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Nonlinear Regge Trajectories in Theory and Practice

The problems related to nonlinear behavior of Regge trajectories (RT) and their renormalization group invariance are discussed.

an argument in favour of the linear and exchange degenerated trajectories, at least as a good first approximation, probably subject to minor nonlinear modifications. Fig. 1 illustrates the Chew-Frautschi plot for ρ-ω meson family, which seemingly confirms such an idea. However, as was stressed in Ref. [1], the models implying both linearity and exchange degeneracy lead to an awfully bad χ 2 /d.o.f. ≥ 100. Nonetheless, one can weaken the demand on RT giving up, say, exchange degeneracy. Does it mean that (in case if the timelike (t > 0) region is described well with approximately linear trajectories) we have to expect the same in the scattering (t ≤ 0) region?

Theory
Prior to search for an apriory answer it is interesting to try to see what simplest field theoretic models, like φ 3 -or φ 4 -theories yield for the RT [2]. Fig. 2 exhibits a typical picture of the RT t-dependence which can be only approximately linear in some interval of positive t, while at negative t (and large enough positive t) flattens and tends to some constant, depending on the model. Certainly, φ 3 and φ 4 models are not very realistic, and afterwards some phenomenological nonlinear models for RT were suggested. We mention only three options (of many). In Ref. [3] the trajectory of the form was considered and applied to quite a successful description of the scattering data. One of the distinctive features of this model is negative and infinite value of α(t) at infinitely large and negative t, which is difficult to interpret physically. This trajectory is devoid of particles (this is not prohibited, though). Equation represents the attempt to account for some QCD-like features, such as vector gluon exchange [4]. The resulting trajectory is negative at large (±t) and again devoid of particles on it. Finally, the equation from Ref. [5] gives us something different: the RT grows linearly at large (+t) and tends to a constant (-1) at large (−t). Linear behavior is normally related to the stringlike behavior of hadrons at higher spins. We remark that α(t) from Eq. (3) has two complex conjugated branch points, t 0 and t * 0 .

QCD
If we turn to QCD -supposedly the ultimate theory of strong interactions -then we find that in the region of applicability of perturbative methods, i.e. large (−t), the pomeron trajectory tends to +1 at large (−t) [6], while the secondary (meson) trajectory tends to 0 in the same direction [7]: Such an asymptotic corresponds to exchanges of non-interacting gluons and quarks. This seems to be quite natural because at small distances (large −t) we expect the effects of asymptotic freedom. However, the approach to these asymptotic values appears to be quite nonperturbative (terms of order g 10/3 for the pomeron trajectory and g 5/3 for the meson one). Both the results were obtained via a Bethe-Salpeter equation with the kernel corresponding to the lowest order in QCD coupling.
What do we get at small t? Leading singularity α gg (0) was calculated in Ref. [8] in two first orders in the QCD coupling g 2 , α P (0) = 1 + 3 ln 2 and does not contain, in contrast to the case of large (−t), nonanalytic terms in g. However, Eq. (6) exhibits a function which depends both on implicit scale dependence of the QCD coupling and the renormalization scheme.

Renormalization group invariance
The last circumstance seems quite worrying because the Regge trajectories are generally related to physical amplitudes and spins and masses of observed particles. Hence they cannot depend on any arbitrary choice of renormalization scale and scheme. From the RG invariance it follows that in case of massless QCD (which fairly admits the existence of massive hadrons due to "dimensional transmutation") RT have a general form with the renormalization scale µ and QCD coupling g. Φ is some function that has to be defined by the dynamics and dK/dg 2 = β(g 2 ) is the usual QCD βfunction: β(g 2 ) = −β 0 g 4 − β 1 g 6 − ... One can immediately derive from Eq. (7) the following important consequences.

If α(t) is analyic at
Eq. (8) shows that the slope of the RT is highly nonperturbative. Let us note that the independence of the intercept of g 2 is self-consistent in the sense that if one takes µ 2 = −t, then α(t) becomes the function of the running coupling at (−t): From the definition of K it follows that so we obtain again Φ(0). In the case of finite g 2 (0) [9] it has to be an IR fixed point: β[g 2 (0)] = 0. So one can conclude that, first, QCD dictates essentially nonlinear RT, and nonlinearity, as will be seen later, is not negligible at quite low t; second, the very RT are not analytic in coupling constants which is a direct consequence of the RG invariance.

Practice
Now we turn to practical implementations of these considerations [10]. Fig. 3 shows the t-dependence of the Nonlinear trajectories with QCD-motivated behavior were used also for the description of the pp andpp scattering. Fig. 4 shows quite a good fit. The dashed lines show what happens if one takes a linear approximation for the RT.
When one deals with hard diffractive processes the soft pomeron appears to be unsufficient. Thereof the need in "hard" pomeron. Fig. 5 demonstrates the use of combined nonlinear hard and soft pomerons with quite a successful quality.

Problems
Certainly, there remain a lot of problems. For instance, the ρ-meson trajectory which according to QCD has to tend to zero at large-t shows quite indefinite picture (Fig. 6). The problem was addressed in Refs. [11,12]. More exacerbated situation occurs with π-meson trajectory which is definitely negative already at t = 0.
If to take the QCD-motivated and physically appealing RT behavior depicted at Fig. 7 (parton exchanges at large (−t) and "stringy" behavior at large t), then we encounter a possible problem with analyticity, as such trajectories have complex t-plane singularities which can be dangerous for the whole amplitude, regular at complex t (on the first sheet). In its turn this can be related to possible microcausality violation [13], one of the pillars of modern physics.

Summary
1. Analyticity of RT in t seems to imply their singular behavior at g 2 ∼ 0; 2. QCD (partonlike) asymptotics at high (−t) and "stringy" asymptotics at high t imply complex singularities of RT in the t-plane. Probable clash with microcausality.
3. QCD behavior at high (−t) does not match with monotony of RT in t for ρ, π,..., heavy quarkonia. I take this opportunity to thank J. Soffer and R. Fiore, the organizers of this marvellous workshop, for their kind hospitality and essential support.