Gluon polarization in the nucleon demystified

0.07 MB·English

by  Yoshitaka Hatta

Gluon polarization in the nucleon demystified
Yoshitaka Hatta
GraduateSchoolofPureandAppliedSciences, UniversityofTsukuba,
Tsukuba, Ibaraki305-8571, Japan
1
1
0 Abstract
2
Recently, X. Chen et al. proposed a new approach to the gauge invariant decomposition of the nucleon
n
a spinintohelicityandorbitalparts.Thekeyingredient intheirconstruction istheseparation ofthegauge
J field into “physical” and “pure gauge” parts. We suggest a simple separation scheme and show that the

1 resulting gluon helicity coincides withthefirstmomentofthe conventional polarized gluon distribution
3
measurable inhighenergyexperiments.

]
h
p
-
p
e
h Despite its intuitive clarity, the decomposition of the nucleon spin into the helicity and or-
[
bital angular momentum of quarks and gluons has remained one of the most elusive problems

1 inQCDspinphysics[1].Thecurrentunsatisfactorysituationmaybeepitomizedbythefollow-
v
ing dilemma: On one hand, continuous efforts have been made both at experimental facilities
9
8 and by the theorists to assess the gluon helicity contribution ∆G [2] defined as the first mo-
9
ment of the polarized gluon distribution.On the other hand, sincethe seminal work of Ji [3], it
5
. has been widely recognized by the community that the gluonic angular momentum cannot be
1
0 decomposed into helicity and orbital parts in a gauge invariant way. This implies that ∆G ex-
1
tracted from measurements and global QCD analyses has unfortunately no natural counterpart
1
: in Ji’s framework, making one wonder what exactly the physical meaning of ∆G is. The case
v
i ofthegluonorbitalangularmomentumisevenmurkiersincethereisnoknownwayofdirectly
X
measuringit, noris itsoperatorrepresentationavailable.
r
a Recently, however, the situation took an interesting turn when Chen et al. proposed a new,
complete decomposition of the nucleon spin [4,5]. The key ingredient in their construction is
theseparationofthetotalgaugefield into“physical”and “puregauge”parts
Aµ = Aµ +Aµ ,
phys pure
Fµν = ∂µAν −∂νAµ +ig[Aµ ,Aν ] = 0, (1)
pure pure pure pure pure
which transformdifferentlyundergaugetransformations
µ µ
A → U A U ,
phys † phys
i
Aµ → U Aµ U − U ∂µU (2)
pure † pure g †
The separation (1) is not unique, and accordingly, the gluon helicity contribution is scheme
PreprintsubmittedtoElsevier 1February2011
dependent. Chen et al. imposed a subsidiary condition that can be used to construct Aµ per-
phys
turbatively, and found that the corresponding gluon helicity indeed differs from ∆G [6]. [See,
also,[7].]Whiletheirschemehassomeattractivephysicalfeatures,howtotesttheirpredictions
inexperimentiscurrentlyunknown.Theexperimentalobservabilityisaprerequisiteforagood
definitionofthegluonhelicity.Onethenasksthequestion:“Isthereaseparationscheme(1)in
which the gluon helicity coincides with ∆G?” In this letter we answer this question positively
in the hope that the conflicting opinions about the nature of the gluon helicity are reconciled
witheach other.
The original proposal by Chen et al. [4,5] achieves a complete decomposition of the QCD
angular momentum operator into quarks’ and gluons’ helicity and orbital angular momentum.
This was further elaborated by Wakamatsu [8,9] where the covariant generalization of the de-
compositionwas derived.Theresult is[9]
1
Mµνλ = ǫµνλσψ¯γ γ ψ, (3)
quark spin 2 5 σ
−
Mµνλ =ψ¯γµ(xνiDλ −xλiDν)ψ, (4)
quark orbit
−
Mµνλ =FµλAνa −FµνAλa , (5)
gluon spin a phys a phys
−
Mµνλ =Fµα xν(Dλ Aphys) −xλ(Dν Aphys)
gluon orbit a pure α a pure α a
−
(cid:16) +(D Fαµ) (xνAλa −(cid:17) xλAνa ). (6)
α a phys phys
whereDµ ≡ ∂µ+igAµ anda,b = 1,2,··· ,8arethecolorindices.Weusetheconvention
pure pure
ǫ = +1. The second term on the right hand side of (6) is gauge–invariant on its own, and
0123
Chen et al. included it in the quark–orbital part. (This would result in the change Dν → Dν
pure
in (4).) Following Wakamatsu [8], we have relocated it to the gluon–orbital part. With this
modification the quark part agrees with Ji’s decomposition and can be extracted from GPD
analyses [3]. The decomposition of the gluon spin into the helicity (5) and orbital (6) parts
has been made possible at the cost of introducing nonlocality: In general, Aµ and Aµ are
phys pure
nonlocallyrelated tothetotalAµ.
Let us focus on the gluon helicity operator (5). We go to the infinite momentum frame and
use the light–cone coordinates x = 1 (x0 ±x3). In the framework of Chen et al., the gluon
± √2
helicity fraction of the nucleon spin is given by the matrix element of the µνλ = +12 tensor
component
−1
hPS|F+µ(0)ǫ+ Aνa (0)|PSi. (7)
2P+ a −µν phys
Ontheotherhand,theconventionalandexperimentallyaccessiblegluonhelicityisgivenby
thefirst momentofthepolarized gluondistribution(see, e.g., [10,11])
1
∆G= dx ∆g(x )
B B
0
Z
1
= ∞ dy ǫ(y )hPS|F+µ(0)
4P+ − − a
Z−∞
0
×P exp −ig A+(y )dy ǫ+ F+ν(y )|PSi, (8)
′− ′− −µν b −
(cid:18) Zy− (cid:19)ab
2
where x is the usual Bjorken variable and the Wilson line is in the adjoint representation. (P
B
denotes pathordering.)
If we insist that the two definitions (7) and (8) are equivalent, we must have that, using the
notationxµ = (x ,~x) with~x = (x+,x1,x2),
−
Aµa (x) =??−1 ∞ dy ǫ(y −x )P exp −ig x− A+(y ,~x)dy F+µ(y ,~x). (9)
phys 2 − − − y− ′− ′−! b −
Z−∞ Z ab
Does this identification make sense? The right hand side obeys the gauge transformation law
(2) as expected for Aµ , but it is far from obvious that the difference Aµ = Aµ −Aµ is
phys pure phys
pure gauge. Remarkably, however, there exists a special, but very simple scheme of separation
(1)in which(9)becomes an identityratherthan adefinition.1
Inorder tofind sucha scheme,wefirst observethat (9)immediatelyimpliesthat
A+ = 0. (10)
phys
Thus wemaywrite, writingfields as matricesin theadjointrepresentation,
i i
A+ = A+ = − VW∂+(VW) = ∂+(VW)(VW) , (11)
pure g † g †
where
x−
+
V(x) = P exp −ig A (x ,~x)dx ,
′− ′−
!
Z±∞
~x
W(~x) = P exp −ig A~(±∞,~x)·d~x . (12)
′ ′
~n !
Z∞
Note that W is evaluated at x = ±∞ where the plus (minus) sign corresponds to the choice
−
x = +∞ (x = −∞) in the lower limit of the integration in V. The path to spatial infinity
− −
(denoted as ‘~x = ∞~n’ with ~n being a constant vector) is arbitrary assuming that the field
strengthvanishesat x = ±∞.
−
Promoting(11)toa four–dimensionalrelation, wedefine
i
Aµ ≡ − VW∂µ(VW) , (13)
pure g †
which guarantees thatFµν = 0, and
pure
Aµ ≡ Aµ −Aµ . (14)
phys pure
In orderforAµ totransformaccording to(2)undergaugetransformation,werequirethat
pure
lim ∂ U(x ,~x) = 0, (15)
µ −
x−=
~x ±∞~n
→∞
1 Wakamatsu [9]discussed the equivalence ofthe matrix elements (7)and (8) in the light–cone gauge.
Hereweintendtoshow(9)asanoperator identityingenericgauges.
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that is, we allow only for global gauge rotations at (x ,~x) = (±∞,∞~n). For consistency,
−
Aµ should vanish there, and this is in fact already implied by (13). Except for this minor
pure
qualification,ourseparationschemeisindependentofthegaugechoice.
Still,itwillbeveryconvenientinthefollowingtoconsiderthelight–conegaugewhichhasa
special status inourschemeand whichcan beaccessed by settingU = VW (consistentlywith
(15)). Denotingfields inthelight–conegaugewithatilde,wefind
A˜µ =(VW) Aµ VW ,
phys † phys
A˜µ =0, (16)
pure
so that A˜µ = A˜µ in thisgauge.Theresidual (x –independent)gaugesymmetryin thelight–
phys −
cone gauge is essentially contained in W(~x). It may seem more natural to let A˜µ carry these
pure
degrees offreedom.However,wehaveabsorbedtheminA˜µ forourpurpose.By usingthese
phys
degrees offreedom, onecan fix theboundary conditionforA˜µ = A˜µ as x → ±∞.
phys −
Wearenowready to prove(9). Thelast factorcan bewrittenas, suppressingcolorindices,
x−
P exp −ig A+(y ,~x)dy F+µ(y ,~x)=VW(x)(VW) (y ,~x)F+µ(y ,~x)
′− ′− − † − −
y− !
Z
=VW(x)F˜+µ (y ,~x)
phys −
∂
=VW(x) A˜µ (y ,~x), (17)
∂y phys −
−
where in the second equality we have used the fact that F˜µν = F˜µν in the light–cone gauge.
phys
[Remember that matrices are in the adjoint representation.] The right hand side of (9) then
becomes
1 ∂
− dy ǫ(y −x )VW(x) A˜µ (y ,~x)
2 − − − ∂y phys −
Z −
1
= VW(x)A˜µ (x)− VW(x) A˜µ (∞,~x)+A˜µ (−∞,~x)
phys 2 phys phys
1 (cid:16) (cid:17)
= Aµ (x)− VW(x) A˜µ (∞,~x)+A˜µ (−∞,~x) , (18)
phys 2 phys phys
(cid:16) (cid:17)
where we integratedby parts. (18) differs from Aµ by thesurface terms at x = ±∞. How-
phys −
ever, these surface terms can be consistently eliminated. To see this, suppose that (9) is valid.
Then
1
Aµ (∞,~x) = dy P exp −ig ∞A+(y ,~x)dy F+µ(y ,~x),
phys 2 Z − (cid:18) Zy− ′− ′−(cid:19) −
1
Aµ (−∞,~x) = − dy P exp −ig −∞A+(y ,~x)dy F+µ(y ,~x). (19)
phys 2 Z − (cid:18) Zy− ′− ′−(cid:19) −
Goingto thelight–conegauge,weget
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A˜µ (∞,~x)=(VW) (∞,~x)Aµ (∞,~x)
phys † phys
1
= W (~x) dy P exp −ig −∞A+(y ,~x)dy F+µ(y ,~x),
† − ′− ′− −
2 Z (cid:18) Zy− (cid:19)
µ
=−W (~x)A (−∞,~x)
† phys
=−A˜µ (−∞,~x), (20)
phys
where, fordefiniteness, wehavechosen x = −∞ as thelowerlimitoftheintegrationin (12).
−
(The other case x = ∞ is a trivial modification.) Therefore the surface terms in (18) cancel
−
and thiscompletestheproofof(9).
Notethatthecancellationwehavejustobservedisnothingbutthewell–knownantisymmet-
ric boundary condition of the gauge field in the light–cone gauge. Thus in the above proof we
have implicitly chosen this boundary condition by adjusting W(~x). This is in accordance with
thesignfunctionǫ(y )in(8),orequivalently,theprincipalvalueprescriptionforthe1/x pole
− B
in ∆g(x )
B
1
∞ dx p.v. eiy−P+xB = iπǫ(y ). (21)
B −
x
Z−∞ (cid:18) B(cid:19)
Different prescriptionsfor the 1/x pole lead to different boundary conditionsfor A˜µ = A˜µ
B phys
at x → ±∞, just like the prescription for the 1/k+ pole of the gluon propagator in the light–
−
cone gauge [12]. It does not matter which prescription oneuses, sincethe difference is propor-
tional to δ(x ) and vanishes under the assumption that the x –integral converges as x → 0.
B B B
However, it does change the appearance of ∆G. Had we chosen a different prescription, say,
1/(x −iǫ), we would have obtained a formula for ∆G similar to (8), but with the step func-
B
tion2θ(y )inplaceofthesignfunctionǫ(y ).Inthelight–conegauge,thiscorrespondstothe
− −
advanced boundary condition A˜µ(∞,~x) = 0. For each different prescription,the surface terms
willbedifferent. Buttheyalways vanishunderthecorrespondingboundarycondition.
In conclusion,thegauge–invariantdecompositionofthegluoniccontributionto thenucleon
spin into helicity and orbital parts is not possible if one restricts to local operators [3]. Once
one allows for nonlocal operators, it becomes possible [4]. We have shown that the traditional
definitionofthegluonhelicity,∆G,canbenicelyaccommodatedinthislatterapproach,thereby
dispellinganyconcernsaboutthephysicalmeaningof∆G.Afterall,∆Gismeasurable,gauge
invariant, and meets the criterion by Chen et al. for a proper definition of the gluon helicity in
QCD.
By using the explicit relation between Aµ and Aµ, one can write down the all–order ex-
phys
pression for the gluon orbital angular momentum (6) as well. While it is measurable as the
difference between thetotal gluon contribution(from the GPD) and ∆G, more direct access to
theorbitalcomponentwouldofcoursebedesirable.
Acknowledgments—I am grateful to Masashi Wakamatsu for keeping me interested in this
problem through his works, and for valuable discussions. I also thank Kazuhiro Tanaka for
helpful correspondence. This work is supported by Special Coordination Funds for Promoting
Science and TechnologyoftheMinistryofEducation,theJapaneseGovernment.
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