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Dual Pomeron Amplitude with External Spin Particles.

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Annelen der Phyaik. 7. Folge, Bend 82, Heft1,1976,8.47-60
J. A. Bsrth, Iaipzig
Dual Pomeron Amplitude with Extwnol Spin Particles
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
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
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
4, A4)
= (01,2,3,4l<Oa,bl DV(%, a29 b ; Pi,P2, -k)
x DV(a,, a 4 , c+;
P39 p43
ZT(b+, c ;
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,.
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
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:
z ( a i J h i=
2 2
i , j - 1 m,n=l
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
2 & $ v v ) a ( m3d w
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
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
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
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
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
$ dq ...
+ n(u + v)) ... can then easily be performed and produce
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].
to a etrip parallel to the imaginary axis and continue afterwards 88 in
Dual Pomeron Amplitude with External Spin Particles
lowing asymptotic expression for 181 --t oo (ap(t)= 3
The form factors are defined as [4]*)
fi$(t) = 2dl)--~r(l) dj(u;a,(t),
a(t))(sin nu)) (cos n u ) ~ - k
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
2 Ji = even.
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
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,@) =
and the form factor integration can be performed explicitely to give
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)*(*).
D.EBERTand H.J. OTn,
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
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)
Dual Pomeron Amplitude with External Spin Particles
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
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
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))
and H.J. OTTO
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
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
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)“
Dual Pomeron Amplitude with External Spin Particles
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
e-" (i
I H ( ~~)) IJ V .
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
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
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
p p
... = 4n974 j- du j-dv [u(l- u) v(1 - v)]"*-'
= - 2ns$
j- dv[u(l - u)v ( l - v)l".--' .I3,
(In the second line we used the JAcoBI-transformed variables eq. (2.10).) Introducing
I = 1 -(1
- w ) u,
p = 1 -(1
- 0 )v
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
The part of eq. (2.5) referring to the particle l-operatoru\:i reads
[M-(1 -
w ) F&
+ 11
M-(1 - a)8,(1 - o)
H!4 =
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)))
= ha
(La) + hb (Lb)
q sin no cos (a
= 2%)
+ nv)
1- u
(In’f(z) means always the logarithmic derivative with respect to the angle variable
of the whole argument z).Similarly we obtain
In w
- 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-.
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
and H.J. on0
eq. (2.9)):
- 4naua(l
A lna w
In" y(A)
Proceeding along the =me lines we get the other functions Hv (i < j)for q -+ 0:
-4322 uv(l
HlS = (1 - u) (1 - v)ln*w In" yd-2)
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)
N -
H, = N
8nauv(l - U )
q cos (a
- U) (1 -
N -
Ap In2 w
8 n * ~ ~-( lU ) (1
+ 2nu)
(1 - w)*
- U)
COB (0
+ v))
~ Z ( U
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
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
simply aa shorthand notation for the sum of all
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
- 4 2 - r1a
= Ja
z,=Js+4-rscz4 = J4
- - ra
!Is= 4s
Pia = T i e
Qla = 2
+ 1- ha -
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
.Iidescribe the sum of all possible partitions among these four
subsets, while the formal factorials in front of each direct product of metric matrices
and H. J. OTTO
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
y4 = z.
a = max (-k, k
b = min (k, 2(y,
- 2(y2 + y,))
+ y4) - k)
The Pomeron form factor describing the coupling to two scalar particles is given by
[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).
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).
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
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Dr. H.J..OTTo
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