Annelen der Phyaik. 7. Folge, Bend 82, Heft1,1976,8.47-60 J. A. Bsrth, Iaipzig Dual Pomeron Amplitude with Extwnol Spin Particles By D. EBERT and H. J. O n 0 Institut fur Hochenergiephysik der Akademie der Wissenschaften der DDR, Berlin-Zeuthen Sektion Physik der Humboldt-UniversitiSt Berlin With 2 Figurea Abstract On the basis of the operator-formulation of the conventional dual rwnance model we derive an expression for the one-bop Pomeron exchange amplitude with four exfern81 excited parficlee lying on the parent h g g e trajectory. Evaluation of the high energy behaviour lea& to the definition of spin-spin-Pomeronform factors. There is a leading Pomeron oontribution only if the sum of external spins is even. The duel Pomemn doea not extubit any helicity conservation propertiea except for 8-channel forward scattering. The extension to multiloop amplitudes is briefly discussed. 1. Introduction One of the major advantages of all dual models is the direct outcome of a “Pomeran"-exchange contribution. The correaponding singularity is produced by the nonplanar orientable loop diagram of Fig. 1which admits only of vacuum quantum numbers in the t-channel if interpreted aa a quark diagram. As is well-known this dual Pomeron is an unpleasant unitarity-violating cut unless we work in the “critical” dimension of space-time (D = 26 for the conventional and D = 10 for the NEVEUSCHWARZ model [l, 21) and have the intercept of the ordinary Itegge trajectory a. = 1 to guarantee a ghost-killing VIRASORO algebra being a t work. The Pomeron intercept comes out in these cases aa a,(O) = 2 whereas one has aJ0) = 1/3 in the original Veneziano model with D = 4. Although the intercept of this dual Pomeron singularity is not yet physical in any dual model a more realistic future dual theory will hopefully reproduce reasonable features like Pomeron-background duality, halfslope of the trajectory and f-dominance of the form factors. Therefore it seems to be useful anyway to study the implications of the dual Pomeron singularity for different physical situations. First attempts in this direction have been made in ref. [3] by calculating the contribution of a non-planar one-loop six-point amplitude to the inclusive reaction a + b +-c X Furthermore, the single diffraction contribution to inclusive crowsections haa been studied on the baais of a non-planar dual two-loop amplitude [43. As has been shown there, the investigation of the two-loop six-point amplitude amounts, by factorization a t fixed missing maw, to a one-loop four-point amplitude with one external spin particle and to the definition of a corresponding Pomeron form factor. Restricting ourselves to the conventional model, we consider in this paper the more general situation of a Pomeron exchange amplitude with four external particles of arbitrary spin. From its asymptotic behaviour we obtain then form factors for the coupling of two spin particles to the Pomeron. I n particular, the external spins shift form factor zero8 so that higher particles on the Pomeron trajectory may couple. Due to the parity propertiea of the Pomeron the leading contribution will be found nonvanishing only if the sum of all external spins is even. As another + D. EBERTand H. J. OTTO 48 application we have investigated the 8-channel helicity amplitudes for scalar-vector and vector-vector scattering. We find that the dual Pomeron does not conserve s-channel helicity except in the case of forward scattering where helicity conservation follows from total angular momentum conservation. Finally, an extension to iiiriltiloop diagranims reproducing the main conclusions for the renormalized dual Pomeron, too, is shortly discussed. Similar investigations have been performed for scalar-vector scattering by other authors, too, using analogue model arguments [5]. ,Although we agree with their main conclusions on helicity non-conservation properties there are differences between their expressions and ours, mainly regarding factorization properties. The paper is organized as follows: in Section 2 we define the amplitude and calculate the asymptotic behaviour, whereas in Section 3 helicity properties are studied. Section 4 involves some results on multiloop amplitudes. Technical details are given in the appendix. 2. Construction of the amplitude and faetorization of the Pomeron Using the fundamental operators ZT(k) and D * V of ref. [6] for the non-planar self-energy and the twisted propagator with a symmetrical vertex attached we write B --f C + D shown in Fig. 1as the Pomeron-exchange amplitude for the process A + A = (01,2,3,4 Iat 4, A4) = (01,2,3,4l<Oa,bl DV(%, a29 b ; Pi,P2, -k) x DV(a,, a 4 , c+; P39 p43 k))Oa,b)l Ai,A$p ZT(b+, c ; k, (2.1) 5, Fig. 1. Four-pointoperator describing Pomeron exchange The operators involved depend on six independent sets of oscillator operators belonging to the four external particles and the neighbouring internal lines. By lAi) we mean an occupation number state representing an excited particle of (incoming)momentum pi at the level ar(pf) = Ji = 2 n - 2,. 00 n-1 We shall always bear in mind that all the Lorentz indices are to be contracted with the direct product of the four spin-helicity eigentensors ei(Ji, p,),,! ...,,j,projecting out pure spin Ji and helicity pi with Dual Pomeron Amplitude with External Spin Particles 49 Straightforward calculation of the 6, c matrix element leads to the following quadratic exponential form in the four sets of operators ai in eq. (2.1) The integration variables u, n are the CHANvariables of the two propagators, while 2 , (0 are the usual loop integrations (see ref. [6]). The measure dp and the coefficients V,, V , and V , are identical with those of ref. [7] so that the case of external scalar particles comes out immediately &s vacuum expectation value of eq. (2.4). All the coefficients V in eq. (2.4) are easily expresssed ;t9 linear combinations of logarithms of elliptic functions w, ( / I ~(see appendix). Besides, this functional part is nothing else than the exponential of a quadratic form in the zero modes a! = pi. We have in eq. (2.4) still an operator part, containing a mixed zero and periodic mode part and a quadratic form in the periodic modes: 4 z ( a i J h i= ) i 0 0 2 2 1 ag)Hp~n)-p$ i , j - 1 m,n=l (2.5) I/. Both sets of matrices H$m,9b)and Hi?) are coinposed of linear combinations of the “elliptic” matrices E , E T and F T defined in ref. [6] multiplied by matrices depending on the CHAN-variables u, 1) from the propagators. Therefore they are expressible as linear combinations of logarithmic derivatives of elliptic-functions y and yT. The problem of calculating the matrix element of Qlbetween the vacuum and an occupation number state /Al, A,, &, A,) (eq. (2.1)) can now be solved as described in a previous paper [4]. As we are only interested in the pure spin Ji-contribution at each external leg, one easily see9 that the diagonal terms H$ and 8, can be dropped due t o eq. (2.3). Due to the energy-momentuni conservation there are this way only two independent momenta in each vector Ihi) in eq. (2.5) and there are no truely quadratic operators of the type (a IHI a). The matrix element in question is reduced then to a vacuum expectation value of the following form : + jzi c 8 2 & $ v v ) a ( m3d w mj-I qr(ir(* )l.7 4.2.3.4) where the powers 1% of Lorentz-vectors (u{y$etc.) are abbreviations for direct products as in eq. (2.2). To simplify the forthcoming calculations we limit ourselves to the contribution of the (non-degenerate) leading trajectory to each external spin J i , i. e., consider only excitations of the first mode for each state I&). This means that all but the first excitation numbers I,, vanish and I!,,= J i . Eq. (2.7) becomes a vacuum matrix element 4 Ann. Physik. 7. Folge, 13d. 32 60 D. EBEUT and H. J. on0 of a product of four factors each being power'' of a linear expression in creation and annihilation operators and can easily be calculated. We get an expression containing a sixfold summation over indices (ria, 4 3 , rsr, ta4,A, z) and rather complicated Lorentz-tensors of degree C Ji built up from direct products of vectors h{!j, and i metric tensors gPiP,(i < j).As mentioned already, each hi contains effectively two of the momenta pi so that the expansion of the "powers" (say zi) of hi!& leads to four new summations (ni).We have then the following result: A = dp(u, v, x , w ) exp ( - f V , - -2 Vl + h o V 0 + z J i V J i )- K t i We have defined The spin coupling tensor, not depending on integration variables, aa well aa the summation region and the dependence of the exponents zi and qij on the summation variables and the selected momenta p(i*a),p(i*b) can be found in the appendix. The functions hf, h4 and Hi, are also explained there. Similar to the scalar case [7] we are now able to compute the contribution of the Pomeron singularity, arising from the end-point w = 1 of the integrand, to the asymptotic behaviour in 8.1). After the usual JAcoBI-transformation for the loop variables lnw=g=- 2n= In9 (2.10) 2n In x In w we expand V , around the critical point q = 0, where it vanishes, and take the other V's a t this point. We have to expand the functions h?, ht, Hij up to the first order in q, too, because the apparent diminution by one unit of the asymptotic a-power by every positive q-power can be compensated when the Lorentz indices are con1 tracted with the spin helicity eigenteneors d ( J i ,pi). The loop integrations 2n and d(u $ dq ... 0 + n(u + v)) ... can then easily be performed and produce 8 the 8-power Regge factor times a r-function as well as the Pomeron signature factor with a coupling B4-function, reapectively. The remaining integrations du ... and / d v ... lead to Pomeron form factors, as they were defined already in ref. [4]. We obtain the foll ) w e f m t limit ref. [7]. 8 to a etrip parallel to the imaginary axis and continue afterwards 88 in Dual Pomeron Amplitude with External Spin Particles 1 lowing asymptotic expression for 181 --t oo (ap(t)= 3 61 +f) (2.11) The form factors are defined as [4]*) j fi$(t) = 2dl)--~r(l) dj(u;a,(t), a(t))(sin nu)) (cos n u ) ~ - k (2.12) where the measure df is a symmetric function of u in [0,13 and reproduces the usual scalar Pomeron form factor for k = J = 0 (ref. 171). The factor v ( t ) and the number functions g and F are given in the appendix. As expression (2.12) vanishes for J k odd, expreesion (2.11) is identically zero unless 2 Ji = even. - i This reflects the parity conservation via the exchanged Pomeron in the t-channel. Obviously, the upper indices of the form factors correspond to the total spins of the two t-channel states (1+ 2) and (3+ 4). As r, is running from 0 to min (Ji,Jk)we have (2.13) I Ji JkI Ji Jk - 2ri, 2 Ji Jk - + + Let U6 briefly discuss our resulta in the case of the critical dimension D = 26 with a, = 1. The Pomeron singularity is then a true Regge pole (not a cut) given by a,@) = 2 +t and the form factor integration can be performed explicitely to give (2.12') Inserting ap(t)- a(t) = - ap(t)+ 3 into the Beta function one easily sees that the external spin coupling leads to a shift of form factor zeros located in the scalar case at ap(t)2 4 and now at ap(t)2 min (4 J , Ja; 4 J, J,). Thus, for higher spins of the external particles higher particle poles on the Pomeron trajectory are able to couple to the amplitude. This generalizes a result obtained for scalar-vector scatform factor tering [2]. I n this cme, factorization yields the usual scalar 0-0-P b i d e s a 1 -1 P form factor. + + + + - For convenience, we have included here a factor 2*(t)*(*). %) 4+ D.EBERTand H.J. OTn, 52 Finally, we mention that the Pomeron amplitude with external spin particles has the same exponential behaviour exp in the fixed angle limit as the amplitude with external scalar particles. This follows easily from eq. (2.8) bearing in mind that the spin modification leads to power factors in the Mandelstam variables only. 3. Helicity conservation properties of the Pomeron exchange amplitude As the problem of helicity conservation in the 8-channel or in the &channel for diffractive processes is answered experimentally for different processes in different ways [8] it is suggestive to investigate the helicity conservation properties of the dual Pomeron exchange amplitude. LOVELACE [9] gave some results for dual Pomeron models with spin 1/2 partons stating that there are longitudinal components conserving 8-channel helicity and transverse components conserving helicity in the t-channel. Looking at the conventional dual model as we did, MOBEL and QUIBOS [ 6 ] found for two external spin-particles no helicity conservation properties at all. As their expressions differ from ours essentially (e. g., they did not obtain factorization of the Pomeron form factors) we have recalculated the matter on the basis of the results given in the first part of this paper. To answer the question which helicity amplitudes contribute to the leading behaviour 8Q(') it is convenient to u8e the unintegrated expression eq. (2.8) after having performed the expansion of the integrand near the Pomeron singularity q = 0 and then integrated over q. Using the relations (A 5, 6,9) given in the appendix and remembering that each power of q corresponds to a factor 8-l we find a new coupling quantity (3.1) where the Tensor FIsNcontains only the leading 8-behaviour of the momenta. Contracting now the tensor FI,Nwith the spin-helicity eigentensors E (cf. eq. (2.3)) for large 8 and using we see that only those par& of (E * 9)lead to a power sadt)in the amplitude which behave as constants in 8. For simplicity we limit ourselves to 8-channel reactions of the type (Fig. 1) and Dual Pomeron Amplitude with External Spin Particles 63 The spin wave factor ~ 1 3 3 . 4is in this case the direct product of 2 or 4 spin-one polarization vectors [(pi, pi) (or E*( -pi, pi)) for incoming (outgoing) momentum pi(-pi) particles with helicity pi given by [101 (3.5) pi = (Ei; lpil sin ei, 0, lpil COB Qi). Dealing now with s-channel helicitiea pi we take eq. (3.5) in the s-channel c.m.8. for a + - . The terms multiplied by p,(l - p;), pa(l - p;) and p, * p, prevent s-channel p, helicity conservation aa they contribute to the leading power s*(O also for p, =I= (integration does not yield compensations). They vanish however for forward scattering ( x = 0) aa should be expected from total angular momentum conservation guaranteed by the Lorentz invariance of the model. Along the same linea we find for four external spin-one particles (see eq. (3.4)) D.EBERT and H.J. OTTO 64 We see that the B-terms provide helicity non-concerving leading contributions for non-forward scattering, being of course also present in A,, As,. The same kind of argument can eaaily be repeated for the investigation of t-channel helicity conservation and again there are leading helicity non-conserving terms present in the amplitude. - 4. Qeneralizationto non-planar multiloop amplitudes Recently, we have studied the non-planar multiloop graphs of Fig.2 in the limit 8+60 [ll]. I n particular, we have found that the asymptotic factorization properties of these diagrams allow for summing up the leading contributions. This provides a new renormalized dual Pomeron trajectory [111.Let us sketch briefly how one can extend the conclusions of the previous chapters to the multiloop caae with external spin particles. Of 2 3 ... N PI p3 Fig. 2. Non-plenar multiloop amplitude deecribing iterated Pomeron exchange course, we would expect from physical reasons that the results obtained for the “bare” Pomeron of Fig. 1remain valid for the renormalized Pomeron, too. The crucial point to deduce this is that the non-planar self-energy operator a N ) ( b +C;, k ) has the same functional dependence on the oscillator operators b+ and c aa in the one-loop caae. Thus all the algebraic steps following eq. (2.1) can be immediately repeated. For the relevant most leading contributions, i. e., in the simultaneous limit where all qi tend to zero (i = 1, ..., N),one can evaluate the elliptic matrices contained in the selfenergy operator [ 111given by where we have used the notation (kl, = Ik), = r / i (2ni)“ n! Dual Pomeron Amplitude with External Spin Particles 66 Further, the integration measure factorizes in this limit, toos) where P(z) is a function explained in ref. [ll]. Performing a11 the integrations and restricting us, for simplicity, to a proceee J1 0 --t Jz 0 we obtain finally + + and + e-" (i I H ( ~~)) IJ V . (4.5) Taking the Mellin transform of eq. (4.4) and summing up, a new Pomeron trajectory + stl7(t, a;(t)) + O(g') a p ) = ap(4 (4.6) is defined as in the case of external scalar particles [ll]. If one transforms back again, the asymptotic behaviour of the sum of all graphs of Fig. 2 is given by an expremion similar to eq. (4.4) without the N-dependent terms and where 7 is replaced by a;@). But this means nothing else than that we have the -me structure as for the bare Pomeron where only a#) is replaced by a;(t). This proves our statement'). We would like to thank Prof. F. KASCHLUHN for his interest in this work. Discussions with Dr. H.Dom are also acknowleged. *) W e disregard l o g * q,-factora in order to obtain log 8-factomleading to a reasonable multipole structure in the angular momentum plane. This a n be justified by working directly in a model with D = 26 and using s "0-2 rule" for the multiloop partition function. ') The integral defining the function (4.6) ia divergent due to a zero at z = 1/2 of the partition function in the denominator [Ill. It has to be suitably mfularized in order to give a:(t) a definite meaning. It is not evident whether than we have yet a re ation a:@)- a(t) = u);(t) 3 necesssry for the discussion of form factor zero8 past eq. (2.12). - + D. EBERTand H.J. Om0 66 Appendix A For completeness we present here the full expressions and formulae contained in abbreviated form in the foregoing part of this paper. The conventions, elliptic functions and matrices used here are the same as in ref. [ 6 ] . The four-point-operator %($, a,, as, u4) with the non-planar self-energy diagram in the t-channel is given in eq. (2.4) IW an Integral. The memure involved haa the explicit form 1 1 p p ... = 4n974 j- du j-dv [u(l- u) v(1 - v)]"*-' 0 1 do 0 1 dx 01 1 1 = - 2ns$ du 6 1 -!2-4/3 j- dv[u(l - u)v ( l - v)l".--' .I3, 0 0 (In the second line we used the JAcoBI-transformed variables eq. (2.10).) Introducing I = 1 -(1 - w ) u, p = 1 -(1 - 0 )v (A2) one has the following representation of the functions of eq. (2.4) including their behaviour near the end-point q = 0 V J ,and V J ,can be obtained from VJ,, VJ,by the interchange u c.)v. For brevity we shall not give here all the matrices contained in the operator part of eq. (2.4) and specified in eqs. (2.6) and (2.6).In the following we present as an example only the treatment of one component in both eq. (2.6) and eq. (2.6) and quote the results for all the others. Dud Pomeron Amplitude with External Spin Particlea 67 The part of eq. (2.5) referring to the particle l-operatoru\:i reads =- (A) [M-(1 - w ) F& + 11 (z) M-(1 - a)8,(1 - o) U H!4 = (v). Limiting ourselves to external excited states on the parent trajectory, i. e., to states generated by only the first mode, we have (aa we shall contract the Lorentz-indices later with spin-helicity-wave functions (eq.(2.3)) we drop Hyl (and a11 H&)from now on (see eq. (2.9))) h(1) 1.P = ha (La) + hb (Lb) 1 IPP N q+o - q-co --18nu -u P, (P(l.a)= q sin no cos (a P(l.b) = 2%) + nv) nu -&gnu 1- u (In’f(z) means always the logarithmic derivative with respect to the angle variable 2nlnx of the whole argument z).Similarly we obtain In w - hi N - nu(1 - u)ctg nu. The corresponding relations for the contributions of particles 3 and 4 can be obtained from (Ab) and ( A 6 ) by interchanging u, v and ( p l , pt), (p3,p4). We now give aa an example the treatment of the contribution of 8 quadratic operator (eqs. (2.6, 2.7)): A,, = 2(-3M-(1 - w ) E(1 - w ) (u)M+(1- u)+ M-. (A7) The first mode component beeing relevant here can eaaily be expressed by logarithmic derivatives of elliptic functions, the behaviour of which for q + 0 is well known (see D.EBBRT and H.J. on0 68 eq. (2.9)): - 4naua(l A lna w N - In" y(A) 7 . (.9*:%)1 Proceeding along the =me lines we get the other functions Hv (i < j)for q -+ 0: -4322 uv(l w)% HlS = (1 - u) (1 - v)ln*w In" yd-2) 8n%uv N q COB a (1 - u ) ( l u) 4 n ' ~ , ( l - V ) (1 w)' H14 = In" yT(-2p) (1 u) p In* UJ 8nauv(l - V ) N q * COB (a 2nv) 1-u - - - - - + N - H, = N 8nauv(l - U ) q cos (a - U) (1 - 4n%~(l - N - Ap In2 w 8 n * ~ ~-( lU ) (1 V) + 2nu) (1 - w)* - U) * In" COB (0 yT ) + + v)) ~ Z ( U - (sinnVnu)a* Obviously each factor hf (all i), Hls, H14, H B or H , gives a power q in the integrand of eq. (2.8) and diminishes the 8-power arising after the q-integration by one. So in 09(t)-qnf - 1 eq. (2.11) the power s come8 out, aa A is the sum of the exponents of the Hi, ((i,j)$: (1, 2) or (3, 4)). Anyway we have calculated the complete leading s-behaviour, aa the contraction of the spin-cou ling tensor with the spin-helicity-f"(eigentensors E may compensate the factor s P Appendix B We present here the complete form of the spin coupling tensor, resulting from the evaluation of the vacuum expectation *value given in equ. (2.7) and the formal binomial expension of powers, i. e. direct products of Lorentz vectors (comp. eqs. (2.8) and (2.9)). Duel Pomeron Amplitude with External Spin Particlee We take ((:)p?p;-") 69 simply aa shorthand notation for the sum of all (:) possible different direct products of n vectors p1 and z - n vectors pa producing a tensor of degree z. Similarly we abbreviate by + a tensor of degree a @ where a Lorentz indices belong to particle i and fi indices belong to particlej. The tensor (B2) is built up by the sum of all different direct products of a metric tensors g,,,,,j and (@ - a) vectors p p j . With these notations we found for the summation region and the coupling tensor in eq. (2.8) the following reault =Jl+4,-rla-A - 4 2 - r1a = Ja z,=Js+4-rscz4 = J4 - - ra and where !Is= 4s Pia = T i e Qla = 2 = q14 + 1- ha - tsc qH =2 -z (B4) Qsr = rsr. Obviously the set of Ji Lorentz indices belonging to the ia external particle have to be spread over four subsets (one direct product of momenta and three direct products of metric tensom). The four symbolical binomial coefficients referring to the iuL set of indices (3. [. .Iidescribe the sum of all possible partitions among these four subsets, while the formal factorials in front of each direct product of metric matrices D. EBERT and H. J. OTTO 60 are to be read as prescription for the summation over all permutations of the first indices of the latter. Finally we present explicit forms of the functions used in eqs. (2.11) and (2.12) yl = z y2 = tw + 1 - tlZ - tw -z =42 y4 = z. Y3 a = max (-k, k b = min (k, 2(y, - 2(y2 + y,)) + y4) - k) 037) - The Pomeron form factor describing the coupling to two scalar particles is given by References [l] For a review of different dual m n a n c a models see J. H. SCHWARZ, Dual Resonance Theory (1973) Phys. Rep. 8 C, Nr. 4 (1973). [2] L. CLAVEL.LI and J. A. SHAFTRO, Nucl. Ph 8. B 67, 490 (1973). [3] V. ALESSANDRINIand D. AMATI, Nuovo &mento 18 A, 663 41973). 141 D. EBEETand H. J. OTTO, CERN-Pmpr. TH 1730 (1973). [6] B. MOEEL and M. QULBOS, Nucl. Phys. B 66, 266 (1973). [6] D. J. GROSSand J. H. SCHWAFLZ,Nucl. Phys. B 28. 333 (1970). [7] V. ALESSANDBINI, D. AMATI and B. MOREL, Nuovo Cimento 7 A, 797 (1972). [8] For a review see B. OTTER, Acta Phys. Pol. B 8, 769 (1972). [9] C. LOVELAOE, Phys. Lett. PO B, 561 (1972). [lo] S. U. C H W N ~Spin , Formalism, CERN Prepr. 71-8 (1971). [ll] H. DOEN,D. EBEETand H.J. OTTO, “High Energy Behaviour of Non-Planar and Planar Dual Multiloop Amplitudes” Prepr. PHE 74-6. Berlin-Zeuthen (1974). Bei der Redaktion eingegangen am 17. Mai 1974. Anechr. d. Verf.: Dr. D. EBEBT Inst. f. Hochenergieph sik d. AdW der DDR DDR-1616 Zeuthen, PLtenenallea Dr. H.J..OTTo Sektion Ph eik d. Humbold-Univ. Berlin DDR-108 &din, Unter den Linden 6

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