Challenges with large transverse momentum in semi-inclusive deeply inelastic scattering

We survey the current phenomenological status of semi-inclusive deeply inelastic scattering at moderate hard scales and in the limit of very large transverse momentum. As the transverse momentum becomes comparable to or larger than the overall hard scale, the differential cross sections should be calculable with fixed order perturbative QCD (pQCD) methods, while small transverse momentum (transverse-momentum-dependent factorization) approximations should eventually break down. We find large disagreement between HERMES and COMPASS data and fixed order calculations done with modern parton densities, even in regions of kinematics where such calculations should be expected to be very accurate. Possible interpretations are suggested.

Figure 1

Momentum labeling in amplitudes for $2\to N$ partonic scattering kinematics. The dashed lines represent partons of unspecified type and flavor. The dot on the end of $k_1$ is to indicate this is the parent parton of the detected hadron. The other momenta are integrated in SIDIS. All final state lines are meant to represent energetic but mutually highly noncollinear massless partons. If two lines become nearly collinear, it should be understood that they merge into a single line. If a line becomes soft, it is simply to be removed. In both cases, $2\to N$ kinematics reduce to $2\to N-1$ kinematics. (a) is a general $2\to N$ amplitude. For perturbatively large $P_{H\mathrm{T}}$, $N\ge 1$, and the lowest order contribution is $O({\alpha }_s)$, corresponding to the $2\to 2$ kinematics in (b). We will consider in addition the $2\to 3$ kinematics in (c), which appear at $O({\alpha }_s^2)$.

Figure 2

Figure 1 reduces to this handbag structure when $|k^2|\sim {\mathrm{\Lambda }}_{\mathrm{QCD}}^2$. The $k$ line becomes part of the target parton, and the lower blob is part of a PDF in the square-modulus amplitude integrated over final states.

Figure 3

Figure 4 from Ref. [24]. The differential cross section was integrated over $x$, $z$, and bins of $Q$ with H1 cuts, calculated with both leading order and next-to-leading order, and compared with ${\pi }^0$ production data from Ref. [23]. Here, $p_T$ corresponds to our $P_{H,\mathrm{T}}$; see Eq. (1). Note the large correction from $O({\alpha }_s^2)$.

Figure 4

Calculation of $O({\alpha }_s)$ and $O({\alpha }_s^2)$ transversely differential multiplicity using code from Ref. [24], shown as the curves labeled DDS. The bar at the bottom marks the region where $q_{\mathrm{T}}>Q$. The PDF set used is cjnlo [33], and the FFs are from Ref. [34]. Scale dependence is estimated using $\mu ={(({\zeta }_{Q}Q{)}^2+({\zeta }_{q_{\mathrm{T}}}{q}_{\mathrm{T}}{)}^2)}^{1/2}$ where the band is constructed point by point in $q_{\mathrm{T}}$ by taking the minimum and maximum of the cross section evaluated across the grid ${\zeta }_Q\times {\zeta }_{q_{\mathrm{T}}}=[1/2,1,3/2,2]\times [0,1/2,1,3/2,2]$ except ${\zeta }_Q={\zeta }_{q_{\mathrm{T}}}=0$. The red band is generated with ${\zeta }_Q=1$ and ${\zeta }_{q_{\mathrm{T}}}=0$. A lower bound of 1 GeV is placed on $\mu$ when $Q/2$ would be less than 1 GeV.

Figure 5

Ratio of data to theory for several near-valence region panels in Fig. 4. The grey bar at the bottom is at 1 on the vertical axis and marks the region where $q_{\mathrm{T}}>Q$.

Figure 6

Calculation analogous to Fig. 4 but for ${\pi }^{+}$ production measurements from Ref. [32].

Figure 7

Ratio calculation analogous to Fig. 5 but for ${\pi }^{+}$ production measurements from Ref. [32].