QCD Instantons and High-Energy Diffractive Scattering

We pursue the intriguing possibility that larger-size instantons build up diffractive scattering, with the marked instanton-size scaleapproximately 0.5 fm being reflected in the conspicuous ``geometrization'' of soft QCD. As an explicit step in this direction, the known instanton-induced cross sections in deep-inelastic scattering (DIS) are transformed into the familiar colour dipole picture, which represents an intuitive framework for investigating the transition from hard to soft physics in DIS at small x_{Bj}. The simplest instanton (I) process without final-state gluons is studied first. With the help of lattice results, the q bar{q}-dipole size r is carefully increased towards hadronic dimensions. Unlike perturbative QCD, one now observes a competition between two crucial length scales: the dipole size r and the size rho of the background instanton that is sharply localized aroundapproximately 0.5 fm. For r exceeding, the dipole cross section indeed saturates towards a geometrical limit, proportional to the area pi^2, subtended by the instanton. In case of final-state gluons, lattice data are crucially used to support the emerging picture and to assert the range of validity of the underlying I bar{I}-valley approach. As function of an appropriate energy variable, the resulting dipole cross section turns out to be sharply peaked at the sphaleron mass in the soft regime. The general geometrical features remain like in the case without gluons.

1. QCD instantons [1] are non-perturbative fluctuations of the gluon fields, with a size distribution sharply localized around ρ ≈ 0.5 fm according to lattice simulations [2] (Fig. 1 (left)). They are well known to induce chirality-violating processes, absent in conventional perturbation theory [3]. Deep-inelastic scattering 1 (DIS) at HERA has been shown to offer a unique opportunity [5] for discovering such processes induced by small instantons (I) through a sizeable rate [6][7][8] and a characteristic final-state signature [5,9,10]. An intriguing but non-conclusive excess of events in an "instanton-sensitive" data sample, has recently been reported in the first dedicated search for instanton-induced processes in DIS at HERA [11].
The validity of I-perturbation theory in DIS is warranted by some (generic) hard momentum scale Q that ensures a dynamical suppression [6] of contributions from larger size instantons with ρ > ∼ O(1/Q). Here, the above mentioned intrinsic instanton-size scale ρ ≈ 0.5 fm is correspondingly unimportant. This paper, in contrast, is devoted to the intriguing question about the rôle of larger-size instantons and the associated intrinsic scale ρ ≈ 0.5 fm, for decreasing (Q 2 , x Bj ) towards the soft scattering regime. A number of authors have focused attention recently on the interesting possibility that larger-size instantons may well be associated with a dominant part of soft high-energy scattering, or even make up diffractive scattering altogether [12][13][14][15][16][17]. We shall argue below that the instanton scale ρ is reflected in the conspicuous geometrization of soft QCD.
There are two immediate qualitative reasons for this idea. First of all, instantons represent truly non-perturbative gluons that naturally bring in an intrinsic size scale ρ ≈ 0.5 fm of hadronic dimension ( Fig. 1 (left)). The instanton size happens to be surprisingly close to a corresponding "diffractive" size scale, R IP = R α ′ IP /α ′ ≈ 0.5 fm, resulting from simple dimensional rescaling along with a generic hadronic size R ≈ 1 fm and the abnormally small IP omeron slope α ′ IP ≈ 1 4 α ′ in terms of the normal, universal Regge slope α ′ . Secondly, we know already from I-perturbation theory that the instanton contribution tends to strongly increase towards the infrared regime [5,7,9]. The mechanism for the decreasing instanton suppression with increasing energy is known since a long time [18,16]: Feeding increasing energy into the scattering process makes the picture shift from one of tunneling between vacua (E ≈ 0) to that of the actual creation of the sphaleron-like configuration [19] on top of the potential barrier of height [5] . In a second step, the action is real and the sphaleron then decays into a multi-parton final state.
The familiar colour dipole picture [20] represents a convenient and intuitive framework for investigating the transition from hard to soft physics (diffraction) in DIS at small x Bj . At the same time, this picture is very well suited for studying the crucial interplay between the qq-dipole size r and the instanton size ρ in an explicit and well-defined manner, as we shall discuss next.
The intuitive content of the colour dipole picture is that at high energies, in the proton's rest frame, the virtual photon fluctuates predominantly into a qq-dipole a long distance upstream of the target proton. The large difference of the γ * → qq-dipole formation and (qq)-P interaction times in the proton's rest frame at small x Bj then generically gives rise to the familiar factorized expression of the inclusive photon-proton cross sections, 1 For an exploratory calculation of the instanton contribution to the gluon structure function, see Ref.
and the qq-dipole -nucleon cross section σ dipole (r, . . .). The variables in Eq. (1) denote the transverse (qq)-size r and the photon's longitudinal momentum fraction z carried by the quark. Ψ L,T (z, r) contains the dependence on the γ * -helicity. The dipole cross section is expected to include in general the main non-perturbative contributions. For small r, however, one finds within pQCD [20,21] that σ dipole vanishes with the area πr 2 of the qq-dipole. Besides this phenomenon of "colour transparency" for small r, the dipole cross section is expected to saturate towards a constant, once the qq-separation r has reached hadronic distances.
The strategy is now to transform the known results on I-induced processes in DIS into this intuitive colour dipole picture. We shall begin with the most transparent case of the simplest I-induced process [6], for one flavour and no final-state gluons. Subsequently, we shall turn to the more realistic case [7] with final-state gluons and n f (= 3) light flavours.
The idea is to consider first large Q 2 and appropriate cuts on the variables z and r, such that I-perturbation theory holds. By exploiting the lattice results on the instanton-size distribution ( Fig. 1 (left)), we shall then carefully increase the qq-dipole size r towards hadronic dimensions.
2. Let us start by recalling the relevant results [6] for the simplest I-induced DIS process (3), corresponding to one flavour (n f = 1) and no final-state gluons ( Fig. 1 right). At small x Bj = Q 2 2 P ·q , the leading I-induced contribution to the respective partonic cross sections comes from the γ * g subprocess. In terms of the gluon density G(x Bj , µ 2 ), the results from Ref. [6] for the γ * N cross sections σ T (x Bj , Q 2 ) and σ L (x Bj , Q 2 ) for transverse (T ) and longitudinal (L) virtual photons, respectively, then take the following form, Eqs. (5), (6) involve the master integral R(Q) with dimensions of a length, The I-size distribution D(ρ) enters in Eq. (7) as a crucial building block of the I-calculus. For small ρ (probed at large Q) D(ρ) is explicitly known within I-perturbation theory [3,23]. Correspondingly, in Ref. [6], the integral (7) was carried out explicitly by specializing on the familiar I-perturbative form (renormalization scale µ r ), in terms of the QCD β-function coefficients, β 0 = 11 − 2 3 n f , β 1 = 102 − 38 3 n f and the known, scheme-dependent constant d MS = C 1 exp[−3C 2 + n f C 3 ]/2 with C1 = 0.46628, C 2 = 1.51137, and C 3 = 0.29175. In this form, it satisfies renormalization-group invariance at the two-loop level [23], In this paper we prefer to adopt a more general attitude concerning the form of D(ρ) and thus leave the integral (7) unevaluated for the time being. For larger I-size ρ (as relevant for smaller Q), D(ρ) is known from lattice simulations ( Fig. 1 (left)). A striking feature is the strong Fig. 1 (left), one finds R(0) to be numerically close 3 to ρ .
By means of an appropriate change of variables and a subsequent 2d-Fourier transformation, Eqs. (4) -(6) may indeed be cast into a colour dipole form, The change of variables used is (t, x) ⇒ (l 2 , z), with l being the quark transverse momentum and z the photon's longitudinal momentum fraction carried by the quark, The subsequent 2d-Fourier transformation then introduces the transverse qq distance r of the colour-dipole picture via Like is usual in pQCD-calculations [21], we throughout invoke the familiar " , for simplicity. In terms of the familiar pQCD wave function (2) of the photon, we then obtain from Eqs. (4) -(6) the following integrands on the r.h.s. of Eqs. (10), As expected, one explicitly observes a competition between two crucial length scales in Eqs. (14), : the size r of the qq-dipole and the typical size of the background instanton of about ρ ≈ 0.5 fm. Like in pQCD, the asymmetric configuration, z ≫ 1−z or 1−z ≫ z, obviously dominates.
The validity of strict I-perturbation theory (D(ρ) ≡ D I−pert (ρ) in Eq. (7)) requires the presence of a hard scale Q along with certain cuts. However, after replacing D(ρ) by D lattice (ρ) (Fig. 1 (left)), these restrictions are at least no longer necessary for reasons of convergence of the ρ-integral (7) etc., and one may tentatively increase the dipole size r towards hadronic dimensions.

3.
We are now ready to turn to the more realistic I-induced inclusive process The corresponding DIS cross sections have been previously worked out in detail [7] and are implemented in the Monte-Carlo generator QCDINS [9] that forms a basic tool in experimental searches for I-induced events at HERA [11].
The differential cross sections 5 entering in Eqs. (4) now take a modified form [7] (Q ′ 2 = −t), The γ * ⇒ q q "flux" factors [7,28], turn out to be directly related to the square of the pQCD photon wave function, | Ψ pQCD L, T | 2 as we shall see explicitly below. Corresponding to the more complex final state, Eqs. (20), (21) now involve an additional integration over the Bjorken- with total cross section σ (I) q * g (x ′ , Q ′ 2 ) that includes the main instanton dynamics (see below). By means of a change of variables like in Eqs. (11), except for the replacement x ⇒ x/x ′ due to x ′ = 1, one now finds approximately (assuming z ≫ (1 − z) throughout without restriction), Since the total c.m. energy √ s ′ of the q * g ⇒ X subprocess (23) is given by √ s ′ = Q ′ 1/x ′ − 1, the x ′ integration above is equivalent to an integration over E ≡ √ s ′ . The functionΨ pQCD L, T (z, l) is just the 2d-Fourier transform (cf. Eq. (12)) of Ψ pQCD L, T (z, r) in Eq. (2). By inserting the known results for σ (I) q * g from Ref. [7] into Eq. (26), one finds the following structure forσ While in case of the simplest I-induced process (3) above, the contribution to the total cross section was obtained by explicitly squaring the scattering amplitude and integrating over the finalstate phase space, the derivation of the DIS results [7] for the inclusive process (19) was based on the optical theorem combined with the IĪ-valley method [24]. In this approach [25,26,7], one most efficiently evaluates the total cross section from the imaginary part of the forward elastic amplitude induced by the IĪ-valley background A (IĪ) µ . This method elegantly accounts for a resummation and exponentiation of the final-state gluons, whose effects are encoded in the explicitly known IĪ-valley interaction [26,27], appearing in Eq. (27). Apart from its dependence on the relative IĪ-orientation U in colour space, the valley action is restricted by conformal invariance to depend only on the dimensionless, "conformal separation" where in Euclidean space, the collective coordinate R (E) µ denotes the IĪ-distance 4-vector, with −R 2 ⇒ R 2 (E) ≥ 0 such that ξ (E) ≥ 2.
In principle, the next step is to transform Eq. (27) further into the (r, z) colour-dipole representation, in generalization of Eq. (18). To this end, however, we first have to locate any possible, additional l 2 = l 2 (Q ′ 2 , x/x ′ , Q 2 ) dependences that might arise from the final-state gluons etc., i.e. from the second line in Eq. (27). Let us begin by exhibiting a number of important features of dσ (I) dipole /dE in Eq. (27) that emerge in the softer Q ′ regime in combination with lattice results. Besides the I-size distribution D(ρ), the IĪ-interaction Ω in Eq. (27) represents a second crucial quantity of the I-calculus, for which we shall exploit independent lattice information that will be instrumental for a transition towards softer Q ′ . Fig. 2 (left) displays (normalized) UKQCD lattice data [2,8,10] of the IĪ-distance distribution versus the (Euclidean) IĪ-distance R ≡ √ R 2 (E) in units of ρ for quenched QCD (n f = 0), along with the prediction of the IĪ-valley approach [8], Note the remarkable similarity in structure of this lattice "observable" and dσ (I) dipole /dE in Eq. (27). This holds notably in the soft Q ′ regime where the exponential suppression of larger size instantons via the K 1 Bessel functions in Eq. (27) tends to vanish, i. e. (ρQ ′ ) K 1 (ρQ ′ ) ∼ 1, and instead ρ ≈ρ ≈ ρ peak ≈ ρ , with ρ peak and ρ being the (close-by) positions of the sharp peaks of ρ 5 D lattice (ρ) and D lattice (ρ), respectively (cf. Fig. 1 (left)).
Indeed, Fig. 2 (left) reveals crucial information concerning the range of validity of the IĪvalley interaction Ω valley . The IĪ-valley approximation appears to be quite reliable down to R ρ min ≈ 1, where the IĪ-distribution shows a sharp peak, while the valley prediction continues to rise indefinitely. According to Eq. (29), with ρ ≈ρ, this peak of the lattice data corresponds to ξ peak ≈ 3 and hence to S (IĪ) valley (ξ peak = 3, U * ) ≈ 1 2 , for the most attractive IĪ colour orientation U = U * that is known to dominate the U-integral in Eqs. (27) at least for sufficiently large values of 4π/α s in form of a saddle point. This important result perfectly matches with previous theoretical claims [29,30], according to which the maximal I-induced (QCD or EW) cross section shows a "square-root" enhancement compared to the pure tunneling behaviour at E = 0 (S  Let us demonstrate next that this marked peak of the lattice IĪ-distance distribution in Fig. 2 (left) in fact corresponds to the top of the potential barrier, i. e. to the sphaleron mass, E ≈ M sph , which may be estimated [30] as the potential energy of the instanton field exactly in the middle of the transition when the instanton passes the N Chern−Simons = 1/2 point, This result for M sph matches with the estimate M sph ∼ Q ′ from Ref. [5] at large Q ′ , where the integrals in Eq. (27) are known to be dominated by a unique saddle-point in all integration variables, notably including ρ =ρ ≈ ρ * (Q ′ ) ∼ 1/(α s Q ′ ).
Outlook: An investigation of the phenomenology associated with the emerging picture of soft high-energy processes induced by instantons is challenging and in progress [31]. Before more quantitative predictions can be made, a careful study of inherent uncertainties are necessary. Let us merely state at this point that the instanton-induced contributions indeed appear significant towards the soft regime. Like in case of the extensively studied DIS processes (HERA) induced by small instantons (cf. e. g. Refs. [15,11]), one expects characteristic final-state signatures. Given the importance of lattice data for the conclusions reached in this paper, further improved lattice results in this direction would be most desirable. While the main intention of this paper was to associate the origin of the conspicuous geometrical scale in diffractive scattering with the average instanton size, clearly, a number of important aspects remain to be investigated. For instance, an understanding of the mechanism that causes the cross section to increase with energy in an instanton framework is of importance.