Nucleon tomography. What can we do better today than Rutherford 100 years ago?

32 Pages·2017·1.5 MB·English

EPJ W eb of C onferences 137, 01003 (2017) DOI: 10 .1051/epjconf/201713701003

XIIthQ uark Confinement & the Hadron Spectrum
Round table: Nucleon tomography. What can we do better
today than Rutherford 100 years ago?
N. G. Stefanis1,a (chair), Constantia Alexandrou2,3,b, Tanja Horn4,5,c, Hervé Moutarde6,d, and
IgnazioScimemi7,e
1InstitutfürTheoretischePhysikII,Ruhr-UniversitätBochum,D-44780Bochum,Germany
2DepartmentofPhysics,UniversityofCyprus,P.O.Box20537,1678Nicosia,Cyprus
3Computation-basedScienceandTechnologyResearchCenter,TheCyprusInstitute,20KavafiStr.,Nicosia
2121,Cyprus
4TheCatholicUniversityofAmerica,Washington,DC20064,USA
5JeffersonLaboratory,NewportNews,VA23606,USA
6IRFU,CEA,UniversitéParis-Saclay,F-91191Gif-sur-Yvette,France
7Departamento de Física Teórica II, Universidad Complutense de Madrid, Ciudad Universitaria, 28040
Madrid,Spain
Abstract.Asurveyispresentedonthecurrentstatusof3Dnucleontomography.Several
researchfrontiersareaddressedthatdominatemodernphysicsfromtheorytocurrentand
future experiments. We have now a much more detailed spatial image of the nucleon
thankstovarioustheoreticalconceptsandmethodstodescribeitschargedistributionand
spindecompositionwhicharehighlightedhere. Theprogressoflatticecomputationsof
thesequantitiesisreportedandtheprospectsofwhatwecancometoexpectinthenear
futurearediscussed. Multi-dimensionalmapsofthenucleon’spartonicstructureappear
nowwithinreachofforthcomingexperiments.
1 Introductory remarks
Thestrivetounderstandmatterintermsofelementaryconstituents(“atoms”)andbringorderintothe
natural world lasts over thousands of years. While the ancient Greeks invented philosophical ideas
abouttheatoms,theirphysicaldiscoverybecamepossibleonlyaftertheinventionoftheRutherford
atomicmodelin1911. Thismodelisbasedontheassumptionthattheatomconsistsofacentrallarge
masswithpositivechargesurroundedbyrotatinglow-masselectrons. Thenextdecisivestepwasthe
observationaround1920thattheHydrogennucleuscanberegardedasthefundamentalblockofall
heaviernuclei,thusgivingrisetothenotionofproton.Inordertocompensatefortherepulsiveeffects
ofthepositivechargesoftheprotons,Rutherfordpostulatedtheexistenceofneutronswhichhaveno
ae-mail:[email protected]
be-mail:[email protected]
ce-mail:[email protected]
de-mail:[email protected]
ee-mail:ignazios@fis.ucm.es
© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative
Common s Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).
EPJ W eb of C onferences 137, 01003 (2017) DOI: 10 .1051/epjconf/201713701003

XIIthQ uark Confinement & the Hadron Spectrum
electric charge but contribute to the nuclear force. They were discovered later in experiment by his
associateJamesChadwick.
Buteventheprotonandtheneutronarecompositeparticlesandhaveaninternalstructurethem-
selves,asitwasshownbyHofstadterin1950bymeansofhigh-energyelectron-scatteringfromnuclei
[1]. Thedifferentialcrosssectionofthehardprocessep −→ e(cid:4)p(cid:4) —illustratedinFig.1(left)—re-
vealsthattheprotonisnotapoint-likeobjectbutbearsaninternalstructurewhichcanbedescribed
in terms of the charge and current form factors F (t = Q2) and F (t = Q2), respectively, where Q2
1 2
isthemomentumtransfer(i.e., thevirtualityoftheexchangedhighlyoff-shellphoton−q2 = Q2)in
thespacelikeregion). Theseformfactorsaredefinedintermsofthehadronicmatrixelementofthe
electromagneticcurrentV intheDiracparametrization
μ
(cid:2) (cid:3)
q
(cid:5)N(p(cid:4),s(cid:4))|V (x)|N(p,s)(cid:6)=u¯(p(cid:4),s(cid:4)) γ F (Q2)−σ ν F (Q2) u(p,s), (1)
μ μ 1 μν 2
2m
N
wherep,s,p(cid:4),ands(cid:4)are,respectively,themomentaandspinsoftheincomingandoutgoingnucleons,
u(cid:4)s their spinors, and m is the nucleon mass. These form factors are related to the magnetic (G )
N M
andelectric(G )Sachsformfactorsenteringtheelectron-protonscatteringcrosssectionbymeansof
E
theRosenbluthformula:
G (Q2) = F (Q2)+F (Q2) (2)
M 1 2
Q2
G = F (Q2)+ F (Q2). (3)
E 1 (2m )2 2
N
The above form factors can be measured in experiments (see Sec. 6) and are also calculable on the
lattice(seeSec.5). Moredetailedtheoreticalanalysisofthenucleonformfactorscomputedinterms
of nonperturbative nucleon distribution amplitudes in convolution with hard partonic subprocesses
amenabletoQCDperturbationtheorycanbefoundin[2–4]. Anextensivereviewwhichcoversthe
comparisonwiththemostrecentexperimentaldataisgivenin[5]. Inthisreport,thefocusisonmore
recenttheoreticalformulations,whichgobeyondthelongitudinaldescriptionofthenucleon,andtheir
verificationbymeasurementsatcurrentandplannedexperiments.
2 Benchmarks of nucleon tomography
Thissectionaddressesthemaintheoreticalframeworktodescribetheinternalstructureofthenucleon
withinQCD.Moredetailedaccountsaregiveninthesubsequentsections.
Thenucleonformfactorsineitherrepresentation—DiracorSachs—parameterizeinsomesense
ourignoranceabouttheinternalbindingeffectsofthenucleonwhichgiverisetoa“diffuse”structure
(representedbyashadedovalintheleftpanelofFig.1)andcausetheelasticscatteringcrosssectionto
decreasewithincreasingQ2,henceindicatingthatthenucleoncannotbepointlike[6]. Indeed,highly
inelasticelectron-protonscattering[7,8]hasrevealedthattheprotoncontainspointlikeconstituents
—partons—whichcoupletotheprobinghighlyvirtualphoton(rightpanelinFig.1).
This naive parton model was later extended to the theory of Quantum Chromodynamics (QCD)
which provides a justification of the parton picture altering its predictions by including corrections
ensuing from the quark-gluon interactions. This interaction is invariant to color SU(3) local gauge
transformations.ThesuccessofQCDinthedescriptionofhadronicprocessesisrootedintheproperty
ofasymptoticfreedom[9,10]whichenablesthesystematiccalculationofshort-distanceprocessesas
aseriesexpansionintermsoftheeffectivecouplingα (Q2)whichvanishesforQ2 −→ ∞whilepre-
s
servingrenormalization.Then,onecanseparateoutallbindingeffects,attributabletononperturbative
physics, and absorb them into universal parton distribution functions (PDF)s, parton fragmentation
2
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XIIthQ uark Confinement & the Hadron Spectrum
γ*
γ*
X
p p' p
Figure1.Leftpanel:illustrationofelastic-electron-nucleonscatteringbymeansoftheinteractionwiththevirtual
photonγ∗(Q2).Rightpanel:inelasticelectron-protonscatteringep−→e(cid:4)Xinnaivepartonmodelapproximation,
wherethevirtualphotoncouplestoasingleparton(quark).Theproducedfinal-statehadronsaredenotedbyX.
functions(PFF)s,lightconedistributionamplitudes(DA)sforhadrons,transverse-momentumdepen-
dent (TMD) PDFs (Sec. 3), generalized parton distributions (GPD)s, (Sec. 4), etc. Their extraction
fromlatticecomputationswillbeconsideredinSec.5,whiletheparticularchannelsandexperiments
toaccessthembymeasurementswillbeaddressedinSec.6andinSec.7whichwillpresentastate
oftheartonthree-dimensional(3D)nucleontomographyanditsfutureprospects.
Let us now introduce the theoretical tools to probe the interior structure of the nucleon in more
detail. Tothisend, consider, forinstance, thelongitudinaldistributionofpartonsinsidethenucleon
N,enteringadeep-inelasticscattering(DIS)processlN −→ l(cid:4)X. Thisprocess,shownfortheproton
intheleftpanelofFig.2,canbeexpressedintermsofthematrixelement1
(cid:4)
f (x,μ)= 1 dy−e−ixp+y−(cid:5)p|ψ¯(0+,y−,0 )γ+W(0−,y−)ψ(0+,0−,0 )|p(cid:6) (4)
q/N 4π T T
where f (x,μ)isthePDFdescribingaquarkqinanucleonNcarryingafractionxofitsmomentum
q/N
pattheresolution(factorization)scaleμ.Gaugeinvarianceofthecorrelatorisensuredbytheinsertion
oftheWilsonline(orgaugelink)operator
⎡ (cid:4) ⎤
W(0−,y−)= Pexp⎢⎢⎢⎢⎣ig y−dz−A+(0+,z−,0 )t ⎥⎥⎥⎥⎦ (5)
a T a
0−
evaluatedinthefundamentalrepresentationofSU(3)andtakenalong√alightlikecontourfrom0− to
y−. Notethathereweareusingthelightconenotationy± =(v0±y3)/ 2foranyvectorvμ.
CollinearPDFs,like f ,wherepartonaisaquark,antiquark,oragluon,inahadronA,represent
a/A
the universal part of the factorized cross section of a collinear process, like DIS, and are related to
leading-twistlightconecorrelatorsofelectroweakcurrentsinthehadronictensor
(cid:4)
(cid:11)
1
Wμν = d4yeiq·y (cid:5)N|jμ(y)|X(cid:6)(cid:5)X|jν(0)|N(cid:6) (6)
4π
X
fortheprocesslN−→l(cid:4)X. ForQ2 largeand xfixed,Wμν canbecastinfactorizedform(see[11]and
referencescitedtherein)toread
(cid:4)
(cid:11)
1 dξ
Wμν(qμ,pμ)= f (ξ,μ)Hμν(qμ,ξpμ,μ,α (μ))+remainder, (7)
ξ a/N a s
a x
μν
wherethecontributionofallshort-distancesubprocessesonthepartonaisdenotedbyH . Byvirtue
a
of universality, the PDFs for the Drell-Yan (DY) process, shown on the right of Fig. 2, should be
1BoldfacedsymbolsdenoteEuclideantwovectorsinthetransverseplane.
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XIIthQ uark Confinement & the Hadron Spectrum
X
l l'
γ* h1 γ* l
h
l'
X
X
p
h
2
Figure 2. Schematic representation of partonic subprocesses in QCD “embedded” within the experimentally
measuredsemi-inclusivedeepinelasticlepton(l)scatteringl+p−→l(cid:4)+h+X,wherepistheincomingproton
andhrepresentsthedetectedhadroninthefinalstate(leftpanel). Therightpanelshowstheanalogoussituation
fortheDrell-Yanprocessh +h −→γ∗+X−→l+l(cid:4)+X,whereh representincominghadrons.Thethicklines
1 2 1(2)
inbothpanelsdenote“eikonalized”,i.e.,Wilson-lineextendedquarkstoaccountforinitial(DY)orfinal(SIDIS)
stateinteractions. Examplesofsinglegluonexchangesemanatingfromtheselinesarealsoshown. Additional
hard-gluonexchangeshavebeenomitted. InbothpanelsthesymbolXrepresentsaninclusivesumoverallfinal
states.
thesameasinDIS—leftpanelofthesamefigure. Moreover, themomentum-scaledependenceof
thesePDFsisgovernedbytheDokshitzer-Gribov-Lipatov-Altarelli-Parisi(DGLAP)[12–14]evolu-
tion equation, so that once determined at an initial scale, they can be evolved in perturbative QCD
to any desired reference momentum to confront theoretical predictions with the experimental data
usingtheappropriateanomalousdimensions(i.e.,splittingfunctions). AlargesetofPDFshasbeen
extracted from global analysis of the existing data, from the low-momentum to the Large Hadron
Collider(LHC)regime,butthisproceduredependsontheaccuracyoftheprocess-dependentpertur-
μν
bativelycalculatedshort-distancepartH ,see[5]forarecentreview.
a
Thus,thefactorizationformalism[15]oftheμdependencecontainsastrongpredictivepowerfor
scattering off a nucleon (hadron). However, its validity on the partonic level, beyond the collinear
approximation,faceschallengeswhicharerelatedtotheappearanceofso-calledrapiditydivergences
ensuingfromWilsonlinesandtheirrenormalization(seeSec.3).Theoretically,theseeffectsoriginate
from the Wilson-line-extended structure of the operator definitionof quark (gluon) correlators, as it
becomesobviousfromthefollowingTMDfieldcorrelator[16–18]
(cid:4)
d(y·P)d2y (cid:12) (cid:13)
Φq[C](x,k ;n)= Teik·y p|ψ¯ (y)W(0,y|C)ψ(0)|p . (8)
ij T (2π)3 j i y·n=0
OnenoticesthepathdependenceofthisexpressionencodedinthecontourC intheexponentialline
integral. It can be resolved by adopting that particular contour which ensures the continuous color
flowintheconsideredpartonicprocess. Asaresult,theDYprocess,shownintherightpanelofFig.
2,containsasignreversalrelativetotheSIDISsituation(leftpanelinFig.2),whichoriginatesfrom
thechangeofafuture-pointingWilsonlinetoonewiththeoppositeorientationasaconsequenceof
CPinvarianceandCPTconservationinQCD.Thisentailsthebreakdownofuniversality,becausethe
factoredoutnonp(cid:14)ertur(cid:15)bativepa(cid:14)rtof(cid:15)theSIDISsetupcannotbeusedwithoutreadjustment(signflip)in
theDYprocess: f⊥ = − f⊥ [19]. Thisintriguingbehaviorconstitutesinfactthelitmus
1Tq DY 1Tq SIDIS
testoftheTMDapproachtosinglespinasymmetries[20]whichrequirethattherescatteringofthe
struckquarkinthefieldoftheremnanthadrongeneratesaninteractionphase. Thisphasewouldbe
forced to vanish by the time-reversal invariance in the absence of the directional dependence of the
Wilson line. Additional phases appear for time-reversal-odd TMD PDFs even at the leading-twist
levelwhenincludesintotheWilsonlinesthePaulitensortermtoaccountforacorrecttreatmentof
thespindegreesoffreedom[21,22].
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XIIthQ uark Confinement & the Hadron Spectrum
Theformalproofoffactorizationemploysadetailedanalysisofsingularitiesthatoriginatefrom
differentsources: (i)ultraviolet(UV)poles,inducedbylargeloopmomenta,thatcanberegularized
dimensionally, (ii) rapidity divergences that originate from the Wilson lines, and (iii) overlapping
UVandrapiditydivergences. Thelatteremergefromgluonsmovingwithaninfiniterapidityinthe
oppositedirectionwithrespecttotheirparenthadronandcannotberegularizedbyinfraredgluonmass
regulators. Whileinthecollinearcaserapiditydivergencescancelinthesumofgraphs,intheTMD
caseoneneedsadditionalregularizationparameters. Wewillhavetosaymoreabouttheseproblems
inSec.3below. InthesamesectionwewillconsiderthecalculationofTMDcorrelatorsbeyondthe
leadingorderinα (μ)anddiscusstheconceptsoftheirevolution. Firstattemptsto“measure”TMD
s
PDFsonthelatticehavebeengivenin[23],whilecurrentinvestigationswillbepresentedinSec.5.
AcompilationofthevariousTMDPDFs(“TMDs”forshort)isgiveninTable1. Notethatasimilar
structureholdsalsoforgluonTMDs.
Table1.Twist-twoTMDsasfunctionsof(x,k )describingcorrelationsbetweenintrinsicspinandtransverse
T
momentumusingthefollowingabbreviations:U(unpolarized),L(longitudinallypolarized),T(transversely
polarized).Theboldfacedelementssurvivethek integration.ThetwotermsinbracketsareT-odd,whereasall
T
elementsinthelastcolumnarechirallyodd.
Nucleon\QuarkPolarization
U L (cid:14) T (cid:15)
U fq hq⊥
1 1
unpolarized Boer-Mulders
L gq hq⊥
1L 1L
(cid:14) (cid:15) helicity worm-gearL
T fq⊥ gq⊥ hq |hq⊥
1T 1T 1T 1T
Sivers worm-gearT transversity|pretzelosity
3 Parton distributions with transverse degrees of freedom — TMDs
2The longitudinal PDFs are based on collinear factorization and provide no information about the
transversestructureofhadrons. Toachievea3Dpictureof thehadronicstructure, onehas toretain
the transverse momenta k of the partons unintegrated, as expressed in Eq. (8). This gives rise to
T
eight k dependent PDFs of leading twist two (see Table 1), which enter various processes as the
T
SIDIS and the DY process, both illustrated in Fig. 2. In SIDIS, one has the convolution of a TMD
withafragmentationfunction,whereasinDYonefacestheconvolutionoftwoTMDs.Inthissection,
wewilldiscussthepropertiesofTMDsfromthetheoreticalpointofviewandaddresstheminmore
detail. Onfocusistheuseof k factorizationtheorems,therenormalizationofrapiditysingularities,
T
andtheTMDevolutionbehaviorinthefactorizeddynamicalregimes(see[24]foracomprehensive
review).
ThecoreoftheTMDfactorizationtheoreminitscurrentform[15,26–28],isbasedontheunder-
standingofthestructureofrapiditydivergencesintheDYand/orSIDIScrosssection.
As shown in Fig. 3, the cross section for SIDIS can be split, by virtue of power counting ar-
guments, intothreepieces: atransverse-momentumdependentinitialstate(called“F”),afinalstate
(termed “D”), and a soft-interaction part (denoted by “S”) which connects the previous two. The
power counting procedure, which defines these states, appears naturally in an effective field-theory
2BasedonthecontributionbyI.Scimemi.
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XIIthQ uark Confinement & the Hadron Spectrum
μ(RGE)
p
l γ∗ B
(Q²)
D
S ζ(CSSE)
X
F
p
A
Figure3.GenericstructureofaSIDIS-likeprocessinvolvingvariousTMDPDFs.Thefactorizationofdynamical
regimesisindicatedbydashedlines.Thelargeμ2evolutioniscontrolledbytherenormalization-groupequation
(RGE),whiletheevolutionwithrespecttoζfollowstheCollins-Soper-Sterman(CSS)equation[25](seetext).
framework, like the Soft Collinear Effective Theory (SCET) [29–31]. However, one can obtain this
dissectionofthecrosssection,byemployingmorestandardQCDargumentsinconnectionwithfac-
torization theorems. Actually, the states so naively identified as above, are not per se well defined.
This can be checked, for instance, by a one-loop calculation, to show that the rapidity divergences
induceamixingofallofthesestatessothatitisimpossibletoarriveatarigorousdefinitionofastate
whoseperturbativecalculationallowstheproperseparationofultravioletandinfrared(IR)scales. In
ordertoachievethisgoal,wehavetoproceedinadifferentwaywhichisexposedbelow.
To this end, we define the bare (unrenormalized and singular in rapidity) quark, anti-quark and
gluonunpolarizedTMDPDFoperatorsasfollows:
Obare(x,b )= 1(cid:11)(cid:4) dξ−e−ixp+ξ−(cid:16)T(cid:14)q¯ W˜T(cid:15) (cid:17)ξ(cid:18) |X(cid:6)γ+(cid:5)X|T¯(cid:14)W˜T†q (cid:15) (cid:17)−ξ(cid:18)(cid:19),
q T 2 2π i n a 2 ij n j a 2
Obare(x,b )= 1(cid:11)X (cid:4) dξ−e−ixp+ξ−(cid:16)T(cid:14)W˜T†q (cid:15) (cid:17)ξ(cid:18) |X(cid:6)γ+(cid:5)X|T¯(cid:14)q¯W˜T(cid:15) (cid:17)−ξ(cid:18)(cid:19),
q¯ T 2 2π n j a 2 ij i n a 2
Obgare(x,bT)= x1p+X(cid:11)(cid:4) d2ξπ−e−ixp+ξ−(cid:16)T(cid:14)F+μW˜nT(cid:15)a(cid:17)2ξ(cid:18)|X(cid:6)(cid:5)X|T¯(cid:14)W˜nT†F+μ(cid:15)a(cid:17)−2ξ(cid:18)(cid:19), (9)
X
whereξ ={0+,ξ−,b }andn,n¯arelightconevectors(n2 =n¯2 =0, n·n¯ =2).Foragenericvectorv,we
T
havev+ =n¯·vandv− =n·v. Repeatedcolorindicesa(a=1,...,N forquarksanda=1,...,N2−1
c c
forgluons)aresummedup. TherepresentationsofthecolorSU(3)generatorsinsidetheWilsonlines
arethesameastherepresentationsofthecorrespondingpartons(i.e.,fundamentalrepresentationfor
quarks and adjoint representation for gluons). The Wilson lines W˜T(x) emerge at the coordinate x
n
and continue to lightcone infinity along the vector n, where they connect to a transverse gauge link
(indicatedbythesuperscriptT)extendingtotransverseinfinity.
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XIIthQ uark Confinement & the Hadron Spectrum
Thehadronicmatrixelementsoftheoperators,definedinEq.(9),providetheunsubtractedTMDs
inaccordancewiththeTMDfactorizationtheorems[15,26,27]:
Φq←N(x,bT) = 21(cid:11)(cid:4) d2ξπ−e−ixp+ξ−(cid:5)N|(cid:16)T(cid:14)q¯iW˜nT(cid:15)a(cid:17)2ξ(cid:18)|X(cid:6)γi+j(cid:5)X|T¯(cid:14)W˜nT†qj(cid:15)a(cid:17)−2ξ(cid:18)(cid:19)|N(cid:6),
Φq¯←N(x,bT) = 21(cid:11)X (cid:4) d2ξπ−e−ixp+ξ−(cid:5)N|(cid:16)T(cid:14)W˜nT†qj(cid:15)a(cid:17)2ξ(cid:18)|X(cid:6)γi+j(cid:5)X|T¯(cid:14)q¯iW˜nT(cid:15)a(cid:17)−2ξ(cid:18)(cid:19)|N(cid:6),
X (cid:4)
Φg←N(x,bT) = x1p+ (cid:11) d2ξπ−e−ixp+ξ−
(cid:16)X (cid:14) (cid:15) (cid:17) (cid:18) (cid:14) (cid:15) (cid:17) (cid:18)(cid:19)
ξ ξ
×(cid:5)N| T F+μW˜nT a 2 |X(cid:6)(cid:5)X|T¯ W˜nT†F+μ a −2 |N(cid:6), (10)
whereN isanucleon/hadron.
Here, the variable x represents the momentum fraction carried by a parton originating from the
nucleon (this refers to the TMD labeling rule f ← N). One notices that at the operator level the
TMDs resemble the integrated parton densities, the only difference being that the parton fields are
additionally separated by the spacelike distance b . In order to renormalize correctly the operators
T
andtherespectivematrixelements,onehastoperformtheregularizationoftheUV,IRandrapidity
divergences. The UV divergences in the TMDs are removed by the usual renormalization factors.
In order to cancel rapidity divergences, one has to consider both the so-called zero-bin subtractions
andthesoftfunction. AccordingtotheSCETterminology,the“zero-bin”representsthesoft-overlap
contributionthathastoberemovedfromthecollinearmatrixelementinordertoavoiddoublecount-
ingofsoftsingularities[32]. Thecombinationofthezero-binsubtractionwiththesoftfunctionhas
a very particular form, which is dictated by the factorization theorem and should be included in the
definitionoftheTMDoperatorsintheformofasingle“rapidityrenormalizationfactor”Rinorderto
completethedefinitionoftherenormalizedTMDoperator. Then,onehas
O (x,b ,μ,ζ)=Z (ζ,μ)R (ζ,μ)Obare(x,b ),
q,q¯ T q q q,q¯ T
O (x,b ,μ,ζ)=Z (ζ,μ)R (ζ,μ)Obare(x,b ), (11)
g T g g g T
whereZ (quark)andZ (gluon)aretheUVrenormalizationconstantsfortheTMDoperatorsandthe
q g
scaleζ emergesasaresultofsplittingthesoftfunctionbetweenthetwoTMDsF andD.
The scales μ and ζ are related to the UV and rapidity subtractions, respectively. While the UV
renormalization factors depend on the UV regularization method and the regularization scale μ, the
“rapidity renormalization factors” depend in addition on the rapidity regularization method and the
rapidityscaleζ aswell. Remarkably,becausethesoftfunctionisprocessindependent,the“rapidity
renormalizationfactors”turnouttobeprocessindependentaswell(see[15, 26–28, 33]forgeneral
argumentsand[34]foranexplicitcalculationatthenext-to-next-to-leadingorder(NNLO)).However,
the particular form of the zero-bin subtractions, contained in the factor R, is regulator dependent.
Therefore, one has to fix the order of how to deal with these singular factors exactly. Following
[35],onecanfirstremoveallrapiditydivergencesandcarryoutthezero-binsubtraction,performing
subsequently the multiplication with the Z factors. In that case, one finds that the factor R contains
not only rapidity divergences, but also explicit UV poles which, however, have already been taken
into account by means of the factor Z. Thus, different subtraction procedures can produce different
intermediateexpressions,whilethefinal(UVfiniteandrapidity-divergences-free)expressionswillbe
thesame.
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XIIthQ uark Confinement & the Hadron Spectrum
ThefinaldefinitionsfortheTMDsenteringtheSIDISprocessinFig.3read
Ff←N(x,bT;μ,ζ) = (cid:5)N|Of(x,bT;μ,ζ)|N(cid:6),
Df→N(z,bT;μ,ζ) = (cid:5)N|†Of(z,bT;μ,ζ)|N(cid:6)†. (12)
This definition in conjunction with the TMD factorization theorem implies the following relation
betweenbareandrenormalizedTMDs
Ff←N(x,bT;μ,ζ) = Zf(μ,ζ)Rf(μ,ζ)Φf←N(x,bT),
Df→N(z,bT;μ,ζ) = Zf(μ,ζ)Rf(μ,ζ)Δf→N(x,bT). (13)
TheTMDfactorizationtheoremdictatestheexplicitformofR aswell. Itisgivenbytheexpression
f
√
S(b )
R (ζ,μ)= T , (14)
f
Zb
whichinvolvesthesoftfunctionS(b )andthezero-bincontributionZb,i.e.,thesoftoverlapofthe
T
collinearandthesoftsectorsenteringthefactorizationtheorem[15,26–28,32].
ThesoftfunctionisdefinedasthevacuumexpectationvalueofacertainconfigurationofWilson
linespertainingtotheprocessunderinvestigation. Consider,forinstance,theSIDISprocess. Then,
onehas
(cid:14) (cid:15) (cid:14) (cid:15)
S(b )= Trc (cid:5)0| T ST†S˜T (0+,0−,b )T¯ S˜T†ST (0)|0(cid:6) . (15)
T N n n¯ T n¯ n
c
TheWilsonlinesaregivenbythefollowingorderedexponentials
ST = T S , S˜T =T˜ S˜ , (16)
n n n (cid:2) (cid:4) n¯ n n¯ (cid:3)
0
S (x) = Pexp ig dsn·A(x+sn) ,
n
−∞
(cid:2) (cid:4) (cid:3)
0
Tn¯(x) = Pexp ig dτ(cid:11)l⊥·A(cid:11)⊥(0+,∞−,(cid:11)x⊥+(cid:11)l⊥τ) ,
(cid:2) (cid:4)−∞ (cid:3)
∞
S˜ (x) = Pexp −ig dsn¯ ·A(x+n¯s) ,
n¯
(cid:2) (cid:4)0 (cid:3)
∞
T˜n(x) = Pexp −ig dτ(cid:11)l⊥·A(cid:11)⊥(∞+,0−,(cid:11)x⊥+(cid:11)l⊥τ) .
0
The transverse Wilson lines T are indispensable in singular gauges, e.g., the lightcone gauge n ·
n
A = 0 (or n¯ ·A = 0) (see [17, 36–40]), while in covariant gauges the T ’s appear only formally in
n
ordertopreservegaugeinvariance, butdonotcontribute. Payattentiontothefactthatthecollinear
Wilson lines WT(x), used in the TMD operators given by Eq. (9), are defined in the same way as
n
the soft Wilson lines ST(x). However, we keep them apart because they behave differently under
n
regularization.
Afewtechnicalremarksarehereinorder. Thezero-bin(oroverlap)subtractionisasubtleissue
anddemandsparticularcaution. (i)Thesubtractionproceduredependsontheregularizationmethod
usedtoremovetherapiditydivergences(see,e.g.,[27]foramorecompletediscussion).(ii)Itmightbe
impossibletodefinethezero-bin(overlap)regionintermsofapropermatrixelementforaparticular
regularization scheme, even if it is calculable. Choosing a convenient rapidity regularization, the
zero-bin subtractions can be related to a particular combination of soft factors. Using, for instance,
themodifiedδ-regularizationscheme,thezero-binsubtractionbecomesequaltothesoftfactor: Zb=
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XIIthQ uark Confinement & the Hadron Spectrum
S(b ). In fact, the modified δ-regularization scheme has been employed with the aim to preserve
T
just this relation, see [34, 35]. A crucial consequence of this relation is that the collinear Wilson
linesW (x)andthesoftWilsonlinesS (x)assumedifferentregularizedforms. Finally,usingthe
n(n¯) n(n¯)
modifiedδ-regularization,theexpressionfortherapidityrenormalizationfactorbecomes
(cid:20)
(cid:20)
(cid:20) 1
Rf(ζ,μ)(cid:20)(cid:20) = (cid:21) , (17)
δ-reg. S(bT;ζ)
which was first explicitly verified at NNLO in [34, 41], and was confirmed for various kinematics
in[35]. Byvirtueoftheprocessindependenceofthesoftfunction[15, 26–28, 33], thefactorR is
f
process independent as well. We complete this discussion by a comment on the TMD formulation
usedinRef.[15]. There,therapiditydivergencesareregularizedbytiltingtheWilsonlinesoff-the-
light-cone. Thisway,thecontributionsoriginatingfromtheoverlappingregionsandthesoftfactors
canberecombinedintheindividualTMDsbyproperlycombiningdifferentsoftfactorswithapartially
removedregulator. ThiscombinationentailsinournotationthefactorRf,i.e.,
(cid:22)
(cid:20)
(cid:20)(cid:20) S˜(y ,y )
Rf(ζ,μ)(cid:20)(cid:20) = n c . (18)
JCC S˜(yc,yn¯)S˜(yn,yn¯)
Thefurtherstepsremainthesameaswiththeδ-regulatortechnique.
The use of the modified δ-regulator brings within reach a perturbative calculation at NNLO. In-
deed,thematrixelementsforthesoftfactorhavebeenevaluatedatthetwo-looporderin[34]. This
hasmadeitpossibletocalculateallunpolarizedTMDsandTMDfragmentationfunctions(FF)satthe
sameorder[35,41].Inaddition,alsothecalculationofthesoftmatrixelementfordouble-partonscat-
tering[42]hasbeencarriedoutatNNLO.TheresultsobtainedfortheTMDsconfirmedtheprevious
QCDcalculationsperformedin[43–47]. Employingsymmetryargumentsforthesoftfactor,theevo-
lutionofallunpolarizedTMDshasbeenperformedatthethree-looporderin[48]findingagreement
with a recent result reported in [49]. These calculations use the operator product expansion of the
TMDs in the lowest order of the power expansion which works sufficiently well for asymptotically
high transverse momenta. However, a complete treatment of the TMDs must also include analysis
beyondthislowestorder,asubjectweexpecttobecomeanactivefieldofresearchinthenearfuture.
Forthetimebeing,thenonperturbativestructureoftheunpolarizedTMDshasbeenmodeledwithina
renormalonanalysisinthelimitofhightransversemomentum,see[50]. Remarkably,itshowsanon-
trivialentanglementbetweenthetransversemomentumandtheBjorkenvariablesbeyondthelowest
orderinthepowerexpansion.
Attheendofthisexposition,weturnourattentiontotheevolutionofTMDs. RecallingEq.(12),
wewrite
d 1 d 1
μ2 O (x,b )= γf(μ,ζ)O (x,b ), μ2 O (z,b )= γf(μ,ζ)O (z,b ). (19)
dμ2 f T 2 f T dμ2 f T 2 f T
TheTMDPDFoperatorandalsotheTMDFFoperatorhavethesameanomalousdimensionγ ,which
f
comessolelyfromtherenormalizationfactorZ andisuniversalonaccountoftheuniversalityofthe
f
hardinteractions[15,26,27]. ApplyingstandardRGEtechniques,weobtain
(cid:24) (cid:25) (cid:24) (cid:25)
γq(μ,ζ)=2A(cid:23)D Z −Z , γg(μ,ζ)=2A(cid:23)D Z −Z , (20)
2 q 3 g
(cid:23)
whereADrepresentstheoperatorwhichextractstheanomalousdimensionfromthecounterterm(i.e.,
gives the coefficient in front of the leading pole in 1/(cid:13) with a n! prefactor, n being the order of the
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XIIthQ uark Confinement & the Hadron Spectrum
perturbativeexpansion).Thefirsttermoftheperturbativeexpansionisthecuspanomalousdimension
Γf [51],givenby
cusp
γf =Γf l −γf , (21)
cusp ζ V
wherewehaveusedthenotation
⎛ ⎞ ! !
L ≡ln⎜⎜⎜⎜⎝ X2b2T ⎟⎟⎟⎟⎠, l ≡ln μ2 , λλλ ≡ln δ+ . (22)
X 4e−2γE X X δ p+
Attheleveloftherenormalizationfactors,thelogarithmicpartofthefactorR canbeunambiguously
f
fixedbymeansoftherelation
d2lnRf (cid:20)(cid:20)(cid:20)(cid:20)(cid:20) = A(cid:23)D⎡⎢⎢⎢⎢⎣Z dlnRf! ⎤⎥⎥⎥⎥⎦=−Γcfusp. (23)
dlnμ2dlnζ(cid:20) f dlnζ 2
f.p s.p
Asimilarrelationwasobtainedin[52,53].
Mostpartoftheworkreportedabove,hascoveredunpolarizedTMDs. However,foracomplete
understanding of the confinement process, the study of polarization effects and twist expansion of
TMDsareoffundamentalimportanceaswell. Progresshasbeenachievedwithrespecttotheunder-
standing of rapidity divergences and their treatment via soft factor functions by virtue of the TMD
factorizationtheorem. ThenonperturbativestructureofTMDsshouldalsobeexploredfromatheo-
reticalandanexperimentalpointofview. Amongsomerecentdevelopments,weliketomentionthe
effortstocombinetheTMDformalismwiththejetanalysisdiscussedin[54–56].
4 3D imaging of the nucleon’s partonic content in terms of GPDs
3In continuation of the previous discussion of the spatial distribution of quarks and gluons, we will
describe in this section distributions which can provide tomographic 3D images of the nucleon [57,
58], termed generalized parton distributions, or (GPD)s for short. These quantities were initially
introducedinconnectionwiththepartonicdescriptionofdeeplyvirtualComptonscattering(DVCS)
by Müller [59], and independently by Ji [60], and Radyushkin [61]. More recently, they have also
beenemployedinthedescriptionofdeeplyvirtualmesonproduction(DVMP)[62,63]andintimelike
Comptonscattering(TCS)[64]. ArecentreviewoftheuseofGPDscanbefoundin[65]andbroader
comprehensivereviewsin[66–70],whileglobalanalysisofavailabledataisperformedin[71]. The
effortstoextractGPDsfromexperimentwillbeaddressedinSec.6andSec.7.
UnlikePDFsandTMDs,GPDsaredefinedintermsofnon-forwardhadronicmatrixelementsof
quarkandgluoncorrelators,i.e., p(cid:4) (cid:2) p,asoneseesfromthegeneralizedformfactorforquarks(see,
e.g.,[65])
(cid:4)
F (x,ξ,t) = dy−e−ixp+y−(cid:5)p(cid:4)|ψ¯(y−)γ+ψ(−y−)|p(cid:6) (24)
q 2π 2 2 2
(cid:2) (cid:3)
≡ H (x,ξ,t)(cid:14)U(p(cid:4))γμU(p)(cid:15) nμ +E (x,ξ,t) U(p(cid:4))iσμν(p(cid:4)−p)νU(p) nμ ,
q p·n q 2M p·n
wherewehavesuppressedtheWilsonlinesneededtoensuregaugeinvariance. Here, thequantities
H (x,ξ,t) and E (x,ξ,t) are the quark GPDs which generalize the nucleon form factors F and F
q q 1 2
in Eq. (1). They depend on the squared hadron momentum transfer t = (p(cid:4) − p)2 and the skewness
3BasedonthecontributionbyH.Moutarde.
10