Joachain, Charles J.

Risk, Clif-ford.

eScholarship - University of California

Lawrence Berkeley National LaboratoryRecent WorkTitleABSORPTION CORRECTIONS AND MULTIPLE SCATTERING IN THE K-MATRIX FORMALISMPermalinkhttps://escholarship.org/uc/item/32p841kzAuthorsJoachain, Charles J.Risk, Clif-ford.Publication Date1970-10-01eScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaSubmitted to Nuclear Physics B "CCR L- 20 114 l-t E C r:: I V E 0 LhWRErKE ({ADIAIION LA30r.!.\TORY NOV 201970 LIBRARY AND DOCUMENTS SECTION Preprint ABSORPTION CORRECTIONS AND MULTIPLE SCATTERING . IN THE K ... MA TRIX FORMALISM Charles J. Joachain and Clifford Risk October 1970 AEC Contract No. W-7405-eng-48 TWO-WEEK LOAN COpy This is a Lihrary Circulating Copy which may be borrowed for two weeks. for a personal retention copy, call T ech.lnfo. Diuision, Ext. 5545 G. c:: () ::0 .t-< I LAWRENCE RADIATION LABORATORY ~ '.I, ! 'ljuNIVERSITY of CALIFORNIA BERKELEY ~:t: ..../ DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California. -iii- UCRL-20ll4 .ABSOHPTIONCO:KRECTIONSm:ID'MtJI,TIPLE SCATTERING IN THE K-MATRIX FoRMALISM * Charles J. Joachain Department of Physics University of California Berkeley, California 94720 and Clifford Riskt Lawrence Radiation Laboratory University of California Berkeley, California 94720 ABSTP.A.C''I' The K-·mat:dx formalism is used to derive the absorption model and to reconcile absorption and rescattering corrections in high-energy hadronic tvJO-bod.y collisions. Detailed calculations are made in the case of elastic proton-proton scattering, for which we compare the K- , matrix approach to the eikonal calculation of Frautschi and I>1argolis. * On leave of absence from Physique Theorique et Mathematique, Facult6 des Sciences, UniversiteLibre de Bruxelles, Belgium. Research supported by the Air Force Office of Scientific Research, Office of Aerospace Research, U. S. Air Force, under grant No. F44620-C-70-0028. t Supported by the United States Atomic Energy Commission. ,~ -1- UCRL-20114 1 . . Introduction Since it.s original introduction by Heitlerl ) t.o analyze radiation damping phenomena, the react.ion or K-mat.rix concept has been widely used to investigate a variety of quantum collision processes. Of particular importance is the case of nuclear reactions where the reaction matrix theory was developed by Wigner2). The advantages of the K-matrixmethod are that it remains valid if the interaction cannot be described by a potential and that it. satisfies the unit.arity require-ments in a particularly simple way. These features of the K-matrix approach have encouraged a number of authors3-9) to use this formalism in the case of high-energy hadronic collisions, where it provides a natural way to incorporate unitarity corrections to the Born diagrams. Another, frequently used unitarization procedure is the eikonal formalism, and the t.wo approaches are of course not unrelated. We shall actually show in Sec. 2 that. t.he results obtained within the eikonal framework can also be derived in the context of the K-matrix formalism for both elastic and inelast.ic t.wo-body processes. In the former case, we recover the multiple scattering interpretation of Arnold6), whereas in the latter one the basic formulae of the absorption modellO- lS ) are obtained. We also show that the K-matrix method provides a simple way of resolving the conflict16 ) between "absorption" and "rescattering" corrections, without invOking17 ) special properties of multiparticle production amplitudes. -In Sec. 3 we carry out a detailed comparison of the eikonal and K-matrix methods for high-energy elastic proton-proton scattering. This process has been analyzed in the eikonal formalism by Frautschi and -2- UCRL-20n4 MargOlislB ) (hereafter referred to as FM) who identified the contribution to the scattering amplitude arising from the (linear) Pomeranchuk trajectory--having a slope -2 ex' = 0.B2 (GeV) --with the Born term and * ~ obtained a differential cross section showing diffraction minima beyond * A similar diffraction pattern was obtained previously by Chou and Yang19) . the forward peak. Using the same Born term as an ansatz·for the K matrix, we find that the resulting differential cross section exhibits only breaks (no dips), . 20) consistent with present experimental evidence • We also analyze the phase of the scattering amplitude and the total cross sections. "," -3- UCRL-20114 ." 2. TheK-matrix Approach to Absorption Corrections Let us .consider a two-body collision process. We denote by T the transition matrix and by K the reaction matrix. In terms of these operators the S matrix is given by S == I - 2iT (2.1) and I - iK S I + iK (2.2) where I is the unit matrix. The T and K matrices are related by the Heitler equations T == K - iKT (2.3) == K - iTK In terms of matrix elements, the Heitler equations for an inelastic transition a ~b read T == ~a -n T na = ~a - i L Tbn Kna n (2.4) where the summation runs over all open channels, and multiplication implies an integration over phase space. Alternatively, we may interpret the Heitler equations (2.4) in terms of partial wave amplitudes, in which case they simply reduce to a set of algebraic equations 2l ). -4- UCRL-20114 We now specialize to high-energy two-body processes. A natural wa.y to proceed in this case is to identify the K-matrix element K -oa with the Born term ~a fUrnishe~ by the peripheral or the Regge model. Before we do this, however, we note that since the Heitler equations automatically lead to a unitary Smatrix, the summation on the right of Eqs. (2.4) provides the desired unitarity correction to the ansatz we have chosen for ~a. Since we expect the initial and final states--"i 22-23) containing diagonal matrix elements--to dominate/'- the sums in Eqs. (2.4), we retain only the contributions from these states7) and write the first of Eqs. (2.4) as Similarly, the second of Eqs. (2.4) yields of Combining these two equations, we have = ~a 1 + 2:.(K 2 aa 1 - 2:.(T 2 aa Before we proceed with Eq. (2.7), let elastic s~attering. The Heitier equations T Kaa - i L K T aa an na n = K - i L Tan Kna aa n (2.6) us now consider the case then read (2.8) ,.. ¥ ,., -5-Keeping only the termn = a in the sum, we now have so that = K - iK T aa aa aa K aa 1 + iK aa If we solve this equation by iteration, we have = Kaa - iK K + ••.• aa . aa UCRL-20114 (2.10) (2.11) We now return to Eq. (2.7), use the first approximation T rv K and T rv K and expand the right-hand side to obtain aa - aa bb - -ob This last equation may be rewritten as where we have expanded 1 S 2 = aa 1 1 Sbb 2 ~a Saa2 1 (1 - 2i T )2 = aa (2.12) (2.13) + ••• (2.14) It is interesting to note that a result similar to Eq. (2.13) has been obtained by Ball and Frazer24 ) who studied absorption corrections to the peripheral model from dispersion theory. We now consider the question of finding a suitable ansatz for the quantities Kaa and ~a appearing in Eqs. (2.10) and (2.12). As mentioned above, a natural way to proceed is to identify these quantities -6- UCRL-20114 with the Born term ~a given by the peripheral or the Regge model. Adopting this point of view, we find that Eq. (2.9) becomes . T. == R - • iR R +. • . aa aa aa aa while Eq. (2.13) yields 1 1 S 2 R S 2 bb --ba aa (2.16) We note that the connection between the elastic and inelastic processes is easily achieved within the K-matrix formalism. Equation (2.14) reproduces--up to second-order terms--the prescription given by Arnold6) to incorporate absorption corrections to elastic processes within the eikonal approximation and leads to a straightforward multiple scattering interpretation of the elastic scattering (see Fig. 1). Equation (2.15) is the standard absorption model formula for inelastic processes. It is important to note that several of the approximations necessary to derive the formulae (2.15) and (2.15) are not required per se within the K-matrix formalism. For example, one could use directly Eq. (2.9) to study elastic scattering pr~cesses. We shall illustrate this point in the next section, where high-energy elastic proton-proton scattering will be analyzed in this way. It is also worth l emphasizing that the K~matrix approach, while preserving unitarity, leads very naturally to the results of the absorption model, and there-fore avoids the conflict between "rescattering" and "absorption" " :;1 -7- UCRL-20ll4 interest is the fact that his conflict is resolved without recourse to absorptive corrections to multiparticle production amplitudes17 ), whose existence is not firmly established. -8- UCRL-20114 3. The K Matrix Applied to Elastic Scattering As an illustration of the methods of Sec. 2, we now apply the K-matrix formalism to study high-energy elastic proton-proton scattering. i, This process has been considered by FM in the eikonal formalism. We shall use their parameters for the single scattering term in order to compare the K-matrix result directly with the eikonal result. A main difficulty of any multiple scattering approach to elastic scattering is identif'ying the Born or "driving" term. That is, should it be a pole or a cut, and fixed or moving? One has some assistance from the shrinking of the diffraction peak, but ambiguities still remain. (In the absorption approach to two-body-inelastic processes, this problem does not arise because all elastic scattering corrections to the inelastic exchange term are grouped into the two blobs displayed in Fig. 2.) We begin by defining the scattering amplitude in terms of the S matrix by 1 s I + iA/[k(S)2J, where A = A(s,t) is the full nonflip helicity amplitude, and k is the c.m. momentum; A is normalized in terms of the total cross section by 1m A(s,O) = We pass to the impact parameter representation via .; .. ' •• -9- UCRL-20114 where b is the impact parameter. and ~ the three-momentum transfer 2 (t = -~). The inverse relation is A(s,b) = 100 q dq A(s,t) JO(qb). From (2.2) and (3.1) we obtain 00 A(s, b) 2ik(s)t. iK = -2ik(S)! \"" (_iK)n. 1 +iK L n=l We now identify the first term of (3.5) with a single scattering term defined by a Regge pole .1 -2ik(s)2 (-iK) whereap(t) = 1 + ta', and c is a constant. This gives -iK(s, b) 1 where ~ = -c(s)2/2ka'SO and j.l = in(s/i}, and then A(s,t) 00 = 2ia'k(s)! ~ L ., .:, t -10- UCRL-2011h Equations (3.5), (3.'7), (3.8) are to be compared with the corresponding equations in the cikonal approach. Instead of (3.5), one uses (3.1) to get A(s, b) This gives and hence = 1 ',ik(s)2(1 - s) = ik(s)t (1 .. exp[ 2i5(b) J) ,00 -il:(SY~ L [2io(b)Jn/n~ 2i5(b) n=l S 2 = - - exp[-b /4a'~J ,~ 00 ?in' k(,I':)t)' 1 . L nn~ n=l (3.5' ) (3.7' ) Using the values of FM, s = 7, a' = 0.82(Gev)-2, we display in Fig. 3 the predictions for dcr/dt. For small t the two methods a,gree because the first two terms of (3.8) are identical with the first two terms of (3.8'). We note, however, that for t "- -1 the dip that occurs in the eikonal case is replaced by a break in the K-matrix cal-culation, even at 1600 GeV. To understand this, we recall that the dip occurs in'the eikonal case because of a net cancellation between the positive first term and negative second term in (3.8'); the positive third term is small (see Fig. 4). In the K-matrix, the third term is larger, and the net result is that the dip is filled in. One can also ,examine the phase 9 = tan-l(Re[A(s,t)J/rm[A(s,t)]}, as a function of t at fixed s. We do this in Fig. 5, where vie .. C -11- UCRL-20114 show a polar plot of get) at 30 and 1600 GeV for the two models. The results are essentially the same. We compare total cross sections in Fig. Ga. For the given a' ,c, the K matrix gives a larger crT because of the larger third term of Eq. (3.8) (see Fig. 6b). The overall rise in as a function of P is maintained, however. This rise, although suggested by the 1969 Serpukhov data, is not confirmed by the 1970 Serpukhov data. The discrepancy of this new data with Fig. 6a can, of course, be accounted for by introducing appropriate secondary poles. Finally, one can also examine the larger angle behavior analyti-cally by converting the sum of (3.8) to an integral on n and performing a steepest descent analysis 27 ) on n, or by performing a steepest descent analysis directly on the variable busing (3.3), (3.5), and (3.7) (see Ref. 28 for this type of analysis for the eikonal case). We comment briefly on the first approach. One converts (3.8)·.to A(s,t) cc s (~.:-dn __ J Sln .nn exp(-na - T!-l/n) where T = ~'t > 0, a = -£n(~/I-l)' and I-l is sufficiently large so that IS/1-l1 < 1 and 11l1» T.The saddle points are at 0 +1.H. 1 11 cos nn = -a - ;.... -2 n sin nn n or, for large n, 1 1 +~ (1.H.)2 (sin ¢)2 - 2 tan ¢ rr n = e n a I' -12-For large enough s, this reduces to the formula of Ref. 27, and eventually to an Orear-tY])e expression A(s,t) cc .L exp([-(2rCTf-l. cotan ~)~-]}. The case T» 1f-l.1 can be dealt with in a similar way. (3.12) ii' • • .. t, -13- UCRL-20111! REFERENCES 1. W. Heitler,Proc. Cambridge Phil. Soc. II (1941) 291; The Quantum Theory of Radiation, Thi:rd Ed. (Oxford University Press, 19'54 ) 2. E. P. Wigner, Phys. Rev. 7~ (1946) 15; for a review of the reaction matrix theory of nuclear collisions, see for example A. M. Lane and R. G. Thomas, Rev. Mod. Phys. ~£ (19'58) 257 -,). R. H. Dalitz, Rev. Mod. Phys';2 (19(,1) 1!71; strange Particles and Strong Interactions (Oxford University Press, 19(2) J+. A. Bialas, T. W. Ruijgrok, and L. Van Hove, Nuovo Cimento;U (196~) 608; A. Bialas andL. Van Have, Nuovo Cimento ~ (1965) 1380; ). K. Dietz and H. Pilkuhn, Nuovo Cimento J1 (1965) l rS61 <: u. R. C. Arnold, Phys. Rev. 136 (1964) B1388 ~H. D. D. watson, Phys. Letters 1 ,-, (196) ) r',','-) :SL 1'--'i'. 8. P. J. O'Donovan, Phys. Rev. ~J22 (1969) 1902 9. w. Dreschler, Phys. Rev. ~ (1970) 364 10. N. J. Sopkovich, Nuovo Cimento ~ (1962) 186 11. L. Durand and Y. T. Chiu, Phys. Re\T. Letters ~~ (1961~) 399, ~~, (1964)45; Phys. Re'/. 137, (1965) B1530; Phys. Rev. 137 (1965) B646 "'vn..,.,~ • "-, ................. l e) '-. 13· 11f • 15. K. Gott.fried and J. D. Jackson, Nuovo Cimento ~ (1964) 75') J. D. Jackson, Rev. Mod. Phys. ~1. (1965) 484 H. C. Arnold,Phys. Hev. ~(1967) 152;; R. C. Arnold and M. L. Blackmon, Phys. Rev. ~ (1968) 2082 F. Henyey, G. L. Kane, J. Pumplin, and M. H. Ross, Phys. Rev. ~Qg, (1969) 1579 16. J. Finkelstein and M. Jacob, Nuovo Ciment;o 'Z~!;. (1968) 681 -14- UCRL-20114 17. L. Caneschi, Phys. Rev. Letters ~ (1969)254 , , 18.s. Frautschi and B. Margolis, Nuovo Cimento ~~, (1968) 1155 • (hereafter referred to as FM); see also K. S. K~lbig and B. Margolis, Phys. Letters31B (1970) 20 , ~19. T. T. Chou and C. N.Yang, Phys. Rev. Letters 20 (1968) 1213 """"' 20. J. V. Al1aby, et al., Phys. Letters 28B (1968) 67 ." ~......."....,. 21. M. L. Goldberger and K~ M. Watson, Collision Theory (J. Wiley and Sons, New York, 1964) Chap. 6 22. A. Dar and W. Tobocman, Phys. Rev. Letters 12 (1964) 511; ibid. 12 (1964) 511; ibid. ~ (1964) 91 23 E. J. S,quires, Nuovo Cimento '~ (1964) 1328; Phys. Letters 26B (1968) 461 24. J. S. Ball and W. R. Frazer" Phys. Rev. Letters ~ (1965) 746 25. J. V. A11aby, et a1.,Phys,' Letters ~ (1969) 500 26. Serpukhov preprint (Kiev Conference, 1970) 27. A. A. AnseDn and I. T.Dyat1ov, Yad. Fiz. £ (1967) 591, 603 [SOY. J. Nuc1. Phys.€ (1968) 430, 439] 28. K. G. Boreskov and Yu. F~ Pirogov, Yad. Fiz. 11 (1970) 210 [SOY. J. Nuc1. Phys. II (1970) 118] """, ' ,I, -15- UCRL-20114 FIGURE CAPTIONS Fig. 1. Elastic scattering viewed asa multiple scattering series. The wav'Y line corresponds to the Born driving term • . Fig. 2. Inelastic tvlo-body scattering with absorption corrections in the initial and. final states. The wavy line corresponds to the exchanged particle. The shaded blobs represent elastic scattering. Fig. 3. Dif'ferentialcross sections in the two models at 30 GeV/c and 1600 GeV/c. Solid lines: K matrix; dashed lines: eikonal. \ ~ Fig. 4. The first three terms of the sums appearing in Eqs. (3.8) and (3.8'); (a) real parts; (b) imaginary parts. Number denotes the corresponding term in the series, 3 refers to the eikonal A nfl ~, t.n t.hp K !'IlAt-ri x. Fig. 5. The phase of the amplitude as a function of t (eikonal) or T (K matrix) for (a) 30 GeV/c and (b) 1600 GeV/c. Fig. 6. The total cross section in the two models (a) aT vs Plab; . (b) individual terms of the real part of the sums appearing in Eqs. (3.8) and (3.8'). \1 -16-.. • • + + II t (/) II V m rt) I en 0 r--...J £Xl X . M . tlll .~ po., UCRL-20114 • \' -17-UCRL-2011L~ • c s-~ a b t D t XBL709-3839 I Fig. 2. -.- .. _- ---i8-30 GeVlc las O~-7--~~L-~--3 -4 -5 XBL709-3640 Fig. 3. UCRL-20114 • J '~ (a ) 10-3L_~_~ t=O 0.5 (1 -19-( b) Im L_~_,I..O . 0.5 t~ XBL 709-3841 UCRL-20114 •. (0) 30 Gev/c l=~0.5 T=-0.5 i _ ... 20-,,,tmA(s,t) I " t=O . --"T' . T=O . I . t= -2.8 t=-0.7 T=-O.7 .. I Fig. 5a. UCRL-20114 t=-1.6 Re A(s t) T=-1.6 X BL 709-3843 'I (} G ( b) ~l 1600 Gev/c t = -0.8 T=-0.8 (i l) -21-1m A(s,O I t=O ____._--t-T=O I t=-2.0 ~--,-T=-5.0 T=-4.8 t=-I.O T=-I.O I Fig. 5b. UCRL-20114 t=-1.9 ReA(s t) T=-1.9 XBL 709-6647 -22- UCRL-20114 ) 45 K - matrix 43 .- ----- ... ..c 41 E -.... b 39 2 4 6 8 10 12 14 ' 16 P1ab (100 GeV/c) XBL 709-3842 Fig. 6a. 1\ -23- UCRL-2011~ I st term 2nd term "" ~---~ 3rd term (Eikonal) - ---- --_...-01 P1ab( 100 GeV/c) XBL709-3838 Fig. 6b. ~­,; , LEGAL NOTICE This report was prepared as an account of Government sponsored work. Neither the United States, nor the Commission, nor any person acting on behalf of the Commission: A. Makes any warranty or representation, expressed or implied, with respect to the accuracy, completeness, or usefulness of the informa-tion contained in this report, or that the use of any information, apparatus, method, or process disclosed in this report may not in-fringe privately owned rights; or B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus, method, or process disclosed in this report. 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ABSORPTION CORRECTIONS AND MULTIPLE SCATTERING IN THE K-MATRIX FORMALISM

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ABSORPTION CORRECTIONS AND MULTIPLE SCATTERING IN THE K-MATRIX FORMALISM

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