A Better Angle on Hadron Transverse Momentum Distributions at the EIC

We propose an observable $q_*$ sensitive to transverse momentum dependence (TMD) in $e N \to e h X$, with $q_*/E_N$ defined purely by lab-frame angles. In 3D measurements of confinement and hadronization this resolves the crippling issue of accurately reconstructing small transverse momentum $P_{hT}$. We prove factorization for $\mathrm{d} \sigma_h / \mathrm{d}q_*$ for $q_*\ll Q$ with standard TMD functions, enabling $q_*$ to substitute for $P_{hT}$. A double-angle reconstruction method is given which is exact to all orders in QCD for $q_*\ll Q$. $q_*$ enables an order-of-magnitude improvement in the expected experimental resolution at the EIC.


I. INTRODUCTION
A deeper understanding of the emergent properties of the nucleon, such as confinement and hadronization, has been a frontier of nuclear and particle physics research since the inception of Quantum Chromodynamics (QCD) five decades ago. An important one-dimensional view of the nucleon is provided by the deep-inelastic scattering (DIS) process e − ( ) + N (P ) → e − ( ) + X, where the scattering is mediated by an off-shell photon of momentum q = − (with Q 2 ≡ −q 2 > 0). Confinement is probed by measurements of x = Q 2 /(2P · q), the momentum fraction carried by the colliding parton inside the nucleon N . A more intricate view is obtained by identifying a hadron h in semi-inclusive DIS (SIDIS), e − ( ) + N (P ) → e − ( ) + h(P h ) + X. Here measurements of the longitudinal momentum fraction z = (P · P h )/(P · q) that the hadron retains when forming from the struck quark give insight into the complex dynamics of hadronization. Measuring the hadron's transverse momentum P hT relative to q gives access to a threedimensional view of the confinement and hadronization processes for N and h, together with spin correlations that probe these processes.
A key challenge in experimental studies of TMDs is that measurements of P hT require reconstructing the photon momentum (or Breit frame) to great accuracy to avoid loss of precision on P hT = | P hT | Q. A misreconstruction of by O(∆) leads to a misreconstruction of q and therefore P hT by O(∆), which is a large uncertainty for P hT Q. For example, for a nominal measurement at P hT /z = 1 GeV with Q = 20 GeV, a typical detector resolution of ∆ = 0.5 GeV leads to a 50% uncertainty. This puts in peril the EIC physics program to unveil the dynamics of hadronization and confinement in the kinematic region with the largest sensitivity.
In this paper we construct a novel SIDIS observable, q * , designed to be maximally resilient against resolution effects while delivering the same sensitivity to TMD dynamics as P hT . The key insight is that while the magnitude of the electron and hadron three momentum is subject to limited detector resolution, modern tracking detectors deliver near-perfect resolution on the angles of charged particle tracks. We will therefore construct q * to satisfy the following three criteria: (i) it is purely defined in terms of lab-frame angles and the beam energies; (ii) at small values q * Q, the differential cross section dσ/dq * , including spin correlations, still satisfies a rigorous factorization theorem in terms of the standard TMD PDFs and FFs; (iii) it does not dilute the statistical power of the available event sample. Our construction is inspired by, but features key differences to, the Drell-Yan φ * η observable in hadron-hadron collisions [23], which has enabled tests of perturbative QCD from permil-level Z-pole measurements at the Tevatron and LHC [24][25][26][27][28][29].
Below we define q * in detail, prove the factorization theorem for q * with standard TMDs, and evaluate the expected detector resolution, statistical power, and resilience against systematic biases of q * versus P hT .
II. CONSTRUCTING q * Consider the target rest frame shown in Fig. 1a where the nucleus N is at rest and the z-axis is along the incoming lepton beam. The lepton momenta and define the lepton plane as the x-z plane. We wish to take advantage of the high-precision reconstruction of polar angles (rapidities) and azimuthal angles in the EIC lab frame. Here Definition of φ rest acop needed to construct q * . Momenta are not to scale. We define the conventional Trento frame [30] for SIDIS, as well as the target rest frame and the EIC frame. φ h is the azimuthal separation between P h and in the Trento frame. The acoplanarity angle in the target rest frame is φ rest acop ≡ π − ∆φ rest , where ∆φ rest is the azimuthal separation between P h and . (b) In-plane geometry for leading-power kinematics P h,⊥ /Q 1, where we can approximate that P h is along the same direction as q. This geometry yields the double-angle formulas in Eq. (2) Q, x, y with angular measurements.
we give results in terms of EIC frame rapidities in the light target mass limit M Q, with full M dependence in Supplement A. The acoplanarity angle in the target rest frame, φ rest acop , is defined by tan φ rest acop = −P h,y /P h,x , where P h,x and P h,y are components of P h . From Fig. 1a, it is obvious that tan φ rest acop ∝ sin φ h P hT , where φ h is the azimuthal angle of P h in the Trento frame. We may thus use φ acop as a precision probe of the hadron transverse momentum P hT . 1 To work out the full relation between φ rest acop and P hT , consider now the leading-power (LP) kinematics illus-1 Lab-frame acoplanarity angles are also useful as a measure of transverse momentum in jet production in DIS [31,32]. Unlike this work, the observable of Refs. [31,32] does not feature the same experimental improvements since traditional jet axis reconstruction is not angular, and it also has nonglobal logarithms that are nonperturbative in the TMD region of interest. trated in Fig. 1 b, where λ ∼ P hT /(zQ) 1. We find We now wish to express Q and y = (P · q)/(P · ) in terms of final-state angles in the EIC frame, which is defined by a 180 • rotation about our rest frame y axis and then a boost along the z-axis, so φ EIC acop = −φ rest acop . From Fig. 1b, momentum conservation gives x = x , q x = − x , and P hT Q implies θ h + θ e + α = π/2. We find y = 1 − sin θ h / cos α and . Boosting to the EIC frame: where η i are the EIC frame pseudorapidities of the outgoing lepton i = e and hadron i = h, ∆η ≡ η h − η e , and s = (P + ) 2 . This construction agrees with the double-angle formula in Ref. [33]. However, Ref. [33] uses the struck quark angle in a tree-level picture, while our Eq. (2) uses the hadron angle and holds to all orders in α s , and up to power corrections in P hT /(zQ), which controls the distance to the Born limit. The O(λ) corrections to Eq. (2) are given in Supplement B.
To exploit the proportionality in Eq. (1) to probe P hT , we define an optimized observable: Expanding in P hT zQ it has a simple LP limit Thus for TMD analyses, Q 2 , x, y, and q * can all be measured from the beam energy P 0 EIC and angular variables. We may also define a dimensionless variable, This is analogous to the setup for the φ * η observable in Drell-Yan [23]. We expect the purely angular observables q * and φ * SIDIS to be measured to much higher relative precision compared to the transverse momentum P hT .
To compute the spectrum differential in x, y, z and q * , we insert the leading-power measurement δ(q * + sin φ h P hT /z) and analytically perform the integral over d 2 P hT = dP hT P hT dφ h . As an explicit example, we work out the contribution from W cos(2φ h ) U U . Using Eq. (7): The φ h integral, which is specific to the structure function, can only depend on q * b T by dimensional analysis, and in this case yields a simple cos(q * b T ). In total, the LP cross section differential in q * is: We stress that the TMD PDFs and FFsf 1 , , . . . are the same as in the standard factorization for the P hT spectrum and TMD spin correlations. This is analogous to the role of the unpolarized [41,42] and Boer-Mulders [43] TMD PDFs in the Drell-Yan φ * η . The factorization theorem can equivalently be written in terms of momentum-space TMDs, see Supplement D. Crucially, definite subsets of these TMDs contribute to the even and odd parts of the spectrum under q * → −q * . The odd parts are accessible through the asymmetry dσ(q * > 0) − dσ(q * < 0). Contributions can be further disentangled experimentally through their unique dependence on λ e , S µ and , i.e., by taking asymmetries with opposite beam polarizations and by measuring cross sections as a function of y . 2 E.g., the double asymmetry for q * → −q * and λ e → −λ e as a function of x and |q * | gives direct access to the worm-gear T functiong ⊥ 1T (x, b T ).

IV. EXPERIMENTAL SENSITIVITY
To show the improvement that q * makes for TMD analyses, we use Pythia 8.306 [44] to simulate e − p → e − X at 18 × 275 GeV 2 , disabling QED corrections. We select on h = π + and apply the following cuts ("SIDIS cuts"): We recommend reconstructing S µ using a rotation by θ h to maintain a purely angular measurement, see Supplement C, which is justified at LP. Note that the transversity and pretzelosity PDFs have a degenerate contribution S T (h 1 + h ⊥ 1T /4) to q * in Eq. (9), while the worm-gear L function h ⊥ 1L drops out, due to q * being even under φ h ↔ π − φ h . Encouragingly, the subleading-power Cahn effect ∝ cos(φ h ), which pollutes standard P hT , also drops out for the same reason. Statistical sensitivity to TMD nonperturbative model coefficients at the 10 fb −1 EIC when measuring P hT (blue) or q * (red) compared to the prior (MAPTMD22 fit [45], gray). Despite its superior resolution, q * enjoys comparable statistical sensitivity to P hT .
Scaled to an integrated EIC luminosity of 10 fb −1 , this results in a sample with N π + = 4.18×10 8 . We first assess the expected detector resolution of q * compared to P hT . We apply Gaussian smearing to the final-state electron and hadron momenta, assuming a tracking detector that matches the performance given in Ref. [16]: a resolution of σ p /p = 0.05% p/GeV ⊕0.5% on the momentum p = | p| of charged particles in the central barrel region |η| < 1, 0.05% p/GeV ⊕1% in the inner endcap 1 < |η| < 2.5, and 0.1% p/GeV ⊕ 2% in the outer endcap 2.5 < |η| < 3.5. A particle is forward (backward) if it has 1 < |η| < 3.5 and η > 0 (η < 0). We assume a fixed angular resolution of σ θ = σ φ = 0.001. We ignore the electromagnetic calorimeter as its e − energy resolution is expected to be a factor of two worse than the tracker [16]. Our key results for the detector resolution on q * compared to P hT are shown in Fig. 2 for the case of a backward e − and a central h, which accounts for the largest share (41%) of the event sample. We see that q * improves over the resolution of P hT /z by an order of magnitude across the strongly confined TMD region q * , P hT /z < ∼ 2 GeV. Similar results are obtained for other detector regions, see Supplement E. It is interesting to compare q * to a direct measurement of sin φ h P hT /z, to which it reduces at leading power. The latter has improved resolution over P hT /z at small values thanks to picking up on the same acoplanarity of the event, which is stable against the electron momentum resolution, but cannot outperform q * since it is not a pure angular variable.
To verify the statistical sensitivity of q * to TMD physics we perform a Bayesian reweighting analysis of the unpolarized cross section dσ/(dx dz dQ 2 dO) for O = P hT /z and |q * |. We assume that the O spectrum is measured in twenty equidistant bins between 0 ≤ O ≤ 4 GeV inside 1000 three-dimensional bins in x, z, Q 2 with equal statistics N π + /1000 in each, and for definiteness consider a bin centered on x = 0.1, z = 0.15, Q = 20 GeV in the following. To account for the fact that the factorized dependence on x, z, Q 2 is determined from all bins at equal x, z, Q 2 simultaneously, we multiply the available statistics by another factor 100, arriving at an effective sample size of N eff = N π + /10 = 4.18 × 10 7 . At fixed x, z, Q 2 , a common model for the nonperturbative TMDs is [45] f NP where the ω i encode the width of the TMDs. We are interested in how much better the three free parameters ω i can be determined at the EIC using either q * or P hT .
(We hold the parameter α fixed for simplicity.) We assume a Gaussian prior probability density π(ω i ) based on the central values and standard deviations from [45]. We combine Eq. (11) with leading-logarithmic TMD evolution and tree-level matching in SCETlib [46], and insert it into Eqs. (6) and (9) to generate EIC pseudodata d n for bin n in O, using the central ω i . In the same way, we generate theory replicas t n (ω i ) distributed according to the prior. By normalizing d n = t n = 1 to the sum over bins at fixed x and z, collinear PDFs and FFs drop out at this order. (For details on the theory calculation, see Supplement F.) We then sample the posterior parameter probability distribution using a standard χ 2 likelihood function, where σ n = d n /N eff is the Poisson error on pseudodata bin n. Our results for the mean and variance of the posterior distribution are shown in Fig. 3 compared to those of the prior. Comparing P hT and q * , we find that the superior experimental resolution of q * only requires giving up a minor amount of statistical sensitivity to the ω i . In particular, there is more than a factor 10 improvement in uncertainty on the dominant fragmentation parameter ω 3 in either case. The choice of binning at small q * should be optimized in the future to exploit its excellent resolution, but we stress that we have not done so here.
The same setup can be used to assess the robustness of q * against systematic uncertainties. We use Pythia [44] to generate biased pseudodata d bias n subject to either (i) a momentum miscalibration, p → (1 + δ p ) p, or (ii) a shape effect from a non-uniform detector response (encoded e.g. in an efficiency) that changes at a slow rate ∆ X as a function of X = {p e , p h , η e , η h } across the x, z, Q 2 bin at hand. (The absolute value of the efficiency cancels in the normalized d bias n .) Repeating the reweighting analysis, we evaluate the partial derivatives of the posterior's mean ω i with respect to the bias parameters, which we dub the "strength" of the bias, as shown in Fig. 4. As anticipated, we find that an analysis using P hT is severely susceptible to the electron momentum calibration δ pe , while the calibration uncertainty using q * vanishes exactly due to its By construction q * (red) is robust against the large calibration uncertainty δp e that impacts P hT (blue). Both q * and P hT exhibit similar susceptibility to non-uniform detector response, modeled by ∆ X . purely angular nature. (Both P hT /z and q * are independent of δ p h at observable level for M h Q.) Figure 4 also shows that P hT and q * have comparable susceptibility to non-uniform detector response, despite the exponential factors of η e,h appearing in q * , demonstrating the robustness of q * against these sources of bias.
By replacing measurements of dσ/dP hT dφ h by dσ/dq * with angular reconstruction of Q, x, y, the prospects for precisely mapping the 3D structure of hadronization and confinement with TMDs are bright. We anticipate a follow-up campaign to aid this endeavor by discovering other useful angular observables that resolve the remaining TMD PDFs and by studying theoretical ingredients, like the convergence of the known higher-order QCD corrections for these cross sections.

A. Constructing q * with finite target mass
In the main text, we present our results in the light target mass limit M Q. Here, we give the corresponding results when fully retaining the target mass M , which can be important when Q is not sufficiently large or when M is large (such as for a nucleus). This amounts to including in our construction the dependence on γ = 2xM Q . (S1) We also have the following variable generalizations: Note that we still take P hT , M h Q (appropriate for example when N is a proton or ion and h is a pion), and hence can still take κ γ = 1. Only receives mass corrections through γ. We will show that the above corrections do not change any of the conclusions regarding the utility of q * .
We continue to work with the leading power (LP) kinematics λ ∼ P hT /(zQ) 1. For the relationship between the acoplanarity angle φ rest acop and the hadron transverse momentum P hT , we now have We now wish to construct an optimized observable q M * for a massive target with M ∼ Q, such that q M * M Q = q * while retaining the desired leading power relation q M *

LP
First of all, we emphasize that the following relations presented in the main text are independent of M : Furthermore, we notice that γ 2 can also be written in terms of angles in a manner that is independent of M : This allows us to define q M * in terms of target rest frame quantities completely free of M dependence: This relation is especially useful for fixed target experiments, where all the quantities can be readily measured. For collider experiments like the EIC, the above equation may be expressed in terms of lab frame quantities simply by substituting tan θi 2 = exp cosh −1 (P 0 EIC /M ) + η i for i = e, h, where η i are the lab frame pseudorapidities, 0 rest = exp cosh −1 (P 0 EIC /M ) 0 EIC , and φ rest acop = −φ EIC acop . We remind the reader that the last relation is due to the different convention for the orientation of the z axis at the EIC, see Fig. 1.
We emphasize that since q M * In the main text, we give the expression of Q 2 and y in Eq. (2) in terms of lab frame angles to leading order in λ ∼ P hT /(zQ). Here, we derive the leading power correction to these kinematic relations. These results can be used to get an idea of the size of power corrections to an analysis, including both i) the size of corrections to the double angle construction for x, y, and Q 2 , and ii) power corrections to the factorization formula for dσ/(dx dy dz dq * ). Note that in this section we still work in the massless target limit M Q. We have the following relations: where η e and η h are lab frame pseudorapidities. Note that we have indicated that Eq. (S7c) is the only equation here that receives corrections in λ ∼ P hT /(zQ) when expanding in this ratio. Solving the above relations for x, y, and Q 2 , we get where ∆η = η h − η e is boost invariant along the z-direction. Notice that in Eq. (S8b), y does not receive linear corrections in λ. An application of Eq. (S8) is to test the size of the power corrections in the expressions for x and Q 2 in a given data set, and thus apply cuts to restrict the data to TMD regions where the leading term is dominant. We can also invert the formulae in Eq. (S8) to define a set of variables x * , y * and Q * that use lab frame measurements. The variables x * , y * and Q * agree with the kinematic invariants x, y and Q up to the determined O(λ) power corrections: The all-order definition of q * in the main text, Eq. (3), can be written as q * ≡ Q * √ 1 − y * tan φ EIC acop . This allows us to easily determine the leading power correction to the formula for q * obtained by expanding in P hT zQ: This kinematic correction to the relationship between variables is the only power correction that would give non-trivial dependence on y and cos φ h to the factorization formula in Eq. (9). Hence it can be unambiguously included in the factorization analysis by using this more complicated relationship in the δ(q * + . . .) when switching variables and integrating over P hT and φ h (cf. the example given in Eq. (8) without these corrections). However, this still does not capture the dynamic hadronic power corrections, which arise from using the P hT zQ expansion when deriving the original factorization theorem for dσ/(dx dy dz d 2 P hT ).
C. Leading-power formulas for target spin vector from angular measurements As mentioned in the main text, the S L , S T and φ S that appear in the factorization formula are defined in the Trento frame by writing the nucleon spin vector as S µ = (0, S T cos φ S , S T sin φ S , −S L ) Trento . Here we give leading-power expressions for these variables in terms of the target rest frame components of S µ = (0, S x , S y , S z ) rest , and the EIC lab frame hadron pseudorapidity η h . Note that we do not assume that the nucleon is in a pure spin state.
We start with expressing S L , S T and φ S using the polar angle θ q of q, At leading power in λ ∼ P hT /(zQ) 1, we may replace θ q by θ h , which can be written in terms of η h . We have Here we define A to be (S15) D. Momentum space factorization formula for the q * spectrum In the main, text our factorization theorem for dσ/(dx dy dz dq * ) was written in terms of b T space TMDs. For completeness, here we give the factorization theorem written in terms of momentum space TMDs, which are Fourier conjugate to those in b T space.
We start with the standard leading power momentum space TMD factorization formulae [36] for the structure functions appearing in Eq. (6): whereF is defined as Here q T = − P hT /z,ĥ = P hT /P hT , and ω( k T , p T ) denotes the weight prefactors in Eq. (S16) that depend on k T and p T .
The leading-power SIDIS cross section differential in q * is Plugging Eq. (6) (S19) Especially notice that W sin 2φ h U L ∼ h ⊥ 1L H ⊥ 1 does not appear in Eq. (S19) since its prefactor sin(2φ h ) is odd under For each term in Eq. (S18), writing the φ h dependent coefficient as κ(φ h ), we have whereŷ is the unit vector in the Trento frame. In the second line, all appearances ofĥ in ω( k T , p T ) are replaced bŷ and κ(φ h ) is replaced by a function of k T and p T according to the rule wherex andŷ are again Trento frame unit vectors.
Then we can write Eq. (S19) as Note that we have used sin(3φ h ) = 3 sin φ h − 4 sin 3 φ h . Here we defineF * as where the weight function ω * ( k T , p T ) includes the full prefactor structures, and can be understood as product of κ( k T , p T ) and ω( k T , p T ) in Eq. (S20). Fig. 2 in the main manuscript was restricted to the case of a backward electron and central pion, which features the largest share of the total pion sample for our selection cuts. In Fig. S1, we provide additional results for the expected detector resolution of different SIDIS TMD observables in all other relevant detector regions. Note the change in vertical scale compared to Fig. 2. We find that a clear improvement in resolution from using q * persists across all detector regions. The case of electrons in the forward detector region (i.e., from backscattered electrons at very large Q 2 → s) has a negligible contribution to the total rate. We note that the improvement of q * in resolution compared to P hT deteriorates slightly in the cases where the hadron is backward for P hT < 5 GeV, and actually features worse resolution for P hT > 5 GeV. This is expected as the fixed angular resolution σ θ EIC h = 0.001 translates to a wider range in pseudorapidity as θ EIC h → π.

E. Resolution curves for all detector regions
F. Details on Bayesian reweighting

Theory templates
In this section we describe in detail how the pseudodata d i and theory replicas t i (ω i ) in the main text are constructed. They are defined as the normalized bin-integrated spectrum for the observable O = P hT /z or O = q * being tested, where the right-hand side is evaluated with the appropriate values of ω i (either central or chosen according to the Monte-Carlo replica) inserted into Eq. (11). Restricting Eqs. (6) and (9) to the unpolarized contribution from W U U,T , the differential leading-power spectrum explicitly reads where µ is the MS scale that the TMD PDF and FF are being evolved to, ζ = Q 2 is the Collins-Soper scale, and

Nonperturbative model parameters and Bayesian priors
We work with the nonperturbative model used in the MAPTMD22 global fit of TMD PDFs and FFs from Drell-Yan and SIDIS data [45]. The model for the CS kernel reads Since our reweighting study is performed at fixed Q, we do not expect to be sensitive to CS evolution between different values of Q and hold the parameter g 2 = (0.248 ± 0.008) GeV fixed at its central value. (A full four-dimensional measurement differential in x, z, Q, and q * would of course exhibit the usual sensitivity to the CS kernel.) For fixed x and z, the nonperturbative models of [45] for the TMD PDF and FF relate to Eq. (11) as where g {1,3,3B} are functions of x and z defined in terms of the underlying model parameters in [45]. In the comparison, we have used that the fit result of [45] down to x ≥ 0.1 is compatible with a single Gaussian for the TMD PDF to a good approximation, and have set λ = λ 2 = 0. For x = 0.1 and z = 0.15 as considered in the main text, the first three parameters evaluate to ω 1 = (0.0791 ± 0.0063) GeV 2 , ω 2 = (0.0167 ± 0.0059) GeV 2 , ω 3 = (0.8153 ± 0.0637) GeV 2 , In our reweighting study, we take the variance of the Gaussian prior probability distributions for {ω 1 , ω 2 , ω 3 } to be numerically equal to the 1σ confidence intervals above. (We ignore nondiagonal entries in the covariance matrix of [45], which were found to be small there.) We hold α = 0.079 ± 0.029 fixed at its central value for simplicity. We note that we have also performed the reweighting study using a simplified version of the nonperturbative model used in [13], which features an approximately exponential term at large distances, and have arrived at similar conclusions.

Parametrizing systematic bias
Here we describe how the bias strengths shown in Fig. 4 in the main text are evaluated. To model the effect of a momentum miscalibration δ p , we use the Pythia sample of pions described in the main text, restrict to a bin 0.085 ≤ x ≤ 0.115, 0.13 ≤ z ≤ 0.17, 400 GeV 2 ≤ Q 2 ≤ 401 GeV 2 centered on our choices for x, z, Q 2 in the main text, and evaluate the O = P hT /z or O = q * spectrum after replacing in the event record. Note that we only use the biased event record to calculate O, but continue to cut on the true values of x, z, Q 2 since we expect the impact of δ pe on the reference Born kinematics to be subleading compared to the direct effect on the reconstructed O. Similarly, to model the effect of a (generic) non-uniform detector response that changes as a function of X = {p e , p h , η e , η h }, we apply an additional weight (X) to each event calculated as where the . . . refers to an average over all events in the current x, z, Q 2 bin. Normalizing the effect of ∆ X to the variance of the distribution ensures that the impacts of ∆ X for different X are comparable even when individual X