The Color Dipole Picture of low-x DIS: Model-Independent and Model-Dependent Results
aa r X i v : . [ h e p - ph ] N ov The Color Dipole Picture of low-x DIS: Model-Independent andModel-Dependent ResultsMasaaki Kuroda ∗ Institute of Physics, Meijigakuin UniversityYokohama, Japan
Dieter Schildknecht † Fakult¨at f¨ur Physik, Universit¨at BielefeldD-33501 Bielefeld, GermanyandMax-Planck Institute f¨ur Physik (Werner-Heisenberg-Institut),F¨ohringer Ring 6, D-80805, M¨unchen, Germany
Abstract
We present a detailed examination of the color-dipole picture (CDP) oflow- x deep inelastic scattering. We discriminate model-independent results,not depending on a specific parameterization of the dipole cross section,from model-dependent ones. The model-independent results include the ra-tio of the longitudinal to the transverse photoabsorption cross section atlarge Q , or equivalently the ratio of the longitudinal to the unpolarizedproton structure function, F L ( x, Q ) = 0 . F ( x, Q ), as well as the low- x scaling behavior of the total photoabsorption cross section σ γ ∗ p ( W , Q ) = σ γ ∗ p ( η ( W , Q )) as log(1 /η ( W , Q )) for η ( W , Q ) <
1, and as 1 /η ( W , Q )for η ( W , Q ) ≫
1. Here, η ( W , Q ) denotes the low- x scaling variable, η ( W , Q ) = ( Q + m ) / Λ sat ( W ) with Λ sat ( W ) being the saturation scale.The model-independent analysis also implieslim W →∞ ,Q fixed σ γ ∗ p ( W , Q ) /σ γp ( W ) → Q for asymptotically largeenergy, W . Consistency with pQCD evolution determines the underlyinggluon distribution and the numerical value of C = 0 .
29 in the expression ∗ Email: [email protected] † Email: [email protected] ( W ) ∼ ( W ) C . In the model-dependent anal-ysis, by restricting the mass of the actively contributing q ¯ q fluctuations byan energy-dependent upper bound, we extend the validity of the color-dipolepicture to x ∼ = Q /W ≤ .
1. The theoretical results agree with the worlddata on DIS for 0 . ≤ Q ≤ . In terms of the (virtual) forward-Compton-scattering amplitude, deep in-elastic scattering (DIS) at low values of the Bjorken scaling variable, x ∼ = Q /W ≪
1, proceeds via forward scattering of massive (timelike) hadronicfluctuations of the photon, much like envisaged by generalized vector dom-inance [1, 2, 3] a long time ago. In QCD, the hadronic fluctuations maybe described as quark-antiquark states that interact with the nucleon in agauge-invariant manner as color-dipole states [5, 6], coupled to the gluon fieldin the nucleon via (at least) two gluons [7]. This is the color-dipole picture(CDP) of low-x DIS. Compare fig. 1. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) γ *p γ *p (qq)p(qq)p } Wqq
Figure 1: The fluctuation of the photon γ ∗ into a massive q ¯ q color-dipole stateand the interaction of the color dipole with the gluon field of the nucleon.A detailed representation of the experimental results on the photoabsorp-tion cross section requires an ansatz for the dipole cross section i.e. an ansatzfor the cross section for the scattering of the color-dipole state on the nu-cleon. Such an ansatz cannot be formulated entirely free from parameters,just as fit parameters are required for the related description of the DIS datain terms of the gluon distribution of the nucleon at low x.In the first part of the present work, we will show that nevertheless muchof the general features of the DIS experimental data [9] on the photoabsorp-tion cross section at low x can be derived in the CDP without a detailed Compare also ref. [4] for a recent review and further references. Compare e.g. ref. [8], Chapter 4, and the bibliography given there. q ¯ q interaction with the nucleon as the interaction of a color-dipolestate. The model-independent results include the ratio of the longitudinal tothe transverse photoabsorption cross section at low x and large Q [10], aswell as the empirically established low-x scaling,the dependence of the pho-toabsorption cross section on a single variable η ( W , Q ) i.e. σ γ ∗ p ( W , Q ) = σ γ ∗ p ( η ( W , Q )) [11]. The empirical dependence on η ( W , Q ), as 1 /η ( W , Q )for η ( W , Q ) ≫
1, and as ln(1 /η ( W , Q )) for η ( W , Q ) ≪
1, is a gen-eral feature of the dipole interaction. Here, η ( W , Q ) denotes the scal-ing variable, η ( W , Q ) ≡ ( Q + m ) / Λ sat ( W ) with m ≃ . GeV , andΛ sat ( W ) denotes the appropriately defined “saturation scale” which risesas a small fixed power, C of the square of the γ ∗ p center-of-mass energy,Λ ( W ) ∼ ( W ) C .A detailed model for the dipole cross section will be analyzed and com-pared with the world experimental data in Sections 3 to 5 of the presentpaper, and conclusions will be presented in Section 6. A model-independent prediction of the longitudinal-to-transverse ratio of thephotoabsorption cross section was recently presented [10]. Based on the gen-eral analysis of the transverse and the longitudinal photoabsorption crosssection in Sections 2.1 and 2.2, we will present a more detailed account ofthe underlying argument in Section 2.3. After a general discussion on theCDP in Section 2.4, we will deal with low-x scaling in Section 2.5 and derivethe functional dependence of the photoabsorption cross section on the scal-ing variable η ( W , Q ). In Section 2.6, we analyze the photoabsorption crosssection in the limit of W → ∞ at fixed values of Q >
0. The η ( W , Q )dependence implies that the photoabsorption cross section for W → ∞ atfixed Q > Q -independent limit that coincides with( Q = 0) photoproduction. In Section 2.7, we will show that the consistencyof the CDP with DGLAP evolution [12] for the sea quark distribution func-tion constrains the energy dependence of the saturation scale, Λ ( W ), andof the structure function F ( x ∼ = Q /W , Q ) for x < .
1. We will also elab-orate on the connection between the CDP and the extraction of the gluon3istribution of the proton. We compare the gluon distribution underlyingthe CDP with the gluon distributions that were extracted from the experi-mental data by directly employing the pQCD improved parton picture in theanalysis of the experimental data. Q , part I. The transverse-position-space representation [13] of the longitudinal and thetransverse photoabsorption cross section [5, 6] σ γ ∗ L,T ( W , Q ) = Z dz Z d ~r ⊥ | ψ L,T ( ~r ⊥ , z (1 − z ) , Q ) | σ ( q ¯ q ) p ( ~r ⊥ , z (1 − z ) , W )(2.1)summarizes in compact form the structure of the x ≈ Q /W ≤ . q ¯ q pair, originating from a γ ∗ L,T → q ¯ q transition, with the gluon fieldof the nucleon. The square of the “photon wave function” | ψ L,T ( ~r ⊥ , z (1 − z ) , Q ) | describes the probability for the occurrence of a q ¯ q fluctuationof transverse size, ~r ⊥ , of a longitudinally, γ ∗ L , or a transversely polarizedphoton, γ ∗ T , of virtuality Q . The variable z , with 0 ≤ z ≤
1, character-izes the distribution of the momenta between quark and antiquark. In therest frame of a q ¯ q fluctuation of mass M q ¯ q , the variable z determines [6] thedirection of the three-momentum of the quark with respect to the photon di-rection. The dipole cross section, related to the imaginary part of the ( q ¯ q ) p forward scattering amplitude, is denoted by σ ( q ¯ q ) p ( ~r ⊥ , z (1 − z ) , W ). For gen-erality, we include a potential dependence on the “ q ¯ q -configuration variable” z (1 − z ). The dipole cross section depends on the center-of-mass energy, W ,of the ( q ¯ q ) p scattering process [6, 11, 14, 15], since it is a timelike massive q ¯ q pair, the photon dissociates or fluctuates into . The interaction of amassive q ¯ q pair with the proton (the integration over d ~r ⊥ corresponding toan integration over fluctuation masses) depends on W and, in particular, isindependent of the photon-virtuality, Q . This point is inherently connectedwith the mass-dispersion relation [1, 2] of generalized vector dominance, andit was recently elaborated upon from first principles of quantum field theoryin ref. [15].The gauge invariance for the interaction of the q ¯ q color dipole with thecolor field in the nucleon requires a representation of the dipole cross section In this respect, we differ from ref. [5], where the dipole cross section is assumed todepend on x ∼ = Q /W . Compare also the discussion on this point in Section 2.4.
4f the form [5, 6] σ ( q ¯ q ) p ( ~r ⊥ , z (1 − z ) , W ) = Z d ~l ⊥ ˜ σ ( ~l ⊥ , z (1 − z ) , W ) (cid:16) − e − i ~l ⊥ · ~r ⊥ (cid:17) , (2.2)where the transverse momentum of the gluon absorbed by the dipole stateis denoted by ~l ⊥ . In the important limit of a small-size dipole, ~r ⊥ →
0, from(2.2) we have σ ( q ¯ q ) p ( ~r ⊥ , z (1 − z ) , W ) = π ~r ⊥ Z d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , z (1 − z ) , W ) . (2.3)A dipole of vanishing transverse size must obviously have a vanishing crosssection (“color-transparency”) as in (2.3), when interacting with the gluonfield. The validity of the approximation (2.3) requires ~r ⊥ ~l ⊥ < ~r ⊥ ~l ⊥ Max ( W ) < , (2.4)where ~l ⊥ Max ( W ) characterizes the W-dependent domain of ~l ⊥ < ~l ⊥ Max ( W )in which ˜ σ ( ~l ⊥ , z (1 − z ) , W ), at a given energy W , by assumption is appre-ciably different from zero. For the subsequent discussion, it will be useful tointroduce the variables ~r ′⊥ = q z (1 − z ) ~r ⊥ and ~l ′⊥ = ~l ⊥ / q z (1 − z ) [16]. Interms of these variables the restriction (2.4) becomes ~r ′ ⊥ ~l ′ ⊥ < ~r ′ ⊥ ~l ′ ⊥ Max ( W ) < . (2.5)The validity of (2.3) to (2.5) is an integral part of the CDP. The ab-sorption of a gluon of transverse momentum squared ~l ⊥ < ~l ⊥ Max ( W ) bya q ¯ q fluctuation (unless the absorbed gluon is re-emitted by the absorbingquark) increases the mass of the q ¯ q fluctuation. At any given squared energy, W , the contributing q ¯ q masses, and consequently the values of ~l ′ ⊥ activelycontributing to the cross section, must be bounded by an upper limit, sinceonly fluctuations of sufficiently long lifetime do contribute to the Comptonforward-scattering amplitude of the CDP.Color transparency (2.3) determines the photoabsorption cross section(2.1) for sufficiently large Q . This will be elaborated upon next.We will consider massless quarks. Inserting the explicit representationof the photon wave function in (2.1), we find the well-known expression The well-known expression for the lifetime of a hadronic fluctuation is given in (2.60)below. Q ≡ √ Q ) [5] σ γ ∗ L,T p ( W , Q ) = 3 α π X q Q q Q · (2.6) · R d ~r ⊥ R dzz (1 − z ) · K ( r ⊥ q z (1 − z ) Q ) σ ( q ¯ q ) p ( r ⊥ , z (1 − z ) , W ) , R d ~r ⊥ R dz (1 − z (1 − z )) z (1 − z ) · K ( r ⊥ q z (1 − z ) Q ) σ ( q ¯ q ) p ( r ⊥ , z (1 − z ) , W ) . Here, r ⊥ ≡ | ~r ⊥ | , and K , ( r ⊥ q z (1 − z ) Q ) denotes modified Bessel functions.A compact and direct way of deriving the large- Q behavior of the crosssections in (2.6) makes use of the strong fall-off of the modified Bessel func-tions at large values of their argument, K , ( y ) ∼ π y e − y , ( y ≫ . (2.7)The integral over R d ~r ⊥ = π R d~r ⊥ in (2.6) is accordingly dominated by r ′⊥ Q ≡ r ⊥ q z (1 − z ) Q < . (2.8)As soon as ~r ′ ⊥ > /Q , the integrand in (2.6) yields negligible contributions.The interval for r ′⊥ defined by the condition (2.8) is contained in the interval(2.5), where color transparency is valid, provided Q is sufficiently large, suchthat ~r ′ ⊥ < Q < ~l ′ ⊥ Max ( W ) , (2.9)or Q > ~l ′ ⊥ Max ( W ) . (2.10)Under this constraint, the photoabsorption cross section (2.6) can be evalu-ated by inserting the ~r ⊥ → σ γ ∗ L,T p ( W , Q ) = 3 α X q Q q Q · (2.11) · R d~r ⊥ ~r ⊥ R dzz (1 − z ) · K ( r ⊥ q z (1 − z ) Q ) R d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , z (1 − z ) , W ) , R d~r ⊥ ~r ⊥ R dz (1 − z (1 − z )) z (1 − z ) ·· K ( r ⊥ q z (1 − z ) Q ) R d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , z (1 − z ) , W ) . In terms of the variable ~r ′⊥ from (2.8) the photoabsorption cross section (2.11)is given by σ γ ∗ L,T p ( W , Q ) = 3 α X q Q q Q ·· ( R dz R d~r ′ ⊥ ~r ′ ⊥ K ( r ′⊥ Q ) R d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , z (1 − z ) , W ) , R dz − z (1 − z ) z (1 − z ) R d~r ′ ⊥ ~r ′ ⊥ K ( r ′⊥ Q ) R d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , z (1 − z ) , W ) . (2.12)6aking use of the mathematical identities [17], Z ∞ dyy K ( y ) = , Z ∞ dyy K ( y ) = , (2.13)the photoabsorption cross section (2.12), valid for Q > ~l ′ ⊥ Max ( W ) from(2.10) (and x ∼ = Q /W ≪ σ γ ∗ L,T p ( W , Q ) = α X q Q q Q ( R dz R d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , z (1 − z ) , W ) , R dz
14 1 − z (1 − z ) z (1 − z ) R d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , z (1 − z ) , W ) . (2.14)According to our derivation, the large- Q result (2.14) is a consequence ofthe transverse-position-space representation (2.1) combined with color trans-parency (2.3) that in turn rests on decent behavior of ˜ σ ( ~l ⊥ , z (1 − z ) , W ) ascharacterized by ~l ′ ⊥ Max ( W ).For the ensuing discussion, it will be useful to represent the contributionof the dipole cross section to the transverse cross section in (2.14) in termsof the contribution to the longitudinal one by introducing the factor ρ W , Z dz
14 1 − z (1 − z ) z (1 − z ) Z d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , z (1 − z ) , W )= ρ W Z dz Z d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , z (1 − z ) , W ) . (2.15)The cross section (2.14) then becomes, σ γ ∗ L,T ( W , Q ) = α X Q q Q Z dz Z d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , z (1 − z ) , W ) (cid:26) , ρ W , (2.16)and the longitudinal-to-transverse ratio, R ( W , Q ), at large Q is given by R ( W , Q ) ≡ σ γ ∗ L p ( W , Q ) σ γ ∗ T p ( W , Q ) = 12 ρ W . (2.17)In (2.15) to (2.17), the index W indicates a potential dependence of ρ W onthe energy W . Actually, we will find that ρ W is a W -independent constant,see Section 2.3. The factor 1 / q ¯ q pairs relative to longitudinal photons,compare (2.13). The additional factor of 1 /ρ W is associated with differentinteractions of q ¯ q fluctuations originating from transverse, γ ∗ T → q ¯ q , andlongitudinal, γ ∗ L → q ¯ q , photons, respectively.7y comparing the representation of the cross section in (2.16) with theone in (2.11), taking into account the ~r ⊥ → σ ( q ¯ q ) p ( ~r ⊥ , z (1 − z ) , W ) → σ ( q ¯ q ) p ( ρ W ~r ⊥ , z (1 − z ) , W ) (2.18)into the longitudinal cross section in (2.11) in conjunction with K ( r ⊥ q z (1 − z ) Q ) → K ( r ⊥ q z (1 − z ) Q ) (2.19)reproduces (2.16), which relates the transverse photoabsorption cross sectionto the longitudinal one, σ γ ∗ L p ( W , Q ) → σ γ ∗ T p ( W , Q ) . (2.20)We thus have arrived at the conclusion that q ¯ q states originating from trans-versely polarized photons, γ ∗ T → q ¯ q , interact with enhanced transverse size, ~r ⊥ → ρ W ~r ⊥ , (2.21)relative to q ¯ q states stemming from γ ∗ L → q ¯ q transitions. Based on theinterpretation of ρ W in (2.21), in Section 2.3, we will show that the absolutemagnitude of ρ W is uniquely determined as ρ W = 4 / σ ( ~l ⊥ , z (1 − z ) , W ), does not depend on the configuration of the q ¯ q state, z (1 − z ). According to (2.14), strict independence of ˜ σ ( ~l ⊥ , z (1 − z ) , W ) from z (1 − z ) implies a logarithmic divergence in the transverse photoabsorptioncross section. The divergence is avoided by a restriction on 0 ≤ z (1 − z ) < given by z (1 − z ) > ǫ. (2.22)This restriction corresponds to adopting an ansatz for ˜ σ ( ~l ⊥ , z (1 − z ) , W ) ofthe form ˜ σ ( ~l ⊥ , z (1 − z ) , W ) → ˜ σ ( ~l ⊥ , W ) θ ( z (1 − z ) − ǫ ) , (2.23)as a “minimal” dependence of the dipole cross section on z (1 − z ). The factorization of the z (1 − z ) dependence in (2.23) strictly speaking amounts toan assumption that does not necessarily follow from (2.14). Finiteness of (2.14) can alsobe achieved by an appropriate correlation of the z (1 − z ) and ~l ⊥ dependences not of theform (2.23). Compare e.g. the specific model (3.3) below.The ansatz (2.23) is explicitlyrealized by (3.17) σ γ ∗ L,T p ( W , Q ) = α X q Q q Q ( R z (1 − z ) >ǫ dz R d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , W ) , R z (1 − z ) >ǫ dz
14 1 − z (1 − z ) z (1 − z ) R d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , W ) . (2.24)It may be rewritten as σ γ ∗ L,T p ( W , Q ) = α X q Q q Q √ − ǫ Z d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , W ) (cid:26) , ρ ( ǫ ) , (2.25)i.e. ρ W in (2.15) becomes ρ W = ρ ( ǫ ) = R z (1 − z ) >ǫ dz − z (1 − z ) z (1 − z ) R z (1 − z ) >ǫ dz = 14 √ − ǫ Z z (1 − z ) >ǫ dz − z (1 − z ) z (1 − z ) . (2.26)Explicitly, one finds ρ ( ǫ ) = 12 √ − ǫ ln (1 + √ − ǫ ) ǫ − √ − ǫ ! ≃
12 ln 1 ǫ . (2.27)We note that in Section 3 we will introduce the parameter a , related to ǫ by ǫ = 1 / a . The ratio R of the longitudinal to the transverse photoabsorptioncross section from (2.17) according to (2.25) is given by 1 / ρ ( ǫ ), R ≡ σ γ ∗ L p ( W , Q ) σ γ ∗ T p ( W , Q ) = 12 ρ ( ǫ ) . (2.28)The ratio R in (2.28) is independent of a particular parameterization of the ~l ⊥ dependence of the dipole cross section, that is for ˜ σ ( ~l ⊥ , W ) in (2.23).With respect to subsequent discussions in Sections 2.2 and 2.3, we notethe origin of the z (1 − z )-dependent factors in (2.14) and (2.24) from thecoupling of the q ¯ q states to the electromagnetic current. The electromagneticcurrent determining the γ ∗ ( q ¯ q ) coupling of a timelike photon of mass squared M q ¯ q = ~k ⊥ /z (1 − z ) is given by [6] X λ = − λ ′ = ± | j λ,λ ′ L | = 8 M q ¯ q z (1 − z ) = 8 ~k ⊥ , (2.29)and X λ = − λ ′ = ± | j λ,λ ′ T (+) | = X λ = − λ ′ = ± | j λ,λ ′ T ( − ) | = 2 M q ¯ q (1 − z (1 − z )) == 2 ~k ⊥ (1 − z (1 − z )) z (1 − z ) , (2.30) Here, with ǫ = const . , we exclude the more general case of ǫ = ǫ ( ~l ⊥ ). γ ∗ L , and a transverse one, γ ∗ T , respectively. Com-parison of (2.29) and (2.30) with (2.14) and (2.15) reveals that the size en-hancement ρ W is related to the difference of the longitudinal and transversephoton couplings of dipole states carrying the transverse momentum ~l ⊥ ofthe absorbed gluon. At large Q , the interaction of the photon according to(2.14) reduces to interactions of fluctuations into q ¯ q dipole states carryinga quark transverse momentum identical to the transverse momentum of theabsorbed gluon, ~l ⊥ .According to (2.29) and (2.30), the normalized z (1 − z ) distributions f L,T ( z (1 − z )) of a q ¯ q pair of fixed mass M q ¯ q originating from a longitudinallyand a transversely polarized photon are given by [10] f L ( z (1 − z )) = 6 z (1 − z ) , (2.31)and f T ( z (1 − z )) = 32 (1 − z (1 − z )) , (2.32)respectively.We end the present Section by stressing the simplicity of the physicalpicture underlying the photoabsorption in DIS at low x and sufficiently large Q . The photon fluctuates into a q ¯ q dipole state. The γ ∗ L,T ( q ¯ q ) transitionstrength is determined by the electromagnetic current in (2.29) and (2.30).The q ¯ q dipole state entering (2.14) and (2.24) carries a quark (antiquark)transverse momentum equal to the transverse momentum of the absorbedgluon, ~l ⊥ . Summation over all fluctuations, the weight function ˜ σ ( ~l ⊥ , W )being characteristic for the transverse momentum distribution of the gluonsin the nucleon, upon multiplication by 1 /Q , determines the photoabsorptioncross section. The representations, (2.14) and (2.24), accordingly, explicitlydemonstrate that the q ¯ q fluctuations directly test the gluon distribution inthe nucleon that is characterized by ˜ σ ( ~l ⊥ , W ). The enhanced transversephotoabsorption cross section, due to 2 ρ W in (2.16) and to 2 ρ ( ǫ ) in (2.25),results from the enhanced transition of transverse photons into q ¯ q pairs,compare (2.13) and (2.14), in conjunction with a ( q ¯ q ) p interaction of the q ¯ q pairs from transverse photons with enhanced transverse size, compare (2.18)and (2.21). 10 .2 The photoabsorption cross section at large Q ,partII, ( q ¯ q ) J =1 L,T states.
In this Section, we will represent the photoabsorption cross section in termsof scattering cross sections for dipole states ( q ¯ q ) J =1 L,T with definite spin J = 1,and longitudinal as well as transverse polarization, L and T , respectively.Upon introducing ~r ′⊥ = ~r ⊥ q z (1 − z ) from (2.8), the photoabsorptioncross section (2.6) becomes [16] σ γ ∗ L,T p ( W , Q ) = 3 α π X q Q q Q · · (2.33) · R d r ′⊥ K ( r ′⊥ Q ) R dz z (1 − z ) σ ( q ¯ q ) p (cid:18) r ′⊥ √ z (1 − z ) , z (1 − z ) , W (cid:19) , R d r ′⊥ K ( r ′⊥ Q ) R dz ( z + (1 − z ) ) σ ( q ¯ q ) p (cid:18) r ′⊥ √ z (1 − z ) , z (1 − z ) , W (cid:19) . The cross section in (2.33) is written in such a manner that the appearanceof the rotation functions, d jj ′ ( z ), is explicitly displayed, i.e. σ γ ∗ L,T p ( W , Q ) = 3 α π X q Q q Q · · (2.34) · R d r ′⊥ K ( r ′⊥ Q ) R dz ( d ( z )) · σ ( q ¯ q ) p (cid:18) r ′⊥ √ z (1 − z ) , z (1 − z ) , W (cid:19) , R d r ′⊥ K ( r ′⊥ Q ) R dz (cid:16) ( d − ( z )) + ( d ( z )) (cid:17) ·· σ ( q ¯ q ) p (cid:18) r ′⊥ √ z (1 − z ) , z (1 − z ) , W (cid:19) . The rotation funtions originate from the γ ∗ ( q ¯ q ) couplings via the electromag-netic currents in (2.29) and (2.30), rewritten as X λ = − λ = ± | j λ,λ ′ L | = 4 M q ¯ q (cid:16) d ( z ) (cid:17) , (2.35)and X λ = − λ ′ = ± | j λ,λ ′ T (+) | = X λ = − λ = ± | j λ,λ ′ T ( − ) | = 4 M q ¯ q (cid:16) ( d − ( z )) + ( d ( z )) (cid:17) . (2.36)Integration over dz in (2.35) and (2.36) defines the total longitudinal andtransverse transition strengths for the γ ∗ L ( q ¯ q ) and γ ∗ T ( q ¯ q ) transitions. Re-quiring factorization of these transition strengths in (2.34), we represent σ γ ∗ L,T p ( W , Q ) in terms of the so-defined cross sections for scattering of( q ¯ q ) J =1 L,T states on the proton, σ ( q ¯ q ) J =1 L,T p ( r ′⊥ , W ), σ γ ∗ L,T p ( W , Q ) = 3 α π X q Q q Q · ( R d r ′⊥ K ( r ′⊥ Q ) R dz ( d ( z )) σ ( q ¯ q ) J =1 L p ( r ′⊥ , W ) , R d r ′⊥ K ( r ′⊥ Q ) R dz (cid:16) ( d − ( z )) + ( d ( z )) (cid:17) σ ( q ¯ q ) J =1 T p ( r ′⊥ , W ) . (2.37)Upon inserting the normalizations Z dz ( d ( z )) = Z dz ( d − ( z )) = Z dz ( d ( z )) = 13 , (2.38)(2.37) becomes σ γ ∗ L,T p ( W , Q ) = απ X q Q q Q Z dr ′ ⊥ K , ( r ′⊥ Q ) σ ( q ¯ q ) J =1 L,T p ( r ′⊥ , W ) . (2.39)By comparing (2.39) with (2.34), we find that the J = 1 dipole cross-sectionsintroduced in (2.37) are explicitly given by σ ( q ¯ q ) J =1 L,T p ( r ′⊥ , W ) = (2.40)= 3 · R dz ( d ( z )) σ ( q ¯ q ) p (cid:18) r ′⊥ √ z (1 − z ) , z (1 − z ) , W (cid:19) , R dz (cid:16) ( d − ( z )) + ( d ( z )) (cid:17) σ ( q ¯ q ) (cid:18) r ′⊥ √ z (1 − z ) , z (1 − z ) , W (cid:19) . We add the comment at this point that the ( q ¯ q ) J =1 L,T p cross sections in(2.37) to (2.40) may be identified as the J = 1 parts of the partial-waveexpansions d ( z ) σ ( q ¯ q ) p r ′⊥ q z (1 − z ) , z (1 − z ) , W == d ( z ) σ ( q ¯ q ) J =1 L p ( r ′⊥ , W ) + d ( z ) σ ( q ¯ q ) J =1 L ( r ′⊥ , W ) + ... (2.41)and d − ( z ) σ ( q ¯ q ) p r ′⊥ q z (1 − z ) , z (1 − z ) , W == d − ( z ) σ ( q ¯ q ) J =1 − p ( r ′⊥ , W ) + d − ( z ) σ ( q ¯ q ) J =2 − ( r ′⊥ , W ) + ... (2.42)as well as d ( z ) σ ( q ¯ q ) p r ′⊥ q z (1 − z ) , z (1 − z ) , W == d ( z ) σ ( q ¯ q ) J =1+1 p ( r ′⊥ , W ) + d ( z ) σ ( q ¯ q ) J =2+1 ( r ′⊥ , W ) + ... (2.43)These partial wave expansions explicitly demonstrate that the cross section(2.40) introduced by the factorization requirement in (2.37) and (2.39) standfor the cross sections for the scattering of ( q ¯ q ) J =1 L,T states on the proton.12IS at low x ≤ Q /W ≪ Q is recognized aselastic diffractive forward scattering of ( q ¯ q ) J =1 L,T fluctuations of the photon onthe proton, compare (2.39).We return to the representation of the dipole cross section (2.2) whichcontains color transparency. Applying the projection (2.40) to representation(2.2), we obtain σ ( q ¯ q ) J =1 L,T p ( ~r ′⊥ , W ) = Z d ~l ′⊥ ¯ σ ( q ¯ q ) J =1 L,T p ( ~l ′ ⊥ , W )(1 − e − i~l ′⊥ · ~r ′⊥ ) . (2.44)The relation between ˜ σ ( ~l ′ ⊥ z (1 − z ) , z (1 − z ) , W ) in (2.2), and ¯ σ ( q ¯ q ) J =1 L,T p ( ~l ′ ⊥ , W )in (2.44), is analogous to (2.40), i.e.¯ σ ( q ¯ q ) J =1 L,T p ( ~l ′ ⊥ , W ) = (2.45)= 3 · ( R dz ( d ( z )) z (1 − z )˜ σ ( ~l ′ ⊥ z (1 − z ) , z (1 − z ) , W ) , R dz (cid:16) ( d − ( z )) + ( d ( z )) (cid:17) z (1 − z )˜ σ ( ~l ′ ⊥ z (1 − z ) , z (1 − z ) , W ) . Expanding (2.44) for ~r ′ ⊥ →
0, in analogy to (2.3), we have σ ( q ¯ q ) J =1 L,T p ( ~r ′ ⊥ , W ) = 14 π~r ′ ⊥ Z d~l ′ ⊥ ~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L,T p ( ~l ′ ⊥ , W ) , ( ~l ′ ⊥ Max ( W ) ~r ′ ⊥ ≪ . (2.46)Substituting (2.46) into (2.39) and integrating over d~r ′ ⊥ with the help of(2.13), we find the large- Q representation σ γ ∗ L,T p ( W , Q ) = α X q Q q Q ( R d~l ′ ⊥ ~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) , R dl ′ ⊥ ~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 T p ( ~l ′ ⊥ , W ) (2.47)in terms of the ( q ¯ q ) J =1 L,T p cross sections, ¯ σ ( q ¯ q ) J =1 L,T p ( ~l ′ ⊥ , W ). The representation(2.47) is also obtained directly from (2.14) by introducing ~l ′ ⊥ and inserting(2.45).The ratio of the integrals over the transverse and the longitudinal ( q ¯ q ) J =1 p cross sections in (2.47) must be identical to the factor ρ W already introducedin (2.15), Z d~l ′ ⊥ ~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 T p ( ~l ′ ⊥ , W ) = ρ W Z d~l ′ ⊥ ~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) . (2.48)According to the proportionality (2.48), the dipole cross sections for trans-versely and longitudinally polarized dipole states in (2.46) become related toeach other via¯ σ ( q ¯ q ) J =1 T p ( ~r ′ ⊥ , W ) = 14 πρ W ~r ′ ⊥ Z d~l ′ ⊥ ~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W )= ¯ σ ( q ¯ q ) J =1 L p ( ρ W ~r ′ ⊥ , W ) , ( ~l ′ ( W ) ~r ′ ⊥ ≪ . (2.49)13ccording to (2.49), for ~r ′ sufficiently small, the cross section for trans-versely polarized ( q ¯ q ) J =1 states on the proton, ¯ σ ( q ¯ q ) J =1 T p ( ~r ′ ⊥ , W ), is obtainedfrom the cross section for longitudinally polarized ( q ¯ q ) J =1 states, ¯ σ ( q ¯ q ) J =1 L p ( ~r ′ ⊥ ,W ), by performing the substitution of ~r ′ ⊥ by ρ W ~r ′ , ~r ′ ⊥ → ρ W ~r ′ ⊥ (2.50)in ¯ σ ( q ¯ q ) J =1 L p ( ~r ′ ⊥ , W ).Upon inserting the proportionality (2.48), the large- Q photoabsorptioncross section (2.47) becomes σ γ ∗ L,T p ( W , Q ) = α X q Q q Q Z d~l ′ ⊥ ~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) (cid:26) ρ W . (2.51)It is tempting to generalize the substitution law (2.50), ~r ′⊥ → √ ρ W ~r ′⊥ ,from its validity for ~r ′ ⊥ → ~r ′⊥ by rewriting (2.44) as σ ( q ¯ q ) J =1 L,T p ( ~r ′ ⊥ , W ) = Z d ~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) ( (1 − e − i~l ′⊥ · ~r ′⊥ ) , (1 − e − i~l ′⊥ · ( √ ρ W ~r ′⊥ ) ) . (2.52)The representation (2.52), in the limit of ~r ′ ⊥ → ∞ implies a helicity-indepen-dent color-dipole cross section that is given by σ ( q ¯ q ) J =1 L,T p ( ~r ′ ⊥ → ∞ , W ) = π Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L ( ~l ′ ⊥ , W ) ≡ σ ( ∞ ) ( W ) . (2.53)The representation (2.52), accordingly, contains the dynamical assumption(2.53). In this respect, (2.53) differs from the representations (2.2) and (2.44)which are based on the gauge invariance of the color-dipole interaction byitself. We will come back to (2.53) in Section 2.5. R ≡ σ γ ∗ L p ( W , Q ) /σ γ ∗ T p ( W , Q ) . The ratio of the longitudinal to the transverse photoabsorption cross sectionat sufficiently large Q , according to (2.16) and (2.51), is determined by theproportionality factor 1 / ρ W . The factor 1 / ~r ′⊥ dependence of the photon wave functions, compare (2.13), for lon-gitudinally and transversely polarized photons. The factor 1 /ρ W , accordingto (2.48) and (2.49), is associated with the enhancement of the transversedipole-proton cross section relative to the longitudinal one in the limit of ~l ′ ⊥ Max ( W ) ~r ′ ⊥ ≪
1. According to (2.21), ρ W is identical to the factor thatis responsible for the enhancement of the size, ~r ⊥ , of q ¯ q states originating14rom γ ∗ T → q ¯ q transitions, relative to the size of q ¯ q states from γ ∗ L → q ¯ q transitions.The enhancement of the transverse relative to the longitudinal q ¯ q -dipole-proton cross section is recognized as a consequence of the enhanced transversesize of transversely relative to longitudinally polarized dipole states. Lon-gitudinally and transversely polarized ( q ¯ q ) J =1 states, ( q ¯ q ) J =1 L and ( q ¯ q ) J =1 T ,determining the cross sections in (2.47), differ in the transverse-momentumdistribution of the quark (antiquark). According to (2.29) to (2.32), as a con-sequence of the γ ∗ L,T → ( q ¯ q ) J =1 L,T transitions, the average value of the squareof the transverse momentum, ~l ⊥ = z (1 − z ) ~l ′ ⊥ , of a quark (antiquark) in the( q ¯ q ) J =1 L,T state is given by h ~l ⊥ i ~l ′ ⊥ = constL,T = ~l ′ ⊥ ( R dzz (1 − z ) = ~l ′ ⊥ , R dz z (1 − z )(1 − z (1 − z )) = ~l ′ ⊥ . (2.54)The q ¯ q states of fixed mass ~l ′ ⊥ from longitudinal photons predominantlyoriginate with z (1 − z ) = 0, in contrast to the q ¯ q states from transversephotons which originate predominantly from z (1 − z ) ∼ = 0, compare (2.29)and (2.30). The average transverse momentum for a ( q ¯ q ) J =1 L state originatingfrom the γ ∗ L → ( q ¯ q ) J =1 L transition, according to (2.54), is enhanced by thefactor 4 / , h ~l ⊥ i ~l ′ ⊥ =const( q ¯ q ) J =1 L = 43 h ~l ⊥ i ~l ′ ⊥ =const( q ¯ q ) J =1 T . (2.55)Longitudinally polarized photons produce ( q ¯ q ) J =1 pairs with (relatively)“large” internal quark transverse momentum, while transversely polarizedphotons lead to ( q ¯ q ) J =1 states of “small” internal quark transverse momen-tum.By invoking the uncertainty principle, ( q ¯ q ) J =1 L states originating from lon-gitudinally polarized photons accordingly have “small” transverse size, while( q ¯ q ) J =1 T states from transversely polarized photons have relatively “large”transverse size. The enhancement factor, when passing from “small-size”longitudinally polarized ( q ¯ q ) J =1 L states to “large-size” transversly polarized( q ¯ q ) J =1 T states, from (2.55) is accordingly given by 4 / ρ W in ~r ′ ⊥ → ρ W ~r ′ ⊥ in (2.50) and (2.21) is equal to 4 /
3, [10] ρ W ≡ ρ = 43 . (2.56) The left-hand and right-hand sides in (2.55) belong to the same value of ~l ′ ⊥ = const. ,but the ratio, 4 /
3, is independent of the specific value chosen for ~l ′ ⊥ . Note that by comparing (2.16) and (2.47), one finds R dz R d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , z (1 − z ) , W ) = R d~l ′ ⊥ ~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ). The right-hand side in the longitudinal photoabsorption ρ W = ρ is independent of the energy, W , since the Lorentz boostfrom e.g. the ( q ¯ q ) J =1 rest frame to the γ ∗ p frame does not affect the ratio ofthe transverse momenta, ~l ⊥ , in the ( q ¯ q ) J =1 T and the ( q ¯ q ) J =1 L state.The ratio R for sufficiently large Q is given by R = σ γ ∗ L p ( W , Q ) σ γ ∗ T p ( W , Q ) = 12 ρ = (cid:26) . ρ = 1 , = 0 .
375 for ρ = . (2.57)In (2.57), for comparison, in addition to the case of transverse-size enhance-ment of ρ = 4 /
3, we have also indicated the case of ρ = 1 obtained fromhelicity independence, i.e. by replacing the transverse-size enhancement bythe simplifying ad hoc assumption of equality of the ( q ¯ q ) J =1 p cross sectionsfor longitudinal and transverse ( q ¯ q ) J =1 states. The transverse-size enhance-ment is responsible for the deviation of R from R = 0 . ρ ( ǫ ) = 4 /
3, one finds ǫ ∼ = 0 . . (2.58)Our examination of the longitudinal-to-transverse ratio R at large Q maybe summarized as follows. The ratio is first of all determined by a factor1 /
2, originating from the ratio of the probabilities to find a q ¯ q with sizeparameter squared, ~r ′ ⊥ = ~r ⊥ z (1 − z ), in a longitudinally and a transverselypolarized photon; compare (2.12) to (2.14), and (2.47). The second factor,1 /ρ in (2.57), results from the different dependence on the configurationvariable z (1 − z ) of q ¯ q states from longitudinally and transversely polarizedphotons implying interactions of ( q ¯ q ) states with different average transversemomenta squared, ~l ⊥ , of the quark (antiquark) in the ( q ¯ q ) J =1 L,T states, compare(2.55). Invoking the uncertainty relation with respect to the scattering ofthese ( q ¯ q ) J =1 L,T states on the proton, one arrives at the fixed value of ρ = 4 / R in (2.57).In terms of the proton structure functions, F L ( x, Q ) and F ( x, Q ), theresult (2.57) for R at large Q becomes F L ( x, Q ) = 11 + 2 ρ F ( x, Q ) = (cid:26) . F ( x, Q ) , ( ρ = 1) , . F ( x, Q ) , ( ρ = ) . (2.59) cross section (2.12) may be rewritten as σ γ ∗ L p ( W , Q ) = 3 α X q Q q · Q Z d~r ′ ⊥ ~r ′ ⊥ K ( r ′⊥ Q ) 16 Z d~l ′ ⊥ ~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L ( ~l ′ ⊥ , W ) , thus explicitly connecting the dipole size ~r ′⊥ with the ( q ¯ q ) J =1 L state of fixed mass ~l ′ ⊥ , asrequired for the above argument. F L = 0 . F is consistent with the experimentalresults from the H1 and ZEUS collaboration. Compare figs. 2 and 3. -0.500.511.5 H1 (Prelim.) H1 PDF 2000Alekhin NNLO (H1 PDF 2000) F × = 460, 575, 920 GeV p E . . . . . . . . . . . . . . . . . x L H1 Preliminary F m e d i u m & h i gh Q ) ( x , Q L F / GeV Q -0.500.511.5 10 Figure 2:
The H1 experimental results for the longitudinal proton structure function, F L ( x, Q ), compared with the prediction of F L ( x, Q ) = 0 . × F ( x, Q ) from transverse-size enhancement of transversely relative to longitudinally polarized ( q ¯ q ) J =1 states. ZEUS -3 & F L F = 24 GeV Q -3 & F L F -3 & F L F = 45 GeV Q -3 & F L F x -3 -2 & F L F = 80 GeV Q x -3 -2 & F L F -3 = 32 GeV Q -3 -3 = 60 GeV Q -3 x -3 -2 F L FZEUS-JETS ZEUS-JETS = 110 GeV Q x -3 -2 Figure 3:
The ZEUS experimental results for the longitudinal proton structure function, F L ( x, Q ), compared with the prediction of F L ( x, Q ) = 0 . × F ( x, Q ) from transverse-size enhancement of transversely relative to longitudinally polarized ( q ¯ q ) J =1 states. .4 Discussion on the Representations of the CDP inSections 2.1 and 2.2 . The CDP of DIS at low x is based on a life-time argument concerningmassive hadronic fluctuations of the photon. The argument is identical tothe one put forward in the space-time interpretation [2, 18] of generalizedvector dominance in the early 1970ies. The life-time in the rest frame ofthe nucleon of a hadronic fluctuation of mass M q ¯ q , given by the covariantexpression [19] 1∆ E = 1 x + M q ¯ q W M p ≫ M p , (2.60)becomes large in comparison with the inverse of the proton mass, M p , pro-vided x ∼ = Q /W ≪ W , is sufficiently large. The γ ∗ p interaction with the nucleon at low x, accordingly, proceeds via the interac-tion of hadronic q ¯ q fluctuations of timelike four-momentum squared identi-cal to M q ¯ q ) . More definitely, the integration over the dipole cross section σ ( q ¯ q ) p ( r ⊥ , z (1 − z ) , W ) in transverse position space in (2.1) describes theinteraction of a continuum of massive q ¯ q states. The dipole cross sectiondepends on W , just as any other purely hadronic interaction cross section.In particular, the dipole cross section does not depend on the virtuality, Q ,of the photon, and consequently, it does not depend on x .The dipole cross section in (2.1) does not refer to a definite spin J ofthe massive q ¯ q continuum states. The interaction with the nucleon, never-theless, proceeds via the spin J = 1 projection of the dipole cross section σ ( q ¯ q ) p ( r ⊥ , z (1 − z ) , W ), compare the discussion in Section 2.2, in particularthe relations (2.39) and (2.40).The W dependence of the dipole cross section explicitly, via ˜ σ ( ~l ⊥ , z (1 − z ) , W ) in (2.16) and ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) in (2.51) with (2.56), enters the large- Q approximation of the photoabsorption cross section. Inserting (2.16) and(2.51) into the proton structure function, F ( x, Q ) = Q (1 − x )4 π α ( Q + (2 M p x ) ) (cid:16) σ γ ∗ L p ( W , Q ) + σ γ ∗ T p ( W , Q ) (cid:17) ∼ = ∼ = Q π α (cid:16) σ γ ∗ L p ( W , Q ) + σ γ ∗ T p ( W , Q ) (cid:17) (2.61) Compare also ref. [15]. F ( x, Q ) = P q Q q π Z dz Z d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , z (1 − z ) , W )(1 + 2 ρ ) , (2.62)and F ( x, Q ) = P q Q q π Z d~l ′ ⊥ ~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) (1 + 2 ρ ) , (2.63)where ρ for Q sufficiently large is given in (2.56). According to the right-hand sides in (2.62) and (2.63), the structure function only depends on W = Q /x in the color-transparency region of sufficiently large Q and sufficientlysmall x < . W dependence in (2.62) and (2.63) can be empirically tested byplotting the experimental data for the proton structure function F ( x ∼ = Q /W , Q ) as a function of 1 /W . ) -2 (GeV -6 -5 -4 -3 -2 -1 F =80-100 GeV Q =50-80 GeV Q =30-50 GeV Q =10-30 GeV Q x -4 -3 -2 -1 F =80-100 GeV Q =50-80 GeV Q =30-50 GeV Q =10-30 GeV Q Fig.4a Fig.4bFigure 4:
In fig.4a we show the experimental data for F ( x ∼ = Q /W , Q ) as a functionof 1 /W , and in fig.4b, for comparison, as a function of x . The theoretical predictionbased on (2.124) and (2.125) is also shown in fig.4a. In fig. 4a, we show the experimental data from HERA for Q in the largerange of 10 GeV ≤ Q ≤ GeV as a function of 1 /W . In the relevantrange of x ∼ = Q /W < .
1, approximately corresponding to 1 /W ≤ − ,the experimental data show indeed a tendency to lie on a single line, quite Figure 4 was kindly prepared by Prabhdeep Kaur (compare thesis, in preparation).
19n contrast to the range of 1 /W ≥ − . The opposite tendency of theexperimental data, approximate clustering around a single line for x ≥ . x ≤ . F ( x, Q ) are plotted in the usual manneras a function of Bjorken x . The replacement of W by W ≃ Q /x , whenpassing from fig. 4a to fig. 4b now yields the well-known increased violationof Bjorken scaling in the diffraction region of x < .
1. Compare Section 2.7for a discussion of the theoretical prediction shown in fig.4a.We summarize: DIS at low x proceeds via the imaginary part of the for-ward scattering amplitude of a continuum of massive ( q ¯ q ) J =1 L,T states. Theinteraction of the q ¯ q color dipole with the gluon field of the proton, by gaugeinvariance, fulfills (2.2) and (2.44), implying color transparency, (2.3) and(2.46). For Q sufficiently large (with x ∼ = Q /W ≪ . F ( x, Q ) only depends on the single variable W . Nodetails of perturbative QCD beyond the gauge-invariant color-dipole interac-tion are needed to deduce the CDP of Sections 2.1 and 2.2, and this (approx-imate) dependence of F ( x, Q ) on the single variable W for Q sufficientlylarge. In particular, no reference to details of the perturbative gluon densityof the proton is needed. In this connection, also compare the derivation ofthe CDP in ref. [6] as well as the formally much more complete and elaboratederivation in ref. [15].By starting from the ~l ⊥ -factorization approach, under certain assump-tions, one may introduce a CDP-like representation [5, 20] for the photoab-sorption cross section containing x instead of W in the dipole cross sectionin (2.1). Such a representation does not factorize the Q dependence inher-ently connected with the photon wave function and the W dependence thatgoverns the ( q ¯ q ) p dipole interaction. As a consequence, the CDP-like repre-sentation is ill-suited to represent the transition from large Q to small Q including the solely W -dependent cross section of Q = 0 photoproduction.Examining and understanding this transition to low x ∼ = Q /W photoab-sorption, however, is the main aim and also the essential achievement of theCDP-representation of DIS at low x . A model-independent analysis of the experimental data on DIS from HERAhas revealed [11, 21] that the photoabsorption cross section, σ γ ∗ p ( W , Q ),20t low x is a function of the low-x scaling variable η ( W , Q ) = Q + m Λ sat ( W ) , (2.64)i.e. a function of the single variable η ( W , Q ), σ γ ∗ p ( W , Q ) = σ γ ∗ p (cid:16) η ( W , Q ) (cid:17) . (2.65)In (2.64) and (2.65), the “saturation scale”, Λ sat ( W ), empirically increasesas Λ sat ( W ) ∼ ( W ) C , with C ∼ = 0 .
27 and m ∼ = 0 . GeV [11, 21]. Theempirical analysis of the experimental data showed that σ γ ∗ p ( η ( W , Q )) forlarge η ( W , Q ) ≫ η ( W , Q ), σ γ ∗ p ( W , Q ) ∼ σ ( ∞ ) ( W ) 1 η ( W , Q ) , (2.66)while for small values of η ( W , Q ) ≪
1, the dependence on η ( W , Q ) islogarithmic, σ γ ∗ p ( W , Q ) ∼ σ ( ∞ ) ( W ) ln 1 η ( W , Q ) , ( η ( W , Q ) ≪ . (2.67)In (2.66) and (2.67) the cross section σ ( ∞ ) ( W ) empirically was found to be ofhadronic size and approximately constant, σ ( ∞ ) ( W ) ≃ const. , as a functionof the energy W .In the present Section 2.5, we will show that not only the existence of thescaling behavior (2.65), but also the observed functional dependence of thecross section, as 1 /η ( W , Q ) for large η ( W , Q ), and as ln(1 /η ( W , Q ))for small η ( W , Q ), in (2.66) and (2.67), respectively, is a general and directconsequence of the color-dipole nature of the interaction of the hadronic fluc-tuations of the photon with the color field in the nucleon. No specific param-eterization of the color-dipole-proton cross section, σ ( q ¯ q ) p ( r ⊥ , z (1 − z ) , W ),must be introduced to deduce the empirically observed functional dependencein (2.66) and (2.67).The ensuing analysis will be based on the representation of the photoab-sorption cross section in Section 2.2 in terms of the scattering of ( q ¯ q ) J =1 L,T states on the proton. Compare (2.39) in particular, as well as the longitudi-nal and transverse dipole cross sections given by (2.44). The representation(2.44) of the dipole cross section, as a consequence of (2.2), is solely based Scaling in terms of a different, x-dependent instead of W-dependent, scaling variablewas found in ref. [22]
21n the gauge-invariant coupling of the color-dipole state to the gluon field inthe nucleon.Upon angular integration, (2.44) becomes σ ( q ¯ q ) J =1 L,T p ( r ′⊥ , W ) = π Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L,T p ( ~l ′ ⊥ , W ) . · − R d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L,T p ( ~l ′ ⊥ , W ) J ( l ′⊥ r ′⊥ ) R d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L,T p ( ~l ′ ⊥ , W ) , (2.68)where r ′⊥ ≡ q ~r ′ ⊥ and J ( l ′⊥ r ′⊥ ) denotes the Bessel function of order zero.We assume that the integrals in (2.68) do exist and are determined bythe integrands in a restricted range of ~l ′ ⊥ < ~l ′ ⊥ Max ≡ l ′ ⊥ Max ( W ), where¯ σ ( q ¯ q ) J =1 L,T p ( ~l ′ ⊥ , W ) is appreciably different from zero. The resulting dipolecross section (2.68), for any fixed value of r ′⊥ , strongly depends on the varia-tion of the phase, l ′⊥ r ′⊥ , in (2.44) and (2.68) as a function of l ′⊥ < l ′⊥ Max ( W ).Indeed, if for a given value of r ′⊥ the phase l ′⊥ r ′⊥ in the relevant range of l ′⊥ < l ′⊥ Max ( W ) is always smaller than unity, i.e.0 < l ′⊥ r ′⊥ < l ′⊥ Max ( W ) r ′⊥ ≪ , (2.69)the second term in the bracket of (2.68) essentially cancels the first one, since J ( l ′⊥ r ′⊥ ) ∼ = 1 −
14 ( l ′⊥ r ′⊥ ) + 14 ( l ′⊥ r ′⊥ ) + · · · . (2.70)Substitution of (2.70) into (2.68) implies the proportionality of the dipolecross section to r ′ ⊥ already given in (2.46). Combining (2.46) with (2.49) and(2.56), we find σ ( q ¯ q ) J =1 L,T p ( r ′ ⊥ , W ) = (2.71)= 14 πr ′ ⊥ Z d~l ′ ⊥ ~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) (cid:26) ,ρ, r ′ ⊥ ≪ l ′ ⊥ Max ( W ) ! . In the limiting case of l ′⊥ Max ( W ) r ′⊥ ≫ , (2.72)alternative to (2.69), the rapid oscillation of the Bessel function under vari-ation of 0 < l ′⊥ < l ′⊥ Max ( W ) at fixed r ′⊥ implies a vanishing contribution ofthe second term in (2.68). The dipole cross section (2.68) in this limit is not22roportional to the dipole size ~r ′ ⊥ , but, in distinction from (2.71), becomesidentical to the ~r ′ ⊥ -independent limit σ ( ∞ ) L,T ( W ) of normal hadronic size, σ ( q ¯ q ) J =1 L,T p ( r ′ ⊥ , W ) ∼ = π Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L,T p ( ~l ′ ⊥ , W ) ≡ σ ( ∞ ) L,T ( W ) , r ′ ⊥ ≫ l ′ ⊥ Max ( W ) ! . (2.73)We note that the r ′⊥ -independent limit on the right-hand side in (2.73)obtained at any fixed value of r ′⊥ for l ′ ⊥ max ( W ) → ∞ coincides with thelimit of r ′⊥ → ∞ at fixed energy, W , or fixed ~l ′ ⊥ max ( W ). A small q ¯ q dipoleat infinite energy yields the same cross section as a sufficiently large dipoleat finite energy W .The gauge-invariant color-dipole interaction with the gluon thus impliesthe emergence of two scales, the helicity-dependent integral over¯ σ ( q ¯ q ) J =1 L,T p ( ~l ′ ⊥ , W ) in (2.73) and the first moment of ¯ σ ( q ¯ q ) J =1 L ( l ′ ⊥ , W ) in (2.71),which determine the dipole cross section for relatively large r ′ ⊥ and relativelysmall r ′ ⊥ , respectively. Whether (2.73) or (2.71) is relevant for a chosenvalue of r ′⊥ depends on the value of l ′ ⊥ Max ( W ) that in turn depends on the W -dependence of the ( q ¯ q ) J =1 L,T p dipole cross section, ¯ σ ( q ¯ q ) J =1 L,T p ( ~l ′ ⊥ , W ).It is appropriate to introduce and use the normalized distribution in ~l ′ ⊥ ,Λ sat ( W ) ≡ R d~l ′ ⊥ ~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) R d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) == 1 σ ( ∞ ) L ( W ) π · Z d~l ′ ⊥ ~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) , (2.74)as the second scale besides σ ( ∞ ) L,T ( W ) from (2.73) . The r ′ ⊥ → σ ( q ¯ q ) J =1 L,T p ( r ′ ⊥ , W ) = 14 r ′ ⊥ σ ( ∞ ) L ( W )Λ sat ( W ) (cid:26) ,ρ, r ′ ⊥ ≪ l ′ ⊥ Max ( W ) ! . (2.75)The cross section σ ( ∞ ) L,T ( W ), as a consequence of the color-dipole interactionin (2.2) and (2.44), according to (2.73) and (2.75), is of relevance for both,the r ′ ⊥ → ∞ as well as the r ′ ⊥ → For generality, we keep the distinction between σ ( ∞ ) L ( W ) and σ ( ∞ ) T ( W ), even thoughthe essential conclusions of this Section do not depend on whether this distinction is keptor replaced by the equality (2.53). The scale Λ sat ( W ) in (2.74) is to be identified with the parameter Λ sat ( W ) in (2.64)that was introduced in the fit [11, 21] to the experimental data. r ′⊥ is determined by the destructive interference originatingfrom the (negative) second term in the bracket in (2.68). At any fixed valueof r ′⊥ , for sufficiently high energy, i.e. with increasingly greater values of ~l ′ ⊥ Max ( W ), the vanishing of this term, due to strong oscillations of theintegrand leads to the ~r ′⊥ -independent limit of a cross section of hadronic sizein (2.73). With increasing energy a transition occurs from the region of colortransparency (2.71), where the cross section is proportional to the dipole size, ~r ′ ⊥ , to the saturation regime (2.73) characterized by a cross section that isindependent of the dipole size, ~r ′ ⊥ ; the interaction of a colorless q ¯ q dipole isin the saturation regime replaced by the interaction of a colored quark and acolored antiquark thus producing a cross section of hadronic size. Both, colortransparency, as well as the transition to the hadronlike saturation behavior,are recognized as a genuine consequence of the gauge-invariant color-dipoleinteraction (2.1). It is a misconception to associate the saturation regimewith an increased density in a small-size region of the proton: in the high-energy limit of (2.73) the cross section is not proportional to the dipole size,and therefore it cannot be interpreted as the product of a (small) dipole sizewith a high-gluon-density region.We turn to the photoabsorption cross section in (2.33). The integrationover d ~r ′⊥ in (2.33) at fixed Q is dominated by ~r ′ ⊥ ≡ r ′ ⊥ ≤ Q . (2.76)Compare (2.7) and (2.8). The resulting photoabsorption cross section forfixed Q then depends on whether the limiting case of either (2.71) (or equiv-alently (2.75)) or of (2.73) is relevant for r ′ ⊥ ≤ /Q .For the case of r ′ ⊥ < Q ≪ l ′ ⊥ Max ( W ) , (2.77)the r ′ ⊥ → Q was treated in Section 2.2. Compare (2.46) and (2.47). Introducing Λ sat ( W )from (2.74) and (2.75) on the right-hand side of (2.51), with ρ W = ρ from(2.56), we find σ γ ∗ L,T p ( W , Q ) = απ X q Q q σ ( ∞ ) L ( W ) Λ ( W ) Q (cid:26) , ρ. (2.78)24he total photoabsorption cross section is given by σ γ ∗ p ( W , Q ) = σ γ ∗ L p ( W , Q ) + σ γ ∗ T p ( W , Q )= απ X q Q q (1 + 2 ρ ) 16 σ ( ∞ ) L ( W ) Λ sat ( W ) Q . (2.79)Unitarity requires the hadronic dipole cross section, σ ∞ L,T ( W ), from (2.73)to only weakly depend on W , σ ( ∞ ) L,T ( W ) ∼ = const . (2.80)Moreover, motivated by quark confinement and/or quark-hadron duality [23],the divergence of r ′ ⊥ → ∞ for Q → r ′ ⊥ < Q + m , (2.81)where m actually depends on the quark flavor. For light quarks, m < ∼ m ρ ,where m ρ is the ρ meson mass, is relevant. Replacing Q → Q + m in (2.79), and identifying the resulting (inverse) ratio with the empiricalparameter η ( W , Q ) in (2.64), we have σ γ ∗ p ( W , Q ) = απ X q Q q (1 + 2 ρ ) 16 σ ( ∞ ) L ( W ) 1 η ( W , Q ) , ( η ( W , Q ) ≫ , (2.82)where η ( W , Q ) ≫ l ′ ⊥ Max ( W ) ≪ Q , (2.83)alternative to (2.77), and relevant in particular for large values of the en-ergy W and relatively small values of Q . In this case of (2.83), within theintegration domain of r ′ ⊥ < /Q from (2.76), we have to discriminate twodifferent regions. For r ′ ⊥ < ~l ′ ⊥ Max ( W ) ≪ Q , (2.84) Actually, a logarithmic increase of σ ( ∞ ) L,T ( W ) is allowed. Actually, realistic values of Λ sat ( W ) fulfill the hierarchy of Λ sat ( W ) ≫ m , suchthat in the relevant range of η ( W , Q ) = ( Q + m ) / Λ sat ( W ) ≈ Q / Λ sat ( W ). Thereplacement of Q → Q + m in the case of (2.82) is of formal nature.
25e have color transparency (2.75). In distinction from (2.77), color trans-parency only holds in a small restricted domain of the full integration interval r ′ ⊥ < /Q . For the remaining integration domain,1 ~l ′ ⊥ Max ( W ) < r ′ ⊥ < Q , (2.85)the r ′⊥ -independent dipole cross section (2.73) becomes relevant.It is useful to split the integration domain into the sum of two differentones. Noting that according to the definition (2.74),Λ sat ( W ) < ∼ l ′ ⊥ Max ( W ) , (2.86)we use Λ sat ( W ) as the splitting parameter of the integral over dr ′ ⊥ . Thephotoabsorption cross section (2.39) then becomes σ γ ∗ L,T p ( W , Q ) = απ X q Q q Q Z sat ( W dr ′ ⊥ + Z Q sat ( W dr ′ ⊥ ·· K , ( r ′⊥ Q ) σ ( q ¯ q ) J =1 L,T p ( r ′ ⊥ , W ) . (2.87)The main contribution to the photoabsorption cross section is due to thesecond term on the right-hand side in (2.87). The first term will subsequentlybe shown to be negligible compared with the second one. Only taking intoaccount the second term, upon introducing the r ′⊥ -independent dipole crosssection from (2.73), we find σ γ ∗ L,T p ( W , Q ) = 2 απ X Q q Q σ ( ∞ ) L ( W ) R Q sat ( W dr ′⊥ r ′⊥ K ( r ′⊥ Q ) ,σ ( ∞ ) T ( W ) R Q sat ( W dr ′⊥ r ′⊥ K ( r ′⊥ Q ) . (2.88)The cross section in the high-energy limit, (2.88), as a consequence ofthe factorization of the dipole cross section (2.73), is directly given by anintegral over the photon wave function, compare e.g. (2.88) with the generalexpression in (2.39).In the integration domain (2.85) of r ′⊥ Q <
1, relevant in (2.87) and (2.88),upon introducing y = r ′⊥ Q , we can approximate K , ( r ′⊥ Q ) = K , ( y ) by K ( y ) ≃ ln y,K ( y ) ≃ y , ( y < . (2.89)We find σ γ ∗ L,T p ( W , Q ) = 2 απ X Q q ( σ ( ∞ ) L ( W ) , σ ( ∞ ) T ( W ) ln Λ sat ( W ) Q . (2.90)26he longitudinal cross section becomes small in this limit of very high energy W and comparatively small values of Q . According to (2.90), the longitu-dinal cross section may be neglected, and the total cross section is givenby σ γ ∗ p ( W , Q ) = απ X Q q σ ( ∞ ) T ( W ) ln Λ sat ( W ) Q . (2.91)With the replacement of Q → Q + m , compare (2.81), and upon introduc-ing η ( W , Q ) from (2.64), we indeed have derived the empirically observedlogarithmic dependence, σ γ ∗ p ( W , Q ) = απ X Q q σ ( ∞ ) T ( W ) ln 1 η ( W , Q ) , ( η ( W , Q ) ≪ , (2.92)where σ ( ∞ ) T ( W ) ∼ = const, compare (2.80).Combining (2.78) and (2.90), the ratio R of the longitudinal and thetransverse parts of the photoabsorption cross section is given by R ( W , Q ) = 12 σ ( ∞ ) L ( W ) σ ( ∞ ) T ( W ) 1ln η ( W ,Q , ( η ( W , Q ) ≪ , ρ , ( η ( W , Q ) ≫ . (2.93)In the limit of η ( W , Q ) ≪
1, i.e. for W → ∞ at fixed Q , the longi-tudinal part of the photoabsorption cross section becomes vanishingly smallcompared with the transverse part. In the limit of η ( W , Q ) ≫
1, we have ρ = 4 / ρ = 1 under the ad hocassumption of helicity independence.We finally have to convince ourselves that the first term in (2.87) canbe neglected relative to the second one. Inserting (2.75) into (2.39), thecontribution of the first term becomes σ ( I ) γ ∗ L,T p ( W , Q ) = απ X q Q q σ ( ∞ ) L ( W ) Λ sat ( W ) Q R Q Λ sat ( W dyy K ( y ) ,ρ R Q Λ sat ( W dyy K ( y ) . (2.94)Evaluation of the integrals upon inserting (2.89) yields σ ( I ) γ ∗ p ( W , Q ) = α P q Q q π σ ( ∞ ) L ( W ) Q Λ sat ( W ) ln Q Λ sat ( W ) + ρ ! ∼ = α P Q q π σ ( ∞ ) L ( W ) ρ, (Λ sat ( W ) ≫ Q ) . (2.95)Since (2.91) is enhanced by ln(Λ sat ( W ) /Q ), we can neglect (2.95) for suf-ficiently large Λ sat ( W ). 27he resulting cross sections (2.82) and (2.92) establish the empiricallyobserved low-x scaling behavior as a consequence of the interaction of the q ¯ q -fluctuations of the (real or virtual) photon as color-dipole states. Low- x scaling is recognized as a genuine consequence of the CDP in the formulationgiven in (2.39) and (2.44) that is based on (2.1) and (2.2). “Saturation”i.e. the slow logarithmic increase as ln Λ sat ( W ) in (2.92), is not based ona specific model assumption. It occurs as a consequence of the transition ofthe ( q ¯ q ) p interaction from the color-transparency region to the hadronic one.This transition occurs for any given Q , or any fixed dipole size, providedthe energy is sufficiently high such that the q ¯ q state does not interact as acolorless dipole, but rather as a system of two colored quarks. W → ∞ at fixed Q > In Section 2.5, we found that the CDP from (2.1) and (2.2) implies that thephotoabsorption cross section at low x ∼ = Q /W ≪ η ( W , Q ) from (2.64) and (2.74). Moreover, the dependenceof σ γ ∗ L,T p ( η ( W , Q )), for small and large values of η ( W , Q ) was found tobe uniquely determined without adopting a specific parameterization for thedipole cross section, compare (2.92) and (2.82), σ γ ∗ p ( W , Q ) = σ γ ∗ p ( η ( W , Q )) = (2.96)= απ X q Q q σ ( ∞ ) T ( W ) ln η ( W ,Q ) , ( η ( W , Q ) ≪ , (1 + 2 ρ ) σ ( ∞ ) L ( W ) η ( W ,Q ) , ( η ( W , Q ) ≫ . where unitarity restricts σ ( ∞ ) L,T ( W ) to being at most weakly dependent on W .In this Section 2.6, we present a more detailed discussion of the important η ( W , Q ) → W → ∞ at fixed values of Q .We explicitly assume Λ sat ( W ) to increase with the energy, W . Thereare convincing theoretical arguments for this assumption, independent ofthe analysis of the experimental data that was referred to in the discussionrelated to (2.64) to (2.67).Note that the absorption of a gluon of transverse momentum ~l ⊥ by a q ¯ q fluctuation leads to “diagonal” as well as “off-diagonal” transitions withrespect to the mass, M q ¯ q , of the q ¯ q fluctuations,( q ¯ q ) M q ¯ q → (cid:26) ( q ¯ q ) M q ¯ q , “diagonal”) , ( q ¯ q ) M ′ q ¯ q = M q ¯ q , (“off − diagonal”) . (2.97)28he mass difference in the second line of (2.97) is proportional to ~l ′ ⊥ = ~l ⊥ /z (1 − z ), or to Λ sat ( W ) from (2.74), on the average,∆ M q ¯ q ≡ M ′ q ¯ q − M q ¯ q ∼ Λ sat ( W ) . (2.98)This connection excludes Λ sat ( W ) = const. unless one is willing to postu-late the mass difference between incoming and outgoing q ¯ q states in hadronicdiffraction to be equal to a fixed value that is W -independent, even for W → ∞ . Constancy, Λ sat ( W ) = const . , would imply a W -dependenceof the photoabsorption cross section (2.96) that is exclusively determinedby the factorized cross section σ ( ∞ ) L,T ( W ) from (2.73), entirely independentof the details of the dynamics of the gluon field in the proton related toΛ sat ( W ) from (2.74). One accordingly can safely dismiss the assumption ofΛ sat ( W ) = const on theoretical grounds, independently of its inconsistencywith the experimental data, compare (2.64) to (2.67). A further argumenton the increase of Λ sat ( W ) with the energy may be based on the consistencyof the CDP with a description of the proton structure function in terms ofsea quark and gluon distributions and their evolution with Q . This will bediscussed below, compare Section 2.7.Considering the limit of η ( W , Q ) →
0, or W → ∞ at fixed Q , weintroduce the ratio of the virtual to the real photoabsorption cross section,and from (2.96) we find [21]lim W →∞ Q σ γ ∗ p ( η ( W , Q )) σ γ ∗ p ( η ( W , Q = 0)) = lim W →∞ Q ln (cid:16) Λ sat ( W ) m m ( Q + m ) (cid:17) ln Λ sat ( W ) m == 1 + lim W →∞ Q ln m Q + m ln Λ sat ( W ) m = 1 . (2.99)At sufficiently large W , at any fixed value of Q , the γ ∗ p cross section ap-proaches the Q -independent ( Q = 0) photoproduction limit. We stressagain that this result (2.99) is independent of any particular parameteriza-tion of the dipole cross section. It is solely based on the CDP (2.1) with thegeneral form of the dipole cross section (2.2) required by the gauge-invarianttwo-gluon coupling of the q ¯ q fluctuation in the forward-Compton-scatteringamplitude.The ( Q , W ) plane corresponding to (2.96) and (2.99) is simple. It con-sists of only two regions separated by the line η ( W ) , Q ) ∼ = 1, compare fig.5.29igure 5: The ( Q , W ) plane showing the line η ( W , Q ) = 1 separating the large- Q and the small- Q region. Below this line i.e. for η ( W , Q ) ≫
1, we have color transparency with σ γ ∗ p ( W , Q ) ∼ Λ ( W ) /Q , while for η ( W , Q ) ≪
1, we have hadron-like saturation behavior. Without explicit parameterization of Λ sat ( W ), therelation (2.99) does not determine the energy scale, at which the limit ofphotoproduction is reached in (2.99). The limit (2.99) was first given [21]under the assumption of a specific ansatz for the dipole cross section in (2.2), σ ( q ¯ q ) p ( r ⊥ , z (1 − z ) , W ) = σ ( ∞ ) ( W )(1 − J ( r ⊥ z (1 − z )Λ sat ( W )) (2.100)that was used in a successful fit [11, 21] to the experimental data from HERA.By extrapolating the fit to the experimental data based on (2.100) to W →∞ at fixed Q , one finds the limiting behavior (2.99). Inserting the fit result[21] Λ sat ( W ) = (0 . ± . W GeV ! C GeV ,C = 0 . ± . , (2.101) m = 0 . GeV ± . GeV , into (2.99) allows one to examine the approach to the photoabsorption limitin (2.99). As expected from the logarithmic behavior in (2.99), exceedinglyhigh energies are needed to approach this limit. Compare Table 1 for aspecific example. The original fit [21] with Λ ( W ) = C ( W + W ) C and W = 1081 ± GeV canin good approximation be replaced by (2.101). [ GeV ] W [ GeV ] σ γ ∗ p ( η ( W ,Q )) σ γp ( W ) . × . × σ γ ∗ p ( η ( W , Q )) to the photoproduction limit, σ γp ( W ) = σ γ ∗ p ( η ( W , Q = 0)), for W → ∞ at fixed Q > σ γ ∗ p ( W , Q ) ∼ l λ eff ( Q ) (2.102)were examined by Caldwell [24], in particular in view of an extrapolation tothe above limit of large W at fixed Q . The ansatz (2.102), with l = 12 M p x ∼ = 12 M p W Q , (2.103)was motivated by the lifetime, or coherence length, of a hadronic fluctuationaccording to (2.60).The particular fit based on σ γ ∗ p ( W , Q ) = σ ( Q ) l λ eff ( Q ) , (2.104)and individually carried out for a series of values of Q in the interval0 . < ∼ Q < ∼ GeV , led to an intersection of the straight lines in therepresentation of the log of σ γ ∗ p ( W , Q ) against the log of the coherencelength l . The intersection, interpreted as indication for the approach to a Q -independent limit at large W , occured at W ∼ = 10 Q , (2.105)to be compared with the (not yet fully asymptotic) results for W from ourapproach in Table 1. It is of interest that the large- W extrapolation of afit to the experimental data based on the simple intuitively well-motivated,but still fairly ad-hoc ansatz (2.104) implies a saturation effect similar to theone predicted from the CDP, the validity of which stands on firm theoreticalgrounds. Not every ansatz for a successful fit in terms of the variable l in (2.103), however, as pointed out in ref. [24], implies an approach to a Q -independent saturation limit. Precise empirical evidence for the limitingbehavior (2.99) presumably requires experiments at energies substantiallyabove the ones explored at HERA. 31 .7 The CDP, the Gluon Distribution Function andEvolution The CDP of DIS corresponds to the low- x approximation of the pQCD-improved parton model in which the interaction of the (virtual) photon occursby interaction with the quark-antiquark sea in the proton via γ ∗ gluon → q ¯ q fusion, compare fig.6. The longitudinal structure function, F L ( x, Q ), in this γ ∗ γ ∗ γ ∗ (a) (b)Figure 6: (a) Photon-gluon fusion.(b) Higher order contributions to photon-gluon → q ¯ q fusion resolving the lower blob infig.1. The lower part of the diagram must be extended to become a gluon ladder. low- x or CDP approximation of pQCD solely depends on the gluon density, g ( x, Q ), [26] F L ( x, Q ) = α s ( Q )3 π X q Q q · I g ( x, Q ) (2.106)with I g ( x, Q ) ≡ Z x dyy xy ! − xy ! yg ( y, Q ) . (2.107)where G ( y, Q ) ≡ yg ( y, Q ). For a wide range of different gluon distributions,independently of their specific form, the integration in (2.107) yields a resultthat is proportional to the gluon density at a rescaled value x/ξ L [26] i.e. F L ( ξ L x, Q ) = α s ( Q )3 π X q Q q G ( x, Q ) . (2.108)The rescaling factor ξ L in (2.108) has the preferred value of ξ L ∼ = 0 .
40 [26].The interaction of the longitudinally polarized photon with the quark (an-tiquark) originating from gluon → q ¯ q splitting, via F L ( ξ L x, Q ), in goodapproximation thus fully determines the x and Q dependence of the gluondistribution function.We turn to the structure function F ( x, Q ). In the DIS scheme of pQCD,at low x and sufficiently large Q , F ( x, Q ) is proportional to the singlet or With respect to the present Section 2.7, compare also ref.[25] P ( x, Q ), X ( x, Q ) = n f X q =1 ( q q ( x ) + ¯ q q ( x )) . (2.109)For four flavors of quarks, n f = 4, and flavor-blind quark distributions, thestructure function is given by F ( x, Q ) = x X ( x, Q ) 14 X q Q q = 518 x X ( x, Q ) . (2.110)In the CDP approximation, γ ∗ gluon → q ¯ q fusion, the evolution of F ( x, Q )with Q is determined by the gluon distribution according to [12] ∂F ( x, Q ) ∂ ln Q = α s ( Q ) π X q Q q Z x dzP qg ( z ) G (cid:18) xz , Q (cid:19) , (2.111)where in leading order of pQCD P qg ( z ) = P (0) qg = 12 ( z + (1 − z ) ) . (2.112)The evolution equation (2.111), again for a wide range of choices for thegluon distribution, may be represented by the proportionality [27] ∂F ( ξ x, Q ) ∂ ln Q = α s ( Q )3 π X q Q q G ( x, Q ) . (2.113)The rescaling factor in this case is given by ξ ∼ = 0 .
50 [27].The validity of (2.108) and (2.113) and the values of the rescaling factors( ξ L , ξ ) = (0 . , .
50) will be reexamined below by evaluating the relations(2.106) for F L ( x, Q ) and (2.111) for F ( x, Q ) for the specific gluon distri-bution to be obtained by requiring consistency with the CDP approach.We introduce the ratio of F ( x, Q ) and F L ( x, Q ) by employing the formof this ratio in (2.59 ), but allowing for a potential dependence of ρ ≡ ρ ( x, Q )on the kinematic variables x and Q , F L ( x, Q ) = 12 ρ + 1 F ( x, Q ) . (2.114)Replacing the right-hand side of (2.113) by F L ( ξ L x, Q ) from (2.108), andsubsequently replacing F L ( ξ L x, Q ) by F ( ξ L x, Q ) according to the definingrelation (2.114), the evolution equation (2.113) becomes(2 ρ + 1) ∂∂ ln Q F ξ ξ L x, Q ! = F ( x, Q ) , (2.115)33r, in terms of the flavor singlet distribution (2.109) according to (2.110),(2 ρ + 1) ∂∂ ln Q ξ ξ L X ξ ξ L x, Q ! = X ( x, Q ) . (2.116)By alternatively replacing F L ( x, Q ) in (2.114) by the gluon distributionfrom (2.108), upon inserting the resulting expression for F ( x, Q ) into theevolution equation (2.113), we find an evolution equation for the gluon den-sity that reads ∂∂ ln Q (2 ρ + 1) α s ( Q ) G ξ ξ L x, Q ! = α s ( Q ) G ( x, Q ) . (2.117)Comparing (2.117) with (2.116), we conclude: if and only if(2 ρ + 1) = const ., (2.118)the evolution of the gluon density multiplied by α s ( Q ) in (2.117) coincideswith the quark-singlet evolution according to (2.115) and (2.116).Identical evolution of the q ¯ q sea originating from γ ∗ gluon → q ¯ q fusion(fig.6a) and the gluon distribution multiplied by α s ( Q ) appears as naturalconsequence of the fact that the q ¯ q state seen by the photon originates fromthe gluon: the evolution of the sea distribution, measured by the interactionwith the photon, directly yields the evolution of the gluon distribution.In the CDP, according to Section 2.4, specifically according to (2.63),and supported by the experimental results in fig.4, the structure function F ( x, Q ) for x < . Q sufficiently large, depends on the single variable W , F ( x, Q ) = F ( W = Q x ) . (2.119)Independently of the specific form of the functional dependence of F ( x, Q )on W , according to (2.119), the Q dependence and the x dependence of F ( x, Q ) are intimately related to each other. This is a consequence of the W dependence of the dipole cross section in (2.1), compare (2.78) and (2.61).In terms of the energy variable W , the evolution equation (2.115) becomes(2 ρ W + 1) ∂∂ ln W F ξ L ξ W ! = F ( W ) . (2.120)Since according to (2.1) the longitudinal as well as the transverse photoab-sorption cross section depend on W , also the potential dependence of ρ on x and Q is restricted to W , and in (2.120) , this is indicated by ρ W .34e assume a power-law dependence for F ( W ) on W , F ( W ) ∼ ( W ) C = Q x ! C . (2.121)We note that the dependence of F ( x, Q ) = F ( W ) in (2.121) on a fixed(i.e. Q -independent) constant power C of 1 /x coincides with the x → /x ) λ inputassumption for the flavor-singlet quark as well as the gluon distribution ( λ =const). A fixed power of 1 /x , as (1 /x ) ǫ , also appears in the Regge approachto DIS based on a linear combination of a “soft” and a “hard” Pomeron, withthe fit parameter of the hard Pomeron contribution being given by ǫ ∼ = 0 . /x dependenceand the Q dependence of F ( x, Q ) (for x < . Q sufficiently large, Q ≥ ) are determined by one and the same constant power C ,compare (2.121).Inserting the power-law (2.121) into the evolution equation (2.120), wefind the constraint (2 ρ W + 1) C ξ L ξ ! C = 1 . (2.122)Consistency of the power law (2.121) for the W dependence with the flavor-singlet evolution (2.120) thus implies the remarkable constraint (2.122) thatconnects the exponent C of the 1 /x dependence with the longitudinal-to-transverse ratio of the photoabsorption cross sections, 2 ρ W , or, equivalentlywith the ratio of F ( x, Q ) and F L ( x, Q ) in (2.114). Constancy of C impliesconstancy of ρ W = ρ = const, and vice versa.In the CDP, from (2.56), ρ has the constant and fixed value of ρ = 4 / ρ = 4 /
3, we find from (2.122) (compare also [25]) C = 12 ρ + 1 ξ ξ L ! C = 0 .
29 (2.123)where the preferred value of ξ /ξ L = 0 . / . .
25 was inserted. We notethat the (ad hoc) variation of this value in the interval 1 ≤ ξ /ξ L ≤ . ξ /ξ L = 1 .
25 yields 0 . ≤ C ≤ .
31. The result C = 0 .
29 accordingly is fairly insensitive under variation of the rescalingfactors ξ and ξ L . Note that (2.123) differs from the result in [30] by taking into account the rescalingfactor ξ L = 0 . ρ = 4 /
35e specify (2.121) by adopting the theoretical result for the exponent C = 0 .
29 from (2.123) and by introducing a proportionality constant, f , F ( W ) = f · W ! C =0 . . (2.124)Via an eye-ball fit to the experimental data for F ( W ) as a function of 1 /W in fig. 4a, we find f = 0 . . (2.125)The theoretical prediction (2.124) with (2.125) is shown in fig.4a. A detailedcomparison with the experimental data, separately for distinct values of Q in the relevant range of 10GeV ≤ Q ≤ shows agreement with thesingle-free-parameter fit (2.124) to the structure function F ( W ) in (2.124)for 10GeV ≤ Q ≤ . Compare the discussion in Section 5, inparticular figs.16 and 17.According to (2.110), the flavor-singlet quark or sea distribution is pro-portional to the structure function F ( W ), x X ( x, Q ) = 185 f · W ! C =0 . . (2.126)Employing the proportionality (2.108) of the gluon distribution to the longi-tudinal structure function F L ( ξ L x, Q ), and expressing F L ( ξ L x, Q ) in termsof F ( ξ L x, Q ) according to (2.114), we find that also the gluon distributioncan be directly deduced from the experimental data for the structure function F ( x, Q ) = F ( W = Q /x ), α s ( Q ) G ( x, Q ) = 3 π P q Q q F L ( ξ L x, Q )= 3 π P q Q q ρ + 1) F ( ξ L x, Q ) (2.127)= 3 π P q Q q (2 ρ + 1) f ξ C =0 . L W ! C =0 . , where (2.124) was inserted in the last step.This is the appropriate point to add a remark, as previously announced,on the validity of the representations (2.108) and (2.113) in terms of therescaling factors ( ξ L , ξ ). It will turn out that indeed without loss of general-ity (2.106) and (2.111) for our gluon distribution may be replaced by (2.108)and (2.113). 36nserting the gluon distribution (2.127) into the representations of F L ( x, Q )and F ( x, Q ) in (2.106) and (2.111), one may explicitly test the validity ofthe proportionalities to the gluon distribution in (2.108) and (2.113) thatoriginate from (2.106) and (2.111). One finds that the above choice of therescaling factors, ( ξ L , ξ ) = (0 . , . F L ( x, Q ) and F ( x, Q ), respectively. The discrepancy is reduced to lessthan 0.5% , for the choice of ( ξ L , ξ ) = (0 . , . C = 0 .
29 to C = 0 .
26 in (2.123), close to the value of C = 0 . ± . C = 0 .
26 and C = 0 .
29 is not very important.We use C = 0 .
29 in fig.4a and in the more extensive comparison with theexperimental data in figs.16 and 17 in Section 5.In fig.7, we compare our gluon distribution from (2.127) with variousgluon distributions obtained in fits to the experimental results for F ( x, Q ).Compare refs.[29, 31] and the Durham Data Base [32] . The gluon distri-butions from the various fits were multiplied by α s ( Q ), where α NLOs ( Q ) = 1 bt " − b ′ ln tbt . (2.128)with b = 33 − n f π , b ′ = 153 − n f π (33 − n f ) , (2.129)and t = ln( Q / Λ QCD ) , n f = 4 and Λ QCD = 340MeV corresponding to α s ( M Z ) = 0 . F ( x, Q ) by different collaborations. The gluon-distribution functioncorresponding to the hard Pomeron of the Regge fit [29] in general lies aboveour result. The results from the so-called global analysis by the CTEQ [33]and MSTW [34] collaborations are lower than ours. The fact that our resultsare fairly close to the results from GRV [30] seems no accident and deservesfurther examination.Our relation (2.127) obtained as a consequence of the low- x pQCD ap-proximations (2.106) and (2.111) and the W dependence of F L, ( x, Q ) = The gluon-distribution functions in fig.7 marked GRV, MSTW and CTEQ were ex-tracted from the Durham Data Base [32]
The gluon-distribution function from (2.127) compared with the gluon dis-tributions from the hard-Pomeron part of a Regge fit [29] to F ( x, Q ), and from the F ( x, Q ) fits GRV [31], CTEQ [33] and MSTW[34]. L, ( W ) from the CDP is transparent and simple as far as the underlying as-sumptions are concerned. The extracted gluon distribution only depends onthe single normalization parameter f that was adjusted to the experimentaldata. The gluon distribution can directly be read off from the experimentaldata for F ( W = Q /x ) shown in fig.4 by multiplication of these data withthe constant given in (2.127).We end this Section with the following summarizing comments:i) The starting point for our extraction of the gluon distribution is thelow- x approximation of the pQCD-improved parton model that relatesthe gluon distribution to the longitudinal structure function, F L ( x, Q ),compare (2.106). This relation is supplemented by the W -dependenceof the structure functions F L ( W = Q /x ) and F ( W = Q /x ) andtheir proportionality via the constant factor of 1 / (2 ρ + 1), both the W dependence and the proportionality being extracted from the CDPand being supported empirically. Finally, a power-law dependence, F ∼ ( W ) C = ( Q /x ) C is inserted, with C = 0 .
29 predicted fromsea-quark evolution. The extraction of the gluon distribution dependson only one fitted normalization constant, f .ii) The gluon distribution resulting from (2.127) lies within the range ofgluon distributions available in the literature. We note that our extrac-tion of the gluon distribution from the data on F ( x, Q ) = F ( W = Q /x ) is not based on a resolution of the ggpp vertex, the lower blobin fig.1. The consistency of our gluon distribution with the ones inthe literature indicates that the gluon distribution does not as sensi-tively depend on details of the structure of the ggpp vertex as usuallyexpected, assumed or elaborated upon. Compare the BFKL approach[35] to DIS at low x , as well as the double asymptotic scaling (DAS)solution [36, 37, 38] of DGLAP evolution [12] based on replacing theunresolved lower part of the diagram in fig.1 by the lower part of thediagram in fig.6b that has to be extended by a gluon ladder. We con-jecture that our gluon distribution nevertheless, in the sense of a nu-merical approximation, is consistent with the DGLAP gluon evolutionequation at low x that supplements the evolution of the flavor-singletquark distribution solely employed in our analysis.iii) As mentioned, our 1 /x dependence (2.127), (2 ρ + 1) α s ( Q ) xg ( x, Q ) ∼ P ( x, Q ) ∼ (1 /x ) C with fixed exponent C , is closely related toDGLAP evolution with the input constraint of a hard Pomeron [28].We differ from ref.[28] insofar, as we have the necessary constraint of(2 ρ + 1) = const, compare (2.122), while the analysis of ref.[28] led to(2 ρ + 1) α s ( Q ) = const . .iv) Our (1 /x ) C dependence is analogous to the (1 /x ) dependence of thehard Pomeron component of the Regge approach [29]. However, wepredict C = 0 .
29 from sea-quark evolution, the value being consis-tent with experiment, while the analogous parameter ǫ ∼ = 0 .
43 in theRegge approach is a pure fit-parameter. Moreover, the CDP contains asmooth transition to low Q , including Q = 0, rather than relying onthe addition of a soft Pomeron. In the language of Pomeron exchange,the CDP only knows of a single Pomeron which is relevant for bothsmall and large values of Q . In Section 2, without adopting a specific parameterization for the dipole crosssection, we found the proportionalities (2.96) of the total photoabsorptioncross section to ln(1 /η ( W , Q )), for η ( W , Q ) ≪
1, and to 1 /η ( W , Q ) for η ( W , Q ) ≫
1. Any specific parameterization of the dipole cross section hasto interpolate between these two limits.In Section 3.1, we will remind ourselves of a previously employed ansatzfor the dipole cross section that implies R ( W , Q ) = 1 / Q forthe ratio of R ( W , Q ) = σ γ ∗ L p ( W , Q ) /σ γ ∗ T p ( W , Q ). In Section 3.2, weintroduce a more general ansatz that allows for the transverse-size reductionand associated enhancement of the transverse relative to the longitudinalphotoabsorption cross section from Section 2.3. R = 0 . The ansatz for the dipole cross section in (2.1), previously employed in a suc-cessful fit to the experimental data on the total cross section, σ γ ∗ p ( W , Q ),is given by [11] σ ( q ¯ q ) p ( ~r ⊥ , z (1 − z ) , W ) = σ ( ∞ ) ( W ) (cid:18) − J (cid:18) r ⊥ q z (1 − z )Λ sat ( W ) (cid:19)(cid:19) , (3.1)40here σ ( ∞ ) ( W ) is of hadronic size and weakly dependent on W , whileΛ sat ( W ) increases as a small power of W . Since the cross section (3.1)depends on the product ~r ′⊥ = ~r ⊥ q z (1 − z ), the longitudinal and transverse J = 1 projections in (2.40) become identical, σ ( q ¯ q ) J =1 L p (cid:16) r ′⊥ Λ sat ( W ) (cid:17) = σ ( q ¯ q ) J =1 T p (cid:16) r ′⊥ Λ sat ( W ) (cid:17) = σ ( ∞ ) ( W ) (cid:16) − J ( r ′⊥ Λ sat ( W ) (cid:17) (3.2)= σ ( ∞ ) ( W ) (cid:26) ~r ′ ⊥ Λ sat ( W ) , for ~r ′ ⊥ Λ sat ( W ) → , , for ~r ′ ⊥ Λ sat ( W ) → ∞ . With respect to momentum space, the ansatz (3.1), according to (2.2),corresponds to˜ σ ( ~l ⊥ , z (1 − z ) , W ) = σ ( ∞ ) ( W ) π δ (cid:16) ~l ⊥ − z (1 − z )Λ sat ( W ) (cid:17) . (3.3)Its J = 1 projections, according to (2.45), are given by¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) = ¯ σ ( q ¯ q ) J =1 T p ( ~l ′ ⊥ , W ) == σ ( ∞ ) ( W ) π δ (cid:16) ~l ′ ⊥ − Λ sat ( W ) (cid:17) . (3.4)Substitution of (3.3) and (3.4) into (2.2) and (2.44), respectively, takes usback to (3.1) and (3.2).We remark that helicity independence, the equality of the cross sectionsfor scattering of the J = 1 projections for longitudinally and transverselypolarized ( q ¯ q ) J =1 states in (3.2) and (3.4), is a general consequence of thedependence of the ansatz (3.1) on the variable r ′⊥ = r ⊥ q z (1 − z ). Any dipolecross section in (2.1) fulfilling σ ( q ¯ q ) p ( ~r ⊥ , z (1 − z ) , W ) = σ ( q ¯ q ) p (cid:18) ~r ⊥ q z (1 − z ) , W (cid:19) , (3.5)together with color transparency (2.2), implies helicity independence and R ( W , Q ) = 1 / Q . Indeed, consistency of (3.5) with (2.2), σ ( q ¯ q ) p ( r ⊥ q z (1 − z ) , W ) = Z d ~l ′⊥ z (1 − z )˜ σ ( ~l ′ ⊥ z (1 − z ) , z (1 − z ) , W ) ·· (cid:16) − e − i~l ′⊥ · ~r ′⊥ (cid:17) , (3.6)requires z (1 − z )˜ σ ( ~l ′ ⊥ z (1 − z ) , z (1 − z ) , W ) to be independent of z (1 − z ).Under this constraint, (2.45) implies helicity independence and R ( W , Q ) =1 / For clarity, in terms of ( q ¯ q ) J =1 helicities, ¯ σ ( q ¯ q ) J =1 L ≡ ¯ σ ( q ¯ q ) J =10 and ¯ σ ( q ¯ q ) J =1 T ≡ (cid:16) ¯ σ ( q ¯ q ) J =1+ + ¯ σ ( q ¯ q ) J =1 − (cid:17) = ¯ σ ( q ¯ q ) J =1+ q ¯ q fluctuations that isbest incorporated by returning from transverse position space to momentumspace. The constraint reads m ≤ M q ¯ q , M ′ q ¯ q ≤ m ( W ) , (3.7)where the notation, i.e. M q ¯ q , M ′ q ¯ q , for the masses of the ( q ¯ q ) dipole states,indicates that incoming and outgoing q ¯ q masses in the forward Comptonamplitude of fig.1 do not necessarily agree with each other. The lower bound, m , depends on the flavor of the actively contributing quarks. For up anddown quarks the value of m must be somewhat below the ρ mass. The upperbound, m ( W ), depends on the available energy. In most applications ofthe CDP, the approximation of m ( W ) → ∞ is employed that restricts thekinematic range 2.of applicability of the CDP. For the present discussion weput m ( W ) → ∞ . We will come back to a finite value of m ( W ) in Section4. According to dimensional analysis, with m ( W ) → ∞ , the photoab-sorption cross section resulting from (3.1) in addition to the dependence on η ( W , Q ) = ( Q + m ) / Λ ( W ) will depend on m / Λ ( W ). For therealistic case of m / Λ ( W ) ≪
1, the total photoabsorption cross section σ γ ∗ p ( W , Q ) = σ γ ∗ T p ( W , Q ) + σ γ ∗ L p ( W , Q ) takes the remarkably simpleexplicit form [11] σ γ ∗ p ( W , Q ) = σ γ ∗ p ( η ( W , Q )) + O m Λ ( W ) ! == αR e + e − π σ ( ∞ ) ( W ) I ( η ) + O m Λ ( W ) ! , (3.8)where I ( η ( W , Q )) = 1 q η ( W , Q ) ln q η ( W , Q ) + 1 q η ( W , Q ) − ∼ = (3.9) ∼ = ln η ( W ,Q ) + O ( η ln η ) , for η ( W , Q ) → m Λ ( W ) , η ( W ,Q ) + O (cid:16) η (cid:17) , for η ( W , Q ) → ∞ , and R e + e − = 3 X q Q q . (3.10)42s expected, since (3.1) fulfills color transparency, compare (3.2), the result(3.8) with (3.9) and σ ( ∞ ) ( W ) ∼ = const constitutes an example for the generalresult in (2.82) and (2.92) from Section 2.5.For further reference, we give the explicit parameterization of the ansatz(3.1) and the values of the parameters obtained in the fit to the experimentaldata. The “saturation scale”, Λ sat ( W ) is given by [11, 21]Λ sat ( W ) = B W W + 1 ! C , (3.11)with B = 2 . ± . GeV ,W = 1081 ± GeV , (3.12) C = 0 . ± . . In good approximation, (3.11) becomesΛ sat ( W ) = C W GeV ! C , (3.13)with C = 0 . ± . (3.14)i.e. Λ ( W ) is in good approximation determined by only two parameters,the normalization scale C and the exponent C .The hadronic cross section, σ ( ∞ ) ( W ), was obtained [11] by requiringconsistency with the Regge fit to the measured Q = 0 photoproductioncross section. It determines the product of R e + e − σ ( ∞ ) ( W ), where R e + e − =3 P q Q q . With three active flavors, R e + e − = 2, and σ ( ∞ ) ( W ) ∼ = 30 mb = 77 . GeV − . (3.15)The value of the lower bound, m , in (3.7) is given by m = 0 . ± . GeV . (3.16)43 .2 The Ansatz for the Dipole Cross Section implying R = 1 / ρ ( ǫ ) Returning to the discussion in Section 2, compare in particular (2.23), wegeneralize (3.3) to become ˜ σ ( ~l ⊥ , z (1 − z ) , W ) = ¯ σ ( ∞ ) ( W ) π δ (cid:18) ~l ⊥ −
16 ¯Λ sat ( W ) (cid:19) Θ( z (1 − z ) − ǫ ) . (3.17)With respect to transverse position space, according to (2.2), we obtain from(3.17), σ ( q ¯ q ) p ( r ⊥ , z (1 − z ) , W ) = ¯ σ ( ∞ ) ( W ) − J ( r ⊥ ¯Λ sat ( W ) √ ! Θ( z (1 − z ) − ǫ ) ∼ = ¯ σ ( ∞ ) ( W )Θ( z (1 − z ) − ǫ ) (
14 ¯Λ ( W )6 ~r ⊥ , for ~r ⊥ → , , for ~r ⊥ → ∞ . (3.18)The δ -function in (3.17), via ¯Λ sat ( W ), specifies the W -dependence of theintegral R d~l ⊥ ~l ⊥ ˜ σ ( ~l ⊥ , W ) that, according to (2.25), determines the photoab-sorption cross section at large Q . The Θ-function in (3.17), compare (2.23),provides the necessary W -dependent cut on ~l ′ ⊥ = ~l ⊥ /z (1 − z ). It forbids q ¯ q fluctuations of infinitely large mass to occur as a result of gluon absorption atfinite energy, W . The J = 1 projections of the ansatz (3.17), by substitutionof (3.17) into (2.45), are found to be given by¯ σ ( q ¯ q ) J =1 L,T p (cid:16) ~l ′ ⊥ , ¯Λ sat ( W ) (cid:17) = f L,T (cid:16) ~l ′ ⊥ , ¯Λ sat ( W ) (cid:17) ·· Θ (cid:18) ~l ′ ⊥ −
23 ¯Λ sat ( W ) (cid:19) Θ (cid:16) a ¯Λ sat ( W ) − ~l ′ ⊥ (cid:17) , (3.19)where f L (cid:16) ~l ′ ⊥ , ¯Λ sat ( W ) (cid:17) = ¯ σ ( ∞ ) ( W )3 π ¯Λ sat ( W ) ~l ′ ⊥ r − sat ( W )3 ~l ′ ⊥ , (3.20)and f T (cid:16) ~l ′ ⊥ , ¯Λ sat ( W ) (cid:17) = 3 ~l ′ ⊥ sat ( W ) −
13 ¯Λ sat ( W ) ~l ′ ⊥ ! f L (cid:16) ~l ′ ⊥ , ¯Λ sat ( W ) (cid:17) . (3.21)The constant a in (3.19) is related to ǫ in (3.18) by ǫ = 1 / a , where a ≫ ~l ′ ⊥ The quantities ¯ σ ( ∞ ) ( W ) and ¯Λ sat ( W ) are proportional to σ ( ∞ ) ( W ) and Λ sat ( W )introduced by the defining relations (2.73) and (2.74). The constant proportionality factorswill be given below. ~l ′ ⊥ = Λ ( W ) in (3.4) is replaced by a broad distribution in the interval(2 /
3) ¯Λ ( W ) ≤ ~l ′ ⊥ ≤ a ¯Λ ( W ). For ~l ′ ⊥ > ¯Λ sat ( W ), the transversepart of the dipole cross section in (3.21) becomes enhanced by a factor of ~l ′ ⊥ / Λ sat ( W ) relative to the longitudinal one.Inserting the J = 1 dipole cross section (3.19), with (3.20) and (3.21),into the large- Q form of the photoabsorption cross section in (2.47), we find(with Q ≫ ¯Λ sat ( W )) σ γ ∗ L,T p ( W , Q ) = απ X q Q q Q
16 ¯ σ ( ∞ ) ( W ) ¯Λ sat ( W ) s − a ( , ρ (cid:16) ǫ = a (cid:17) , (3.22)where 2 ρ (cid:16) ǫ = a (cid:17) coincides with the result given in (2.27). Here, we assumed m ( W ) → ∞ . The generalization to finite values of m ( W ) will be givenin Section 4, compare (4.28).The photoabsorption cross section (3.22) may be expressed in terms of thecross section σ ( ∞ ) L ( W ) and the scale Λ sat ( W ) introduced in Section (2.5)in terms of integrals over the longitudinal part of the J = 1 dipole crosssection. Compare (2.73) and (2.74). Evaluating (2.73) and (2.74) for theansatz (3.19), we find σ ( ∞ ) L ( W ) = ¯ σ ( ∞ ) ( W ) (cid:18) a (cid:19) s − a (3.23) ∼ = ¯ σ ( ∞ ) ( W ) (cid:18) − a (cid:19) , ( a > sat ( W ) = ¯Λ sat ( W ) 11 + a . (3.24)The photoabsorption cross section (3.22) may accordingly be written in termsof σ ∞ L ( W ) and Λ sat ( W ) to become σ γ ∗ L,T p ( W , Q ) = απ X q Q q σ ( ∞ ) L ( W ) Λ sat ( W ) Q ( ρ (cid:16) ǫ = a (cid:17) , ( Q ≫ Λ sat ( W )) . (3.25)The result (3.25) correctly coincides with the general result (2.78).A comparison of (3.25) with (3.8) and the η ( W , Q ) → ∞ limit in(3.9) shows that the large- Q cross section (3.25) formally corresponds tothe polarization-dependent replacement in (3.1) ofΛ sat ( W ) → (cid:26) Λ sat,L ( W ) = Λ sat ( W ) , Λ sat,T ( W ) = ρ ( ǫ )Λ sat ( W ) . (3.26)45ombined with the substitution σ ( ∞ ) ( W ) → σ ( ∞ ) L ( W ) (3.27)The justification of the resulting cross section (3.25) rests on the ansatz(3.18), since the dipole cross section in (2.1), and accordingly in (3.1) must beindependent of the polarization indices T and L of q ¯ q dipole fluctuations. Thereplacement (3.26) with (3.27) is nevertheless illuminating for an intuitiveunderstanding of the transition from (3.1) to the ansatz (3.17). For the evaluation of the ansatz for the photoabsorption cross section pre-sented in Section 3, we return to momentum space. Inserting the represen-tation for the longitudinal and the transverse part of the J = 1 dipole crosssection (2.44) into (2.39), and employing the momentum-space representationof the modified Bessel functions K , ( r ′⊥ Q ), one finds (compare Appendix A) σ γ ∗ L p ( W , Q ) = αR e + e − π Q Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) ·· Z dM Z dM ′ w ( M , M ′ ,~l ′ ⊥ ) Q + M ) − Q + M )( Q + M ′ ) ! (4.1)and σ γ ∗ T p ( W , Q ) = αR e + e − π Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 T p ( ~l ′ ⊥ , W ) ·· Z dM Z dM ′ w ( M , M ′ ,~l ′ ⊥ ) M ( Q + M ) − M + M ′ − ~l ′ ⊥ )( Q + M )( Q + M ′ ) . (4.2)In the transition from (2.39) to (4.1) and (4.2), we introduced the q ¯ q masses, M = ~k ⊥ z (1 − z ) ≡ ~k ′ ⊥ , (4.3)in terms of the quark transverse momentum, ~k ⊥ , and M ′ = ( ~k ⊥ + ~l ⊥ ) z (1 − z ) , (4.4)46n terms of the transverse momentum of the quark upon absorption of thegluon.In (4.1) and (4.2), R e + e − = 3 P q Q q , where the sum runs over the activelycontributing quarks. The Jacobian w ( M , M ′ ,~l ′ ⊥ ) in (4.1) and (4.2) is givenby [6] w ( M , M ′ ,~l ′ ⊥ ) = 12 M M ′ √ − cos φ = 12 M q ~l ′ ⊥ q (1 − cos ϑ ) , (4.5)where φ denotes the angle between ~k ⊥ and ( ~k ⊥ + ~l ⊥ ), and ϑ denotes the anglebetween ~k ⊥ and ~l ⊥ . Sincecos φ = 14 M M ′ (cid:16) M + M ′ − ~l ′ ⊥ (cid:17) (4.6)is symmetric under exchange of M and M ′ , also w ( M , M ′ ,~l ′ ⊥ ) in (4.5) issymmetric under exchange of M and M ′ . The integrands in (4.1) and (4.2)may be cast into a form that is fully symmetric under exchange of M and M ′ , thus explicitly displaying the symmetry of the virtual forward-Compton-scattering amplitude from fig. 1. It describes the process γ ∗ p → γ ∗ p in termsof the “diagonal” transitions M ( q ¯ q ) → M ( q ¯ q ) and M ′ ( q ¯ q ) → M ′ ( q ¯ q ) and the “off-diagonal” ones M ( q ¯ q ) ↔ M ′ ( q ¯ q ) , in a symmetric manner.The integrations in (4.1) and (4.2) have to fulfill the restrictions m ≤ M , M ′ ≤ m ( W ) . (4.7)The lower bound, m , in (4.7) corresponds to vanishing γ ∗ → q ¯ q transitions,as soon as ~k ⊥ (and ( ~k ⊥ + ~l ⊥ ) ) become sufficiently small. A vanishing value of ~k ⊥ would imply contributions to the Compton-forward-scattering amplitudeof states of unbounded transverse size that do not occur as a consequence ofquark confinement. Via quark-hadron duality in e + e − annihilation, the valueof m must be somewhat below the ρ mass . The upper limit, m ( W ),in (4.7) follows from the restriction on the lifetime, (2.60), of a hadronic q ¯ q fluctuation that requires M and M ′ to be strongly bounded for any finitevalue of the energy, W . Quantitatively, for a typical HERA energy of W =225 GeV , the crude estimate of M q ¯ q /W = 0 .
01 requires m ( W ) = 22 . GeV .This value is approximately consistent with the mass range of the diffractive A refined treatment has to discriminate between the masses of the different quarkflavors, and, in particular, has to introduce a larger lower limit for the charm contributionto the cross section. q ¯ q fluctuations relevantfor the total photoabsorption cross section. Obviously, the mass bound, m = m ( W ), increases with increasing energy.For the evaluation of (4.1) and (4.2) with the restriction of (4.7) on M and M ′ , it is convenient to replace the integration over dM ′ by an integra-tion over dϑ . Noting that M ′ ( M ,~l ′ ⊥ , cos ϑ ) = M + ~l ′ ⊥ + 2 M q ~l ′ ⊥ cos ϑ, (4.8)and ∂M ′ ( M ,~l ′ ⊥ , cos ϑ ) ∂ϑ = − w ( M , M ′ ,~l ′ ⊥ ) , (4.9)upon incorporating the restrictions in (4.7), the integrations in (4.1) and(4.2) simplify to become Z dM Z dM ′ w ( M , M ′ ,~l ′ ⊥ ) == Z m ( W ) m dM Z π dϑ − Z ( √ ~l ′ ⊥ + m ) ( √ ~l ′ ⊥ − m ) dM Z πϑ ( M ,~l ′ ⊥ ) dϑ − Z m ( W )( m ( W ) − √ ~l ′ ⊥ ) dM Z ϑ ( M ,~l ′ ⊥ )0 dϑ. (4.10)2. The first term in (4.10) takes care of the bound on M in (4.7), ignoring,however, the restriction on ϑ induced by the bound on M ′ . The secondand the third term in (4.10) correct for this ignored restriction on M ′ . Thebounds on the angles, ϑ ( M ,~l ′ ⊥ ) and ϑ ( M ,~l ′ ⊥ ) in (4.10), are obtainedfrom the lower and the upper bound on M ′ ( M ,~l ′ ⊥ , cos ϑ ) implied by (4.8)and are given by cos ϑ , ( M ,~l ′ ⊥ ) = m , − M − ~l ′ ⊥ M q ~l ′ ⊥ . (4.11)Here m stands for m ≡ m ( W ). In terms of the dM dϑ integration (4.10),the photoabsorption cross sections in (4.1) and (4.2) become σ γ ∗ L p ( W , Q ) = αR e + e − π Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) · (4.12) · Z dM Z dϑ Q ( Q + M ) − Q ( Q + M )( Q + M ′ ( M ,~l ′ ⊥ , cos ϑ )) ! σ γ ∗ T p ( W , Q ) = αR e + e − π Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 T p ( ~l ′ ⊥ , W ) · (4.13) · Z dM Z dϑ M ( Q + M ) − M + M ′ ( M ,~l ′ ⊥ , cos ϑ ) − ~l ′ ⊥ Q + M )( Q + M ′ ( M ,~l ′ ⊥ , cos ϑ )) . The integrations in (4.12) and (4.13), according to (4.10), lead to a sum ofthree terms, σ γ ∗ L,T p ( W , Q ) = σ domγ ∗ L,T p ( W , Q ) + ∆ σ ( m ) γ ∗ L,T p ( W , Q ) + ∆ σ ( m ( W )) γ ∗ L,T p ( W , Q ) . (4.14)The first term will be dominant. The correction due to the lower bound m will turn out to be small, of order 1 %. The third term in (4.14) will befound to yield a somewhat larger contribution, of order 10 %, dependent onthe values of the kinematical variables.For the dominant term, the integration of (4.12) and (4.13) with theintegration domain given by the first term in (4.10), can be carried out ana-lytically. We concentrate on the dominant term, and for the correction termsrefer to Appendix B.Upon integration over dϑ of (4.12) and (4.13), the dominant contributionsto the photoabsorption cross section become [16] σ domγ ∗ L p ( W , Q ) = αR e + e − Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) · (4.15) · Z m ( W ) m dM Q ( Q + M ) − Q ( Q + M ) q X ( M ,~l ′ ⊥ , Q ) , and σ domγ ∗ T p ( W , Q ) = αR e + e − Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 T p ( ~l ′ ⊥ , W ) ·· Z m ( W ) m dM Q + M ) − Q ( Q + M ) − q X ( M ,~l ′ ⊥ , Q ) +2 Q + ~l ′ ⊥ ( Q + M ) q X ( M ,~l ′ ⊥ , Q ) , (4.16)where X ( M ,~l ′ ⊥ , Q ) ≡ ( M − ~l ′ ⊥ + Q ) + 4 Q ~l ′ ⊥ . (4.17)Carrying out the integration over dM in (4.15) and (4.16), we finally obtain σ domγ ∗ L,T p ( W , Q ) = αR e + e − Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L,T p ( ~l ′ ⊥ , W ) · (cid:16) I L,T ( ~l ′ ⊥ , m ( W ) , Q ) − I L,T ( ~l ′ ⊥ , m , Q ) (cid:17) , (4.18)where I L,T ( ~l ′ ⊥ , M , Q ) denotes the indefinite integrals over dM in (4.15)and (4.16). They are given by I L ( ~l ′ ⊥ , M , Q ) = − Q Q + M + Q q ~l ′ ⊥ ( ~l ′ ⊥ + 4 Q ) · (4.19) · ln q ~l ′ ⊥ ( ~l ′ ⊥ + 4 Q ) q X ( M ,~l ′ ⊥ , Q ) + ~l ′ ⊥ (3 Q − M + ~l ′ ⊥ ) Q + M and I T ( ~l ′ ⊥ , M , Q ) = (4.20)= Q Q + M + 12 ln Q + M q X ( M ,~l ′ ⊥ , Q ) + M − ~l ′ ⊥ + Q − Q + ~l ′ ⊥ q ~l ′ ⊥ ( ~l ′ ⊥ + 4 Q ) ·· ln q ~l ′ ⊥ ( ~l ′ ⊥ + 4 Q ) q X ( M ,~l ′ ⊥ , Q ) + ~l ′ ⊥ (3 Q − M + ~l ′ ⊥ ) Q + M . The representation (4.18) of the (dominant part of the) photoabsorption crosssection does not depend on a specific ansatz for the dipole cross section. Therepresentation (4.18) only relies on the general form of the CDP given by(2.1) with (2.2) and by (2.39) with (2.44) that follow from (2.1) and (2.2).In other words, (4.18) only rests on the low-x kinematics and the formationof q ¯ q color-dipole fluctuations that interact as color dipoles with the gluonfield in the nucleon. In most applications of the CDP one considers the limitof m ( W ) → ∞ that restricts the kinematic range of validity of the CDP.In this limit of ∆ σ ( m ( W )) γ ∗ L,T p ( W , Q ) = 0, the photoabsorption cross section iswell represented by the dominant term (4.18) evaluated for m ( W ) → ∞ ,since ∆ σ ( m ) γ ∗ L,T p ( W , Q ) can be neglected.The evaluation of (4.18) for the case of the ansatz (3.4) of the dipolecross section with helicity independence is straightforward. For the sum ofthe longitudinal and the transverse cross section, for m ( W ) → ∞ , theresult is given in (3.8) with (3.9).For the evaluation of the more general ansatz (3.19), it will be convenientto replace the integration variable ~l ′ ⊥ by y = 23 ¯Λ sat ( W ) ~l ′ . (4.21)50he J = 1 dipole cross sections (3.20) and (3.21) then become f L ( y, ¯Λ sat ( W )) = 98 ¯ σ ( ∞ ) ( W ) π ¯Λ sat ( W ) y √ − y , . (4.22)and f T ( y, ¯Λ sat ( W )) = (1 − y ) y f L ( y, ¯Λ sat ( W )) . (4.23)Explicitly, the photoabsorption cross section (4.18) for the ansatz (3.19) isthen given by σ domγ ∗ L p ( W , Q ) = αR e + e − σ ∞ ( W ) π Z / a dy y √ − y · (4.24) · I L
23 ¯Λ sat ( W ) y , m ( W ) , Q ! − I L
23 ¯Λ sat ( W ) y , m , Q !! and σ domγ ∗ T p ( W , Q ) = αR e + e − σ ( ∞ ) ( W ) π Z / a dy − y/ √ − y · (4.25) · I T
23 ¯Λ sat ( W ) y , m ( W ) , Q ! − I T
23 ¯Λ ( W ) y , m , Q !! . We note that the replacements¯ σ ( ∞ ) ( W ) → σ ( ∞ ) ( W ) ,
23 ¯Λ sat ( W ) y → Λ sat ( W ) , (4.26)and the formal replacements Z / a dy y √ − y → , Z / a dy − y √ − y →
43 (4.27)in (4.24) and (4.25) take us back to the photoabsorption cross section forthe dipole cross section (3.4) with helicity independence that is obtained bysubstitution of (3.4) into (4.18).The correction terms ∆ σ ( m ) γ ∗ L,T p ( W , Q ) and ∆ σ ( m ) γ ∗ L,T p ( W , Q ) from (4.14)that are to be added to the dominant parts of the cross sections (4.24) and(4.25) are explicitly given in Appendix B, compare (B.9) and (B.10).The evaluation of the cross sections in (4.24) and (4.25) together with thecorrection terms (B.9) and (B.10) in general requires numerical integration. A computer program can be provided on request.
51 simple analytic approximation of the cross sections can be derived,however, for the limit of Q ≫ ¯Λ sat ( W ) = Λ ( W ) · (1 + 1 / a ) ∼ = Λ ( W ),or η ( W , Q ) ≫
1. Ignoring the negligible contribution from ∆ ( m ) γ ∗ L,T p ( W , Q ),the analytic approximation for the sum of σ domγ ∗ L,T p ( W , Q ) and ∆ σ ( m ) γ ∗ L,T p ( W , Q )is given by σ γ ∗ L,T p ( W , Q ) = σ γ ∗ L,T p ( η ( W , Q ) , ξ ) = (4.28)= αR e + e − σ ( ∞ ) L ( W ) π η ( W , Q ) (cid:26) G L ( η ( W , Q ) , ξ ) , ρ ( ε = a ) G T ( η ( W , Q ) , ξ ) , where G L ( η ( W , Q ) , ξ ) = G L ξη ( W , Q ) ! (4.29)= (cid:16) ξη (cid:17) + 3 (cid:16) ξη (cid:17) (cid:16) ξη (cid:17) = , for ξη → ∞ , . , for ξη = 10 , . , for ξη = 1 , and G T ( η ( W , Q ) , ξ ) = G T ξη ( W , Q ) ! (4.30)= 2 (cid:16) ξη (cid:17) + 3 (cid:16) ξη (cid:17) + 3 (cid:16) ξη (cid:17) (cid:16) ξη (cid:17) = , for ξη → ∞ , . , for ξη = 10 , . , for ξη = 1 , and ρ ( ǫ = a ) is given by (2.27). Compare Appendix C for the derivation of(4.28) to (4.30). In (4.28) to (4.30), η ≡ η ( W , Q ) = ( Q + m ) / Λ sat ( W )denotes the low- x scaling variable defined by (2.64), and the parameter ξ specifies m ( W ) via m ( W ) = ξ Λ sat ( W ) = ξη ( W , Q ) ( Q + m ) . (4.31)where the approximation of m ∼ = 0 is valid, since we are concerned with Q ≫ Λ sat ( W ) ≫ m . With (4.28), we have obtained the generalization of(3.25) to the case of a finite upper bound, m ( W ), for the masses of the q ¯ q fluctuations. The limit of ξ/η → ∞ , or ξ → ∞ at fixed η ( W , Q ), yieldsthe frequently employed approximation of the CDP that ignores the upperbound on the masses of the contributing q ¯ q fluctuations. Since ξ must befinite, compare (3.7) and (4.31), this approximation of the CDP breaks downas soon as η ( W , Q ) becomes sufficiently large.52ccording to (4.28), the ratio of the longitudinal to the transverse pho-toabsorption cross section for Q ≫ Λ sat ( W ) is given by R ( W , Q ) = σ γ ∗ L p ( η ( W , Q ) , ξ ) σ γ ∗ T p ( η ( W , Q ) , ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η ( W ,Q ) ≫ = 12 ρ (cid:16) ǫ = a (cid:17) G T ( ξη ) G L ( ξη ) . (4.32)The ratio R ( W , Q ) in (4.32), compared with (2.57) is modified by the fac-tor of G T ( ξ/η ) /G L ( ξ/η ). The transverse-size enhancement of transverselypolarized relative to longitudinally polarized ( q ¯ q ) J =1 fluctuations from Sec-tion 3.3 must be applied for realistic values of m ( W ), sufficiently largesuch that the CDP, approximately unmodified by the finiteness of m ( W ),becomes applicable. We accordingly consider R ( W , Q ) for ξ/η ≥ η ( W , Q ) in the interval of 5 < η ( W , Q ) <
10, this corresponds to50Λ ( W ) < m ( W ) < ( W ) and 5Λ ( W ) < Q < ( W ) ,and m ( W ) ≫ Q ≫ Λ sat ( W ) . (4.33)Taking into account the transverse-size enhancement in the denominator of(4.32) according to (2.57) and (2.56) requires ρ (cid:18) ǫ ≡ a (cid:19) G T (cid:16) ξη ∼ = 10 (cid:17) G L (cid:16) ξη ∼ = 10 (cid:17) = 43 . (4.34)With ρ ( ǫ ≡ / a ) from (2.27), and the numerical values of G T ( ξ/η = 10) and G L ( ξ/η = 10) from (4.29) and (4.30), G T ( ξ/η ∼ = 10) /G L ( ξ/η ∼ = 10) ∼ = 0 . a ∼ = 7 . . (4.35)With this uniquely determined value of a = 7 .
5, our ansatz (4.17) forthe dipole cross section yields a concrete realization of the transverse-sizeenhancement that implies R ( W , Q ) = · ( ) = 0 . m ( W ) = ξ Λ sat ( W ) by examining the behavior of the large- Q approximation of thecross section in (4.28) under variation of ξ . In particular, we first of all chosethe value of ξ required by consistency with the experimental results in therange of η ( W , Q ) ≥
10. This value of ξ , compare Section 5, is given by ξ = ξ exp = 130 . (4.36) At HERA energies, we approximately have 3GeV ≤ Λ ( W ) ≤ . A value of a = 7 is applied in the analysis of the experimental data in Section 5.
53e illustrate the effect of ξ , by comparing the theoretical results for thephotoabsorption cross section obtained for the choice of (4.36) with the onesfor ξ → ∞ and for various values of ξ < ξ exp = 130.Figure 8: The photoabsorption cross section σ γ ∗ p ( η ( W , Q ) , ξ ) for different values of ξ = m ( W ) / Λ sat ( W ). In fig.8, we show the cross section for σ γ ∗ p ( η ( W , Q ) , ξ ) = σ γ ∗ L p ( η ( W , Q ) , ξ ) + σ γ ∗ T p ( η ( W , Q ) , ξ ) (4.37)obtained by numerical evaluation of (4.24) and (4.25) together with (B.9)and (B.10). The numerical input for Λ sat ( W ) and m is identical to whatwill be used in Section 5, when comparing with the experimental data.The main features of the behavior of σ γ ∗ p ( η ( W , Q ) , ξ ), in fig.8 can beunderstood by looking at the analytic approximations in (4.28) to (4.30),which hold for η ( W , Q ) sufficiently large compared with unity, η ( W , Q ) > ξ = ξ exp = 130 and ξ/η >
10, or η < η exp = 13, the effect ofthe finite upper bound of m ( W ) = 130Λ sat ( W ) becomes negligible.The corresponding range of Q and W is given by Q < η exp Λ sat ( W ) ∼ = (cid:26) , forΛ sat ( W ) = 3GeV , , forΛ sat ( W ) = 7GeV . (4.38)The result (4.38) gives the domain, where at HERA energies the fre-quently employed approximaton of the CDP with m ( W ) → ∞ is54pplicable .ii) For fixed ξ = ξ exp = 130, and ξ/η <
10, or η > η exp = 13, theapproximation of m ( W ) → ∞ breaks down, and large correctionsof order 0 .
5, according to (4.29) and (4.30), depending of the value of η ( W , Q ), are necessary. Compare fig.8. The finite value of ξ = ξ exp =130 explicitly excludes high-mass fluctuations that have too short alifetime to actively contribute to the cross section.iii) In fig.8, we also show the theoretical results for the photoabsorptioncross section for values of ξ between ξ = 7 and ξ = ξ exp = 130. Thepredicted cross sections for η ( W , Q ) sufficiently below η ( W , Q ) = η exp = 13, dependent on the chosen value of ξ , coincide with both theresults for ξ = ξ exp = 130 and ξ = ∞ . This is consistent with theanalytic result, G T,L ( ξ/η ) ∼ = 1 for ξ > η , compare (4.29) and (4.30).The actively contributing masses M q ¯ q are actually bounded by ξ < η or M q ¯ q < η Λ sat ( W ) = 10 Q . (1 < η < η exp ∼ = 13) (4.39)Compare Table 2. The upper bounds on the masses of the q ¯ q fluctu-ations, M q ¯ q , contributing to σ γ ∗ p ( η ( W , Q )) according to Table 2 ap-proximately coincide with the upper bounds of the q ¯ q masses in whichthe dominant contributions to diffractive production are observed atHERA [9]. η Λ sat ( W )[GeV ] Q [GeV ] M q ¯ q [GeV ]13 3 39 3907 91 9105 3 15 1507 35 350Table 2 The upper limit of the masses of the actively contributing ( q ¯ q ) fluctuations, M q ¯ q for values of η ∼ = Q / Λ sat ( W ) and Λ sat ( W ) relevant for HERA ener-gies. We return to the cross section in (4.25) and (4.16), as well as (4.24) and(4.15) and consider the approximation of η ( W , Q ) ≪ The notation η exp for η exp = 13 results from the choice of ξ = ξ exp = 130 necessaryfor agreement with the experimental data for x ≤ . W → ∞ at fixed Q , and specifically thelimit of Q = 0. In this limit the longitudinal cross section vanishes, whilethe transverse cross section (4.16) is given by σ domγ ∗ T p ( W , Q = 0) = αR e + e − Z d~l ′ ⊥ σ ( q ¯ q ) J =1 T p ( ~l ′ ⊥ , W ) ·· Z m ( W ) m dM M − M − ~l ′ ⊥ M | M − ~l ′ ⊥ | = (4.41)= αR e + e − Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 T p ( ~l ′ ⊥ , W ) ln ~l ′ ⊥ m Since according to (3.19) the cross section ¯ σ ( q ¯ q ) J =1 T ( ~l ′ ⊥ , W ) is non-vanishingonly for ~l ′ ⊥ < a Λ ( W ), the upper bound m ( W ) = ξ Λ ( W ) in (4.41)may be replaced by m ( W ) = a Λ ( W ). With a = 7, and 2GeV ≤ Λ ( W ) ≤ at HERA energies, this implies 14GeV ≤ M q ¯ q ≤ .Only q ¯ q fluctuations in a strongly limited range of masses, bounded by ap-proximately a value between 3 . W , areresponsible for the photoabsorption cross section when Q approaches thephotoproduction limit of Q →
0. This analytic estimate is confirmed bythe numerical results for η = a = 7 shown in fig.8. For η ( W , Q ) < q ¯ q fluctuations with masses squared larger than m ( W ) = 7Λ ( W ) do notcontribute to the interaction.Inserting the dipole cross section (3.19) and passing to the variable y according to (4.21) and (4.23), the photoproduction cross section (4.41) be-comes σ γp ( W ) = σ domγ ∗ T p ( W , Q = 0) = (4.42)= αR e + e − σ ( ∞ ) ( W ) π Z a dy − y √ − y ln 2 ¯Λ sat ( W )3 ym . The substitutions (4.26) and (4.27) take us back to (3.8) and (3.9).
The total photoabsorption cross section from (4.24) and (4.25) together with(B.9) and (B.10) depends on the saturation scale Λ sat ( W ), or rather thelow- x scaling variable, η ( W , Q ) = ( Q + m ) / Λ sat ( W ), the lower andthe upper bounds, m and m ( W ) = ξ Λ sat ( W ), on the masses of the q ¯ q fluctuations, and the total ( q ¯ q ) p cross section σ ( ∞ ) ( W ), where from (3.24) σ ( ∞ ) ( W ) ≡ σ ( ∞ ) L ( W ) ∼ = ¯ σ ( ∞ ) ( W ). 56he numerical results to be shown subsequently are based on the set ofparameters that is specified as follows. The saturation scale is parameterizedby ¯Λ sat ( W ) = Λ sat ( W ) (cid:18) a (cid:19) = ¯ C W W + 1 ! C (5.1)with ¯ C = 2 . ,W = 1081GeV , (5.2) C = 0 . . The lower and the upper bound on the masses of the q ¯ q fluctuations are givenby m = 0 . , (5.3)and m ( W ) = ξ ¯Λ sat ( W ) = 130 ¯Λ sat ( W ) . (5.4)The total cross section, σ ( ∞ ) ( W ), is determined by requiring [11] consistencyof the CDP at Q = 0 from (4.42) with the Regge parameterization given by σ Regge γp ( W ) = A P ( W ) α P − + A R ( W ) α R − , (5.5)where W is to be inserted in units of GeV , and A P = 63 . ± . µb,α P = 1 . ± . , (5.6) A R = 145 . ± . µb,α R = 0 . . Since both the CDP and the Regge parameterization have similar (soft) en-ergy dependence, one finds that the variation of σ ( ∞ ) ( W ) in the HERAenergy range is restricted to about 10%. Quantitatively, since the total pho-toabsorption cross section is dependent on the product of R e + e − σ ( ∞ ) ( W ),we have σ ( ∞ ) ( W ) ∼ = (cid:26) mb, (for 3 active flavors , R e + e − = 2)18 mb, (for 4 active flavors , R e + e − = ) (5.7) A computer program is available on request. For the connection between Λ ( W ) and ¯Λ ( W ), compare (4.35). The value of C = 0 .
27 is taken from the previous fit in refs.[11, 21]. The difference between this valueof C = 0 .
27 and C = 0 .
29 from (2.123) is not significant in the relevant kinematic range. -2 -1
10 1 10 b ) µ ( * p γ σ -3 -2 -1 ZEUSH1EMCE665BCDMSNMC(W=275) σ (W=10) σ Figure 9:
The theoretical prediction for the photoabsorption cross section σ γ ∗ p ( η ( W , Q ) , ξ ) for ξ = 130 compared with the experimental data on DIS. Comparing the above parameters with the ones in (3.11) to (3.16), fromrefs. [11, 21], one notes the smaller value of ¯ C = 2 .
04 that is required asa consequence in the change of the longitudinal-to-transverse ratio R from R = 0 . R = 0 . ξ = ξ exp = 130 was determinedfrom an eye-ball fit to the experimental data. Compare fig.8 for the variationof the total photoabsorption cross section under variation of ξ .In fig.9, we show the total cross section, σ γ ∗ p ( W , Q ) = σ γ ∗ p η ( W , Q ) , m Λ sat ( W ) , ξ = ξ exp = 130 ! (5.8)as a function of the low- x scaling variable η ( W , Q ). The upper and thelower theoretical curve in fig.9 refer to the variation of σ ( ∞ ) ( W ) under vari-ation of the energy W , i.e. σ ( W = 275GeV) ≡ σ ( ∞ ) ( W = 275 GeV ) and σ ( W = 10GeV) ≡ σ ( ∞ ) ( W = 100GeV ). It is interesting to note that theviolation of scaling in η ( W , Q ) of the order of about 10%, as a consequenceof the W dependence of the ( q ¯ q ) p dipole cross section σ ( ∞ ) ( W ), is seen inthe experimental data: the high-energy data from ZEUS and H1 lie abovethe data obtained at lower energies. Figure 10 is relevant for the discussion ofthe limit of W → ∞ for fixed values of Q given in Section 2, compare(2.99)and Table 1. In terms of the structure function F ( x ∼ = Q /W , Q ) the58igure 10: The approach to the saturation limit of F ( η ( W , Q ) , Q ) /σ γp ( W ) for η ( W , Q ) ≪ W → ∞ limit in (2.99) becomeslim W →∞ Q F ( x ∼ = Q /W , Q ) σ γp ( W ) = Q π α . (5.9)Higher energies are required to uniquely experimentally verify the expectedsaturation property for a larger range of η ( W , Q ) ≪ Q .In figs.11 to 13, we show our predictions from the CDP for the protonstructure function F ( W , Q ) as a function of Q for fixed values of W , andas a function of W for fixed values of Q . For comparison, we also show theresults of a very precise fit to the world experimental data for F ( x, Q ) for x < .
025 (and Q >
0) carried out by Caldwell [24]. In particular, we showthe results from the so-called 2P-fit that is based on the simple ansatz [24] σ γ ∗ p = σ M Q + M ll ! ǫ +( ǫ − ǫ ) q Q Q (5.10)where l = 12 x bj M p . (5.11)The curves in figs.11 to 13 use the mean values of the six fit parameters σ , M , l , ǫ , ǫ and Λ given in Table 5 of ref.[24]. There is acceptable agree-ment of the predictions of the CDP with the results of the 2P-fit.59igure 11: The proton structure function F ( W , Q ) as a function of Q for variousvalues of W . The theoretical prediction of the CDP is compared with the Caldwell 2 P-fitas a representation of the experimental data. Figure 12:
As in Fig.11, but as a function of 1 /W for various values of Q ≤ . As in Fig.12, but for 30GeV < Q < . In figs.14 and 15, we directly compare the theoretical results for F ( W , Q )from the CDP (shown in figs.11 to 13) with the world experimental data [32]. As expected from figs.11 to 13, there is consistency between the CDPand the experimental data in the full range of 0 . ≤ Q ≤ .The theoretical curves are restricted by the condition of x ∼ = Q /W < . F ( x ∼ = Q /W , Q ) to Q according to(5.9) becomes visible, however, when comparing the experimental data infig.14 for the very low values of Q = 0 . and Q = 0 . witheach other. According to the proportionality in (5.9), for sufficiently large W we have F ( W , Q = 0 . ) = Q Q F ( W , Q = 0 . )= 2 . F ( W , Q = 0 . ) . (5.12) W [GeV − ] F ( W , Q = 0 . ) Q Q F ( W , Q = 0 . )2 · − ∼ = 0 .
055 0.1510 − ∼ = 0 .
04 0.11Table 3
The (approximate) validity of the proportionality (5.12). The results in the secondcolumn were read off from fig.14. The predictions from (5.12) in the third column(approximately) agree with the experimental results in fig.14. We thank Prabhdeep Kaur for providing the plots of the experimental data in figs.14to 17. -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W =0.036 GeV Q ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W =0.1 GeV Q ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W =0.31 GeV Q ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W =1. GeV QE665EMCNMC BCDMSH1ZEUSCDP ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W =3.16 GeV Q ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W =10 GeV Q Figure 14:
The predictions from the CDP for the structure functionn F ( W , Q ) com-pared with the experimental data for 0 . ≤ Q ≤ . ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W Q2=31.6 GeV ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W Q2=100 GeVE665EMCNMCBCDMSH1ZEUSCDP ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W Q2=316 GeV
Figure 15:
As in fig. 14, but for 31 . ≤ Q ≤ . F ( W , Q = 0 . ) obtained from (5.12)and shown in Table 3 are consistent with the experimental results in fig.14.In figs.16 and 17, in addition to the theoretical results in figs.14 and 15,we show the prediction (2.124) of F ( W ) = f · ( W / ) . , where f is the fitted normalization constant of f = 0 .
063 from (2.125), and W ∼ = Q /x . As expected from the analysis in Section 2.7 and fig.4a, there isagreement between theory and experiment for 10GeV ≤ Q ≤ and disagreement for values of Q outside of this range.Equation (2.127) may be inverted and read as a prediction for F ( W = Q /x ) from the pQCD-improved parton picture in terms of a suitable gluondistribution i.e. as a prediction for the flavor-singlet quark distribution,according to F ( W = Q x ) = 518 x X ( x, Q ) = (2 ρ + 1) P Q q π ξ C L α s ( Q ) G ( x, Q ) . (5.13)In (5.13), the numerical values for the gluon-distribution function have tobe inserted, which are obtained by evaluating the right-hand side of thesecond equality in (2.127). The resulting gluon distributions were shown infig.7. Since (5.13) coincides with (2.127), the resulting structure function F ( W , Q ) is identical to the one given by (2.124) and shown in figs.16 and17. The present interpretation of the results for F ( W , Q ) is different, how-ever. The agreement with experiment in figs.16 and 17 shows that a suitablechoice of the gluon distribution, compare fig.7, yields agreement with exper-iment for F ( W , Q ) in the relevant range of 10GeV ≤ Q ≤ .The results in figs.16 and 17 thus explicitly display the agreement with thepQCD-improved parton picture based on the gluon distribution function offig.7 in Section 2.7. For the ensuing discussion, we note the proportionalityof the gluon distribution function to the saturation scale, α s ( Q ) G ( x, Q ) ∼ W ! C =0 . ∼ Λ ( W ) σ ( ∞ ) L (5.14)that follows from comparing (5.13) with the representation of F ( W , Q ) interms of the saturation scale, Λ ( W ), in (2.63) with (2.74) and σ ( ∞ ) L ∼ =const . Compare also (2.101) for the approximation of (5.1) by the propor-tionality to ( W / ) used in (5.14).The pQCD-improved parton picture in (5.13) with the power-like W dependence (5.14) fails as soon as η ( W , Q ) <
1, or Q < , compare63 -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W =0.036 GeV Q ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W =0.1 GeV Q ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W =0.31 GeV Q ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W =1. GeV QE665EMCNMCpQCD BCDMSH1ZEUSCDP ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W =3.16 GeV Q ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W =10 GeV Q Figure 16:
In addition to the prediction from the CDP, also the prediction of F ( W ) = f · ( W / ) . from (2.124) and (2.125) (valid for 10GeV ≤ Q < ) for Q ≤ . ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W Q2=31.6 GeV ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W Q2=100 GeVE665EMCNMCBCDMSH1ZEUSCDPpQCD ) -2 (GeV -6 -5 -4 -3 -2 -1 ) , Q F ( W Q2=316 GeV
Figure 17:
As in fig.16, but for 31 . ≤ Q ≤ . W ,at any fixed value of Q , it leads to a logarithmic dependence of σ γ ∗ p ( W , Q ),and of F ( W , Q ), on the energy W , or on Λ ( W ) as given in (2.96), (2.99)and (5.9), F ( W , Q ) ∼ Q σ ( ∞ ) L ln Λ ( W ) Q + m (5.15) ∼ Q σ ( ∞ ) L ln α s ( Q ) G ( x, Q ) σ ( ∞ ) L ( Q + m ) ! , (for η ( W , Q ) ≪ . In distinction from the pQCD-improved parton picture in (5.13), for η ( W , Q ) <
1, the structure function in (5.15) depends logarithmically onthe gluon distribution function.The CDP with its W -dependent ( q ¯ q )-dipole-proton cross section is uniquein providing a smooth transition from the region of η ( W , Q ) >
1, withpieceful coexistence between the CDP and the pQCD-improved parton pic-ture, to the saturation region of η ( W , Q ) <
1, exclusively governed by theCDP. The pQCD-improved parton picture is not allowed to invade the regionof η ( W , Q ) <
1. The suppressed gluon distribution function at x < − ,occasionally with even negative results, from global fits (compare fig.7) ispresumably related to the inclusion of experimental data for F ( x, Q ) atvery low values of Q , where saturation must actually be taken into account,compare (5.15).The CDP, in distinction from the discrimination between a soft and a hardPomeron of the low- x Regge picture [29], only knows of a single Pomerongoverning both the regions of η ( W , Q ) > η ( W , Q ) <
1. Thetransition from η ( W , Q ) > η ( W , Q ) <
1, or to W → ∞ at fixed Q , is not associated with the transition to a (first or second) soft-Pomeronexchange. The transition corresponds to Λ ( W ) → ln Λ ( W ), or equiva-lently to α s ( Q ) G ( x, Q ) → ln( α s ( Q ) G ( x, Q )). The single Pomeron of theCDP has a less strong increase of the corresponding gluon distribution infig.7 with decreasing x , when compared with the hard Pomeron of the Reggefit. In figs.18 and 19, we show a comparison of our predictions for the longi-tudinal structure function F L ( x, Q ) with the experimental data. Since ouransatz for the dipole cross section incorporates transverse-size enhancement, ρ = const = 4 /
3, the theoretical results in figs.18 and 19 agree with the onesin figs. 2 and 3. 65igure 18:
The experimental results on the longitudinal structure function F L ( x, Q )from the H1 collaboration[39] compared with the prediction from the CDP. Figure 19:
As in fig.18, but showing the experimental results from the ZEUScollaboration[40]. Conclusion
In the present paper we reexamined and reanalysed DIS at low values of theBjorken scaling variable x ∼ = Q /W < . W -dependent color-dipole cross section. We explicitly showed that all essentialfeatures of the experimental data on the longitudinal and the transversephotoabsorption cross section can be understood as a consequence of thecolor-gauge-invariant q ¯ q -dipole-proton interaction, without relying on anyspecific parameterization of the dipole-proton cross section.We also examined the consistency between the description of the exper-imental data in the CDP and the description in terms of q ¯ q -sea and gluondistributions of the pQCD-improved parton picture within its range of valid-ity. The resulting ( Q , W ) plane of DIS at low x consists of only two regions,separated by the line η ( W , Q ) ∼ = 1.For η ( W , Q ) ∼ = Q / Λ ( W ) ≫ Q , colortransparency of the color-dipole-proton cross section becomes relevant: thestrong destructive interference among different dipole-proton scattering am-plitudes originating as a consequence of color-gauge invariance implies a( q ¯ q )-proton interaction that vanishes proportional to the transverse dipolesize, ~r ⊥ . The photoabsorption cross section correspondingly behaves asΛ ( W ) /Q , and the proton structure function (for 10GeV ≤ Q ≤ )as F ( x, Q ) = F ( W = Q /x ).The experimental data for η ( W , Q ) > q ¯ q )-sea-quark and the gluon distribution of the pQCD-improved parton picture. Consistency of the pQCD approach with the CDPrequires the gluon distribution function to be proportional to the satura-tion scale, Λ ( W ), and implies a definite value for the exponent C in therepresentation of the saturation scale, Λ ( W ) ∼ ( W ) C . The resultingprediction, C ∼ = 0 .
27 to C ∼ = 0 .
29, is consistent with the experimental data.The formulation of the CDP in terms of a W -dependent ( Q -independent)color-dipole-proton cross section is essential to arrive at this conclusion.With increasing energy, W , for any fixed dipole size, ~r ⊥ , again due tocolor-gauge invariance, the destructive interference among different ampli-tudes contributing to the q ¯ q interaction with the color field of the nucleondies out and leads to an ~r ⊥ -independent limit for the ( q ¯ q )-proton cross sec-tion. The q ¯ q -proton cross section “saturates” in this high-energy limit to67ecome identical to a cross section of hadronic size.The limit of increasingly larger energy, W , at fixed dipole size in the pho-toabsorption process is realized by W → ∞ at fixed Q , or η ( W , Q ) ≪ ( W ), and for W → ∞ at any fixed value of Q , itreaches the limit of ( Q = 0) photoproduction. The pQCD-improved par-ton picture fails, insofar as the photoabsorption cross section in this limitdepends logarithmically on the ( W -dependent) gluon distribution function.A concrete parameterization of the dipole cross section is necessary forthe interpolation between the regions of η ( W , Q ) > η ( W , Q ) <
1. We refined previous work in several respects, the representation of thelongitudinal-to-transverse ratio of the photoabsorption cross section by tak-ing into account the transverse-size enhancement of q ¯ q fluctuations origi-nating from transversely polarized photons, the extension of the CDP toinclude the region of x increasing to values close to x = 0 .
1, among oth-ers. We found agreement with the available DIS data in the full range of0 . ≤ Q ≤ for x ≤ . Acknowledgement
Useful discussions with Allen Caldwell and Reinhart K¨ogerler, as wellas the help of Prabdeep Kaur for providing plots of experimental data, aregratefully acknowledged. 68 ppendix A. Derivation of (4.1) and (4.2)
In this Appendix, we derive the photoabsorption cross section in the mo-mentum space (4.1) and (4.2) from the coordinate representation (2.39). Westart with the integral representation of the modified Bessel function K ( r ′⊥ Q ) = 12 π Z d ~k ′⊥ Q + ~k ′ ⊥ e − i~r ′⊥ · ~k ′⊥ (A.1)where r ′⊥ = | ~r ′⊥ | , Q = q Q . (A.2)(A.1) can be easily verified from the following equations, Z π dθ exp ( − iz cos θ ) = 2 πJ ( z ) , (A.3) Z ∞ dx xQ + x J ( r ′⊥ x ) = K ( r ′⊥ Q ) . (A.4)We compute the following quantity I L ( ~l ′ ⊥ ) ≡ Z d ~r ′⊥ K ( r ′⊥ Q ) e − i~r ′⊥ · ~l ′⊥ (A.5)Inserting (A.1), we find I L ( ~l ′ ⊥ ) = 1(2 π ) Z d ~r ′⊥ Z d ~k ′⊥ Z d ~k ′′⊥ Q + ~k ′ ⊥ )( Q + ~k ′′ ⊥ ) e − i~r ′⊥ · ( ~k ′⊥ + ~k ′′⊥ + ~l ′⊥ ) = Z d ~k ′⊥ Q + ~k ′ ⊥ )( Q + ( ~k ′⊥ + ~l ′⊥ ) )= Z d~k ′ ⊥ Z π dϑ Q + ~k ′ ⊥ )( Q + ( ~k ′⊥ + ~l ′⊥ ) ) (A.6)where ϑ is an angle between ~k ′⊥ and ~l ′⊥ . Recalling (4.9), we have dϑ = − ω ( M , M ′ ,~l ′ ⊥ ) dM ′ . (A.7)Inserting (2.44) and (A.6) and using(A.7), the integral in (2.39) becomes Z dr ′ ⊥ K ( r ′⊥ Q ) σ ( q ¯ q ) J =1 L p ( r ′⊥ , W )= Z dl ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( l ′ ⊥ , W ) (cid:16) I L (0) − I L ( l ′ ⊥ ) (cid:17) (A.8)= Z dl ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( l ′ ⊥ , W ) Z dM Z dM ′ ω ( M , M ′ ,~l ′ ⊥ ) Q + M ) − Q + M )( Q + M ′ ) ! , ~r ′⊥ , one finds ~r ′⊥ r ′⊥ q Q K ( r ′⊥ Q ) = i π Z d k ′⊥ ~k ′⊥ Q + ~k ′ ⊥ e − i~r ′⊥ · ~k ′⊥ (A.9)The integral I T ( l ′ ⊥ ) ≡ Z d ~r ′⊥ K ( r ′⊥ Q ) e − i~r ′⊥ · ~l ′⊥ (A.10)can be evaluated as I T ( l ′ ⊥ ) = 1(2 π ) Z d ~r ′⊥ Z d ~k ′⊥ Z d ~k ′′⊥ − ~k ′⊥ · ~k ′′⊥ Q ( Q + ~k ′ ⊥ )( Q + ~k ′′ ⊥ ) e − i~r ′⊥ · ( ~k ′⊥ + ~k ′′⊥ + ~l ′⊥ ) = 1 Q Z d ~k ′⊥ ~k ′⊥ · ( ~k ′⊥ + ~l ′⊥ )( Q + ~k ′ ⊥ )( Q + ( ~k ′⊥ + ~l ′⊥ ) )= 1 Q Z dk ′ ⊥ Z π dϑ ~k ′⊥ · ( ~k ′⊥ + ~l ′⊥ )( Q + ~k ′ ⊥ )( Q + ( ~k ′⊥ + ~l ′⊥ ) ) (A.11)Inserting (2.44) and (A.11), the integral in (2.39) becomes Z dr ′ ⊥ K ( r ′⊥ Q ) σ ( q ¯ q ) J =1 T p ( r ′⊥ , W )= Z dl ′ ⊥ ¯ σ ( q ¯ q ) J =1 T p ( ~l ′ ⊥ , W ) (cid:16) I T (0) − I T ( ~l ′ ⊥ ) (cid:17) (A.12)= 1 Q Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 T p ( ~l ′ ⊥ , W ) Z dM Z dM ′ ω ( M , M ′ ,~l ′ ⊥ ) M ( Q + M ) − M + M ′ − ~l ′ ⊥ Q + M )( Q + M ′ ) , which leads to (4.2). Appendix B Correction terms
In this Appendix, we will give the explicit expressions for the correctionterms, ∆ σ ( m ) γ ∗ L,T p ( W , Q ) and ∆ σ ( m ) γ ∗ L,T p ( W , Q ) in (4.17), which in conjunctionwith the dominant term guarantee the required bound on M ′ that is givenby m ≤ M ′ ≤ m ( W ) ≡ m from (4.7).With the splitting of the integrand (4.10) as applied to the dominantterm, the integrations over dϑ in (4.12) and (4.13) yield the following results70or the correction terms in (4.14),∆ σ ( m ) γ ∗ L p ( W , Q ) + ∆ σ ( m ) γ ∗ L p ( W , Q ) = − αR e + e − Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W ) · Z ( √ ~l ′ ⊥ + m ) ( √ ~l ′ ⊥ − m ) dM S L, ( M ,~l ′ ⊥ , Q , m )+ Z m ( m − √ ~l ′ ⊥ ) dM S L, ( M ,~l ′ ⊥ , Q , m ) ! , (B.1)and ∆ σ ( m ) γ ∗ T p ( W , Q ) + ∆ σ ( m ) γ ∗ T p ( W , Q ) = − αR e + e − Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 T p ( ~l ′ ⊥ , W ) · Z ( √ ~l ′ ⊥ + m ) ( √ ~l ′ ⊥ − m ) dM S T, ( M ,~l ′ ⊥ , Q , m )+ Z m ( m − √ ~l ′ ⊥ ) dM S T, ( M ,~l ′ ⊥ , Q , m ) ! , (B.2)where S L, ( M ,~l ′ ⊥ , Q , m ) = Q ( Q + M ) π − ϑ ( M ,~l ′ ⊥ , m ) π (B.3) − Q ( Q + M ) √ X (cid:18) − π arctan q Y ( M ,~l ′ ⊥ , Q , m ) (cid:19) S L, ( M ,~l ′ ⊥ , Q , m ) = Q ( Q + M ) ϑ ( M ,~l ′ ⊥ , m ) π (B.4) − Q ( Q + M ) √ X π arctan q Y ( M ,~l ′ ⊥ , Q , m ) S T, ( M ,~l ′ ⊥ , Q , m ) = M − Q ( Q + M ) π − ϑ ( M ,~l ′ ⊥ , m ) π (B.5) − M − ~l ′ ⊥ − Q ( Q + M ) √ X (cid:18) − π arctan q Y ( M ,~l ′ ⊥ , Q , m ) (cid:19) S T, ( M ,~l ′ ⊥ , Q , m ) = M − Q ( Q + M ) ϑ ( M ,~l ′ ⊥ , m ) π (B.6) − M − ~l ′ ⊥ − Q ( Q + M ) √ X π arctan q Y ( M ,~l ′ ⊥ , Q , m ) In (B.3) - (B.6), ϑ ( M ,~l ′ ⊥ , m , ) = arccos m , − M − ~l ′ ⊥ M q ~l ′ ⊥ , (B.7) X ( M ,~l ′ ⊥ , Q ) = ( M − ~l ′ ⊥ + Q ) + 4 Q ~l ′ ⊥ ,Y ( M ,~l ′ ⊥ , Q , m , ) = Q + ( M − q ~l ′ ⊥ ) Q + ( M + q ~l ′ ⊥ ) · − cos ϑ ( M ,~l ′ ⊥ , m , )1 + cos ϑ ( M ,~l ′ ⊥ , m , )71 − Q + ( M − q ~l ′ ⊥ ) Q + ( M + q ~l ′ ⊥ ) · ( q ~l ′ ⊥ + M ) − m , ( q ~l ′ ⊥ − M ) − m , > q ~l ′ ⊥ > m , For photoproduction, Q = 0, from (B.2) and (B.5), (B.6), we have thesimplified expression ∆ σ ( m ) γ ∗ T p ( W , Q = 0) + ∆ σ ( m ) γ ∗ T p ( W , Q = 0) == − αR e + e − Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 T p ( ~l ′ ⊥ , W ) · (B.8) · Z ( p ~l ′ ⊥ + m ) ( p ~l ′ ⊥ − m ) dM M π − ϑ ( M ,~l ′ ⊥ , m ) π − M − ~l ′ ⊥ | M − ~l ′ ⊥ | (cid:18) − π arctan q Y ( M ,~l ′ ⊥ , Q = 0 , m ) (cid:19)! + Z m ( m − p ~l ′ ⊥ ) dM M ϑ ( M ,~l ′ ⊥ , m ) π − π arctan q Y ( M ,~l ′ ⊥ , Q = 0 , m ) !! for m > q ~l ′ ⊥ .Specializing the dipole cross section in (B.1) to the ansatz (3.13) and its J = 1 projections in (3.14), the longitudinal cross section in (B.1) becomes∆ σ ( m ) γ ∗ L p ( W , Q ) + ∆ σ ( m γ ∗ L p ( W , Q ) = − αR e + e − σ ( ∞ ) π Z a dy y √ − y Z ( √ ~l ′ ⊥ + m ) ( √ ~l ′ ⊥ − m ) dM S L, ( M ,~l ′ ⊥ , Q , m )+ Z m ( m − √ ~l ′ ⊥ ) dM S L, ( M ,~l ′ ⊥ , Q , m ) ! , (B.9)while for the transverse cross section, we have∆ σ ( m ) γ ∗ T p ( W , Q ) + ∆ σ ( m γ ∗ T p ( W , Q ) = − αR e + e − σ ( ∞ ) π Z a dy (1 − y ) √ − y Z ( √ ~l ′ ⊥ + m ) ( √ ~l ′ ⊥ − m ) dM S T, ( M ,~l ′ ⊥ , Q , m )+ Z m ( m − √ ~l ′ ⊥ ) dM S T, ( M ,~l ′ ⊥ , Q , m ) ! , (B.10) ~l ′ ⊥ on the right-hand side in (B.9) and (B.10) is to be replaced by the inte-gration variable y , ~l ′ ⊥ = 2 ¯Λ sat ( W )3 y (B.11)72or photoproduction, from (B.8), we have∆ σ ( m ) γ ∗ T p ( W , Q = 0) + ∆ σ ( m γ ∗ T p ( W , Q = 0) = (B.12)= − αR e + e − σ ( ∞ ) ( W ) π Z a dy − y √ − y ·· Z ( √ ~l ′ ⊥ + m ) ( √ ~l ′ ⊥ − m ) dM M π − ϑ ( M ,~l ′ ⊥ , m ) π − M − ~l ′ ⊥ | M − ~l ′ ⊥ | (cid:18) − π arctan q Y ( M ,~l ′ ⊥ , , m ) (cid:19) + Z m ( m − √ ~l ′ ⊥ ) dM M ϑ ( M ,~l ′ ⊥ , m ) π − π arctan q Y ( M ,~l ′ ⊥ , , m ) Appendix C. Derivation of (4.28), (4.29) and(4.30)
In this Appendix, we derive the approximate expression for σ domγ ∗ L/T p in thelarge Q region. we expand the integrand I L/T ( ~l ′ ⊥ , M , Q ) in (4.19) and(4.20) in terms ofˆ x = ~l ′ ⊥ Q + m , ˆ y = ~l ′ ⊥ m , ˆ z = m ~l ′ ⊥ , (C.1)all of which are small in the limit Q ≫ ~l ′ ⊥ ≫
1. Each term in the integrandbecomes − Q M + Q (cid:12)(cid:12)(cid:12) m m = ˆ x ˆ x + ˆ y + o (ˆ x ˆ z ) . (C.2) Q q ~l ′ ⊥ ( ~l ′ ⊥ + 4 Q ) ln q ~l ′ ⊥ ( ~l ′ ⊥ + 4 Q ) √ X + ~l ′ ⊥ (3 Q − M + ~l ′ ⊥ ) Q + M i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m m = − ˆ x ˆ x + ˆ y + ˆ x + 3ˆ x ˆ y + 6ˆ y x + ˆ y ) ˆ x + · · · . (C.3)12 ln M + Q √ X + M − ~l ′ ⊥ + Q (cid:12)(cid:12)(cid:12) m m = ˆ x h ˆ x ˆ y (ˆ x + ˆ y ) + · · · i , (C.4) − Q + ~l ′ ⊥ q ~l ′ ⊥ ( ~l ′ ⊥ + 4 Q ) ln q ~l ′ ⊥ ( ~l ′ ⊥ + 4 Q ) √ X + ~l ′ ⊥ (3 Q − M + ~l ′ ⊥ ) Q + M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m m = ˆ x ˆ x + ˆ y + 2ˆ x + 3ˆ x ˆ y − y x + ˆ y ) ˆ x + · · · (C.5)73nserting (C.2)-(C.5) into (4.18), we find σ domγ ∗ L p ( W , Q ) = αR e + e − Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W )ˆ x h ˆ x + 3ˆ x ˆ y + 6ˆ y x + ˆ y ) ˆ x + · · · i , (C.6)and σ domγ ∗ T p ( W , Q ) = αR e + e − Z d~l ′ ⊥ ¯ σ ( q ¯ q ) J =1 L p ( ~l ′ ⊥ , W )ˆ x h ˆ x + 3ˆ y x + ˆ y ) ˆ x + · · · i . (C.7)Recalling ˆ x ˆ y = ξη , (C.8)and introducing the integration variable y defined by (4.21),ˆ x = 23 ηy , (C.9)we find σ domγ ∗ L p ( W , Q ) = αR e + e − (cid:16) σ ( ∞ ) ( W ) π (cid:17) Z / (3 a ) dy y √ − y ˆ x G L ( ξη ( W , Q ) )= αR e + e − (cid:16) σ ( ∞ ) ( W ) π (cid:17) η ( W , Q ) AG L ( ξη ( W , Q ) ) , (C.10)and σ domγ ∗ T p ( W , Q ) = αR e + e − (cid:16) σ ( ∞ ) ( W ) π (cid:17) Z / (3 a ) d ˆ y − ˆ y/ √ − ˆ y ˆ x G T ( ξη ( W , Q ) )= αR e + e − (cid:16) σ ( ∞ ) ( W ) π (cid:17) η ( W , Q ) × h log 1 + A − A − A ] G T ( ξη ( W , Q ) ) . (C.11)Here A ≡ s − a = 0 .
951 for a = 7 (C.12)and the functions G L ( ξη ( W ,Q ) ) and G T ( ξη ( W ,Q ) ) are defined by (4.29) and(4.30). Noting that A ∼ A − A − A = 2 Aρ ( ǫ = 16 a ) ∼ ρ ( ǫ = 16 a ) (C.13)we reach the approximate expression for the dominant parts given in (4.28).74 eferenceseferences