orcid: 0000-0002-1535-6208 arxiv:2111.01170v1 [hep-ph] 1

JLAB-THY-21-3507

Positivity and renormalization of parton densities

John Collins∗

Department of Physics, Penn State University, University Park PA 16802, USA

Ted C. Rogers†

Department of Physics, Old Dominion University, Norfolk, VA 23529, USAJefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, USA and

ORCID: 0000-0002-0762-0275

Nobuo Sato‡

Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, USA andORCID: 0000-0002-1535-6208

(Dated: November 1, 2021)

There have been recent debates about whether MS parton densities exactly obey positivity bounds(including the Soffer bound), and whether the bounds should be applied as a constraint on globalfits to parton densities and on nonperturbative calculations. A recent paper (JHEP 11 (2020)129) appears to provide a proof of positivity in contradiction with earlier work by other authors.We examine their derivation and find that its primary failure is in the apparently uncontroversialstatement that bare pdfs are always positive. We show that under the conditions used in thederivation, that statement fails. We provide some elementary calculations in a model QFT thatshow how this situation can generically arise in reality. In addition, we observe that the methodsused in the derivation are in common with much, but not all, other work where factorization isderived. Our examination pinpoints considerable difficulties with these methods that render themeither wrong or highly problematic. The issue of positivity highlights that these methods can leadto wrong results of phenomenological importance. From our analysis we identify the restrictedsituations in which positivity can be violated.

I. INTRODUCTION

Central to many phenomenological applications ofQCD is the concept of a parton density (or distribution)function (pdf). An issue that has become particularlyimportant recently is whether or not pdfs are always pos-itive. Although they generally obey positivity, there hasbeen disagreement on whether it is possible for some pdfsto be slightly negative under some conditions.

In the literature, one can find cautionary statementsto the effect that negative pdfs are possible, at least atlow scales [1], and some fitting procedures do allow forslightly negative pdfs (e.g., Ref. [2]). The reason given isthat while the most elementary definition of a pdf doesmanifestly obey positivity, it also has ultraviolet (UV)divergences. The necessary UV renormalization coun-terterms are not guaranteed to preserve positivity, as wewill explain in Sec. VIII with explicit counterexamples topositivity.

However, it has also been recently argued, notably byCandido, Forte and Hekhorn [3], that positivity is anautomatic and general property of pdfs defined in the MSscheme. Since this result is in contradiction with explicitcalculations, it creates an apparent paradox that needsto be resolved. We will find that the problem is that

∗ [email protected]† [email protected]‡ [email protected]

certain simple and apparently uncontroversial assertionsin Ref. [3] are in fact false, but for non-trivial reasons.We will give more details later, but we summarize whatgoes wrong here.

Fundamental to the argument in Ref. [3] is the posi-tivity of bare pdfs and of partonic cross sections in a the-ory dimensionally regulated in the UV and infrared (IR).These positivity properties result from standard proper-ties of quantum mechanical states, notably the positivityof the metric on state space. However, once a theory isregulated, the state-space metric need not be positive. Aclassic case of such a violation of positivity is given bythe Pauli-Villars method. In the case of Ref. [3], dimen-sional regularization is used for both the UV and collineardivergences. (Bare pdfs have UV divergences, and mass-less partonic cross sections have collinear divergences.)In that case, positivity properties fail, as we will showexplicitly.

If instead one were to use regulators that preservedpositivity, we will show that another of the foundationsof Ref. [3] fails. This is the commonly made assertionthat a DIS structure function on a target factors into anunsubtracted massless partonic structure function and abare pdf on the target: F (Q, xbj) = F partonic ⊗ fbare,B.

We stress that the failures just identified do not in factaffect the final factorization into renormalized pdfs andsubtracted coefficient functions. But they do break theargument for the absolute positivity of MS pdfs, as wewill see, and this observation motivates greater scrutinyof the properties of pdfs more generally. Moreover, fail-

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ure of a widely asserted factorization property deserves acloser analysis, which is the main purpose of this paper,and we use the positivity issue to motivate it.

The failure of positivity of MS pdfs in some situationsoccurs despite the fact that the original concept of a pdf,within Feynman’s parton model [4], entailed positivity;that was simply because a parton density was intendedto be the number density of a particular flavor of partonin a fast-moving hadron. But, as is well-known and aswe will review below, the situation in real QCD requiresmodification of the parton model.

Since factorization gives predictions for cross sections,and cross sections are intrinsically positive, the scope fornegative pdfs is severely limited. For each parton fla-vor, one can construct a DIS-like process in which thelowest-order term in the hard scattering is initiated byonly the chosen parton flavor. (This can be done by re-placing the currents in the hadronic part of deep inelasticscattering by suitable operators containing only fields forthe chosen flavor.) At a high scale Q, the effective cou-pling αs(Q) is small. Therefore, the lowest-order termtypically dominates, so that positivity of a cross section(or other quantity) entails positivity of the pdf. The onlyway this can be avoided is if some other pdf is sufficientlymuch larger in magnitude for flavors and/or regions thatdo not contribute at lowest order, such that perturbativecorrections to the cross section dominate the lowest-orderpart. That is, any negative pdf must be small in magni-tude relative to other pdfs, which are necessarily positive,by the argument involving a lowest-order approximationto a hard scattering.

This argument gradually loses its force as Q getssmaller, since then perturbative corrections are no longerso suppressed. This leads to the expectation that nega-tive pdfs can occur at most at low scales. (Later, we willsee support for this in our calculations.)

It is desirable to have a treatment of the positivity is-sue in terms of the pdfs themselves and their definition,rather than indirectly through factorization and the posi-tivity of cross sections. We will present the basics of suchan analysis in Sec. VIII.

One important implication of the possibility of neg-ative pdfs arises in phenomenological fits of pdfs, sinceoften the scale Q0 used for the initial scale for evolutionis rather low, and may even be below scales at which itis reasonable to use factorization. One aim of a low Q0

is to ensure that the fitted pdfs are provided at all scaleswhere factorization could conceivably be usefully applied.Therefore, if a positivity constraint were applied to fit-ted pdfs, especially at a low initial scale Q0, it is likelyto introduce excessive theoretical bias.

Note that positivity constraints, if they are valid, notonly apply directly to unpolarized pdfs, but also giveconstraints on polarized pdfs, with a particularly non-trivial case being the Soffer bound [5–7].

Another important situation where the same issuearises is when one is making calculations of pdfs fromQCD by non-perturbative methods. A common method

is to calculate a quasi-pdf by lattice Monte-Carlo meth-ods, and to infer the pdfs themselves by a factorizationproperty [8–13], similarly to the way global fits of pdfsare made to experimental scattering data. Such calcula-tions typically give results for a low value of Q0, and insome lattice QCD calculations positivity is imposed as aconstraint [11–13] on parametrizations, especially in thelimit of large momentum fractions, ξ → 1. It is critical toknow whether it is correct to apply positivity constraintsin this situation. Another similar example is in calcula-tions based on the Dyson-Schwinger equation, where ascale as low as µ0 = 0.78 GeV [14] is used.

As regards applications that use perturbative calcula-tions, a circumstance where pdfs do definitely becomenegative is in the treatment of heavy quarks [15]. Insuch treatments, one uses perturbative calculations tomatch the versions of pdfs with different numbers of ac-tive quark flavors. The pdf for a heavy quark which is ac-tive is perturbatively related to pdfs defined with a lowernumber of active quarks. The lowest order calculationexpresses the heavy quark density in terms of the gluondensity, with a coefficient that is just the MS pdf of theheavy quark in a massless on-shell gluon at perturbativeorder αs. It contains a factor of ln(µ/mh), where mh isthe heavy-quark mass; in this particular instance there isno non-logarithmic term. The calculation of a pdf for anactive heavy quark can be applied where the MS scale µis somewhat less than the heavy-quark’s mass. Becauseof the factor of ln(µ/mh), the result is a negative heavy-quark pdf in this region; the negative pdf is essential topreserving the momentum sum rule. The smallness of theeffective coupling at the heavy quark scale implies thatthe perturbative calculation of the negative heavy quarkpdf is reliable,1 given a specific definition of the pdf. Inthat situation the size of the negative heavy-quark pdfis substantially less than the gluon pdf, since to a firstapproximation it is given in terms of the gluon pdf by aone-loop calculation. Hence in the application of factor-ization the O(αS) gluon-induced term can be comparableto the lowest order heavy-quark-induced term, therebyallowing preservation of positivity of the cross section.2

However, when using factorization it is generally onlyimportant to treat a heavy quark as active when aprocess’s physical scale is significantly larger than thequark’s mass. In that case the heavy quark pdf is evolvedfrom its calculated value at the scale of the quark’s massto a substantially higher scale; it is then positive.

Let us now return to examining the differing ap-proaches to the positivity question. We will show thatthe differences originate in a long-standing divergence in

1 In contrast, light-quark pdfs in QCD cannot be usefully com-puted by low order perturbation theory.

2 Note that the coefficient function for the gluon-induced processincludes a subtraction to avoid double-counting heavy quark con-tributions. If the heavy quark term is negative, that results inan increased value for the gluon term.

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views about certain conceptual foundations for QCD fac-torization and the definitions of pdfs. We trace this topioneering QCD literature of the 1970s, written whilemuch of the technical framework of factorization was be-ing developed. One track, which we call track-A, origi-nated in efforts to give the earliest parton ideas a concreterealization in quantum field theory, with inspiration forderivations coming from those for the operator-productexpansion, which was already applied in QCD to deep in-elastic scattering. The second track (track-B) arose earlyout of a practical desire to perform partonic calculations.In the absence at that time of a full track-A treatment,conjectures were made as to appropriate methods.

First, we will review the basics of track A in Sec. II.Then, in Sec. III we will explain track B and present acritique of it. We will argue that track-A is actually thecorrect one. In Sec. IV we will examine what is necessaryfor track B expressions in dimensional regularization toreproduce the DIS cross section. In Sec. V we point outpitfalls with using dimensional regularization simultane-ously for UV and IR divergences; these pitfalls break thepositivity argument of Ref. [3]. In Sec. VI we will com-bine the observations of the previous sections to summa-rize the reasons why the argument in [3] fails and arguethat it is traceable to the use of track B. In Sec. VIII, weillustrate that an MS pdf can turn negative in the con-crete example of a Yukawa field theory where everythingis calculable in fixed, low order perturbation theory.3 Weend with concluding remarks in Sec. IX.

In our examination of DIS, we will use a fairly standardnotation: P is the momentum of the target, q is theincoming momentum at the current in the hadronic part,and M is the target mass. We use light front coordinates,with the inner product of two vectors being given bya · b = a+b− + a−b+ − aT · bT, and the components of avector being written as a = (a+, a−,aT). The coordinateaxes are chosen such that the components of P and q are

P =

(P+,

M2

2P+,0T

), q =

(−xP+,

Q2

2xP+,0T

), (1)

where x is the Nachtmann variable, which agrees withthe Bjorken xBj up to power-suppressed corrections.

II. TRACK-A: RENORMALIZATION ANDLIGHT-CONE PDFS

One of the motivating points of track-A was workto provide a definite field-theoretical implementation ofthe original pdf concept. At the beginning, this led

3 The reason for presenting an example in Yukawa theory is thatthe arguments in [3] are independent of the theory. So we choosea theory and coupling where low-order perturbative calculationson a massive target give a sufficiently accurate calculation todetermine the validity of the methods for proving positivity.

to the insight that light-front quantization provides asuitable candidate definition as the expectation of alight-front number operator [16–18], provided that nodifficulties arise. But difficulties do arise. The dif-ficulties are particularly notable when one includes atreatment of transverse-momentum-dependent pdfs, as inSoper [19] and Collins [20]. For the case of the transverse-momentum-integrated pdfs and fragmentation functionsin full QCD, the elementary definition of these quantitiesneeds to be modified [21] to allow for UV renormaliza-tion. At least for collinear pdfs, this form of the defi-nition has continued to be used to the present, withoutmodifications. It is the pdfs with this definition that onenow studies using lattice QCD and other nonperturbativetechniques.

A second motivation was the realization that the meth-ods that led to operator-product expansion (OPE) couldbe generalized. For DIS, the OPE applies to certain in-teger moments of structure functions. The methods canbe extended to obtain the large-Q asymptotics of thestructure functions themselves. The factorization andOPE derivations have overall structures that are verysimilar. In fact, when one takes the appropriate mo-ments of DIS structure functions, one recovers the resultsfrom the OPE. Although the factorization work drewon the derivation of the OPE by Wilson and Zimmer-mann (e.g. [22, 23]), there were some important enhance-ments/modifications that we will make more explicit inSec. IV.

For unpolarized quark pdfs, a bare quark pdf is definedby

fbare,Aj/H (ξ) ≡

∫dw−

2πe−iξp

+w−

〈p| ψj,0(0, w−,0T)γ+

2W [0, w−]ψj,0(0, 0,0T) |p〉 . (2)

where ψj,0 is the bare field for a quark of flavor j asappears in the Lagrangian density that defines the the-ory. The factor W [0, w−] is a light-like Wilson line,also defined with bare field operators and the bare cou-pling. The label H denotes the kind of target particlethat is used for the state |p〉. This definition is actuallythe expectation value of a quark number density oper-ator [17, 18], [24, Chap. 6], expressed in terms of barefields in a gauge-invariant form. The “A” superscriptis to distinguish this track-A bare pdf from a differenttrack-B concept of the same name, as discussed later inSec. III.

In the bare pdf there is a logarithmic UV divergenceassociated with the bilocal operator in Eq. (2). This isdistinct from the UV divergences that are canceled bycounterterms in the QCD Lagrangian, and hence the barepdf is UV divergent.

An MS renormalized pdf is defined in terms of the barepdf by including a renormalization factor ZA,

f renorm,A(ξ) ≡ ZA ⊗ fbare,A , (3)

4

with the product being in the sense of a convolution inξ and a matrix in flavor space. In the MS scheme, ZA isdefined by analogy with renormalization factors in othercases, and in perturbation theory it has the form:

ZAij(ξ) = δ(1− ξ)δij +

∞∑n=1

Cn,ij(ξ, αs)

(Sεε

)n. (4)

Here the Cn,ij are the coefficients necessary to subtractonly powers of Sε/ε, where Sε ≡ (4π)ε/Γ(1 − ε) '(4πe−γE )ε, as in Ref. [24, Eq. (3.18)]. The dimensionof spacetime is n = 4 − 2ε. The significance of the sec-ond formula for Sε is that many authors use the secondformula to define Sε, or an equivalent method. In thecases we are interested in, the difference does not affectphysical quantities [24, Eq. (3.18)].

The definition of the convolution over collinear mo-mentum fraction is

[A⊗B] (ξ) ≡∑j

∫ 1

ξ

dξ′

ξ′Aj(ξ/ξ

′)Bj(ξ′) . (5)

Notice that we carefully distinguish parton momentumfraction (ξ) from process-specific kinematic variables likeBjorken xbj, although we will frequently drop ξ argu-

ments for brevity. Associated with the use of the MSscheme and dimensional regularization is the renormal-ization scale µ that is defined to appear as a factor µε inthe bare coupling in the Lagrangian density of the theory.

It is the renormalized version of the pdf in Eq. (3) thatenters into the derived factorization formulas for physi-cal observables like cross sections. For a DIS structurefunction F , for example,

F (Q, xbj) = CA ⊗ f renorm,A + error . (6)

Here, CA is a perturbatively calculable coefficient func-tion, and the error is suppressed by a power of 1/Q.

A convenient technique for perturbative calculationsof CA arises from recognizing that it is independent ofwhich target is used for the structure function F ; all thetarget dependence is in the parton density f . So we canwork with perturbative calculations of pdfs and struc-ture functions on partonic targets. The results all followfrom definite Feynman rules. Since the coefficients CAare independent of light-parton masses, it is sufficient tosimplify the calculations by setting the mass parametersfor all fields to zero. Then the hard coefficient is effec-tively an inverse of Eq. (6) when a partonic target is used,i.e.,

CA =F partonic(Q, xbj)

fpartonic,renorm,A, (7)

and this gives a perturbative calculation of CA. Heredivision is in the sense of an inverted convolution inte-gral. Although there are collinear divergences in both thestructure function and pdfs with a massless partonic tar-get, they necessarily cancel in the hard scattering coeffi-cient CA. When the CA coefficients are used phenomeno-logically, the renormalization group scale µ is generally

fixed numerically to be proportional to a physical hardscale (e.g., Q), thereby ensuring that CA is perturbativelywell-behaved, i.e., that useful calculations can be madeby expanding it to low orders in the effective couplingαs(Q).

Equation (7) can be regarded as equivalent to a versionof the R∗ operation [25–27] that was devised to subtractboth IR and UV divergences in Feynman graphs. As tothe situation with hadronic targets, the definition of apdf in Eqs. (2) and (3) is complete enough to be used forcalculations from first principles QCD with nonperturba-tive techniques like lattice QCD, at least given sufficientlyadvanced methods, and to some approximation.

In view of later discussions, it is important to observethat because all actual hadrons in QCD have mass, thereare no actual soft or collinear divergences in structurefunctions and pdfs for a hadronic target. There is sensi-tivity to the collinear region in these quantities, but noactual divergence. When collinear divergences do appearin calculations, it is at intermediate stages of a calcula-tion of a hard scattering; they are artifacts of havingperturbatively calculated structure functions and pdfswith massless, on-shell partonic targets before applyingEq. (7).

It is perfectly possible to do the calculations for theright-hand side of (7) with quark masses kept non-zero. Then some of the collinear divergences4 in the all-massless calculation correspond to logarithms of mass/Qin perturbative calculations in the limit of zero mass(es).Given the common situation where all the masses aresmall compared with Q, one would then take the limit ofzero mass to obtain CA, an operation which would needto be inserted on the right-hand side of (7).

However, considerable simplifications in Feynmangraph calculations occur in the massless case, especiallywith dimensional regularization, so it is normal to useonly massless calculations. One of the simplifications isthat perturbative corrections to bare pdfs on partonictargets are zero to all orders of massless perturbationtheory. That is simply because all the integrals are scalefree, and therefore vanish in dimensional regularization;there is an exact numerical cancellation of the quantifiedIR and UV divergences with no remaining finite part.Therefore, a bare track-A pdf for a parton in a masslessparton is just

fpartonic,bare,Ai/j (ξ) = δ(ξ − 1)δij , (massless, dim. reg.)

(8)where i and j label parton flavors. Hence, the renormal-ized pdf on a massless partonic target is exactly equal tothe MS renormalization factor:

fpartonic,renorm,Ai/j = ZAi/j . (massless, dim. reg.) (9)

4 The masslessness of the gluon in perturbation theory in QCDcontinues to provide some collinear divergences.

5

It is important that the conceptual status of the diver-gences in ZA as ε→ 0 has changed dramatically, betweenEq. (3) and Eq. (9). The poles in ZA in Eq. (3) are allUV poles, to cancel UV divergences in the bare pdf. Ona hadronic target, this results in finite renormalized pdfs,of course. But in (9), the numerically identical poles areactually collinear divergences in a UV-finite pdf on a par-tonic target.

Although working with massless partonic pdfs is a use-ful technique for calculating quantities like hard factors,what phenomenology ultimately needs is the set of pdfsfor hadrons. It is Eq. (3) that is relevant for these pdfs.There are situations where non-zero quark masses areneeded in perturbative calculations. An important oneis where one deals with heavy quarks whose masses arecomparable to or bigger than Q. Then the heavy-quarkmasses need to be retained in the calculations, and equa-tions like (8) and (9) are no longer true.

III. TRACK-B: COLLINEAR ABSORPTION

Next we contrast the above with an alternative waythat factorization is often described and used to deriveproperties of pdfs and other parton correlation functions.We will ultimately critique this approach, which we willcall track B.

A. Content of track B

The starting point of track B, is the assertion that astructure function on a hadronic target is the convolutionof the corresponding massless on-shell partonic structurefunction with bare pdfs on the same target:

F (Q, xbj) = F partonic ⊗ fbare,B . (10)

In contrast with the similar-looking factorization formula(6) in track A, the first factor is an unsubtracted partonicstructure function and has collinear divergences, unlikethe corresponding quantity CA in (6). Although the pdffactor fbare,B is called a “bare” pdf, it must in generalbe different from the bare pdf in track A, as we will see,and we have therefore distinguished it by a label “B”.

To deal with the collinear divergences in F partonic, itis then proved [28, 29] that the partonic structure func-tion can be written as a convolution of a finite coefficientfunction CB with a factor ZB containing the collineardivergences,

F partonic = CB ⊗ ZB . (11)

When the collinear divergences are quantified as poles indimensional regularization, ZB can be defined to be ofthe MS form, similarly to the UV renormalization fac-tor in (4). Commonly this is modified by the use of afactorization scale µF which is distinct from the renor-malization scale µ. The exact MS form is obtained when

µF = µ. The pole structure can be modified by ex-tra finite contributions, the choice of which defines thescheme. In all cases, the collinear-divergence factor isindependent of which hard process is considered, e.g.,which DIS structure function is treated, or whether DISor Drell-Yan is treated.

Process-independence of ZB permits the final step,which is the absorption of the collinear divergences intoa redefinition of the pdfs:

F (Q, xbj) =(CB ⊗ ZB

)⊗ fbare, b

= CB ⊗(ZB ⊗ fbare,B

)= CB ⊗ f renorm,B , (12)

where the renormalized pdf is defined to be

f renorm,B = ZB ⊗ fbare,B . (13)

The final line of (12) has the same form and nature asthe factorization formula (6) in track A.

In standard phenomenological applications to scatter-ing processes, the coefficients CB or CA and the corre-sponding quantities for other processes are computed per-turbatively, while the pdfs at some initial scale are ob-tained from fits to data. Scale-evolution of renormalizedpdfs is implemented by the DGLAP equations, with theirperturbatively calculable kernels.

B. Equality of coefficient functions andrenormalized pdfs between tracks A and B

Although, as we will see shortly, there are importantreasons to at least question the starting point (10) oftrack B, nevertheless the structure of the final factoriza-tion formula (last line of (12)) for the standard applica-tions agrees with that of track A and is correct.

In fact, when the MS prescription is used and µF = µ,as is common, the coefficient functions in the two tracksare equal: CA = CB . This is because in both cases, thepartonic structure function and the coefficient functiondiffer by a factor of an MS form, and the poles can beuniquely determined by the requirement that the coeffi-cient function be finite. In track A, this follows from Eq.(7) and the form for the renormalized pdf on a masslesstarget given by Eqs. (9) and (4).

It follows that the collinear -divergence factor ZB intrack B equals the UV-renormalization factor ZA in trackA. The renormalized pdfs also have to agree, since theycan be fit to the same cross sections with hadronic tar-gets, and the coefficient functions are equal.

From ZA = ZB it follows that the bare pdfs in bothschemes are equal. However, this result relies on the useof dimensional regularization, massless partonic calcula-tions, and the consequent vanishing of scale free integrals.It is these properties that led to Eq. (9) for the values ofthe massless partonic pdfs in track A.

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C. Critique

Completely essential to track B is the statement (10)that a structure function on a hadron is the convolu-tion of an unsubtracted partonic structure function andbare pdfs. Let us call this statement “bare factoriza-tion”. However, as far as we can see, bare factorizationis merely asserted and never actually derived. In addi-tion, the bare pdfs are commonly not defined. Especiallyin the early literature, the assertion of bare factorizationappears with a reference to Feynman’s parton model —e.g., see Refs. [28, 30] — perhaps as the natural general-ization of the parton model to QCD.

But when the statement of bare factorization is ex-amined in more detail, it becomes highly implausible.The parton model itself [4] can be motivated by examin-ing relevant space-time scales for DIS in the Breit framewith a hadronic target of high energy. The hadron istime dilated from its rest frame, and therefore the natu-ral scales for internal processes in the hadron are the largeranges of time and longitudinal position that arise fromthe boost to a high energy. The scattering with the vir-tual photon involves much smaller scales, of order 1/Q.To obtain the parton model, it was hypothesized thatthe transverse momenta and virtualities of constituentsin the hadron are limited. Then a factorization formulaarises in which the virtual-photon-quark interaction isrestricted to lowest-order in strong interactions.

However this motivation does not extend to the gen-eralization from the parton model to bare factorizationin QCD. This can be seen from the collinear divergencesin massless partonic cross sections. The divergences in-volve infinitely long times and distances (in the longitu-dinal direction, the same direction as the hadron). Suchscales are much longer than the scales for hadrons, sincehadrons are massive and their actual interactions there-fore do not have collinear divergences. This indicates thatthe collinear divergences in the partonic structure func-tion are a property of intermediate results in a methodof calculation, rather than a property of full QCD.

Another way to see the problems is to examine the na-ture of the divergences in the three quantities in (10).The hadronic structure function on the left-hand side ismeasurable and finite. In particular it has no collinear di-vergences because all true particles in QCD are massive.Possible UV divergences are canceled by renormalizationcounterterms in the Lagrangian.

On the right-hand side, the massless partonic structurefunction also has no UV divergences, for the same rea-son. But it does have perturbative collinear divergencesbecause of the masslessness of the partons.

As to the bare pdf, let us copy Candido, Forte andHekhorn [3] and say that the track-B bare pdf is given bythe standard operator formula, as in their Eq. (2.2), es-sentially the same as our (2) for track A. When a hadronictarget is used, the pdf has no collinear divergences, be-cause of the massiveness of hadrons. But it does have UVdivergences associated with the operator; these are be-

yond those canceled by counterterms in the Lagrangian.

So we have a mismatch of divergences: The right-hand side of (10) has both collinear and UV divergences,whereas the left-hand side has none. The obvious con-clusion is that (10) is wrong, despite the fact that it is sowidely quoted in the literature.

Table I summarizes the divergence properties of thevarious quantities we have been discussing.

However, there is in fact a loophole in the argumentfor the mismatch of divergences. This is that it mighthappen that the two kinds of divergence cancel. Withinthe context of dimensional regularization and masslesson-shell partonic calculations this does happen. There-fore it is useful to examine more carefully the situationsin which bare factorization holds, which we will do inSec. IV.

But, as we will show in Sec. VI an almost immediateconsequence of relying on the cancellation of UV andcollinear divergences is that the positivity arguments inRef. [3] fail.

D. An alternative definition of a bare pdf

A rather different definition of a bare pdf in track Bwas given by Curci, Furmanski, and Petronzio in Ref.[29], in their Eq. (2.46). This quantity is represented di-agrammatically by the bottom-most object in their Fig.3, labeled “qB.H”. Their definition is obtained by modi-fying (2) so that the quark-antiquark Green function in ahadron is restricted to its two-particle-irreducible (2PI)part in light-cone gauge. The two particle irreducibil-ity implies, by standard power-counting arguments, thatthis bare pdf has no UV divergences. The massivenessof hadrons ensures that this kind of bare pdf also has nocollinear divergences.

We now have a real contradiction, since the sole re-maining divergences on the right-hand side of (10) arethe collinear divergences in the partonic structure func-tions, and there is nothing to cancel them to make a finiteleft-hand side.

In Sec. VII, we will explain the rather trivial reasonsthat Curci et al.’s assertion of bare factorization (at thestart of their Sec. 2.7) cannot be true with their defini-tion. Their remaining derivations are rather clear, and itis quite simple to modify their arguments to make a cor-rect derivation. For the result, see Sec. 8.9 of [24], whichitself is based on an earlier paper [31]. The announcedfocus of Ref. [31] was heavy quark effects, but its argu-ment is not so restricted. The derivation is definitely ofthe track-A kind, and the definition of a bare pdf is thatof track A, not that of Ref. [29].

7

Object Hadron structure function, Partonic hard part Bare hadronic pdf Renormalized hadronic pdf

Ultravioletbehavior

Standard Lagrangiancounterterms

UV finite

Lagrangiancounterterms &operator counterterms

UV finite

Bare pdf, so no counterterms

UV divergent


UV finite

Collinearbehavior

Non-massless, finite range theory

Collinear finite

Double countingsubtractions

Collinear finite


Collinear finite


Collinear finite

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Track A

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Object Partonic structure function Partonic hard part Bare hadronic pdf

Ultravioletbehavior

Standard Lagrangiancounterterms

UV finite


UV finite

Track B bare pdf must be UV finite to be consistent with hadronic structure function

UV finite

Collinearbehavior

Massless partons

Collinear divergent

Collinear divergence absorbed into pdf redefinition

Collinear finite

Track B bare pdf must be collinear divergent to cancel collinear divergence in partonic structure function

Collinear divergent


Track B

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TABLE I. A summary of the divergences that appear with hadron targets in track A and track B treatments. (See text formore details.) In the track A table, the only divergences (apart from those involved in renormalizing the QCD Lagrangian) arethe UV divergences in the bare pdf. These get removed by operator counterterms when the pdf is renormalized. The hadronstructure function is the same for track A and track B and so is not repeated in the track B table. But F partonic and fbare,B

appear in Eq. (10), and both contain collinear divergences that must cancel once all factors are combined to reproduce thephysical structure function. (So fbare,B is not actually the bare pdf in Eq. (2).)

IV. RECONSTRUCTING TRACK B PARTONDENSITIES

We have questioned the validity of bare factorization,Eq. (10), which is the starting point of track B. The prob-lematic issue was that an unsubtracted partonic structurefunction is used. In this section, we start from the ob-servation that our argument in Sec. III against the va-lidity of bare factorization appears to be undermined bythe formulation and proof of the OPE that was givenby Wilson and Zimmermann [22, 23]. Their form of theOPE is rather like bare factorization, in that their coeffi-cient function, the analog of CB in Eq. (10), also has nosubtractions for what in this case are low-momentum re-gions (instead of collinear regions). In this it differs fromCB only in that all parton masses are preserved insteadof being set to zero, and so there are no actual collineardivergences. The Wilson-Zimmermann derivation relieson the use of a particular subtraction scheme.

Now the OPE applies in a short-distance asymptote:qµ →∞ at fixed hadron momentum P ; the operators inthe analog of pdfs are local. In contrast, factorizationapplies in the Bjorken asymptote, where Q→∞ at fixed

Q2/P · q. A generalization of the Wilson-Zimmermannmethod should apply. This would suggest that one canin fact derive bare factorization, as used in track B. How-ever, it is important that in the OPE the local operatorsused are UV-renormalized, not bare. Correspondingly,the pdf in bare factorization should be a renormalizedquantity, contrary to assertions in track-B literature.

The purpose of this section is therefore to reverse en-gineer what definition of a pdf is needed in order for barefactorization, Eq. (10), to be correct.

For the following discussion, we will find that we needto modify the notation for bare factorization, Eq. (10),to allow for modified definitions:

F (Q, xbj) =[F partonic

]R1,IRR

⊗[fbare,B

]R2,IRR

. (14)

Here R1 and R2 are the UV renormalization schemes forF partonic and fbare,B respectively, and IRR is an IR reg-ulator scheme. R1 is simply the renormalization in theQCD Lagrangian because there are no other UV diver-gences to deal with in the partonic structure function. Aseparate UV scheme is allowed for fbare,B. In the gen-eral case, some such scheme must be present in fbare,B

because for the nature of the divergences on the left and

8

right of Eq. (10) or (14) to match, the pdf cannot con-tain UV divergences. IRR needs to be defined such thatcollinear divergences cancel between F partonic and fbare,B

in Eq. (14) in order to recover the physical structure func-tion on the left side of the equation. In general, thechoices of IRR and R2 need to be carefully adjusted tomaintain the overall correctness of Eq. (14). In Sec. IV Cwe will show an explicit example of how this works forthe specific case of dimensional regularization and MS.

The next three subsections form a rather technical de-tour, but they are important because the will allow usto state very rigorously how each factor in Eq. (14) mustbe defined for a track B approach to be consistent. Thisin turn will allow us to make a truly apples-to-applescomparison with the corresponding factors defined in thetrack A approach. Once this is done, the origin of anydifferences between the two approaches regarding ques-tions like positivity will be clear and easy to diagnose.

A. The Wilson-Zimmermann treatment of theOPE, generalized to DIS

To understand the relation between the Wilson-Zimmermann approach to the OPE and the track-Btreatment of factorization, it is useful to summarize theWilson-Zimmermann approach as it would apply to a DISstructure function, with the aid of some of the methodsof Curci, Furmanski and Petronzio [29].

The methods of Wilson and Zimmermann can be char-acterized by the observation that there is a close similar-ity between the operations needed to extract the largeQ asymptotics of some Green function and those neededto extract UV divergences and thereby obtain renormal-ization counterterms. Moreover, when zero-momentumsubtraction is used, as they do, the operations are iden-tical except for the characterization of what subgraphsthey are applied to. The proofs, as written, work toall orders of perturbation theory. Zero-momentum sub-tractions can be applied to the integrands of Feynmangraphs. Then no regulator is needed and the scheme islabeled “BPHZ”.

The limit involved for the OPE is the short-distancelimit where q → ∞ at fixed P , or, equivalently, Q → ∞with x ∝ Q. It applies to the uncut amplitude for DIS,which is the expectation value in the target state of thetime-ordered product of two currents. The short-distancelimit entails x → ∞, which is not in the physical regionfor actual physical DIS, but is related to it by a dispersionrelation, giving results for certain integer moments of DISstructure functions. But the structure of the derivationof the OPE itself applies equally to DIS in the physicalregion, and that is what we will present here.

One begins by examining situations where all the mo-menta k inside a subgraph are large while the momental attaching it to the rest of the graph are small. TheUV divergences or the leading power of Q can be quan-tified by expanding to the relevant order in powers of l

relative to k. In the renormalization of UV divergences,using the first term in this expansion to construct coun-terterms amounts to defining the counterterms by zeromomentum subtraction.

For the arguments in their simplest form to work ina gauge-theory, light-cone gauge is used.5 In this gauge,the Wilson line in the operator in the definition of thebare pdf equals unity and can therefore be omitted. Mostimportantly, in the leading power for the large Q asymp-totics of a structure function, the relevant regions of loopmomentum space are as denoted in

l

P

hard, U

collinear, L

q

(15)

At the top, there is a subgraph U (the “hard” subgraph)with large transverse momenta; it has two parton linesat its lower end. The lower part L (the “collinear” sub-graph) has low transverse momentum. Each graph typ-ically has multiple possible regions of this form, and forthe purposes of this discussion we omit details of howintermediate regions are accounted for by a suitable re-cursive subtraction scheme.

Since all the possible hard subgraphs are nested withrespect to each other, the treatment can be simplifiedcompared with the general treatment by Zimmermann.We then have the algebraic structures that were foundin [29] and that we will treat below. In a more generalcase, there can be non-trivial overlaps between differentpossible hard subgraphs, and the full Zimmermann for-est formula, or some equivalent, would be needed. (Thetreatment of UV divergences renormalized by countert-erms in the Lagrangian similarly does not break the al-gebraic structure, and need not be treated explicitly.)

A similar graphical structure applies for the regionsthat give UV divergences in the pdfs:

k

l

P

UV

collinear

(16)

where UV divergences arise when the transverse mo-menta in the upper subgraph go to infinity, and the

5 There are certain problems with the use of light-cone gauge,which we will describe later, but it is sufficient to ignore themfor our present purposes.

9

crosses denote the factor corresponding to the operator in the definition of the pdf. Again, any single graph canhave many different regions of this form.

Therefore to extract the largeQ asymptotics of DIS, we use an expansion in two-particle-irreducible (2PI) subgraphs,as was done by Curci, Furmanski and Petronzio [29]. For the DIS structure functions on a hadron this gives

PF

q

= PB

A

q

+ BP

K

A

q

+ BP

K

A

q

K

+ . . . . +

+ PBγ

q

=∞∑n=0

AKnB +Bγ = A1

1−KB +Bγ . (17)

Here, F denotes the full matrix element of two currents in a state of a target of momentum P . The subgraphs A, K,and B are two-parton irreducible in the vertical channel, with K and B including full parton propagators on theirtop two lines, but excluding the propagators on the lower lines. Finally the quantity Bγ is completely two-partonirreducible in the vertical channel; it turns out to be power suppressed compared with the contributions from the2PI graphs in the top line. Generally, a hadronic target state will entail the use of some kind of bound-state wavefunction. The definition that the lower parton lines of K are amputated can be notated by short lower lines:

K. (18)

In Eq. (17), there is a sum over the number of K rungs from zero to infinity, and the products of the different factors,as in AKnB, are defined to entail integration over the loop momenta connecting the factors and the appropriate sumsover any spin indices. We define each 2PI subgraph to include all the appropriate counterterms from the Lagrangianfor UV renormalization.

For the actual DIS structure functions, a final-state cut should be inserted in the graphical structures in Eq. (17).But essentially all the analysis and factorization apply equally to the corresponding matrix elements in a target stateof a time-ordered product of two currents, as well as to the corresponding structure-function-like objects.

With a hadronic target, the bottom 2PI rung B in Eq. (17) never participates in the hard part. Similarly, in the2PI expansion, (19) below, for a pdf, B never participates in the UV divergence of the pdf.

Similar but simpler expansions in 2PI graphs apply for a bare pdf (in the track-A sense) on the same target:

Pf

k

= PB + P

B

K

+ PB

K

K

+ . . .

= T1

1−KB. (19)

Here the crosses and T correspond to the operator in the defining matrix element (2) of a pdf (where the case of thepdf of a quark is shown). Given that we are working in the light-cone gauge here, the Wilson line is simply unity.But note that because we constructed K and B to be UV finite, the quark fields in (2) must now be renormalizedfields, not bare fields.

10

The explicit definition of T , in the case of an unpolarized quark pdf, is

Tf = Pf

k

=

∫dk− d2−2εkT

(2π)4−2εTr

Dirac, color

γ+

2P

f

k

, (20)

with k+ ξP+.Compared with the expansion of a DIS matrix element, the main change in (19) is that the two currents at the top

are replaced by partonic fields and a suitable integral over the parton momentum k. In addition, there is no special2PI subgraph like Bγ containing both pdf vertex and the target.

Next we write the corresponding expansions when the target is a parton instead of a hadron. Since the target stateis elementary, the 2PI graphs B and Bγ are no longer needed. Instead we just need an external line factor for eachof the two lines for the incoming parton target. Then we have for DIS:

p

F partonic

q

=A

q+

K

A

q

+K

A

q

K

+ . . .

=

∞∑n=0

AKn = A1

1−K , (21)

where each term has the external parton propagators amputated, as in the definitions of A and K. The external-linefactors are the same for all the terms; they play no role in the rest of our treatment, so we have omitted them.

The similar expansion for a pdf is

fpartonic

p

= + K + K

K

+ . . .

=

∞∑n=0

TKn = T1

1−K . (22)

In a gauge theory, like QCD, when the Feynman gaugeis used, the graphical specification of the leading regionsis more complicated than given above [24]. The use oflight-cone gauge gives the simpler results stated above.However it comes with the penalty that the 1/n·k term inthe gluon propagator gives what are now known as rapid-ity divergences. These lead to considerable complicationsin the case of transverse-momentum-dependent pdfs andof the cross sections for which they are used [32]. Butfor the case we consider here, the rapidity divergencescancel, although general proofs, as opposed to examples,are hard to find. Although these problems are very non-trivial, they are essentially orthogonal to the issues wediscuss here, and so we will ignore them.

In the generalization of the Wilson-Zimmermann ar-gument, the first step is to construct for each graph Γits remainder rem(Γ), which is Γ with the subtraction of

both UV divergences and of the behavior at large Q tosome power, which for us is the leading power.

In the original case, the OPE, both the countertermsfor UV divergent subgraphs and the subtractions for thelarge Q behavior of hard subgraphs are constructed byzero momentum subtraction, i.e., of an appropriate poly-nomial in the external momenta of the subgraph in ques-tion. A slightly different expansion is needed in the DIScase, with a generic leading region shown in (15). The ex-pansion for the leading power of Q in the hard subgraphinvolves neglecting the relatively small minus and trans-verse components of the momentum l connecting the twosubgraphs. In contrast, for the OPE, all components ofl would be neglected in the hard subgraph.

For the DIS structure functions, we use a modification

11

of the notation of Ref. [29], and obtain

rem(F ) =

∞∑n=0

A(1− T ) [K(1− T )]nB +Bγ . (23)

Here T is what is in fact a generalization of the objectof the same name defined in (20), that corresponded toa pdf operator. In (23), T is defined to be an operationthat extracts the leading asymptotics when the factoron its left has large transverse momenta relative to thefactor on its right. Given that a product like AK means

AK =

∫dnl

(2π)nAab(Q, l;m)Kab(l, . . . ;m), (24)

ATK is defined to be

ATK =∫dl+Aab(Q, l;m)Tab;cd

dl− dn−1lT(2π)n

Kcd(l, . . . ;m),

(25)

where

l = (l+, 0,0T) (26)

and Tab;cd is a matrix that projects out the terms in thesum over spin indices that are needed for the leadingpower. In the case of unpolarized DIS with quark linesconnecting the two subgraphs, we have

Tab,cd =γ−ab2

γ+cd

2. (27)

The factor γ+/2 here corresponds to the same factor inthe definition of a quark pdf. Similarly, the integralsover l− and lT in (25), correspond to the integrals in thedefinition of a pdf. Thus the result of inserting T is theconvolution product of an approximation to the factor onits left with some kind of pdf vertex applied to the factoron its right. Hence it is useful to overload the semanticsof T : If there is no factor to its left, it denotes simplythe operator for the pdf. If instead there is a factor toits left, an approximation is applied to that factor. Inits uses in factorization, a pdf is always multiplied by acoefficient function that is obtained by an approximantapplied to some graph.

Observe that, as in the Wilson-Zimmermann treat-ment of the OPE, masses are left unchanged in all quan-tities.

In a term like

A(1− T )KB, (28)

there is a UV divergence where transverse momenta inK go to infinity; these correspond to UV divergences ina pdf. But the corresponding term in rem(F ) is

A(1− T )K(1− T )B. (29)

Then the second factor of 1 − T removes not only theleading large Q asymptotics of A(1 − T )K, but also theUV divergence in −ATKB.

Subtracting the remainder Eq. (23) from the originalstructure function in Eq. (17), and performing some al-gebraic manipulations gives

F =

( ∞∑n=0

AKn

){T

∞∑n′=0

[K(1− T )]n′B

}+ rem(F ).

(30)Let us define a renormalized pdf by

f ren., BPHZ = T

∞∑n′=0

[K(1− T )]n′B . (31)

Then, since rem(F ) is suppressed by a power Q, Eq. (30)has the form of a factorization property, with the coeffi-cient function being

CBPHZ(q, x/ξ) =

( ∞∑n=0

AKn

)l→l=(ξP+,0,0T)

. (32)

That is, it is an unsubtracted parton DIS structure func-tion, with the external parton lines amputated, and witha light-like external parton momentum. It is simplyEq. (21). Note that it can be shown that the renormalizedpdf defined above is a renormalization factor convolutedwith a bare pdf, just as in track A. That is, it is a fullyrenormalized version of Eq. (19).

Thus we have a bare factorization just like that in trackB, except that

• The masses of internal lines of the (unsubtracted or“bare”) coefficient function are unchanged insteadof being set to zero.

• The pdf is definitely a UV renormalized quantitywith a particular scheme, not a bare quantity likethat in definition (2).

In this approach, the renormalization scheme for the pdfsis implemented by counterterms that are obtained by thesame operation T as for extraction of large-Q asymp-totics. It is in fact the same as the BPHZ scheme usedby Wilson and Zimmermann, except for being extendedfrom the pure zero-momentum renormalization schemefor local operators to a version suitable for the renormal-ization of the bilocal operators in pdfs. The subtractionsare at zero values of only the minus and transverse com-ponents of momentum.

In a renormalizable non-gauge theory with non-vanishing masses, the above procedure works as is, butin a gauge theory modifications of the counterterms areliable to be needed (a) to preserve gauge-invariance, and(b) to avoid IR and collinear divergences associated witha massless gluon when external momenta are zero orlight-like.

If one wanted to use MS renormalization for the pdfs,then the above treatment needs to be modified so as to

12

decouple the subtraction operation for UV divergencesfrom that for extraction of large Q asymptotics. Thiswas done in [24, 31], and leads directly to the track-Aformulation with its subtracted coefficient function.

B. Relation of track B to the Wilson-Zimmermanntreatment of OPE

It would be natural to expect that the coefficient func-tion in the OPE or factorization is a short-distance quan-tity, so that in QCD asymptotic freedom implies thatuseful perturbative calculations can be made, since theeffective coupling αs(Q) is small.

However, in the Wilson-Zimmermann form of the OPEthe construction of the coefficient function does not in-clude subtractions for the collinear region, and so it doesnot obey the purely short-distance property. Thus it is anon-perturbative quantity in QCD.

As regards the validity of the OPE itself, Wilson andZimmermann point out that it is sufficient that all de-pendence on Q is in the coefficient function.6 Thus theshort-distance part is correctly contained solely in thecoefficient function.

But for the OPE to be valid a suitable definition of theoperators must be made. An important finding of Wilsonand Zimmermann was that zero momentum subtractions(with unchanged masses) accomplish this, in the BPHZscheme. In effect, the OPE can be regarded as givinga definition of the composite local operators used in theOPE.

Bare factorization in track B also has an unsubtractedcoefficient function, but with masses set to zero. Soit is natural to expect that a variation on the Wilson-Zimmermann approach would lead to bare factorizationtogether with a suitable definition of the pdfs. In thenext section, we will implement this idea. It will re-quire a change of scheme to what we will call the BPHZ′

scheme.It is far from obvious that the initial papers for track

B intended to use the Wilson-Zimmermann method. Forexample, Politzer in Ref. [30] does not mention it. Hismotivation seems to be entirely different, arising from anattempt to generalize the parton model.

Much of the early work that applied the OPE inQCD appears not to use the actual Wilson-Zimmermannmethod with its non-perturbative coefficient functions,even when the Wilson-Zimmermann papers are referredto. Instead the composite operators were often definedby MS renormalization for UV divergences. This is ex-actly like track-A factorization. The corresponding coef-ficient function is then perturbative. The evolution equa-tions are then standard renormalization-group equations,

6 As they point out, essentially the same observation for a similarpurpose had been made much earlier by Valatin [33–36].

rather than the Callan-Symanzik equations that apply tothe coefficient functions in the Wilson-Zimmermann ap-proach, where the pdfs (or their analogs in the OPE) donot evolve at all.

C. The BPHZ′ scheme

With inspiration from the Wilson-Zimmermann pa-pers, we will now show how to define the track B “bare”pdfs in a way that ensures that (a) the track B equationsare correct, (b) the pdfs have a definite relationship tostandard operator matrix elements such as given in (2),and (c) the definition applies independently of the choiceof dimensional regularization for both UV and IR diver-gences.

Given how Wilson and Zimmermann derive the OPE,as summarized in Sec. IV A, this amounts to recog-nizing that the track-B “bare” pdfs are actually UV-renormalized, and to determining which renormalizationscheme is needed. Recall that, in general, from theUV finiteness of structure functions, both partonic andhadronic, it follows that fbare,B is also UV finite in orderfor Eq. (10) to be true.

Renormalization counterterms for pdfs are used forsubgraphs of the form of the hard subgraphs specified in(16). So we can infer the UV renormalization scheme forthe pdfs, by examining DIS on an on-shell partonic tar-get, with all parton masses set to zero. Then F (Q, xbj)in Eq. (10) is the same as F partonic. Hence the partondensities for massless partonic targets must be exactlythe lowest order, or free-field values:

fmassless partonic,bare,Bi/j (ξ) = δ(ξ − 1)δij . (33)

The unique renormalization scheme that achieves this istherefore the one where renormalization counterterms ex-actly remove all perturbative contributions to the mass-less partonic pdf. We call it “BPHZ′”.

To see the nature of this scheme, it is useful to ex-amine the structure of one-loop calculations of pdfs ona partonic target, but with non-zero masses, and withthe external parton permitted to be off-shell7. As seenin many examples, e.g., in Sec. VIII, the basic form canbe written as an integral over transverse momentum:

Ibare(m2, p2, x) =

∫d2kT

(2π)2

1

k2T + C(x)

, (34)

multiplied by an overall factor that depends on x butnot on kT. This integral will need a regulator to cutoffits UV divergence. The quantity C(x) summarizes thedependence on parton mass and external virtuality:

C(x) = m2 ×A(x) + p2 ×B(x), (35)

7 I.e., we are really treating a Green function with the pdf operatorand two parton fields; a pdf-like Green function.

13

with the coefficients A and B depending on x but notkT. We could, of course, use dimensional regularizationfor the UV divergence, but that would obscure certainconceptual issues. Instead for a one-loop integral we cansimply use an upper cutoff Λ, so that

Ibare(m2, p2, x) =

∫ Λ2

0

dk2T

4π

1

k2T + C(x)

. (36)

With purely a zero-momentum subtraction, i.e., theBPHZ scheme, the renormalized value is

Iren. BPHZ =

∫ ∞0

dk2T

4π

(1

k2T + C(x)

− 1

k2T +m2A(x)

),

(37)in which the counterterm is applied in the integrand, sothat the UV regulator can be removed to give a finiteresult.

But for the BPHZ′ scheme we need for track B, thesubtraction is of the value of the integrand when both p2

and m2 are zero:

Iren. BPHZ′ =

∫d2−2εkT

(2π)2−2ε

(1

k2T + C(x)

− 1

k2T

). (38)

Although the integral is UV convergent, it has a collineardivergence at kT = 0. As such, the BPHZ′ scheme is notreally completely defined until an IR regulator schemeis chosen. (This is in contrast to the standard BPHZscheme.) We have chosen dimensional regularization asthe IR regulator.

The UV divergence is from the asymptote 1/k2T of the

integrand, which can be characterized by saying that it isobtained by setting to zero both of the quantities m2 andp2 that are negligible with respect to k2

T when it goes toinfinity. Hence BPHZ′ is actually a very natural scheme,in a sense more so than BPHZ. It is a kind of minimalsubtraction. By its motivation, this scheme is exactly anduniquely what is needed to give a renormalized value ofzero when the parton mass is zero and the external partonis on-shell.

The penalty for this subtraction is the introductionof a collinear divergence that was not at all present inthe original integral, but that is present when we restorethe parton mass and/or the external parton is off shell.Recall the “IRR” subscript in Eq. (14). The off-shelland massive case applies to a graph that appears as asubgraph in the pdf on a hadronic target.

A simple lowest-order example with the application ofrenormalization is

PB

l

k

−

BP

l

k

.

(39)The sole UV divergence is in the upper loop of the left-hand graph. The box denotes the operation that replaces

the integrand of the upper loop by a quantity like the1/k2

T term in Eq. (38), whose negative gives the coun-terterm for the subdivergence.

Since the renormalization counterterm for the upperloop has a collinear divergence, the full hadronic pdfalso has this collinear divergence, even though there isno collinear divergence in the bare pdf. Here “bare” isin the track-A sense that the graphs for the pdf are ob-tained purely from graphs for the quantity defined in Eq.(2) without extra counterterms or a renormalization fac-tor.

However, it could be argued that the counterterm iszero, because of the vanishing of the dimensionally regu-lated integral over the counterterm’s integrand:

−∫

d2−2εkT

(2π)2−2ε

1

k2T

= 0. (40)

However, this is quite misleading. Suppose we used acutoff Λ to regulate the UV divergence, and then useddimensional regularization with ε < 0 only to regulatethe collinear divergence. Then the integral is not onlynonzero, but is power-law divergent as Λ→∞:

−∫kT<Λ

d2−2εkT

(2π)2−2ε

1

k2T

= −∫ Λ2

0

dk2T(k2

T)−ε

Γ(1− ε) (4π)1−ε1

k2T

=Λ−2ε

εΓ(1− ε) (4π)1−ε . (41)

Of course, this UV divergence cancels the correspondingUV divergence in the integral over the first term in Eq.(38). As we will discuss in Sec. V, the construction ofa dimensionally regulated integral with a UV divergencehas effectively and implicitly introduced a countertermlocalized at kT =∞ to give the result in Eq. (40).

Given that ε < 0 to regulate the collinear divergence,the collinear divergence in the counterterm is actuallynegative. Therefore the supposed positivity of the “bare”track-B pdfs is actually violated.

We summarize our results in this section, together withimmediate implications;

1. The so-called bare pdf fbare,Bj/H in track-B is actually

a pdf renormalized to remove its UV divergence,

but in the BPHZ′ scheme: fbare,Bj/H = f ren, BPHZ′

j/H

2. The renormalization counterterms entail collineardivergences in all pdfs on a hadronic target.

3. This choice of scheme and the use of dimensionalregularization for collinear divergences amounts toa particular choice of the schemes labeled R2 andIRR in Eq. (14),

4. In the bare factorization formula (10), thesecollinear divergences cancel against collinear diver-gences in the massless partonic structure function,so that there are no divergences in the hadronicstructure function on the left-hand side.

14

V. DIMENSIONAL REGULARIZATION ANDPOSITIVITY

Under conditions such as superrenormalizable theories,where it is possible to construct pdfs as literal (light-front) number densities, the normal properties of posi-tivity follow automatically from positivity of the metricof quantum-mechanical state space. But this is a prop-erty that is not necessarily true when there is a regu-lator. The Pauli-Villars regulator for UV divergences isthe classic case. Here, we will explain how dimensionalregularization, when simultaneously applied to UV andIR divergences, violates positivity.

A. Dimensional regularization

In Wilson’s original argument [37] for defining integra-tion in n = 4− 2ε dimensions for arbitrary continuous ε,integrals are uniquely determined (aside from normaliza-tion) by (a) linearity in the integrand, (b) scaling behav-ior, (c) invariance under translations. In addition, appli-cations require an extension to the definition: (d) Ana-lytic continuation in n is applied to extend the range ofn from where an integral is convergent by normal math-ematical criteria. (It is not even necessary to requireagreement with ordinary integrals in integer dimensionwhen those are convergent; that follows from the postu-lates and a choice of normalization.)

One can then give a construction of the dimensionallyregulated integrals we need for Feynman graphs in termsof ordinary integration and analytic continuation in n.It is unique given the natural choice of normalization,which is that the integral of a Gaussian in a Euclideanspace obeys∫

dnx e−∑

j x2j =

(∫dy e−y

2

)n. (42)

However, dimensional regularization does not preserveall properties of standard integration. For example, ingeneral it is not allowed to exchange the order of a limitand integration, e.g., for the massless limit of the inte-gral for a massive Feynman graph. Most importantly forus, standard integrals obey positivity, of which a triv-ial example is that the integral of a positive function ispositive: That is, if f(x) is strictly positive, then so isI[f ] =

∫dnx f(x). If the integrand is merely required to

be non-negative, the integral is also non-negative; more-over, it is zero if and only the integrand is zero every-where.

In dimensional regularization, those properties do ap-ply if the integral is convergent by the standard mathe-matical criteria. Otherwise, it is often violated whenevercontinuation in n is used in the construction of the inte-gral.

The vanishing of scale free integrals violates positiv-ity very much. Thus, in dimensional regularization, the

Euclidean integral ∫dnk

1

k2(43)

is zero but has an integrand that is positive everywhere.The integrand is rotationally invariant, so that vanishingof the n-dimensional integral is equivalent to vanishingof the following one-dimensional integral:∫ ∞

0

dk kn−1 1

k2= 0. (44)

With the standard mathematical definition this integralis unambiguously positive infinite. Technically, we couldsay that in dimensional regularization the integrationmeasure is not positive everywhere, unlike standard in-tegration. Whenever the degree of UV divergence of theintegral is non-negative, i.e., n ≥ 2, there is a negativecontribution which we can treat as localized at k = ∞.Similarly, whenever the degree of IR divergence is non-negative, i.e., n ≤ 2, there is a negative contributionlocalized at k = 0. These two ranges of n overlap atn = 2, thereby preventing the integral from existing atany n, with the ordinary mathematical definition.8

By themselves, the above statements about non-positivity of the integration measure can prevent a naiveapplication of the standard positivity argument for pdfswhenever we use dimensional regularization for both UVand IR/collinear divergences. The positivity argumentfor pdfs involves sums and integrals over final states ofthe absolute square of matrix elements.

An interesting mathematical example is the integral∫d2−2εkT

(k2T −Q2)2

k2T(k2

T +Q2)2. (45)

The integrand is positive definite everywhere, but theintegral evaluated in dimensional regularization is −4πin the limit that ε→ 0. For general ε, the integral equals−4πΓ(1 + ε)(πQ2)−ε, which is negative for all ε > −1.

None of above is to deny that dimensional regulariza-tion is an extremely useful and elegant method for doingmany calculations. But one has to be careful in goingbeyond those properties and manipulations that followfrom its definition and construction.

B. Application to pdfs

In light of these results for dimensional regularization,we examine the basic argument that pdfs are positive.

8 The proof that a scale-free integral is defined (and zero) relies ondefining the integral as a sum of a term with no UV divergenceand one with no IR divergence. Each term is defined by ordinaryintegration for values of n where it is convergent by the ordinarycriterion, and is then analytically continued to all n except forn = 2. In the sum, the poles at n = 2 cancel, so the sum is alsodefined at n = 2, by analytic continuation.

15

Now the operator definition of a bare pdf is equivalentto the expectation value of a light-front number opera-tor integrated over transverse momentum [17, 18], [24,Chap. 6]. Specifically, the pdf is an integral over trans-verse momentum of the following expectation value:9

f(x, kT ) = lim|ψ〉→state of definite P

〈ψ|a†kak|ψ〉〈ψ|ψ〉 . (46)

A minor complication is that the expectation value of anoperator in a state requires the state to be normalizable,and so the state cannot be a momentum eigenstate. Sowe define the pdf in terms of a limit as the target’s statebecomes a momentum eigenstate. Elementary manipu-lations [24, Chap. 6] convert that to a standard form likeEq. (2).

Now the numerator in (46) is given by a sum/integralover intermediate states:

〈ψ|a†kak|ψ〉 =∑X

〈ψ|a†k|X〉〈X|ak|ψ〉

=∑X

|〈X|ak|ψ〉|2 . (47)

This is non-negative, provided that the sums and inte-grals have their standard meanings. Hence the TMD pdfis also non-negative.

The definition of a collinear pdf has an insertion of anintegral over kT , and the result is similarly non-negative:

fbare(x) =

lim|ψ〉→state of definite P

∫d2kT

∑X

|〈X|ak|ψ〉|2〈ψ|ψ〉 ≥ 0. (48)

The UV divergence arises from kT → ∞, and hencewhere also the final state X has large transverse momen-tum. When we consider a pdf in the massless partoniccase, divergences similarly arise when kT → 0, with cor-responding final states. The divergences are not in theintegrand itself, |〈X|ak|ψ〉|2 /〈ψ|ψ〉, but in the integralover certain limits of it.

Then when we perform the integral and use dimen-sional regularization to construct a bare pdf in a masslesspartonic target, our analysis of Eqs. (43) and (44) showsthat the integration measure acquires negative terms inany limit of the integration variables that would other-wise give a divergence. For a massless integral for a pdf,divergences are present for all n, and hence the dimen-sional regulated integral also violates positivity for all n.

Now positivity of the integrand in Eq. (48) results frompositivity of the metric on a normal quantum-theoreticstate space. So we could interpret a violation of positivityof the dimensionally regulated integral as correspondingto a non-negative metric in some kind of extended statespace.

9 For the purposes of this section, we ignore the added com-plications in giving a fully correct definition of a transverse-momentum-dependent pdf in QCD.

VI. FAILURE OF AN ARGUMENT FORPOSITIVITY OF MS PDFS

We now show how Ref. [3]’s argument for positivity ofMS pdfs breaks down.

A number of critical steps are in the first part of theirSec. 2. It begins with a statement of track B bare factor-ization (to use our terminology) in Eq. (2.1). It has thesame form as our Eq. (10), except for changed notationand normalizations.

One factor is an unsubtracted structure function(F partonic in our notation) for DIS on an on-shell par-tonic target with all masses set to zero. The other factoris of a pdf, whose operator definition was given in Eq.(2.2) of [3]. That definition agrees with the definitionthat we gave in Eq. (2) for a bare pdf fbare,A in track A.(The fields and coupling in [3] must be bare quantities inorder that (a) the pdf is a number density in the light-front sense, and (b) the operator is gauge-invariant.)

In our formula for bare factorization, the bare par-ton density is notated fbare,B rather than fbare,A. HenceEqs. (2.1), (2.2) and (2.6) of [3] are equivalent to an as-sertion of our Eq. (10) together with an assertion thatfbare,B = fbare,A. We have shown in previous sectionsthat these assertions are valid provided that dimensionalregularization is used for both UV and collinear/IR diver-gences, but not in general. In Ref. [3], only dimensionalregularization is used.

The derivation of positivity of MS pdfs relies on pos-itivity both of bare pdfs and of partonic cross sections.The derivation is indirect, with the definition and use ofsubtraction schemes named DPOS and POS, followed bya scheme change to MS. Primarily the argument is givenin terms of the results of one-loop calculations.

The intermediate subtraction schemes were motivatedby the fact that the standard MS subtraction of acollinear divergence from a partonic structure functionis actually an oversubtraction. It results in negative sub-tracted partonic structure functions. The definitions ofthe DPOS and POS scheme remove the oversubtraction,so that the subtracted partonic structure functions re-main positive. This was needed because one part of theargument—in the early part of Sec. 3 of [3]—requiredpositivity of all four quantities in the first line of our Eq.(12), including the subtracted partonic structure func-tion CB . But to show that the change of scheme to MSpreserves positivity of the pdf did not need any propertiesof CB .

We can gain an overall view of the derivation fromthe statement [3, p. 5]: “If all contributions which arefactored away from the partonic cross section and into thePDF remain positive, then the latter also stays positive.”In our notation, what is factored away from the partoniccross section is ZB in Eq. (11). We absorbed it into aredefinition of the pdf in Eq. (12).

Now dimensional regularization with space-time di-mension n = 4 − 2ε was used, so we apply the resultsof our discussion in Sec. V A.

16

To regulate collinear divergences, a space-time dimen-sion above 4 is needed, i.e., ε < 0. So, when ε → 0 frombelow, the collinear divergences are positive, Hence thecollinear-divergence factor ZB in Eqs. (11) and (13) onlyobeys positivity in space-time dimensions above 4. Pos-itivity of MS pdfs would then follow were the bare pdfsalso positive for negative ε, i.e., for space-time dimensionsabove 4.

Positivity of bare pdfs appears to be almost trivial toprove—e.g., Sec. V B—and as such it seems that it shouldbe an uncontroversial statement. However, we also sawthat the argument only applies if the integrals giving thepdfs are convergent; failures can occur when dimensionalregularization is used for both UV and IR/collinear diver-gences. Now a pdf defined by Eq. (2) has UV divergences,As we have seen in Sec. V that implies that the contri-bution from the UV region is necessarily positive only ifthe degree of UV divergence is negative, i.e., in space-time dimensions below 4, i.e., ε > 0. But that is wherethe collinear divergence factor does not obey positivity.

If we go in the opposite direction in dimension, i.e.,ε < 0, to make the collinear contribution positive, thenthe UV contribution obtained by analytic continuationfrom positive ε does not obey positivity.

Hence there is no value of ε for which positivity isobeyed by both the factors in the track-B formula forrenormalized pdfs, Eq. (13). A proof of positivity ofrenormalized pdfs from positivity of the collinear diver-gence factor ZB and the bare pdfs fails.

Alternatively one could use methods of regulation orcutoff other than dimensional regularization, at the priceof losing the simplicity that goes with its use. We haveseen that bare factorization then generally fails if the pdfis still the bare pdf defined by its standard formula.

The only way of recovering the validity of the formulafor bare factorization is to replace the bare pdf by arenormalized pdf with the UV divergences removed bythe BPHZ′ scheme. As we have seen these pdfs acquirecollinear divergences not present in the bare pdf itself –recall Eq. (41). The subtractions violate positivity of theresulting pdfs.

To summarize, there are three assertions that need tobe true simultaneously for the derivation of strict posi-tivity in Ref. [3] to be valid

1. Bare factorization Eq. (10) is valid.

2. Each bare pdf, as given by the standard operatordefinition (2), obeys positivity.

3. Partonic cross sections F partonic obey positivity.

As we have shown, at least one of items 1–3 must befalse. As a result, negative pdfs are not excluded in theMS scheme.

VII. CURCI ET AL.

The treatment in Sec. IV A now enables us to critiquethe derivation by Curci et al. [29]. They combine a formof bare factorization and a version of the derivation inSec. IV A, but applied to a massless partonic structurefunction. They use light-cone gauge just as we did in Sec.IV A.

Their definition of a bare pdf is modified from the onegiven in Eq. (2). The matrix element is restricted to 2PIgraphs:

fbare, CFP = PB = TB. (49)

It has no UV divergences, and so is a purely collinear ob-ject. Moreover, the factor B is in full QCD, with massivehadrons, so there are also no collinear divergences in thispdf. Its definition is clearly different from the standardone that is given by Eq. (19), which transcribes Eq. (2)in light-cone gauge.

In their Fig. 3, they assert (but do not prove) a barefactorization formula, which is exactly the same as ourEq. (10), except that the bare pdf is not given by Eq.(19) but by Eq. (49). In the notation of Sec. IV A, thisfactorization is

F =

(A

1

1−K

)m=0

T B + power-suppressed

= Fmassless, partonic ⊗ fbare, CFP + power-suppressed.(50)

This equation cannot be correct: When a hadronic tar-get is used, both the bare pdf used here and the hadronicstructure function have no divergences, but the unsub-tracted massless partonic structure function does havecollinear divergences.

Despite this problem, it is interesting that, as we sawin Sec. IV A, their methods can be used very easily toprovide a correct derivation, either in the BPHZ version,the MS version (track-A) or even the BPHZ′ version.

VIII. EXAMPLES

So far, the discussion has been general but abstract.Concrete examples illustrate the issues very clearly.

The most direct way to test whether pdfs obey proper-ties like positivity would be to simply calculate a suitablesample of them directly from Eq. (3) from first principlesin QCD. But this requires a calculation of their non-perturbative behavior at a level which is beyond currentabilities.

However, neither the derivation of factorization nor thederivation of positivity of MS pdfs in Ref. [3] is specificto QCD. Instead they apply generally to all theories withthe standard desirable properties like renormalizability,etc. Therefore, it is convenient to stress-test proposed

17

general features by examining them in a theory where itis straightforward to perform appropriate reliable calcu-lations, i.e., a model QFT in a parameter region wherelow-order perturbative calculations are accurate for pdfsand structure functions.

We will do this in a Yukawa theory, with all particlesmassive, and with weak coupling. Of course, even thoughfactorization is still valid, its utility is much less than inQCD, which is asymptotically free and where we alwayshave substantial non-perturbative contributions to pdfsand structure functions.

Perturbative results in this theory provide a counter-example to any general theorem that MS pdfs are alwayspositive. Examining the details of the calculation willalso indicate that there is the limited range of low scalesover which positivity can be violated. A primary impacton QCD is then that it is incorrect to impose a prioripositivity constraints on MS pdfs at a low initial scalewhen making phenomenological fits to data. Equally, itis incorrect to impose positivity on fits to the results ofnon-perturbative calculations at low scales. One caution,though, is that the MS scheme is defined within pertur-bation theory, so that it is not at all clear how to ensureit is sufficiently well defined at low scales that are tooclose to where QCD is clearly non-perturbative.

In addition, a careful examination in the model theoryof both how the results for pdfs arise and for where fac-torization is valid will suggest some conclusions for QCDitself.

A. Calculation of pdf

We will use a scalar Yukawa field theory with two sep-arate fermion fields and the following interaction term:

Lint = −λΨN ψq φ + H.C. . (51)

Here, ΨN is a field that we will refer to as correspondingto a “nucleon” or a “proton”, of spin-1/2 and mass mp;we will use the particle as a target in our calculations. Inaddition, there is a spin-1/2 “quark” field ψq with massmq, and a zero charge scalar “diquark” field φ with amass ms.

10 We will use the notation aλ(µ) ≡ λ2/(16π2)in analogy with a similar notation, as = g2

s/(16π2) fromperturbative QCD. Keeping all masses nonzero ensuresthat the theory is finite range in coordinate space, likefull QCD but not massless perturbative QCD. We maychoose the coupling λ small enough that low order graphsin perturbation theory approximate DIS structure func-tions across a wide range of scales to sufficient accuracyfor any given xbj < 1, with controllable sizes of error.

10 The sole purpose of these names is to indicate how we will usethe model theory to construct analogs of what in QCD are thestandard pdfs on hadronic targets.

p

k

FIG. 1. Lowest order contribution to f renorm,Aq/p (ξ;µ).

The bare fermion pdf is Eq. (2), but without the Wil-son line, i.e.

fbare,Ai/p (ξ) =

∫dw−

2πe−iξp

+w−

〈p| ψ0,i(0, w−,0T)

γ+

2ψ0,i(0, 0,0T) |p〉 , (52)

where the i label indicates either the ψq or the ΨN field.The renormalized collinear parton density has the form

f renorm,Ai/p (ξ;µ) = ZAi/i′ ⊗ fbare,A

0,i′/p

≡∑i′

∫dz

zZA(z, aλ)i/i′ f

bare,A0,i′/p (ξ/z) .

(53)

We will work with the quark-in-proton pdf, for whichthe lowest order value is at order aλ, from the graph inFig. 1. A direct computation gives

f renorm,Aq/p (ξ;µ)

= aλ(µ)(1− ξ)(

(mq + ξmp)2

∆(ξ)+ ln

[µ2

∆(ξ)

]− 1

)+O

(a2λ

), (54)

where

∆(ξ) ≡ ξm2s + (1− ξ)m2

q − ξ(1− ξ)m2p . (55)

(Note that ∆(ξ) is positive if the target state is stable,i.e., mp < mq+ms.) The MS counterterm used to obtainEq. (54) is

MS C.T. = −aλ(µ)(1− ξ)Sεε. (56)

This gives ZAq/p = −aλ(µ)(1 − ξ)Sε/ε + O(a2λ), thereby

matching the general form of Eq. (3).By choosing aλ(µ0) small enough at some reference

scale µ0, we ensure that the one-loop renormalized pdfin Eq. (54) is a good approximation to the exact pdf tosome given accuracy over a range of µ. Since the ef-fective coupling does not increase out of the perturbativerange at small scales, unlike QCD, the calculation retainsits accuracy when µ is of order particle masses. It only

18

0.2 0.4 0.6 0.8 ξ

−0.5

0.0

0.5

1.0

1aλ(µ)

f renorm,aq/p (ξ, µ) a)

µ = 2.5 GeV

µ = 1.2 GeV

µ = 0.8 GeV

0.2 0.4 0.6 0.8 ξ−0.4

−0.2

0.0

0.2

0.4

µ2 = Q2(1−xbj)

4xbj

Q = 2.5 GeV

b)

xbj = 0.4 µ = 1.5 GeV

xbj = 0.5 µ = 1.2 GeV

xbj = 0.6 µ = 1.0 GeV

xbj = 0.7 µ = 0.8 GeV

xbj = 0.8 µ = 0.6 GeV

FIG. 2. (a) An example of the MS quark-in-proton pdf in Eq. (54) for several values of µ, with mq = 0.3 GeV, mp = 1.0 GeV,and ms = 1.5 GeV. (b) Similar curves but in a form applicable to use in factorization for DIS with µ2 = s/4, given Q = 2.5 GeVand several values for xbj. The pdfs are only used in the range xbj ≤ ξ ≤ 1, so the curves are restricted to this region.

loses accuracy when µ is so large11 that the logarithms ofµ/mass in higher orders of perturbation theory compen-sate the smallness of the coupling, and use of DGLAPevolution becomes necessary; that is not a concern here.

So that the results of calculations give suggestions asto what happens in QCD, we choose mass parametersto be in a range reminiscent of masses in QCD: mq =0.3 GeV, mp = 1.0 GeV and ms = 1.5 GeV. Thus thequark mass is similar to the “constituent mass” [38] ofa light quark in QCD, and similarly the hadron massis similar to a nucleon mass. But we choose the diquarkmass to be somewhat larger than might be expected werewe to treat the calculation as an actual model for a pdfin non-perturbative QCD; this diquark mass allows us toillustrate that more than one mass scale could be relevantin a ξ-dependent way.

From Eq. (54), it immediately follows that for anygiven value of ξ, the pdf is negative for low enough µand positive for large µ. This is illustrated in Fig, 2(a)which shows the ξ-dependence of the quark pdf for threedifferent values of µ. The values are chosen to be repre-sentative of the low end of the range of µ used in QCDfits. At fairly low values of µ, there is a range of moder-ately large ξ where the pdf is negative. As µ increases,the range of negativity shrinks and eventually disappears.Later we will interpret these results in terms of scales inthe shape of the transverse momentum distribution.

One might worry that the strong negativity might beincompatible with the momentum and flavor sum rules,which entail that some of the pdfs are sufficiently posi-tive. This issue is resolved by observing that the nucleon

11 The calculation also loses accuracy when µ is very small. Butthat is irrelevant to the uses of pdfs, which are in factorizationfor hard processes where µ is chosen to be proportional to a largescale Q.

in our Yukawa model is a possible parton, and that a firstapproximation to the corresponding pdf fp/p (diagonalin parton/particle labels) is the free-field value δ(ξ − 1),which is positive, and does not involve any UV renor-malization. Thus the quark-in-nucleon pdf that we havecalculated is a minority contribution, i.e., much smallerthan the other pdf at large ξ.

B. Systematics of why the pdf becomes negative

To understand how and where the MS pdf becomesnegative, we relate it to an integral over transverse mo-mentum of the corresponding transverse momentum de-pendent (TMD) pdf. Calculating Eq. (54) involves cal-culating the following integral in dimensional regulariza-tion:

aλ(µ)

∫ ∞0

dk2T k−2εT

(1− ξ)[k2

T + (mq + ξmp)2][

k2T + ∆(ξ)

]2 . (57)

Suppose that instead of using dimensional regularizationand subtracting the MS pole to define a scale-dependentpdf, we simply applied a cutoff on transverse momentumin the unregulated integral:12

aλ(µ)

∫ k2cut,T

0

dk2T

(1− ξ)[k2

T + (mq + ξmp)2][

k2T + ∆(ξ)

]2 . (58)

The new pdf is an ordinary integral with a positive inte-grand everywhere when 0 < ξ < 1, with no further sub-traction term. So, this pdf is guaranteed to be positive.It can be verified that when k2

cut,T is set to equal µ2, the

12 Such a definition is used by Brodsky and collaborators [39, 40].

19

cut-off integral matches the MS result in Eq. (54) exceptfor corrections by a power of m2/µ2, which are small athigh scales. Its evolution is of the DGLAP form, but witha power-suppressed inhomogeneous term. In a gauge the-ory, there are problems with a naive use of a cutoff intransverse momentum, since the most natural definitionof a TMD pdf suffers from rapidity divergences [32] as-sociated with light-like Wilson lines. These necessitatechanges in any method of working with TMD pdfs13 inQCD. But the same problem does not occur in a non-gauge theory.

Equation (58) is a very intuitive way to represent ascale-dependent pdf in terms of hadron structure: It con-tains the contributions from transverse momenta up tothe cutoff. In particular, when k2

cut,T is large, it includes,among other things, all the physics associated with in-trinsic hadron structure. Then the coefficient function infactorization for DIS at high Q can simply be character-ized as whatever else contributes to the correct structurefunction. When k2

cut,T is of order Q2, the coefficient func-tion is only concerned with physics at the scale Q.

Up to terms of order m2/k2cut,T, the pdf in Eq. (58) is

related to the MS pdf at scale µ by subtracting the term

aλ(µ)(1− ξ)∫ k2cut,T

µ2

dk2T

k2T

, (59)

as can be verified by explicit calculation. The integrandis the large kT asymptote of the integrand in Eq. (58),as is appropriate to the minimal subtraction used in theMS scheme.

By taking k2cut,T to infinity, we find that the MS pdf

equals

aλ(µ)(1− ξ)∫ ∞

0

dk2T{

k2T + (mq + ξmp)

2[k2

T + ∆(ξ)]2 − θ(kT − µ)

k2T

}, (60)

i.e., there is a subtraction of the asymptote of the inte-grand, with a lower cutoff.

At high kT, the subtraction term closely matches thefirst term. When µ is large, there is a logarithmic contri-bution from kT in the range ∆ <∼ kT < µ, since the sub-traction term vanishes there. The positive logarithmiccontribution overwhelms any negativity in the remainingnon-logarithmic contributions to the integral.

Now when kT goes to zero the 1/k2T term goes to in-

finity, unlike the first term in (60), giving a strong mis-match. Hence when µ is small, there is a large nega-tive logarithmic contribution to the subtracted integral.Therefore, where the MS pdf becomes negative is gov-erned by the mass scale(s) below which the mismatch isstrong.

13 Note that the rapidity divergences cancel in MS pdfs [21].

When ξ → 0, the first term in the integrand becomes1/(k2

T +m2q). Then the pdf becomes negative when µ <

mq, which is a low scale, well under a GeV.When ξ → 1, the term is

k2T + (mq +mp)

2

(k2T +m2

s)2

. (61)

Where the integral becomes negative is now governed bya combination of the scales mq +mp and ms, which arearound a GeV or larger.

The existence of these two scales differing by a modestfactor qualitatively explains Fig. 2. For µ in the mid-dle of the range between the quark mass and about aGeV, the pdf is positive at low ξ but not at high ξ. Onlywhen µ is well above a GeV does the pdf become posi-tive everywhere. The effect is enhanced by our choice of asomewhat large value of ms, but it illustrates the effectsof different scales of transverse momentum in differentranges of ξ in a way that can easily happen in QCD. In-deed in recent fits, by the NNPDF group [41], do produceresults for sea-quark distributions at µ = 1.65 GeV sug-gestive of this, when a fit is not constrained by positivity.

C. Relation to factorization

As to the impact of the negative values of pdfs on possi-bly predicting a negative cross section from a factorizedform, we recall first that the factorized cross section isindependent of µ. But in finite order approximations,µ-independence is valid only up to errors of the order ofuncalculated higher order corrections. One should chooseµ to avoid large logarithms in the perturbative expansionof the coefficient functions.

Furthermore, the factorization theorem itself is onlyvalid up to errors of size m2

q/Q2, xbjm

2s/Q

2 and m2p/Q

2.These arise from mismatches between the exact inte-grands for graphs for a structure function and the approx-imations used in obtaining the factorized form. Once theerrors are too large compared with the factorized valueof a cross section, negative values from the factorizedcross section are irrelevant. In addition, when we workin QCD, the coupling becomes too large at low scalesto allow low-order perturbative calculations of the coef-ficient functions to be useful.

In our model the second issue does not arise, and weexamine the sizes of the power-law errors and their im-pact on the effect of negative pdfs, especially in relationto the mass scales involved. Because of our use of a weakcoupling, it is sufficient to work at order aλ. Some detailsof our calculations are given in Appendix A.

In Fig. 3, we compare an exact calculation of F1 at or-der aλ with the factorized approximation, for three quitelow values of Q. These are within the range of a num-ber of experiments, e.g., Ref. [42]. The lowest value isQ = 0.8 GeV, which is below where one normally usesfactorization in QCD. But even there, there is a rangeof xbj, viz. <∼ 0.2, where the factorized approximation

20

0.2 0.4 0.6xbj−0.004

−0.002

0.000

0.002

0.004

0.006

0.008

0.0101λ2F1(xbj, Q)

Q = 2.5 GeV

Q = 1.2 GeV

Q = 0.8 GeV

Exact

Factorized (µ = Q)

FIG. 3. A comparison between Eq. (A1) and the unfactorizedgraphs in Fig. 4 for the F1 structure function (droppingO

(a2λ

)terms), for three values of Q. The graphs are independent ofthe choice of µ. The zero in the blue curve just above xbj = 0.5is the kinematical upper bound on Bjorken-xbj.

is reasonably good. Recall that as xbj gets small, theinvariant mass of the final state gets large, so that thecollision is quite inelastic and there is another larger scalein the problem than Q.

In contrast, at the larger xbj values, the factorized ap-proximation is not merely a poor approximation, but insome places gives an unphysical negative value to F1,while the true value is zero because the final state massis below threshold.

As Q increases, the range of xbj where the factorizedapproximation is good increases, while the range of neg-ative values for the approximation decreases. But therange of negative values does not closely match the rangeof negative values for the pdf. This is to be expected,since the calculation contains contributions from the one-loop coefficient function as well as directly from the neg-ative pdf. At the highest value Q = 2.5 GeV, the factor-ized approximation is positive everywhere. But at largexbj it increases. Much of the increase is where the truevalue F1 is zero because the final state is below the quark-diquark threshold, so that the factorized approximationis very incorrect.

From our calculation of the pdf in Sec. VIII B, wesaw that at small values of parton momentum fractionξ, the width of the internal transverse momentum inte-gration in the pdf is governed by the quark mass, whichis 0.3 GeV, a typical value for a constituent light quarkmass, whereas at large ξ the width is governed by thelarger diquark mass. The approximation that leads tofactorization involves neglecting quark transverse mo-mentum in the hard scattering; in addition kinematicapproximations involve neglecting xbjmp with respect toQ.

This suggests that at small enough xbj, the power errorin factorization involves a relatively low mass, while at

larger xbj it involves a relatively high mass.

Now a standard choice of scale is µ = Q. But, as ob-served in [3] for example, this is quite inappropriate atlarge xbj. The partonic transverse phase space is limitedby s/4 = Q2(1 − xbj)/4xbj, which produces large loga-rithms of s/Q2 at larger values of xbj in the coefficientfunction. A consequence [3] is that choosing µ = Q givesconsiderable oversubtraction in the coefficient function.A more sensible would be µ2 = s/4, at least at large xbj.For a given value of Q, this gets very small as xbj → 1.Although this choice of µ should improve the perturba-tive treatment of hard parts, it substantially exacerbatesthe possibility that the value of a pdf in a calculationwill turn negative, as illustrated in Fig. 2(b). This is thesame as Fig. 2(a) but with µ2 = s/4 and a sequence oflarge values for xbj, with Q2 = 2 GeV. But with such lowvalues of the final state’s invariant mass, we must also ex-pect that factorization also gives a poor approximationto the actual cross section.

Regardless of the details, we see that, given the massscales mp, mq, and/or ms, low values of µ result in a pdfthat turns negative. The example is enough, therefore,to show that a pdf defined in the MS scheme is not con-strained by general principles to be positive everywhere.As mentioned in the Introduction, the result of calcula-tions of heavy-quark pdfs provides one example of thisphenomenon in QCD.

We have also seen that when the negative pdf occurs ina region of momentum fraction ξ where power correctionscause factorization to be a poor approximation if ξ = xbj.Recall, however, that pdfs in the range xbj ≤ ξ ≤ 1 entercalculations at higher orders. So ξ ∼ 1 pdfs are relevanteven for small xbj calculations.

D. “Bare” pdf of track B

We now find the corresponding bare pdf in track B.From the treatment in Sec. IV, we know that fbare,B isjust Eq. (54), but before the MS pole in Eq. (56) hasbeen subtracted:

fbare,Bq/p (ξ;µ) = f renorm,A

BPHZ′

= aλ(µ)(1− ξ)(

(mq + ξmp)2

∆(ξ)+ ln

[µ2

∆(ξ)

]− 1

)+ aλ(µ)(1− ξ)Sε

ε

∣∣∣∣IR

+O(aa2λ

), (62)

with terms of O(ε) ignored, and with the pole identifiedas collinear rather than UV. In the methods of track B,dimensional regularization is used to regulate collinear(and IR) divergences, so that ε < 0. Then the bare pdfin Eq. (62) is negative for small negative ε.

21

E. Valence v. non-valence pdfs

The argument about MS pdfs becoming negative atsmall µ might appear to be quite general. It may seemthat all pdfs should become entirely negative at smallenough µ for all ξ. But this would be in contradictionwith the sum rules obeyed by the pdfs, notably the mo-mentum sum rule. These entail that the negativity atlow µ cannot be a general property.

That worry is removed by observing that not all graphsfor pdfs have UV divergences. In the model, the simplestcase is for the nucleon-in-nucleon pdf, for the which thelowest order value, including transverse momentum de-pendence is a simple delta function: δ(ξ−1)δ(2)(kT); thisobviously has no UV divergence when integrated over kT.

We characterize the situation by saying that in themodel, the nucleon is itself the analog of a valence quarkin QCD, while the quark in the model is an analog ofa sea quark in QCD, given that the target particle is anucleon. (For our rough purpose here, we characterizea valence quark, or, more generally, a valence parton asone that at large and moderately large momentum frac-tion corresponds at low scales to the largest pdfs andcorresponds to the basic structure of the target state.)

The generalization to other theories, including QCD,can be explained by using the expansion (19) of a pdf interms of 2PI graphs. The part without a UV divergenceis the first term in the expansion, i.e., the term withoutany rungs in the ladder. We notated it as TB. All theremaining terms in (19) have UV divergences. Thosestatements follow from the standard power counting forUV divergences in pdfs. (Our Yukawa model provides adegenerate case where the lowest order 2PI graph is justa lowest order disconnected graph.)

The 2PI term is exactly the bare pdf fbare, CFP thatwas defined by Curci et al. [29]. This term is necessarilypositive, by a version of the argument given in Sec. V B,but without any complications arising due to the needfor a regulator.

It is natural to suppose that TB is dominated by thevalence quark terms, which essentially correspond to sim-ple quark models of hadrons. Any other terms, for non-valence partons, are presumably substantially smaller.

The argument leading to possible negative MS pdfsapplies to the graphs with rungs. We then get a two-component picture: A positive valence contribution fromthe 2PI term for the relevant pdfs, and then potentiallynegative contributions from the graphs with rungs. Fornon-valence partons, the first component is small, andso the potentially negative component from the renor-malization of the UV divergent terms can dominate. Ofcourse in QCD the coupling is not particularly small inthe low µ region. So there is the potential for higher-order terms to modify the results.

As we observed, the negative contribution to a pdf goesaway once µ is enough larger than the scale setting thewidth of the kT distribution at small kT. But this is asoft constraint.

This pattern appears compatible with the uncon-strained fit in [41]. At the initial scale µ = 1.65 GeV,all the sea-quark distributions (u, d, s, s) have a rangeof ξ where a negative value is allowed within the errorestimates. These are small values compared with valencequark distributions in the same range. This implies thatin a process chosen to probe the sea quark distributions,NLO terms in factorization need not be small comparedwith LO, and so a negative non-valence pdf does not itselfentail a negative cross section.

It is perhaps surprising that the negative pdfs in thefits are at a scale 1.65 GeV that is a modest factor largerthan the obvious relevant mass scales.

IX. CONCLUSIONS

One of our main goals with this article has been todraw attention to gaps in one particularly common ap-proach to QCD factorization. However, the issues areabstract and it is tempting to view them as only for-mal, with no impact on practical phenomenology. Wehave focused on the positivity question, therefore, to il-lustrate how those gaps can influence practical consid-erations. The positivity example shows that the trackA/track B dichotomy is especially relevant to questionsabout the properties of the pdfs themselves. Other inter-esting examples likely exist.

Past approaches to pdf phenomenology have mainlyfocused on the universal nature of pdfs to constrainthem from experimental scattering data. But increas-ingly sophisticated methods are being used to study pdfsand their properties directly using nonperturbative tech-niques like lattice QCD, and to combine those approacheswith more traditional phenomenological approaches. It isimportant to check the derivation of properties that orig-inally relied on a track B view before they are adoptedunchecked into larger phenomenological programs. Moregenerally, the pitfalls of the track B approach need tobe taken into consideration in nonperturbative QCD ap-proaches. Many of these calculations are performed atrather low scales where, as illustrated by the positivityexample, the problems with track B are most prominent.

As regards the positivity issue itself, there are severalfinal points to make. First, we emphasize that we havenot argued that MS pdfs must be negative for any par-ticular choice of scales or µMS, only that nothing in thedefinition of pdfs or in the factorization theorems them-selves excludes negativity as a possibility, especially atlow or moderate input scales. Answering the question ofwhether a particular pdf turns negative depends on itslarge distance/low energy nonperturbative properties, asthe sensitivity of the question to mass scales in the exam-ple of Sec. VIII illustrates. Also, the failure of absolutepositivity in the MS scheme does not necessarily implythat other schemes do not exist which exactly preservepositivity.

Instead, we argue that imposing strict positivity on

22

MS pdfs as an absolute phenomenological constraint isan excessive theoretical bias. As our examples show, thisis especially relevant to the question of how low Q maybe before factorization theorems become unreliable. Ap-plications of factorization to low or moderate Q are oftenimportant for studies of hadron structure. Note, for ex-ample, that past Jefferson Lab DIS measurements are inthe range of Q ∼ mp [43], with Q > 1 GeV identified asthe “canonical” DIS range. It is possible that for factor-ization theorems to continue to hold at these scales andwith the desired precision, exact positivity constraintsneed to be relaxed. Indeed, if positivity constraints arerelaxed, it is plausible that DIS factorization theoremscan be extended to significantly lower Q than might oth-erwise be expected.

As we have seen, once the MS scale µ is large enoughcompared with intrinsic transverse momentum scales, thepdfs indeed become positive.

Further refinements in knowledge about transverse mo-mentum mass scales will likely help to sharpen estimatesof where in kinematics a positivity assumption begins tobe warranted. We leave such considerations to futurework.

Given that positivity is not absolutely guaranteed,there is the possibility that cross sections predicted byfactorization can be negative, as observed in Ref. [41].Indeed, we illustrated by calculation that a factorizedcross section can be unambiguously negative, withoutcontradicting positivity of physical cross sections. A fac-torized cross section is only an approximation to the truecross section. If the difference between a measured andcalculated cross section is within the expected error infactorization, then there is no contradiction.

We conclude that positivity should not be imposed asan absolute constraint in fits for pdfs, neither on the pdfsnor on the calculated cross sections. Instead, any findingof a negative pdf or especially a negative cross sectionshould be simply treated as a focus of attention. Does anegative pdf occur in a region where it can be expected?For a negative cross section, what are the expected errorsand uncertainties? Which data are critical to obtaining

the negative results?

ACKNOWLEDGMENTS

We thank Markus Diehl, Pavel Nadolsky, Duff Neill,Jianwei Qiu, Juan Rojo and George Sterman for usefuldiscussions. T.R. was supported by the U.S. Departmentof Energy, Office of Science, Office of Nuclear Physics,under Award Number DE-SC0018106. The work of N.S.was supported by the DOE, Office of Science, Office ofNuclear Physics in the Early Career Program. This workwas also supported by the DOE Contract No. DE- AC05-06OR23177, under which Jefferson Science Associates,LLC operates Jefferson Lab.

Appendix A: The full cross section

Since the MS pdf only turns negative for rather lowµ, it is worthwhile to consider whether the scales mustbe so low that factorization theorems are badly violated.To test this, we can work out the exact, unfactorizedlowest order cross section for deep inelastic scattering inthe Yukawa theory of Sec. VIII and compare the resultswith the factorized expression that uses Eq. (54). Thegraphs contributing to the DIS cross section at lowestorder and for xbj < 1 are shown in Fig. 4. (The Hermitianconjugate of Fig. 4(c) is also needed, but for brevity wedo not show it.)

The steps for deriving the factorization theorem [24]can be applied directly to these graphs. Namely, a se-quence of approximations that neglect small, intrinsicmass scales relative to Q lead (order-by-order in aλ) tothe separation into factors in Eq. (6). Errors are sup-pressed by powers of mass2/Q2. In the case of the F1

structure function, the C becomes a partonic F1, and thequark-in-hadron pdf for the Yukawa theory is just the re-sult in Eq. (54), while the lowest order hadron-in-hadronpdf is a δ function. The expression for the factorized F1

structure function in DIS is

F1(xbj, Q)xbj<1

=∑f

∫ 1

xbj

dξ

ξ×

× 1

2

δ(

1− xbj

ξ

)δqf + aλ(µ)

(1− xbj

ξ

)ln (4)−

(xbj

ξ

)2

− 3xbj

ξ + 32(

1− xbj

ξ

)2 − ln4xbjµ

2

Q2(ξ − xbj)

δpf×

×{δ (1− ξ) δfp + aλ(µ)(1− ξ)

[(mq + ξmp)

2

∆(ξ)+ ln

(µ2

∆(ξ)

)− 1

]δfq

}︸︷︷︸

ff/p(ξ;µ)

+O(aλ(µ)2

)+O

(m2/Q2

). (A1)

23

p

q

kp

q k + q

p

q

k

(a) (b) (c)

FIG. 4. Contributions to DIS structure functions from Eq. (51) at O (aλ). Graph (a) is the “handbag” diagram that contributesat leading power and small transverse momentum. Graphs (b) and (c) contribute at leading power to large kT. (the Hermitianconjugate for (c) is not shown).

Both the hadron-in-hadron pdf and the order-aλ quark-in-hadron pdf appear in the braces on the third line. Notethat the µ-dependence cancels between the second andthird lines up to order a2

λ, as expected. The expressionfor the unfactorized structure function is straightforward

but rather complex and so we do not display it here.Using the same numerical values for parameters as

were used in Fig. 2, we now compare Eq. (A1) (drop-ping both the O

(a2λ

)terms and the O

(m2/Q2

)terms)

with the exact O (aλ) F1 in Fig. 3 for several values ofQ.

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