Probing the transversity spin structure of a nucleon in neutrino-production of a charmed meson

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by  B. Pire

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Probing the transversity spin structure of a nucleon in neutrino-
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production of a charmed meson
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F
2 B.Pire1,a,L.Szymanowski2,b,andJ.Wagner2,c

1Centre de physique théorique, École Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau,
]
h France
p 2NationalCentreforNuclearResearch(NCBJ),Warsaw,Poland
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p
e
h Abstract.IncludingO(m )termsinthecoefficientfunctionsand/orO(m )twist3contri-
c D
[
butionsintheheavymesondistributionamplitudesleadstoanon-zerotransverseampli-

2 tudeforexclusiveneutrinoproductionofaDpseudoscalarcharmedmesononanunpolar-
v izedtarget. WeworkintheframeworkofthecollinearQCDapproachwherechiral-odd
6 transversitygeneralizedpartondistributions(GPDs)factorizefromperturbativelycalcu-
6 lablecoefficientfunctions.
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1 Introduction
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v The now well established framework of collinear QCD factorization [1–3] for exclusive reactions
i mediatedbyahighlyvirtualphotoninthegeneralizedBjorkenregimedescribeshadronicamplitudes
X
using generalized parton distributions (GPDs) which give access to a 3-dimensional analysis [4] of
r
a theinternalstructureofhadrons. Neutrinoproductionisanotherwaytoaccess(generalized)parton
distributions [5]. Although neutrino induced cross sections are orders of magnitudes smaller than
thoseforelectroproductionandneutrinobeamsaremuchmoredifficulttohandlethanchargedlepton
beams, they havebeen veryimportant toscrutinizethe flavorcontent ofthe nucleonand theadvent
ofnewgenerationsofneutrinoexperimentsopensnewpossibilities. Inparticular,theflavorchanging
character of the electroweak current allows charmed quark to be produced in processes involving
light quark partonic distributions [6]. This in turn allows helicity flip hard amplitudes to occur at
the O(mc) level where Q is the typical large scale allowing QCD collinear factorization. Such a
Q
coefficient function has to be attached to a chiral-odd generalized parton distribution, the elusive
transversity GPDs [7–9]. The transverse character of these GPDs select the transverse polarization
oftheW−boson,whichphenomenologicallyallowsaseparationofthisinterestingamplitudethrough
theazimuthaldistributionofthefinalstateparticles[6].
ae-mail:[email protected]
be-mail:[email protected]
ce-mail:[email protected]
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2 Kinematics
Fordefiniteness,weconsidertheexclusiveproductionofapseudoscalarD−mesonthroughthereac-
tion(seeFig. 1):
ν(k)N(p ) → l−(k(cid:48))D+(p )N(p ), (1)
l 1 D 2
where N is a proton or a neutron, in the kinematical domain where collinear factorization leads to
a description of the scattering amplitude in terms of nucleon GPDs and the D−meson distribution
amplitude,withthehardsubprocesses:
W+(ε,q)d → D+(p )d. (2)
D
Ourkinematicalnotationsareasfollows(mand M arethenucleonand D−mesonmasses, m will
D c
denotethecharmedquarkmass):
q=k−k(cid:48) ; Q2 =−q2 ; ∆= p −p ; ∆2 =t;
2 1
1m2−∆2/4 ∆µ 1m2−∆2/4 ∆µ
pµ =(1+ξ)pµ+ T nµ− T ; pµ =(1−ξ)pµ+ T nµ+ T ; (3)
1 2 1+ξ 2 2 2 1−ξ 2
Q2 M2 −∆2
qµ =−2ξ(cid:48)pµ+ nµ ; pµ =2(ξ−ξ(cid:48))pµ+ D Tnµ−∆µ ,
4ξ(cid:48) D 4(ξ−ξ(cid:48)) T
with p2 = n2 = 0 and p.n = 1. As in the double deeply virtual Compton scattering case [10], it is
meaningfultointroducetwodistinctmomentumfractions:
(p −p ).n q.n
ξ =− 2 1 , ξ(cid:48) =− . (4)
2 2
Neglectingthenucleonmassand∆ ,theapproximatevaluesofξandξ(cid:48)are
T
Q2+M2 Q2
ξ ≈ D , ξ(cid:48) ≈ . (5)
4p .q−Q2−M2 4p .q−Q2−M2
1 D 1 D
Tounifythedescriptionofthescalingamplitude,wedefineamodifiedBjorkenvariable xD ≡ Q2+MD2
whichallowstoexpressξandξ(cid:48)inacompactform: B 2p1.q
xD x
ξ ≈ B , ξ(cid:48) ≈ B . (6)
2−xD 2−xD
B B
Ifthemesonmassistherelevantlargescale(forinstanceinthelimitingcasewhereQ2vanishesasin
thetimelikeComptonscatteringkinematics[11]):
τ M2
Q2 →0 ; ξ(cid:48) →0; ξ ≈ ; τ= D . (7)
2−τ s −m2
WN
3 The transverse amplitude
IntheFeynmangauge,thenon-vanishingm −dependentpartoftheDiractraceinthehardscattering
c
partdepictedinFig. 1areads:
m −g 2(Q2+M2) m 1
Tr[σpiγνpˆ γ5γν(cid:48) cεˆ(1−γ5) νν(cid:48)]= D ε [(cid:15)µpin+igµi] c , (8)
D D D ξ µ ⊥ D D
1 2 1 2
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Figure1.Feynmandiagramsforthefactorizedamplitudefortheν N →µ−D+N(cid:48)ortheν N →µ−D0N(cid:48)process
µ µ
involvingthequarkGPDs;thethicklinerepresentstheheavyquark.IntheFeynmangauge,diagram(a)involves
convolutionwithboththetransversityGPDsandthechiralevenones,whereasdiagram(b)involvesonlychiral
evenGPDs.
where ε is the polarization vector of the W±boson (we denote pˆ = p γµ for any vector p). The
µ
fermionictracevanishesforthediagramshownonFig. 1bthankstotheidentityγρσαβγ = 0. The
ρ
denominatorsofthepropagatorsread:
Q2 Q2+M2
D =k2−m2+i(cid:15) = (x+ξ−2ξ(cid:48))−m2+i(cid:15) = D(x+ξ)−Q2−m2+i(cid:15), (9)
1 c c 2ξ(cid:48) c 2ξ c
Q2+m2
D =k2+i(cid:15) =z¯[z¯m2 + D(x−ξ)+i(cid:15)],
2 g D 2ξ
where k (k ) is the heavy quark (gluon) momentum. The transverse amplitude is then written as
c g
(τ=1−i2):
 
TT = iC√ξ2((mQc2−+2MMD2D))N¯(p2)HTφiσnτ+H˜Tφm∆2Nτ +EφTnˆ∆τ2+mN2ξγτ −E˜φTmγτNN(p1), (10)
withC = 2πC α V ,intermsoftransverseformfactorsthatwedefineas:
3 F s dc
(cid:90) φ(z)dz(cid:90) Fd(x,ξ,t)dx
Fφ = f T , (11)
T D z¯ (x−ξ+βξ+i(cid:15))(x−ξ+αz¯+i(cid:15))
where Fd is any d-quark transversity GPD, α = 2ξMD2 , β = 2(MD2−m2c) and we shall denote E¯φ =
T Q2+M2 Q2+M2 T
D D
ξEφ −E˜φ . Inthefollowing,weshallputβto0.
T T
TheprefactorinEq.(10)showsthetwosourcesofthetransverseamplitude:m signalsthecontri-
c
butionfromthehelicitychangingpartoftheheavyquarkpropagator,while M signalsthecontribu-
D
tionfromthetwist3heavymesondistributionamplitudewhichweparametrize(omittingtheWilson
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lines)as:
f (cid:90) 1
(cid:104)0|c¯(y)γ5d(−y)|D−(PD)(cid:105)=−i 4DMD dzei(2z−1)PD.yφ(Ds)(z), (12)
0
andforsimplicityweidentifyφswiththeleadingtwist2pseudoscalarcharmedmesonDAφ defined
D
as:
f (cid:90) 1
(cid:104)0|c¯(y)γµγ5d(−y)|D−(PD)(cid:105)=−i 4DPµD dzei(2z−1)PD.yφD(z).
0
4 The azimuthal dependence of neutrinoproduction.
Thedependenceofaleptoproductioncrosssectiononazimuthalanglesisawidelyusedwaytoana-
lyzethescatteringmechanism.Thisprocedureishelpfulassoonasonecandefineanangleϕbetween
aleptonicandahadronicplane,asfordeeplyvirtualComptonscattering[12]andrelatedprocesses.
Intheneutrinocase,itreads:
d4σ(νN →l−N(cid:48)D)
= (13)
dx dQ2dtdϕ
B √
(cid:110) 1+ 1−ε2 √ √ √ (cid:111)
Γ˜ σ + εσ + ε( 1+ε+ 1−ε)(cosϕReσ +sinϕImσ ) ,
−− 00 −0 −0
2
with
G2 1 1 1 Q2
Γ˜ = F (cid:113) ,
(2π)416xB 1+4x2m2/Q2(s−m2N)21−(cid:15)
B N
andthe“cross-sections”σ = (cid:15)∗µW (cid:15)ν areproductofamplitudesfortheprocessW((cid:15))N → DN(cid:48),
lm l µν m l
averaged (summed) over the initial (final) hadron polarizations. In the anti-neutrino case, one gets
√ √ √
a√similar ex√pression√with σ−− → σ++ , σ−0 → σ+0, 1+ 1−ε2 → 1− 1−ε2 and 1+ε+
1−ε→ 1+ε− 1−ε.Weusethestandardnotationsofdeepexclusiveleptoproduction,namely
y= p .q/p .kand(cid:15) (cid:39)2(1−y)/[1+(1−y)2]. Theazimuthalangleϕisdefinedintheinitialnucleon
1 1
restframeas:
(cid:126)q·[((cid:126)q×(cid:126)p )×((cid:126)q×(cid:126)k)]
sinϕ= D , (14)
|(cid:126)q||(cid:126)q×(cid:126)p ||(cid:126)q×(cid:126)k|
D
whilethefinalnucleonmomentumliesinthexzplane(∆y =0).
Thequantityσ isdirectlyrelatedtotheobservables<cosϕ>and< sinϕ>through
−0
(cid:82) √ √ √
cosϕdϕd4σ ε( 1+ε+ 1−ε)Reσ
<cosϕ> = (cid:82) = √ −0 ,
dϕd4σ 2(cid:15)σ +(1+ 1−ε2)σ
00 −−
(cid:82) √ √ √
sinϕdϕd4σ ε( 1+ε+ 1−ε)Imσ
< sinϕ> = (cid:82) = √ −0 . (15)
dϕd4σ 2(cid:15)σ +(1+ 1−ε2)σ
00 −−
Estimatingthecountingratesandtheangularobservablesdefinedaboveisinprogress[13]. Asa
firststep,wecalculateσ whichisbilinearintransversityquarkGPDs. Atzerothorderin∆ ,σ
−− T −−
reads:
4ξ2C2(m +2M )2(cid:26) ξ2 (cid:27)
σ = c D (1−ξ2)|Hφ|2+ |E¯φ|2−2ξRe[HφE¯φ∗] . (16)
−− (Q2+M2)2 T 1−ξ2 T T T
D
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10(cid:45)4
s = 20 GeV2
10(cid:45)5 y = 0.7
(cid:68)
4V
e
G
b (cid:144)10(cid:45)6
p
(cid:64)
dt
2
Σ Q 10(cid:45)7
d d
B
x
d
10(cid:45)8
10(cid:45)9
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Q2 GeV2
(cid:64) (cid:68)
Figure2. Contributionofσ tothedifferentialcrosssection dσ ofneutrino-productionofaD+mesonon
−− dxBdQ2dt
aproton(dashedline)oraneutron(solidline)ats=20GeV2,y=0.7andt=t ,asafunctionofQ2.
0
Using the model of Ref [14] for the D+ meson distribution amplitude and the parametrization of
the dominant transversity GPD H (x,ξ,t) from Ref [15] (and neglecting for the time being other
T
chiral-oddGPDscontributions),wecomputethecontributiontothedifferentialcrosssectiongivenin
Eq.(13)integratedoverϕ. TheresultisshowninFig. 2asafunctionofQ2 for s=20GeV2,y=0.7
andt = t . Sincetheprocessselectsthed−quarkcontribution,theprotonandneutrontargetcases
min
allowtoaccess Hd and Hu respectively. Althoughsmall,thecross-sectionsareofthesameorderof
T T
magnitudeasthosefortheneutrinoproductionofπor D mesonsestimatedin[5]. Thisshowsthat
s
theseprocessesshouldbemeasurableinintenseneutrinobeamfacilities.
Let us remind the reader that we allow Q2 to be quite small since the hard scale governing our
processisM2 +Q2.
D
5 Conclusion.
Collinear QCD factorization has allowed us to calculate neutrino production of D−mesons in terms
ofGPDs. Gluonandbothchiral-oddandchiral-evenquarkGPDscontributetotheamplitudefordif-
ferentpolarizationstatesoftheW± boson. Theazimuthaldependenceofthecrosssectionallowsto
separatedifferentcontributions. Plannedhighenergyneutrinofacilities[16]whichhavetheirscien-
tificprogramorientedtowardtheunderstandingofneutrinooscillationsorelusiveinertneutrinosmay
thusallow-withoutmuchadditionalequipment-someimportantprogressintherealmofhadronic
physics.
ThisworkwaspartiallysupportedbytheCOPIN-IN2P3AgreementandbythefrenchgrantANR
PARTONS(GrantNo. ANR-12-MONU-0008-01);L.SzwaspartiallysupportedbygrantofNational
ScienceCenter,Poland,No. 2015/17/B/ST2/01838.
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