Double handbag description of proton-antiproton annihilation into a heavy meson pair

30 Pages·2013·0.24 MB·English

Double handbag description of proton-antiproton annihilation
into a heavy meson pair
Alexander T. Goritschnig∗ and Bernard Pire†
Centre de Physique The´orique, E´cole Polytechnique, CNRS, 91128 Palaiseau, France
Wolfgang Schweiger‡
3
1
Institut fu¨r Physik, Universita¨t Graz, A-8010 Graz, Austria
0
2
(Dated: October22,2013)

t
c
O Abstract

1
We propose to describe the process pp¯ D0D0 in a perturbative QCD motivated framework where
2
→

] a double-handbag hard process udu¯d¯ c¯c factorizes from transition distribution amplitudes, which are
h →
p quasiforward hadronic matrix elements of Ψ Ψ Ψ operators, where q denotes light quarks and c denotes
q q c
-
p
e theheavy quark. Weadvocate thatthecharm-quark massactsasthelarge scaleallowingthisfactorization.
h
[ Wecalculate thisprocess inthesimplifiedframeworkofthescalardiquark modelandpresent theexpected

2
crosssectionsforthePANDAexperimentatGSI-FAIR.
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0 PACSnumbers:13.75.-n,13.85.Fb,25.43.+t
8
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∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
1
I. INTRODUCTION
The collinear factorization framework allows us to calculate a number of hard exclusive am-
plitudes in terms of perturbatively calculable coefficient functions and nonperturbative hadronic
matrix elements of light-cone operators. The prime example is the calculation of the deeply vir-
tual Compton-scattering amplitude in the handbag approximation with generalized parton distri-
butions, nonforward matrix elements of a quark-antiquark nonlocal operator Ψ(z)Ψ(0) between
an incoming and an outgoing baryon state. Strictly speaking, this description is only valid in
a restricted kinematical region, called the generalized Bjorken scaling region, for a few specific
reactions, and in the leading-twist approximation. It is, however, suggestiveto extend this frame-
work to the description of other reactions where the presence of a hard scale seems to justify
the factorization of a short-distance dominated partonic subprocess from long-distance hadronic
matrix elements. Such an extension has, in particular, been proposed in Ref. [1] for the reaction
pp¯ Λ Λ¯ with nucleon to charmed baryon generalized parton distributions. The extension of
c c
→
thecollinearfactorizationframeworktothebackwardregionofdeeplyvirtualCompton-scattering
anddeepexclusivemesonproduction[2,3]leadstothedefinitionoftransitiondistributionampli-
tudes (TDAs) as nonforward matrix elements of a three-quark nonlocal operator Ψ(z)Ψ(y)Ψ(0)
between an incoming and an outgoing state carrying a different baryon number. Here, too, this
descriptionis likelyto be validin arestricted kinematicalregion,forafew specificreactions, and
in the leading twist approximation. We propose here to extend the approach of Ref. [1] to the
reaction pp¯ D0D0 which will be measured with the PANDA [4] detector at GSI-FAIR. For
→
this process the baryon numberexchanged in the t-channel implies that hadronic matrix elements
with Ψ(z)Ψ(y)Ψ(0) operators enter the game. Let us stress that we have no proof of the validity
of this approximation but take it as an assumption to be confronted with experimental data. For
this approach to be testable, one needs to model the occurring nucleon to charmed meson TDAs.
In contrast to the N π TDAs, which have been much discussed [5], we do not have any soft
→
mesonlimittonormalizetheseTDAs. Wewillratherusean overlaprepresentationin thespiritof
Ref. [6].
2
x
p(p,µ) D0 p′!
θ
z
D0 q′! p¯(q,ν)
FIG.1: Kinematicsofpp¯ D0D0 inthesymmetricCMS.
→
II. HADRONKINEMATICS
The kinematical situation for pp¯ D0D0 scattering is sketched in Fig. 1. The momenta and
→
helicities of the incoming proton and antiproton are denoted by p, µ and q, ν and the momenta
of the outgoing D0 and D0 by p′ and q′, respectively. The mass of the proton is denoted by m
and that of the D0 by M. We choose a symmetric center-of-momentum system (CMS) in which
the longitudinal direction is defined by the average momentum of the incoming proton and the
outgoing D0, respectively. The transverse momentum transfer is symmetrically shared between
theincomingand outgoinghadrons.
Inlight-conecoordinatesthehadronicmomentaare parameterized as follows,
m2 +∆2/4 ∆ M2 +∆2/4 ∆
p = (1+ξ)p¯+, ⊥ , ⊥ , p′ = (1 ξ)p¯+, ⊥ , + ⊥ ,
2(1+ξ)p¯+ − 2 − 2(1 ξ)p¯+ 2
(cid:20) (cid:21) (cid:20) − (cid:21)
m2 +∆2/4 ∆ M2 +∆2/4 ∆
q = ⊥ , (1+ξ)p¯+, + ⊥ , q′ = ⊥ , (1 ξ)p¯+, ⊥ , (1)
2(1+ξ)p¯+ 2 2(1 ξ)p¯+ − − 2
(cid:20) (cid:21) (cid:20) − (cid:21)
wherewehaveintroducedsumsanddifferences ofthehadronmomenta,
1 1
p¯ := (p+p′), q¯ := (q +q′) and ∆ := p′ p = q q′. (2)
2 2 − −
The minus momentum components can be obtained by using the on-mass shell conditions p2 =
q2 = m2 and p′2 = q′2 = M2. Theskewness parameter ξ givestherelativemomentumtransferin
3
theplusdirection,i.e.,
p+ p′+ ∆+
ξ := − = . (3)
p+ +p′+ −2p¯+
TheMandelstamvariable sis givenby
s = (p+q)2 = (p′ +q′)2 . (4)
InordertoproduceaD0D0pair,smustbelargerthan4M2. TheremainingMandelstamvariables,
tand u,read
t = ∆2 = (p′ p)2 = (q q′)2 (5)
− −
and
u = (q′ p)2 = (p′ q)2 , (6)
− −
sothat s+t+u = 2M2 +2m2. Forlaterconveniencewealsointroducetheabbreviations
Λ := 1 4m2/s and Λ := 1 4M2/s. (7)
m M
− −
p p
Forfurtherrelationsbetween thekinematicalquantities, seeAppendixA.
III. DOUBLEHANDBAGMECHANISM
The double handbag mechanism which we use to describe pp¯ D0D0 is shown in Fig. 2.
→
It is understood that the proton emits an S[ud] diquark with momentum k and the antiproton a
1
p¯(q, ν) D0 q
′
!
S[ud](k2) c k2′, λ′2
!
S[ud](k1) c¯ k1′, λ′1!
p(p, µ) D0 p
′
!
FIG.2: Thehandbag contribution totheprocess pp¯ D0D0. Themomentaandhelicities ofthebaryons
→
andquarksarespecified.
S[ud]-diquark with momentumk . They undergo a scattering witheach other, i.e. they annihilate
2
4
in ourcase into a gluon which subsequently decays into the heavy c¯c pair. Thoseproduced heavy
partons, characterized by k′, λ′ and k′, λ′, are reabsorbed by the remnants of the proton and the
1 1 2 2
antiproton to form the D0 and the D0, respectively. One could, of course, also think of vector-
diquark configurations in the proton and V [ud]V [ud] annihilation to produce the c¯c pair. But in
common diquark models of the proton it is usually assumed that the probability to find a V [ud]
diquark is smaller than the one for the S[ud] diquark. Further suppression of V [ud] diquarks as
comparedtoS[ud]diquarksoccursinhardprocessesviadiquarkformfactorsatthediquark-gluon
vertices[7]. WethusexpectthatourfinalestimateoftheD0D0crosssectionwillnotbedrastically
altered by theinclusionofvector-diquark contributionsand we stickto thesimplerscalardiquark
model.
The whole hadronic four-momentum transfer ∆ is also exchanged between the active partons
inthepartonicsubprocess
S[ud](k )S[ud](k ) c¯(k′,λ′)c(k′,λ′). (8)
1 2 → 1 1 2 2
In Eq. (8) we neglect the mass of the S[ud] (anti)diquark, but take into account the heavy (anti-)
charm-quark mass m . In order to produce the heavy c¯c pair, the Mandelstam variable sˆ of the
c
partonicsubprocesshas to be
sˆ 4m2, (9)
≥ c
where 4m2 6.5GeV2. We have taken the (central) value for the charm-quark mass from the
c ≈
ParticleDataGroup[8], whichgivesm = 1.275 0.025GeV. Thus,theheavy-quarkmassm
c c
±
isanaturalintrinsichardscalewhichdemandsthattheintermediategluonhastobehighlyvirtual.
This allows us to treat the partonic subprocess perturbatively, even at small t, by evaluating the
−
corresponding Feynman diagram. All the other non-active partons inside the parent hadrons are
unaffected bythehard scatteringand thusact as spectators.
Forthedoublehandbag mechanismthehadronic pp¯ D0D0 amplitudecan bewrittenas
→
d4z d4z
M = d4k¯ θ(k¯+) 1 eik¯1z1 d4k¯ θ(k¯−) 2 eik¯2z2
µν 1 1 (2π)4 2 2 (2π)4
Xa(i′) Xα′i Z Z Z Z
×hD0 : p′|T Ψca′1α′1(−z1/2)ΦaS1[ud](+z1/2)|p : p,µi H˜a(i′)α′i(k¯1,k¯2) (10)
D0 : q′ ΦS[ud]†(+z /2)Ψ¯c ( z /2) p¯: q,ν ,
×h |T a2 2 a′2α′2 − 2 | i
5
where the assignment of momenta, helicities, etc., can be seen in Fig. 2. a(′) and α′ denote color
i i
and spinor indices, respectively. In analogy to the hadronic level we have introduced the average
partonic momenta k¯ := (k +k′)/2, i = 1,2, of the active partons. We note once more that
i i i
thefullhadronicmomentumtransferisalsotransferred between theactivepartons,i.e. k k′ =
1 − 1
p p′ = k′ k = q′ q. Thehardscatteringkernel,denotedby H˜ (k¯ ,k¯ ),describesthehard
− 2− 2 − a(i′)α′i 1 2
S[ud]S[ud] c¯c subprocess. The soft part of the p D0 transition is encoded in the Fourier
→ →
transform of a hadronic matrix element which is a time-ordered, bilocal product of a quark and a
diquarkfield operator:
d4z
(2π)14eik¯1z1hD0 : p′|T Ψca′1α′1(−z1/2)ΦaS1[ud](+z1/2)|p : p,µi. (11)
Z
In Eq. (11) ΦS[ud](+z /2) takes out an S[ud]diquark from theproton state p : p,µ at the space-
1
| i
timepointz /2. TheS[ud]diquarkthentakespartinthehardpartonicsubproces. The Ψc( z /2)
1 1
−
reinsertsthec¯quarkat z /2intotheremnantoftheprotonwhichgivesthedesiredfinalhadronic
1
−
D0 state D0 : p′ . Atthisstagetheappropriatetime-orderingofthequarkfieldoperators(denoted
| i
by thesymbol )has to be taken intoaccount. The remnant oftheproton, which does notpartic-
T
ipate in the hard partonic subprocess, constitutes the spectatorsystem. For the p¯ D0 transition
→
wehavetheFouriertransform
d4z
(2π)24eik¯2z2hD0 : q′|T ΦaS2[ud]†(+z2/2)Ψca′2α′2(−z2/2)|p¯: q,νi, (12)
Z
whichcan beinterpreted inaway analogoustoEq.(11). The pp¯ D0D0 amplitude(10)isthus
→
a convolution of a hard scattering kernel with hadronic matrix elements Fouriertransformed with
respect totheaveragemomenta k¯ and k¯ oftheactivepartons.
1 2
Fortheactivepartonswecan nowintroducethemomentumfractions
k+ k′+
x := 1 and x′ := 1 . (13)
1 p+ 1 p′+
Forlaterconveniencewealso introducetheaverage fraction
k+ +k′+ k¯+
x¯ = 1 1 = 1 , (14)
1 p+ +p′+ p¯+
whichis related to x and x′ by
1 1
x¯ +ξ x¯ ξ
x = 1 and x′ = 1 − , (15)
1 1+ξ 1 1 ξ
−
respectively.
6
As for the processes in Refs. [1, 9], due to the large intrinsic scale given by the heavy quark
mass m , the transverse and minus (plus) components of the active (anti)parton momenta in the
c
hard scattering kernel H˜ are small as compared to their plus (minus) components. Thus, the
parton momenta can be replaced by vectors lying in the scattering plane formed by the parent
hadron momenta. For this assertion one only has to make the physically plausible assumptions
thatthemomentaarealmoston mass-shellandthattheirintrinsictransversecomponents[divided
by the respective momentumfractions (15)] are smallerthan a typical hadronic scale of the order
of1 GeV. Wethusmakethefollowingreplacements:
x2∆2 ∆
k k+, 1 ⊥ , x ⊥ with k+ = x p+,
1 → 1 8k+ − 1 2 1 1
(cid:20) 1 (cid:21)
m2 +x′2∆2/4 ∆
k′ k′+, c 1 ⊥ ,x′ ⊥ with k′+ = x′p′+,
1 → 1 2k′+ 1 2 1 1
(cid:20) 1 (cid:21)
x2∆2 ∆
k 2 ⊥ , k−, x ⊥ with k− = x q−,
2 → 8k− 2 2 2 2 2
(cid:20) 2 (cid:21)
m2 +x′2∆2/4 ∆
k′ c 2 ⊥ ,k′−, x′ ⊥ with k′− = x′q′−. (16)
2 → 2k′− 2 − 2 2 2 2
(cid:20) 2 (cid:21)
As a consequence of these replacements it is then possible to explicitly perform the integrations
over k¯−, k¯+, k¯ and k¯ . Furthermore, the relative distance between the (anti-)S[ud]-diquark
1 2 ⊥1 ⊥2
andthe(anti-)c-quarkfield operatorsinthehadronicmatrixelementsisforcedtobelightlike,i.e.,
theyhaveto lieonthelightcone and thusthetimeordering of thefield operators can bedropped.
Afterthesesimplificationsonearrivesatthefollowingexpressionforthepp¯ D0D0 amplitude:
→
Mµν = dk¯1+θ(k¯1+) d2zπ1−eik¯1+z1− dk¯2−θ(k¯2−) d2zπ2+eik¯2−z2+
a(X′),α(′) Z Z Z Z
i i
×hD0 : p′|Ψca′1α′1(−z¯1/2)ΦaS1[ud](+z¯1/2)|p : p,µi H˜a(i′)α′i k¯1,k¯2 (17)
D0 : q′ ΦS[ud]†(+z¯ /2)Ψc ( z¯ /2) p¯: q,ν . (cid:0) (cid:1)
×h | a2 2 a′2α′2 − 2 | i
From now on we will omit the colorand spinorlabels wheneverthis does not lead to ambiguities
and replacethefield-operatorarguments z¯ andz¯ bytheirnon-vanishingcomponentsz− and z+,
1 2 1 2
respectively. Furthermore, if one uses k¯+ = x¯ p¯+ and k¯− = x¯ q¯− to rewrite the k¯+ and k¯−
1 1 2 2 1 2
integrations in the amplitude (17) as integrations over the longitudinal momentum fractions x¯
1
7
and x¯ ,respectively,onearrivesat,
2
Mµν = dx¯1p¯+ dz1−eix¯1p¯+z1− dx¯2q¯− dz2+eix¯2q¯−z2+
2π 2π
Z Z Z Z
D0 : p′ Ψc( z−/2)ΦS[ud](+z−/2) p : p,µ H˜ x¯ p¯+,x¯ q¯− (18)
×h | − 1 1 | i 1 2
D0 : q′ ΦS[ud]†(+z+/2)Ψc( z+/2) p¯: q,ν . (cid:0) (cid:1)
×h | 2 − 2 | i
−
As in Ref. [1] for p Λ+ (p¯ Λ ), the p D0 (p¯ D0) transition matrix element is
→ c → c → →
expectedtoexhibitapronouncedpeakwithrespecttothemomentumfraction. Thepositionofthe
peak isapproximatelyat
m
x = c = 0.68. (19)
0
M
From Eq. (9) one then infers that the relevant average momentum fractions x¯ and x¯ have to
1 2
be larger than the skewness ξ. This means that the convolution integrals in Eq. (18) have to be
performed onlyfrom ξ to1 and notfrom 0to 1.
Inthefollowingsectionwewillanalyzethesofthadronicmatrixelementsinsomemoredetail.
IV. HADRONICTRANSITIONMATRIXELEMENTS
Compared to Eq. (10) the Fourier transforms of the hadronic matrix elements for the p D0
→
and p¯ D0 transitions are rendered to Fourier integrals solely over z− and z+, respectively.
→ 1 2
Hencewehavetostudytheintegral
p¯+ dz1−eix¯1p¯+z1− D0 : p′ Ψc( z−/2)ΦS[ud](+z−/2) p : p,µ , (20)
2π h | − 1 1 | i
Z
overthe p D0 transitionmatrixelementand theintegral
→
dz+
q¯− 2 eix¯2q¯−z2+ D0 : q′ ΦS[ud]†(+z+/2)Ψc( z+/2) p¯: q,ν , (21)
2π h | 2 − 2 | i
Z
overthe p¯ D0 transitionmatrixelementinsteadofEqs. (11)and (12), respectively.
→
We will first concentrate on the p D0 transition (20) and investigate the product of field
→
operators Ψc( z−/2)ΦS[ud](+z−/2). Forthispurposeweconsiderthe c quark field operator Ψc
− 1 1 −
in the hadron frame of the outgoing D0, cf. e.g., Refs. [10, 11], where the D0 has no transverse
momentumcomponent. ItcanbereachedfromoursymmetricCMS byatransverseboost[12,13]
withtheboostparameters
∆
b+ = (1 ξ)p¯+ and b = ⊥ . (22)
⊥
− 2
8
In this hadron-out frame we write the field operator in terms of its “good” and “bad” light cone
components,
1
Ψc = (γ−γ+ +γ+γ−)Ψc Ψc +Ψc , (23)
2 ≡ + −
bymeansofthegoodandbadprojectionoperators + = 1γ−γ+ and − = 1γ+γ−, respectively.
P 2 P 2
After doing that we eliminate the γ− appearing in + and − by using the c¯ quark energy
P P −
projector
v(k′,λ′)v¯(k′,λ′) = k′ γ m . (24)
1 1 1 1 1 · − c
λ′
X1
In thehadron-outframeitexplicitelytakes ontheform
m2
v(kˆ′,λ′)v¯(kˆ′,λ′) = k′+γ− + c γ+ m , (25)
1 1 1 1 1 2k′+ − c
λ′ 1
X1
sincetherethec¯ quark momentumis
−
m2
kˆ′ = k′+, c ,0 . (26)
1 1 2k′+ ⊥
(cid:20) 1 (cid:21)
Withthosereplacements the c quark field operatorbecomes
−
1
Ψc = v(kˆ′,λ′) v¯(kˆ′,λ′)γ+Ψc
2k′+ 1 1 1 1
1 Xλ′1 n (cid:16) (cid:17)
+ γ+ v(kˆ′,λ′)(v¯(kˆ′,λ′)Ψc)+2m Ψc . (27)
1 1 1 c
h io
−
As in the case of pp¯ Λ+Λ in Ref. [1], one can argue that the contribution coming from
→ c c
v¯(kˆ′,λ′)γ+Ψc dominates over the one in the square brackets, and thus the latter one can be
1 1
(cid:16)neglected. Sinc(cid:17)e this dominant contribution can be considered as a plus component of a four-
vector,onecan immediatelyboostback tooursymmetricCMSwhereit thenstillholdsthat
1
Ψc( z /2) = v(k′,λ′) v¯(k′,λ′)γ+Ψc( z /2) . (28)
− 1 2k′+ 1 1 1 1 − 1
1 λ′
X1 (cid:0) (cid:1)
Furthermore, one can even show that in v¯(k′, λ′)γ+Ψc( z−/2) on the right-hand side of
1 1 − 1
Eq.(28)onlythegoodcomponentofΨc( z(cid:0)−/2)isprojected out,sin(cid:1)ce
− 1
v¯(k′, λ′)γ+Ψc( z−/2) = v¯(k′, λ′)γ+ +Ψc( z−/2). (29)
1 1 − 1 1 1 P − 1
Finally, we note that such manipulations are not necessary for the scalar field operator ΦS[ud] of
theS[ud]diquark.
9
Puttingeverythingtogethergivesforthe p D0 transitionmatrixelement(20)
→
p¯+ dz1−eix¯1p¯+z1− D¯0 : p′ Ψc( z−/2)ΦS[ud](+z−/2) p : p,µ =
2π h | − 1 1 | i
Z
p¯+ dz1−eix¯1p¯+z1− D¯0 : p′ v(k′, λ′) v¯(k′, λ′)γ+Ψc ( z−/2) ΦS[ud](+z−/2) p : p,µ .
2k′+ 2π h | 1 1 1 1 + − 1 1 | i
1 λ′ Z
X1 (cid:0) (cid:1)
(30)
Proceeding in an analogous way for the p¯ D0 transition matrix element, where the role of
→
the+and componentsare interchanged,weget forEq.(21)
−
dz+
q¯− 2 eix¯2q¯−z2+ D0 : q′ ΦS[ud]†(+z+/2)Ψc( z+/2) p¯: q,ν =
2π h | 2 − 2 | i
Z
q¯− dz+
2 eix¯2q¯−z2+ D0 : q′ ΦS[ud]†(+z+/2) Ψc ( z+/2)γ−u(k′,λ′) u¯(k′,λ′) p¯: q,ν .
2k′− 2π h | 2 + − 2 2 2 2 2 | i
2 λ′ Z
X2 (cid:0) (cid:1)
(31)
Alsohereonly thegoodcomponentsofthequark field are projectedouton theright-handside.
Using now Eqs. (30) and (31) and attaching the spinors v(k′, λ′) and u¯(k′,λ′) to the hard
1 1 2 2
subprocessamplitudeH˜ byintroducing
Hλ′ ,λ′ (x¯1,x¯2) := u¯(k2′,λ′2)H˜ x¯1p¯+,x¯2q¯− v(k1′, λ′1) , (32)
1 2
(cid:0) (cid:1)
weget forthepp¯ D0D0 amplitude(18)
→
1 1 1
Mµν =4(p¯+)2 dx¯1 dx¯2 Hλ′1,λ′2 (x¯1,x¯2) x¯ ξ x¯ ξ
λ′,λ′ Z Z 1 − 2 −
X1 2
v¯(k′, λ′)γ+ p¯+ dz1− eix¯1p¯+z1− D¯0 : p′ Ψc ( z−/2)ΦS[ud](+z−/2) p : p,µ
× 1 1 2π h | + − 1 1 | i
Z
dz+
q¯− 2 eix¯2q¯−z2+ D0 : q′ ΦS[ud]†(+z+/2)Ψc ( z+/2) p¯: q,ν γ−u(k′,λ′) .
× 2π h | 2 + − 2 | i 2 2
Z
(33)
Introducingtheabbreviations
Hλc¯S′1µ := v¯(k1′, λ′1)γ+ p¯+ d2zπ1− eix¯1p¯+z1−hD¯0 : p′|Ψc+(−z1−/2)ΦS[ud](+z1−/2)|p : p,µi (34)
Z
and
dz+
HλcS′2¯ν := q¯− 2π2 eix¯2q¯−z2+hD0 : q′|ΦS[ud]†(+z2+/2)Ψc+(−z2+/2)|p¯: q,νi γ−u(k2′,λ′2) , (35)
Z
10