Consistency tests for the extraction of the Boer-Mulders and Sivers functions

At present, the Boer-Mulders (BM) function for a given quark flavour is extracted from data on semi-inclusive deep inelastic scattering (SIDIS) using the simplifying assumption that it is proportional to the Sivers function for that flavour. In a recent paper we suggested that the consistency of this assumption could be tested using information on so-called difference asymmetries, i.e. the difference between the asymmetries in the production of particles and their anti-particles. In this paper, using the SIDIS COMPASS deuteron data on the $\langle\cos\phi_h\rangle$, $\langle\cos 2 \phi_h\rangle$ and Sivers difference asymmetries, we carry out two independent consistency tests of the assumption of proportionality, but here applied to the sum of the valence-quark contributions. We find that such an assumption is compatible with the data. We also show that the proportionality assumptions made in the existing parametrizations of the BM functions are not compatible with our analysis, which suggests that the published results for the Boer-Mulders functions for individual flavours are unreliable. The $\langle\cos\phi_h\rangle$ and $\langle\cos 2 \phi_h\rangle$ asymmetries receive contributions also from the, in principle, calculable Cahn effect. We succeed in extracting the Cahn contributions from experiment (we believe for the first time) and compare with their calculated values, with interesting implications.


I. INTRODUCTION
There is a major effort at present to progress beyond a knowledge of collinear parton distribution functions (PDFs) and fragmentation functions (FFs) and to obtain information about the transverse momentum dependent (TMD) versions of these functions. In extracting these distributions from data, a standard parametrization is usually adopted (see for example [1]), which involves various simplifying assumptions. In addition, because of lack of sufficient data, additional relations between different TMD-functions are sometimes assumed. We focus on, and examine, the particular assumption that the BM functions for a particular flavor are proportional to the Sivers functions of the same flavor.
In our recent paper [2], we showed that the difference asymmetries in SIDIS allow the determination of the valence quark TMDs in a model-independent way, without any assumptions about the sea quark or gluon densities. Also, that using the difference asymmetries, one can test many of the basic assumptions in the standard parametrization, such as factorization of the x B and z h dependencies, the Gaussian flavor-and hadron-independent k ⊥ behavior etc.
In [2], we derived two types of relations-between the hcos ϕ h i, hcos 2ϕ h i and Sivers asymmetries-that allow tests of the simplifying assumption used in extracting the Boer-Mulders (BM) function i.e. its proportionality to the Sivers function [3,4], an assumption motivated by model calculations [5]. In addition, present analyses make a further assumption concerning the Q 2 evolution of these functions for a given quark flavor, which, as explained in the next section, is theoretically inconsistent.
Our previously published tests [2] were formulated without taking into account the Cahn effect, which inevitably contributes to these asymmetries. In this paper we show how these tests are modified when the Cahn effect is included.
We then use COMPASS SIDIS measurements of the hcos ϕ h i, hcos 2ϕ h i and Sivers asymmetries on a deuteron target to test for the consistency of the assumed relation between BM and Sivers functions.
We work with the so-called difference asymmetries of the following general structure. If the asymmetries for h þ and h − have the form where σ h þ ;h − and Δσ h þ ;h − are the unpolarized and polarized cross sections, respectively, then The difference asymmetries are expressed in terms of the usual asymmetries A h þ ;h − and the ratio of the corresponding multiplicities [6], where r is the ratio of unpolarized SIDIS cross sections for production of h − and h þ : As shown in Ref. [2], the advantage of using the difference asymmetries is that, based only on charge conjugation (C) and isospin (SU(2)) invariance of the strong interactions, they are expressed purely in terms of the best known valence-quark distributions and fragmentation functions; sea-quark and gluon distributions do not enter. For a deuteron target there is the additional simplification that, independently of the final hadron, only the sum of the valence-quark distributions enters.
The paper is organized as follows: the notation and conventions for the various TMD functions and the used experimental asymmetries are explained in Secs. II and III; in Sec. IV, we formulate the two tests for the assumed relation between the BM and Sivers functions. They are based on the hcos ϕ h i and hcos 2ϕ h i azimuthal asymmetries of the final hadrons in unpolarized SIDIS, and the Sivers asymmetry for unpolarized leptons on transversely polarized nucleons. Because the above two unpolarized asymmetries receive contributions from both the BM and Cahn effects, we are able also to extract information about the Cahn effect; in Sec. V, we apply these tests using the COMPASS SIDIS data on deuterons.

II. PARAMETRIZATION OF THE TMD DISTRIBUTIONS
A. The polarized parton distribution functions Conventionally, a typical spin-dependent TMD density Δfðk ⊥ ; x B ; Q 2 Þ has been parametrized following several simplifying assumptions: (1) The transverse-momentum dependence on k ⊥ is factorized from the x B dependence. (2) The k ⊥ dependence is flavor and hadron independent, and usually assumed to be a Gaussian. We adopt these two simplifications. (3) An additional simplifying assumption is that TMD functions are proportional to the related collinear parton distribution functions (PDFs) and fragmentation functions (FFs). The Q 2 -evolution is usually assumed to be given via the collinear PDFs and FFs, i.e. making the ansatz: This is, however, physically unacceptable because it leads to gluons contributing to the evolution of nonsinglet combinations of quark densities. Since we deal here only with valence quark densities we replace this simplification by an ansatz for the valencequark densities. Hence we take the Q 2 evolution to be controlled via: Note, however, that we do not think this difference in approximating the evolution is important when assessing the impact of our tests on the published BM data.
In this paper we consider only the difference asymmetries on a deuteron target. As mentioned earlier, in these asymmetries only one combination of parton density enters-the sum of the valence-quark TMD functions: Below we present the parametrizations of the valencequark Q V unpolarized, BM and Sivers distributions and the Collins FFs following the above simplifying anzatz. We work in the approximation Oðk ⊥ =QÞ, neglecting terms of the order Oðk 2 ⊥ =Q 2 Þ.

B. The unpolarized TMD parton distributions and fragmentation functions
The unpolarized TMD PDFs and FFs are parametrized proportional to the corresponding collinear functions times a Gaussian-type, flavor-and hadron-independent k 2 ⊥ =p 2 ⊥ dependence [7]. In accordance with this for the valencequark unpolarized TMD PDFs f Q V =p ðx B ; k 2 ⊥ ; Q 2 Þ and TMD FFs D h=q V ðz h ; p 2 ⊥ ; Q 2 Þ we adopt the parametrizations [8]: where Q V ðx B ; Q 2 Þ is the sum of the collinear valence-quark PDFs: hk 2 ⊥ i and hp 2 ⊥ i are parameters extracted from study of the multiplicities in unpolarized SIDIS.
The parameters hk 2 ⊥ i (and hp 2 ⊥ i are basic as they enter in the normalization functions in all TMD asymmetries. At present the experimentally obtained values are controversial: (1) hk 2 ⊥ i ≈ 0.25 GeV 2 and hp 2 ⊥ i ≈ 0.20 GeV 2 [9], extracted from the old EMC [10] and FNAL [11] SIDIS data (2) hk 2 ⊥ i ¼ 0.18 GeV 2 and hp 2 ⊥ i ¼ 0.20 GeV 2 [12], derived from the P T -spectrum of HERMES data and confirmed by Monte Carlo calculations. The extraction of the BM functions in [4] utilized these values. An analysis [7] of the more recent available data on multiplicities from HERMES [13] and COMPASS [14] separately, gives quite different values: These values are obtained using a kinematical cut on z h < 0.6 and they change slightly on placing the cut at z h < 0.7.
Further we shall be able to comment on this controversial situation, since the Cahn effect, which contributes to the asymmetries which we study and extract from data, is calculable, and depends sensitively on hk 2 ⊥ i and hp 2 ⊥ i.

C. The BM and Sivers distributions
The Sivers function describes the correlation between the spin of the nucleon S, its momentum P, and the momentum of the quark k ⊥ , via a term proportional to S · ðk ⊥ × PÞ [15], while the BM function describes the correlation between the spin of the quark s q and the momentum of the quark k ⊥ , via a term proportional to s q · ðk ⊥ × PÞ [16].
The k ⊥ , x B dependence of the valence-quark BM and Sivers distribution functions Δf Q V J ðx B ; k ⊥ ; Q 2 Þ, (J ¼ BM, Sivers), is assumed to factorize [1,4] in the form with Here, the N Q V J ðx B Þ are unknown functions, and M J , or equivalently hk 2 ⊥ i J , where are unknown parameters. As mentioned earlier, hk 2 ⊥ i is supposed to be known from multiplicities in unpolarized SIDIS.

The
Collins fragmentation functions (FFs) Δ N D h=q↑ ðz; p ⊥ Þ describe phenomenologically the spindependent part of the fragmentation functions of transversely polarized quarks, with transverse spin s q and 3-momentum p q , into hadrons h with momentum p ⊥ , transverse to the direction of the initial quark [17]: It relates the transverse momentum of the produced hadron to the transverse spin of the quark and leads to nonuniform azimuthal distribution of final hadrons around the initial quark direction. The valence-quark Collins functions Δ N D h=u V ↑ ðz h ; p ⊥ ; Q 2 Þ are parametrized [2] proportional to the corresponding unpolarized valence-quark collinear fragmentation functions D h u V ðz h ; Q 2 Þ: where The unknown quantities are N h=u V C ðz h Þ and M C (often M C is denoted by M [18] or M h [1,19]), or equivalently hp 2 ⊥ i C : which characterizes the p ⊥ dependence. As mentioned earlier, hp 2 ⊥ i is known from multiplicities in unpolarized SIDIS.

III. THE UNPOLARIZED AZIMUTHAL AND SIVERS ASYMMETRIES
The general expression for the difference cross section in SIDIS, for unpolarized leptons on transversely polarized nucleons, with polarization S T , l þ N ↑ → l þ h þ X, in the kinematic region P T ≃ k ⊥ ≪ Q, is given in terms of the   [2,8].
Here P T is the transverse momentum of the final hadron in the γ Ã -nucleon c.m. frame, and z h , Q 2 and y are the usual measurable SIDIS quantities: with l and l 0 , P and P h the 4-momenta of the initial and final leptons, and initial and final hadrons. Note that where M is the target mass (in this paper the deuteron mass) and E the lepton laboratory energy. Throughout the paper we follow the notation and kinematics of Ref. [1]. In current analyses [3,4], in extracting the BM function, an additional simplifying assumption is made, namely, the BM function is taken proportional to its chiral-even partner-the Sivers function. Clearly the resulting BM function depends critically on the validity of this assumption. Our fundamental aim is to check this key assumption using only measurable quantities-the difference asymmetries-and without requiring any knowledge about the TMD functions.
The difference azimuthal cosϕ h , cos2ϕ h and sinðϕ S −ϕ h Þ, Sivers, asymmetries that single out these terms are: The corresponding x B -dependent asymmetries, integrated over P 2 T ; z h and Q 2 , that we shall work with are

IV. TESTS FOR THE RELATION BETWEEN THE BM AND SIVERS FUNCTIONS ON A DEUTERON TARGET
In the difference asymmetries on deuterium, only the sum of the valence-quarks Q V ¼ u V þ d V enters for any final hadron h. Therefore, in contrast to the currently used assumption of proportionality between BM and Sivers functions for each quark and antiquark flavor, we assume the simpler relation: where λ Q V is a constant. Using the parametrizations (11), Eq. (27) implies that the k ⊥ dependencies in BM and Sivers functions are the same, while the x B dependencies are proportional: The hcos ϕ h i and hcos 2ϕ h i azimuthal asymmetries in unpolarized SIDIS receive contributions from both the BM function and the purely kinematic Cahn effect. The connection (27) between the BM and Sivers functions leads to relations between the BM induced contributions in hcos ϕ h i or hcos 2ϕ h i and the Sivers asymmetries. Here we present the resulting relations between the x B -dependent hcos ϕ h i or hcos 2ϕ h i and Sivers asymmetries. These relations are particularly simple and predictive if the bins in x B are small enough, so as to neglect the Q 2 -evolution of the collinear functions inside the bins.
A. Tests based on the asymmetry A cos ϕ h UU Here we present the relation between the x B -dependent cos ϕ h and Sivers asymmetries on a deuteron target, when the Q 2 -evolution of the collinear parton densities and fragmentation functions can be neglected inside the considered x B -bin. The standard parametrizations (7), (8) and (11), (12) are used.
(1) The asymmetry A cos ϕ h UU has two twist-3 contributions of 1=Q-order from the BM function and from the Cahn effect. For the x B -dependent difference asymmetry on a deuteron A cos Here C h Cahn and C h BM are constants, given by: The function Φðx B Þ is completely fixed by kinematics, the same for all final hadrons: hQi 2 is some mean value of Q 2 for each x B -bin (see Appendix A), M d is the mass of the deuterium target.
The notation ½D h q V is shorthand for the following: and Analogously for ½Δ N D h q V ↑ ðz h Þ we have: (2) Following the same path, for the x B -dependent Sivers difference asymmetry on a deuteron : which expresses the unknown x B dependence of the BM-distribution in terms of the measurable x Bdependent Sivers asymmetry. The assumed relation (27) between the BM and Sivers functions then leads to the following relation between the x B -dependent azimuthal cos ϕ h -asymmetry A cos ϕ h UU ≡ hcos ϕ h i and the Sivers asymmetry on a deuteron target: Here the function Φðx B Þ and the constant C h Cahn are given by (33) and (30) where There are two important consequences of Eq. (42), which we shall use further: (1) It represents a direct and simple test of the relation (27) between the BM and Sivers TMD-functions, in which only measurable quantities enter, and no knowledge about the TMD functions is required. (2) The different x B dependences of the Cahn and BM contributions, allow us to disentangle the Cahn contribution from the BM one in our fits to the experimental data.
B. Tests based on the asymmetry A cos 2ϕ h UU (1) The asymmetry A cos 2ϕ h UU has two contributions: the leading twist-2 contribution from BM function and the twist-4 contribution of 1=Q 2 -order from the Cahn effect.
Following the same path as in obtaining Eq. (29) (details are given in Appendix B), we obtain the x Bdependent difference asymmetry on a deuteron, The only difference is that the integration from the convolution in k ⊥ , in the contribution from the Cahn effect, cannot be carried out analytically and it remains in the final expressions-these are the integrals over ϕ and k ⊥ in Eq. (48). Here we give only the final expression.
For the x B -dependent difference asymmetry on a deuteron A cos 2ϕ h ;h−h UU;d ðx B Þ, when the Q 2 -dependence in Q V and in the FFs can be neglected, we obtain whereΦðx B Þ is a completely fixed kinematic function, the same for all final hadrons: The contribution from the Cahn effect is of order 1=Q 2 compared to the BM contribution. The con-stantsĈ h BM andĈ h Cahn are: (2) The Sivers asymmetry is given in (39).
(3) The assumed relation (27) between the BM and Sivers functions leads to the following relation between the x B -dependent azimuthal cos 2ϕ h asymmetry A cos 2ϕ h UU ≡ hcos 2ϕ h i and the Sivers asymmetry on a deuteron target: This relation and Eq. (45) were previously obtained in [2] without including the 1=Q 2 -Cahn contribution. However, as present measurements are performed at rather low Q 2 , now we have included the 1=Q 2 -suppressed Cahn contribution as well. This is important for comparing to existing data, which we shall do in the next section.
The constantsĈ h f BM is expressed in terms of the parameter λ Q V and the TMD-fragmentation functions: The . We perform the fits in three steps. First, we form the difference asymmetries A h þ −h − J ; J ¼ hcos ϕ h i; hcos 2ϕ h i; Siv from the corresponding usual asymmetries A h þ j and A h − j for positive and negative charged hadron production [6]: Here r is the ratio of the unpolarized x B -dependent SIDIS cross sections for production of negative and positive hadrons r ¼ σ h − ðx B Þ=σ h þ ðx B Þ measured in the same kinematics [6]. As the available data for the different asymmetries is in different x B bins, which do not match we need to interpolate the data. It turns out that a linear interpolation is adequate. Hereafter we work with the interpolation functions A h AE J ðx B Þ only. When we determine the errors of the difference asymmetries we assume that data is not correlated.
Second, we choose the Q 2 interval where the Q 2 dependence of the collinear PDF's and FFs can be neglected. In the COMPASS kinematics to each value of hQ 2 i corresponds one definite value of hx B i, thus fixing the Q 2 interval we fix also the x B -interval. Using the available CTEQ parametrizations for the PDFs [22], we see that there is almost no Q 2 dependence in the valence-quark distributions u V and d V in the whole Q 2 -range covered by COMPASS, Q 2 ≃ ½1-17 GeV 2 , i.e. in the whole x B interval. To get some feeling for the Q 2 dependence of the fragmentation function D h þ u V to charged hadrons, bearing in mind that h AE production is strongly dominated by π AE production, in Fig. 1 we plot the dependence of D π þ u V on Q 2 for different values of z h . We use the parametrization in [23] obtained using the recent HERMES [24] and preliminary COMPASS data [25] on multiplicities. This parametrization is in qualitative agreement with the one obtained from analysis of the latest COMPASS data [26]. We see that, aside from the small values of Q 2 ≲ 1.8 GeV 2 , the Q 2 dependence is weak. We thus consider it reasonably safe to use the following fitting interval x B ∈ ½0.014; 0.13 corresponding to Q 2 ∈ ½1.77; 16.27 GeV 2 .
Third, we fit the parameters in Eqs. (42) and (50) using χ 2 analysis. There are two ways to utilize (42) and (50), we shall follow both of them:  The problem with this approach, however, is that the Cahn constants depend both on the chosen parametrizations for the FFs, which don't differ so much, and on the values of the parameters hk 2 ⊥ i, hp 2 ⊥ i, which, as discussed in Sec. II B, are rather poorly known and vary considerably. Consequently the main interest in this second approach will be to compare the calculated Cahn constants with those determined by fitting the parameters as in (A) above. The used χ 2 for the hcos ϕ h i h AE and hcos 2ϕ h i h AE asymmetries are: which take into account the different widths of x B -bins in which the data is collected.
In this way the tested relations are put in the standard form "experiment" ¼ "theory". Note however, that the situation here is rather peculiar because the errors of experimental data ΔF exp ðx B Þ and ΔF exp ðx B Þ contain not only the errors of the asymmetries ΔA cos ϕ h UU;d ; ΔA cos 2ϕ h UU;d and ΔA Siv UU;d , but the fitting parameter as well. We have: In (53) the upper limit x f ¼ 0.13 is fixed by the existing data for both A cos ϕ h UU;d and A cos 2ϕ h UU;d asymmetries, and x i is determined by the requirement that it is safe to ignore Q 2variation.
To test quantitatively the applicability of Eqs. (42) and (50) for small x B we have made series of fits with increasing x i starting with x iðminÞ ¼ 0.006 and going up to x iðmaxÞ ¼ 0.025 and we introduce the quantity χ 2 =Δx, which is χ 2 normalized to the length of the fitting interval Δx ¼ x f − x i . (It is the continuum analogue of χ 2 per degree of freedom in the discrete case.) The obtained χ 2 =Δxðx i Þ functions for both asymmetries are plotted on In the next two subsections, we present the obtained values and standard deviations of the fitted parameters. The values correspond to the best fit of the available data with χ 2 defined as above. We use Monte Carlo simulation in order to estimate the deviations of the fitting parameters. On the basis of the experimental data and assuming they have a Gaussian distribution we construct 10 3 sets of "virtual experimental data". For each virtual experimental data set we determine corresponding best-fit parameters.  Fig. 3, panel (a). Note that the Sivers asymmetry A Siv;h þ −h − UT;d ðx B Þ is almost zero and rather poorly determined, which suggests, and is proven in our fits, that the corresponding fitting parameter C h f BM will be poorly determined. (ii) ðBÞ In approach (B) we need an expression for C h Cahn with integration over the measured P T interval in COMPASS: where the limits of integration are ½P T;min ; P T;max ¼ ½0.1; 1.0 GeV and z h ¼ ½0.2; 0.85 [20]. (If the integration over P T is in the interval P T ∈ ½0; ∞ we recover Eq. (30).) We need also the FF for unidentified charged hadrons ½D h þ q V . To estimate this, we neglect the contribution from produced protons, (about 1%) and use: where we have used SU(2)-invariance for the pions, implying: and D K þ d V ¼ 0, which follows from the quark content of kaons; this assumption is used in all present analyses in extracting the kaon FFs.
We use two of the available parametrizations for the FFs: AKK [27] and LSS [23] and find that the value of C h Cahn is not sensitive to the used parametrization; also, as expected, it is not sensitive to the chosen hQ 2 i. However it is very sensitive to the values hk 2 ⊥ i and hp 2 ⊥ i. We find that the quality of the fits in the approach B, with one exception, are considerably worse than in the approach A when both C h Cahn and C h f BM are fitted-see Fig. 4 This can be verified also in Table I Note the different scales in the panels.
(B) are presented. The presented errors correspond to 1 standard deviation. Note that from the analytic expression Eq. (30), it follows that C h Cahn should be negative, which is in agreement with the value obtained from the fit.
To the best of our knowledge, this is the first time that the Cahn contribution C h Cahn has been determined from data and it is puzzling that its value is in agreement with a calculated result based on the early values of the Gaussian parameters hk 2 ⊥ i ¼ 0.18; hp 2 ⊥ i ¼ 0.20 GeV 2 , which are supposed to be ruled out by later measurements. Fig. 5a. Note that now both asymmetries are poorly determined with large relative   C. Comparison to the existing published extraction of the BM functions [3,4] In this paper we have tested the assumption of proportionality of the BM and Sivers functions for the sum of valence quarks Q V ¼ u V þ d V , [Eq. (27)]. However, in Refs. [3,4] the BM functions have been extracted from the cos 2ϕ asymmetry assuming proportionality for each quark and antiquark flavor q separately:

B. Test using the COMPASS data on
A legitimate question arises as to the compatibility of the two approaches i.e. whether Eqs. (60) and (27) are compatible. Here we study this question. Under the assumption of Eqs. (60) one obtains: where Equation (61) is compatible with our assumption of proportionality Eq. (27) if Note that, at we have Δ ¼ 0 and we obtain Eq. (27). The values for λ u;d are those obtained in [4] assuming λū ¼ −1; λd ¼ þ1 for the antiquarks i.e.
The parametrization of the Sivers function for each quark flavor is taken from [28]: with where: As the dependence on k 2 ⊥ is the same for both the BM and Sivers functions, in Fig. 6 we compare only the dependence on x B of the two functions ðλ u þ λ d þ λū þ λdÞΔf Q V Siv ðx B Þ and Δðx B Þ. For the unpolarized PDFs the CTEQ6 parametrization was used.
From this figure it is clear that, even accounting for the enormous errors induced by the errors of the Sivers functions, jΔj is much bigger than jðλ u þ λ d þ λū þ λdÞΔf Q V Siv j, which is just the opposite to Eq. (63). This suggests that the extraction of the Boer-Mulders function in the literature [3,4] is unreliable.

VI. CONCLUSIONS
We had shown previously [2] that data on difference asymmetries allow one to test the assumed relation of proportionality between the BM and Sivers functions, which is currently used in the extraction of the BM function from data. In the present paper we perform two independent tests of this assumption applied, however, to the sum of the valence-quark TMD distributions, (27), using the COMPASS SIDIS data [20,21]  However, in the published extractions of the BM functions [3,4], obtained in a completely different kind of analysis, based on the available parametrizations of both Sivers and Collins functions, it is assumed that BM and Sivers functions are proportional for each quark and antiquark separately [Eq. (60)]. This would agree with our result, based only on measurable quantities, if λ u ≈ λū ≈ λ d ≈ λd ≈ λ Q V , which does not correspond to the values and their errors obtained in [3,4].
We have also determined the kinematical Cahn contribution, both directly from a fit to the data (as far as we know for the first time) and from a calculation.

ASYMMETRY
The structure function F cos ϕ h UU that determines the azimuthal A cos ϕ h UU asymmetry, Eq. (24), has two twist-3 contributions of 1=Q-order from the Cahn effect and the BM TMDs: For the difference cross sections ðh −hÞ on the deuteron target, it is only the sum of the valence-quark parton densities Q V enter these functions and for h ¼ π þ ; K þ ; h þ they read [8] F cos ϕ h ;h−h FIG. 6. A comparison between jðλ u þ λ d þ λū þ λdÞΔf Q V S j (the black curve which is almost 0) and jΔj (white curve). The shaded areas are the corresponding statistical errors. The parametrization of f q Siv is taken from [28] and the values of λ q are from [4].
Further, after neglecting Q 2 dependence in the collinear FFs, and replacing the integration over Q 2 by ΔQ 2 times the function evaluated at some average value hQi (or equivalentlyȳ) for each x B bin, we obtain the simple x B -dependent expression for the asymmetry: The function Φðx B Þ is given in Eq. (33), it is completely fixed by kinematics, the same for all final hadrons.   (25), has two contributions-the leading twist-2 contribution from the BM functions and the twist-4 contribution of 1=Q 2 order from the Cahn effect: Again we shall consider only difference cross sections ðh −hÞ on the deuteron target. In this case, it is only the sum of the valence-quark parton densities Q V that enter these functions. For the BM contribution on the deuteron target for h ¼ π þ ; K þ ; h þ , we have [8] F Equation (B4) implies that if we can neglect Q 2 dependencies in Q V and in the FF, the x B and z h dependencies will factorize, and ½Δ N D h u V ↑ is given in Eq. (38). The Cahn contribution to the asymmetry looks more complicated as the integration over k ⊥ that comes from the convolution of the TMD PDFs and FFs cannot be fulfilled analytically. Nevertheless, it has the same structure: