ECCE unpolarized TMD measurements

The study of transverse momentum dependent distribution and fragmentation functions originated with the first nonzero single transverse spin asymmetries that were discovered in the past by the E704 experiment in fixed-target proton-proton collisions [1]. Both of the most famous effects, the Sivers [2] effect and the Collins [3] effect that were initially suggested to describe these asymmetries require an intrinsic transverse momentum dependence on the distribution and fragmentation sides. While those two effects are also explicitly spin dependent, unpolarized transverse momentum dependent, TMD, functions are furthermore of importance in many processes. From the low-$x$ processes where the transverse momentum dependence of the gluon distribution function may affect potential saturation effects to the transverse momentum dependence of PDFs that may affect the actual transverse momentum dependent cross sections of Higgs or heavy boson production at the LHC, TMDs play an important role.

For the most part the information on explicitly transverse momentum dependent distribution functions originates from Drell-Yan, DY, and heavy boson production measurements, predominantly at the Tevatron and the LHC, as well as fixed target DY. However, due to the nature of these processes, only a very limited knowledge on the flavor structure of TMDs can be obtained this way. On the other hand, in semi-inclusive deep inelastic scattering, SIDIS, measurements, one mostly obtains cross sections that are convolutions of TMD distribution and fragmentation functions. The fragmentation functions provide an additional flavor sensitivity that is neither available in the Drell-Yan type measurements nor in any jet type DIS measurements.

At the moment the information on the transverse momentum dependent fragmentation functions is also rather limited, with essentially only one pure measurement from $e^{+}e^{-}$ annihilation [4] available and a few SIDIS measurements from fixed target experiments [5, 6].

Some permutations of all this data have been combined in global fits that try to extract the flavor and transverse momentum dependent distribution and fragmentation functions simultaneously. The most recent of those are refs.  [7, 8, 9, 10], but none have so far extracted the TMDs from all of this data simultaneously.

Closely related are various theoretical questions that are not entirely answered, such as which regions in phase space can actually be interpreted via TMDs and how that may limit the available data [11]. The authors identify regions where TMD factorization should be applicable in their approach and regions where other treatments may be relevant, such as collinear factorization. One finds, that at lower $x$ and scales $Q^{2}$, only very small transverse momenta can be interpreted that way while higher transverse momenta likely involve higher order collinear processes. At higher scales and $x$ more of the phase space appears to be applicable for TMD interpretations. The question which regions are applicable in various factorization frameworks is currently under active discussion among theorists, where the EIC data help cover a much larger phase space than so far. However, not only the TMD region is relevant, but also the collinear region and the transition between those are of great interest. There are calculations that show how those can be related [12, 13]. This relation is commonly used when taking into account TMD evolution of TMD related cross sections and asymmetries. However, some aspects of the scale evolution of TMDs are non-perturbative in itself and two fits mentioned above, despite having used the same data, come to rather different evolution effects of even the unpolarized TMD PDFs.

The EIC in general and ECCE in particular can help answer most of these questions as a very large range in phase space is covered from the low scales of most fixed target DIS experiments to the high scales of the DY and even close to some heavy boson production measurements. The particle identification capabilities of ECCE furthermore may allow the flavor decomposition of TMDs and may answer the question whether valence and sea quark intrinsic transverse momenta in the nucleon are distributed differently. Also the regions of applicability of TMDs and collinear PDFs/FFs can be explored in detail.

Naturally, any unpolarized TMDs are also the baseline for any polarized TMDs such the as the Sivers and Collins functions and the related Tensor charge of the nucleon. So, improving the knowledge and precision on the unpolarized TMDs in turn will improve the precision of these polarized functions as well.

Given that the cross sections vary quite drastically with both $x$ and $Q^{2}$, it is often useful not to display the actual cross sections for SIDIS measurements but rather multiplicities. In multiplicities the yield of SIDIS events in a particular $x$, $Q^{2}$, $z$ and $P_{T}$ bin is normalized by the yield of DIS events in the same $x$ and $Q^{2}$ bin. Effectively, this normalizes the SIDIS cross sections with their matching DIS cross section and rather highlights the fragmentation aspects of the measurements (although the transverse momentum is of course still a convolution of the participating intrinsic transverse momenta from distribution and fragmentation functions while in the TMD factorization regime). For the full 4-dimensional theoretical extraction, however, cross sections differential in all 4 variables are necessary.

The multiplicities can be seen in Fig. 1 extracted as a function of the transverse momentum for pions, kaons and protons in three example $x$ and $Q^{2}$ bins to highlight their behavior. The differences between true and reconstructed multiplicities are assigned as a systematic uncertainties which are displayed as uncertainty boxes. This procedure highlights the amount of smearing in the kinematic variables and clearly overestimates the actual uncertainties as smearing and particle mis-identification would in reality be unfolded. However, the regions where large smearing occurs will eventually also show increased uncertainties due to the unfolding while not necessarily being as large as shown in this crude estimation. One can see peaking and then rapidly falling multiplicities that resemble the Gaussian behavior generally seen in fixed target SIDIS measurements at relatively small transverse momenta. In this example figure, therefore a simple double-Gaussian is fitted to the multiplicities (taking only the statistical uncertainties into account) and shows a reasonable description of the pseudo-data. It is interesting to note that pions have the narrowest distribution in all three sample bins while both kaons and protons appear to be wider. Such a feature has been seen also in the Belle data [4] and is in part described by pythia, but the ordering is different, since in the $e^{+}e^{-}$ case (in data and simulation) the differences were most pronounced between mesons and baryons. With the EIC also the transition into the higher transverse momenta, where collinear factorization is needed can be studied in detail.

In Fig. 2 the pion, kaon and proton multiplicities are shown for all $x,Q^{2}$ kinematic bins as a function of $P_{T}$. As can be seen, the scale of the multiplicities is very similar for all $x$ and $Q^{2}$ bins, highlighting the normalization of the rapidly changing cross sections in these variables. A change in the shapes is also visible where generally lower $x$ and $Q^{2}$ bins show narrower distributions than the at higher $x,Q^{2}$ bins where the distributions appear to be wider. This figure also highlights the large range that can be covered at the EIC.

As the multiplicities highlight the fragmentation aspects of the TMD cross sections, it is also very interesting to study the $z$ dependence together with the $P_{T}$ dependence. This is shown for pions in Fig. 3 in a few $z$ bins and for three example $x,Q^{2}$ bins. As the transverse momentum that is available in the fragmentation strongly depends on $z$, one should see that the shapes are different in different $z$ bins, appearing narrower at low and high $z$ with a wider distribution at intermediate $z$. In reality this is relatively hard to infer directly from this figure as the scale for the various $z$ bins is vastly different due to the rapidly changing fragmentation functions. As such also double Gaussians are fitted to these sample bins. The widths of the narrower Gaussian, that most likely represents the TMD part of the distribution, indeed shows an increase with $z$ as has been found both in fixed target SIDIS experiments as well as by Belle as is displayed in Fig. 4 for all three hadron types. One also sees that the supposedly larger widths from kaons and protons really only appear at low $z$ where especially for protons contributions from target fragmentation may cause these differences.

The whole range for of the covered phase space is shown for pions in various $z$ bins in Fig. 5 where again it can be seen that the multiplicities can extracted in all four kinematic variables with high precision.

Once the pseudo-data is analyzed one can obtain the relevant improvements on the TMD PDFs and FFs within the framework of the existing global fits, as discussed in the previous section. As an example the current and expected uncertainties following the unpolarized extraction of Ref. [7] are shown in Fig. 9. These show for selected $x$ and $z$ slices the expected impact on the intrinsic transverse momentum dependence of the distribution and fragmentation functions when including the EIC pseudo-data. One can see that the impact is quite substantial. It also shows some of the limitations of the re-weighting approach as in the distribution functions a node-like feature is visible that originates from the functional form and the overall normalization of this particular global fit. Nevertheless, it highlights the impact the EIC data will have on the TMDs and their theoretical understanding, including the scale evolution of TMDs, eventually their flavor dependence and finally the three-dimensional momentum picture of the nucleon for various parton flavors.

In conclusion, the unpolarized transverse momentum dependent hadron multiplicities as well as cross sections can be extracted over a large range in the DIS kinematic variables $x$ and $Q^{2}$ and the semi-inclusive variables $z$ and $P_{T}$. The ECCE detector configuration is well suited to obtain a comparable precision as the reference detector parametrization used in the studies of the Yellow Report. From these measurements the precision on the transverse momentum dependent distribution and fragmentation functions can be greatly increased, likely allowing for a detailed, flavor dependent extraction of these TMDs. The lever arm of these measurements will definitely remove the uncertainties that currently exist on the TMD evolution. Also, further theoretical insights into the regions of applicability of TMD factorization, collinear factorization, etc can be explored in a large area of phase space.