Semi-inclusive Deep Inelastic Scattering at Small-x

We study the semi-inclusive hadron production in deep inelastic scattering at small- x . A transverse momentum dependent factorization is found consistent with the results calculated in the color-dipole framework in the appropriate kinematic region. The transverse momentum dependent quark distribution can be studied in this process as a probe for the small- x saturation physics. Especially, the ratio of the quark distributions as functions of transverse momentum at diﬀerent x demonstrates strong dependence on the saturation scale. The Q 2 dependence of the same ratio is also studied by applying the Collins-Soper-Sterman resummation method.

There have been compelling theoretical arguments and experimental evidence that saturation [1,2] plays a very important role in high-energy hadronic scattering processes, and an effective theory called Color-Glass-Condensate emerges to describe the relevant physics [3,4].
In particular, the parton distributions at small-x (x is the longitudinal momentum fraction of the hadron carried by the parton) and/or of large nucleus can be calculated from this effective theory, and they all demonstrate a saturation behavior.The rapidity (Y = ln 1/x) evolution of these distributions are controlled by a nonlinear JIMWLK equation [5,6,7], which has been thoroughly studied in the last decade.By employing the saturation physics, the deep inelastic scattering (DIS) structure function measured by the HERA experiments can be very well described [8,9,10], as well as the diffractive structure functions [11,12,13,14] and vector-meson production [15,16,17].Forward hadron suppression in d + Au collisions at RHIC experiments also indicates the importance of saturation in the small-x region [18,19,20,21].All these successes have encouraged rapid developments in small-x physics in the last few years [22].
One of the key predictions of this effective theory is the transverse momentum dependence of the parton distributions in big nucleus at small-x, especially the gluon distribution [3,23,24].In the inclusive DIS process, the gluon distribution is convoluted into a dipole cross section, which only provides an indirect probe.In this paper, we argue that the transverse momentum dependent parton distributions can be directly probed in semi-inclusive processes, for example, in the semi-inclusive hadron production in DIS (SIDIS) [25].In this process, there are separate momentum scales: the momentum transfered by the virtual photon squared Q 2 and the transverse momentum of the observed hadron in the final state p ⊥ .Because of the additional hard momentum scale Q 2 , the final state hadron transverse momentum can be directly related to that of the parton distribution in the nucleon/nucleus when Q 2 is much larger than p 2 ⊥ .The relevant QCD factorization theorem [26,27,28] has been rigorously studied for the leading-power contribution to the differential cross section.
In the following calculations, we will extend this factorization argument to the case that involves saturation physics, and we argue that the transverse momentum dependent factorization formula is still valid in the so-called geometric scaling regime [29,30,31,32], when Q 2 is much larger than the saturation scale Q 2 s , but saturation effects are still important.As an example, we will demonstrate this factorization for the semi-inclusive DIS process at small-x.
FIG. 1: Semi-inclusive DIS at small-x, where the cross represents the quark fragmenting into the final-state hadron.The quark carries the momentum fraction ξ of the virtual photon, and transverse momentum k ⊥ .
In the semi-inclusive DIS process, we observe the final state hadron with characteristic kinematic variables, such as the longitudinal momentum fraction z h of the virtual photon and transverse momentum p ⊥ .The usual DIS kinematics variables are defined as and z h = P h • P A /q • P A , where P h , ℓ, P A and q are momenta for the final state hadron, incoming lepton and nucleon (nucleus), and the exchanged virtual photon, respectively.The transverse momentum p ⊥ is usually defined in the center of mass frame of the virtual photon and the incoming hadron.In Fig. 1, we show the schematic diagram for this process in the dipole framework at small-x [24], where the virtual photon splits into a quark-antiquark dipole, then scatters off the nucleon/nucleus target before the quark (antiquark) fragments into a final state hadron.In the current fragmentation region (forward direction of the virtual photon), the quark-fragmentation contribution will dominate the cross section.
The differential cross section for the above process can be calculated in the dipole formalism [24] or in the classical Yang-Mills effective theory approach [23], and we readily have where D(z) is the quark fragmentation function into the final state hadron, F (q ⊥ , x B ) the unintegrated gluon distribution defined below, ξ = z h /z, and the fragmenting quark's trans-verse momentum is k ⊥ = p ⊥ /z. 1 The variable b here is a suppressed variable in F (q ⊥ , x B ) which is defined as the impact parameter with respect to the center of the nucleus.If one assumes that the nucleus is cylinder like and nucleons are uniformly distributed inside, one can easily see that the b dependence is trivial and evaluate d2 b which yields πR 2 with R being the effective radius of the nucleus.The phase space factor dP is defined as , and H reads as where We have also taken the massless-quark limit in the above formula for simplicity, and the first term is the contribution from transversely polarized photons while the second one corresponds to longitudinally polarized photons.The unintegrated gluon distribution is defined through the Fourier transform of the dipole cross section: where T q q is the scattering amplitude, and is characterized by the saturation scale Q 2 s which depends on x.This unintegrated gluon distribution contains the saturation physics, which diagrammatically represents the multiple scattering of the quark-antiquark dipole on nucleon/nucleus target.When integrating over transverse momentum p ⊥ and the fragmentation function using h dzD q→h (z) = 1, the above formula will reproduce the total DIS cross section in ep(A) → eX 2 .
In this paper, we are interested in the factorization property of the above differential cross section in the kinematic region where Q 2 is much larger than the final-state hadron transverse momentum p 2 ⊥ .In the current fragmentation region, z h is of order 1.Therefore the quark transverse momentum k ⊥ is of the same order as p ⊥ .Furthermore, we assume that Q 2 is also much larger than the saturation scale Q 2 s which sets the transverse momentum q ⊥ of the unintegrated gluon distribution.Under these limits, we will be able to study the transverse-momentum-dependent factorization, where we can separate the transverse momentum dependence of the final-state hadron into the incoming quark distribution and fragmentation function and/or soft factor [26,28].An important advantage to utilize the above limits is that we can apply the power counting to analyze the leading power contribution, and neglect the higher order corrections in terms of p 2 ⊥ /Q 2 where p ⊥ stands for the typical transverse momentum (p ⊥ ∼ k ⊥ ∼ q ⊥ ).
Moreover, we notice that the integral of Eq. ( 2) is dominated by the end point contribution of ξ ∼ 1 where ǫ 2 f is in order of k 2 ⊥ [24].In order to extract the leading power term from this equation, we can introduce a delta function in Eq. ( 2): , and integrate out ξ first.This delta function can be further expanded in the limit of where a logarithmic term in the above expansion is power suppressed and has been neglected.
The contribution from the second term is also power suppressed.To see this more clearly, we can substitute ǫ 2 f = ξΛ 2 /(1 − ξ) into Eq.( 3), and the hard coefficient H will have an overall factor (1 − ξ) 2 .Combining this with the delta function expansion, we will find that the second term is the above expansion is power suppressed relative to the first one.Applying the delta function expansion in Eqs. ( 2) and (3), we will obtain the leading contribution to the differential cross section in the limit of where We noticed that the longitudinal photon contribution is power suppressed and has been dropped.
On the other hand, a transverse momentum dependent factorization can also be used to describe the SIDIS process when the hard scale (Q 2 ) is much larger than the transverse momentum scale p 2 ⊥ .To leading power of p 2 ⊥ /Q 2 , for example, we will have following factorization formula for the differential cross section for the semi-inclusive DIS [26,27,28], where q(x B , k ⊥ ), D(z h , p 1⊥ ), S(λ ⊥ ), and H are the transverse-momentum-dependent (TMD) quark distribution, fragmentation function, soft factor, and hard factor, respectively.We emphasize that the above factorization is valid in the leading power of p 2 ⊥ /Q 2 , and all power corrections have been neglected. 4The energy dependent parameter ζ, ζ and ρ have been introduced to regulate the light-cone divergences in the associated functions.In a special frame, we can simplify them as . The transverse momentum resummation can be performed by studying the evolution equation in terms of these variables.We notice that, since the TMD quark distribution starts with nontrivial leading order expansion, the TMD fragmentation and soft factor in Eq. ( 8) are trivial at this order: (2) (λ ⊥ ).However, at higher order, for example, the gluon radiation contribution to the SIDIS process in Fig. 1, we need to take into account nontrivial expansion of the fragmentation function and the soft factor up to α s order [28].
The above factorization formalism was studied without considering the small-x resummation effects [26,27,28].Here, we assume that the factorization argument can still hold when the hard momentum scale Q 2 is much larger than the saturation scale Q 2 s and we can use the power counting method to study the leading contribution in this process.On the other hand, if Q 2 s is the same order as Q 2 (or even larger), the power counting used to argue the TMD factorization is no longer valid, and we will not have a TMD factorization.
Similar studies for the heavy quark-antiquark production in pA (AA) collisions have also been discussed in [34].As an important check, in the following we will compare the prediction from the TMD formula Eq. ( 8) to the dipole result Eq. ( 6) in the same kinematic region, Q 2 ≫ p 2 ⊥ (Q 2 s ).To do that, we need to calculate the TMD quark distribution in nucleon/nucleus at small-x.This quark distribution is defined as [26] q(x, k ⊥ ) = 1 2 where P is the momentum for the hadron, x and k ⊥ are longitudinal momentum fraction of the hadron and transverse momentum carried by the quark.In the above equation, L is the gauge link introduced to guarantee the gauge invariance of the above definition [26,28].At this particular order, the gluon splitting contribution to the TMD quark distribution can be calculated in the k t -factorization approach at small x.We show the relevant Feynman diagrams in Fig. 2, where (b-d) diagrams come from the gauge link contributions.These diagrams have to be taken into account because the gauge field connecting to the hadron state (nucleon/nucleus) are dominated by the A + component in the k t -factorization calculations.
Their contributions are important to obtain a consistent and gauge-invariant result.The derivation is straightforward, and we have where A has been defined in Eq. ( 7).This is the quark distribution calculated in the k t -factorization.In order to compare to the results we obtain above in the color-dipole formalism, we need to extrapolate in the leading logarithmic approximation at small-x, i.e., replacing the unintegrated gluon distribution F (q ⊥ , x ′ ) by F (q ⊥ , x) in the above equation.
Following this replacement, we will reproduce the differential cross section Eq. ( 6) calculated in the dipole framework at the leading order of p 2 ⊥ /Q 2 .Therefore, we have demonstrated that the small-x calculation of the differential cross section for the SIDIS process is consistent with the TMD factorization at this particular order.At even higher order, we will have to take into account the contributions from the fragmentation function and soft factor.At this order, they are trivial: D(z h , p 1⊥ ) = D(z h )δ (2) (p 1⊥ ) and S(λ ⊥ ) = δ (2) (λ ⊥ ) where D(z h ) is the integrated fragmentation function.We further argue that the TMD factorization will work at higher orders as well, because the power counting is valid when s ) as we mentioned above.
In the leading logarithmic approximation at small-x, we can further integrate out ξ in Eq. ( 10): which is consistent with the result calculated before [23].A number of interesting features of this quark distribution have been discussed in the literature [23,24].For example, in the small k ⊥ limit, the quark distribution saturates: These two features will be manifest if we employ the GBW model for the unintegrated gluon distribution: and λ = 0.28 [8].Note that while the large q ⊥ behavior of the unintegrated gluon distribution F (q ⊥ , x) is incorrect in the GBW model (it falls exponentially instead of a power law), this bad feature does not translate to the TMD quark distribution: the convolution with the splitting kernel in Eq. ( 7) insures the proper leading-twist behavior.In Fig. 3 (left panel), we show the ratio of the TMD quark distribution xq(x, k ⊥ ) relative to that at x = 10 −2 as a function of k ⊥ for x = 10 −4 and x = 10 −3 , respectively.From this figure, we can clearly see that the ratio remains unchanged when k ⊥ goes to 0, whereas the ratio is proportional to the ratio of Q 2 s at different x when k ⊥ is large.This clearly demonstrates that the transverse momentum dependence provides an important information on the saturation physics.We have shown that these TMD quark distributions can be studied in semi-inclusive DIS process.
Furthermore, the transverse momentum dependence is also sensitive to the QCD dynamics of the small-x evolution.In the above example, we took the simple parameterization from the GBW model [8].This result shall be modified by the nonlinear evolution.For example, at large q ⊥ , the unintegrated gluon distribution scales as (q 2 ⊥ /Q 2 s ) −λc where λ c is the anomalous dimension [18,30,35,36].In the DGLAP domain, we have λ c = 1 whereas in the BFKL domain it is λ c = 0.5.By solving the BK equation, it was found that λ c = 0.63 for large rapidities Y = ln 1/x [30,37].With this modification, the ratio of the TMD quark distribution at large k ⊥ will approach (Q 2 s ) λc instead of Q 2 s .Another important QCD dynamics effect is the transverse momentum resummation [27], which will affect the Q 2 dependence of the k ⊥ spectrum.In the results we plotted in the left panel of Fig. 3, this effect was not considered, which correspond to the low Q 2 results, for example, at Q 2 = Q 2 0 = 10GeV 2 .This effect can be studied by applying the Collins-Soper-Sterman resummation method [27].There have been great applications of this method to various high-energy processes, in particular, in the semi-inclusive DIS at HERA [38] where important consequences have been observed.To demonstrate this effect in the TMD quark distribution at small-x we calculated above, we take the double leading logarithmic approximation (DLLA) to solve the evolution equation for the quark distributions.Under this approximation, we can write down quark distribution at higher Q 2 in terms of that at lower Q 2 0 [39, 40]5 : where the Sudakov form factor at the DLLA is defined by where we have also included a non-perturbative form factor contribution c 0 r 2 ln Q 2 /Q 2 0 [41].This resummation effect will shift the transverse momentum distribution to higher end when Q 2 increases.As an example, in Fig. 3 (right panel) we show the typical changes for the quark distributions at Q 2 = 20, 50, 100GeV 2 as compared to the lower Q 2 0 = 10GeV 2 , with the following parameters: fixed coupling α s = 0.3, and c 0 = 0.1 for the non-perturbative input for the form factor [38].From this plot, we can see that indeed, the transverse momentum distribution becomes harder when Q 2 is larger.We notice that neglecting the resummation effects in the left panel of Fig. 3 will introduce additional theoretical uncertainties in the predictions.However, we expect that much of the resummation effects will be cancelled out in the ratios of the quark distributions at the same Q 2 .
In conclusion, we have studied the semi-inclusive DIS processes at small-x, and found that the quark distribution studied can be used as a probe of saturation physics.Especially, the ratio of the quark distributions is crucially depending on the saturation scale.We also studied the quark distribution at different Q 2 values, and found that the resummation effects shift the distribution to larger k ⊥ with larger Q 2 .An ideal place to study this physics will be an electron-ion collider in the near future [42], where large nucleus target will provide an additional direction to study saturation.Meanwhile, we notice that the ratios plotted in Fig. 3 qualitatively agree with the experimental data from HERA [43].Of course, in order to compare to these data, we have to take into account the fragmentation contributions to calculate the differential cross sections.We also notice that an extension to a study on the gluon transverse momentum distributions [44] will have to consider both small-x and transverse momentum resummations.The result from this paper shall provide us confidence to carry out these important studies.

FIG. 2 :
FIG. 2:Transverse-momentum-dependent quark distribution calculated from small-x gluon splitting.The double line represents the gauge link contribution from the TMD quark distribution definition.