity of T implies the following important relations30: -a?M, 1- (a^)2 = ^^^_z_i^■M. (A24) We now propose to show that 1 - (a^M)2=0^/3_!_rXAr =0 (A25) whenever X=±(H-l+») or X=±( - /+«'), i.e., at the "kinematical" poles of the representation functions. We show in Sec. 2 of this appendix that this property of a produces the necessary cancellation of the kine­matical poles in the 0(3, 1) completeness relation.'The proof of (A25) makes use of the fact26 that Km*M(l, s;j) has poles at \-\-l=n and X- I- l = n [see (A13)] but no other poles which move in X as a function of I. Sciarrino and Toller have defined the residues at these poles lim (l-\+n+l)Km™(l,+-j)^Wim™n, (A26) l-*\-n-1 from which it follows [see (A12c)] that lim (l+\-n)Km™(l,+-,j) = - Um-^nUMx-n-lW}™n. (A27) From (A9), (All), (A12b), and (A18) we obtain the 80 Explicit calculation from (A9) gives, for M = 0, ffixo = sinjr//sinxX, and/3ixo= - T(X) r(X+l)/[r(7-t-l-|-X) T(A-/)]. The properties (A23)-(A25) can be explicitly verified for these expressions.1 0 ( 2 , 1 ) D E C O M P O S I T I O N OF THE E Q U A L -MA S S 2925 relation +; j) = + ,j) r1,_!_rx'M + +,i)r2._l_rx"w]. (A28) If we require that Eq. (A28) be consistent with the left-hand side having poles at -\-l-l = n and - X+ l=n it follows that rli,_;_r'x'M or r2,_;_i_x,Af or both must have poles at these points. Let us denote the residues by rlrr*M and r2n~*M in each case. With these definitions we write the residue of (A28) for both signs of M at -\-l-l = n: W3-„rx'-M'n= +,j)rln-** + +,j)r*r*-Ml, (A29) where we have used (A10) and (A19). Sciarrino and Toller have given the identity W^~M-n = (-) (A30) With this identity and the orthogonality property (A16) [with (A12b)] we conclude that ' rln^-M/r2n-^={-Y{-y\ (A31) Consequently, from the definition (A15), lim (a;XM)2= lim (ri,_i_fx'J7r2,_!-rx'M)2= 1. Z-»\+l+n Z->X+l+n. (A32) By the same methods one may verify that (A32) holds for - X+Z=w. From (A23) and (A24) it follows that (A31) is also valid in all cases when X->-X, which proves (A25). ~2. Asymptotic 0(3, 1) Decomposition in Noncompact Basis In this part we derive an expansion of the incomplete absorptive part in terms of 0(3, 1) representation func­tions, which is asymptotic in the sense of Sec. II. From our final formula (A41) one may also obtain a simple expression for the leading 0(3, 1) pole contribution. Into the completeness relation for 0(3, 1) partial waves, Bmam(a) = £ f d[X] f d[f\ Bu,vi'MDjamajs^M{a), M,s J fl J (A33) we substitute the following identities, derived from Eq. (A21) and the definition of BXM: Bu-,m™=al™Bl+,™+frMBl+,m-*'-M (U^M) *, I-,mXM= l+JM + (frx^) *Djama; ^--*11,"*. (A34) The integrand then reads, schematically, {Bn>MDlSMll+a?M(a?M)*2 + {Bl+-*‘-MDi+-*--Mpi™(p?M) * + Bi+-X,-MZ)(+XM( t/XM) *(a;XM) _ (A35) By making use of (A22) and (A23), one may readily verify that the terms grouped in the first set of curly brackets are the same as those in the second after replacing (X, M) by (-X, - M). Consequently, we extend the limits of integration over X, and keep only the first two terms in (A35): /*■+•*co f r-l/2+ioo 1 Bmam(a)= 23 I S | M J-ioo (V-1/2 l=k+i + Uh™Dhma., n.lB-x'-"(a) (A36) We now proceed to shift the contour of integration in I in (A36) so as to collect the residues*at the input poles of Bi+,^M at l=ay and l=-ay- 1. If we write a=rvg, where rG0(3), g=R2{n)Bx{$)Rz{v)£0{2, 1), and i] is a z boost, then = £ Dhma,lsm^(rr,)Dm,J(g), (A37) and we seek an expansion of Sm„m(ri]g) as an asymptotic series in coshf. Following Toller,15,25 we first write Dm.J{g) = am,J{g) + Um(A38) and then substitute (A37) into (A36). By making use of identities for the reflection I-*-1-1, we obtain ~ /•+*» f /■-l/2+»'oo 1 5m„m(a) = £ / 4X] I <C0+ £ [ M •'-ioo IV-1/2-too l-k±J + U3™Dian,, l+,m-^-M{a)al^, (A39) where XUm,lam,m-l-\g)Um-l-\ (A40) The partial-wave amplitude Bi+,J'M has a pole on the left-hand side at l = a_ in addition to X-dependent kine- matical poles. The D function contributes additional X-dependent poles and also contains "nonsense" poles in I arising from the a function in (A40) ,15 However, when the I contour is shifted to the left, the first term in the curly brackets in (A39) contributes a residue only at 1= a. The kinematical poles are canceled because of (A25), and the nonsense poles are canceled in the usual way by the contributions of the discrete series. The kinematical poles in the second term are not all canceled. However, the residues of these poles are regular in X and, since Di+~x~mBi+xm vanishes exponen­ 2926 M. C I A F A L O N I A N D C. D e T A R 1 tially as ReA->- oo ,31 they give vanishing contributions to the X integral. The upshot of this analysis is that, asymptotically in g, we may simply replace the integration over I and summation over the discrete series by the pole contri­bution at l=a, as follows: r+iao _ M,S J Q = £ M J-ioo (A41) Equation (A41) is also suitable for obtaining, for 77 large, the asymptotic contributions coming from dy­namical singularities in X of 2?w. Shifting the X contour to the right31 for the first term and to the left for the second, we see that the a-dependent poles in X of the first term are canceled as before, whereas the second term has no such poles on the left and no dynamical poles in &+xitf, either. Except for an additional complica­tion, one would replace the integral over X by a sum over the residues at the Lorentz singularities of bct+1'M of the first term only. The complication is that in general it is possible that aXM has extra poles in X 32 that are absent in basXM. It can be shown, from arguments based on the absence of such poles in and ba-M in (A34) that, should such extra poles occur, they must be canceled by contributions from similar poles in in the first term in the curly brackets. This circumstance has a precedent in the Mandelstam-Sommerfeld-Watson transform.32-34 3. 0(1, 1) Decomposition of 0(2, 1) Representation Functions Note first that manipulations analogous to those of Sec. 2 of this appendix can be performed in the case of the 0(2, 1) expansion of Bm,nr(a). If we parametrize a=<t>y£, 4>-Rz{4>), v=Bx(ti), i=By{£), (A42) and substitute the conjugation relations (Al) and (A3) 31 We follow here the conventions of Sciarrino and Toller (Ref. 25) and MM1 for the sign of X in the induction construction of the representation in the mixed basis. This implies that ~M is well behaved in the left-half X plane. Note that this convention is opposite to the usual one for the I plane. 32 The explicit expression given in Ref. 30 has such poles at integer values of X. They correspond to the half-integer values in the I plane. At such values ba+XM has a symmetry analogous to the Mandelstam (Ref. 33) symmetry, and called "gemel symmetry" by Gatto and Menotti (Ref. 34). 33 S. Mandelstam, Ann. Phys. (N.Y.) 19, 254 (1962). 34 R. R. Gatto and P. Menotti [Phys. Letters 28B, 668 (1969); 29B, 592 (1969)] have studied this symmetry in the case « = 0 where aoxo = 0, and therefore the poles at the integers do not appear in our expression. When a^O, the absence of such poles in can be used in Eq. (A34) to prove the gemel symmetry very easily. into the 0(2, 1) expansion /-1/2-f »oo d[l] -1/2 X f ( - i)dfi BnT,v.rl'Dm,v.r(a), (A43) iao we get /-1/2+ioo r- d\J] { - i)dixBnrtli+lplrl~'L -1/2-«oo icc X[/VlDm,ll+l(4>ri)-\-aillDmtli+-~l-1(4>ri) (A44) The first term in the integrand has the ^-dependent poles in the 11 plane canceled by the factor /V/SjT*-1, whereas the second term still has the poles n= ± ( - /+«), but their residues are analytic and well behaved in the right-half I plane, so they give vanishing contribution to the integral. Therefore, the asymptotic series of (A44) in e,fl is simply obtained by picking up the contribution at the "dynamical" pole -r(a-n) =aw, which is nonvanishing only when r£> 0. We have finally Bm ,nr ,anTr^ (<M) r X exp( -ow£)0(t£), (A45) where bm,nrrl are the residues of BnT:IJrl at the poles fl OLnT. We want now to obtain the 0(1, 1) decomposition of the function amm -"-1(f), which occurs in the produc­tion amplitudes (2.22) and (3.1) as single Regge-pole contribution. The main purpose is to prove factoriza­tion at the 0(1, 1) poles. We start from the relation /+100 (-i)dv.KmJe-«K,rm\ (A46) . -ICO which follows from the definition (A8) of the trans­formation functions Km:lirl. After substitution of the conjugation relation (A7) in the form (Al), the right-hand side of Eq. (A46) can be written in the form (A47) where f { - i)dfi 03fr!-1.K')B,,t+-M J-ioo - a,‘Km ,„+* t^r*-1) (A48) Note that the first term in the integrand has no I- dependent poles in the n plane, as usual. The second term has only the poles ju==b il-ri) coming from Km:li+l. By displacing the p contour either to the right or to the left according to whether and neglectingthe background integrals, we get 1 0 ( 2 , 1 ) D E C O M P O S I T I O N OF THE E Q U A L -MA S S 2927 where X exp\T${l-n)y(T$)Wnr,m'1, (A49) Vm,nrl=2iv lim ( -t)[/1 + t(/- H~>-ril- n) Wn r,m'l= [ - CtJfiJK. ^ ,m "l_1 Um '_J_13/i=-r(i-») • ( A50) Since the asymptotic expansion (A49) contains only the powers (exp|f|)i_n, we can identify a1 with Toller's a1, and we finally get = E Fm,„rzexp[Tf(^-»)]fl(rf)PF„r>m-!, (A51) nr which exhibits the factorization at the 0(1,1) pole contributions. For convenience of the reader, we quote finally the result36 Km,»rl = [r (- l+») - /lirVT (- 21)2 X exp\jir(l-{-m-n)/2~]F( - l-rm, - l+n; -21; 2+j‘O). (A52) APPENDIX B: 0(3, 1) DIAGONALIZATION OF *=0 EQUATION We present here a simplified form for the diagonalized t-Q equation of Mueller and Muzinich.1 The simplifica­tion parallels the method of CDM for the <<0 equation. Rather than using the Andrews- Gunson E function36 for the BCP amplitude, we prefer to use the a function (2.10) of the text. We require the 0(2, 1) decomposi­tion of Toller's a function, which reads /-1/2+ too dlli}Kmm,(lhh)Dmm-^(g) -1 '-1/2 + E &±) Dmm'k±(g) k± . for ReZi = ■ X (2/i+l) r(w'+fe+l) (h- k) (W-fe+1) and for m< m!, Kmm> (/i, 12) = (/j, I2). A property of the K function which we will find useful may be deduced from the orthogonality relationship between two D functions and the expression given by Toller: Dmm'l(g) = Omm'l(g) + UJ<W_M(g) (B3) where UJ=T(l+m+l)/T(m-l). That property is lim 4) + V- li- 1 -e, l2) Um'11 £->0+ = 6{h,h), (B4) where lm/i>0 and Im4>0, and Re/i = Refe=- \. We have defined -1/2+100 t-2- •'-1/2 d[i{]&(iuh)m =m- With this form for the decomposition of the a func­tion and the form (3.1) for the unitarity integrand, the t=0 equation as diagonalized by MM1 reads = (0)^wx"(O+E [ <*[/]+ E 1 (Bl) where the a function and the D functions for the con­tinuous and discrete series have been given by Toller.16 From Andrews and Gunson's formulas (3.3), (2.1), and (7.12) we find that with d[l2=v(l)dl for m>m', Kmm, (k, h) = [2riv (k) j"1 T(h+ m+1) , (B2) X f dtsrnhqbim^M(t)Rm>(t,t') J-00 X4.M.,m'W(f1)[-^ml(-a-1, -I- 1)], (B5) where bims'*11 (t) = E bi'n's^M(t)£- Knm( - a- 1, -/-l)] n (B6) is the amplitude of MM1, and we have suppressed the 7 index for the moment. Recall that37 dv+,i+,v^M(q~v)= J d cosh«[^+Af,m"(a)]* X (coshg+ sinhg cosha)x_1d+jtf,OTz(a/) for q> 0, (B7) cosha'^ (sinhg+ coshg cosha)/ (coshg+ sinhg cosha), and similarly for the other representation functions (cf. MM1). Because q is always positive, the d function in Eq. (B5) vanishes for /=+ and s= - . As with the t<0 equation, the system of equations in s reads 35 Equation (AS2) is obtained from the^complex conjugate of Eq. (A20) of CDM after multiplication by the phase factor exp (iiirm). The reason is that Mukunda's convention for the 0(2) basis differs from Toller's. We are now using Toller's basis, whereas we used Mukunda's basis in Eqs. (A19)-(A20) of CDM. [See N. Mukunda, J. Math. Phys. 8, 2210 (1967).] 36 M. Andrews and J. Gunson, J. Math. Phys. 5, 1391 (1964). = (o)*+w+ b+™K. bJ \M + + ! = mbJu+b+™K- (B8) As in Eq. (Al), one can make use of the equivalence of the representations (X, M) and ( - X, -M) [Eq. (A9)3 37 We keep the conventions of Sciarrino and Toller and MM1 for the sign of X (see Ref. 26).2928 M . C I A F A L O N I AND C. D e T A R 1 Fig. 7. Location of the poles in the I plane in the integration of Eq. (B13). to reduce the second equation to the form where = [ d cosho a+M,m~l'~Ka)d+M,ml(a') Ji X (coshg+sinhg cosh«)x~1. (B14) We have constructed biJ'n so that it has X-dependent poles in I at - \+n (i.e., in the right-half I plane) only. It also has poles and zeros contributed by in the separation (B12). These poles and zeros cancel poles and zeros in the weight function tj (I) in the usual way,16 and the resultant /-plane singularity structure of the integrand in Eq. (B13) is indicated in Fig. 7. If we shift the contour to the left, we collect the residues at the nonsense poles at /= -1, - 2, ..., - N.3S These cancel the contributions of the discrete series, as usual, and we are left with the contribution from the "dynam­ical" pole at a. In terms of the values at the dynamical poles bayJ'M, the equation reads b+ x> M=wb+ *+b+ x> MK+ + x' " (B9) = E f dt sinhg , .i ^ . .* • / Tv \ • ii. m.y *'-oo Therefore, only the first equation in (B8) is needed to determine the locations of the Lorentz poles. We shall henceforth restrict our attention to this equation. We now wish to present a scheme for shifting the I contour in Eq. (B5) so as to collect only those residues arising from the input Regge poles. As the equation now stands, we are prevented from doing this by the presence of X-dependent "kinematical" poles in I in the function b, which lie on^both sides of the contour. They appear at the same locations as the poles of dv+,i+,^M in V, which, from Eq. (B7), appear at V- -X+», -r-i=-x+« for « = 0,1,___ (BIO) The two sets of poles are additive with respect to each other, as may be seen by substituting Eq. (B3) into Eq. (B7). This offers the possibility of writing b in Eq. (B5) as a sum of two terms, each of whicl/has only one set of singularities in I. We define -1/2-j-ioo d[f\+ £ -1 L-7-1/2 l=k±. bim+XMKMm{l, I) for ReZ= - i--e. (Bll) This function has kinematical poles at Z+ X = w but none at - I- 1+X = ». Moreover, because of Eq. (B4), h^M= b!m™+ Uu-^b^^UJ. (B12) Substituting Eq. (B12) into (B5) and the result into Eq. (Bll), we obtain S^-XM(0=<o)Ww(0 ~ r-1/2+*co /*0 + E d\J]+ E dt sinhg m \_J-m-ioo J-<x> Xh™{t)l-Kmm,{-a- 1, -1-1)1 XRtn '^Ol '^iq -1), (B13) m,y oo XbayJM{t)Rm,^'{t, Odcya^iq-1), (B15) from which an w-independent equation for ba7XM= Em ba~imM can be obtained, having as the kernel Sinhg E Rmy7'(t, Oday'aym^iq-1). (B16) m Owing to Eqs. (B12) and (B6), the residue functions basXM appearing in the modified completeness relation (A41) are given by39 f«7+u,= E[Cw+^-"'-1L7-i,BXJ,M- (B17) APPENDIX C: REPRESENTATION FUNCTION NEEDED IN TEXT We derive here an integral representation for an 0(3, 1) representation function required in Eqs. (3.12) and (3.15) of the text. That function is the matrix element of a y rotation in the noncompact 0(2, 1) basis: Di'S',„'r>-,is,^M(&)= (Vs', n'r' | exp(- idJy) | Is, fir) = «(/-/i)ij..M.:^,X"(fl). (Cl) The index s = ± represents the required doubling of the 0(2, 1) basis and theindexr=± the analogous doubling of the 0(1, 1) basis for the representations of 0(2, 1). The procedure for constructing matrix elements of the Lorentz group in the 0(2, 1) basis by the method 88 iV = min( ] m \ , ] M j ) for miW>0jand?iV = 0 for mM<0 (see Ref. 15). - 39 In the 0(2, 1) case, we were able to remove the kinematical poles from the incomplete absorptive part explicitly by factoring out a B function. Since we have not been able to do the same in the|0(3, 1) case, we do not have an expression analogous to (2.25) relating baXM to fe„+XM'. Hence in practice one must substitute (B14) into (B17) to relate ba+XM to baXM, although we believe the relationship is not fundamentally a dynamical^one.0 ( 2 , 1 ) D E C O M P O S I T I O N OF THE E Q U A L -MA S S 2929 of induced representations has been summarized nicely by MM1, who give further references. We shall merely sketch those points which must be altered in their treatment for these special representations. The parametrization of the 0(2, 1) elements approp­riate to the basis required is gs= exp( - i4>Jz) exp( - iasKx) exp( - i\Ky), (C2) where 0<<#><4ir, - °o and - t»<X<+oo spans the group. Note in particular that both signs of as are required here. The mapping on g induced by the rotation 6 leaves X and 4> unchanged. The mapping on as is , scos^exp(as)-sin^ J "P(" - s sin^fl exp(os) + co,S# ' (C3) The mapping on the elements in the Hilbert space 3C=£2M®£2~M is Z7(x'")[exp(-j0/!/)]{0+(g+), <£_(g_)} = IE X+,a(x)(0, a*)<l>s(g+), Z X-,s<x)(0, as)<h(g-)}, 8 S (C4) where g's' = exp( - exp (- ia's'Kx) exp ( - i\Ky) for a's> as defined in (C3). We have defined Xs's{y>{6, a-s) = (s' sinfl sinha8-f-s's cos0)x_1 X 8 (s' sin0 sinha8+s's cosd). (C5) We use, as a basis for the Hilbert space 3C, the repre­sentations of 0(2, 1) in the mixed 0(2) X0(1, 1) basis, described in MM2 and CDM: (S+, S- \l+,»r)= {DM^rl(g+), 0}, (g+, g- I l~,nr)= {0, D_M,Mr!(g-)}. In this basis we have, from (C4), the final result di'S',u-,u.r'^M(6)= f dsmhas\_dS'M.y.r<l'(as)Y J - CO Xxs',s<x)(0, as)dsM,p,rl(a'sr). (C6) APPENDIX D: AFS-TYPE MODEL AS EXAMPLE It has been shown in CDM that the unitarity model of Fubini et al.40 (AFS-type model) can be described easily with the three-dimensional BCP variables. Analogous treatment holds in the t= 0 case. Since spin- less particles are exchanged, the kernel of the multi­peripheral equation is g-independent (a7= 0 throughout, and no Clebsch-Gordan coefficients are needed); and it contains the off-shell tt-tt cross section3 ^(coshg) = ^(coshg) as a factor replacing the S function which appears in the single-ladder approximation. The t=0 equation can be obtained from (B15)41 noting that, apart from the factor ^(coshg), we require -^(/'-M2)-2, (Dl) Ro^'aw-oy'-^g') - Cty,Ctyf-* 0 and since aoo~"_1(g)->1 as a->0, we have R»yy'(l, O-^'-M2)-2. (D2) Substituting (D2) into (B15), and noting that dooo^r1) = ^*(coshg+x sinhg)x-1 = e^/X sinhg, (D3) we get the equation M(O = <0)W+ r tdtb\t)V\t,t')(t'-v?)-\ (D4) J-00 where sinhg d coshq A2( cosh§) eX9 V\t,t')^ f J Zi *o(M') ^ sinh2 z^(W-t-t')/2(tt'yi\ (D5) sinh$0= (s-m?-t')/2m( - t')112, (o)P(t') =^2(smhg0) (f-M2)-2- Note that bx=bx, because 6_ix = 0, since Ooo°=0. The 0(3,1) expansion now reads rioo H-ioo B(a)= d[\JbxDsx(a) = I d[X] bxD+x (a), v q •'-ioo (D6) where bx=b+\ D+X(a) = Ao;o+,oxo(®) = D-~x(a). (D7) Since h in Eq. (3.5) is an 0(2, 1) transformation and a=0, the relation between the two incomplete absorp­tive parts is rather trivial42: B(aff)=B(a), (D8) 40 D. Amati, A. Stanghellini, and S. Fubini, Nuovo Cimento 26, 896 (1962). and the indices m and nr are not needed. [More precisely,43 B(a) = Bor(a) contains both (+) and ( -) 0(1, 1) poles.] The partial-wave amplitude bl = bo±t+l is then easily obtained, either by direct applica­tion of group theory to (D8), or from (3.15) in the limit a=0. We have V(k, w) = E M>] (Kia,<m™)*bx(t)du,s\d), (D9) s,sf where the relevant functions, according to (3.17), are43 dl+ +x(0) = A+.o+^/'W, dt+ _x(fl) = di+ +x(tt-0). _________ (DIO) 41 The t = 0 equation can be obtained directly in a much simpler way [see S. Nussinov and J. Rosner, J. Math. Phys. 7, 1670 (1966)]. Here we want simply to show how the a = Q limit is reached with our formalism. 42 The absence of the factor sinf) of Eq. (3.5) is consistent with the form of Eq. (D4) and of Eq. (4.12) of CDM. 43 From Eq, (ASO) and (A52) one can verify that Fq,„,° = W O,nr0 = ^»0.2930 M. C I A F A L O N I A ND C. D e TAR 1 From Eq. (C6) we get the explicit expression44 di± +x(0) =7T_1 j dx il+iQi{ix) (sin0 x- cos0)x_1 *'cot 0 r(X)r(/+l) (sin0)1 „ = 2 "-TT-TV Cx-J-im(cos0), r (/+ X+ 1) Sin7T (X - I) (Dll) where Cn" are the Gegenbauer functions.46 After the manipulations of the end_of Sec. Ill, we can explicitly calculate the Regge-pole eigenfunctions /k(/, 6) of the isTth daughter ^ = Xo- ^- 1 correspond­ing to a given Lorentz pole of eigenfunction /o(/). The result is, apart from inessential factors, f a f r(Zx+l)ix hit, e) «:/„(/) r(J-K)r(i-ix+Xo) X (sin0) lK+1CKlK+1(cosB). (D12) 44 Bateman Manuscript Project, Higher Transcendental Func­tions, edited by A. Erdelyi (McGraw-Hill, New York, 1953), Vol. I, Eqs. 3.7 (31), 3.3 (13), and 3.15 (4). 46 Reference 44, Sec. (3.15). Note that the odd daughters are absent because, due to (D7) and (DIO), bl is even under 0<->7r-0 (ux->-w). Note also that (Dll) gives a result similar to the Bethe- Salpeter calculation8 when the initial particles are put on-shell. The latter circumstance explains why only amplitudes even in w are obtained in this simple case.