A N N A L E N D E R PHYSIK 7. Folge. Band 32. 1975. Heft 2, S.81-160 DAMA tor Merodoryon Scattering By L. L. JENKOVSZKY, N. A. KOBYLLNSKY, and A. B. PBOONIMAE Academy of Sciences of the Ukrainian SSR, Institute for Theoretical Physics, Kiev (USSR) With 1 Figure A bstraet The generalization of DAMA (dual amplitude with Mandeletam analyticity) to the w e of meaon-baryon scattering L studied. To this end trajectoriea with left-hand cut are introduced in the model under consideration. Threshold and asymptotic bounds on theae trajectories are obtained. Combinations suitable for boson-fermion and fermion-fermionamplitudes are constructed. A s cific example, namely nN-scattering, is considered, for which dual models of invariant amplitug are constructed. Introduction For the further study of the properties of DAMA (dual amplitudes with Mandelstam analyticity) and for their applications the inclusionsof fermion trajectories and the generalization of DAMA to meson-baryon processes are necessary. Usually, when considering Regge trajectories it is assumed that they are even functions of W = r/i.There are no serious rewons in favour of this aseumption. This form is chosen rather for convenience, and i t is based on the observed approximate linearity of the trajectories in s in the domain of the first few resonances. However, the assumed parity of the trajectories in W leads to serious difficulties for the processes involving fermions. For instance, the McDowell symmetry for nN -scattering predicts in the case of trajectories even in W = the existence of unobserved states degenerate in parity with the states lying on the trajectories Nu,Ny,A d . This degeneracy can be removed in the following manner [l]: 1. the residue has dynamical zeros a t the appropriate points. 2. the residue contains a cut in the J-plane removing the unwanted trajectory from the physical sheet. 3. the trajectory ie not even function in W. Experimentally baryon resonances are observed which are parity partners of the nucleon N, (940) or dd (1240) but are not degenerate with them in mass, namely, N a ! 1520, 1/2-) and A , (1670, 3/2-). The mass difference of resonances with opposite parity can be fitted by the trajectories [2]. fi * 0,83 We + 0,60 W - 0,70, (Nu) 0,78 w 2+ 0934 w - 0,10, ( 4 6 ) o~N(s) Ori(S)= When changing the sign before W ,theae trajectories describe the states Na (1520) and A , (1670). Thus, experimental data support case 3. We shall m u m e that the fermion trajectories are not even functions in W.Trajecsymmetry [3]: tories with opposite parity are related by the MCDOWELL &(+)(W) = &(-)(-W), 6 Ann. PbyriL. 7. Fdg* Bd (1) L.L.JBNKOVSZKY, N. A. KOBYLINSKY, end A. B. PROQNIMAK 82 where (f) indicate the parity and w &(W)= a($), = vi. Thus, the fermion trajectories have a root branch point in 8 at s = 0 and they can be presented in the form (2) where a,,&) are even functions in W with right-hand cut in 8 only. As to the meson trajectories, no relation of the type (1)exists for them and experimental data are also pure. The presence of a left-hand cut in meson trajectories does not contradict the postulates concerning the analytic properties of the scattering amplitude. A possible mechanism giving rim to this cut is discussed in [4, 51. I n what follows, we shall consider both possibilities: meson trajectories with and without a left-hand cut. By studying the pole structure of DAMA in sec. 1 we define the boson-boson, fermiom-boson and fermion-fermion amplitudes. Then, in sec. 2 we write down expressions for the invariant nN-scattering amplitudes including the contributions from As and Ad aa well as the meson trajectories e and f. the baryon trajectories N,, N,,, The paper contains also two Appendices, in which restrictions on the threshold (Appendix A) and the asymptotic behaviour of the dermion trajectories (Appendix B) are studied. 1. Pole Structure of DAMA in tho Casc of Trajectories with Left-Hand Cud Consider one, say the (at) - term of the scattering amplitude. I n the pole approximation (only poles in the J - plane!) it is natural to consider this amplitude as a functional of two Regge trajectories, u(8)and a ( t ) , specifying the motion of these poles. This functional can be chosen as follows (for details see [6]): where p,q=zkf, 8‘ = 8(1 - X), t’ = tx, g = con& > .l. (4) In expression (3) wc have assumed that both trajectories, a(8)and or@), contain a left-hand cut, i. e., they can be presented in the form (2). Therefore, expression (3) actually describes 4 terms differing by the choice of the signs in trajectories before and V i . The physical amplitudes will be represented by combinations of these four terms whose form will becomes clear after the study of each term. Consider now the pole structure in s of the exprwion (3) for definite values of indices p and q (for brevity we drop for some time the index p and put q = +).Now we use the analytic continuation of D(s, t ) into the domain Reu(8) 2 0. It is given by the expremion vi The integration contour C is shown in Fig. 1. The twofold encircling of the point x = 0 is motivated by the square root branch point of a + ( t ‘ )at x = 0 (see (2), (4)). DAMA for Yeeon-Baryon Scattering 83 The integrand in (b) has poles at x = x,, where x,, is defined by the equations n 4 4 1 - z n ) ) 4 ( W y l / l - xn) 2’ 3 : n = 0,f l , &2, ... (6) The poles of D+(8, 1 ) in 8 are generated by a c o b i o n of the singu~aritiesx = zn (n 2 0) and x = 0. Therefore, in what follows we shall consider the residues at the poles in x only. k I Fig. 1 Integration contour C in expreeeion (6), used for the study of the pole structure of the amplitude. The double encircling of the point z 0 is due to the square mt branch point of the function a ( f x ) at this point. Dotted line shows the motion with 8 of the point 2., which is a pole in the integrand 5 : For small values of x, we have Remembering that the pole at x = x,, crosses the contour C twice (on different sheets) we write the contribution of this pole to the amplitude D+(8, 1 ) as follows [(1 9.xn)-”+(f“’-1 -a-(l,,)-l where By using (?), (8) and keeping in (8) only those terms which affect the coefficient near the pole (in W )with maximum multiplicity for the given level, we get - The functions f*(&) are regular in y = f z in the vicinity of y = 0. Let us write them in the form L. L. JENKOVSZKY, N. A. KOBYLINSKY, and A. B. PROONIMAK 84 @+ t, ( Y) -+const, y --f 0. ,f+2 It is clear from (l),(2) and (11) that uf ( t ) = (- l ) k a; (t). Therefore, the singular part in 8 of the expression (9) can be presented aa Expressions (12) and (13) reflect the pole structure of the amplitude Dp+(8Jt ) . When using Dp-(8, t ) , the relation H,-)= (-1)- an+', (14) must be utilized. The sign (j-)in R, corresponds to the value of q in a,@). Thus, the amplitudea Dp+ and Dp- have poles both at integer and half-integer values of a&) (similarly in a&)). However, because of (12-14) the sum (the difference) of the amplitudes Dp+ and Dp- will have poles only for integer (half-integer) values of %48). Now we can construct the following combinations of the amplitudes Dpq(8,t ) : Here the indices 0 and 112 in b show for which valuea of a(a) and a(t) - integer or half-integer - the poles of the amplitude arise, further i/i and are introduced to regularize the amplitude a t 8, t = 0. We note that the fermion-boson amplitude (16) possesses the necessary properties only when the boson trajectory haa a left-hand cut. By using the above analysis, the fermion-boson amplitude, valid for arbitrary trajectories, can be easily written : v? DAMA for Meson-Baryon Scattering 85 2. Dual Model for zN-Scattering I n this section we write a DAMA for nN-scattering and obtain restrictions on parameters and on the trajectories. (In the cwe of the Veneziano amplitude this problem waa studied in [7, 81). The problem under consideration is described by four invariant amplitudes A(*) and with the following symmetry properties a*) For the description of this process we shall include the contributions from the baryon trajectories Nu, N,, A,, and Ad, and from the meson trajectories e and f (we do not consider here the contribution from the Pomeron; this problem was treated in [9]). According to (20), the e-pole will contribute to the amplitudes A ( - ) and B-1, and thef-pole to A ( + )and B+). We shall use expressions (18) and (19) for the fermion-boson and fermion-fermion amplitudes. These expressions, however, should be modified since the amplitudes A(*) 1 1 do not contain nucleon yolea a t a(8)= -and a(u)= 3,and the poles in the t-channel 2 start a t a(t)= 1. For the unification of the notations we introduce - bl=oL-r), where = 0 for boson trajectories 1 11 = 2 for fermion trajectories. r) Let us define four amplitudes with different first poles: where the curly brackets indicate the functional dependence of @) on Z(v) and Z(w). Here 0 , W = 8, t , U , v' = v(1 z), w' = w2, h(')(v',w ' ) = 1, P ( v ' , w ' ) = a'(v'), h(*)(V', w ' ) = E(w'), h(')(v',w') = E(v') E(w'). - In the 8- and the u-channels the X-and Y-trajectories (X, Y = Nu, N,, A,, A, and their parity partners N,, Nd, A,, dd) contribute, whereas in the t-channel the bltrajectory (M = e, f) does. In what follows we shall use the notation X(4, M ( t ) insteads of ax@), a&). To write the amplitudesA(*), @*I, the following combinations with definite (8 - u)-qmrnetry will be necessary (x,M)? = Si){Ex(8), g ~ ( t f ) } f i ) { E x ( U ) ,Ex(t)}, (x,Y)$) = 3){zx(8), a r ( U ) } f3 ) { g x ( U )a , 'y(8)}, (Xfi) = @){Ex(8), E ; x ( U ) } . (22) (23) (24) L. L.JBNKOVBZKY, N. A. KOEYLINBKY. and A. B.PROGNIMAK 86 On account of (20-24) the amplitudei A(*) and B(f)can be written in the forin . . Amplitudes with definite isospin in the 8-channel can be easily related to A ( * ) and B(*).For example, -.- -.- and similarly for B*). The amplitude A& should not contain the contribution from the a&) trajectories. This is possible if the following relation is estisfied . .- , I. I - - \ ,, . ," , ,. I, ,\ ' I We get another relation by requiring that the trajectory ad(8)doea not contribute to Af,: DAMA for Meson-BaryonSaettering 87 We shall need also the asymptotic expressions of the amplitudes (21). They read (25) (26) where ~ The asymptotic expressions for B){y(8), ?(u)) are similar to (25-28) with and t --f u. By demanding correct signature of the leading trajectories one can significantly reduce the number of the free parametem. In the construction of the amplitudes the asymptotic behaviour and ( 8 u)-symmetry were explicitely used, therefore the contribution from &channel trajectories automatically has correct signature. Below, we present the asymptotic expressions of the amplitudes for 8 3 00, t = const li? 3 - The asymptotic form of the amplitudes in the u-channel u-channel is more amplitudes A(*) and B(f) cumbersome. cumbersome. By extracting here the signature signature factors and by setting setting zero zero the contricontri- 88 L.L.JBNKOVSZKY. N.A. KOBYLINSKY, and A. B. PBOONIMAIC Conelusions We have demonstrated that fermion trajectories can be included in DAMA in 8 consistent way. A h , the problem of constructing nN-acattering amplitude haa been treated. 89 DAMA for Meeon-Baryon Scattering These results fill partly a gap in the important application problem, namely, how to “attach” DAMA to realistic processes (the earlier DAMA’s were constructed for the idealistic cme of scalar meson systems). The model presented in this paper doea not resolve completely all the problems connected with the generalization of DAMA to nN-scattering. In fact, i t includes consistently fermions and it is a correct model for the scattering of scalar mesons on fermions. The transition from scalar mesons to pseudoscalar ones is far from trivial, aa can be seen from attempts to construct a model for nn-scattering (see ref. [6]). This problem will be treated in forthcoming publications. We thank A. I. BUGRIJ,G. P. DBMCHENKO, E. S. MARTYNOV,and B. V. $mufor their activity in discussing the problems presented in this paper. MINSKY Appendix A :Thrcshold Behavionr of Fermion Trajectories In the case of spinless particles, when the equation for elastic unitarity is of simple form, the threshold behaviour of boson Regge trajectories is [lo]: If the particles carry spin, the equation for elastic unitarity becomes cumbersome (for n N see, e.g. [ll]),and the study of threshold behaviour of trajectories is more difficult. It is, however, known that elastic unitarity for partial nN-scattering amplitudes is written in the same form aa for spinlees particles [12]. Hence, the threshold behaviour of fermion trajectories should have the form analogous to (A.l), where instead of &(so) the position of the pole of the partial amplitude in orbital momentum is to be set. Since a(e)indicates the position of the pole in total angular momentum, fermion trajectories should behave near the threshold a8 follows I m a ( s ) N ( 8 - 80)a(8*). (A3 L +a* Consider now the threshold behaviour of DAMA, for example, that of its ferniionboson term (18).The discontinuityof the amplitude on the physical cut is defined by the expression 1 zr -a,(89-1/2 1 -a#)-l (33 1 (f) - 1, = Im,P.Q’2f 0 dz By evaluating the imaginary part using the method presented in [6],we get (here we drop those functions which have a logarithmic branch point at the threshold; a discussion of this problem can be found, for example, in [13]) Im, D1,2o(82 ):( -a+(*.) 4 &, +112 + Im a+@) e)- -u-b.)+ 112 The discontinuity of the amplitude will be proportional to Im Or *(8 N (8 80)af(8b), - f8 Im &-(a). - 80, if which is in agreement with (A.2). Note that if in expression (2) a1(sO)$= 0, then a+@,,) &-(so) and the behaviour of the trajectories &+(a) and ~ ( 8a t) the threshold d l be different. + Appendix B: Bounds on the Asymptotic Growth of Fermion Trajectories Polynomial boundedness of DAMA for 8 + 00 requires (see, for example, [6]), the following condition R Im a(8)2 Re a(8)In 8 (B.1) L. L. JENKOVSZKY, N. A. KOBYLIBSKY, and A. B. h O O N l M A K 90 to be satisfied. In the case under consideration two trajectories ark@) of the form (2) contribute to the amplitude, and they are to satisfy condition (B.l) simultaneously. Hence, an additional bound on the asymptotic behaviour of the trajectories follows. Aesume the trajectories to rise indefinitely in modulus and parametrize the functions Ul,3(8) in the asymptotic region as follows: U1,2(8) -yl,2( -e).l*s[ln( - 8 ) p 3 -y1,28vi.s e-im1,s (In - in)”l*s, - (B.2) where v1,2 and p1,2are real parameters. Then 1 +> va condition (B.l) cannot be satisfied for both 2 1 trajectories ar* (8) simultaneously. I n the cme vl + - < v, a&) dominates and nothing 2 1 new happens. The case v1 + - = v2 = v will be treated in detail. 2 One easily finds that for vl Condition (B.l) restricts the interval of possible values of v , namely 0 5 v < 1 P’ Consider three different c m . 1 1. 0 < v < T . Here (B.l) together with (B.3) and (B.4) gives: F ny, (In a ) p l COB nv + ny, (In a)”* sin nv 2 7 y1 (In s ) ” s + l sin nv - ys (In a ) p ~ +cos ~ nv. From (B.5)we get p1< pe. If, however, p, = A,then the condition 03.5) Y Z l Y l 2 tan nv must be satisfied. 1 2 . v y Here (B.l) assumes the form f ny, (In - np, + ny, (In (B.6) 8)” 2 F y1 (In a ) ” ~ + l+ Ya (In 8)”s * rips. (B.7) Condition (B.?) implies p, 1 5 p,. The case p, 1 < pa is similar to that consider1 = p,, then we get from (B.7) ed in [6], and it gives pa 5 1. If p, + 3. v = 0. Here one can write F ny, (In 8)ps ny, (In a)”*-, + + + * np, 2 y1 (In a)”l np, - y, (In (B.9) DAMA for Meeon-Baryon Scattering 91 + instead of (B.1). Condition (B.9) is satisfied for p, < pa 1. Note that for trajectories rising in modulus one of the following conditions must be srttisfied: p1 > 0 or > 0. By means of these conditions one can achieve the simultaneous rise of the real and imaginary parta of the trajectory, the increasing real part with the imaginary part bounded and vice verss. Consider now (B.9) with pl = p, 1. The following condition + ’ l - a < p l g + AY 1 Ye (B.lO) ZYl can be w i l y obtained. Besidee (B.10), one of the restrictions pl > 0 or pa > 0 is also neceseary. Let us now study the restrictions on the growth of the trajectories following from the Mandelstam reprentation. It has been shown in [6] that the polynomial boundednem of the double spectral function requires (B.ll) Re a ( 8 ) 5 Const, 8 +- 00. In this caae two trajectories a,@) must satisfy condition (B.ll) simultaneously. By + using (B.3) and putting v1 + -12 = v, = v we consider the following three caaes: 1 < 5. Here 1. 0 < v - Re Ork ( 8 ) ,+Foo 7 y18’ (In 8)” sin AP ya8’ (In 8)lraCOB Z V . (B.12) In order that (B.12) satisfy (B.ll), it is necessary that pl5 p,, while for p, = p, condition (B.6) must be satisfied. 1 2. v = 8.Here + ’ye81’2(In 8)p8-1 Ap,. (B.13) Condition (B.13) will now satisfy (B.ll) if f i + 12 p, < 0: If pl + 1 = pa, then Re a*(8),+Ym Y18’” (In (B.14) 3. v = 0. Here R e a * ( 8 ) r + ~ a r ~ y l ( I n 8 ) C 1 - 1 . A- ~p,l( l n 8 ) ” a . Now p1 - 12 pa is necessary to satisfy (B.11). If p, = p, Ye ZY1 P1 I (B.15) + 1, then (B.16) is necessary. The lower limit for pl in (B.16) (and, hence, also for p a )is determined by the requirement of the indefinite rise of the trajectories in modulo. Thus, condition (B.l), necessary for the polynomial boundedness of DAMA, in fact, gives an upper bound on the increaae of the trajectories. By using the parametrization (B.2) of the functions U1,2(8)in expression (2) we get restrictions on the allowed values of the parameters. These restrictions must be taken into account for a specific parametrization of the trajectories. One more observation deserves attention. In this model it is possible, in principle to use trajectories growing in modulo as a linear function or even a bit faster. For this purpose i t is necessary to introduce a mechanism for removing from the physical sheet one of the trajectories of the form (2). Then for the second trajectory of this pair the condition (6.1) will result in an upper bound for the increase of the modulus 8 In 8 . L. L. JENKOVSZKY, N.A. KOBYLINSKY, and A. B. PROONIMAK 92 Here, the real part of the trajectory can increase also linearly. The following form can serve as an example of such a trajectory: - - For this trajectory Re &(a) 8 whereas Im a(8) 8 In 8 for 8 + 00. The behaviour of Re 4 8 ) and I m a+) is such that condition (B.l) is satisfied. Also, trajectories satisfying (B.l) and growing in modulo like 8 can be eaeily constructed. In such a trajectory I m ( ~ ( 8 ) 8 and Re a(8)<,s/ln 8 . The increase of the allowed power in the growth of the trajectory is achieved here by means of a left-hand cut. In the absence of the left-hand cut condition (B.l) gives Ia(8)I 2 11/’.1n 811, whereas in the presence of square root branch point at 8 = 0 Ia(8)I 5 18 In 81 follows from (B.1). However, as already mentioned, if there is a branch point at 8 = 0 in the trajectory, then trajectories must appear in pairs related by (2) and condition (B.l) must be satisfied for each of them. Hence, both o[+(Y) and a&) must increase not faster than In 8. Ultimately, if we want in the model trajectories increasing linearly or almost linearly, then it is necessary to introduce a cut in the J-plane in such a way that for 8 > 0 one of the poles would go away from the physical sheet. Irrespective of DAMA, models with two complex conjugate poles, and a cut were treated, for example, in [4, 51. - References [l] D. SIVERS,Ph 8. Rev. D 4, 1444 (1971). [2] M. IDA,Lett. h o v o Cimento 4, 707 (1972). [3] S. MCDOWELL, Phye. Itev. 116. 774 (1969). [4] P. KAUSand F. ZACHARIAZEN,Phys. Rev. D 1, 2962 (1970). [6] R. OEHNE, Phys. Rev. D 2, 801 (1970). [6] A. BUORIJ, G. COHEN-TANNOUDJI, L. JENKOVSZKY and N. A. KOBYLINSKY, Fortachr. Phys. 21, 427 (1973). [7] K. 101, Phye. Lett. 2SB. 330 (1968). [8] E. L. BEROERand G. C. FOX,Phye. Rev. 188, 2120 (1969); S. FPNSTERand K. C. WALI, Phye. Rev. D 1.1409 (1970); G. C. JOSH[and A. PAONAMENTA, Phys. Rev. D 1,3117 (1970). [9] A. I. BIJORIJ,L. L. JENKOVSZKY, N. A. KOBYLINSKY and V. P. SHELEST,Lett. Nuovo C i e n t o 6, 577 (1973). [lo] A. 0. BARUTand D. E. ZWANZIOER,Phye. Rev. 137, 974 (1962). [ll] S. MANDELSTAY,Phye. Rev. ll%, 1344 (1968). (121 R. J. EDEN,in “Hi h Energy Collisions of Elementary Particles”, ch. V (Cambridge, 1967). L. L. ~ B N K O V S Z K Yand N. A. KOBYLINSKY, Preprint ITP-72-44E, Kiev (1972). [13] A. I. BUORIJ, h i der Redaktion eingegangen am 7. Februar 1974. Anschr. d. Verf.: Dr. L. L. JENKOVSZKY, Dr. N. A. KOBYLINSKY. and Dr.A. B. ~ O O N I Y A K Academ of Sciencea of the Ukrainian SSR, Institute for Theoretical Physics Kiev-138 (USSR),Metrologicheakaya St. 14-b

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