The gauge-invariant canonical energy-momentum tensor

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by  Cédric Lorcé

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The gauge-invariant canonical energy-momentum tensor
n
a
J
8 CédricLorcé1,a

1CentredePhysiqueThéorique,Ecolepolytechnique,CNRS,UniversitéParis-Saclay,F-91128Palaiseau,
]
h France
p
-
p Abstract.Thecanonical energy-momentum tensorisoftenconsideredasapurelyaca-
e demic object because of itsgauge dependence. However, ithas recentlybeen realized
h
that canonical quantities can in fact be defined in a gauge-invariant way provided that
[
strictlocalityisabandoned, thenon-localaspectbeingdictactedinhigh-energyphysics

1 by thefactorization theorems. Using thegeneral techniques for the parametrizationof
v non-localpartoncorrelators,weprovideforthefirsttimeacompleteparametrizationof
8 the energy-momentum tensor (generalizing the purely local parametrizations of Ji and
3 Bakker-Leader-Truemanusedforthekineticenergy-momentumtensor)andidentifyex-
9
plicitly the parts accessible from measurable two-parton distribution functions (TMDs
1
andGPDs).Asby-products,weconfirmtheabsenceofmodel-independentrelationsbe-
0
tweenTMDsandpartonorbitalangularmomentum,recoverinamuchsimplerwaythe
.
1 Burkardtsumruleandderivethreesimilarnewsumrulesexpressingtheconservationof
0 transversemomentum.
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:
v
i 1 Introduction
X
r FollowingNoether’sprocedure,oneobtainsacanonicalenergy-momentum(EMT)Tµν whichisusu-
a
ally neither symmetric1 nor gauge invariant. These properties are often considered as pathological
andcanbecuredbyaddingtotheEMTa superpotentialtermoftheform∂ f[αµ]ν [2–4], wherethe
α
squarebracketsstand for antisymmetrization. Thisamountsto a redefinitionof the localdensity of
energyandmomentum[5,6]whileleavingthetotallinearandangularmomentaunchanged.
Superpotentialtermshavealsobeenusedtodecomposetheangularmomentumintospinandor-
bitalcontributions[7,8]. Accordingtostandardtextbookslike e.g.[9,10], onecannotperformthis
decomposition for the gauge field in a gauge-invariantway. Nonetheless, it appears that the pho-
tonspinandorbitalangularmomentum(OAM)areroutinelymeasured,seee.g.[11]andreferences
therein. Similarly, a gauge-invariantquantity called ∆G, interpreted in the light-front gauge as the
gluonspin[7],hasbeenmeasuredinpolarizeddeepinelasticandproton-protonscatterings,see[12]
for a recent analysis. Since physicalobservablesare gauge invariant, Chen et al. [13] claimed that
the textbookswere wrong and proposeda formalgauge-invariantdecompositionof the photonand
ae-mail:[email protected]
1WestressthatthesymmetryrequirementessentiallycomesfromGeneralRelativitywheretorsionisassumedtovanish.
Inmoregeneraltheoriesofgravitationlikee.g.Einstein-Cartantheoryandmetric-affinegaugetheory,torsionisallowedtobe
nonzeroleadingtoasymmetricEMTsandanaturalcouplingbetweengravitationandspin.Theseeffectsarehoweverexpected
tobeextremelysmallandtoshowuponlyunderextremeconditions,seee.g.[1]andreferencestherein.
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gluonangularmomentum. Theirworkreceivedstrongcriticismsandtriggerednumeroustheoretical
papers on the subject, summarized in recent reviews [14, 15]. It is now understood that textbooks
implicitly referred to local quantities only, whereas the formal construction of Chen et al. and the
above-mentionedquantitiesextractedfromexperimentaldataareintrinsicallynon-local[16–19].
Typical examples of measurable non-localquantities in Quantum ChromoDynamics(QCD) are
parton distribution functions (PDFs) whose gauge invariance is ensured by a Wilson line along a
path determinedby the factorizationtheorems[20]. GeneralizedTransverse-Momentumdependent
Distributions (GTMDs) [21] are natural generalizations of the PDFs and provide a natural way to
access the canonicalOAM [22–24] and other angular momentum correlations[25]. Unfortunately,
it is not known so far how to access them directly in experiments. They can however be accessed
indirectlyusinge.g.realisticmodels[22,26–29],orlatticeQCDintheinfinite-momentumlimit[30–
33].
We present here the first complete parametrization of the EMT obtained in [34] which can be
applied to virtually any form of the EMT discussed in the literature, providedthat the non-locality
appearsonlyalongthelight-front(LF)directionn.
2 A basis of gauge-invariantenergy-momentumtensors
MostoftheEMTsthatappearedintheliteraturecanbedecomposedinabasisoffivegauge-invariant
tensors
↔
Tµν =ψγµiDνψ, Tµν =−2TrGµαGν +gµν 1Tr GαβG ,
1 2 2 α 2 αβ
Tµν =−ψγµgAν ψ, Tµν = 1ǫµνα(cid:2)β∂ ψγ(cid:3)γ ψ , h i (1)
3 phys 4 4 α β 5
h i
Tµν =−2∂ Tr GµαAν ,
5 α phys
h i
↔ ↔ ↔ → ←
where iDµ = i∂µ+gAµ isthehermitiancovariantderivativewith ∂µ = ∂µ− ∂µ,andǫ = +1. For
2 2 0123
example,thekineticformofthequarkandgluonEMTsaregivenbyTµν andTµν, respectively,and
1 2
the correspondingcanonical forms are given by Tµν +Tµν and Tµν −Tµν +Tµν, respectively. The
1 3 2 3 5
superpotentialsTµν andTµνtogetherwiththeQCDequationsofmotion
4 5
↔
ψ(r)γ[µiDν]ψ=−ǫµναβ∂ ψγ γ ψ ,
α β 5
(2)
2 D Gαβ c =−gψ γβψhc, i
α c′ c′
h i
wherec,c′arecolorindicesinthefundamentalrepresentationandD =∂ −ig[A , ]istheadjoint
µ µ µ
covariantderivative,canbeusedtorelatethevariousEMTs[14,34]. Becauseofthefirstidentityin
Eq.(2),wehaveTµν =−1T[µν]whichcanthenbediscardedinthefollowingdiscussions.
4 2 1
Inordertoobtaingauge-invariantcanonicalEMTs,oneneedstodecomposethegaugepotential
A intopure-gaugeand“physical”(orcovariant)contributions
µ
Apure ≡ i W∂ W−1, Aphys ≡ A −Apure, (3)
µ g µ µ µ µ
whereWisanon-integrablephasefactortransformingasW 7→UWundergaugetransformations.
InthegaugewhereW(r) = 1, thegauge-invariantcanonicaldecompositionformallyreducestothe
Jaffe-Manohardecomposition[7],andcanthereforebeconsideredasagauge-invariantextensionof
thelatter[14,17,35,36].
POETIC2015
3 Generalparametrization
Since our aim is to relate the matrix elements of the EMT to measurable parton distributions, we
identify the non-localphase factor W with a straightWilson line runningalong the LF direction n
tor = r+∞ηn,andtheninthetransversedirectionto∞ . Itisthenclearthat,besidetheaverage
n ⊥
targetmomentumP = (p′+ p)/2andthemomentumtransfer∆ = p′− p,thematrixelementsofthe
genericLFEMTalsodependinprincipleonN = M2n foratargetofmassM,andonthedirectionof
P·n
theWilsonlineη= ±1. ThescalarfunctionsparametrizingthegenericLFEMTarecomplex-valued
functionsofξ =−(∆·N)/2(P·N)andt=∆2,whicharetheonlytwoindependentscalarsthatcanbe
formedwithP,∆andN.
In Ref. [34], we used the techniques presented in the Appendix A of Ref. [21] and found
that the generic LF EMT for a spin-1/2 target can be parametrized as hp′,S′|Tµν(0)|p,Si =
a
u(p′,S′)Γµν(P,∆,N;η)u(p,S)witha=1,··· ,5andthematrixΓµν givenby
a a
PµPν ∆µ∆ν Pµiσν∆ Pνiσµ∆
Γµν = MgµνAa+ Aa+ Aa+ Aa+ Aa
a 1 M 2 M 3 2M 4 2M 5
NµNν PµNν PνNµ Nµiσν∆ Nνiσµ∆ ∆µiσνN ∆νiσµN
+ Ba+ Ba+ Ba+ Ba+ Ba+ Ba+ Ba
M 1 M 2 M 3 2M 4 2M 5 2M 6 2M 7
PµPν ∆µ∆ν NµNν PµNν PνNµ iσN∆
+ MgµνBa+ Ba+ Ba + Ba + Ba + Ba
" 8 M 9 M 10 M 11 M 12 M 13# 2M2
Pµ∆ν Pν∆µ ∆µNν ∆νNµ M ∆νiσµ∆
+ Ba + Ba + Ba + Ba + iσµνBa + Ba
M 14 M 15 M 16 M 17 2 18 2M 19
PµiσνN PνiσµN NµiσνN NνiσµN
+ Ba + Ba + Ba + Ba
2M 20 2M 21 2M 22 2M 23
Pµ∆ν Pν∆µ ∆µNν ∆νNµ iσN∆
+ Ba + Ba + Ba + Ba . (4)
" M 24 M 25 M 26 M 27# 2M2
Forconvenience,weintroducedthenotationiσµb ≡iσµαb andthefactorsofisuchthattherealpart
α
ofthescalarfunctionsisη-evenwhiletheimaginarypartisη-odd
Xa(ξ,t;η)= Xe,a(ξ,t)+iηXo,a(ξ,t) (5)
j j j
asaconsequenceoftime-reversalsymmetry. Thehermiticitypropertythenimpliesthattherealpart
ofBawith j≥14isξ-oddandtheimaginarypartisξ-even.Fortheotherfunctions,itistheopposite.
j
Thefactthatonly32independentstructuresexistcanbeobtainedfromanaivesimplecounting: the
EMTTµνhas4×4=16components;thetargetstatepolarizations±S and±S′bringanotherfactorof
a
2×2=4;time-reversalandhermiticityhavingbeenusedtofixthefactorsofiandtheξ-dependence,
we are left with parity which reduces the number of independent polarization configurations by a
factor2. Asannounced,thisleadstoatotalof32independentcomplex-valuedamplitudes.
Manifestly,theEMTsTµνandTµνdonotdependonNorη.Allthecorrespondingscalarfunctions
1 2
mustthenvanishexceptforAe,a(0,t)witha = 1,2and j = 1,··· ,5,whicharelinearlyrelatedtothe
j
standardenergy-momentumformfactors[8,14,37]asfollows
A (t)= Ae,1(0,t), A (t)= Ae,2(0,t),
q 2 G 2
B (t)= Ae,1(0,t)+Ae,1(0,t)−Ae,1(0,t), B (t)= Ae,2(0,t)+Ae,2(0,t)−Ae,2(0,t),
q 4 5 2 G 4 5 2
C (t)= Ae,1(0,t), C (t)= Ae,2(0,t), (6)
q 3 G 3
C¯ (t)= Ae,1(0,t)+ t Ae,1(0,t), C¯ (t)= Ae,2(0,t)+ t Ae,2(0,t),
q 1 M2 3 G 1 M2 3
D (t)= Ae,1(0,t)−Ae,1(0,t), 0= Ae,2(0,t)−Ae,2(0,t).
q 4 5 4 5
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4 Linear and angular momentum constraints
The parametrization(4) is only constrainedby space-time symmetries. We briefly present here the
additionalconstraintsarisingfromtheconservationoftotallinearandangularmomentum.
Theaveragefour-momentumintheLFformofdynamicscanbeobtainedbycontractingtheEMT
with 1 N intheforwardlimit∆→0
2M2 µ
1 η
hpνi≡ hP,S|TNν(0)|P,Si= PνAe,a+Nν(Ae,a+Be,a)+δa3 ǫνS (Bo,3−Bo,3). (7)
a 2M2 a 2 1 2 2 T 18 20
Interestingly, the last term in Eq. (7) originating from the potential EMT Tµν is η-odd and can be
3
interpretedasthespin-dependentcontributiontothemomentumarisingfrominitialand/orfinal-state
interactions,seee.g.[38]andreferencestherein.ItcomeswiththestructureǫνS ≡ǫνµαβS n n¯ where
T µ α β
n¯ isanotherlightlikefour-vectorsatisfyingn·n¯ = 1andsuchthatPµ = (P·n)n¯µ+(P·n¯)nµ,which
meansthatatransversetargetpolarizationisrequired.Thetotalfour-momentumbeingPν,werecover
fromthesumoverallpartonsthewell-knownmomentumconstraints
Ae,a(0,0)= C¯ (0)=0, Ae,a(0,0)= A (0)=1. (8)
1 a 2 a
aX=1,2 aX=q,G aX=1,2 aX=q,G
Thanks to the complete parametrization (4), we can easily compute the matrix elements of the
correspondingOAM tensors Lµνρ = rνTµρ −rρTµν. Because of the explicitfactors of r, the matrix
a a a
elementsofthegenericLFOAMtensorneedtobehandledwithcare[14,37]. Foralongitudinally
polarizedtarget,wefoundasimpleexpressionforthelongitudinalcomponentofOAM
ǫ ∂
hLai≡ Tαβ i hp′,S′|TNβ(0)|p,Si = Ae,a(0,0). (9)
L 2M2 " ∂∆ a # 4
α ∆=0
Similarly,thequarkandgluonspincontributionsSµνρ = 1ǫµνρσψγ γ ψandSµνρ = −2Tr Gµ[νAρ]
1 2 β 5 2 phys
canbeexpressedintermsofLµνρ andLµνρ,respectively.Wethenfound h i
4 5
1 1
hSqi≡ hP,S|1ǫ SNαβ(0)|P,Si=− Ae,1(0,0)−Ae,1(0,0) ,
L 2M2 2 Tαβ 1 2 4 5
h i (10)
1
hSGi≡ hP,S|1ǫ SNαβ(0)|P,Si=−Ae,5(0,0),
L 2M2 2 Tαβ 2 4
where Eq. (2) has been used to express Lµνρ in terms of Tµν. Adding together the spin and OAM
4 1
contributions,wenaturallyrecovertheJirelationfortotalangularmomentum[8]
hJqi=hSqi+hLqi= 1 Ae,1(0,0)+Ae,1(0,0) = 1 A (0)+B (0) ,
L L L 2 4 5 2 q q
h i h i (11)
hJGi=hSGi+hLGi= 1 Ae,2(0,0)+Ae,2(0,0) = 1[A (0)+B (0)].
L L L 2 4 5 2 G G
h i
Finally,combiningthefactthatthetotalangularmomentumis1/2andthemomentumconstraints(8),
werecoveralsotheanomalousgravitomagneticmomentsumrule[39,40]
[Ae,a(0,0)+Ae,a(0,0)−Ae,a(0,0)]= B (0)=0. (12)
4 5 2 a
aX=1,2 aX=q,G
POETIC2015
5 Link with measurableparton distributions
ThematrixelementsoftheEMTweareinterestedincaneasilybeexpressedintermsoftheGTMD
correlator[18]
hp′,S′|Tµν(0)|p,Si= d4kkνWµ . (13)
Z S′S
Byconsideringappropriateprojections[21,41,42],wecanattheendrelatesomeofthescalarfunc-
tionsappearinginourgenericparametrization(4)tomeasurablepartondistributions,likee.g.Gener-
alizedPartonDistributions(GPDs)accessedinexclusivescatterings[43]andTransverse-Momentum
dependentDistributions (TMDs) accessed in semi-inclusive scatterings[20]. The detailed relations
betweentheEMTscalarfunctionsandtwo-partonGPDsandTMDsofanytwistcanbefoundin[34].
Amongtheinterestingresults,letusjustmentionthatwederivedthefollowingsumrules
dxd2k kT2 Fa(x,k2)=0 (14)
Z T 2M2 T
aX=q,G
with Fa = f⊥a, f⊥a, f⊥a, f⊥a. They expressthe fact that the total transverse momentum (w.r.t. the
1T L 3T
targetmomentum)hastovanish.Theleading-twistrelation(i.e. theoneinvolvingtheSiversfunction
f⊥a) is known as the Burkardtsum rule [44, 45]. The other three are new and much harder to test
1T
experimentally,butitwouldbeveryinterestingtotestthemusingphenomenologicalmodels,Lattice
QCDandperturbativeQCD.
6 Conclusions
It is possible to define a gauge-invariantcanonical energy-momentumtensor once one relaxes the
assumption of strict locality. We argued that the canonical energy-momentum tensor can be con-
sideredasa physicalobjectanda priorimeasuredexperimentallyviaparticularmomentsof parton
distributionsextractedfromvariousphysicalprocesses.
We presentedherea completeparametrizationforthe matrix elementsof the genericlight-front
energy-momentumtensoranddiscussedsomeoftheconstraintsarisingfromlinearandangularmo-
mentum conservation. We showed that this energy-momentum tensor can be related to particular
moments of the generalized and transverse-momentumdependent parton distributions. Among the
interestingresults,werederivedinasimplerwaytheBurkardtsumruleandobtainedthreenewsum
rulesinvolvinghigher-twistdistributions,allexpressingbasicallytheconservationoftransversemo-
mentum. We expectin a near futureexciting new developmentsin these matters comingfrom new
experimental data obtained in existing and future facilities, and explicit investigations using phe-
nomenologicalmodels,LatticeQCDandperturbativeQCD.
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