Elastic scattering of a quark from a color field: longitudinal momentum exchange
Jamal Jalilian-Marian

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    11_R11 trz_tx_ty_tp_tq_tk_tu̅q̅ Elastic scattering of a quark from a color field: longitudinal momentum exchange
             Jamal Jalilian-Marian^1,2,3
        ^1Department of Natural Sciences, Baruch College,
            CUNY,
         17 Lexington Avenue, New York, NY 10010, USA
        ^2CUNY Graduate Center, 365 Fifth Avenue, New York, NY 10016, USA
        ^3Centre de Physique Théorique, École
            Polytechnique, CNRS,
         Université Paris-Saclay, 91128 Palaiseau, France
            August 24, 2017
        =============================================================================================================================================================================================================================================================================================================================
        
        
        
        Perturbative QCD in the small Bjorken x limit can be formulated as an 
        effective theory known as the Color Glass Condensate (CGC) formalism. The 
        CGC formalism takes into account the dynamics of large gluon densities 
        at small x and has been successfully applied to Deep Inelastic 
        Scattering (DIS) and particle production in high energy hadronic and 
        nuclear collisions in the small x kinematic region. The effective 
        degrees of of freedom in CGC are Wilson lines which enter in the 
        effective quark (and gluon) propagators and re-sum multiple soft 
        scatterings from the small x gluon field of the target. It is 
        however known that the CGC effective theory breaks down when one probes 
        the moderately large x (high p_t) kinematics where collinear factorization 
        and DGLAP evolution of parton distribution functions should be 
        the right framework. Here we propose a general framework which 
        may allow one to eventually unify the two approaches and to 
        calculate pQCD cross sections in both small and large Bjorken x 
        regions. We take the first step towards this goal by deriving an 
        expression for the quark 
        propagator in a background field which includes scatterings from 
        both small and large x modes of the gluon field of the target. 
        We describe how this quark propagator can be used to calculate 
        QCD structure functions F_2 and F_L at all x and thus 
        generalize the dipole model of DIS. We outline this approach can 
        also be used to extend the so-called hybrid approach to 
        particle production in the forward rapidity region of high energy 
        hadronic and nuclear collisions to all x and p_t regions and 
        speculate on how one may apply the same techniques to extend 
        the McLerran-Venugopalan effective action used in high energy 
        heavy ion collisions to include high p_t physics.  
        
        
        
        
        § INTRODUCTION
        
        
        Parton (gluon) distribution functions are known to 
        grow very fast at small Bjorken x which would lead to a violation 
        of the Froissart bound on growth of physical cross sections with energy. 
        Gluon saturation was proposed <cit.> as a dynamical mechanism which can 
        tame this growth and restore perturbative unitarity. The Color Glass 
        Condensate (CGC) formalism <cit.> is an effective action approach 
        to gluon saturation in QCD that includes coherent multiple scatterings from 
        the gluon fields of the target, which are essential when gluon density 
        of the target is large, i.e. at small x. 
        It also includes high energy effects by re-summing large logs of 1/x via 
        the JIMWLK/BK evolution equation <cit.> and as such it differs from 
        the collinear factorization approach to particle production in pQCD where 
        parton distribution functions evolve according to the DGLAP evolution 
        equation <cit.>.
        
        The CGC formalism has been developed significantly since its inception 
        both in terms of precision (higher order corrections) as well as the range of 
        physical processes considered (see <cit.> for references). Perhaps 
        the most important aspect of the CGC 
        formalism is the emergence of a dynamically generated and potentially large 
        scale, the saturation scale Q(x, A, b_t), which allows one to understand 
        a wide range of phenomena in high energy QCD by making controlled approximations 
        and quantitatively reliable calculations. 
        
        Despite many successes of the CGC formalism, its kinematic domain of 
        applicability remains limited; at best it may be applied to processes where the 
        cross section is dominated by small x gluons (as a rough guide one can take  
        small x to mean x < 0.01). Recalling that in particle production in high 
        energy hadronic/nuclear collisions x and p_t are kinematically correlated, 
        the dominant contribution to high p_t processes is from the not so small x 
        region. Therefore high p_t particle production is currently not computable 
        in CGC formalism where one must have log 1/x ≫log p_t^2. Furthermore it's 
        been realized quite recently that CGC 
        calculations of higher order corrections to particle production cross sections 
        are not stable and can become negative <cit.> at transverse 
        momenta slightly higher 
        than the saturation scale Q_s (x). While there has been 
        various remedies proposed to cure this problem <cit.> in 
        case of single inclusive 
        hadron production in asymmetric collisions (such as proton-proton and 
        proton-nucleus collisions in the forward rapidity region in the hybrid 
        approach <cit.>), 
        the general situation remains unsatisfactory and is likely an 
        indication that at high 
        p_t important physics is missing. One such deficiency is the physics of 
        DGLAP evolution which re-sums potentially large logs of p_t^2 in parton 
        distribution functions in the context of collinearly factorized cross sections 
        in pQCD in the leading twist approximation. This is the dominant effect at high 
        p_t where log p_t^2 ≫log 1/x and is not included in the CGC formalism. 
        The two approaches coincide when log p_t^2 ∼log 1/x, known 
        as the double log limit. 
        
        Ideally one would like to have a unified formalism for particle production
        in QCD which reduces to collinear factorization and DGLAP evolution equation 
        at intermediate/large x and/or p_t and to CGC and JIMWLK equation at 
        small x, p_t (see <cit.> for instance). In collinear factorization/DGLAP 
        formalism quarks and gluons and their distribution functions are the degrees 
        of freedom used while in the small x limit it is most efficient to use 
        (fundamental or adjoint) Wilson lines, path ordered exponentials of gluon 
        field which re-sum coherent multiple scatterings of a quark or gluon on a 
        soft color field. This Wilson line
        is very closely connected to the quark/gluon propagator in a background color
        field <cit.>. One may then 
        naturally ask, what would be the effective degrees of freedom in a unified 
        approach? In other words, what could be the building blocks of a physical cross
        section in a more general approach which has both small x and large x
        combined?
        
        In the small x limit one makes a drastic approximation by taking
        the x→ 0 limit, for example, in parton splitting functions whereas
        in DGLAP approach the full splitting function is kept. Clearly to have any
        hope of a unified approach, one must relax the x→ 0 limit. As
        x is the ratio of longitudinal momenta or energies of the partons in the
        parent proton/nucleus, this means that one must consider scattering of a
        projectile parton not only from the small x modes of the target 
        proton/nucleus but
        also from the more energetic modes at large x.
        Furthermore, DGLAP is a leading twist formalism so that multiple exchanges 
        that involve extra powers of the hard scale are dropped. Keeping these two
        points in mind, we propose a new approach which may enable us to 
        eventually "unify" the two formalisms such that one recovers the CGC 
        formalism and JIMWLK evolution equation at small x and pQCD collinear 
        factorization and DGLAP evolution equation at large x. 
        
        Toward this end we consider here scattering of a quark from a background color 
        field which is, unlike in the small x limit, allowed
        to have both large and small x modes. We re-sum the multiple scatterings from
        the fields with small x modes to all orders while keeping only one scattering 
        from large x modes of the field. Therefore the scattered quark can in general
        carry any energy and transverse momentum and be deflected by large angles, unlike 
        the small x limit where the scattering is eikonal (energy of the quark remains 
        the same) and limited to small transverse momentum exchanges. From the 
        calculated quark 
        scattering amplitude we extract the effective quark propagator in this more 
        general background field. We show how this new quark propagator can be used to 
        generalize the dipole model of the QCD structure functions F_2, F_L 
        at small x. We expect that the resulting expressions would form the starting 
        point for calculating the one-loop correction to the structure functions 
        which would then result in a more general evolution equation containing 
        JIMWLK evolution at small x and DGLAP evolution at intermediate/large x. 
        We outline how this new approach may be used to generalize 
        the CGC framework to include particle production at high p_t.
        
         
        
        
        § MULTIPLE SCATTERING AT SMALL X: EIKONAL APPROXIMATION
        
        
        In this section we briefly review multiple scattering in the eikonal 
        limit and its use in small x physics. There are already excellent 
        reviews covering eikonal scattering <cit.> so everything 
        in this section is already well-known. Therefore here we use the example of 
        eikonal scattering to remind the reader of the approximations and 
        assumptions made at small x, and to set up our formalism. In this
        section we closely follow the approach of Casalderrey-Solana and
        Salgado in section 4.1 of <cit.> (see also section 4
        of the first reference in <cit.>).
        
        
        The light cone coordinates are defined as
        
        x^+ ≡t + z √(2)   ,    
        x^- ≡t - z √(2)
        and similarly for momenta and fields. We start
        by considering scattering of a high energy quark moving in the positive 
        z direction, on a high energy target (proton/nucleus) moving to the left
        as shown in Fig. (<ref>).
        The quark has momentum p^μ = (p^+ ∼√(s),  p^- = 0,  p_t =0) 
        whereas the target has momentum 
        P^μ = (P^+ = 0,  P^- ∼√(s),  P_t = 0) even though 
        what's really meant in the eikonal approximation is that p^+ ≫ p^-, p_t 
        and likewise for the target momentum. 
        
        
        
        In the CGC formalism one thinks of the target as a current of color charges which has 
        only one large component, J^μ_a ≃δ^μ -ρ_a. The color current 
        is covariantly conserved and satisfies 
        
        D_μ  J^μ = D_-   J^- = 0
        
        If we now choose to work in the light cone gauge A^+ = 0, we get ∂_- J^- =0
        which means the color current (charge) is independent of the coordinate x^-,
        
        J^  - = J^  - (x^+, x_t)
        
        To find the color fields generated by this color current one can solve the classical
        equations of motion 
        
        D_μ  F^μν = J^ν
        in the light cone gauge A^+ = 0 and find the classical solution A^μ in 
        terms of the color charge ρ. The only non-zero component of the field is
        A^- which is independent of the coordinate x^-,
        
        A^- = A^- (x^+, x_t)
        
        so that in momentum space   
        
        A^- = A^- (p^+ ∼ 0, p^-, p_t).
         
        This is important in eikonal scattering, the target 
        field carries almost no plus momentum so that it can not impart any plus 
        momentum to the projectile. As a result the plus momentum of the projectile 
        is conserved as we will see in detail below. Also related, the small x 
        momentum modes of the target (moving in the negative z direction) carry 
        negligible P^- momentum and therefore can not impart any sizable minus
        component of momentum to the projectile. So basically the only momentum 
        exchanged in the eikonal limit is small transverse momenta. 
        
        An equivalent 
        way to understand this to consider the coupling of the gauge field A^μ 
        to the fermion current u̅ (p_f) γ_μ  u (p_i). In the 
        eikonal approximation one takes p_f ≃ p_i and using 
        u̅ (p) γ_μ  u (p) ∼ p_μ we see that if the projectile
        quark has a large p^+ it couples only to A^- component of the gauge
        field in eikonal scattering. This is a general result; the quark moving
        along the "+" direction in its light cone frame will couple only to the
        "-" component of the filed in that frame. This is extremely important 
        and will be used extensively here.
          
        Clearly to include large x modes of the target one needs to 
        allow exchange of longitudinal momentum which amounts to 
        scattering of a quark not only from the small x modes of the 
        target but also from the large x modes carrying a finite 
        fraction of the target energy P^-. We will consider this in 
        the next section. Therefore to study scattering of a quark on a 
        target at small x we consider scattering (propagation) of a 
        quark in a background field A^- which is independent of 
        coordinate x^-. Each scattering costs a power of g so that if 
        A^- ∼ 1/g then g  A^- ∼𝒪 (1) 
        and one needs to re-sum all such scatterings. Note that we haven't made 
        any assumption about the x^+ dependence of the fields which in the 
        extreme limit is usually taken to be a delta function due to Lorentz 
        contraction of the target. Here we will keep the x^+ dependence 
        general to allow for an extended target. To proceed it is useful to 
        define a light-like vector n^μ which points in the x^- direction 
        so that n^μ = (n^+ = 0, n^- = 1, n_t = 0) and n^2 = 0 and that 
        n· A = 0 defines our gauge with = γ^+. Using this 
        light-like vector one can also write 
        A^-_a (x^+, x_t)= n^-  S_a (x^+, x_t) so that 
        =   S = S. So basically the Lorentz 
        index of the field A^- is carried by the vector n^μ. 
        
        To start we consider multiple scattering of a quark on the target 
        color fields one scattering at a time; momentum of the incoming 
        quark is denoted p, while q is its outgoing momentum. To make 
        the approximations made clear (and so that we don't have to repeat the 
        details in the next section) we will show all the details of the 
        calculation here. The amplitude for one scattering is shown in 
        Fig. (<ref>) 
        
        where solid line denotes a quark while the dashed line denotes the 
        soft gluon field of the target at coordinate x_1^μ = (x_1^+, x_1t).
        The target is represented by point x_1 and not shown for brevity. 
        The amplitude is 
        
        i ℳ_1   =    (i g) ∫ d^4 x_1  e^i (q - p) x_1  (q) [  S (x_1)]  u(p)   =   
        (i g)
        (2π) δ (p^+ - q^+)∫ d^2 x_1t  d x_1^+  
        e^i (q^- - p^-) x_1^+  e^- i (q_t - p_t) x_1t  (q) 
        [  S (x_1^+, x_1t)] u(p)  .       
        Since the field S does not depend on coordinate x^- the integration over x^- is
        trivial and gives an overall delta function 2 πδ (p^+ - q^+) so that the plus 
        momentum of the quark is conserved. Including a second scattering is depicted in 
        Fig. (<ref>) and gives
        
        i ℳ_2 = (i g)^2 ∫ d^4 x_1  d^4 x_2  ∫d^4 p_1  (2 π)^4  
        e^i (p_1 - p) x_1  
        e^i (q - p_1) x_2  (q) [   S (x_2) i  p_1^2 + i ϵ   S (x_1) 
        ]  u(p)  .
        
        Once again one can perform the x_1^-, x_2^- integration giving two delta 
        functions 2 πδ (p_1^+ - q^+)  2 πδ (q^+ - p_1^+) one of which can be
        used to perform the p_1^+ integration setting p_1^+ = p^+ = q^+ with an overall 
        2 πδ (p^+ - q^+) remaining. We get
        
        i ℳ_2    =    (i g)^2   2 πδ (p^+ - q^+) ∫ d^2 x_1t  d x_1^+ d^2 x_2t  d x_2^+  ∫d^2 p_1t (2 π)^2 d p_1^-  (2 π) 
        e^i (p_1^- - p^-) x_1^+  e^- i (p_1t - p_t) · x_1t       e^i (q^- - p_1^-) x_2^+  e^- i (q_t - p_1t) · x_2t  (q) [ S (x_2^+, x_2t)  i  p_1^2 + i ϵ   S (x_1^+, x_1t) 
        ]  u(p)
        
        and p_1^+ = p^+ = q^+ in the propagator. The next step is to perform the integral
        over p_1^-. This can be done via contour integration noticing that  in the
        numerator is next to  and that p_1^- γ^+ = p_1^-   = 0.
        Therefore the integration over p_1^- can be performed by closing the contour below
        the real axis and gives
        ∫d p_1^-  (2 π)e^i p_1^- (x_1^+ - x_2^+) 
        2 p^+ [p_1^- - p_1t^2 - i ϵ 2 p^+] =
        - i  2 p^+ θ (x_2^+ - x_1^+)   
        e^i p_1t^2  2 p^+ (x_1^+ - x_2^+)
        There are two points which we should mention here, first an incoming quark with 
        positive p^+ forces ordering along the direction of motion x^+, enforced 
        via the theta function, and second, in the strict eikonal approximation one 
        disregards the exponential on the right hand side by using the fact that 
        p_1t p^+≪ 1. It is possible to go beyond this strict eikonal 
        limit and keep these exponentials factors (for example see <cit.>), 
        however we will not do that here. We then have
        
        i ℳ_2    =    (i g)^2   (- i )   2 πδ (p^+ - q^+) ∫ d^2 x_1t  d x_1^+ d^2 x_2t  d x_2^+  θ (x_2^+ - x_1^+)  
        e^- i p^- x_1^+  e^i q^- x_2^+      ∫d^2 p_1t (2 π)^2 
        e^- i (p_1t - p_t) · x_1t  e^- i (q_t - p_1t) · x_2t  (q) [ S (x_2^+, x_2t)  i  2 p^+   S (x_1^+, x_1t) 
        ]  u(p)      
        There are two further approximations we need to make before we get the final 
        result, first, we note that p^- = p_t^2  2 p^+≪ 1 and 
        q^- = q_t^2  2 q^+≪ 1 and one therefore can drop the exponential 
        terms involving p^-, q^- above. More importantly, there is still a p_1t
        term in the intermediate propagator which we also neglect since it is of the form 
        p_1t p^+≪ 1  and can be dropped (it is possible to keep 
        these terms, order by order, see <cit.> for example). After this last 
        step, one can perform the integration over transverse momentum p_1t 
        which gives
        ∫d^2 p_1t (2 π)^2  e^- i p_1t· (x_1t - x_2t)
        = (2 π)^2 δ^2 (x_1t - x_2t)
        
        and enables one to perform one of the transverse coordinate integrations to 
        get
        
        i ℳ_2    =    (i g)^2   (- i ) ( i )   2 πδ (p^+ - q^+) ∫  d x_1^+   d x_2^+  θ (x_2^+ - x_1^+)  ∫ d^2 x_1t 
        e^- i (q_t - p_t) · x_1t       (q) [ S (x_2^+, x_1t)   2 p^+   S (x_1^+, x_1t) 
        ]  u(p)
        
        where  is now understood to be p_1^+  γ^-. The last step is
        to use 2 p^+  = since p_1^+ = p^+.
        We note that the two small x fields S are at the same transverse coordinate x_1t
        which was made possible due to neglecting terms p_1t p^+ in both 
        the exponentials and in the intermediate propagator. The second point to be kept in
        mind is the ordering of the scatterings which was caused by positivity of p^+ of
        the incoming quark and that contour integration over the propagator pole puts the
        intermediate quark line on shell. 
        
        The approximations and steps needed to extend this procedure to include any number of scatterings from the small x fields S should now be clear; the energetic 
        projectile moving in a given direction will couple to the component of the gluon 
        field Lorentz-conjugate to that direction, for example p^+   :  A^-. One
        neglects terms of the order p_t  p^+ everywhere and that the p^+ 
        component of the 
        projectile momentum is conserved. The generic diagram for n soft scatterings is
        shown in Fig. (<ref>), 
        
        The amplitude is then
        
        i ℳ_n    =    2 πδ (p^+ - q^+)  (q)  ∫ d^2 x_t  e^- i (q_t - p_t) · x_t      {
        (i g)^n   (- i )^n ( i )^n ∫  d x_1^+   d x_2^+  ⋯ d x_n^+  θ (x_n^+ - x_n - 1^+)  ⋯θ (x_2^+ - x_1^+)      [ S (x_n^+, x_t)  S (x_n - 1^+, x_t) ⋯ S (x_2^+, x_t)  S (x_1^+, x_t) 
        ]}  u(p)
        
        which after summing over all n scatterings 
        (iℳ = ∑_n=1^∞ i ℳ_n) gives 
        the standard result for quark-target scattering amplitude in the 
        eikonal limit <cit.>,
        
        i ℳ (p,q) = 2 πδ (p^+ - q^+)  (q)  ∫ d^2 x_t  e^- i (q_t - p_t) · x_t [V (x_t) - 1]  u(p)
        
        where V (x_t) is the Wilson line in the fundamental representation, 
        defined as [Here we are being sloppy about path ordering vs 
        anti-path ordering of the fields. The Wilson lines appearing in the 
        propagators are path ordered, i.e., the soft fields increase in their 
        x^+ argument from left to right, whereas the Wilson lines in the 
        amplitudes are anti-path ordered.
        As introducing different symbols for path-ordered vs anti path-ordered 
        will result in clutter and because it will not have any bearing on 
        our result we will ignore this distinction.]
        V (x_t) ≡P̂ exp{i g ∫_- ∞^+∞ d x^+   S^-_a (x^+, x_t)  t_a}
        and S^- (x^+, x_t) = n^-  S (x^+, x_t) is our small x gluon field.  
        
        One can extract an effective quark propagator using the result for 
        scattering amplitude. To do so we recall that the incoming 
        projectile was a quark with positive p^+ which is all we need in 
        case of scattering of a physical particle. However, to construct a 
        Feynman propagator one also needs to consider the case when a 
        particle with - p^+ is moving backward (what we have computed so far
        would give us the retarded and not the Feynman propagator). This is 
        practically trivial to implement since the only change in the 
        derivation above is the locations of p^- poles of the intermediate 
        propagators which will be above 
        the real axis as compared to before where the poles were below the 
        real axis. This 
        reverses the ordering of projectile propagation in x^+ direction. 
        It also results in a relative minus sign between the positive and 
        negative p^+ contributions due to the reversal of the direction 
        of the integration contour when the contour is closed above vs 
        below the real axis.
        
        Defining (recall that in calculating the amplitude one goes from right to left
        in a Feynman diagram while for the propagator we go from left to right)
        τ_F (p,q) ≡  2 πδ (p^+ - q^+)  ∫ d^2 x_t  e^- i (q_t - p_t) · x_t {θ (p^+)  [V (x_t) - 1] -
        θ (- p^+)  [V^† (x_t) - 1]
        }
        the quark propagator is then given by 
        
        S_F (p,q) = (2 π)^4 δ^4 (p - q)  S_F^0 (p) +  
        S_F^0 (p) τ_F (p,q)   S_F^0 (q)
        
        where we have added the free propagator for completeness
        and the scattering amplitude is related to τ_F via 
        
        i ℳ (p,q) =  (q) τ_F (p,q)  u (p)  .
        
        
        § MULTIPLE SCATTERING AT SMALL AND LARGE X
        
        
        
        It should be clear from the derivation in the last section 
        that the target fields denoted by S have only small P^+ modes. 
        Furthermore, since we are considering scattering from the 
        small x fields of the target, they carry small fraction of 
        the target energy P^- and therefore can not impart large P^- 
        into the projectile. So the only momentum exchanged is 
        transverse momentum which is 𝒪 (p_t p^+).
        Therefore these soft multiple scatterings  
        can not cause a big deflection of the projectile which 
        continues to travel along a straight line in x^+ direction 
        and remains at the same transverse coordinate x_t. 
        
        This fact allows one to re-sum 
        the soft multiple scatterings into a Wilson line. Physical observables 
        in DIS and the hybrid formulation of particle production in 
        proton-proton and proton-nucleus collisions involve multi-point 
        correlators of these Wilson lines; the most prominent one being 
        the dipole scattering cross section which is the trace of two 
        Wilson lines in the fundamental representation and satisfies the 
        JIMWLK/BK evolution equation. In this sense Wilson lines are the 
        effective degrees of freedom at small x.
        
        Clearly to have any hope of merging the 
        full DGLAP physics, i.e. exact splitting functions, with the JIMWLK/BK 
        evolution which takes the small x limit of the splitting functions, 
        one needs to relax the small x approximation and include scattering 
        from the large 
        x modes of the target. One would also need to figure out what the 
        effective degrees of freedom are in this more general kinematics where
        large x modes of the target are included. This is our main goal in 
        this paper. To this end we propose to go beyond scattering from small 
        x fields by including one general field which can carry any momentum 
        fractions x. Such a general field will clearly depend on all 
        4-coordinates and all of its components (subject to the gauge condition) 
        will participate in the scattering. Therefore we consider scattering of 
        a quark on a target when in addition to the small x fields one "all x" 
        field participates. Since this general field will carry large
        (as well as small) P^-, it can impart a large p^- momentum to the 
        projectile and cause a large angle deflection so that the scattered quark 
        may have parametrically large transverse momentum 
        (p^- = p_t^2  2 p^+) and lose some (or even all) of its p^+ 
        momentum (rapidity). We expect 
        that we may not have to include more than one scattering from an all x 
        field and that higher number of such scatterings would be suppressed by 
        the coupling constant. In this paper we will ignore higher number of 
        scatterings from the all x field but intend to come back to this 
        point in the future as it may be relevant for gauge invariance at large x. 
        
        
        
         §.§ A hard scattering after multiple soft scatterings
        
           
        We therefore consider scattering of a quark from an all x field
        which we denote A^μ = A^μ (x^+, x^-, x_t), the single scattering
        amplitude is shown in Fig. (<ref>). From now on we 
        will refer to the all x field as hard and the small x fields 
        as soft so that hard or soft refers to momentum fraction x carried
        by a field.  
        
        
        
        The result is
        
        i ℳ_1    =    (i g) ∫ d^4 x  e^i (q - p) x  (q) [  (x)]  u(p)    =   
        (i g) ∫ d^4 x  e^i (q - p) x  (q) [  (x)  2 p^+ ]  u(p)
        
        where we have used the Dirac equation to arrive at the second line.
        We can now successively include more scatterings from the small x 
        fields S. Let us consider the case that all such soft scatterings 
        happen before the hard scattering. Including one such soft scattering 
        gives
        
        i ℳ_2 = (i g)^2 ∫ d^4 x  d^4 x_1  ∫d^4 p_1  (2 π)^4  
        e^i (p_1 - p) x_1  
        e^i (q - p_1) x  (q) [  (x)  i  p_1^2 + i ϵ   S (x_1) 
        ]  u(p)
        
        shown in Fig. (<ref>).
        
        
        As before we can perform the x_1^- integration since the soft field S 
        does not depend on it, and use the resulting delta function to set 
        p_1^+ = p^+. Next integration over p_1^- can be done using contour 
        integration which gives
        
        ( - i)  θ (x^+ - x_1^+)  1  2 p^+  
        e^i p_1t^2  2 p^+ (x_1^+ - x^+)  .
        
        We get
        
        i ℳ_2    =     (i g)^2 ∫ d^4 x  d^2 x_1t  d x_1^+  θ (x^+ - x_1^+) ∫d^2 p_1t (2 π)^2  
        e^i (q - p_1) x  e^- i (p_1t - p_t) · x_1t       (q) [  (x)   2 p^+   S (x_1^+, x_1t) 
        ]  u(p)
        
        where p_1^- = p_1t^2  p^+ and p_1^+ = p^+. This procedure can now be repeated
        to include any number of soft scatterings which happen before the hard one. For example, 
        inserting two soft scatterings before the hard one is shown in Fig. 
        (<ref>) 
        
        and gives
        
        i ℳ_3    =     (i g)^3 ∫ d^4 x  d^2 x_1t  d x_1^+   d x_2^+  θ (x^+ - x_2^+)  θ (x_2^+ - x_1^+)∫d^2 p_2t (2 π)^2  e^i (q - p_2) x  
        e^- i (p_2t - p_1t) · x_1t       (q) [  (x)   2 p_2^+   S (x_2^+, x_1t)  2 p_1^+   S (x_1^+, x_1t) ]  u(p)
        
        where the soft fields are now at the same transverse coordinate as before. 
        It should be now clear how to repeat this procedure by adding more and more
        soft scatterings. The amplitude for n soft scatterings followed by a 
        hard one is shown in Fig. (<ref>),  
        
        After re-summing all the contributions from n soft
        scatterings followed by a hard one we get
        
        i ℳ =  (i g) ∫ d^4 x  d^2 z_t  ∫d^2 k_t  (2 π)^2  e^i (q - k) x  
        e^- i (k_t - p_t) · z_t (q) [  (x)   2 k^+   V (z_t, x^+) 
        ]  u(p)  .
        
        where k^+ = p^+, k^- ≡k_t^2  k^+ (momentum k is on shell) 
        and the Wilson line now extends to x^+,
        
        V (z_t, x^+) ≡P̂ exp{i g ∫_- ∞^x^+ d z^+   S^-_a (z_t, z^+)  t_a}
        and S^- (z_t, z^+) = n^-  S (z_t, z^+) as before. 
        
         This has a simple interpretation; the quark 
        propagates through the soft
        field and scatters eikonally from the target starting from 
        z^+ = - ∞ until the point x^+ where the hard scattering takes place.  
        As before multiple soft scatterings do not change the transverse coordinate of 
        the projectile quark and re-sum into a Wilson line.
         
        
        
         §.§ Multiple soft scatterings after a hard one
         
        In the eikonal approximation used for the soft multiple scatterings the
        final state quark can gain some transverse momentum but not much as 
        p_t ≪ p^+. Furthermore its p^+ is conserved and does not change which
        also means it has negligibly small p^-. However, a hard scattering such
        as the one considered above can in principle impart large p^- and p_t to the
        projectile (and P^+, p_t to the target) so that after the hard scattering
        the quark will not necessarily propagate along its original trajectory but
        can be deflected by a large angle and lose some or even all of its p^+. This
        means we must take into account the possibility that the scattered quark is 
        not moving along the original (positive) z direction but can now move in an
        arbitrary direction with projections on all x, y, z coordinates. Equivalently
        the scattered quark will have momenta p^+, p^-, p_t without any restriction
        on their magnitude (still subject to being on shell). For example, in the extreme
        case of back scattering p^+ → 0 while p^- will be very large. Or 
        in the case of scattering at right angle we will have p^+ ∼ p^- ∼ p_t. 
        So it seems that we can not use our eikonal methods after the hard scattering 
        any more.
        
        However one can still our eikonal results  if we work in the light cone 
        frame of the scattered quark so that light cone frame of the scattered quark is
        rotated (in 3 dimensions) with respect to the light cone frame of the incoming
        quark. We define the direction of motion of the scattered quark to be the new 
        positive z axis and label it as z̅. The direction of propagation of 
        the scattered quark z̅ will now be related to it x,y,z in the original 
        frame (defined by the incoming quark) via the standard rotation matrix 
        𝒪 in 3 dimensions.
        ([ x̅; y̅; z̅ ])
         = 
        𝒪 ([ x; y; z ])
        
        The same rotation matrix will allow us to express the momentum vector 
        of the scattered quark in the rotated frame in terms of quantities 
        in the original frame. The rotation matrix 𝒪 will depend 
        on azimuthal and polar angles of the scattered quark with respect to the
        original frame which can be related to energy and rapidity of the 
        final state quark. 
        
        In the new coordinate system the scattered quark is moving along
        positive z̅ so that it will have a large p_z̅ but 
        no p_x̅ or p_y̅. However it will possibly have 
        all momentum components when expressed in terms old frame quantities 
        p_x, p_y, p_z. One can then define the new light cone coordinates 
        x̅^+, x̅^-, x̅_t and light cone momenta 
        p̅^+, p̅^-, p̅_t in the new frame in the standard way. 
        To facilitate this we define a new light cone vector n̅ which 
        projects out the plus component in the new frame, so that 
        n̅·p̅ = p̅^+.
        
        The scattered quark now moves along the new plus direction x̅^+ and has only
        one large momentum component p̅^+. Therefore it will couple only to the minus
        component of the field expressed in the new coordinates 
        S̅^- = n̅^-   S(x̅^+, x̅_t). It is important to realize that
        even though only the - component of the field in the new coordinate system is 
        involved, the rotation matrix will generate the other Lorentz components if one 
        expresses the soft field S̅^- in terms of the fields in the old coordinate 
        system via S̅^μ = Λ^μ_  ν  S^ν where the only non-zero 
        components of the Lorentz transformation matrix Λ^μ_  ν are the
        elements of the 3-dimensional rotation sub-group. 
        
        
        To illustrate this we consider the case when there is a hard scattering first, followed
        by multiple soft scatterings. The amplitude for one soft scattering after the hard
        one, shown in Fig. (<ref>), can be written as
        
        i ℳ_2 = (i g)^2 ∫ d^4 x  d^4 x̅_1  ∫d^4 p̅_1  (2 π)^4  
        e^i (p̅_1 - p) x  
        e^i (q̅ - p̅_1) x̅_1  (q̅) [ 
          S (x̅_1)  i p̅_1^2 + i ϵ  (x) ]  u(p)  .
        
        All the steps we went through in section (<ref>) can now be repeated; 
        the field S (x̅_1) is independent of x̅_1^- so that integration over
        x̅_1^- can be performed leading to a delta function of 
        2 π δ (p̅_1^+ - q̅^+) which allows one to perform the 
        p̅_1^+ integration setting p̅_1^+ = q̅^+. One then performs 
        the contour integration over p̅_1^ - noticing that it is next to 
        which puts p̅_1 on shell and gives a factor of θ (x̅_1^+ - x^+),
         we get
        
        i ℳ_2    =     (i g)^2 ∫ d^4 x  d^2 x̅_1t  d x̅_1^+ 
         θ (x̅_1^+ - x^+) ∫d^2 p̅_1t (2 π)^2  e^i (p̅_1 - p) x  
        e^- i (q̅_t - p̅_1t) ·x̅_1t       (q̅) [ 
          S (x̅_1^+, x̅_1t)   2 p̅_1^+  (x)
        ]  u(p)
        
        with 
        p̅_1^+ = q̅^+ and p̅_1^- = p̅_1t^2  2 p̅_1^+.
        By repeating this procedure for further soft scatterings as shown in 
        Fig. (<ref>),
        
        and re-summing them we get
        
        i ℳ =  (i g) ∫ d^4 x  d^2 z̅_t  ∫d^2 k̅_t  (2 π)^2  e^i (k̅ - p) x  
        e^- i (q̅_t - k̅_t) ·z̅_t (q̅) [ 
        V (x^+, z̅_t)    2 k̅^+  (x)
        ]  u(p)
        
        where 
        k̅_1^+ = q̅^+ and k̅_1^- = k̅_t^2  2 k̅_1^+
        and the Wilson line is now 
        V (x^+, z̅_t) ≡P̂ exp{i g ∫_x^+^+∞ d z̅^+   S^-_a 
        (z̅_t, z̅^+)  t_a}
        
        Again this expression has a simple interpretation; the projectile quark 
        undergoes a hard scattering which changes its direction (possibly in all 
        3 dimensions) after which it undergoes multiple soft scatterings when 
        expressed in terms of coordinates in the new frame.
        
        
        
        
         §.§ Multiple soft scatterings before and after a hard scattering
        
        We now have all the ingredients to write the general expression for 
        scattering of a projectile quark from the target including soft (small x)
        scatterings to all order and hard (all x) to the first order. This is shown
        in Fig. (<ref>) where the final state quark may now
        have a large q^- or q_t as measured in the incoming quark's frame (but has 
        only a large momentum in z̅^+ direction in the rotated frame).
        
        
        
        
        The procedure
        is the same as above, one considers multiple scatterings from the soft fields
        which are independent of the "-" coordinate leading to conservation of the 
        "+"
        momentum across a soft scattering, and then doing the contour integration 
        over the "-" component of the intermediate momentum which puts the quark
        line on shell. All the terms of the form p_t  p^+  x^+ or 
        p̅_t p̅^+ x̅^+ are neglected which allows one to 
        do the transverse coordinate integration over the soft fields. The only exception
        is if there is a "mis-match" of barred and un-barred momenta and coordinates
        in the exponentials which are then kept, for example exponents like 
        p̅^-  x^+ may have large components when expressed fully in terms 
        of quantities in the old frame. The rest of details are the same as before;
        during soft multiple scatterings the transverse coordinate of the quark 
        remains the same and
        there is ordering in the direction of x^+ or x̅^+ which allows
        one to re-sum the multiple scatterings into a Wilson line. As a result we
        have x_1t = x_2t = ⋯ = x_it= z_t and 
        x̅_1t = x̅_2t = ⋯ = x̅_nt= z̅_t. Therefore 
        the full amplitude can be re-summed and written as
        
        i ℳ   =     (i g) ∫ d^4 x  d^2 z_t   d^2 z̅_t  ∫d^2 k_t  (2 π)^2 d^2 k̅_t  (2 π)^2  
        e^i (k̅ - k) x  
        e^- i (q̅_t - k̅_t)·z̅_t  
        e^- i (k_t - p_t)· z_t       (q̅) [ 
        V (x^+, z̅_t)    2 k̅^+  (x)  2 k^+   V (z_t, x^+)  
        ]  u(p)
        
        with k^+ = p^+, k^- = k_t^2  2 k^+ and 
        k̅^+ = q̅^+, k̅^- = k̅_t^2  2 k̅^+. 
        
        As before the scattering amplitude can be written in terms of the 
        interaction part of the propagator via
        
        i ℳ (p,q̅) =  (q̅) τ_F (p,q̅)  u (p)
        
        in terms of which the full (Feynman) propagator is 
        
        S_F (p,q̅) = (2 π)^4 δ^4 (p - q̅)  S_F^0 (p) +  
        S_F^0 (p) τ_hard (p,q̅)   S_F^0 (q̅)
        
        with interacting part of the propagator is then given by 
        (including the contribution where quark is moving backward 
        in x^+ which is needed for the Feynman propagator)
        τ_hard (p,q̅)    ≡     (i g) ∫ d^4 x ∫d^2 k_t  (2 π)^2 d^2 k̅_t  (2 π)^2  
        d^2 z_t   d^2 z̅_t   e^i (k̅ - k) x  
        e^- i (q̅_t - k̅_t)·z̅_t  
        e^- i (k_t - p_t)· z_t       {θ (p^+)  θ (q̅^+)   V (z_t, x^+)   2 k^+  (x)   2 k̅^+  V (x^+, z̅_t) -       θ (- p^+)  θ (- q̅^+)   
         V^† (z_t, x^+)   2 k^+  (x)   2 k̅^+  V^† (x^+, z̅_t) 
        }  .
         
        It is straightforward to check that this expression re-sums all the 
        diagrams shown in Fig. (<ref>) by expanding 
        the Wilson lines. The dashed lines correspond to scatterings 
        from the soft fields carrying small x fractions of the target 
        energy whereas the wavy line corresponds to scattering from a hard field 
        carrying arbitrary x fraction of the target energy. It describes 
        propagation of an energetic quark along longitudinal direction z^+ and 
        undergoing multiple soft interactions described by V (z_t, x^+) until 
        the location x^μ at which point it undergoes a potentially hard 
        (large x) scattering which changes it direction. After the hard 
        scattering it propagates again along the new longitudinal direction 
        z̅^+ and undergoing multiple scatterings described by 
        V (x^+, z̅_t). The location of the hard scattering 
        x^μ is integrated from - ∞ to + ∞ (or from - R to
        R where 2 R is the diameter of the nucleus/size of the medium) 
        so that the hard scattering can happen at any x^+, at the front, 
        back or middle of the nucleus/medium, followed or preceded by multiple 
        soft scatterings. 
        
        
        
        There is one more technical but essential point that we need to address; 
        the coupling 
        between the quark and gluon fields we have considered is of the form
        Ψ̅  Ψ which is not gauge invariant. Instead one
        needs to use the full covariant coupling Ψ̅  Ψ
        in the gauge invariant Lagrangian.
        This does not matter in the eikonal limit since replacing any of the 
        soft fields by the normal derivative gives a vanishingly small correction. 
        However in our more general case it can act on the Wilson line in the 
        rotated frame and give
        contributions which are of the same order as considered here. Therefore
        to have the interacting part of the effective propagator transform 
        covariantly one needs to include it. To do so is trivial and amounts 
        to replacing the hard field  in our step by step derivation by 
        the covariant derivative . Therefore our expression in eq. 
        (<ref>) is modified to
        τ_hard (p,q̅)    ≡   ∫ d^4 x ∫d^2 k_t  (2 π)^2 d^2 k̅_t  (2 π)^2  
        d^2 z_t   d^2 z̅_t   e^i (k̅ - k) x  
        e^- i (q̅_t - k̅_t)·z̅_t  
        e^- i (k_t - p_t)· z_t       {θ (p^+)  θ (q̅^+)   V (z_t, x^+)   2 k^+ _x   2 k̅^+  V (x^+, z̅_t) -       θ (- p^+)  θ (- q̅^+)   
         V^† (z_t, x^+)   2 k^+ _x   2 k̅^+  V^† (x^+, z̅_t) 
        }  .
            
        This is our final result for the interacting part of the quark propagator.
        This expression has the interesting property that it includes both 
        partially and fully coherent multiple scatterings of the quark 
        from the target and as such generalizes the fully coherent multiple 
        scattering in eikonal approximation. 
        
        It is very interesting to consider the soft limit of this
        expression, i.e. the limit when the all x field  carries 
        a small x fraction of the target energy. To do so we need to replace 
        (x^+, x^-, x_t)→  S(x^+, x_t) and take all 
        barred quantities to be un-barred, for example, 
        → and k̅→ k. It is easy then
        to show eq. (<ref>) vanishes in this limit due to action of
        the covariant derivative on a Wilson lines under a parallel transport
        along the direction of motion (n· D_x  V (x^+, x_t) = 0). 
        This property of eq. (<ref>) suggests that to have a quark 
        propagator which can be used for construction of physical quantities
        such as structure functions F_2 and F_L valid for any x one should 
        include both contributions from purely eikonal scattering as given by eq. 
        (<ref>) and hard scattering given by eq. (<ref>). 
        Therefore we take the most general interaction part of the effective 
        propagator to be
        τ_all   x≡τ_F + τ_hard  
        which is shown symbolically in Fig. (<ref>) 
        
        and where the thick solid lines denote a Wilson line representing multiple
        soft scatterings from the target to all orders.
        
        It should be noted that while the projectile quark scatters from both 
        low and high x modes of the target color fields, we have not included
        interactions between the different x modes. In this sense the small and
        large x fields are non-interacting with respect to each other. For 
        instance it should be possible for a soft field to couple to a hard field 
        as shown in Fig. (<ref>)
        
        
        
        
        A 
        preliminary study shows that the first coupling of 
        the soft field to the hard field forces the two fields to be at the same 
        position in coordinate space. Contributions of diagrams like this are 
        currently under investigation and will be reported elsewhere.
        
        
        
        
        § DISCUSSION AND SUMMARY
        
        
        We have constructed an effective quark propagator in the background
        of color fields which can carry both small and large x energy 
        fractions of the target from which the quark scatters. This generalizes 
        the known results for the quark propagator in the background of 
        a color field carrying small x fraction of the target energy based
        on eikonal approximation. 
        
        The constructed effective propagator can be useful in many ways; first,
        it includes the possibility that a projectile quark can scatter from the 
        more energetic (large x) partons of the target and get deflected by 
        a large angle (high p_t). This may already give significant contributions 
        to single inclusive particle production in high energy collisions <cit.> 
        when one uses the hybrid 
        formalism. It may also lead to significant final state momentum anisotropy
        at high p_t if one couples it to realistic nuclear geometries that 
        include fluctuations in the target nucleus/medium <cit.>.
        
        Furthermore, one can use this propagator to calculate the QCD structure 
        functions F_2 and F_L following the procedure suggested in <cit.>.
        This would generalize the standard dipole picture of DIS at small x by
        including the contributions of the large x gluons in the target. This work
        is in progress and will be reported elsewhere <cit.>.
        
        To proceed further, one can use this method to calculate the effective 
        gluon propagator in a similar fashion. One can then generalize the 
        hybrid formalism for particle production in asymmetric high energy 
        proton-proton and proton-nucleus collisions and extend the validity 
        of the hybrid formalism to high p_t, keeping in mind that high p_t in
        this context is equivalent to large x. Hence it would include 
        contributions of both fully and partially coherent multiple scatterings.
        
        It will be interesting to see if one can use our results to generalize 
        and extend the McLerran-Venugopalan (small x) effective 
        action <cit.> to include contributions of large x gluons. 
        If so one would then be able to apply the MV model to not only low <cit.>
        but also high p_t particle production in high energy heavy ion collisions. 
        This would allow us to 
        investigate and quantify the contributions of early time dynamics to 
        jet energy loss in Quark-Gluon Plasma formed in high energy heavy ion 
        collisions.
        
        Perhaps most significantly, we hope the physical quantities computed  
        using this effective propagator will serve as the Leading Order 
        expressions which can then be used for deriving Leading Log evolution 
        equations which would have DGLAP and JIMWLK evolution equations in 
        appropriate limits <cit.>.
             
        
        
        
        
        
        
        
        § ACKNOWLEDGMENTS
        
        We acknowledge support by the DOE Office of Nuclear Physics
        through Grant No. DE-FG02-09ER41620 and by the Idex Paris-Saclay 
        though a Jean d'Alembert grant. We would like to thank N. Armesto, 
        F. Gelis, E. Iancu, C. Lorcé, C. Marquet, A.H. Mueller, S. Munier, 
        B. Pire, C. Salgado, G. Soyez and B. Xiao for critical questions, 
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