Mapping the Kinematical Regimes of Semi-Inclusive Deep Inelastic Scattering

TL;DR

This paper develops a framework to identify and analyze different kinematical regimes in semi-inclusive deep inelastic scattering, especially at moderate to low Q, linking them to partonic models and QCD factorization.

Contribution

It introduces a language and methods for mapping kinematical regions to underlying partonic pictures in semi-inclusive deep inelastic scattering.

Findings

Provides a systematic way to identify kinematical regimes

Links kinematical regions to partonic models and QCD factorization

Offers methods to test the consistency of kinematic regions with partonic pictures

Abstract

We construct a language for identifying kinematical regions of transversely differential semi-inclusive deep inelastic scattering cross sections with particular underlying partonic pictures, especially in regions of moderate to low $Q$ where sensitivity to kinematical effects outside the usual very high energy limit becomes non-trivial. The partonic pictures map to power law expansions whose leading contributions ultimately lead to well-known QCD factorization theorems. We propose methods for estimating the consistency of any particular region of overall hadronic kinematics with the kinematics of a given underlying partonic picture. The basic setup of kinematics of semi-inclusive deep inelastic scattering is also reviewed in some detail.

Equations

$$V^{\mu }=\left(V^{+},V^{-},{\mathbf{V}}_{\text{T}}\right),$$

$$V^{+}=\frac{V^0+V^z}{\sqrt{2}},V^{-}=\frac{V^0-V^z}{\sqrt{2}},{\mathbf{V}}_{\text{T}}=\left(V^x,V^y\right).$$

$$y=\frac{1}{2}\ln \left(\left|\frac{V^{+}}{V^{-}}\right|\right).$$

$$V=\left(\frac{M_{\mathrm{T}}}{\sqrt{2}}{e}^y,\frac{M_{\mathrm{T}}}{\sqrt{2}}{e}^{-y},{\mathbf{V}}_{\text{T}}\right),$$

$$M_{\mathrm{T}}=\sqrt{\left|M^2+{\mathbf{V}}_{\text{T}}^2\right|}.$$

$$V=\left(\frac{M_{\mathrm{T}}}{\sqrt{2}}{e}^y,-\frac{M_{\mathrm{T}}}{\sqrt{2}}{e}^{-y},{\mathbf{V}}_{\text{T}}\right).$$

$$V_{\mathrm{b},\mathrm{c}}^{\mathrm{a}},$$

$${\mathbf{V}}_{\mathrm{b},\mathrm{c},\mathrm{T}}.$$

$$\text{lepton}(l)+\text{proton}(P)\to \text{lepton}(l^{\prime })+\text{Hadron}(P_{\mathrm{B}})+X.$$

$$Q^2$$

$$y$$

$$W_{\mathrm{tot}}^2$$

$$x_{\mathrm{h}}=\frac{q\cdot P_{\mathrm{B}}}{P\cdot q}$$

$$Q^2$$

$$q_{\gamma }$$

$$P_{\gamma }$$

$$P_{\mathrm{B},\gamma }$$

$$P_{\mathrm{B},\gamma }^{-}=\frac{z_{\mathrm{h}}{Q}^2}{4x_{\mathrm{Bj}}{P}_{\gamma }^{+}}\left(1\pm \sqrt{1-\frac{4x_{\mathrm{Bj}}^2{M}^2\left({\mathbf{P}}_{\mathrm{B},\gamma ,\mathrm{T}}^2+M_{\mathrm{B}}^2\right)}{z_{\mathrm{h}}^2{Q}^4}}\right)\approx \frac{z_{\mathrm{h}}{Q}^2}{2x_{\mathrm{Bj}}{P}_{\gamma }^{+}},$$

$$q_{\mathrm{b}}$$

$$P_{\mathrm{b}}$$

$$q_{\mathrm{H}}$$

$$P_{\mathrm{H}}$$

$$P_{\mathrm{B},\mathrm{H}}$$

$$P_{\mathrm{H}}^{+}$$

$$P_{\mathrm{H}}^{-}$$

$${\mathbf{P}}_{\mathrm{H},\mathrm{T}}$$

$$P_{\mathrm{B},\mathrm{b}}=\left(\frac{M_{\mathrm{B}}^2+z_{\mathrm{N}}^2{\mathbf{q}}_{\mathrm{T}}^2}{\sqrt{2}{z}_{\mathrm{N}}Q},\frac{z_{\mathrm{N}}Q}{\sqrt{2}},-z_{\mathrm{N}}{\mathbf{q}}_{\mathrm{T}}\right)=\left(\frac{M_{\mathrm{B},\mathrm{T}}}{\sqrt{2}}{e}^{y_{\mathrm{B},\mathrm{b}}},\frac{M_{\mathrm{B},\mathrm{T}}}{\sqrt{2}}{e}^{-y_{\mathrm{B},\mathrm{b}}},{\mathbf{P}}_{\mathrm{B},\mathrm{b},\mathrm{T}}\right),$$

$$z_{\mathrm{N}}\equiv \frac{P_{\mathrm{B},\mathrm{b}}^{-}}{q_{\mathrm{b}}^{-}},$$

$${\mathbf{q}}_{\mathrm{T}}\equiv -\frac{{\mathbf{P}}_{\mathrm{B},\mathrm{b},\mathrm{T}}}{z_{\mathrm{N}}}=-\frac{q_{\mathrm{b}}^{-}{\mathbf{P}}_{\mathrm{B},\mathrm{b},\mathrm{T}}}{P_{\mathrm{B},\mathrm{b}}^{-}}.$$

$$z_{\mathrm{N}}=\frac{Q^4{x}_{\mathrm{N}}{z}_{\mathrm{h}}\left(1\pm \sqrt{1-\frac{4M^2{M}_{\mathrm{B}}^2{x}_{\mathrm{Bj}}^2(Q^4+x_{\mathrm{N}}^2{M}^2{\mathbf{q}}_{\mathrm{T}}^2)}{Q^8{z}_{\mathrm{h}}^2}}\right)}{2x_{\mathrm{Bj}}(Q^4+x_{\mathrm{N}}^2{M}^2{\mathbf{q}}_{\mathrm{T}}^2)}\stackrel{\mathrm{Fixed}{\mathrm{x}}_{\mathrm{N}},{\mathrm{z}}_{\mathrm{h}},{\mathrm{q}}_{\mathrm{T}}}{=}{z}_{\mathrm{h}}\left(1+O\left(\frac{m^4}{Q^4}\right)\right).$$

$$\kappa \equiv \sqrt{z_{\mathrm{N}}^2{\mathbf{q}}_{\mathrm{T}}^2+\frac{M^2{x}_{\mathrm{N}}^2{\left(M_{\mathrm{B}}^2+{\mathbf{q}}_{\mathrm{T}}^2{z}_{\mathrm{N}}^2-\frac{Q^4{z}_{\mathrm{N}}^2}{M^2{x}_{\mathrm{N}}^2}\right)}^2}{4Q^4{z}_{\mathrm{N}}^2}}=O\left(\frac{Q^2}{m}\right).$$

$$V_{\mathrm{H}}^{+}$$

$$V_{\mathrm{H}}^{-}$$

$${\mathbf{V}}_{\mathrm{H},\mathrm{T}}$$

$$V_{\mathrm{H}}^{+}$$

$$V_{\mathrm{H}}^{-}$$

Full text

††preprint: JLAB-THY-19-2920

1 Introduction

Deep inelastic scattering (DIS), and especially semi-inclusive deep inelastic scattering (SIDIS) are headlining processes in most programs to the study of partonic (quark and gluon) degrees of freedom. It is a cornerstone process of, for example, the Jefferson Lab 12 GeV program to study partonic structure in hadrons, and is one of the important processes for study in a possible future Electron-Ion Collider (EIC)  Dudek:2012vr ; Accardi:2012qut ; Aschenauer:2014twa ; Avakian:2014hga ; Aschenauer:2016our ; Avakian:2016rst . Interest in SIDIS arises from a variety of considerations. Well-established collinear factorization theorems for SIDIS provide access to the flavor dependence of standard parton distribution functions (PDFs) and fragmentation functions (FFs). In the target fragmentation function region, different kinds of objects, called fracture functions Trentadue:1993ka ; Graudenz:1994dq , are involved and these are sensitive to still other novel QCD phenomena. Beyond collinear factorization, transversely differential SIDIS at low transverse momentum is sensitive to the properties of transverse momentum dependent (TMD) PDFs and FFs.

Many DIS experiments are performed at medium $Q$ (roughly $1$-$3$ GeV), where it is reasonable to expect some sensitivity to intrinsic properties of hadron structure and other non-perturbative effects to be significant. By “moderate-to-low $Q$,” we will mean SIDIS measurements with $Q$ roughly between $1$ GeV and $3$ GeV and Bjorken-$x_{\rm Bj}$ not too far below the valence region. This includes JLab 6 GeV and 12 GeV SIDIS cross section measurements Dudek:2012vr ; Avakian:2014hga ; Avakian:2016rst .

So, the moderate-to-low $Q$ region has some obvious advantages in the mission to refine the current view of hadron structure. If all energies and hard scales are extremely large, then asymptotic freedom means that pictures of partonic interactions rooted in perturbation theory can usually be applied confidently and with very high accuracy and precision. But, with the large relative fraction of the hard process contributions and perturbatively produced radiation involved, it becomes less clear to what extent observables are truly sensitive to the intrinsic properties of the actual hadron constituents. This further points to moderate-to-low $Q$ measurements as ideal sources of information about partonic hadron structure. However, there are also unique challenges to interpreting moderate-to-low $Q$ cross sections, particularly for less inclusive versions of DIS like SIDIS. With lower hard scales, access to intrinsic effects of constituents may be more direct, but this also comes with less confidence in the reliability and accuracy of perturbative and/or parton-based descriptions. Moreover, the average final state hadron multiplicity in such measurements is typically less than about 3 in the valence region of Bjorken-$x_{\rm Bj}$. In long term efforts to establish intrinsic properties for partons, the trade-off in advantages at large and small $Q$ needs to be confronted systematically, and such that knowledge of one complements the other.

Sophisticated theoretical frameworks, usually involving some form of QCD factorization and perturbation theory Muta:1987mz ; Greiner:2002ui ; Collins:2011qcdbook have long existed for describing specific underlying physical mechanisms in many highly differential processes over many regions, including in SIDIS, in terms of partonic degrees of freedom. However, they always assume specific kinematical limiting cases, e.g. very large or very small transverse momentum, or very large or very small rapidity. The interface between different physical regimes remains somewhat unclear in practice, and most especially when the hard scales involved are not especially large. Estimating the kinematical boundaries of any specific QCD approach or approximation beyond very rough orders of magnitude is difficult and subtle. It requires at least some model assumptions, e.g. about the role of parton virtuality and/or the onset of various non-perturbative or hadronic mechanisms generally. Monte Carlo simulations can help, but these also involve physical assumptions whose impact needs to be understood systematically. Future phenomenological and experimental efforts will hopefully clarify the location of region boundaries, and discriminate between competing hypotheses.

The aim of the present article is to discuss how such questions can be posed in a systematic way. To this end, we will refrain from discussing specific theoretical frameworks or QCD models and instead enumerate the steps needed to map any given set of assumptions concerning exact intrinsic partonic/constituent properties to a corresponding kinematical region of $x_{\rm Bj}$, $Q$, $z_{\rm h}$, and transverse momentum in a cross section. The goal is to organize an interpretation strategy applicable with any model of underlying non-perturbative dynamics for exact parton momentum, independent even of assumptions about factorization. Our final result is a sequence of tests that probe the proximity of any given kinematical configuration to a conventional partonic region of SIDIS, and that also probe the sensitivity to the various model assumptions needed to make such an assessment. A convenient web interface for implementing these tests can be found at Ref. sidistool .

We also review the basics of the SIDIS process itself, in some cases translating past results into an updated language, motivated by current research efforts. For other general introductions to SIDIS in pQCD see, for example, Refs. Meng:1991da ; Levelt:1993ac ; Meng:1995yn ; Mulders:1996dh ; Nadolsky:1999kb ; muldersnotes . See especially Refs. Anselmino:2005nn ; Bacchetta:2006tn ; Bacchetta:2008xw ; Anselmino:2011ch for another review of the basics of SIDIS that includes a full catalogue of spin and azimuthal dependences. For general treatments of SIDIS in the context of fracture functions and target fragmentation, see Ref. Graudenz:1994dq . Finally, see Chapters 12-13 of Ref. Collins:2011qcdbook , which influences much of the language and notation of this article.

The first half of this paper reviews many of the basics of SIDIS: Sec. 2 and Sec. 3 provide an overview of our notation and setup, Sec. 4 discusses the various reference frames commonly used, Sec. 5 explains the kinematical characterization of final state hadron momentum, and Sec. 6 explains our conventions for the decomposition of cross sections into structure functions. We then begin the discussion of standard approximations in the second half of the paper, starting with the purely kinematical approximations in Sec. 7. We explain the characterization of partonic kinematics, and the typical approximations associated with them, in Sec. 8, with a focus on the current and large transverse momentum regions. In Sec. 9 we translate these considerations into the language of rapidity. In Sec. 10 and Sec. 11 we discuss the target and soft regions. We provide examples of the region characterization in Sec. 12, with experimentally reasonable kinematics. Finally, we make concluding remarks in Sec. 13.

2 Light-Cone Variables

Light-cone variables are defined as follows: for a four-vector $V^{\mu}$,

[TABLE]

where

[TABLE]

For a four-momentum $V$, rapidity is defined as usual:

[TABLE]

In terms of rapidity, light-cone momentum is:

[TABLE]

where $V^{2}=M^{2}$ and transverse mass is

[TABLE]

For a virtual momentum, $M^{2}<0$ and either the plus or minus light-cone component is negative, e.g.,

[TABLE]

In labeling a four-momentum component of $V$, we will write:

[TABLE]

where $a$ is the contravariant component, $c$ specifies the reference frame, and $b$ is any other necessary subscript depending on the given context. A two-dimensional transverse momentum is

[TABLE]

The frame subscripts $b,c$ on a four-momentum indicate in which frame its components will be expressed.

3 The Process

We consider the process:

[TABLE]

The final state hadron has type $B$. The “$X$" is an instruction to sum over all unobserved particles including other $B$ hadrons. Note that each $B$ in an event is counted. A sketch is shown in Fig. 1111In this figure we followed the so-called Trento conventions Bacchetta:2004jz ..

The proton has momentum $P$, the virtual photon has momentum $q$, the produced hadron has momentum $P_{\rm B}$, and the incoming and scattered leptons have momenta $l$ and $l^{\prime}$ respectively. The mass of the target hadron is $M$ and the mass of the produced hadron is $M_{\rm B}$.

Observables like cross sections or structure functions are conventionally parameterized by a combination of the following kinematical variables:

[TABLE]

In the light-cone ratios that define $x_{\rm N}$ and $z_{\rm N}$, momentum components $q^{\pm}$, $P^{+}$ and $P_{\rm B}^{-}$ are defined in a photon frame (see Section 4), where the incoming proton is in the positive $z$-direction with zero transverse momentum and the virtual photon is the the negative $z$-direction with no transverse momentum. (See the discussion of photon frames below.) Since boosts along the $z$-axis do not affect light-cone ratios, the exact photon frame does not matter. $x_{\rm N}$ is the kinematical variable usually called Nachtmann-$x$. It is often labeled by a $\xi$, but for us $\xi$ will label a partonic momentum fraction, so we use $x_{\rm N}$ instead, with the subscripts on $x_{\rm N}$ and $x_{\rm Bj}$ distinguishing between Bjorken and Nachtmann $x$-variables. For descriptions of fragmentation, the light-cone fraction $z_{\rm N}$ is the analogue of $x_{\rm N}$, and the ${\rm N}$ subscript is meant to emphasize this analogy. Our $x_{\rm Bj}$, $z_{\rm h}$ and $P_{\rm B}$ correspond, respectively, to $x$, $z$ and $P_{h}$ from Bacchetta:2006tn . Our $z_{\rm N}$ corresponds to $\zeta_{h}$ of Ref. Guerrero:2015wha . A variable

[TABLE]

is useful if the target fragmentation region is being described.

The deep inelastic limit is $m/Q\to\infty$ with fixed $x_{\rm N}$ and $z_{\rm N}$. The “$m$" symbol will always represent a generic mass scale in this paper, considered to be very small relative to $Q$, such as a small hadron mass or $\Lambda_{\rm QCD}$. The kinematical variables obey

[TABLE]

The "$\approx$" symbol will always mean "dropping $m/Q$ power-suppressed corrections" with $x_{\rm N}$ and $z_{\rm N}$ fixed.

4 Reference Frames

It is useful and common to switch between photon and hadron frames. Here we describe the SIDIS kinematics in these frames.

• •

Photon frame:

In a photon frame, the virtual photon and the initial proton both have zero transverse momentum, while the final state produced hadron has non-zero transverse momentum:

[TABLE]

The $\gamma$ subscript signals the use of components in the photon frame, following the notation of Eq. (7). In the photon frame

[TABLE]

where the approximation symbol shows the limit of zero hadron masses for the solution corresponding to the current fragmentation region. Note that Eq. (15) fixes $x_{\rm N}$ to be defined as in Eq. (10).

The angles $\psi$ and $\phi$ are the azimuthal angles of the final state lepton and produced hadron respectively in a photon frame. Note that ratios of plus and minus components are independent of boosts in the $z$-direction, and so are the same in all photon frames.

• •

Breit frame:

A particular case of the photon frame is the Breit (Brick Wall) frame, see Fig. 2(a), where

[TABLE]

The small ${\rm b}$ indicates that components are in the Breit frame. This will be our default frame, so any four-momentum components without a subscript should be assumed to be in the Breit frame.

• •

Hadron Frame:

In the hadron frame, see Fig. 2(b), labeled by “$\rm H$,” the incoming hadron and final state hadron are exactly back-to-back (zero relative transverse momentum) while the virtual photon generally has non-zero transverse momentum. It is an especially useful frame for setting up factorization. (See (Collins:2011qcdbook, , Sec.13.15.5).) The components of the four-momenta are:

[TABLE]

For definiteness, define the hadron frame such that the components of the incoming target momentum are exactly the same as in the Breit frame:

[TABLE]

• •

Varying Conventions:

For us, the hadron frame has zero transverse momentum for the produced hadron and non-zero transverse momentum for the virtual photon Mulders:1996dh ; Collins:2011qcdbook . Note that this is opposite the situation in the hadron frame of Meng-Olness-Soper (MOS) Ref. Meng:1991da . The MOS hadron frame corresponds to the photon frame of Collins Collins:2011qcdbook . MOS define a Lorentz invariant four-vector ((Meng:1991da, , Eq. (10))) that measures the deviation from the back-to-back configuration. From (Meng:1991da, , Eq. (11)) and (Collins:2011qcdbook, , Eq. (13.104)), the Ref. Collins:2011qcdbook hadron frame ${{\bf q}^{{2}}_{\rm H,\rm{T}}}$ is the same as the MOS ${{\bf q}^{{2}}_{\rm H,\rm{T}}}$ if hadron masses are neglected.

Restricting to the MOS hadron frame, MOS use (Meng:1991da, , Eq. (11)) and (Meng:1991da, , Eq. (13)) and $P_{\rm B}^{2}=0$ to find (Meng:1991da, , Eq. (12)), which in light-cone coordinates is Eq. (27) with $M_{\rm B}^{2}=0$ and with the MOS ${\bf q}_{\text{\scriptsize T}}$ defined to point along the positive $x$-axis. In the MOS hadron frame, the transverse part of $P_{\rm B}$ is always in the $x$ direction and is always positive.

Mulders and Tangerman (Mulders:1996dh, , Eqs. (15-17)) give general expressions for four vector components that include the effects of hadron masses, and the reference frames used correspond to the hadron and/or photon frames defined above. References such as Bacchetta:2004jz ; Diehl:2005pc ; Bacchetta:2006tn specialize the photon frame to the target rest frame rather than the Breit frame. ${{\bf P}_{{\rm B},\rm b,\rm{T}}}$ is invariant, however, with respect to boosts along the $z$-axis. Other conventions use some combination of the above. Refs. Ji:2004xq ; Idilbi:2004vb ; Ji:2004wu use a hadronic tensor with an extra $1/4z_{\rm h}$ relative to the above and Refs. Mulders:1996dh ; muldersnotes have an extra $1/2M$. The notation of Ref. Koike:2006fn is similar to Ref. Meng:1991da .

5 Variables for the final state momentum

Here we treat the final state momentum in terms of the the light-cone momentum fraction $z_{\rm N}$ Eq. (11) variable, and also express it in both the photon and hadron frames; in turn we relate it to the hadron frame photon transverse momentum.

First, define the exact Breit frame ${P}_{{\rm B},\rm b}$ in terms of $z_{\rm N}$

[TABLE]

where ${{\bf q}_{\rm{T}}}$ is so far only a symbol used to define the Breit frame transverse component, and it has yet to be related to physical quantities. In other words, first define

[TABLE]

in accordance with Eq. (11), and then define

[TABLE]

Note the minus sign. The momentum fraction $z_{\rm N}$ is related to the kinematical parameter $z_{\rm h}$ by

[TABLE]

The expansion after the far right equals sign is for the "$+$" solution, which in conventional treatments of SIDIS corresponds to the current fragmentation region. Note that the relationship between $z_{\rm N}$ and $z_{\rm h}$ is generally double valued. Parameterizing final hadron momentum as in Eq. (27) is convenient for some purposes such as in factorization derivations.

It is often useful to switch back and forth between the photon (e.g., Breit) and hadron frames. For this, define

[TABLE]

The law to transform a vector $V$ from the Breit frame to the hadron frame is then222The simplest sequence of transformations to get this are: 1) boost from the Breit frame to the proton rest frame 2) rotate until the momentum of the final state hadron is along the negative $z$-axis 3) boost along the $z$-axis to a frame where the proton has a light-cone plus component equal to that of Breit frame $P^{+}_{\rm b}=Q/x_{\rm N}\sqrt{2}$.

[TABLE]

In the limit that masses are small relative to $Q$, these become

[TABLE]

Also, using Eq. (19) in Eq. (37),

[TABLE]

Comparing this with Eq. (27) and using Eq. (30) confirms that

[TABLE]

As usual, the $\approx$ symbol means "neglecting $m/Q$-suppressed corrections." Thus, ${{\bf q}_{\rm{T}}}{}$ has the physical meaning in the limit of $m/Q\to 0$ of the hadron frame photon transverse momentum ${{\bf q}_{\rm H,\rm{T}}}{}$, which in turn is $-{{\bf P}_{{\rm B},\rm b,\rm{T}}}/z_{\rm h}$.

It is also useful to write the hadron momentum directly in terms of its Breit frame transverse momentum:

[TABLE]

The momentum fraction $z_{\rm h}$ is related to the kinematical parameter $z_{\rm N}$ by

[TABLE]

The inverse is

[TABLE]

See also (Guerrero:2015wha, , Eq. (2.12)). Note that $x_{\rm N}$ is a function of $x_{\rm Bj}$, $Q$, and $M$, but we will sometimes keep it to minimize the size of expressions as in Eq. (42) rather than writing everything explicitly in terms of $x_{\rm Bj}$, $Q$, and $M$.

6 Cross Sections and Structure Functions

A cross section differential in $N$ final state particles for particle $A$ scattering from particle $B$ is related to modulus-squared matrix elements $|M|^{2}$ in the usual way:

[TABLE]

with the triangle function

[TABLE]

The total DIS cross section is

[TABLE]

This fixes the normalization convention for the hadron and lepton tensor combination $L_{\mu\nu}W_{\rm tot}^{\mu\nu}$, where the "${\rm tot}$"-subindex indicates that this is conventional DIS: totally inclusive in all final state hadrons. Recall Fig. 1 and Eq. (9) for our momentum labeling. The leptonic tensor is defined in the usual way:

[TABLE]

So, the totally inclusive hadronic tensor is

[TABLE]

Here, the $\sum_{X}$ symbol is a sum over all possible final states $|X\rangle$, including invariant integrals,

[TABLE]

over the momentum of each final state particle $p_{i}$. The incoming hadron has a polarization specified by $S$.

The SIDIS cross section is differential in the momentum of one observed final state hadron of type $B$:

[TABLE]

This fixes our normalization conventions for SIDIS, and gives a SIDIS hadronic tensor:

[TABLE]

The same meaning applies to $\sum_{X}$ as in the totally inclusive case. $|P_{\rm B},X\rangle$ is a final state with at least one identified hadron of type $B$. The sum over $X$ includes a sum over any number of other final state particles, including other type-$B$ hadrons. Each separate type-$B$ hadron in an event is counted, in accordance with the definition of an inclusive cross section.

$W^{\mu\nu}_{\rm tot}(P,q)$ and $W^{\mu\nu}_{\rm SIDIS}(P,q,P_{\rm B})$ are the most convenient objects to work with theoretically because they are Lorentz tensors directly related to hadronic matrix elements of the electromagnetic current operator, and they are defined without reference to conventions associated with choices of reference frames etc, so we will organize our structure function analysis around them.

The relationship between the semi-inclusive and totally inclusive cross sections follows from the definition in Eq. (43) (see, e.g., (Byckling:1971vca, , Chapt. VII)):

[TABLE]

where $\langle N\rangle$ is the total average particle multiplicity, and the sum is over all particle types. Thus,

[TABLE]

Note that the integration measure in Eq. (50) is Lorentz invariant, although we will continue to specify a photon frame for the components, both for definiteness and because $z_{\rm N}$ is defined in terms of a photon frame momentum fraction.

The usual structure function decompositions on $W^{\mu\nu}_{\rm tot}$ and $W^{\mu\nu}_{\rm SIDIS}$ are

[TABLE]

“Pol. Dep.” is a place holder for polarization and azimuthal angle dependent terms, which we leave unspecified for now. The structure functions’ explicit dependence on $M$ and $M_{\rm B}$ has been dropped for brevity. While $x_{\rm Bj}$ and $z_{\rm h}$ are shown as the independent variables for the structure functions, it is useful to view them as being themselves functions of $x_{\rm N}$, $z_{\rm N}$, $M$ and $M_{\rm B}$. We have not done this here in order to avoid over-complicating notations, but it is useful for making kinematical approximations clear, as discussed in Moffat:2019qll . The differential SIDIS cross section in the Breit frame (or any photon frame) is then

[TABLE]

In the last two lines

[TABLE]

which are definitions generalized from the inclusive case to SIDIS. To match with other common notational conventions, we have used

[TABLE]

along with the identities (see (Bacchetta:2006tn, , Eqs. (2.8-2.13))),

[TABLE]

After changing variables from $z_{\rm N}$ to $z_{\rm h}$ Eq. (6) becomes

[TABLE]

This Jacobian factor can be expressed in terms of combinations of $z_{\rm N}$ and $z_{\rm h}$, but we keep the square root factors explicit to highlight the dependence on transverse momentum via $M_{\rm B,T}^{2}$ at fixed $z_{\rm h}$.

A convenient recipe for calculating structure functions is to contract with Lorentz covariant extraction tensors, ${\rm P}_{\Gamma}^{\mu\nu}$, defined as

[TABLE]

Then

[TABLE]

where

[TABLE]

From Eq. (50)

[TABLE]

where $j$ labels a structure function, i.e., $j\in\{1,2,T,L\}$. To reproduce the equations of Ref. Bacchetta:2006tn , we have define barred structure functions:

[TABLE]

Substitute into Eq. (58) to get

[TABLE]

Now (Bacchetta:2006tn, , Eq. (2.7)) can be used to fill in the remaining polarization and $\phi$-dependent structure functions. The barred normalization convention in Eq. (64) is defined so that structure functions exactly obey a particularly convenient energy sum rule (Appendix A) found in (Bacchetta:2006tn, , Eqs. (2.18-2.21)):

[TABLE]

Note that a factor of hadron multiplicity does not appear. Equation (63) becomes

[TABLE]

The forms of Eqs. (66)–(67) are only valid if the barred normalization conventions from Eq. (64) are used for the structure function normalizations, making them very useful for some practical applications. An advantage of the unbarred convention is that the structure functions have a direct connection to the matrix elements of current operators via Eq. (48), and their Lorentz covariant structure function decomposition with the standard normalization conventions in Eq. (47) and Eq. (52).

The unobserved invariant mass-squared in inclusive DIS is

[TABLE]

In SIDIS it is

[TABLE]

Note that if both $z_{\rm h}$ and $x_{\rm Bj}$ are close to 1, then $|{{\bf P}_{{\rm B},\rm{T}}}|$ cannot be much greater than zero without hitting the resonance region of $W_{\rm SIDIS}^{2}\approx 0$.

7 Purely Kinematical Approximations

Since we have not discussed the theory underlying the structure functions, all small mass approximations mentioned so far are unambiguously kinematical. For example, the usual $x_{\rm N}\approx x_{\rm Bj}$ and $z_{\rm N}\approx z_{\rm h}$ follow from expanding in $x_{\rm Bj}^{2}M^{2}/Q^{2}$:

[TABLE]

If hadron masses are neglected, then $P$ and $P_{\rm B}$ become the approximate $\tilde{P}$ and $\tilde{P_{\rm B}}$, which we define as

[TABLE]

that is, Eq. (20) and Eq. (27) but with all hadron masses set equal to zero. In the most common treatments of SIDIS, $P$ and $P_{\rm B}$ in Eqs. (51)–(52) are replaced with $\tilde{P}$ and ${\tilde{P}}_{{\rm B},}$, and $x_{\rm N}$ and $z_{\rm N}$ are replaced with $x_{\rm Bj}$ and $z_{\rm h}$ inside the structure functions, which is a good approximation in the $m/Q\to 0$ limit as long as the structure functions are reasonably smooth functions of $x_{\rm N}$ and $z_{\rm N}$. In Moffat:2019qll that was called the massless target approximation (MTA) for inclusive DIS, and an obvious extension applies to SIDIS. In that case, Eqs. (51)–(52) become

[TABLE]

Extracting the structure functions in Eqs. (74)–(75) requires, instead of Eqs. (61)–(62),

[TABLE]

where ${\rm\tilde{P}}_{PP}^{\mu\nu}=\tilde{P}^{\mu}\tilde{P}^{\nu}$. Our Eq. (74) is Eq. (18) from Moffat:2019qll , with the calligraphic notation explained there. Equation (75) is the analogous approximation for the SIDIS cross section. The MTA greatly simplifies kinematical relations at large $Q$.

The ratios

[TABLE]

are measures of the quality of the MTA. They must not deviate too much from $1$ if the standard massless approximations are to be considered valid. (See Sec. 12 for some examples.)

This exhausts the approximations that can be assessed entirely independently of questions about the partonic dynamics responsible for the behavior of the structure functions themselves.

Refs. Accardi:2014qda ; Guerrero:2015wha made first attempts to incorporate kinematical improvements to collinear QCD factorization by keeping $M_{\rm B}$ in kinematical factors. They point out the importance of this for moderate-to-low $Q$ SIDIS. However, they explicitly drop ${{\bf P}^{{2}}_{{\rm B},\rm b,\rm{T}}}$-dependence in an attempt to stay within a collinear factorization framework. Note from Eq. (71), however, that it is not consistent with collinear factorization power counting to simultaneously retain $M$ and $M_{\rm B}$ dependent kinematical power corrections while neglecting ${{\bf P}^{{2}}_{{\rm B},\rm b,\rm{T}}}$ dependent corrections, even for ${{\bf P}^{{2}}_{{\rm B},\rm b,\rm{T}}}\sim m^{2}$. The first non-vanishing $M_{\rm B}$-dependent correction term

[TABLE]

is the same size as the

[TABLE]

term when ${{\bf P}^{{2}}_{{\rm B},\rm b,\rm{T}}}$ is small. And, Eq. (79) is actually the dominant power-correction term when ${{\bf P}^{{2}}_{{\rm B},\rm b,\rm{T}}}/z_{\rm h}^{2}$ approaches order $Q$. The difficulty is that collinear factorization methods only characterize dependence on light-cone momentum fractions of the final state hadron, like $z_{\rm N}$, with only $z_{\rm h}$, $Q$, and $x_{\rm Bj}$ known. This is not a problem if keeping only the first term in the expansion on the right of Eq. (71) is valid. But the exact $z_{\rm N}$ requires knowledge not just of $M_{\rm B}$ and $M$, but also of (both small and large) ${{\bf P}^{{2}}_{{\rm B},\rm b,\rm{T}}}/z_{\rm h}^{2}$. So if it turns out that final state mass effects are large enough that they have to be accounted for, then it must be done in combination with an account of small transverse momentum dependence effects (e.g., TMD factorization), not independently of it.

8 Partons

So far, we have only discussed definitions and relativistic kinematics, with no mention at all of partons or dynamics. The question now is the following: Assuming that the configuration of initial and final hadrons is the result of scattering and fragmentation by small-mass constituents (i.e., partons), what are the possible kinematical configurations of those constituents, given a set of assumptions about their intrinsic properties? For now, we do not necessarily identify these partons with a particular theoretical approach or even real QCD, though ultimately we have that in mind.

This kind of very general partonic picture is illustrated in Fig. 3. We start by exploring the possibility that the produced hadron is collinear to an outgoing parton (a “current” hadron). We need clear steps for asking how reasonable it is to assume that a given external kinematical configuration for measured hadrons maps to current region partonic kinematics. The incoming hadron and its remnants are represented by the lower blob while the final state hadron emerges from a final state blob at the top of the diagram. Dashed lines represent the flow of momentum. It is very important for the discussion below to understand that they do not necessarily represent single quarks or gluons, and in reality they may correspond to groups of particles. What is important for us is only the flow of four-momentum through the process. Moreover, it is assumed that the momenta of these lines is known exactly and are never approximated. Although the word “parton” often implies a massless on-shell approximation for single particle lines, to keep language reasonably simple, we will nevertheless continue to call these dashed lines "partons." The picture in Fig. 3 does imply that quantities like $|k_{\rm i}^{2}|$ and $|k_{\rm f}^{2}|$ are small, and much of the discussion in this section will be about addressing the question of what is meant by “small.” So to summarize, “partonic" dashed lines represent the flow of a momentum with small invariant energy. In practical situations, they will often turn out to refer to actual quark and/or gluon lines, but they do not need to generally.

The partonic subprocess in Fig. 3 is marked off in a blue box. A black dot indicates the parton we associate with an observed hadron. The momentum $k_{\rm i}$ is the incoming struck parton momentum, and there is at least one hadronizing parton $k_{\rm f}$. The $k_{\rm X}$ momentum labels the total momentum of all other unobserved partons combined. Outside the box in Fig. 3, the position of the hadron implies a current region picture, though an analogous picture of course applies to the target region case. We ask questions about partonic regions in the context of the steps needed to factorize graphical structure in a manner consistent with particular partonic pictures. Our general view of factorization is based on that of Collins Collins:2011qcdbook ; Collins:1988gx and collaborators, though the same statements apply to most other approaches.

We are interested in the kinematics of the $k_{\rm i}+q\to k_{\rm f}+k_{X}$ subprocess and how closely it matches the overall $P+q\to P_{\rm B}+X$ process under very general assumptions. Specific realizations of the partonic subprocess, each of which can contribute to a different kinematical region, are shown in Fig. 4.

We will analyze the subprocess in the Breit frame and write

[TABLE]

Hats always indicate a partonic kinematical variable, whereas $\xi$ and $\zeta$ are momentum fractions (see below). We will write the transverse momentum as

[TABLE]

In the hadron frame, Eq. (32) gives

[TABLE]

so $\delta{\bf k}_{\text{\scriptsize T}}$ is good for characterizing an intrinsic relative transverse momentum in the large $Q$ limit; in Eq. (80) intrinsic transverse momentum is $\delta{\bf k}_{\text{\scriptsize T}}$ when $q_{\rm{\scriptsize T}}=0$. For nearly on-shell partons,

[TABLE]

In the limit where $m\ll Q$ and $x_{\rm Bj}$, $z_{\rm h}$, $q_{\rm{\scriptsize T}}$ are fixed, the outgoing parton is exactly aligned with the observed hadron so long as

[TABLE]

We have defined the Breit frame momentum fractions and Breit frame $\hat{x}_{\rm N}$, $\hat{z}_{\rm N}$ analogous to $x_{\rm N}$ and $x_{\rm Bj}$:

[TABLE]

For fixed $\hat{x}_{\rm N}$, $\hat{z}_{\rm N}$ and $q^{2}_{\rm{T}}$, $k_{\rm X}^{2}$ is calculable from momentum conservation,

[TABLE]

It will also be useful to define a momentum variable

[TABLE]

It is sometimes useful to have $k$ in terms of $k_{X}^{2}$ instead of $\hat{z}_{\rm N}$. For example, in the special case that $k_{\rm i}^{2}=k_{\rm f}^{2}={\bf k}_{\rm i,b,T}^{2}=\delta{\bf k}_{\text{\scriptsize T}}^{2}=0$

[TABLE]

On the second line, the "$\cdots$" represents higher powers in an expansion in small $q^{2}_{\rm{T}}/Q^{2}$ and $k_{\rm X}^{2}/Q^{2}$. When $q^{2}_{\rm{T}}/Q^{2}\to 0$ and $k_{\rm X}^{2}/Q^{2}\to 0$, the kinematics of the struck parton approach the kinematics of TMD factorization, or the handbag contribution in collinear factorization, with the errors in each component proportional to $q^{2}_{\rm{T}}/Q^{2}$.

The most basic of partonic approximations is that the masses and off-shellness of partons is small relative to the hard scale:

[TABLE]

On top of these, other approximations are normally needed. For instance, in the current region $k_{\rm f}$ is aligned with the final state hadron and

[TABLE]

Beyond these, still further approximations apply to different specific partonic subprocesses. First, in the $2\to 1$ process of Fig. 4(a), $k_{\rm i}\to k$, and the $1/Q^{2}$-suppressed terms in equations like Eqs. (88)–(90) are dropped. For a hard $2\to 2$ process shown in Fig. 4(b), $|k^{2}|\sim Q^{2}$ while $k_{\rm X}^{2}/Q^{2}\to 0$. If both $|k^{2}|$ and $k_{\rm X}^{2}$ are large, then at least three partons (e.g., Fig. 4(c)) are ejected at wide angles from the hard collision. For fixed $x_{\rm N}$, $z_{\rm N}$, $Q^{2}$, and ${{\bf P}_{{\rm B},\rm{T}}}{}{}$, only certain $k_{\rm i}$ and $k_{\rm f}$ are consistent with any given picture in Fig. 4.

For example, say we wish to interpret a particular SIDIS region with a partonic configuration like Fig. 4(a), corresponding to the current fragmentation region. For a partonic description to hold at all, a minimum requirement is that ratios like Eq. (91) are very small. So define a ratio

[TABLE]

and consider regions of $Q$ where $R_{0}$ is less than a certain numerical size for a given set of estimates for $k_{\rm i}^{2}$ and $k_{\rm f}^{2}$. Next, since scattering is assumed to be in the current region in Fig. 4(a), the ratio

[TABLE]

must also be small. See Ref. Boglione:2016bph for more discussion – $R_{1}$ corresponds to $R$ from that reference. The expression for $R_{1}$ in terms of the variables in Eq. (27) and Eq. (80) is straightforward, but slightly cumbersome and not instructive, so we will not write it explicitly here.

The $2\to 1$ partonic kinematics only apply if $k^{2}/Q^{2}\approx 0$, an approximation that fails if transverse momentum is too large. So define another ratio,

[TABLE]

$R_{2}$ is small for $2\to 1$ partonic kinematics. From Eq. (80),

[TABLE]

Note that this suggests $q_{\rm{\scriptsize T}}$ from Eq. (29) as the most useful transverse momentum for quantifying transverse momentum hardness relative to $Q$; if $q^{2}_{\rm{T}}/Q^{2}\sim 1$, then $R_{2}\sim 1$ for both large and small $\hat{z}_{N}$ while if $q^{2}_{\rm{T}}/Q^{2}\ll 1$ and $\zeta\sim z_{\rm N}$ (as in the current fragmentation region with TMDs) then $R_{2}\ll 1$ (see also discussion in Ref. Gonzalez-Hernandez:2018ipj ).

If the SIDIS region corresponds to $2\to 2$ hard partonic kinematics, then $R_{2}$ must be large ($\sim 1$). However, then the ratio $k_{\rm X}^{2}/Q^{2}$ must be small since there is only one unobserved parton, and its invariant mass must be small relative to hard scales to qualify as a single massless parton. (See Fig. 4(b).) If $k_{2}$ is a massless on-shell quark or gluon, then $k_{2}^{2}=0$ and this places a strong kinematical constraint on relationship between the momentum fractions $\xi$ and $\zeta$. See, for example, Eq.(83) of Nadolsky:1999kb . So define one more ratio,

[TABLE]

Large $R_{2}$, but small $R_{3}$, corresponds to $2\to 2$ parton kinematics. Large $R_{2}$ and large $R_{3}$ corresponds to partonic scattering with three or more final state partons, such as Fig. 4(c).

To see that the size of $R_{2}$, Eq. (96), reflects the importance of transverse momentum, we repeat an argument very similar to that on page 4 of Gonzalez-Hernandez:2018ipj . Note that Feynman graphs corresponding to the inside of the box in Fig. 4 contain propagator denominators of the form

[TABLE]

where the denominators with $+O\left(Q^{2}\right)$ arise in corrections to the virtual photon vertex or internal propagators from the emission of wide-angle $k_{\rm X}$ partons. Note also that $k\cdot q\sim q\cdot P=O\left(Q^{2}\right)$. The possible approximations to these denominators are representative of the approximations needed in derivations of factorization. If $|k^{2}|\sim Q^{2}$, the $O\left(m^{2}\right)$ terms in the denominators are negligible so that the part of the graph inside the box can be calculated in perturbative QCD using both $Q^{2}$ and $k^{2}$ as equally good hard scales. In this case, and $k_{\rm X}^{2}\ll Q^{2}$, then Fig. 4(b) becomes the relevant picture. However, if $|k^{2}|\ll Q^{2}$, the $O\left(m^{2}\right)$ terms in the first of the denominators in Eq. (98) must be kept. Then, a $|k^{2}|/Q^{2}\ll 1$ approximation in the second denominator can be used, and it is this type of approximation that leads to TMD factorization at small transverse momentum. This is the handbag topology in Fig. 4(a). Note that the $k$ line has become the target parton. Using Eq. (80) and Eq. (87) for $k^{2}$ gives Eq. (96).

In perturbative QCD, the lowest order (in $O\left(\alpha_{s}\right)$) contribution to large transverse momentum is the partonic $2\to 2$ process. Again, all partons are massless and on-shell, and the picture is Fig. 4(b). Since there is only one unobserved massless parton in this region, it correspond to $k_{X}^{2}=0$. To see that it is the ratio $R_{3}$ in Eq. (97) that must be small in this region, consider how the size of $k_{X}^{2}$ affects the denominators in Eq. (98) at fixed $\hat{x}_{N}$, large $q_{\rm{\scriptsize T}}$, and $Q^{2}$ by expressing $|k^{2}/Q^{2}|$ in terms of $k_{X}^{2}$ instead of $\hat{z}_{N}$:

[TABLE]

To get a simple form, we have already assumed here that $k_{\rm i}^{2}$ and $k_{\rm f}^{2}$ are negligible. In propagators, therefore, the size of $k^{2}$ is independent of $k_{X}^{2}$ at large $k^{2}_{\rm{T}}$ if $k_{X}^{2}/Q^{2}\ll 1$ and $\hat{x}_{N}$ is not too close to $1$. Otherwise, if $R_{3}$ becomes large, the $2\to 3$ or greater cases are likely the more applicable partonic subprocesses. In pQCD this means that $O\left(\alpha_{s}^{2}\right)$ or higher calculations are needed.

Different combinations of sizes for the above ratios correspond to other regions. For example, the target fragmentation region handles cases where $R_{1}$ gets large – see Sec. 10 below. All of the approximations discussed above are intertwined in potentially complicated ways, especially when $Q$ is not especially large and mass effects may be non-negligible. This can make even crude, order-of-magnitude estimates of their effects nontrivial, although the influence of model assumptions should diminish rapidly at large $Q$. The catalogue of ratios represented by the $R_{0}$-$R_{3}$ is meant to make this more straightforward to check.

A choice concerning acceptable ranges of $R_{0}$, $R_{1}$, $R_{2}$, and $R_{3}$ translates into a choice about the range of possible reasonable values for the components of $k_{\rm i}$ and $k_{\rm f}$. In practice, this might be more conveniently stated in reverse. That is, one starts with general expectations regarding the sizes of the partonic components of $k_{\rm i}$ and $k_{\rm f}$ based on models and/or theoretical considerations. The question then becomes whether the resulting $R_{0}$, $R_{1}$, $R_{2}$, and $R_{3}$ are consistent with a particular region of partonic kinematics (hard, current region, large transverse momentum, etc).

Our aim here is not to address any particular theoretical framework for estimating intrinsic properties of partons, or to estimate exactly acceptable ranges for the above ratios, but only to demonstrate how, once these choices are made, they fix the relationship between external kinematics and the region of partonic kinematics.

9 Rapidity

It is often useful to express results in terms of rapidity instead of $z_{\rm N}$ or $z_{\rm h}$. In the Breit frame,

[TABLE]

The boost invariant rapidity difference is

[TABLE]

If $x_{\rm N}\approx z_{\rm N}$ and $M_{\rm B,T}\approx M$, then the produced hadron rapidity is approximately the negative of the proton rapidity. For fixed $z_{\rm N}/x_{\rm N}$, fixed $M_{\rm B,T}$ and large $Q$

[TABLE]

$z_{\rm h}$ in terms of $y_{\rm B,b}$ is Boglione:2016bph

[TABLE]

In terms of $z_{\rm h}$, the rapidity of the hadron in the Breit frame is double valued:

[TABLE]

The “$+$” solution corresponds to a hadron with large rapidity in the direction of $P$, while the “$-$” solution corresponds to a rapidity in the opposite direction, and thus is more consistent with current region factorization. The approximation after the “$\approx$” corresponds to the $m^{2}/Q^{2}\to 0$ limit of the “$-$” solution.

Expressing the plus and minus components in Eq. (80) in terms of rapidity,

[TABLE]

Then, values of $\hat{z}_{\rm N}$, $\hat{x}_{\rm N}$, $k_{\rm i}$, $k_{\rm f}$, ${\bf k}_{\rm i,T}$, ${\bf k}_{\rm f,T}$ can be mapped, along with values of $R_{0}$-$R_{3}$, to regions of a $q_{\rm{\scriptsize T}}$ versus rapidity map like Fig. 8. If $\hat{z}_{\rm N}q_{\rm{\scriptsize T}}=O\left(Q\right)$, then $y_{\rm f}^{\rm b}\approx\ln\left(\frac{q_{\rm{\scriptsize T}}}{Q}\right)\approx 0$, while if $\hat{z}_{\rm N}q_{\rm{\scriptsize T}}=O\left(m\right)$, then $y_{\rm f}^{\rm b}\approx\ln\left(\frac{m}{Q}\right)$. In the handbag configuration, wherein all partonic transverse momenta are zero, the parton four-momenta may be written,

[TABLE]

Since $k_{\rm i}^{+}\approx-{q}^{+}_{\rm b}=Q/\sqrt{2}$ and $k_{\rm f}^{+}\approx{q}^{-}_{\rm b}=Q/\sqrt{2}$ in the handbag configuration, then $y_{\rm i}^{\rm b}\approx-y_{\rm f}^{\rm b}=O\left(\ln\left(Q/m\right)\right)$. Therefore, partons in the handbag configuration are centered roughly on $y\approx 0$ in the Breit frame.

Note also that if $x_{\rm N}$ and $z_{\rm N}$ are small, then according to Eq. (100) both the target and produced hadrons will tend to be skewed toward larger rapidities in the Breit frame. Therefore, hadrons measured in the final state will tend to be at larger rapidities than the corresponding handbag-configuration partons.

10 Target Remnant Hadrons

If, in contrast to the discussion in Sec. 8, the hadron is in the target fragmentation region (see Fig. 5), then

[TABLE]

In the target region, $z_{\rm h}$ is no longer as useful for parameterizing the process since it no longer necessarily describes a momentum fraction – see Eq. (30) and note that the quantity under the square root diverges as $z_{\rm h}\to 0$. In terms of $x_{\rm h}$, $z_{\rm N}$ is:

[TABLE]

where we have kept the solution that gives exactly $z_{\rm N}=0$ when $P_{\rm B}{}{}$ is exactly massless and collinear to $P$. Now,

[TABLE]

Equation (109) is no larger than $O\left(m^{2}\right)$ if $z_{\rm N}\sim m^{2}/Q^{2}$ and $q^{2}_{\rm{T}}z_{\rm N}^{2}/Q^{2}\ll 1$. So for the target region, Eq. (107) with Eqs. (108)–(109) means

[TABLE]

The “Big $\Theta$” symbol is used because the first term in Eq. (109) puts a lower limit on acceptable sizes for $z_{\rm N}$. In other words, the target region criterion fails both when $z_{\rm N}\ll m^{2}/Q^{2}$ as well as when $z_{\rm N}\gg m^{2}/Q^{2}$. From Eq. (108), this means the target fragmentation criterion in terms of $x_{\rm h}$, $x_{\rm Bj}$ and ${{\bf P}_{{\rm B},,\rm{T}}}{}$ is

[TABLE]

To translate Eq. (107) into a dimensionless ratio, define

[TABLE]

Therefore, the target region criterion is

[TABLE]

In Boglione:2016bph , it was $1/R_{1}$ that was used to characterize the target region, and that is another acceptable definition, but Eq. (112) has the advantage of working even when $k_{\rm i}$ differs significantly from $P$ and of being simpler to calculate.

11 Soft-Central Hadrons

It is possible that for some hadrons, $R_{2}\ll 1$, while neither $R_{1}$ nor $R^{\prime}_{1}$ is small. We call this the soft region since such hadrons are not a product of hard scattering but do not associate in any obvious way with a quark or target direction.

12 Specific Examples

For illustration, let us insert some specific numbers into the above system of formulas. First, consider the purely kinematical ratios in Sec. 7 for realistic experimental scenarios. In Fig. 6 we display the ($Q,x_{\rm Bj}$) kinematic coverage of three SIDIS experiments: JLab 12 (11 GeV electron beam), HERMES (27.5 GeV electron beam) and COMPASS (160 GeV muon beam). The shaded regions are obtained by applying the appropriate experimental cuts in each case, as reported in Refs. Avakian:2016rst ; Airapetian:2012ki ; Aghasyan:2017ctw . Notice that the JLab 12 kinematics covers a very wide range of $x_{\rm Bj}$ values, well above 0.6, but it is limited to intermediate/small values of $Q$. Instead, the COMPASS kinematics reaches up to much larger values of Q, but the accessible range of $x_{\rm Bj}$ is confined to values no larger than 0.4. In each plot, the values of the ratio $x_{\rm N}/x_{\rm Bj}$, Eq. (10), are color coded: darker shades represent regions where $x_{\rm N}/x_{\rm Bj}$ deviates from $1$ and the MTA approximation deteriorates. As expected mass corrections are more important at large values of $x_{\rm Bj}$ and small values of $Q$.

Fig. 7 shows the ratio $z_{\rm N}/z_{\rm h}$, over the $(z_{\rm h},P_{B,T}/Q)$ kinematic coverage of the three experiments. Again darker shades represent larger deviations from $1$ which, in this case, are more significant than for $x_{\rm Bj}/x_{\rm N}$, especially at JLab kinematics.

It is helpful to sketch the landscape of possible scenarios in a transverse momentum versus rapidity map like the one shown in Fig. 8. Each of the regions discussed in Sec. 8, Sec. 10, and Sec. 11 is represented there as a colored blob, and the task is to determine the sizes of the blobs, their borders, and their degree of overlap. The relevant power suppression factors are shown. (Recall, for example, Eq. (96).)

To give more detailed examples than the above, a few assumptions about non-perturbative properties of partons are necessary. $300$ MeV is a typical estimate of non-perturbative mass scales so we try $k_{\rm i}=k_{\rm f}=\delta k_{\rm{\scriptsize T}}=300$ MeV. Also, to start with we assume that ${\bf q}_{\text{\scriptsize T}}\cdot\delta{\bf k}_{\text{\scriptsize T}}=q_{\rm{\scriptsize T}}\delta k_{\rm{\scriptsize T}}$. (Azimuthal effects may be added later.)

In addition, the particular partonic kinematics of interest need to be specified. Say, for example, that the goal is to examine target partons in the valence region (such as discussed on page 3 of Avakian:2016rst ). Then the focus should be on momentum fraction values of $\xi$ roughly around $0.3$. For $\zeta$, we might reasonably focus on values where collinear fragmentation functions are large but have reasonably small uncertainties, say $\zeta\approx 0.3$. From Fig. 6, JLab12 measurements at $x_{\rm Bj}\approx 0.2$ may reach to as large as about $2$ GeV in $Q$.

First let’s consider overall kinematics. Contour plots of $W_{\rm SIDIS}^{2}$, Eq. (69), are shown for a pion mass in Fig. 9 for (a) $q_{\rm{\scriptsize T}}=0$ and (b) $q_{\rm{\scriptsize T}}=2.0$ GeV, giving a sense of what is kinematically possible for the SIDIS remnant at different $q_{\rm{\scriptsize T}}$ and for lower $Q$. The expectation is that the area near the kinematically forbidden region, where the final state phase space vanishes, does not readily separate into distinct regions as in Fig. 8. So in the below we will focus on kinematics away from those boundaries. Also, for now we will restrict to large enough $Q$ that $R_{0}$ in Eq. (93) is negligible, so $R_{1}$ is the first of the $R_{0}$-$R_{3}$ that we will consider here.

For the representative values discussed above ($\xi=0.3$, $z_{\rm h}=0.25$, $\zeta=0.3$ and a small $q_{\rm{\scriptsize T}}=0.3$ GeV), values of $R_{1}$ are shown on the $Q$ vs. $x_{\rm Bj}$ contour plot in Fig. 10. The trend is as expected: at large $Q$ and not-too-large $x_{\rm Bj}$, $R_{1}$ remains small for all transverse momenta, while corrections might be necessary at smaller $Q$ and larger $x_{\rm Bj}$.

In addition to confirming the current-region approximation, which holds valid where collinearity $R_{1}$ is small, it is necessary to map out the applicability of large and small transverse momentum approximations. For this we turn to $R_{2}$. Fig. 11 is an example that corresponds to the same kinematics as Fig. 10. It confirms basic expectations, such as that what constitutes “large-$q_{\rm{\scriptsize T}}$” grows with $Q$. It also shows that, while the hadron is in the current region for most $q_{\rm{\scriptsize T}}$ as in Fig. 10 (a,b), the small transverse momentum region shown in Fig. 11 (a) is much more restrictive. For $q_{\rm{\scriptsize T}}\lesssim 0.5$ GeV, $R_{2}$ is firmly in the small transverse momentum region for most of the $Q$ shown, while for $q_{\rm{\scriptsize T}}\gtrsim 1.5$ GeV $R_{2}$ indicates that we are well in the large transverse momentum region. There is a broad intermediate region where the situation is not clear. The flavor of the final state hadron is a decisive factor in determining the relevant factorization region. For example, comparing the plots of $R_{1}$ in Fig. 10 for (a) $M_{B}=m_{\pi}$, and (c) $M_{B}=m_{K}$, shows a completely different behaviour of the collinearity ratio $R_{1}$. For $Q=1.5$ GeV and $x_{Bj}=0.1$, $R_{1}\approx 0.1$ for pions and $R_{1}\approx 0.8$ for kaons. If $R_{1}\approx 0.8$ is taken to be large, then confidence that one is in the current region deteriorates.

The flavor of the final state hadron has little effect on the transverse momentum hardness, $R_{2}$, from Eq. (95). From Fig. 11 (a) and Fig. 11 (c) flavor dependence is only noticeable at low Q and even then the effect is small. To summarize, the produced hadron mass affects collinearity $R_{1}$ significantly, but does not appear to be a primary factor in determining transverse hardness $R_{2}$.

Within a specific example, collinearity $R_{1}$ and transverse hardness $R_{2}$ have helped us to map out the current kinematic region (small $R_{1}$) and to separate the "small" from the "large" transverse momentum regions (small $R_{2}$ vs large $R_{2}$). The former will reasonably correspond to a region where we expect TMD factorization to apply, while for the latter a collinear factorization will be appropriate. At this stage, one might wonder whether a LO calculation could be enough or whether higher order perturbative corrections are necessary. This is where $R_{3}$ comes into the game: large $R_{3}$ coupled with large $R_{2}$ signal a large $q_{T}$ region where presumably higher order pQCD corrections are relevant, while small $R_{3}$ together with small $R_{2}$ clearly indicate a TMD current region, which requires a TMD factorization scheme.

Clearly the above indications only apply to the specific example we have chosen, corresponding to specific values of the kinematic variables ($\xi=0.3$, $z_{\rm h}=0.25$, $\zeta=0.3$, $q_{\rm{\scriptsize T}}=0.3$ GeV) and of the non perturbative parameters ($k_{\rm i}=k_{\rm f}=\delta k_{\rm{\scriptsize T}}=300$ MeV, ${\bf q}_{\text{\scriptsize T}}\cdot\delta{\bf k}_{\text{\scriptsize T}}=q_{\rm{\scriptsize T}}\delta k_{\rm{\scriptsize T}}$). A web tool which allows to compute $R_{1}$-$R_{3}$ for any kinematic configuration can be found in Ref. sidistool .

13 Conclusion

Since the early work in presented in Refs. Meng:1995yn ; Nadolsky:1999kb ; Graudenz:1994dq there has been a large number of studies on unpolarized SIDIS cross sections Koike:2006fn ; Anselmino:2006rv ; Anselmino:2013lza ; Signori:2013mda ; Sun:2013dya ; Sun:2013hua ; Bacchetta:2017gcc . Unpolarized SIDIS is, however, only one component in a broad program of phenomenological studies where the universality of parton correlation functions plays a central role in testing pictures of nucleon structure Anselmino:2016uie ; Anselmino:2015sxa ; Bacchetta:2015ora ; Scimemi:2017etj ; Kang:2017btw ; Kang:2014zza ; Landry:2002ix ; Sun:2016kkh ; Sun:2011iw ; Echevarria:2015uaa ; Guzzi:2013aja ; Echevarria:2016scs ; Echevarria:2014xaa ; Boglione:2018dqd ; Boglione:2017jlh ; Kang:2015msa ; Lin:2017stx ; Ye:2016prn ; Bacchetta:2019tcu ; Bastami:2018xqd ; Bertone:2019nxa . This demands a clear language for identifying kinematical regions of transversely differential deep inelastic scattering cross sections with particular underlying partonic pictures, especially in regions of moderate to low Q where sensitivity to kinematical effects outside the usual very high energy limit becomes non-trivial.

In this paper, we have outlined the ways that the questions about the boundaries between different partonic regimes of SIDIS can be posed systematically, based on the power-law expansions that apply in each region (recall Fig. 8). As the ratios $R_{0}$-$R_{3}$ in Sec. 8 show, quantifying the separation between different SIDIS regions requires at least some rough model assumptions for the intrinsic properties of partons. Hence, our position is that region mapping should be viewed as one of the aspects of SIDIS that is to be determined with guidance from data, rather than being treated as well-known input. Nevertheless, the $R_{0}$-$R_{3}$ can already be useful for querying the reasonableness of some region assumptions. For example, if collinearity $R_{1}$ is found to be approximately $10$ for a wide range of even rough models, then a current region assumption could be viewed with skepticism. Conversely, very small values of collinearity $R_{1}$ might be considered a strong signal that one is deep in a regime where a current region fragmentation function picture is appropriate. If, in addition, there is a small transverse hardness ratio $R_{2}$ it may be taken to signal the close proximity to small transverse momentum, where a TMD factorization scheme would be appropriate. If transverse hardness ratio $R_{2}$ and spectator virtuality ratio $R_{3}$ are both large, then high order pQCD corrections are likely important. In a fitting context, the $R_{0}$-$R_{3}$ can be utilized to fix Bayesian priors. Conversely, the success or failure of theoretical predictions can be used to constrain the ranges of $R_{0}$-$R_{3}$ that are acceptable for particular regions in future theoretical predictions.

In developing a picture of the likelihood that a particular kinematical region corresponds to a particular partonic picture, one should of course consider a wide range of multiple non-perturbative models for the values of $k_{\rm i}$, $k_{\rm f}$, etc., in addition to sampling from a range of $\zeta$, $\xi$, and azimuthal angles, and track the values of $R_{0}$-$R_{3}$, in addition to $x_{\rm N}/x_{\rm Bj}$, $z_{\rm N}/z_{\rm h}$, $W_{\rm tot}^{2}$, $W_{\rm SIDIS}^{2}$ to assess the validity of various purely kinematical approximations. This could be done, perhaps, at the level of computer simulations, where the values of $R_{0}$-$R_{3}$ can be tracked. For now, the effect of changing quantities like $k_{\rm i}^{2}$ and $k_{\rm f}^{2}$ can be examined directly with our web tool Ref. sidistool .

In the future we plan to incorporate this view into phenomenological procedures, particularly in situations with not-too-large $Q$. We hope that this will ultimately contribute to a clearer picture of the borders between different regions and an improved understanding of the transition between hadronic and partonic degrees of freedom.

Acknowledgements.

We thank Eric Moffat, Wally Melnitchouk and Jian-Wei Qiu for useful discussions. We are especially thankful to Markus Diehl for discussions of Eq. (66) that led to Appendix A. T. Rogers, A. Dotson, S. Gordon and N. Sato were supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under Award Number DE-SC0018106. This work was also supported by the DOE Contract No. DE- AC05-06OR23177, under which Jefferson Science Associates, LLC operates Jefferson Lab. L. Gamberg was supported by the U.S. Department of Energy under grant No. DE-FG02-07ER41460. A. Prokudin was supported by the National Science Foundation under Grant No. PHY-1623454 and the DOE Contract No. DE- AC05-06OR23177, under which Jefferson Science Associates, LLC operates Jefferson Lab. J. O. Gonzalez-Hernandez work was partially supported by Jefferson Science Associates, LLC under U.S. DOE Contract DE-AC05-06OR23177 and by the U.S. DOE Grant DE-FG02-97ER41028.

Appendix A Integration over $z_{\rm N}$ and ${\bf P}_{B,\text{\scriptsize T}}{}$.

In the appendix, we will work in the target hadron rest frame (a photon frame). Start with the elementary relation

[TABLE]

Change the $z_{\rm N}$ variable on the left side to $z_{\rm h}$. The $\mathop{\mathrm{d}\mskip-0.59999muz_{\rm h}}\nolimits{}$ appears in both the integral and the derivative and Jacobian factors cancel:

[TABLE]

Expressed in differential form, and for one particular hadron type $B$, this is

[TABLE]

where $\mathop{\mathrm{d}\mskip-0.59999mu\langle N_{B}\rangle}\nolimits$ is the number of particles of type $B$ in the differential volume $\mathop{\mathrm{d}\mskip-0.59999mu^{2}{{\bf P}_{{\rm B},\gamma,\rm{T}}}}\nolimits\,\mathop{\mathrm{d}\mskip-0.59999muz_{\rm h}}\nolimits{}$. Let $E_{B}$ be the energy per particle of type $B$ (in the target rest frame), and multiply both sides of Eq. (116) by $E_{B}$:

[TABLE]

$E_{B}\mathop{\mathrm{d}\mskip-0.59999mu\langle N_{B}\rangle}\nolimits$ is the energy per $B$-particle times the number of $B$ particle in the differential volume, so it is the total energy of all $B$-particles in the differential volume. Therefore, we have defined it as $\mathop{\mathrm{d}\mskip-0.59999mu\langle E_{B}^{\rm all}\rangle}\nolimits$ in the last equality. Integrating it and summing over all types of final state particles produces the total energy of the entire final state:

[TABLE]

Note that the sum over $B$ is a sum over all types of particles, not a sum over actual particles. Divide both sides of Eq. (117) by $q^{0}$:

[TABLE]

Integrate over both sides, restore the sum over particle types $B$, and use Eq. (118) for the right side:

[TABLE]

Now, in the target rest frame,

[TABLE]

Also,

[TABLE]

From energy conservation,

[TABLE]

so

[TABLE]

So, Eq. (120) becomes

[TABLE]

Now we need to use this to relate the SIDIS and the total DIS structure functions. For the total DIS cross section, the structure function decomposition with standard notational conventions uses Eq. (44), Eq. (46), and Eq. (51). The cross section is thus

[TABLE]

Substituting Eq. (126) into the right side of Eq. (125), and substituting Eq. (58) into the left side gives

[TABLE]

Substituting Eq. (42) for $z_{\rm N}$ gives the factor in Eq. (58). Thus, the normalization of $F_{T/L}$ needs to be redefined as in Eq. (64) in order to get the integration/sum rule in Eq. (66) and (Bacchetta:2006tn, , Eqs. (2.18-2.21)).