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Complex Regge poles
Nuclear Physics B33 (1971) 200-213. North-Holland Publishing Company L--_____J
COMPLEX REGGE POLES V.BARGER* Physics Dept., University of Wisconsin, Madison, Wisconsin, USA R.J.N.PHILLIPS Rutherford High Energy Laboratory, Chilton, Berkshire, England Received 1 March 1971
Abstract: The practical consequences of introducing conjugate complex pairs of Regge poles, in place of the usual real boson Regge poles, are considered. A range of properties are discussed systematically.
1. INTRODUCTION Recently various authors have proposed to introduce complex conjugate pairs of Regge poles [ 1 - 4 ] , in place of the single boson Regge poles with real trajectories that are commonly used in high energy phenomenology. There are a variety of theoretical arguments for this. Apart from possible fundamental justifications, complex Regge poles may also be regarded as a simple parametric way to introduce oscillatory elements into high energy scattering, consistent with the real analyticity and crossing properties of the amplitude (a question first considered in ref. [5] ). It is therefore interesting to study the practical implications. Several different applications of complex boson poles have already been made [ 1 , 4 , 6 - 8 ] . There is a brief discussion of several physical effects in ref. [3], but hitherto no systematic discussion has been presented: this we try to provide here. Complex conjugate pairs of Regge poles are not new. They already occur in the conventional treatment o f baryon exchanges [9], where however the complexity of a is also related to the spin-dependence and special consequences ensue. Boson exchanges correspond to different physical situations, and it is helpful to spell out the new implications explicitly.
* Supported in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation, and in part by the US Atomic Energy Commission under contract AT(11-1)-881, COO-294.
201
V.Barger and R.J.N.Phillips, Complex Regge poles 2. FORMALISM A typical t-channel boson Regge pole contribution, e.g. for a t-channel helicity amplitude, can be written in various ways: f = / 7 1 + exp (-i~'a) (vlvo)O~ sin ~'~
(1)
= ")' i exp (-~ina)(V/Vo)
(1)
c<
for
(
eoV2d},
signature
(2)
Here cfft) is the trajectory, ¢3(t) is the residue function, and ~,(t) is simply related to c~,/3 and the signature, v = (s-u)/4M is the usual crossing-antisymmetric energy variable, M being the target mass. For a conjugate pair of poles, c~,/3 and 7 are all complex conjugate pairs. The amplitude, given by summing the two, looks rather complicated in general. Can we simplify it, to gain some insight? In the formulae above, the energy scale vo is a redundant parameter, and it can be chosen so that either/3 or 7 is real at given t. Arranging for 7 to be real, and writing a = a R + icq, the conjugate poles contribute
f=7
{l}( ;q °" { 2 c ° s h ( l n a I ) C°S(XCq) i \-~o! 1
.
-
+ 2i sinh (~ncq) sm (Xal) } ,
(3)
where X = In (V/Vo). This looks like a con'¢entional (oq=0) Regge pole term, multiplied by the final factor in curly brackets, that rotates logarithmically round a fixed ellipse in the Argand diagram. Compared to a real Regge pole term, the modulus is multiplied by the fluctuating factor A = 2(cosh 2 (~-Tro~i) 1 -- sin 2 (Xal))7 ,
(4)
and the phase is augmented by the logarithmically increasing phase q~= arctan [tanh (½7rai) tan (Xal) ] ,
(s)
with X = In (V/Vo) as before. Note that the maxima and minima of A as a function of X are correlated with the values q~= nn and q~~- (n+~)lr (n = 1,2,...), where the modifying factor in eq. (3)is pure real and pure imaginary, respectively. If 13or 3, happen to be real, we still have dramat|c new properties. But if a is real, the formulae reduce to a single Regge pole with real residue, It is essential that c~ be complex.
202
V.Bargerand R.J.N.Phillips, Complex Regge poles
3. PROPERTIES We discuss a range of properties and possibilities, some overlapping, under separate sub-headings for convenience. Points to bear in mind are the size o f l m a needed to produce a given effect, and the extent to which different effects are compatible with one another and with experiment.
3.1. Exchange degeneracy In certain reaction channels, duality arguments [ 10] suggest exchange degeneracy between Regge poles of opposite signatures such that I m f = 0 (we call these "nonplanar" processes, following the language of duality diagrams [ 11,12] ). This condition is easily met for complex conjugate pairs of trajectories too. In the notation of eq. (2), and using labels " o d d " and "even" to denote signature, we require, with common Vo, a(odd) = o~(even) ,
(6)
~(odd) =fl(even).
(7)
Hence,
f(even) + 3"(odd) = sin rra
+
sin ha* \Vo ]
f(even) - f ( o d d ) = e -ilra - sin ;rot
= real,
+ e -ilr~*
(8)
(9) sin no~* \Vo !
Here eq. (8) is for the non-planar process in question; eq. (9) refers to the line-reversed process, that we discuss below. Note that Re [f(even)+f(odd)] = 0 also - i.e. the non-planar amplitude vanishes completely - at the energies where arg {O/sin got)(V/Vo)a } = (n + I N . These zeroes have no counterpart for real Regge poles. Their positions as functions of t change with p. 3.2. Line reversal For a single conjugate pair of poles, line reversed amplitudes have the same simple relation as for a single real pole: f(ab--*cd) = +f(~b-+~d),
(lO)
where the sign + depends on charge-conjugation in the t-channel (a~-~bd), and hence on the signature.
V.Bargerand R.J.N.Phillips, Complex Regge poles
203
For real Regge poles, there are also simple relations for the contributions o f an
exchange-degenerate pair, namely, If(ab~cd)[ = If(gb~d)l arg f ( a b - ~ c d ) = 0 arg f ( ~ b ~ d )
-Tra
if
a(odd) = a(even),
(lib)
/
J
(1 la)
if also
fl(odd) = fl(even) . (llc)
With complex Regge poles, however, only relation (11 b) survives (see subsect. 3.1). Hence there are no simple relations between the two channels in general. Eq. (1 la) is replaced by the inequality i
If(gb~d)l/lf(ab-+cd)[ = x/1 + sinh 2 7rc~i/cos2
0 /> 1 ,
(12)
where 0 = arg {(fl/sin na)(V/Vo)a} is energy-dependent. Thus the amplitude in the non-planar channel ab -+ cd is always less than or equal to the line-reversed amplitude, with the ratio oscillating logarithmically between 1 and o~. The non-planar amplitude is sometimes zero; the line-reversed amplitude is never zero, unless fl = 0. Experimentally, the non-planar process often has a larger cross section than the crossed process; never smaller [ 13]. Clearly, just making exchange-degenerate Regge poles complex goes in the wrong direction to explain this effect*.
3.3. Total cross sections In some models Im a = 0 at t = 0; in such cases, through the optical theorem, total cross sections behave as for real Regge poles. I f l m a 4 : 0, o T = const. A sin ($-q~o) PAR-1 ,
(13)
where A and q~ are as defined in eqs. (4) and (5) and (bo = ~Trc~ I R and ½7r(aR+l ) for even and odd signature, respectively. The Pomeron clearly cannot be a complex pole, for this would give negative cross sectionsat some energies. These oscillations in o T are one o f many suggested explanations for the Serpukhov data [1,6]. The period o f oscillation depends o n oq In v: t O get an appreciable effect in the 5 - 5 0 GeV range, for example, we need O~I to be large (say a I > 0.5) at t = 0. To fit the Serpukhov data with significant oscillations from a complex P' pole, a value oq = 1.4 was used in ref. [6]. The effect is more marked in total cross section differences. If we assume that * It appears to be possible to reverse the inequality, in certain complex-pole models, by carefully taking account of accompanying J-plane cuts: private communication from D.P.Roy.
V.Barger and R.J.N.Phillips, Complex Reggepoles
204
OT(Op) -- OT(pp ) and OT(K-p) -- OT(K+p) are dominated by a complex w exchange, they oscillate with periodic sign changes (not necessarily at the same energies in the two cases). In fact there are no sign changes - nor any promise of sign changes - in the 3 - 3 0 GeV data; this certainly rules out very large values o f a l , but is not inconsistent with a I ~< 0.5. Duality arguments suggest w - P' exchange degeneracy in pp and Kp scattering. If so, the net contributioias to OT(K+p) and aT(pp ) are zero, and the contributions to aT(K-p) and aT(~p ) oscillate about zero. We have fitted available data with this kind of model, but find that some additional ingredient (such as multi-pomeron cuts, or Pomeranchuk dipole) seems to be needed also. Fig. 1 illustrates the oscillations for an example from ref. [14], with aw = ap, = 0.4 + 0.4i, but with non-exchange-degenerate residues.
25
t K-p
t +~
2C
J
4
10
40 PLAB (G • Vie)
0
z,O0
I000
Fig. 1. Oscillations compatible with present a T data. K+p data from refs. [ 2 9 - 3 1 ] are compared with a fit from ref. [14}, using non-exchange-degenerate w and P' parameters.
3.4. do~dr and shrinking Single real Regge-pole exchange gives the simple energy-dependence do/dt (real pole) = F(t)u 2~(t)-2 .
(14)
205
V.Barger and R.J.N.Phillips, Complex Regge poles A complex pair of poles introduce the further factor A 2 (see eq. (4)):
do~dr (complex pair) = F(t)v 2aR(t)-2 [cosh 2 ~nc~ i I _ sin 2 (a I in V/Vo)] ,
(15)
assuming a single helicity amplitude is d o m i n a n t * . If several amplitudes are important, we must add their c o n t r i b u t i o n s , each of the form of eq. (15) and each with an i n d e p e n d e n t value of v o. In principle, b o t h ct R and a I can be measured from the s-dependence of do/dt. But if a 1 is very small, the oscillations in d o / d t have a very long period in Ins and will be impossible to discriminate in practice. Also, if a I is very large, the percentage fluctuations in A 2 are very small, and again the effect will be hard to measure. The best example of single-Reggeon exchange is ~r-p ~ n ° n , d o m i n a t e d b y the t channel/9 pole. Fig. 2 shows the energy-dependence of do/dr ( T r - p ~ n ° n ) at t = - 0 . 1 7 with data from ref. [16] and a complex-p fit by Desai et al. [4] using a I = 0.088.
1000
u
v
100 CL
[0.088---
\,,
I0 10
PL GeV/c
100
Fig. 2. Oscillations in da/dt. Data for do/dt 0r-p~rOn) at t = -0.175 from ref. [ 16] are compared with solution 2 of ref. [6], that uses a I = 0.088, and with two other parametric fits using al = 0.5 and 1.0. * For example, in meson-nucleon scattering at t = 0.
206
V.Barger and R.J.N.Phillips, Complex Regge poles
(At this t-value the t-channel helicity-flip amplitude dominates). The departure of this theoretical curve from a straight line on the log-log plot is scarcely perceptible, and well below the level of experimental precision. For comparison, parametric fits to data with a I = 0.5 and a I = 1.0 are also shown. It appears that values of the order a I = 0.5 could be detected by present-day experiments, especially when 70 GeV/c data are available, but that much smaller or much larger values would be hard to measure this way. Note that the complex Regge poles in conventional baryon exchange do not give oscillations in do/du (see e.g. ref. [ 16] ). This is because the complexity of a is linked also to the spin dependence in this case. Summing over flip and non-flip amplitudes, there is no net interference between the conjugate Regge poles. If we regard the factor A 2 as averaging to a constant, the mean shrinking of da/dt is the same as for a real Regge pole with trajectory a R.
3.5. Dips in da/dt With real Regge poles, the residue/3(t) must vanish at right-signature integral values of a(t) with t < 0, to prevent unphysical singularities. Exchange degeneracy and factorization then imply zeros of/3(0 at both right- and wrong-signature points, and hence zeros of the Regge pole contribution at wrong-signature points. These can give dips in do~dr. With complex Regge poles, the unphysical singularities are at unphysical (complex) values of t; however, unless cq is large they will give unwanted bumps in cross sections, so we probably need to keep the zeros of t3. The dips in do/dt will then remain, but become less pronounced as a I increaseS. As an illustration, consider a typical amplitude f = i (constant) {a(-iv) ~ + a*(-iv)~*} with a = 0.5 + t + ia I. Fig. 3 shows the cross sections do/dt = Ifl2v -2 at v = 10 for a I = 0.1, 0.3 and 0.6. The dip associated with a = 0 is effectively removed for values a I > 0.3. 3.6. Crossover effect The 7r+p and 7r-p differential cross sections intersect near t = - 0 . 2 ; (similarly K+p and K-p, pp and pp, near t = - 0 . 1 5 ) . The cross section difference comes from (oddsignature) - (even-signature) interference. Since even-signature exchanges are dominated by the pomcron amplitude, that is imaginary and non-flip at small t, the crossover occurs approximately where Im (odd-signature nonflip) = 0 .
(16)
With real Regge poles this effect is attributed to a residue zero of the dominant odd-signature trajectory, and/or pole-cut cancellation in the imaginary part. The imaginary part of a pole term would not otherwise change sign, except at a rightsignature point. With complex Regge poles, the phase of an amplitude is not fixed by a alone. By
V.Bargerand R.J.N.Phillips, Complex Regge poles
\
\\
\"k ol
207
0.1
k
\
\\\
~I
\\
=
f
0.1 0.3
0.6 ......
\\
\\\\
C
\\
x\ x\Nx \\
0.01
0.005
0
0.2
0.4 0.6 -t (GeVlc) 2
0.8
1.0
Fig. 3. The filling-in o f dips in do/dt. A typical e x a m p l e (described in the text) has a R = 0.5 + t and a wrong-signature zero at c~ = 0. Successive curves illustrate the filling-in o f the dip as aI increases.
suitably choosing arg (/3), the imaginary part can be made to vanish at any given value of t, at given energy. (This property is stressed in refs. [3,4] .) However the phase rotates with energy, as shown in eq. (5), so the crossover point moves. We know from o T and Coulomb interference data that the phase of the forward odd-signature nonflip amplitude is near rr/4, in the first quadrant, at t = 0. The crossover point, being very near to t = 0, presumably has phase zero not n. Hence, since the phase increases with increasing energy, the crossover point moves outwards to larger t, as In v increases. This contrasts with the residue-zero explanation, that gives a fixed crossover, and pole-cut cancellation, that gives a crossover moving logarithmically inwards. In solution 2 of ref. [4], with a I = 0.088, the np crossover defined by eq. (16) moves from t = - 0 . 2 7 at 10 GeV/c to t = - 0 . 3 6 at 300 GeV/c.
208
V.Bargerand R.ZN.Phillips, ComplexReggepoles
3.7. Real part or forward scattering amplitude The real part of a forward elastic amplitude can often be measured, through Coulomb interference. For conventional (ai=0) Regge poles, the real parts have smooth sa behaviour. For complex Regge poles, the real parts oscillate - just as the imaginary parts do. These oscillations will appear most clearly in amplitude differences, where the pomeron and some of the Regge poles are eliminated. The elastic regeneration amplitude is a particular case, where the real part can be measured directly: see below.
3.8. Regeneration phase The amplitude for K L ~ K s regeneration on protons, f(KLP~KsP ) = } [f(K°P)-f(K°P)] ,
(17)
has been studied at high energies, using both coherent and incoherent regeneration techniques [ 17-21 ]. The phase of the forward (t=0) amplitude is known to be near -4S-lr in the 5 GeV/c region. Preliminary results in the 1 6 - 3 6 GeV/c range [20] suggest - within large errors - that the phase rotates as energy increases, passing through -~Tr 1 near 20 GeV/c. The regeneration amplitude is commonly supposed to be dominated by co exchange. If ct~ is complex, the phase will indeed rotate in the direction suggested. To get a rotation o f ¼7r as s increases by a factor 4 requires a I In 4 -~ 7r/4 i.e. cq ~ 0.6 (see eq. (5)). In fact a simultaneous fit to this and other relevant KN data has been made [7] with cq = 0.35, giving a somewhat weaker rotation: this effect is illustrated in fig. 4. With the parameters of ref. [7] the cross section difference OT(K°p) OT(K.°p) does not change sign until 300 GeV/c. 3.9. Polarization For a single real Reggeon exchange, all helicity amplitudes have a common phase given by the signature factor, and all first-rank polarization tensors that depend on interference terms of the form Im (flf~') must vanish. For a single complex pair, however, the phases of the different amplitudes do not depend on.the signature factor alone. In general, if arg (/31) ~ arg (/~2) we find Im 0rtf~') ~ 0, and polarization does not vanish. The p-exchange analysis of 7r-p ~ ~r°n by Desai et al. [4] is a good example. In their solution 2, a I = 0.088 at all t. Even this small imaginary part is enough to allow substantial polarizations, of order 15-20%. Polarization is thus very sensitive to cq (but note that complex/3 is essential also, otherwise all amplitudes again have a common phase). The sign is not constrained. Polarization does not tend to zero as u ~ 0% but fluctuates indefinitely. For np charge-exchange it has the form
209
V.Barger and R.J.N.Phillips, Complex Regge poles
/
0 X o
-
r e f . 17 r e f . 20
-30
.c
-6C
ul -9(3 £;k J ol
-120
-150
-180
I
3
I
I
10
30
I
100
I
300
1000
Plab G ¢ V / c
Fig. 4. Rotation of the regeneration phase. Data from refs. [ 17,20] are compared with the complex-t~ model of ref. [7].
5~ = 2 Im (h++h*_)/(lh++ 12+lh+.12),
(18)
where h+_ and h++ are s-channel helicity-flip and nonflip amplitudes, each with highenergy expansions following eqs. (1)-(5). Then in a region where h+. dominates do/dt, we have = const. (A++/A+_) sin (~b++-q%_) ,
(19)
in an obvious notation, following eqs. (4)-(5). With exchange degenerate complex poles, non-planar amplitudes are purely real: see eq. (8). For these cases we get the same prediction as for real Regge poles [10], (non-planar processes) = O.
(20)
Polarization in elastic scattering is complicated by the presence of the pomeron, and several Regge trajectories, in general. However, we note that in conventional theory the characteristic phases of Regge pole terms have often been important in explaining polarization. For example, 7rp elastic polarization is explained by vacuumrho intereference, and the double zero of polarization near at) = 0 is intimately re1 ( a + l ) of the p term. With a complex p pole, the phase is no lated to the phase -~-rr
210
V.Barger and R.J.N.Phillips, Complex Regge poles
longer given by a alone, and the previous explanations no longer work. It seems this is a difficulty for the complex O models of ref. [4] *
3.10. Finite energy sum rules Complex Regge poles contribute relatively simply to finite energy sum rules (FESR) [ 2 2 - 2 4 ] , because the v-dependence is a simple power, albeit complex. Take for example the 7rN invariant amplitude A ' - , that has an odd-signature Regge pole expansion of the form of eq. (2). The FESR with integer moment 2m reads:
fN ImA'-v2mdu = (-)m+l(uo)Zm+l 0
T ( - i N / v o )C~+2m +l ~ Im poles
+ (~*,')'* term)
o~ + 2m + 1
(21)
Continuous moment sum rules (CMSR) [25] also have a simple form if we replace the factors/(-iv) ~ in A ' - by v(v2-v2) ~-(~-1), where v t is the threshold, following ref. [26]. The CMSR with continuous moment/3 then reads: / T(u2-p2)}(a+/3+ 1)
N
f
Im A'-(v2--v2)~¢ dv = .
~ ] lm [
~o (a+/3+l)
) I- (a*,y* term)
.
(22)
0 Hitherto there has been little attention to FESR or CMSR in the complex-pole context. The importance of these constraints is shown by two examples, however. In fig. 5 we compare the FESR integrals for A ' - and uB'-, evaluated at a range of tvalues from the CERN II [27] and Saclay [28] 7rN phase shift analyses (cut-off at 2.1 GeV/c) with the complex-~ models of Desai et al. [4]. These models agree qualitatively with the A ' - sum rules but disagree with the B - sum rules in sign beyond t = --0.4. As a second example, we evaluate CMSR at t = 0 for the A '÷ amplitude, using the same phase-shift sets as low energy input, and comparing with the complex - P ' model of ref. [6]. Here P' was made strongly complex with a I = 1.4 to introduce an oscillation in o(~) as a possible explanation of the Serpukhov data. Fig. 6 shows that this model gives a poor fit to CMSR. (However the P' contribution is rather small, and the fault may lie in other components.)
* To be precise, if we add a dominantly imaginary s-channel helicity-nonflip vacuum exchange contribution (without zeros) to the o-exchange contributions from ref. [4], the predicted 10 GeV elastic n-+p polarizations have single zeros near t = -0.5. This difficulty applies to particular models, not to complex poles in general.
211
V.Barger and RJ.N.Phillips, Complex Regge poles
20'
,
~,4
,
0,~ -t IG,= Ic) 2
-I
?
continuous rnorncnt p
,2
3
./
o CERN II
o
x Bc=reyr¢ [
/
800
E
~
0
/
->
-10
o~
400
Ocsai 1 .... Descti 2 o o o CERN 1] ~ ~t "~ Borcyr¢ I
•
-
1504 ~, ~
~
8
o
T ..._,
100
-400 o x\ "o ->
E
/
50
C
o
-8oo
-5C
~
o #
-100
i
I
i
0.8_t~t~ ; - ' - ev - l c"~
Fig. 5. FESR integrals for the nN chargeexchange amplitudes A'- and vB-, evaluated from CERN II [271 and Saclay [28] analyses, are compared with predictions from the complex-p models of Desai et al. [4] at a range of t-values. Saclay solution I is used here: solution 1I gives very similar results. The upper limit of integration is 2.1 GeV/c.
Fig. 6. CMSR integrals for the nN amplitude A '+ at t = 0, evaluated from CERN II and o Saclay analyses, are compared with the complex P' model of ref. [6] for a range of (continuous) moments.
4. C O N C L U S I O N S
The most striking properties o f complex Regge poles are various kinds o f oscillation, coming from the final factor of eq. (3), that rotates round an ellipse in the Argand diagram as In v increases. If cq is small (<~ 0.1 say) these oscillations are hard to detect. I f a I is relatively large ( 0 . 3 - 0 . 5 , say) the oscillations in OT, do/dt, forward real parts and regeneration phase are all quite substantial, and should be detectable.
V.Bargerand R.J.N.Phillips, Complex Regge poles
212
Complex Regge poles also have some negative properties, in the sense that they avoid some predictions o f real Regge poles. Dips coming from residue zeros at ct = integer tend to be t'flled in. If for instance we wish to explain the well-known dip in do/dt (3,p-->Tr°p) by a zero o f the co contribution at a = 0, we must keep ctI small; but if we wish to explain a rotating regeneration phase by complex co, we need a I large. The phase o f a Regge pole contribution is no longer given by c~ alone. This makes it easy to adjust crossover points of do/dt, and to explain non-zero polarization in 7r-p -->n°n. We simply have more parameters, more freedom, and less predictive power. Exchange degeneracy goes through to the complex pole case. For non-planar processes, we get real amplitudes and the usual predictions. For the line-reversed processes, the real-pole relations for 9 and d o / d t are lost, but an inequality for d o / d t remains. FESR present no difficulty, and should be taken into account as additional constraints on c o m p l e x ~ models whenever possible. We have explicitly considered complex Regge poles, without significant cuts. The authors of ref. [3] have also considered other situations, where the complex poles are accompanied by significant cuts, and may themselves lie on an unphysical sheet. In these situations it is argued [3] that a complex conjugate pair of poles still provides a good parametrization, but only in a limited energy region. Some predictions, such as oscillations as cq Ins becomes large, may then be invalid. We are grateful to G.F.Chew, B.R.Desai and D.P.Roy for helpful conversations at various times.
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COMPLEX REGGE POLES V.BARGER* Physics Dept., University of Wisconsin, Madison, Wisconsin, USA R.J.N.PHILLIPS Rutherford High Energy Laboratory, Chilton, Berkshire, England Received 1 March 1971
Abstract: The practical consequences of introducing conjugate complex pairs of Regge poles, in place of the usual real boson Regge poles, are considered. A range of properties are discussed systematically.
1. INTRODUCTION Recently various authors have proposed to introduce complex conjugate pairs of Regge poles [ 1 - 4 ] , in place of the single boson Regge poles with real trajectories that are commonly used in high energy phenomenology. There are a variety of theoretical arguments for this. Apart from possible fundamental justifications, complex Regge poles may also be regarded as a simple parametric way to introduce oscillatory elements into high energy scattering, consistent with the real analyticity and crossing properties of the amplitude (a question first considered in ref. [5] ). It is therefore interesting to study the practical implications. Several different applications of complex boson poles have already been made [ 1 , 4 , 6 - 8 ] . There is a brief discussion of several physical effects in ref. [3], but hitherto no systematic discussion has been presented: this we try to provide here. Complex conjugate pairs of Regge poles are not new. They already occur in the conventional treatment o f baryon exchanges [9], where however the complexity of a is also related to the spin-dependence and special consequences ensue. Boson exchanges correspond to different physical situations, and it is helpful to spell out the new implications explicitly.
* Supported in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation, and in part by the US Atomic Energy Commission under contract AT(11-1)-881, COO-294.
201
V.Barger and R.J.N.Phillips, Complex Regge poles 2. FORMALISM A typical t-channel boson Regge pole contribution, e.g. for a t-channel helicity amplitude, can be written in various ways: f = / 7 1 + exp (-i~'a) (vlvo)O~ sin ~'~
(1)
= ")' i exp (-~ina)(V/Vo)
(1)
c<
for
(
eoV2d},
signature
(2)
Here cfft) is the trajectory, ¢3(t) is the residue function, and ~,(t) is simply related to c~,/3 and the signature, v = (s-u)/4M is the usual crossing-antisymmetric energy variable, M being the target mass. For a conjugate pair of poles, c~,/3 and 7 are all complex conjugate pairs. The amplitude, given by summing the two, looks rather complicated in general. Can we simplify it, to gain some insight? In the formulae above, the energy scale vo is a redundant parameter, and it can be chosen so that either/3 or 7 is real at given t. Arranging for 7 to be real, and writing a = a R + icq, the conjugate poles contribute
f=7
{l}( ;q °" { 2 c ° s h ( l n a I ) C°S(XCq) i \-~o! 1
.
-
+ 2i sinh (~ncq) sm (Xal) } ,
(3)
where X = In (V/Vo). This looks like a con'¢entional (oq=0) Regge pole term, multiplied by the final factor in curly brackets, that rotates logarithmically round a fixed ellipse in the Argand diagram. Compared to a real Regge pole term, the modulus is multiplied by the fluctuating factor A = 2(cosh 2 (~-Tro~i) 1 -- sin 2 (Xal))7 ,
(4)
and the phase is augmented by the logarithmically increasing phase q~= arctan [tanh (½7rai) tan (Xal) ] ,
(s)
with X = In (V/Vo) as before. Note that the maxima and minima of A as a function of X are correlated with the values q~= nn and q~~- (n+~)lr (n = 1,2,...), where the modifying factor in eq. (3)is pure real and pure imaginary, respectively. If 13or 3, happen to be real, we still have dramat|c new properties. But if a is real, the formulae reduce to a single Regge pole with real residue, It is essential that c~ be complex.
202
V.Bargerand R.J.N.Phillips, Complex Regge poles
3. PROPERTIES We discuss a range of properties and possibilities, some overlapping, under separate sub-headings for convenience. Points to bear in mind are the size o f l m a needed to produce a given effect, and the extent to which different effects are compatible with one another and with experiment.
3.1. Exchange degeneracy In certain reaction channels, duality arguments [ 10] suggest exchange degeneracy between Regge poles of opposite signatures such that I m f = 0 (we call these "nonplanar" processes, following the language of duality diagrams [ 11,12] ). This condition is easily met for complex conjugate pairs of trajectories too. In the notation of eq. (2), and using labels " o d d " and "even" to denote signature, we require, with common Vo, a(odd) = o~(even) ,
(6)
~(odd) =fl(even).
(7)
Hence,
f(even) + 3"(odd) = sin rra
+
sin ha* \Vo ]
f(even) - f ( o d d ) = e -ilra - sin ;rot
= real,
+ e -ilr~*
(8)
(9) sin no~* \Vo !
Here eq. (8) is for the non-planar process in question; eq. (9) refers to the line-reversed process, that we discuss below. Note that Re [f(even)+f(odd)] = 0 also - i.e. the non-planar amplitude vanishes completely - at the energies where arg {O/sin got)(V/Vo)a } = (n + I N . These zeroes have no counterpart for real Regge poles. Their positions as functions of t change with p. 3.2. Line reversal For a single conjugate pair of poles, line reversed amplitudes have the same simple relation as for a single real pole: f(ab--*cd) = +f(~b-+~d),
(lO)
where the sign + depends on charge-conjugation in the t-channel (a~-~bd), and hence on the signature.
V.Bargerand R.J.N.Phillips, Complex Regge poles
203
For real Regge poles, there are also simple relations for the contributions o f an
exchange-degenerate pair, namely, If(ab~cd)[ = If(gb~d)l arg f ( a b - ~ c d ) = 0 arg f ( ~ b ~ d )
-Tra
if
a(odd) = a(even),
(lib)
/
J
(1 la)
if also
fl(odd) = fl(even) . (llc)
With complex Regge poles, however, only relation (11 b) survives (see subsect. 3.1). Hence there are no simple relations between the two channels in general. Eq. (1 la) is replaced by the inequality i
If(gb~d)l/lf(ab-+cd)[ = x/1 + sinh 2 7rc~i/cos2
0 /> 1 ,
(12)
where 0 = arg {(fl/sin na)(V/Vo)a} is energy-dependent. Thus the amplitude in the non-planar channel ab -+ cd is always less than or equal to the line-reversed amplitude, with the ratio oscillating logarithmically between 1 and o~. The non-planar amplitude is sometimes zero; the line-reversed amplitude is never zero, unless fl = 0. Experimentally, the non-planar process often has a larger cross section than the crossed process; never smaller [ 13]. Clearly, just making exchange-degenerate Regge poles complex goes in the wrong direction to explain this effect*.
3.3. Total cross sections In some models Im a = 0 at t = 0; in such cases, through the optical theorem, total cross sections behave as for real Regge poles. I f l m a 4 : 0, o T = const. A sin ($-q~o) PAR-1 ,
(13)
where A and q~ are as defined in eqs. (4) and (5) and (bo = ~Trc~ I R and ½7r(aR+l ) for even and odd signature, respectively. The Pomeron clearly cannot be a complex pole, for this would give negative cross sectionsat some energies. These oscillations in o T are one o f many suggested explanations for the Serpukhov data [1,6]. The period o f oscillation depends o n oq In v: t O get an appreciable effect in the 5 - 5 0 GeV range, for example, we need O~I to be large (say a I > 0.5) at t = 0. To fit the Serpukhov data with significant oscillations from a complex P' pole, a value oq = 1.4 was used in ref. [6]. The effect is more marked in total cross section differences. If we assume that * It appears to be possible to reverse the inequality, in certain complex-pole models, by carefully taking account of accompanying J-plane cuts: private communication from D.P.Roy.
V.Barger and R.J.N.Phillips, Complex Reggepoles
204
OT(Op) -- OT(pp ) and OT(K-p) -- OT(K+p) are dominated by a complex w exchange, they oscillate with periodic sign changes (not necessarily at the same energies in the two cases). In fact there are no sign changes - nor any promise of sign changes - in the 3 - 3 0 GeV data; this certainly rules out very large values o f a l , but is not inconsistent with a I ~< 0.5. Duality arguments suggest w - P' exchange degeneracy in pp and Kp scattering. If so, the net contributioias to OT(K+p) and aT(pp ) are zero, and the contributions to aT(K-p) and aT(~p ) oscillate about zero. We have fitted available data with this kind of model, but find that some additional ingredient (such as multi-pomeron cuts, or Pomeranchuk dipole) seems to be needed also. Fig. 1 illustrates the oscillations for an example from ref. [14], with aw = ap, = 0.4 + 0.4i, but with non-exchange-degenerate residues.
25
t K-p
t +~
2C
J
4
10
40 PLAB (G • Vie)
0
z,O0
I000
Fig. 1. Oscillations compatible with present a T data. K+p data from refs. [ 2 9 - 3 1 ] are compared with a fit from ref. [14}, using non-exchange-degenerate w and P' parameters.
3.4. do~dr and shrinking Single real Regge-pole exchange gives the simple energy-dependence do/dt (real pole) = F(t)u 2~(t)-2 .
(14)
205
V.Barger and R.J.N.Phillips, Complex Regge poles A complex pair of poles introduce the further factor A 2 (see eq. (4)):
do~dr (complex pair) = F(t)v 2aR(t)-2 [cosh 2 ~nc~ i I _ sin 2 (a I in V/Vo)] ,
(15)
assuming a single helicity amplitude is d o m i n a n t * . If several amplitudes are important, we must add their c o n t r i b u t i o n s , each of the form of eq. (15) and each with an i n d e p e n d e n t value of v o. In principle, b o t h ct R and a I can be measured from the s-dependence of do/dt. But if a 1 is very small, the oscillations in d o / d t have a very long period in Ins and will be impossible to discriminate in practice. Also, if a I is very large, the percentage fluctuations in A 2 are very small, and again the effect will be hard to measure. The best example of single-Reggeon exchange is ~r-p ~ n ° n , d o m i n a t e d b y the t channel/9 pole. Fig. 2 shows the energy-dependence of do/dr ( T r - p ~ n ° n ) at t = - 0 . 1 7 with data from ref. [16] and a complex-p fit by Desai et al. [4] using a I = 0.088.
1000
u
v
100 CL
[0.088---
\,,
I0 10
PL GeV/c
100
Fig. 2. Oscillations in da/dt. Data for do/dt 0r-p~rOn) at t = -0.175 from ref. [ 16] are compared with solution 2 of ref. [6], that uses a I = 0.088, and with two other parametric fits using al = 0.5 and 1.0. * For example, in meson-nucleon scattering at t = 0.
206
V.Barger and R.J.N.Phillips, Complex Regge poles
(At this t-value the t-channel helicity-flip amplitude dominates). The departure of this theoretical curve from a straight line on the log-log plot is scarcely perceptible, and well below the level of experimental precision. For comparison, parametric fits to data with a I = 0.5 and a I = 1.0 are also shown. It appears that values of the order a I = 0.5 could be detected by present-day experiments, especially when 70 GeV/c data are available, but that much smaller or much larger values would be hard to measure this way. Note that the complex Regge poles in conventional baryon exchange do not give oscillations in do/du (see e.g. ref. [ 16] ). This is because the complexity of a is linked also to the spin dependence in this case. Summing over flip and non-flip amplitudes, there is no net interference between the conjugate Regge poles. If we regard the factor A 2 as averaging to a constant, the mean shrinking of da/dt is the same as for a real Regge pole with trajectory a R.
3.5. Dips in da/dt With real Regge poles, the residue/3(t) must vanish at right-signature integral values of a(t) with t < 0, to prevent unphysical singularities. Exchange degeneracy and factorization then imply zeros of/3(0 at both right- and wrong-signature points, and hence zeros of the Regge pole contribution at wrong-signature points. These can give dips in do~dr. With complex Regge poles, the unphysical singularities are at unphysical (complex) values of t; however, unless cq is large they will give unwanted bumps in cross sections, so we probably need to keep the zeros of t3. The dips in do/dt will then remain, but become less pronounced as a I increaseS. As an illustration, consider a typical amplitude f = i (constant) {a(-iv) ~ + a*(-iv)~*} with a = 0.5 + t + ia I. Fig. 3 shows the cross sections do/dt = Ifl2v -2 at v = 10 for a I = 0.1, 0.3 and 0.6. The dip associated with a = 0 is effectively removed for values a I > 0.3. 3.6. Crossover effect The 7r+p and 7r-p differential cross sections intersect near t = - 0 . 2 ; (similarly K+p and K-p, pp and pp, near t = - 0 . 1 5 ) . The cross section difference comes from (oddsignature) - (even-signature) interference. Since even-signature exchanges are dominated by the pomcron amplitude, that is imaginary and non-flip at small t, the crossover occurs approximately where Im (odd-signature nonflip) = 0 .
(16)
With real Regge poles this effect is attributed to a residue zero of the dominant odd-signature trajectory, and/or pole-cut cancellation in the imaginary part. The imaginary part of a pole term would not otherwise change sign, except at a rightsignature point. With complex Regge poles, the phase of an amplitude is not fixed by a alone. By
V.Bargerand R.J.N.Phillips, Complex Regge poles
\
\\
\"k ol
207
0.1
k
\
\\\
~I
\\
=
f
0.1 0.3
0.6 ......
\\
\\\\
C
\\
x\ x\Nx \\
0.01
0.005
0
0.2
0.4 0.6 -t (GeVlc) 2
0.8
1.0
Fig. 3. The filling-in o f dips in do/dt. A typical e x a m p l e (described in the text) has a R = 0.5 + t and a wrong-signature zero at c~ = 0. Successive curves illustrate the filling-in o f the dip as aI increases.
suitably choosing arg (/3), the imaginary part can be made to vanish at any given value of t, at given energy. (This property is stressed in refs. [3,4] .) However the phase rotates with energy, as shown in eq. (5), so the crossover point moves. We know from o T and Coulomb interference data that the phase of the forward odd-signature nonflip amplitude is near rr/4, in the first quadrant, at t = 0. The crossover point, being very near to t = 0, presumably has phase zero not n. Hence, since the phase increases with increasing energy, the crossover point moves outwards to larger t, as In v increases. This contrasts with the residue-zero explanation, that gives a fixed crossover, and pole-cut cancellation, that gives a crossover moving logarithmically inwards. In solution 2 of ref. [4], with a I = 0.088, the np crossover defined by eq. (16) moves from t = - 0 . 2 7 at 10 GeV/c to t = - 0 . 3 6 at 300 GeV/c.
208
V.Bargerand R.ZN.Phillips, ComplexReggepoles
3.7. Real part or forward scattering amplitude The real part of a forward elastic amplitude can often be measured, through Coulomb interference. For conventional (ai=0) Regge poles, the real parts have smooth sa behaviour. For complex Regge poles, the real parts oscillate - just as the imaginary parts do. These oscillations will appear most clearly in amplitude differences, where the pomeron and some of the Regge poles are eliminated. The elastic regeneration amplitude is a particular case, where the real part can be measured directly: see below.
3.8. Regeneration phase The amplitude for K L ~ K s regeneration on protons, f(KLP~KsP ) = } [f(K°P)-f(K°P)] ,
(17)
has been studied at high energies, using both coherent and incoherent regeneration techniques [ 17-21 ]. The phase of the forward (t=0) amplitude is known to be near -4S-lr in the 5 GeV/c region. Preliminary results in the 1 6 - 3 6 GeV/c range [20] suggest - within large errors - that the phase rotates as energy increases, passing through -~Tr 1 near 20 GeV/c. The regeneration amplitude is commonly supposed to be dominated by co exchange. If ct~ is complex, the phase will indeed rotate in the direction suggested. To get a rotation o f ¼7r as s increases by a factor 4 requires a I In 4 -~ 7r/4 i.e. cq ~ 0.6 (see eq. (5)). In fact a simultaneous fit to this and other relevant KN data has been made [7] with cq = 0.35, giving a somewhat weaker rotation: this effect is illustrated in fig. 4. With the parameters of ref. [7] the cross section difference OT(K°p) OT(K.°p) does not change sign until 300 GeV/c. 3.9. Polarization For a single real Reggeon exchange, all helicity amplitudes have a common phase given by the signature factor, and all first-rank polarization tensors that depend on interference terms of the form Im (flf~') must vanish. For a single complex pair, however, the phases of the different amplitudes do not depend on.the signature factor alone. In general, if arg (/31) ~ arg (/~2) we find Im 0rtf~') ~ 0, and polarization does not vanish. The p-exchange analysis of 7r-p ~ ~r°n by Desai et al. [4] is a good example. In their solution 2, a I = 0.088 at all t. Even this small imaginary part is enough to allow substantial polarizations, of order 15-20%. Polarization is thus very sensitive to cq (but note that complex/3 is essential also, otherwise all amplitudes again have a common phase). The sign is not constrained. Polarization does not tend to zero as u ~ 0% but fluctuates indefinitely. For np charge-exchange it has the form
209
V.Barger and R.J.N.Phillips, Complex Regge poles
/
0 X o
-
r e f . 17 r e f . 20
-30
.c
-6C
ul -9(3 £;k J ol
-120
-150
-180
I
3
I
I
10
30
I
100
I
300
1000
Plab G ¢ V / c
Fig. 4. Rotation of the regeneration phase. Data from refs. [ 17,20] are compared with the complex-t~ model of ref. [7].
5~ = 2 Im (h++h*_)/(lh++ 12+lh+.12),
(18)
where h+_ and h++ are s-channel helicity-flip and nonflip amplitudes, each with highenergy expansions following eqs. (1)-(5). Then in a region where h+. dominates do/dt, we have = const. (A++/A+_) sin (~b++-q%_) ,
(19)
in an obvious notation, following eqs. (4)-(5). With exchange degenerate complex poles, non-planar amplitudes are purely real: see eq. (8). For these cases we get the same prediction as for real Regge poles [10], (non-planar processes) = O.
(20)
Polarization in elastic scattering is complicated by the presence of the pomeron, and several Regge trajectories, in general. However, we note that in conventional theory the characteristic phases of Regge pole terms have often been important in explaining polarization. For example, 7rp elastic polarization is explained by vacuumrho intereference, and the double zero of polarization near at) = 0 is intimately re1 ( a + l ) of the p term. With a complex p pole, the phase is no lated to the phase -~-rr
210
V.Barger and R.J.N.Phillips, Complex Regge poles
longer given by a alone, and the previous explanations no longer work. It seems this is a difficulty for the complex O models of ref. [4] *
3.10. Finite energy sum rules Complex Regge poles contribute relatively simply to finite energy sum rules (FESR) [ 2 2 - 2 4 ] , because the v-dependence is a simple power, albeit complex. Take for example the 7rN invariant amplitude A ' - , that has an odd-signature Regge pole expansion of the form of eq. (2). The FESR with integer moment 2m reads:
fN ImA'-v2mdu = (-)m+l(uo)Zm+l 0
T ( - i N / v o )C~+2m +l ~ Im poles
+ (~*,')'* term)
o~ + 2m + 1
(21)
Continuous moment sum rules (CMSR) [25] also have a simple form if we replace the factors/(-iv) ~ in A ' - by v(v2-v2) ~-(~-1), where v t is the threshold, following ref. [26]. The CMSR with continuous moment/3 then reads: / T(u2-p2)}(a+/3+ 1)
N
f
Im A'-(v2--v2)~¢ dv = .
~ ] lm [
~o (a+/3+l)
) I- (a*,y* term)
.
(22)
0 Hitherto there has been little attention to FESR or CMSR in the complex-pole context. The importance of these constraints is shown by two examples, however. In fig. 5 we compare the FESR integrals for A ' - and uB'-, evaluated at a range of tvalues from the CERN II [27] and Saclay [28] 7rN phase shift analyses (cut-off at 2.1 GeV/c) with the complex-~ models of Desai et al. [4]. These models agree qualitatively with the A ' - sum rules but disagree with the B - sum rules in sign beyond t = --0.4. As a second example, we evaluate CMSR at t = 0 for the A '÷ amplitude, using the same phase-shift sets as low energy input, and comparing with the complex - P ' model of ref. [6]. Here P' was made strongly complex with a I = 1.4 to introduce an oscillation in o(~) as a possible explanation of the Serpukhov data. Fig. 6 shows that this model gives a poor fit to CMSR. (However the P' contribution is rather small, and the fault may lie in other components.)
* To be precise, if we add a dominantly imaginary s-channel helicity-nonflip vacuum exchange contribution (without zeros) to the o-exchange contributions from ref. [4], the predicted 10 GeV elastic n-+p polarizations have single zeros near t = -0.5. This difficulty applies to particular models, not to complex poles in general.
211
V.Barger and RJ.N.Phillips, Complex Regge poles
20'
,
~,4
,
0,~ -t IG,= Ic) 2
-I
?
continuous rnorncnt p
,2
3
./
o CERN II
o
x Bc=reyr¢ [
/
800
E
~
0
/
->
-10
o~
400
Ocsai 1 .... Descti 2 o o o CERN 1] ~ ~t "~ Borcyr¢ I
•
-
1504 ~, ~
~
8
o
T ..._,
100
-400 o x\ "o ->
E
/
50
C
o
-8oo
-5C
~
o #
-100
i
I
i
0.8_t~t~ ; - ' - ev - l c"~
Fig. 5. FESR integrals for the nN chargeexchange amplitudes A'- and vB-, evaluated from CERN II [271 and Saclay [28] analyses, are compared with predictions from the complex-p models of Desai et al. [4] at a range of t-values. Saclay solution I is used here: solution 1I gives very similar results. The upper limit of integration is 2.1 GeV/c.
Fig. 6. CMSR integrals for the nN amplitude A '+ at t = 0, evaluated from CERN II and o Saclay analyses, are compared with the complex P' model of ref. [6] for a range of (continuous) moments.
4. C O N C L U S I O N S
The most striking properties o f complex Regge poles are various kinds o f oscillation, coming from the final factor of eq. (3), that rotates round an ellipse in the Argand diagram as In v increases. If cq is small (<~ 0.1 say) these oscillations are hard to detect. I f a I is relatively large ( 0 . 3 - 0 . 5 , say) the oscillations in OT, do/dt, forward real parts and regeneration phase are all quite substantial, and should be detectable.
V.Bargerand R.J.N.Phillips, Complex Regge poles
212
Complex Regge poles also have some negative properties, in the sense that they avoid some predictions o f real Regge poles. Dips coming from residue zeros at ct = integer tend to be t'flled in. If for instance we wish to explain the well-known dip in do/dt (3,p-->Tr°p) by a zero o f the co contribution at a = 0, we must keep ctI small; but if we wish to explain a rotating regeneration phase by complex co, we need a I large. The phase o f a Regge pole contribution is no longer given by c~ alone. This makes it easy to adjust crossover points of do/dt, and to explain non-zero polarization in 7r-p -->n°n. We simply have more parameters, more freedom, and less predictive power. Exchange degeneracy goes through to the complex pole case. For non-planar processes, we get real amplitudes and the usual predictions. For the line-reversed processes, the real-pole relations for 9 and d o / d t are lost, but an inequality for d o / d t remains. FESR present no difficulty, and should be taken into account as additional constraints on c o m p l e x ~ models whenever possible. We have explicitly considered complex Regge poles, without significant cuts. The authors of ref. [3] have also considered other situations, where the complex poles are accompanied by significant cuts, and may themselves lie on an unphysical sheet. In these situations it is argued [3] that a complex conjugate pair of poles still provides a good parametrization, but only in a limited energy region. Some predictions, such as oscillations as cq Ins becomes large, may then be invalid. We are grateful to G.F.Chew, B.R.Desai and D.P.Roy for helpful conversations at various times.
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