Low x Scattering as a Critical Phenomenon

We discuss deep inelastic scattering at low $x$ as a critical phenomenon in 2+1 space-time dimensions. QCD (SU2) near the light cone becomes a critical theory in the limit of $\lim x \to 0$ with a correlation mass $m(x) \propto x^{\nu/2}$. We conjecture that the perturbative dipole wave function of the virtual photon in the region $1/Q<x_{\bot}<1/m$ obeys correlation scaling $\Psi \propto (x_{\bot})^{-(1+n)}$ before exponentially decaying for distances larger than the inverse correlation mass. This behavior combined with an $x$ -independent dipole proton cross section gives a longitudinal structure function which shows the dominant features of the experimental data. For SU3 QCD a similar second order phase transition is possible in the presence of quark zero modes on the light cone.


Introduction
Perturbative QCD has partially been successful to explain low x physics. The BFKL-equation [1] gives an increasing high energy x cross section. Especially interesting is the formulation of the BFKL-pomeron with the help of Wilson lines [2]. A perturbative evolution equation of Wilson line correlation functions with respect to their slope against the light cone reproduces the BFKL pomeron. Higher orders [3] need special resummation techniques [4] to extract reasonable information. There are also unitarity corrections from the multiple scattering of the numerous evolved dipoles which may be equally important as the NLO-BFKL-corrections [5,6]. In addition the characteristic transverse momenta in the BFKL equation drift to smaller values making a perturbative calculation questionable.
We follow the very promising Wilson line method, but use a nonperturbative lattice approach for an effective near light cone Hamiltonian. Wilson loops already have been a very fruitful tool to calculate the soft photon hadron cross section [7]. In standard light cone theory a trivial vacuum is the starting basis [8]. Our alternative approach [9,10] based on the work of refs. [11,12] includes the important nonperturbative physics near the light cone in a solvable form related to the transverse dynamics of Wilson line operators which in modified light cone gauge are functions of the x − -independent zero mode fields a − = A − (x ⊥ ). High energy scattering where the incoming particles propagate near the light cone and interact mainly through particle exchanges with transverse momenta is reduced to an effective 2 + 1 dimensional theory.
In this paper we would like to demonstrate the relevance of the near light cone formulation of QCD to the low x limit of structure functions in high energy deep inelastic scattering. QCD (SU2) has an infrared stable fixed point in the limit when the longitudinal light like interval determining the boundary conditions approaches infinity and the Bjorken variable x goes to zero. Near this critical point the photon wave function is characterized by two length scales : the inverse photon virtuality 1/Q and a correlation length 1/m. Critical behavior is expected when the correlation length is larger than the intrinsic size of the perturbative qq dipole in the photon. By setting up high energy scattering as a theory of Wilson lines near the light cone, we are able to determine the low x behavior of the correlation mass m and thereby get a handle on the photon wave function at large distances where perturbative treatments experience infrared instabilities.
Near light cone QCD has a nontrivial vacuum in the light cone limit. The zero mode Hamiltonian depends on an effective coupling constant containing an additional parameter η which labels the coordinate system. The light cone is approached when the parameter η goes to zero. Generally, if the zero mode theory has massive excitations, these masses diverge like 1 η in the light cone limit η → 0 and decouple. Massless excitations in the zero mode theory, however, influence the light cone limit. Genuine nonperturbative techniques must be used to investigate their behavior. We claim that the increasing high energy cross sections are a directly observable consequence of these massless excitations. We choose the following near light cone coordinates which smoothly interpolate between the Lorentz and light front coordinates : The transverse coordinates x 1 , x 2 are unchanged; x t = x + is the new time coordinate, x − is a spatial coordinate. As finite quantization volume we take a torus with extension L in spatial "-" direction, as well as in "1, 2" direction. The scalar product of two 4-vectors x and y is given with x ⊥ y ⊥ = x 1 y 1 + x 2 y 2 as Obviously, the light cone is approached as the parameter η goes to zero. Now consider high energy photon proton scattering at small x = Q 2 /s, where s = W 2 is the cm energy squared. Using the photon vector q, q 2 = −Q 2 and the proton vector p, p 2 = m 2 ≈ 0 we can define two light like vectors For finite energies the vector of the photon q can be calculated as linear combination of the light like vector e 1 with a small amount of e 2 admixed .
In the limit of infinite energies the mixing η related to the Bjorken variable x vanishes as η 2 2 = x. Therefore it is very natural to formulate high energy scattering in near light cone coordinates.
The appropriate gauge fixing procedure near the light cone is modified light-cone gauge ∂ − A − = 0 which allows zero modes dependent on the transverse coordinates. These zero mode fields carry zero linear momentum p − in near light cone coordinates, but finite amount of p 0 + p 3 . They correspond to "wee" or low x -partons in the language of Feynman. In colour SU(2) the zero-mode fields a 3 − (x ⊥ ) can be chosen colour diagonal proportional to τ 3 . The use of an axial gauge is ideally suited to the light-cone Hamiltonian. The asymmetry of the background zero mode coincides with the asymmetry of the space coordinates on the light cone. The zero-mode fields describe disorder fields. Depending on the effective coupling the zero-mode transverse system is in a massive or massless phase. In a previous paper [10] we calculated the transverse correlation length of the reduced 2 + 1 dimensional theory as a function of η describing the nearness to the light cone. Since the light cone limit is synchronized with the continuum limit a considerable simplification of the QCD Hamiltonian can be achieved.

Near Light Cone QCD Hamiltonian
In ref. [9] the near light cone Hamiltonian has been derived. We refer to this paper for further details. The light cone Hamiltonian on the finite light like x − interval of length L has Wilson line or Polyakov operators similarly to QCD formulated on a finite interval in imaginary time at finite temperature.
Recently the importance of the Wilson line phase operators for deep inelastic scattering has been demonstrated in the context of shadowing corrections [13]. Contrary to naive light cone gauge A − = 0 the rescattering of the quark in the photon changes the deep inelastic cross section. The dynamics of these Polyakov operators is determined by the QCD SU2 Hamiltonian H which is a functional of the fields A i , a − and the canonical conjugate momenta Π i , p − , which represent the electric fields. Here we give only the gluonic part, we discuss the quarks later in connection with QCD SU3 : with The prime indicates that the summation is restricted to n = 0 if p = q. The operator G ⊥ gives the right hand side of Gauss' law: with ρ m the matter density. The above Hamiltonian shows rather clearly that a naive limiting procedure η → 0 does not work. There are severe divergences in this limit. The diverging terms reappear in the usual light cone Hamiltonian as constraint equations which are extremely difficult to solve on the quantum level in 3 + 1 dimensions. The zero-mode part h(a − (x ⊥ ), p − (x ⊥ )) of the full Hamiltonian H is coupled to the three-dimensional modes of the Hamiltonian. In the following we will concentrate on universal properties of the zero-mode Hamiltonian which survive the renormalization of the (2 + 1) transverse dynamics due to the coupling to the (3 + 1) dimensional residual Hamiltonian. The zero-mode Hamiltonian contains the Jacobian J(a − ) from the Haar measure of SU(2) J (a − ( x ⊥ )) = sin 2 gL 2 a − ( x ⊥ ) . The measure stems from gauge fixing and also appears in the functional integration volume element for calculating matrix elements. It is convenient to introduce dimensionless variables ϕ which vary in a compact domain 0 ≤ ϕ ≤ π and characterize the Wilson lines P .
We regularize the zero-mode Hamiltonian by introducing a lattice with lattice constant a in transverse directions. Next we appeal to the physics of the infinite momentum frame and factorize the reduced true energy from the Lorentz boost factor γ = √ 2/η and the cut-off by defining h red For small lattice spacing we obtain the reduced Hamiltonian with the effective coupling constant In the continuum limit of the transverse lattice theory the lattice size a goes to zero and/or the extension L of the lattice to infinity. This limit combined with the light cone limit η → 0 leads to an indefinite behavior of the effective coupling constant. The critical behavior of the zero-mode theory resolves this ambiguity. In ref. [10] we have done a Finite Size Scaling (FSS) analysis obtaining a second order transition as a function of the coupling g 2 eff between a phase with massive excitations at strong coupling and a phase with massless excitations in weak coupling. The critical effective coupling is calculated as g * 2 = 0.17 ± 0.03, which is, however, not a universal quantity and subject to radiative corrections from the (3+1) dimensional modes Π i , A i . A calculation in the epsilon expansion [14] gives the zero of the β-function as an infrared stable fixed point. Therefore, the combined limit of large dimensions L/a and light like coordinates η → 0 is well defined. Using the running coupling constant, we get the critical exponent for the correlation mass m with ν = 0.63. The critical theory is in the same universality class as the 3dim Ising model. Therefore, we use in the following the critical exponents of the Ising model.
To match the Hamiltonian lattice calculation with scattering we consider a lattice with a lattice constant a ≈ 1 Q and an extension L which is larger than the colour coherence length of the qq state in the photon-proton cm system by an amount L 0 . The photon and proton move on almost light like trajectories and the coherence length grows with x as: We demand therefore: Identifying the x and a independent limit of the running coupling g 2 ef f as the fixed point coupling g * 2 one finds that the correlation mass decreases with x towards the critical point: Near the critical point the Wilson lines experience long range correlations, which means that dipoles in the photon wave function are correlated over distances 1/m. In the intermediate range where 1 Q < x ⊥ < 1 m the correlation function of Wilson lines is power behaved: where n = 0.04 in Ising like systems. For even larger distances x ⊥ > 1 m the correlation function decreases exponentially: In the following we construct an effective photon wave function which has these length scales built in and calculate the resulting longitudinal γ -proton cross section.

Longitudinal Photon Proton Scattering and Critical Behavior
The longitudinal photon is ideal to investigate high energy cross sections of hadronic objects with a variable size. Experimental longitudinal cross sections are until now less accurate but they are much better suited to investigate the complex scattering dynamics of small size objects at high energies.
Since the light cone wave function of the qq in the longitudinal photon is peaked at equal momentum fractions z = 1/2 an increasing virtuality of the photon guarantees a decreasing size of the perturbative photon dipole wave function. In the course of x-evolution this wave function will develop many dipoles which in general diffuse into distance scales beyond the original size 1/Q. This increase in parton density and/or size of the photon wave function is generally believed to be the origin of the increasing high energy cross section. In this work, we do not follow the development of the photon dipole state in detail, we only give a qualitative description of the effective photon size as a function of x using the results of the 2 + 1 dimensional critical QCD SU2 theory as a guiding principle. We parameterize the longitudinal photon probability density as: If the correlation mass is larger than the perturbative momentum scale ε, then the function F (εx ⊥ ) = K 0 (εx ⊥ ) 2 . This is the uninteresting case for small x. The photon density is modified only in the small x region when x < x 0 . The parameter x 0 is not given by the critical theory, we choose x 0 = 10 −3 . Thereby we set the scale for the running of the correlation mass m with the effective momentum ε = Q 2 z(1 − z) and x.
Here ν = 0.63 is the Ising critical index determining the Wilson line correlations.
For x < x 0 i.e. for m < ε we replace the perturbative photon density using the correlation functions of the critical theory eqs. (17,18).
The above parametrization extends the photon density into the scaling region using the scaling index n = 0.04. For distances beyond the scaling region the density decays exponentially. The connections are made in such a way that the density is continuous at the respective boundaries of each region.
The photon density combined with an energy independent dipole-proton cross section determines the longitudinal structure function F L and the photonproton cross section: where we take for definiteness the Golec-Biernat-Wüsthoff [15] dipole cross section at fixed x = x 0 The numerical values entering the above formulas are taken over from the original reference. Note, the value R 0 = 0.24f m is independent of x.
Let us for simplicity first consider the photon wave function at a fixed virtuality e.g. Q 2 = 12GeV 2 and fixed momentum fraction of the quark and antiquark 1 2 = z = (1 − z). The respective contributions to F L as a function of x are shown in fig.1. A tiny (see thin line) part of F L comes from the very short range part of the photon (eq. 22). The scaling region in the photon density (see thick line in fig.1 and eq. 23 ) leads to a structure function which behaves as F L ∝ x ν , but is subleading in the interval x 0 > x > 10 −5 . Scaling of this contribution to the structure function F L at small x is a consequence of the scaling photon wave function depending only on εx ⊥ and a dipole cross section ∝ r 2 for small distances. With decreasing x one traverses this scaling region finally having strong non scaling physics at smaller x. The region of the photon density with the exponential decay (cf. dashed curve in fig.1 and eq. 24) dominates the total cross section until x ≈ 10 −5 where the correlation length has increased to about 1 m ≈ 3R 0 . A saturating cross section is building up which asymptotically for x < 10 −5 decreases as σ tot ∝ x νn . This contribution to the longitudinal structure function is approximately proportional to Q 2 , i.e. it does not scale.
In fig. 2 we present the total longitudinal structure function as a function of x for Q 2 = 12GeV 2 . One sees that effective power of the x -increase in the currently accessible region is smaller than the asymptotic critical index ν. In the region 10 −5.5 < x < 10 −3.5 the effective power is about 0.33 dominated by the saturating dipole cross section, which is scanned by the exponential tail of the photon wave function. In the next section we discuss the realistic case taking into account the integration over quark longitudinal momenta, which reduces the effective power mixing length scales of different ε for the same photon virtuality Q.
An interpretation of low x scattering using the language of the gas liquid transition goes as follows: At moderate x the partons are in a gas phase the density of which increases with decreasing x. The increasing parton density exhibits large fluctuations which produce critical opalescence at the parton gas-liquid phase transition where an infinite correlation length develops. Because of the QCD Hamiltonian cf.eq. (11) one can directly see how this critical behavior arises from gluon dynamics. At large effective coupling g 2 eff = g 2 Lη 4a the correlation mass is large, few finite sites with excited electric  Figure 2: Total longitudinal structure function F tot L (x, Q 2 = 12GeV 2 ) calculated with fixed z = 1/2 momentum fraction of the quark (dashed curve) and with integration over z-momentum fraction (thick line). The data with error bars are from the H1-experiment [16] fields p − (x ⊥ ) exist. The dynamics is dominated by a large excitation energy for the half rotors 0 < ϕ(x ⊥ ) < π . With decreasing coupling when the system approaches the light cone these rotors become strongly coupled in transverse space and massless excitations develop around aligned configurations.

Can Critical Dynamics Be Observed?
In a theory of total cross sections the nonperturbative infrared dynamics in the transverse plane is essential. In this section we will discuss three obstacles which may make it difficult to observe critical behavior in high energy scattering: • SU3 dynamics • z-smearing of the photon wave function

• unitarity effects versus transverse growth
It is important to question whether real QCD with N c = 3 really can lead to critical behavior. We know from comparable finite temperature studies that the critical behavior depends on the number of colours. SU3 pure glue QCD is in the same universality class as the Z(3) Potts model and generates a 1-st order phase transition. The same is expected for the Wilson line zero mode theory, which would contain now two diagonal zero mode fields a 3 − and a 8 − after Abelian gauge fixing. As shown in ref. [9], the full Hamiltonian contains a piece where the the zero mode field is coupled to the negative energy quark field ψ − The coupling term can act as an "external magnetic field" in the language of spin models for the a − dynamics. A possible quark zero mode ψ † − ψ − would represent the magnetic field for the Potts spin. If such a zero mode exists, the second order phase transition in SU2 will become a pseudo-critical cross over and a first order transition point in SU3 goes over in a first order transition line in the coupling constant-quark fermion zero mode plane. This line ends at a second order end point, where similar critical dynamics as described above occurs. One must admit that the existence of such a coloured zero mode in ψ †3 − ψ 3 − and ψ †8 − ψ 8 − is puzzling and more theoretical work is needed to study its possibility. If there are zero mode coloured gluon fields on the light cone, it is not excluded that also the negative energy states have zero modes on light like trajectories, comparable to the filled Dirac sea in the normal vacuum.
The second question about the averaging of distance scales through quark z-motion can be easily solved by numerical computation. In the full calculation of the structure function the integration over z smears the clear transverse boundaries in x ⊥ space of the effective photon wave function c.f.eqs. (21-23). In fig.2 we compare the fully z-integrated structure function F L (x, Q 2 = 12GeV 2 ) with the approximate one taken at fixed z = 1/2. One sees the expected flattening of the x-behavior , in the interval 10 −5.5 < x < 10 −3.5 the power decreases from 1/x 0.33 to 1/x 0.29 . For larger Q 2 = 45GeV 2 the effective power increases to 1/x 0.45 and 1/x 0.41 , respectively. An even smaller value of x decreases the power further, because the scaling part of the wave function giving the dominant part of the structure function at very small x multiplies a saturating dipole proton cross section and produces therefore a small x-decrease. Only at moderate 10 −5.5 < x < 10 −3.5 when the scaling part of the wave function overlaps with the r 2 dependent dipole cross section, the critical index ν could be observed in this part of the cross section increasing ∝ ( ε m ) 2−2n ∝ 1/x ν(1−n) . Unfortunately this part of the cross section from the integration over [1/ε, 1/m] is smaller than the dipole cross section integrated over the interval [1/m, ∞]. In fig.2 the data points from ref. [16], illustrate how far the theory extrapolates beyond the currently accessible experimental region.
This finding leads us to the third question about unitarity effects. The numerical calculation clearly shows that saturation effects in the dipole proton cross section tame the growth of the photon proton cross section due to an increasing effective photon size at very small x. It has been conjectured that the saturating dipole cross section in the above parametrization is due to unitarity effects. For a final proof of such a hypothesis it is, however, necessary to study the dependence of dipole proton scattering on the impact parameter. Only the S-matrix S(b) can teach us about unitarity. We refer to such a study in the context of the stochastic vacuum model in ref. [17]. In this work saturation effects are clearly seen in proton-proton scattering at W = 2.5T eV , but no indication in photon-proton scattering at the same energy is visible. Preliminary phenomenological analysis of the data in vector meson production also do not show big effects [18].
Clearly on the theoretical side it is necessary to take into account the target at x − = 0 in a calculation with the dynamical gluon fields A ⊥ (x − , x ⊥ ). Recently important progress has been made using the assumption that the low x problem can be studied in perturbation theory with an effective theory of Wilson lines. The lattice studies can in principle also treat the problem with an external source at x − = 0 with a given structure (e.g. dipole state or "spin-glass" [19]) in transverse space. Ultimately they always will be the best tool to handle the infrared diffusion of the low x wave function. In this paper we have made a first step extending the picture of a dilute parton gas into a dense liquid like phase at small x. Starting from the QCD Hamiltonian a critically diverging correlation length in transverse space leads to increasing cross sections in qualitative agreement with the observed growth for high Q 2 deep inelastic scattering.