# On the definition of fragmentation functions and the violation of sum rules

John Collins<EMAIL_ADDRESS>Department of Physics, Penn State University,
University Park PA 16802, USA
ORCID: 0009-0000-8494-6992 Ted C. Rogers<EMAIL_ADDRESS>Department of
Physics, Old Dominion University, Norfolk, VA 23529, USA  Jefferson Lab, 12000
Jefferson Avenue, Newport News, VA 23606, USA
ORCID: 0000-0002-0762-0275

(September 5, 2023)

###### Abstract

We point out a problem with the formulation and derivations of sum rules for
quark fragmentation functions that impacts their validity in QCD, but which
potentially points toward an improved understanding of final states in
inclusive hard processes. Fragmentation functions give the distribution of
final-state hadrons arising from a parton exiting a hard scattering, and the
sum rules for momentum, electric charge, etc express conservation of these
quantities. The problem arises from a mismatch between the quark quantum
numbers of the initial quark and the fact that all observed final-state
hadrons are confined bound states with color zero. We point that, in a
confining theory like QCD, the Wilson line in the operator definition of a
fragmentation function entails that the final state in a fragmentation
function includes a bound state in the external field generated by the Wilson
line. We justify this with the aid of general features of string
hadronization. The anomalous bound states are restricted to fractional
momentum $z=0$. They tend to invalidate sum rules like the one for charge
conservation when applied to the fragmentation functions inferred from
experimental data, but not the momentum sum rule. We propose to exploit our
ideas in future studies as a way to relate the ffs extracted from inclusive
cross sections to more detailed non-perturbative descriptions of final state
hadronization. We also describe scenarios wherein the traditional sum rules
might remain approximately valid with a reasonably high degree of accuracy.

††preprint: JLAB-THY-23-3909

##  I Introduction

Factorization theorems for inclusive processes in QCD are the theoretical
basis for much existing QCD phenomenology. When factorization is applied to
distributions of final-state hadrons, fragmentation functions (ffs) are used
for the distributions of hadrons that arise from partons exiting the hard
scattering. In analogy with sum rules for parton densities, sum rules for ffs
have been written down [1, 2, 3, 4, 5, 6], including for cases with more than
one detected final-state hadron, and with proofs that ostensibly apply non-
perturbatively. The momentum sum rule in particular is sometimes proposed as a
constraint in phenomenological extractions [7, 8]. Ideally, one could combine
the use of ffs in studies of scattering data with theoretical non-perturbative
studies of hadronization. However, one finds that a contradiction arises
concerning the final state sum/integral that appears in the operator ff
definition, Eq. (2) below.

If we ignore complications, the definition of an ff involves an initial
partonic state created by applying a light-front creation operator for a quark
(or gluon) field to the vacuum. Then an ff is a distribution of particles in
the final state (i.e., at asymptotically large time). The sum rules are just
expressions of conservation laws in QCD. Now, all observed final-state
particles in QCD have hadronic quantum numbers. In contrast, the initial state
of a quark ff has quark quantum numbers, e.g., fractional baryonic and
electric charge, and so the final state must also have the same quantum
numbers. This not possible for a normal hadronic final state.

In this paper, we propose a resolution of this paradox. It involves the Wilson
line needed for a gauge-invariant definition of the operator creating the
partonic state. Since the Wilson line extends out to infinity, it requires a
modified concept for parts of the final state. By using the successful string
scheme for non-perturbative hadronization, we find that the final state of a
quark ff should contain a particle that is effectively the bound state of
quark(s) with the Wilson line, but that this is localized essentially at zero
fractional momentum, $z=0$. This does not affect factorization and does not
contribute to the momentum sum rule. But it modifies how other sum rules are
to be applied phenomenologically.

Factorization applies in a general quantum field theory, not just QCD. So part
of our paper will contrast the situation in a QCD-like theory with the
situation in a non-gauge model theory where no Wilson lines are used and there
is no quark confinement.

##  II The definition

The standard definition of a “bare” quark ff
$d_{(0),h/j}(z,\boldsymbol{p}_{\text{T}})$ is (e.g., Eq. (12.35) of [9]):

$d_{(0),h/j}(z,\boldsymbol{p}_{\text{T}})~{}\langle j,k_{1}|j,k_{2}\rangle\\\
\equiv\frac{\sum_{X}\langle j,k_{1}|h,X,\text{out}\rangle\langle
h,X,\text{out}|j,k_{2}\rangle}{2z(2\pi)^{3-2\epsilon}}\,.$ (1)

Here, $|j,k_{1}\rangle$ and $|j,k_{2}\rangle$ are initial quark states with
flavor $j$, which are obtained by applying a light-front creation operator to
the vacuum. We define light-front variables by $k^{\pm}=(k^{0}\pm
k^{z})/\sqrt{2}$, and a two dimensional transverse momentum. The initial quark
states have given values of $k_{1}^{-}$ and $k_{2}^{-}$, and transverse
momenta (with the transverse momenta taken to zero later). The variable $z$ is
the fractional minus component of the quark momentum carried by the observed
final state hadron $h$, $p^{-}=zk^{-}$, and $\boldsymbol{p}_{\text{T}}$ is its
transverse momentum. The out-states are $|h,X,\text{out}\rangle$ with $h$
labeling the measured particle species and the sum/integral over $X$
representing a complete sum over all other contributions to the final state.
Because states of definite momentum are non-normalizable, we work indirectly
to get a number density, which entails the factor $\langle
j,k_{1}|j,k_{2}\rangle$ on the left of Eq. (1).

We start by considering nongauge theories, so we have omitted from Eq. (1) the
color factor in the corresponding equation for QCD given in [9]. We require
that the theory be renormalizable and, to show the contrast with QCD, we
require it to contain elementary Dirac fields that we call quarks, with at
least one flavor. The definition is stated in $4-2\epsilon$ dimensions to
anticipate the existence of ultraviolet divergences that need to be regulated
and renormalized. Equation (1) actually defines a bare transverse-momentum-
dependent ff. The bare _collinear_ ff is obtained by integrating over all
$\boldsymbol{p}_{\text{T}}$. It can be re-expressed in the more familiar form,

$\displaystyle{}d_{(0),h/j}(z)\equiv\int\mathop{\mathrm{d}^{2-2\epsilon}\mskip-0.59999mu\boldsymbol{p}_{\text{T}}}\nolimits
d_{(0),h/j}(z,\boldsymbol{p}_{\text{T}})$ $\displaystyle=$
$\displaystyle{}\frac{{\rm
Tr}_{D}}{4}\sum_{X}z^{1-2\epsilon}\int\frac{\mathop{\mathrm{d}\mskip-0.59999mux^{+}}\nolimits}{2\pi}e^{ik^{-}x^{+}}\gamma^{-}\times$
$\displaystyle{}\,\times\langle
0|\psi_{j}^{(0)}\left(x/2\right)|h,X,\text{out}\rangle\langle
h,X,\text{out}|\overline{\psi}_{j}^{(0)}\left(-x/2\right)|0\rangle\,,$ (2)

with the fields separated in the $+$-direction. Some of the manipulations
needed to give Eq. (2) are just to deal with the fact that partonic states
with definite $k^{-}$ and $\boldsymbol{k}_{\text{T}}$ are not normalizable, so
the definition of probability densities in terms of quantum mechanical states
is obtained indirectly. In addition, there is a Lorentz transformation, to
take the transverse momentum of the hadron to zero while preserving the minus-
components of momenta.

Finally, we define $|j,k\rangle=b_{k,j}^{\dagger}|0\rangle$, where
$b_{k,j}^{\dagger}$ is a light-front creation operator. It is obtained from an
expansion of the Fourier transform of the good components of the quark field
on a light-front, and obeys anticommutation relations

$\left\\{b_{k,\alpha},b^{\dagger}_{l,\alpha^{\prime}}\right\\}=(2\pi)^{3}2k^{-}\delta\left(k^{-}-l^{-}\right)\delta^{(2)}\left(\boldsymbol{k}_{\text{T}}-\boldsymbol{l}_{\text{T}}\right)\,.$
(3)

In terms of the quark field of a specific flavor, the $b_{k\alpha}$ operator,
for example, is

$b_{k,\alpha}(x^{+})=\int\mathop{\mathrm{d}\mskip-0.59999mux^{+}}\nolimits\mathop{\mathrm{d}^{2}\mskip-0.59999mu\boldsymbol{x}_{\text{T}}}\nolimits
e^{ik^{-}x^{+}-i\boldsymbol{k}_{\text{T}}\cdot\boldsymbol{x}_{\text{T}}}\bar{u}_{k,\alpha}\gamma^{-}\psi(x)\,.$
(4)

See, for example, Sec. 6.6 of [9] for a more detailed review of light-cone
quantization. Our reason for discussing it here is to emphasize that there are
two different types of particle state involved in ffs: those like the
fragmenting quark states $|j,k\rangle$, and those for stable particles in the
final states at large time scales, with correspondingly different operators to
create them.

##  III Sum rules

Now we briefly review the essential steps in the derivation of the momentum
sum rule for ffs. For a more complete treatment see Sec. 5.4 of [1].

We start with the following expression for the single inclusive final state
sum in terms of creation and annihilation operators for out-particles: 111We
emphasize the need to make clear conceptual and notational distinctions
between the (off-shell) light-front creation operators and those for on-shell
asymptotic final-state particles, and similarly for the states they create.
The issue is particularly acute in a theory without quark confinement, where
one finds a non-trivial ff for a quark to a quark, as we will illustrate
calculationally in the Appendix. Such a concept is paradoxical if the use of
two different types of quark state is not made explicit.

$\displaystyle\sum_{X}|h,X,\text{out}\rangle\langle h,X,\text{out}|$
$\displaystyle\qquad\equiv\sum_{X}a_{h,p,\text{out}}^{\dagger}|X,\text{out}\rangle\langle
X,\text{out}|a_{h,p,\text{out}}$
$\displaystyle\qquad=a_{h,p,\text{out}}^{\dagger}a_{h,p,\text{out}}\,,$ (5)

so that, for example, Eq. (2) is equivalent to

$d_{(0),h/j}(z)=\frac{{\rm
Tr}_{D}}{4}z^{1-2\epsilon}\int\frac{\mathop{\mathrm{d}\mskip-0.59999mux^{+}}\nolimits}{2\pi}e^{ik^{-}x^{+}}\gamma^{-}\times\\\
\times\langle
0|\psi_{j}^{(0)}\left(x/2\right)a_{h,p,\text{out}}^{\dagger}a_{h,p,\text{out}}\overline{\psi}_{j}^{(0)}\left(-x/2\right)|0\rangle.$
(6)

In terms of $a_{h,p,\text{out}}^{\dagger}$ and $a_{h,p,\text{out}}$, the
operators for components of momentum are

$\mathcal{P}^{\mu}=\sum_{h}\int_{0}^{\infty}\frac{\mathop{\mathrm{d}\mskip-0.59999mup^{-}}\nolimits}{2p^{-}}\int\frac{\mathop{\mathrm{d}^{2-2\epsilon}\mskip-0.59999mu\boldsymbol{p}_{\text{T}}}\nolimits}{(2\pi)^{3-2\epsilon}}\,a_{h,p,\text{out}}^{\dagger}p^{\mu}a_{h,p,\text{out}}\,,$
(7)

where the sum over $h$ is over all species and spin states of stable particle.
We substitute this into the matrix element $\langle
j,k_{1}|\mathcal{P}^{-}|j,k_{2}\rangle$, which equals $k_{1}^{-}\langle
j,k_{1}|j,k_{2}\rangle$, and then apply Eq. (5) to get an integral over $z$
and $\boldsymbol{p}_{\text{T}}$ of the right-hand side of Eq. (1). This gives
the momentum sum rule

$\sum_{h}\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits\,z\,d_{(0),h/j}(z)=1\,,$
(8)

for each quark flavor $j$.

The remaining nontrivial step is to show that Eq. (8) is preserved after
renormalization. It is well known that this works for standard renormalization
schemes like $\overline{\rm MS}$, so from here forward we will drop “$(0)$”
subscripts in equations like Eq. (8) and assume that renormalization has been
implemented.

##  IV The final states

The above sum rule derivation relies, for its validity, on the use of a
complete set of basis states $|X,\text{out}\rangle$ whose sum/integral obeys

$\sum_{X}|X,\text{out}\rangle\langle X,\text{out}|=\widehat{1}$ (9)

where $\widehat{1}$ is the unit operator on the physical state space
$\mathcal{E}$.222The non-vacuum part of the space is then stratified in a one-
particle inclusive form as in Eq. (2), with $h$ being summed over all possible
kinds of single-particle state, including a spin sum. The derivation also
relies on the sum over $h$ in Eq. (2), etc, being over all kinds of possible
stable single-particle states, such that the momentum operators obey Eq. (7).

Now all observed final-state particles in QCD have integer baryon number and
electric charge. But the initial state $b^{\dagger}|0\rangle$ in a quark ff
has quark quantum numbers, notably fractional electric charge, and so is
orthogonal to all purely hadronic final states. This appears to give zero for
the matrix elements in Eq. (2) and hence for the ffs.

The paradox does not arise in a non-gauge theory: There, we can apply a
locally smeared quark field to the vacuum to create a normal physical state
with quark quantum numbers. By general principles of QFT, there must be final
states of quark quantum numbers, and hence at least one stable particle (bound
or not) of the appropriate quantum numbers, including electric charge.

But in a gauge theory, local quark fields are not gauge-invariant physical
operators. Applied to the vacuum, they do not give an unambiguously physical
state. Instead, the field used to define the ff of a quark is multiplied by a
Wilson line, which goes out to infinity in an appropriate light-like or almost
light-like direction.333Note that when one treats transverse-momentum
dependent ffs there are some complications associated with the details of the
Wilson line — see Ch. 13 of [9] and references therein. These complications do
not affect the basic ideas being explained here. A Wilson line is effectively
a source of color charge, so a Wilson line going out to infinity changes the
nature of the possible final-state particles. In a confining theory like QCD,
we must have, in addition to normal hadrons, states that are bound to the
Wilson line. For a quark ff, the Wilson line is in a color anti-triplet
representation. So we can have a meson-like state with a quark bound to the
Wilson line in a color-singlet configuration. Other possibilities include an
antibaryon-like state of two antiquarks bound to the Wilson line, again in a
color-singlet configuration. Since the Wilson line has a rapidity with respect
to the fragmenting quark that is infinite, or at least large, we can expect
the bound states to be at a fractional momentum $z$ that is zero, or close to
zero.

We therefore extend the normal QCD state space $\mathcal{E}$, with its Fock
basis of out states, to include these extra bound states. That is, we replace
the hadronic state space $\mathcal{E}$ by

$\mathcal{E}\otimes\mathcal{B},$ (10)

where $\mathcal{B}$ is the space of bound states for the Wilson line.
Correspondingly there is a modified momentum operator, with corresponding
consequences for the sum rules for ffs.

Let $H$ be the set of kinds of normal hadrons, and $B$ be the set of bound
states of the Wilson line. We propose that the momentum operator should be
modified from the one given in Eq. (7) to

$\mathcal{P}^{\mu}=\mathcal{P}_{H}^{\mu}+\mathcal{P}_{B}^{\mu}\,,$ (11)

where

$\displaystyle\mathcal{P}_{H}^{\mu}$ $\displaystyle\equiv\sum_{h\in
H}\int_{0}^{\infty}\frac{\mathop{\mathrm{d}\mskip-0.59999mup^{-}}\nolimits}{2p^{-}}\int\frac{\mathop{\mathrm{d}^{2-2\epsilon}\mskip-0.59999mu\boldsymbol{p}_{\text{T}}}\nolimits}{(2\pi)^{3-2\epsilon}}a_{h,p,\text{out}}^{\dagger}p^{\mu}a_{h,p,\text{out}}\,,$
(12) $\displaystyle\mathcal{P}_{B}^{\mu}$ $\displaystyle\equiv\sum_{b\in
B}\int_{0}^{\infty}\frac{\mathop{\mathrm{d}\mskip-0.59999mup^{-}}\nolimits}{2p^{-}}\int\frac{\mathop{\mathrm{d}^{2-2\epsilon}\mskip-0.59999mu\boldsymbol{p}_{\text{T}}}\nolimits}{(2\pi)^{3-2\epsilon}}a_{b,p,\text{out}}^{\dagger}p^{\mu}a_{b,p,\text{out}}\,.$
(13)

The conflict about the initial state with quark quantum numbers relies on the
theory being QCD-like, with color confinement. So we should expect similar
issues to arise in QED in low space-time dimensions ($1+1$, $2+1$), where the
classical Coulomb potential rises linearly or logarithmically with distance.
In contrast, QED in $3+1$ dimensions does not have electron confinement, so
that electrons do appear as possible final-state particles. Even so, the
definition of an electron ff in QED still needs a Wilson line going out to
infinity, for which bound states can exist.444The need for a Wilson line is
likely to be related to the complications in defining states of charged
particles in QED — for a review, see Ref. [10]. So some version of the QCD
issues does arise there, probably only in a minor way.

Once we have modified the momentum operator, we get a corresponding modified
form for the momentum sum rule, compared with Eq. (8):

$\sum_{h\in
H}\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits\,z\,d_{h/j}(z)=1-\sum_{b\in
B}\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits\,z\,d_{{b}/j}(z)\,.$
(14)

The left-hand side is the same quantity as before, and corresponds to the ffs
that can be inferred from scattering data, with their purely hadronic final-
state particles. The second term on the right-hand side represents a deficit
with respect to the standard value. To the extent that the parts of the final-
state with a particle of quark quantum numbers give a term that is a delta
function at $z=0$, which is the natural expectation, the deficit term is zero,
because of the explicit factor of $z$ in the integrand.

Observe that in a non-confining theory, the term with final-state particles of
quark quantum numbers is not restricted to $z=0$; indeed it can in part give a
term proportional to $\delta(z-1)$, as in perturbative calculations in model
QFTs. Then the set $H$ should be defined to include such terms, and the set
$B$ is to be restricted to bound states with the Wilson line.

The situation changes for the flavor sum rules, such as were formulated in
[1]. For the charge sum rule, our modified derivation gives.

$\sum_{h\in
H}\mathcal{Q}_{h}\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits\,d_{h/j}(z)=\mathcal{Q}_{j}-\sum_{b\in
B}\mathcal{Q}_{b}\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits\,d_{{b}/j}(z)\,.$
(15)

A $\delta(z)$ term for $d_{b/j}(z)$ no longer gives zero, and we can no longer
expect the original formulation [1] of the sum rule to be necessarily
accurate. A closer and non-perturbative analysis of the dynamics is needed to
get a prediction for the right-hand side; this we can do with the aid of the
string model, to a useful approximation.

For the total hadron number sum rule [3], we similarly have

$\sum_{h\in
H}\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits\,d_{h/j}(z)=\langle
N\rangle-\sum_{b\in
B}\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits\,d_{{b}/j}(z)\,,$
(16)

where $\langle N\rangle$ is the average multiplicity of _all_ final states
produced by the fragmenting quark, including those with nonhadronic quantum
numbers.555It is a slight abuse of language to refer to Eq. (16) as a “sum
rule” since the multiplicity associated with a single ff is not a known
measurable or conserved quantity. Moreover, its exact numerical value depends
on an arbitrary choice of renormalization scheme or of a cut-off. We will
continue to call it the “hadron number sum rule,” however, to remain
consistent with existing literature, where identities like Eq. (16) guide
interpretations.

In the Appendix, we illustrate how the the sum rules apply in a renormalizable
nongauge theory. The examples there indicate the importance of keeping terms
with final-state particles that have quark quantum numbers.

##  V Interpretation in QCD

The nature of the final state in a quark ff in QCD is illustrated by the
strong-model account of hadronization in $e^{+}e^{-}$-annihilation. We present
this as a qualitative perturbative schema and approximation in Fig. 1, as in
the paper by Casher, Kogut and Susskind [11]. At center-of-mass energy $Q$, an
electron and positron annihilate over a short distance scale $1/Q$ to make an
outgoing quark-antiquark pair. A color field between them, that ends up as a
flux tube, is created by gluon emission. Quark-antiquark pairs are generated
in the flux tube, and recombine into color-singlet mesons. In space-time, the
meson production is roughly localized around a space-like hyperbola. The
slowest mesons, of low center-of-mass rapidity, are formed first, and the
fastest, high rapidity, mesons are formed on a long time-dilated scale. The
result is a two jet structure, with each jet created from one or other of the
quark and antiquark. Between the leading particles in each jet, mesons fill in
the rapidity region with an approximately uniform distribution in rapidity.

Figure 1: A schematic picture of the production of a final state in the
process $e^{+}e^{-}\to\text{hadrons}$. Hadrons on the left hand side move with
large negative rapidity, and those on the right hand side move with large
positive rapidity. To split the figure into two ffs like Fig. 2 with quark
number 1, the central boxed hadron must be split into an orphan quark and an
orphan antiquark.

The quark generating a jet preferentially ends up in the leading particle in
its jet, with a corresponding bias in the leading particle’s charge. For
example, the leading particle in a $u$-quark jet is more often a $\pi^{+}$
than a $\pi^{-}$. But there is, at the same time, no hadron of the fractional
charge of the $u$-quark. This contrasts with the situation in the simple, low-
order model in the Appendix, of fragmentation in a non-gauge model. In that
model, the dominant leading particle in the fragmentation of a quark-analog is
exactly an on-shell particle of quark quantum numbers.

The distribution of hadrons in each jet is given by the corresponding
fragmentation function. One can propose splitting the final state in Fig. 1
between the two jets. For symmetry, this split is in the middle of the box
labelled “orphans”, thereby leading to a left-over quark or antiquark in each
part, which we call an orphan quark or antiquark. This is illustrated in Fig.
2. In this approximation, whether the orphan is a quark or an antiquark is
fully determined by the corresponding property of the jet-initiating parton.
This gives a long-range correlation, given that the orphan is at low momentum.
But no such correlation applies to which flavor the orphan has (e.g., $u$
versus $d$).

Figure 2: A schematic picture of an allowed final state in a calculation of
the quark ffs. Hadrons are labeled by $H_{1}$ through $H_{3}$. These belong in
the set of particles labeled $H$ in the text. A final orphan quark with very
small $z$, labeled by a red dot, always remains. It is part of the set of
states labeled $B$.

In Fig. 2, only mesons are shown in the final state. More complex arrangements
with multiple leftover quarks and antiquarks can arise to give, for example,
baryons in the final state, as in Fig. 3.

Figure 3: Like Fig. 2 but with a baryon in the final state, represented by the
oval. The result now has a pair of orphan antiquarks.

Note that the quantitative description of the hadron distribution by a
fragmentation function should only be accurate for the faster hadrons, and the
just-mentioned split is at the lowest end of the $z$ range for which
fragmentation is relevant. Given a process at some energy, we can choose a
$Q$-dependent value $z_{\rm min}$, below which we do not attempt to describe
hadron distributions by a fragmentation function of a given parton. This
implies that at a given value of $Q$, the portion of the integrals in the sum
rules in the range $0<z<z_{\rm min}$ is not accessible to experimental
measurements. This range decreases approximately like $1/Q$ as $Q$ gets large.
The orphan quark or antiquarks at low $z$ carry quantum numbers that are able
to leak out of the valid factorization region $z>z_{\rm min}$. Thus, we will
call the $d_{{b}/j}(z)$ that appear in the correction terms for sum rules like
Eqs. (15)–(16) the “deficit” ffs.

We can also apply the string model to the actual ff we defined, that includes
a Wilson line. This is shown in Fig. 4. The string model indicates that the
orphan quark should simply combine with the Wilson line, and make a bound
state, as we proposed earlier. The bound state appears at low $z$ in a region
where an ff is not intended to be accurate as a description of a real process.
In contrast, the top part of Fig. 4, for the faster hadrons, matches the
corresponding part of Fig. 1 for the physical process. The bound state of the
orphan quark has no need to correspond to any simple observable.

Figure 4: A graph contributing to the calculation of a quark-to-quark ff. The
observed hadron is in the thick-dashed black oval. The zig-zag line at the
bottom is an orphan quark binding to the Wilson line.

The simplest way to match ffs to the full process is to implement the afore-
mentioned split at central rapidity by replacing the orphan quark by a bound
state to a Wilson line that has zero rapidity in the overall center-of-mass
frame. In Fig. 1, this would correspond to an insertion of a time-like Wilson
line, with rapidity zero, between the central orphan quark and antiquark. That
would match the definition of a transverse-momentum-dependent ff by Collins
and Soper – see Eq. (5.1) of Ref. [1] – with its use of an axial gauge and
dependence on an auxiliary vector $n$. In contrast, for an integrated ff, the
Wilson line is of _infinite_ rapidity (opposite to that of the jet). Given the
physical picture just discussed, that would appear at first sight to include
too much of the final state. However, this problem is resolved by a careful
analysis of the definitions in the context of a factorization proof, as in
[9]. First, transverse-momentum dependent ffs are defined, and include “soft
factors” defined in precisely such a way as to compensate incorrect treatment
and overcounting of emission at low rapidity. When one makes the transition to
integrated ffs, the soft factors give unity, because of an exact cancellation
between real and virtual emission. This type of cancellation between graphs
with different kinds of final states highlights the importance of the level of
inclusivity in determining the validity of a particular factorization formula.

##  VI Implications for sum rules

Determining the orphan quark ffs requires information from nonperturbative
QCD, and a careful treatment is beyond the scope of this paper. However,
general physical considerations lead to some reasonable conjectures that may
be useful points of departure for more sophisticated treatments in the future.

The Wilson line that needs to be included with each field in Eq. (2) is in the
light-like plus direction, opposite the nearly light-like minus momentum of
the hadronizing quark. It carries the memory of the oppositely moving
antiquark (and its associated orphan partons) that appeared in the full
process before factorization. It has color opposite to the original quark and
ensures that the entire system is color neutral. Therefore, the Wilson line
forms a type of bound state with an orphan quark, as discussed in the previous
section. (See Fig. 4.) The orphan quark will always have a $z$ momentum
fraction almost equal to zero, so the deficit ffs should be approximated by

$d_{{b}/j}(z)\approx c_{j,b}\delta(z)\,,$ (17)

with constant coefficients $c_{j,b}$ that can be different for each type of
$b$. In cases with baryon production like Fig. 3, it would be a pair of orphan
antiquarks that bind with the Wilson line. In this picture, the Wilson line
acts as an external source. Because the ffs need to satisfy a DGLAP equation,
a literal $\delta$-function may not be appropriate here, but for now Eq. (17)
should be interpreted simply as saying that an orphan ff is only non-
negligible in a very narrow region around $z\approx 0$ and is normalized to a
constant $c_{j,b}$. This also matches the physical intuition provided by the
discussion of Fig. 1. In this picture, the last term vanishes for the momentum
sum rule in Eq. (14), giving back the original momentum sum rule of [1]. But
the number sum rules need corrections.

To interpret the meaning of the deficit ffs, it is also useful to consider the
consequences of the approximation where baryons contribute a negligible amount
to the total number of final states. Then the hadronic sector of the final
state always has quark number exactly zero, and the orphan contribution is
always just a single quark of flavor $j$ as in Fig. 2. We express this below
by writing $\\{b\\}\approx\\{q_{j}\\}$ where $q$ represents just a single
isolated orphan quark of flavor $j$, bound to the Wilson line. In such a
scenario, the deficit ffs satisfy a simple number sum rule,

$\sum_{j^{\prime}}\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits
d_{{q_{j^{\prime}}}/j}(z)=1\,,$ (18)

where now $j^{\prime}$ runs over the flavors of a single orphan quark. Then
the modified hadron number sum rule in Eq. (16) also takes the very simple
form,

$\sum_{h\in\left\\{H\right\\}}\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits\,d_{h/j}(z)=\langle
N\rangle-1\,.$ (19)

To satisfy Eq. (18), the coefficients in Eq. (17) must obey

$\sum_{j}c_{j,j^{\prime}}=1\,.$ (20)

The corrected expression for the hadron number sum rule in Eq. (19) highlights
an ambiguity in the interpretation of the total number of hadrons associated
with an ff as in Fig. 1: It is ambiguous whether the central, boxed hadron
should be grouped with the left (quark) or right (antiquark) ff. If we insist
that it goes with the left side ff, then the number of total particles counted
in the quark ff includes the hadron with the orphan quark, and $\langle
N\rangle$ would be the appropriate quantity to associate with the total
particle multiplicity in the quark ff. However, if the central hadron goes
with the right side antiquark ff, then the left ff only includes three
hadrons, and $\langle N\rangle-1$ would be the appropriate measure of
multiplicity in the quark ff. If $\langle N\rangle\gg 1$, then Eq. (19)
recovers the hadron number sum rule in, for example, [3] because the effect of
adding or removing just one particle becomes negligible. This picture becomes
more complicated once we allow for baryons. However, the no-baryon
approximation illustrates how the intuitive hadron number sum rule
interpretation might be recovered at high enough energies that the average
hadron multiplicity is very large.

To extend this model still further, and still using the meson dominance model
above, we note that it is plausible that the $c_{j,j^{\prime}}$ coefficients
in Eq. (17) are approximately equal for all active partons $n_{f}$, given that
the orphan quark will be very separated in rapidity from the initial quark. In
that case, Eq. (20) gives

$c_{j,j^{\prime}}\approx 1/n_{f}$ (21)

and the charge sum rule in Eq. (15) becomes simply

$\sum_{h\in\left\\{H\right\\}}\mathcal{Q}_{h}\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits\,d_{h/j}(z)\approx\mathcal{Q}_{j}-\frac{1}{n_{f}}\sum_{j^{\prime}\in\text{active}}\mathcal{Q}_{j^{\prime}}\,.$
(22)

In this approximation, if the only active quark flavors are only $u$,$d$, and
$s$ then the sum of charges is zero and the subtracted term in Eq. (22)
vanishes. Then the standard charge sum rule from [1] is recovered. If the
active flavors include the charm quark, then Eq. (23) becomes

$\sum_{h\in\left\\{H\right\\}}\mathcal{Q}_{h}\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits\,d_{h/j}(z)\approx\mathcal{Q}_{j}-\frac{e}{6}\,.$
(23)

##  VII Comments

The simple model above helps to clarify the meaning of the orphan quark ffs,
and it shows how the standard ff sum rules might be approximately true despite
the mismatch of final state quark numbers. But it is important to keep in mind
that a deeper understanding of the final states and the nonperturbative
properties of the deficit ffs is necessary before this picture can be placed
on a very firm footing. As it currently stands, it is possible that the range
of $z$ where the orphan ffs are non-negligible extends to somewhat higher
values than might be expected on the basis of the intuition sketched above,
such that there are non-negligible violations of the momentum sum rule. That
possibility is especially relevant at moderate hard scales where the
kinematical range of validity of factorization is more limited than at very
high scales, and the number of final state hadrons is smaller.

The ff momentum sum rule is rarely ever used _directly_ to constrain ffs
phenomenologically since to do so requires knowledge of ffs for all hadron
flavors, and these are not known with enough precision and over a wide enough
range of $z$ for the momentum sum rule to be practically useful [12, 13, 14,
15, 16]. To the extent that it is used, it is typically only in the form of an
upper bound,

$\sum_{h\in
H}\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits\,z\,d_{h/j}(z)\leq
1$ (24)

used to test general consistency, usually with a lower bound on the $z$
integration. Our analysis indicates that this bound remains valid.
Nevertheless, given that the sum rule is widely quoted as a fundamental
property of ffs (see, for example, Eq. (19.3) of [17]), it is important to
recognize that, from a theory standpoint, it is not a guaranteed identity.
Unless a model like Eq. (17) is at least approximately valid, then the true
upper bound in Eq. (24) might be more or less restrictive. It is possibly
relevant for existing phenomenology that DGLAP evolution only preserves the
momentum sum rule if all the relevant final states appear in the sum,
including the orphan quark ffs. An incomplete sum over final state particles
is not guaranteed to be preserved under evolution. Practical difficulties with
implementing momentum sum rules in combination with DGLAP evolution have been
noted in the literature [18].

To our knowledge, the charge sum rule has not been used in applications to
phenomenological extractions.

With regard to the hadron number sum rule in Eq. (19), it may seem that
subtracting 1 is a minor modification. However, in semi-inclusive deep
inelastic scattering measurements at moderate $Q$ at facilities like Jefferson
Lab, typical hadron multiplicities are around $5$, and at a future EIC are
expected to be about $12$ to $13$ [19]. Adding or removing a single hadron
could significantly impact a hadron number interpretation of ffs in scenarios
like these.

What is perhaps more relevant than the impact on existing phenomenology is the
role of sum rules in guiding the formulation of new types of ffs and establish
their interpretation. For example, specific definitions for dihadron ffs and
even $n$-hadron ffs were proposed in [6] according to a requirement that they
satisfy extended versions of the hadron number sum rule. There, the problems
discussed in this paper are exacerbated because the $n$-hadron ffs occupy a
larger part of the final state phase space, and therefore adding or removing a
single particle has a larger impact. Likewise, [20, 21] suggests relating ff
sum rules to the dynamical generation of quark masses and jet functions. In
[22, 5], momentum sum rules are derived that involve first taking transverse
moments of transverse momentum dependent ffs. For these types of sum rules,
the orphan quark problem is compounded by ultraviolet divergent transverse
momentum integrals. Finally, the sum rules for fracture functions [23, 24]
suffer from the same complication as ffs. In all the of above applications, a
careful look at the issues discussed in this paper is warranted.

Models used in Monte Carlo event generators (e.g., Refs. [25, 26, 27]) could
potentially provide frameworks for clarifying what is needed. There, one is
forced to deal directly with descriptions of complete final states. For
example, in Ref. [28], the authors find that their model only preserves the
momentum sum rule exactly in the limit of infinite final state cascades.

Finally, we propose that understanding the nonperturbative features of the
orphan quark ffs through their operator definitions will help to clarify the
connection between ffs and full descriptions of the final state hadronization
process, and that this may help with the development of applications like
those listed above.

###### Acknowledgements.

We thank J. O. Gonzalez-Hernandez for very useful comments on the text. The
ideas in this paper were originally inspired by a reading of the recent work
in [6], and we thank the authors for a discussion of their paper. T. Rogers
also thanks the Jefferson Lab QCD study group for helpful discussions of
relevant topics. This work was supported by the U.S. Department of Energy,
Office of Science, Office of Nuclear Physics, under Award Number DE-SC0018106.
It was also supported by the DOE Contract No. DE-AC05-06OR23177, under which
Jefferson Science Associates, LLC operates Jefferson Lab.

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##  Appendix A Calculations in non-gauge model

To make the sum rules in Eqs. (14)–(16) less abstract, it is instructive to
validate them in a renormalizable nongauge theory. We will use a real scalar
Yukawa theory with one flavor of Dirac fermion “quark” and one pion. Both
fields will have nonzero masses, but we will set the masses equal to simplify
calculations, $m_{\text{quark}}=m_{\pi}=m$. We will consider the case of a
single quark flavor $j$. Normal Feynman graph calculations give the pion-in-
quark ff from the definition in Eq. (5) in $\overline{\rm MS}$ renormalization
at lowest nontrivial order,

$\displaystyle d_{\pi/j}(z;\mu)\;$
$\displaystyle{}=\begin{array}[]{c}\includegraphics[scale={0.35}]{FFgreal}\end{array}+\text{$\overline{\rm
MS}$\; C.T.}$ (26)
$\displaystyle=a_{\lambda}(\mu)z\left[\ln\left(\frac{1}{(1-z)+z^{2}}\right)+\ln\frac{\mu^{2}}{m^{2}}-1+\frac{(2-z)^{2}}{(1-z)+z^{2}}\right]\,,$
(27)

where $a_{\lambda}(\mu)$ is a coupling constant and $\mu$ is the usual
$\overline{\rm MS}$ scale from dimensional regularization. The term with a
final-state quark is, to the same order,

$\displaystyle d_{q_{j}/j}(z;\mu)$
$\displaystyle{}=\begin{array}[]{c}\includegraphics[scale={0.35}]{FFlo}\end{array}+\left[\begin{array}[]{c}\includegraphics[scale={0.35}]{FFbubble}\end{array}+\text{H.C.}\right]+\begin{array}[]{c}\includegraphics[scale={0.35}]{FFqreal}\end{array}+\text{$\overline{\rm
MS}$\; C.T.'s}$ (31)
$\displaystyle{}=\delta(1-z)-a_{\lambda}(\mu)\delta(1-z)\left[\pi\frac{\sqrt{3}}{2}-2+\ln\frac{\mu}{m}\right]+a_{\lambda}(\mu)(1-z)\left[\ln\left(\frac{1}{(1-z)^{2}+z}\right)+\ln\frac{\mu^{2}}{m^{2}}-1+\frac{(1+z)^{2}}{(1-z)^{2}+z}\right]\,.$
(32)

Taking the zeroth and first moments gives

$\displaystyle\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits
d_{q_{j}/j}(z;\mu)=1-a_{\lambda}(\mu)\left[\pi\frac{\sqrt{3}}{2}-2+\ln\frac{\mu}{m}\right]+a_{\lambda}(\mu)\left[\pi\frac{\sqrt{3}}{2}-2+\ln\frac{\mu}{m}\right]=1\,,$
(33) $\displaystyle\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits
d_{\pi/j}(z;\mu)=a_{\lambda}(\mu)\left[\pi\frac{\sqrt{3}}{2}-2+\ln\frac{\mu}{m}\right]\,,$
(34) $\displaystyle\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits
zd_{q_{j}/j}(z;\mu)=1-a_{\lambda}(\mu)\left[-\frac{13}{9}+\frac{\pi}{\sqrt{3}}+\frac{1}{3}\ln\frac{\mu^{2}}{m^{2}}\right]\,,$
(35) $\displaystyle\int_{0}^{1}\mathop{\mathrm{d}\mskip-0.59999muz}\nolimits
zd_{\pi/j}(z;\mu)=a_{\lambda}(\mu)\left[-\frac{13}{9}+\frac{\pi}{\sqrt{3}}+\frac{1}{3}\ln\frac{\mu^{2}}{m^{2}}\right]\,.$
(36)

Equation (33) demonstrates that the quark number sum rule in Eq. (18) is
indeed satisfied. Noting that the pion has zero charge, Eq. (33) also confirms
the charge sum rule in Eq. (15). Finally, substituting Eq. (35) and Eq. (36)
into Eq. (14) shows that the momentum sum rule in Eq. (14) is satisfied, and
it appears as

$\left\\{a_{\lambda}(\mu)\left[-\frac{13}{9}+\frac{\pi}{\sqrt{3}}+\frac{1}{3}\ln\frac{\mu^{2}}{m^{2}}\right]\right\\}=1-\left\\{1-a_{\lambda}(\mu)\left[-\frac{13}{9}+\frac{\pi}{\sqrt{3}}+\frac{1}{3}\ln\frac{\mu^{2}}{m^{2}}\right]\right\\}\,.$
(37)

To match the formula in QCD, with its deficit term that involves final-state
particles with quark quantum numbers, we have put the quark term on the right-
hand side.

The sum rules are only satisfied here if the contribution of an on-shell
final-state quark to the ff is included. Note that if this term were dropped
from the right-hand side of Eq. (37), not only would the momentum sum rule be
violated, but the amount of the violation would depend on the evolution scale
$\mu$. An important part of this contribution is from large $z$, as in the
delta-function terms in Eq. (32). Therefore, in this theory, the quark term is
an important contribution to the measurable hadronization of a quark-induced
jet. In terms of our notation for QCD, quark particles would be included in
the set we denoted by $H$, and there would be no orphan particles to put in
the set $B$.

The hadron number sum rule in Eq. (19) becomes, using Eq. (34),

$a_{\lambda}(\mu)\left[\pi\frac{\sqrt{3}}{2}-2+\ln\frac{\mu}{m}\right]=\langle
N\rangle-1\,.$ (38)

If the coupling is negligible, the left side of Eq. (38) vanishes and the sum
rule is just $\langle N\rangle=1$, i.e. the final state is always just a
single quark.