Production of pentaquarks in $pA$-collisions

0.94 MB·

by  Ivan Schmidt

Production of pentaquarks in pA-collisions
Iván Schmidt, M. Siddikov
Departamento de Física, Universidad Técnica Federico Santa María,
y Centro Científico - Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, Chile
Wearguethatahidden-charmpentaquarkrecentlyobservedinweakdecaysofΛ canbeproduced
b
in proton-nucleus collisions without electroweak intermediaries. We analyze the production cross-
section for several scenarios of internal structure and find that a cross-section is sizable. This
process can be studied both in collider as well as in fixed-target experiments. In the former case,
thepentaquarksareproducedatveryforwardrapidities,whereasinthelattercase,pentaquarksare
produced with relatively small rapidities and can be easily detected via invariant mass distribution
of a forward J/ψ and a comoving proton. Additionally, the suggested process allows to check
the existence of a neutral pentaquark P0 (an isospin partner of P+) predicted in several models.
c c
The rapidity and transverse momentum distributions of pentaquarks could provide comprehensive
6 information about the c¯c component of this exotic baryon.
1
0 PACSnumbers: 14.20.Pt,11.10.St,13.85.Ni
2

n
a
I. INTRODUCTION
J

1
2 The recent discovery of hidden charm pentaquarks, Pc+(4380) and Pc+(4450), in weak decays of Λb hyperons [1] is
withoutanydoubtsanimportantstepinthestudyofexoticbaryonspredictedbyGell-Mann[3]. Theexistenceofthe

] P+ pentaquark agrees with early observations that there is an attractive Van der Waals like interaction between c¯c
h c
and light quark matter, which could lead to formation of bound states [4, 5] and could provide a natural explanation
p
for the large intrinsic charm of the proton suggested in [6, 7] as a phenomenological description of certain charm
-
p production processes. As of now, the following facts have been established experimentally: the existence of P+,
c
e its mass, decay width and that J/ψp is one of the possible decay channels. Albeit the latter fact might indicate a
h
large Fock component with c¯c in a singlet state, this does not exclude other color-spin-flavour arrangements, like for
[
exampleaweaklyboundstateofoneoftheD-mesonsandcharmedΛ /Σ -hyperonsuggestedin[8–15],aboundstate
c c
1 ofχ andpsuggestedin[16],aweaklyboundstateofJ/ψ andp[17],ψ(2S)andp[18],orastronglyboundsystemof
c
v colored c¯c with light quarks [19]. The formation of pentaquark P+ in low-energy phenomenological models indicates
1 c
itsexistence[16,20],albeitdoesnotexcludeapossibilitythatitisakinematiceffectduetoaLandaupolesingularity
2
in a triangle diagram [21, 22]. For this reason, it was concluded that a pentaquark existence could be confirmed only
6
5 after a study of additional decay channels [23], as well as confirmation of the existence of other pentaquarks from
0 SU(3) flavor symmetry decuplet [25–27].
. Anotherwaytoavoidtheabove-mentionedanomaloustrianglesingularityistostudyotherproductionmechanisms
1
0 and instead of weak Λb-decays consider photoproduction of pentaquarks in γp collisions [28–30] or πp collisions [31].
6 Fromatheoreticalpointofview,acrucialadvantageoftheP+ comparedtoaputativeΘ+ pentaquark(see[32]for
c
1 a review of Θ+ studies) is that it possesses two massive c¯c quarks, whose dynamics can be described by perturbative
v: QCDmethods. Additionally,inprocesseswherec¯cisproduceddiffractively,thetypicaldistancebetweenthequarksis
i small, 1/mc,andincertaincasesthereisanadditionalsuppression ΛQCD/mc duetoadestructiveinterferenceof
X ∼ ∼
interactionamplitudesfromcandc¯. Thisfacthasbeenusedextensivelyinthestudyofcharmoniumandbottomonium
r production, where different models successfully predict the cross-section (see [33–35] for a review of the current
a
experimental and theoretical situation).
In this paper we suggest that P+ might be produced in proton-nucleus collisions in forward kinematics, as a two-
c
stage process discussed in the next Section II. We assume that a c¯c pair needed for formation of a pentaquark is
produced diffractively, so we can apply the above-mentioned formalisms developed for charmonium dynamics and
study dynamics of a c¯c pair in P+. According to our estimates, this process has a sizable cross-section, both when c¯c
c
is in color singlet and in color octet states.
Potentially pentaquarks can be produced diffractively also in pp collisions, when a c¯c pair from extrinsic charm
after interaction with the projectile leads to the formation of a c¯c. However, typical cross-sections of this process
are smaller than in case of pA collisions, and production occurs at smaller rapidities. A full analysis of diffractive
pentaquark production in pp collisions will be presented elsewhere.
The paper is structured as follows. In Section II we discuss a suggested mechanism of pentaquark production and
a framework which we use for its description. For the sake of simplicity in this section we consider proton-deuteron
collisions, and take into account only the contribution of extrinsic charm. In Section III we generalize our framework
forrealisticlarge-Anuclei. ThecontributionoftheintrinsiccharmisdiscussedinSectionIV.InSectionVwediscuss
2
(a) (b)
p uu ~ri p uu
d d
A A
N X P+c R~l N X P+c
g c¯ R~cc g c¯
~rcc
c c
~b
g
N X p X
Figure 1. Perturbative diagrams contributing to the pentaquark production if the c¯c pair is in color singlet state. Diagrams
shownintheleft(right)paneprobeP-wave(S-wave)componentofthec¯cdipoleinapentaquark. Eachsquaredblockimplies
a sum of diagrams with a gluon attached to each of the quarks. In a diagram (b) both emissions before and after interaction
with a target are possible.
(c) (d)
u u
p u p u
d d
A A
N X P+c N X P+c
g c¯ g c¯
c c
g
p X p X
Figure2. Perturbativediagramscontributingtothepentaquarkproductionifthec¯cpairisincoloroctetstate. Bothdiagrams
can probe a c¯c dipole both in S- and P-wave. Each squared block implies a sum of diagrams with a gluon attached to each of
the quarks. In case of S-wave production, there is an additional diagram with t-channel gluon attached to a projectile gluon
instead of c¯c (not shown).
parametrizations of the pentaquark and proton light-cone wave functions, and in Section VI we present numerical
estimates for the pentaquark production cross-sections.
II. PENTAQUARK PRODUCTION MECHANISMS
In this section we analyze P+ production in proton-deuteron collisions at high energies, postponing a discussion
c
of nuclear effects until Section III. The dominant production mechanism depends on the internal structure of this
baryon. As was mentioned in Section I, at this moment several competing phenomenological models assume that a
pentaquark is a bound state of a hidden charm meson with a proton or a charmed D-meson and Σ /Λ -hyperon.
c c
Fromthe QCD pointof view, thedifferencebetweenthe two classes ofmodels isin the colorstate of the c¯c state: the
former case implies that the c¯c pair is in a color singlet state, whereas in the latter case the c¯c pair is mostly in the
color octet state. However, we would like to note that in QCD, if we assume dominance of a state with definite color
or spin of the c¯c-component, due to exchange of virtual gluons between light and heavy quarks this Fock state will
certainly contain a small admixture of all possible color-spin combinations.
In the leading order over α (m ), the pentaquark production might proceed in a process schematically shown in a
s c
diagram (a) of Fig 1. This process probes a color singlet c¯c component of the pentaquark in a P-wave, as suggested
in[16]. Ac¯c-pairnecessaryforapentaquarkformationisproducedviaagluonsplittinginpAcollisions. Theproduced
c¯c has a negative invariant mass, so in order to be able to produce a near-onshell c¯c, it should interact at least once
with the target. To probe the c¯c pair inside a pentaquark in a color singlet S-wave, the c¯c pair should emit at least
one gluon (jet), as shown in the diagram (b) of the Figure 1.
Ifthec¯cpairinsideapentaquarkisinthecoloroctetstate,assuggestedin[9–13,19],itsproductionmightproceed
via processes shown in the Figure 2. In both cases the c¯c pair can be produced both in S- and P-wave.
3
Inallfourdiagramsthetypicalvaluesofalight-conefractionx carriedbyaprojectilegluonfromaprotoncannot
1
beverysmall: otherwiselightquarksandac¯cpairseparatedbyalargerapiditygapcannotformaboundstate. Atthe
same time, typical values of a variable x M /x √s 1. In this kinematics, application of perturbative QCD
(cid:104) 2(cid:105) ∼ Pc 1 (cid:28)
mightbequestionableduetosaturationeffects. Forthisreason,weadoptadipolemodelwhichnaturallyincorporates
allsaturationeffectsandhasbeenappliedtothedescriptionofc¯cproductionin[36–40]. Sincetheproduceddipolesare
smallandhaveatypicalsize r m−1,upto (1/m )-correctionsthisapproachisequivalenttoak -factorization
(cid:104) cc(cid:105)∼ c O c T
approach suggested in [41–43], taking into account the relation of the dipole cross-section and the unintegrated gluon
PDF (x, k ) suggested in [44].
⊥
F
The cross-section for the diagram (a) in the Figure 1 has the form
ˆ
dσ(a) 1+x (cid:16) (cid:17) (cid:16) (cid:17)
= 1 x g(x ) d2R(1)d2R(2)dα(1)d2r(1)dα(2)d2r(2)Φµ¯µ α(1),(cid:126)r(1) Φν¯ν∗ α(2),(cid:126)r(2) (1)
dy x 1 1 cc cc c cc c cc c¯c c cc c¯c c cc
1
(cid:18) M (cid:19) (cid:18) M (cid:19) (cid:16) (cid:17) (cid:16) (cid:17)∗
Φ Pc R(cid:126)(1) Φ∗ Pc R(cid:126)(2) µ¯µ α(1), x ,(cid:126)r(1), R(cid:126)(1) ν¯ν α(2), x ,(cid:126)r(2), R(cid:126)(2)
× D −M 2m cc D −M 2m cc H c 1 cc cc H c 1 cc cc
Pc − c Pc − c
1 (cid:104) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105)
σ α(1)(cid:126)r(1)+α¯(2)(cid:126)r(2) +σ α¯(1)(cid:126)r(1)+α(2)(cid:126)r(2) σ α(1)(cid:126)r(1) α(2)(cid:126)r(2) σ α¯(1)(cid:126)r(1) α¯(2)(cid:126)r(2)
× 16 c cc c cc c cc c cc − c cc − c cc − c cc − c cc
where x is the light-cone fraction of the nucleon carried by the gluon, x g(x ) is the gluon density in a projectile
1 1 1
proton,superscriptindices1and2refertothenormalandcomplexconjugateamplitudes,α and(cid:126)r arethelight-cone
c cc
fraction and dipole size of the c-quarks in a c¯c pair, and we also use variables R(cid:126) for the distance between the center
cc
of mass of the pentaquark and the c¯c pair, (cid:126)r for the distance between the light quarks w.r.t. is center of mass 1.
i
Theimpactparameter(cid:126)bofthepentaquarkdoesnotappearinthep -integratedcross-section. Thenotation µ¯µ in1
T
H
stands for the overlap of proton and pentaquark wave functions,
ˆ 3 (cid:32) (cid:33) (cid:32) (cid:33)
(cid:16) (cid:17) (cid:89) (cid:88) (cid:88)
µ¯µ α , ξ, (cid:126)r , R(cid:126) = (dα dr )δ2 (cid:126)r δ 1 α dα (4)
c cc cc i i i i c
H −
i=1 i i
(cid:18) α α ξ α¯ ξ (cid:19)ν1ν2ν3µ¯µ
Ψ† i ,(cid:126)r +R(cid:126) ; c ,R(cid:126) α (cid:126)r , c ,R(cid:126) +α¯ (cid:126)r Ψν1ν2ν3(α ,r ),
× Pc 1+ξ i l 1+ξ c¯c− c c¯c 1+ξ c¯c c c¯c p i i
where Ψ and Ψ are the light-cone wave functions of the proton and pentaquark, µ¯/µ are spinor indices of the c
p Pc
and c¯, and ξ = (P+ p+)/p+ is the ratio of light-cone momenta of c¯c pair and light quarks. For the wave function
c −
of heavy c¯c dipole we use the well-known perturbative expression [45]
Φµ¯µ(α , r)= √αs ξµ(m σ e+i(1 2α )(σ n) (e )+[n e] )ξµ¯K ((cid:15)r), (5)
c¯c c 2π√2 c · − c · ·∇ × ·∇ 0
where
(cid:113)
(cid:15)= m2 α (1 α )m2, (6)
c − c − c G
and m is the effective gluon mass. In the dipole cross-section σ(r) for the sake of brevity here and in what follows
G
we suppress the dependence on the Bjorken variable x , tacitly assuming that a value x M2/(x s) is used. In (1)
2 2 ≈ cc 1
we assume that the light cone momentum of the nucleus is equally distributed among the nucleons: as we will show
in the next Section III, the light-cone distribution of nucleons is indeed very narrow.
Theterm(1+x )/x intheprefactorof(1)hasapoleatsmallx ,whichimpliesthatthecorrespondingamplitude
1 1 1
shouldvanishinthelimitx 0inordertoprovideafiniteintegratedcross-section. Aswewillseebelow,ifarealistic
1
→
parametrization is used for a pentaquark wave function, indeed such behavior is observed: a c¯c pair separated by a
large rapidity gap from the proton cannot form a pentaquark.
The variable x is related to the light-cone components of the pentaquark momentum P in the nucleon-nucleon
1 c
center of mass system as
1 ThissetofvariablesismoreconvenientthanthestandardsetofJacobivariables(cid:126)rcc,R(cid:126)cc,ρ(cid:126)1,ρ(cid:126)2,whereρ(cid:126)1,2 aredefinedas
ρ(cid:126)1= 2mcR(cid:126)cc+mq(cid:126)r1 R(cid:126)cc= mq (cid:0)(cid:126)r1 R(cid:126)cc(cid:1), (2)
2mc+mq − 2mc+mq −
ρ(cid:126)2= (2mc+mq)ρ(cid:126)1+mq(cid:126)r2 ρ(cid:126)1= mq ((cid:126)r2 ρ(cid:126)1), (3)
2mc+2mq − 2mc+2mq −
usedfordescriptionofaclassicalmany-bodysystems.
4
(cid:18) M2 +P2 (cid:19)
P = (1+x )√s, Pc ⊥ , P(cid:126) , (7)
c 1 ⊥
(1+x )√s
1
where √s is a collision energy per nucleon, x is the fraction of momentum carried by the gluon coming from one
1
of the projectile protons, M is a pentaquark mass, for which we use a value M 4.4GeV both for P+(4380)
Pc Pc ≈ c
and P+(4450), and P(cid:126) is the transverse momentum of the produced pentaquark. From 7 we can extract a relation
c ⊥
between the rapidity y of the produced pentaquark and x . This relation, in the nucleon-nucleon center of mass
1
system, is given by
 
1 (cid:18)P+(cid:19) (1+x )√s
y = 2ln Pcc− =ln(cid:113)M2 1+P2, (8)
Pc ⊥
i.e. in a collider experiments pentaquarks cannot be produced with rapidities smaller than a threshold value
 
√s
ymin(s, P⊥)=ln(cid:113) , (9)
M2 +P2
Pc ⊥
and are concentrated in narrow bins
y (y (s, P ), y (s, P )+ln2). (10)
min ⊥ min ⊥
∈
In the nucleus rest frame, relevant for fixed target experiments, the results might be obtained by a shift
 
(cid:18) (cid:19)
√s (1+x )m
yRF =y ln =ln(cid:113) 1 N . (11)
− mN M2 +P2
Pc ⊥
Thecross-sectioncorrespondingtodiagram(b),whichisthedominantmechanismifthec¯cpairinsideapentaquark
is in S-wave, has the form
ˆ
dσ(b) 5 1+x
= x g(x ) 1 d2R(1)d2R(2) d2ρdα dα d2r(1)dα d2r(2) (12)
dy 8 1 1 x cc cc G 1 cc 2 cc
1
(cid:18) (cid:19) (cid:18) (cid:19)
M M
Φ Pc R(cid:126)(1) Φ∗ Pc R(cid:126)(2)
× D −M 2m cc D −M 2m cc ×
Pc − c Pc − c
6
(cid:88) (cid:104) (n) (n) (cid:105) (cid:104) (n(cid:48)) (n(cid:48)) (cid:105)
ηnηn(cid:48)Tr ΛMΦc¯c ((cid:15)n,(cid:126)rn)ΦcG (δn, ρ(cid:126)n) Tr ΛMΦc¯c ((cid:15)n(cid:48),(cid:126)rn(cid:48))ΦcG (δn(cid:48), ρ(cid:126)n(cid:48)) ,
n,n(cid:48)=1
(cid:16) (cid:17) (cid:18) α (cid:19) (cid:18) α (cid:19)
×σ (cid:126)bn−(cid:126)bn(cid:48) Hµ¯µ 1 1α , x1(1−αG),(cid:126)rc(1c), R(cid:126)c(1c) Hν¯ν 1 2α , x1(1−αG),(cid:126)rc(2c), R(cid:126)c(2c)
G G
− −
whereinadditiontothenotationsthatappearin(1)weintroducedα andρ(cid:126)forthelight-conefractionandtransverse
G
coordinateoftheemittedgluon,α forthelight-conefractionsoftheincidentgluonmomentumcarriedbythec-quark
1,2
in the amplitude and its conjugate. The gluon emission wave function Φ in (12) has the form
cg
Φ (β,ρ) i√αs ξ† ˆξ K (δ ρ(cid:126)), (13)
cg ≈ π√3 µD µ¯ 0 | |
where
(cid:18) (cid:19)
β
ˆ =2 1 1 e +im β2(n e ) σ iβ ( e ) σ, (14)
D − 2 f ·∇ c 1 × f · − 1 ∇× f ·
(cid:113)
δ = β2m2+(1 β)m2, (15)
c − G
5
e is a polarization vector of the emitted gluon, µ and µ¯ are helicities of the quark before/after emission, and β is the
f
ratio of the gluon light-cone fraction to the light cone fraction of the initial quark. The vector (cid:126)r , ρ(cid:126) ,(cid:126)b , as well as
n n n
the coefficients (cid:15), δ in (5,13) are given by 2
(cid:15)2 =(cid:15)2 =m2 (1 α α )(α+α )λ2 (16)
1 3 c − − − G G
δ2 =δ2 =α2m2+α(α+α )λ2 (17)
1 5 G c G
(α+α )(1 α )(cid:0)α m2+α(1 α α )λ2(cid:1)
δ2 =(cid:15)2 = G − G G c − − G (18)
3 5 1 α α
G
− −
δ2 =δ2 =α2m2+(1 α)(1 α α )λ2 (19)
2 6 G c − − − G
(1 α )(1 α)(cid:0)α m2+α(1 α α )λ2(cid:1)
δ2 =(cid:15)2 = − G − G c − − G (20)
4 6 α
(cid:15)2 =(cid:15)2 =m2 (1 α)αλ2 (21)
2 4 c − −
α(cid:126)r α ρ(cid:126) ρ(cid:126)+(1 α α )(cid:126)r
(cid:126)r =(cid:126)r =(cid:126)r = − G , ρ(cid:126) =ρ(cid:126) =ρ(cid:126) = − − G (22)
1 3 5 1 3 5
α+α − α+α
G G
(1 α α )(cid:126)r+α ρ(cid:126) ρ(cid:126) α(cid:126)r
(cid:126)r =(cid:126)r =(cid:126)r = − − G G , ρ(cid:126) =ρ(cid:126) =ρ(cid:126) = − (23)
2 4 6 2 4 6
− 1 α − 1 α
− −
α α(1 α α ) α α(1 α α )
(cid:126)b =(cid:126)b+ G ρ(cid:126) − − G (cid:126)r, (cid:126)b =(cid:126)b+ G ρ(cid:126)+ − − G (cid:126)r, (24)
1 2
α+α − α+α 1 α 1 α
G G
− −
(cid:126)b =(cid:126)b =(cid:126)b (1 α α )(cid:126)r, (cid:126)b =(cid:126)b =(cid:126)b+α(cid:126)r. (25)
3 6 G 4 5
− − −
and η = 1,1, 1, 1, α , α .
n G G
{ − − − − }
Ifthec¯cpairisinacoloroctetstate,itsproductionmayproceedviaanyofthemechanismsshownintheFigure(2).
The diagram (c) has the cross-section
ˆ ˆ ˆ
dσ(c) 1+x (cid:18) M (cid:19)
= 1 x g(x ) dα d2ρ d2R(1)d2R(2)dα(1)d2r(1)dα(2)d2r(2)Φ Pc R(cid:126)(1) (26)
dy x 1 1 G c¯c c¯c c cc c cc D −M 2m cc
1 Pc − c
(cid:18) M (cid:19) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Φ∗ Pc R(cid:126)(2) Σ(L) α(1), r(1),α(2), r(2) Ψ¯µ¯µ∗ α(1), r(1) µ¯µ α(1), α , x ,ρ(cid:126), R(cid:126)(1)
× D −MPc −2mc cc c cc c cc c cc OPc c G 1 c¯c
(cid:16) (cid:17) (cid:16) (cid:17)
Ψ¯ν¯ν α(2), r(2) ν¯ν∗ α(2), α , x , ρ(cid:126), R(cid:126)(2) ,
× c cc OPc c G 1 c¯c
where we introduced the shorthand notation Σ(L) for the cross-section of the c¯c with internal orbital momentum L,
(cid:16) (cid:17) 9 (cid:104) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Σ(L=0) α(1), r(1),α(2), r(2) = σ α(1)(cid:126)r(1)+α¯(2)(cid:126)r(2) +σ α¯(1)(cid:126)r(1)+α(2)(cid:126)r(2) σ α(1)(cid:126)r(1) α(2)(cid:126)r(2) (27)
c cc c cc 16 c cc c cc c cc c cc − c cc − c cc
(cid:16) (cid:17)(cid:105)
σ α¯(1)(cid:126)r(1) α¯(2)(cid:126)r(2) ,
− c cc − c cc
(cid:16) (cid:17) 5 (cid:104) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Σ(L=1) α(1), r(1),α(2), r(2) = σ α(1)(cid:126)r(1)+α¯(2)(cid:126)r(2) +σ α¯(1)(cid:126)r(1)+α(2)(cid:126)r(2) +σ α(1)(cid:126)r(1) α(2)(cid:126)r(2) (28)
c cc c cc 16 c cc c cc c cc c cc c cc − c cc
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
+σ α¯(1)(cid:126)r(1) α¯(2)(cid:126)r(2) 2σ α(1)(cid:126)r(1) 2σ α¯(1)(cid:126)r(1) 2σ α(2)(cid:126)r(2)
c cc − c cc − c cc − c cc − c cc
(cid:16) (cid:17)(cid:105)
2σ α¯(2)(cid:126)r(2) ,
− c cc
and
µ¯µ(cid:16)α , α , ξ, ρ(cid:126), R(cid:126) (cid:17)=3ˆ (cid:89)3 (cid:0)dα d2r (cid:1)δ2(cid:32)(cid:88)(cid:126)r (cid:33)δ(cid:32)1 (cid:88)α (cid:33)Φ (cid:18)ρ(cid:126)−αG(cid:126)r1(cid:19) (29)
OPc c G c¯c k k k − k cG 1 α1
k=1 k k −
(cid:18)(cid:26) (cid:27) (cid:19)
α α δ α ξ α¯ ξ
Ψµ¯µ† k− G k1,(cid:126)r +R(cid:126) ; c , R(cid:126) α (cid:126)r , c , R(cid:126) +α¯ (cid:126)r
× Pc 1+ξ αG k l 1+ξ αG c¯c− c c¯c 1+ξ αG c¯c c c¯c
− − −
ψ ( α ,(cid:126)r )
p k k
× { }
2 Ifweuseacoordinatespaceevaluation,naivelywecouldexpectthatthecoefficients(cid:15)fordiagrams(5,6)shouldfulfill(cid:15)5=(cid:15)1,(cid:15)6=(cid:15)2,
butthisisnotso. Thishappensbecauseintheevaluationof(5)atleastoneofthequarksshouldbeonshell, whichisnolongertrue.
Inthesameway,offshellnessofthefinalquarkintheevaluationofdiagrams(3,4)leadstoδ3=δ1 andδ4=δ2.
(cid:54) (cid:54)
6
u
u
A p d A
X P+c P+c X
N
g c¯
c
N N
u
A p ud A
X P+c P+c X
N
g c¯
c
N N
Figure3. (coloronline)Generalmultipomeronconfigurationswhichcontributetoadiagram(d)[upperplot]cannotbeexpressed
intermsofadipolecross-section. However,thelargestintercepthasatwo-pomeronconfiguration[lowerplot](seethetextfor
explanations).
for the overlap, numbering the active quark from which the emission takes place with the index 1. As we will see
below, this diagram is small since the light quarks both in a proton and in a pentaquark are almost onshell.
The evaluation of diagram (d) in the general case is challenging, since under reggeization potentially we may get
multipomeron contributions as shown in the upper part of the Figure 3. These contributions are not reducible to a
mere dipole cross-section and their evaluation presents a complicated problem. However, as was shown in [46, 47], in
thelarge-N limittheinterceptsofthesecontributionsaresmaller,sothedominantcontributionatveryhighenergies
c
is a two-pomeron contribution shown in the lower part of Figure 3. In this limit the corresponding cross-section is
given by
ˆ ˆ ˆ ˆ
dσ(d) 1+x
=3 1 x g(x ) dα d2ρ d2R(1) d2R(2)dα(1)d2r(1)dα(2)d2r(2) (30)
dy x 1 1 G c¯c c¯c c cc c cc
1
(cid:18) (cid:19) (cid:18) (cid:19)
M M
Φ Pc R(cid:126)(1) Φ∗ Pc R(cid:126)(2)
× D −M 2m cc D −M 2m cc
ˆ Pc − c Pc − c
(cid:16) (cid:17) (cid:16) (cid:17)
d2b α(1)(cid:126)r(1) α(2)(cid:126)r(2),(cid:126)b+R(cid:126)(12) (L) α(1), r(1),α(2), r(2),(cid:126)b R(cid:126)(12)
× N 1 1 − 1 1 l S c cc c cc − cc
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Ψ¯µ¯µ∗ α(1), r(1) Ψ¯ν¯ν α(2), r(2) µ¯µ α(1), α , x ,ρ(cid:126), R(cid:126)(1) ν¯ν∗ α(2), α , x , ρ(cid:126), R(cid:126)(2) ,
× c cc c cc OPc c G 1 c¯c OPc c G 1 c¯c
where we introduced the notation
(cid:16) (cid:17) 9 (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)
(L=0) α(1), r(1),α(2), r(2),(cid:126)b = α(1)(cid:126)r(1)+α¯(2)(cid:126)r(2),(cid:126)b + α¯(1)(cid:126)r(1)+α(2)(cid:126)r(2),(cid:126)b (31)
S c cc c cc 16 N c cc c cc N c cc c cc
(cid:16) (cid:17) (cid:16) (cid:17)(cid:105)
α(1)(cid:126)r(1) α(2)(cid:126)r(2),(cid:126)b α¯(1)(cid:126)r(1) α¯(2)(cid:126)r(2),(cid:126)b ,
−N c cc − c cc −N c cc − c cc
7
(cid:16) (cid:17) 5 (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)
(L=1) α(1), r(1),α(2), r(2),(cid:126)b = α(1)(cid:126)r(1)+α¯(2)(cid:126)r(2),(cid:126)b + α¯(1)(cid:126)r(1)+α(2)(cid:126)r(2),(cid:126)b (32)
S c cc c cc 16 N c cc c cc N c cc c cc
(cid:16) (cid:17) (cid:16) (cid:17)
+ α(1)(cid:126)r(1) α(2)(cid:126)r(2),(cid:126)b + α¯(1)(cid:126)r(1) α¯(2)(cid:126)r(2),(cid:126)b
N c cc − c cc N c cc − c cc
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105)
2 α(1)(cid:126)r(1),(cid:126)b 2 α¯(1)(cid:126)r(1),(cid:126)b 2 α(2)(cid:126)r(2),(cid:126)b 2 α¯(2)(cid:126)r(2),(cid:126)b ,
− N c cc − N c cc − N c cc − N c cc
(cid:16) (cid:17)
whereListheorbitalmomentumofinternalmotionofthec¯cpair,and (cid:126)r,(cid:126)b isthedipolescatteringamplitude[48]
N
related to the dipole cross-section as
ˆ
(cid:16) (cid:17)
σ(r)= d2b (cid:126)r,(cid:126)b . (33)
N
III. EFFECTIVE WAVE FUNCTION FROM 2N CORRELATIONS
InthepreviousSectionIIweconsideredpentaquarkproductioninproton-deuteroncollisions. Inthecaseofaheavy
nucleus, we should replace the deuteron wave function with a two-nucleon correlator. It was realized some time ago
thatthisobjecthaspropertiessimilartoadeuteronwavefunction[49,50],namelyvanishesifthedistanceRbetween
the two nucleons is smaller than 1 fm, and is suppressed at R 2 fm. A discussion of two-nucleon correlations is
(cid:29)
usually done in terms of the two-particle density
ˆ
A
ρ ((cid:126)x ,(cid:126)x )=A(A 1) (cid:89) Ψ (x ,...,x )2 A(A 1)ρ((cid:126)x )ρ((cid:126)x )(1 C((cid:126)x ,(cid:126)x )), (34)
2 1 2 A 1 A 1 2 1 2
− | | ≈ − −
i=3
where ρ((cid:126)x) is a normalized to unity (one-particle nuclear density), and the function C((cid:126)x ,(cid:126)x ) interpolates smoothly
1 2
from 1 at distances smaller than 1 fm to 0 at distances larger than 2-3 fm. This object has been extensively studied
in various theoretical models of nuclear structure [51–55], yielding similar results. For infinite nuclear matter, as well
as inside a finite nuclei far from a nuclear border, C((cid:126)x ,(cid:126)x ) depends only on the distance between the two nucleons,
1 2
C((cid:126)x ,(cid:126)x ) C(r = (cid:126)x (cid:126)x ). (35)
1 2 ||(cid:126)x1,2|(cid:28)RA ≈ | 1− 2|
Experimentally,anonzerovalueofthefunctionC revealsitselfinshort-rangecorrelations(SRC)oftwonucleonpairs
knocked out with large momentum, k > k [56, 57], where k is a Fermi momentum. As was found in [58–60], this
F F
effect depends on the isospin of the nucleons, and the dominant contribution (>90%) comes from pn correlations.
However, SRCs can appear not only in specially crafted observables: as was demonstrated in [61], even the total
quasielastic nucleon-nucleus cross-section is reduced by 15% due to SRCs.
From (34) we may deduce a probability to find two nucleons separated by a distance r, provided at least one of the
nucleons is a proton,
ˆ
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:126)r (cid:126)r
ρ ((cid:126)r) (AZ 1) d3Xρ X(cid:126) ρ X(cid:126) + (1 C(r)). (36)
2N
≈ − − 2 2 −
In what follows it is convenient to introduce an effective wave function of relative motion of the 2N system, which
we define as
ˆ
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)1/2
(cid:126)r (cid:126)r
Φ ((cid:126)r) ρ1/2((cid:126)r) (AZ 1) d3Xρ X(cid:126) ρ X(cid:126) + (1 C(r)) , (37)
2N ≡ 2N ≈ − − 2 2 −
anduseaparametrizationofC(r)takenfrom[51], withaWoods-Saxonparametrizationfornucleardensities. Asone
can see from the Figure 4, the shape of the wave function Φ ((cid:126)r) inside lead is very similar to the realistic deuteron
2N
wave function evaluated with an Argonne v -potential [62]. In our evaluations the wave function (37) contributes in
18
convolution with the pentaquark wave function, so the dominant contribution stems from the region r 2 3 fm.
∼ −
The two-nucleon wave function (37) is given in the nucleus rest frame, whereas in equations (1,12,26,30) we need
a wave function in the light-cone formalism. In the general case a relation between the two objects is a nontrivial
dynamical problem, however, under assumption that a motion of nucleons inside the nucleus is nonrelativistic, we
may use the relation suggested in [63, 64]
(cid:115) (cid:32) (cid:115) (cid:33)
k2 +m2 k2 +(2α 1)2m2
Φ˜ (α,k )= 4 ⊥ N Φ˜ k = ⊥ − N , (38)
2N ⊥ 4[α(1 α)]3 2N 4α(1 α)
− −
8
u(r)
w(r)
0.6
Φ (r)
2N
0.5
0.4
)
r
(
Φ
0.3
0.2
0.1
1 2 3 4 5 6
r, fm
Figure 4. (color online) Comparison of the effective 2-nucleon wave function Φ (37) (green solid lines) with components of
2N
realistic deuteron wave function evaluated with Argonne v potential [62] (blue dashed lines). The function Φ is scaled by
18 2N
arbitrary factor to match the peak value of the deuteron wave function component u(r).
where Φ˜ stands for a Fourier transform of the corresponding wave function. As follows from (38) and is seen from
2N
the left pane of the Figure 5 , the α-dependence of the wave function is strongly peaked near a value α 1/2, for
≈
this reason in Equations (1,12,26,30) we do not write a convolution over a light-cone fraction α carried by a nucleon,
tacitly assuming that for all physical observables
ˆ ˆ (cid:18) (cid:19) (cid:32)(cid:42)(cid:18) (cid:19)2(cid:43)(cid:33)
1 1
dαΦ (α, r ) (α,...) dαΦ (α, r ) α , ... + α , (39)
2N ⊥ 2N ⊥
A ≈ A ≈ 2 O − 2
i.e. in Equations (1,12,26,30) we should replace the deuteron wave function Φ with
D
ˆ
Φ2D (r )= dαΦ (α, r ) (40)
LC ⊥ 2N ⊥
In the right pane of the Figure 5 we compare the Φ2D (r ) with a rest frame wave function (37). As we can see,
LC ⊥
the difference between the two objects is relevant only at small-r.
IV. ON INTRINSIC CHARM CONTRIBUTION
In addition to the extrinsic charm contribution discussed in a previous Section II, an additional contribution might
beduetointrinsiccharmofaprojectileproton,firstsuggestedin[6,7]. Thisintrinsiccharmisknownwithconsiderable
uncertainty (see e.g. [24] for a short review). Its contribution to forward pentaquark production might be obtained
by a simple replacement of extrinsic charm with intrinsic charm wave function in (1,12,26,30),
(cid:18) (cid:19)
x g(x )Ψ¯µ¯µ∗(α , r )Φ MPc R(cid:126) Ψν1ν2ν3(α ,(cid:126)r ) (cid:112)P Ψµ¯µν1ν2ν3(α ,(cid:126)r ), (41)
1 1 c cc D −M 2m cc N i i → 5 IC i i
Pc − c
where P is a normalization coefficient which takes into account the amount of intrinsic charm inside a proton, and
5
Ψ isthe(unknown)wavefunctionoftheuudcc¯Fockcomponent. Usuallytheamountofintrinsiccharmisquantized
IC
9
Φ (α,r = 1fm)
2N
Φ (α,r = 2fm)
2N
0.25
8 Φ (α,r = 4fm)
2N
Φ (α,r = 6fm)
] 2N 2]
2 / 0.2
/ 3
3 6 V
V
e
Ge G 0.15
[
[
4 )
) r
r ( 0.1
( Φ
Φ
2
0.05
√m Φ2D(r)
N LC
Φ (r)
2N
1 2 3 4 5 6
0.35 0.4 0.45 0.5 0.55 0.6 0.65
r, fm
α
Figure 5. (color online) Left: α-dependence of the light-cone correlator Φ (α, r ) for several fixed values of r. As we can
2N ⊥
see, the peak is very narrow. Right: Comparison of the light-cone wave function Φ2D (r ) from (40) (solid line) and 3D wave
√ LC ⊥
function (37) (dashed line). m is added to match dimensions.
N
in terms of the fraction of the momentum of a proton carried by c¯c, and current phenomenological estimates for this
quantity indicate a very small amount, x 0.15 0.5% [24]. In view of this, in what follows we neglect the
IC
(cid:104) (cid:105) ∼ −
contribution of the intrinsic charm, tacitly assuming that its contribution could only increase the extrinsic charm
cross-sections. Besides, in case of pA collisions we expect that the extrinsic charm cross-sections will be enhanced by
a factor A1/3 compared to the intrinsic charm contribution.
∼
V. PARAMETRIZATION OF PENTAQUARK AND NUCLEON WAVE FUNCTION
While there are very detailed theoretical models for internal structure of a putative Θ+(see e.g. [65, 66]), as of now
there is no parameterizations of the light-cone wave functions of the P -pentaquark known from the literature. From
c
phenomenological models it is expected that a pentaquark P could be a “molecular” state of J/ψp, χ p Σ D¯ or
c c c
Λ D¯. This implies that a c¯c can be predominantly either in a color singlet or in a color octet state. In the general
c
case, wave functions obtained in effective models are in the rest frame of the baryon. Boosting them to a light-cone
frame presents a complicated dynamical problem and mixes Fock components with different number of quarks. For
this reason, a direct modeling might be a better approach. For this pioneering study, we completely disregard the
spin structure of light quarks both in the pentaquark and in the proton, tacitly assuming that it is the same. We
assume that the proton wave function in its rest frame has the form [37]
(cid:12)
Ψ ( α ,(cid:126)r )=f (α ,α ,α ) √3 exp(cid:18) 1 (cid:0)r2+r2+r2(cid:1)(cid:19)(cid:12)(cid:12) , (42)
p { i i} 3 1 2 3 πRp2 −2Rp2 1 2 3 (cid:12)(cid:12)(cid:80)
i(cid:126)ri=0
where the parameter a is fixed from the charge radius of the proton r2 and equals a=3/(cid:0)2r2 (cid:1). For modelling the
ch ch
light-cone dependence of a pentaquark, we follow the general receipt suggested long ago in [6, 7] for a baryon with n
constituents and choose
(cid:12)
(cid:12)
fn(α1,...,αn)= (cid:16)MB2 −N(cid:80)nni=1 mαi2i(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:80)αi=1, (43)
where M is a mass of a baryon and m is the mass of constituent quarks. This function has a smooth behavior
B i
(cid:80)
provided M < m . For the masses of light and heavy quarks in what follows we take m 0.35 GeV and
B i l
≈
10
Figure 6. (color online) Dependence of the overlap of proton and pentaquark wave functions on x and α .
1 c
m 1.8 GeV respectively. The normalization constant N is fixed from the normalization condition
c n
≈
ˆ (cid:32) (cid:33)
(cid:89)dα δ 1 (cid:88)α f( α )2 =1. (44)
i i i
− | { } |
i i
As was discussed in [6, 7], this function provides a correct endpoint behavior of parton PDFs near α 1. Also, it
i
≈
vanishes when a light-cone fraction of any parton goes to zero, which guarantees that the total cross-section is finite,
as was discussed in the previous Section II.
For the pentaquark, modeling is a bit more complicated due to a myriads of possible spin-orbital arrangements of
all quarks. For us it is important to distinguish the case when c¯c pair is either in S-wave or in a P-wave. In what
follows, we assume that it has a structure similar to (42),
(cid:12)
Ψ ( α ,(cid:126)r )=f (α )vµ¯(p )uµ(p ) √3 exp(cid:18) 1 (cid:0)r2+r2+r2(cid:1)(cid:19)(cid:12)(cid:12) (45)
Pc { i i} 5 i c¯ c πRp2 −2Rp2 1 2 3 (cid:12)(cid:12)(cid:80) ×
i(cid:126)ri=0
Uˆ((cid:126)r ) (cid:18) r2 R2 (cid:19)
cc exp c¯c c¯c ,
× π r R −2 r2 − 2 R2
(cid:104) cc(cid:105)(cid:104) cc(cid:105) (cid:104) cc(cid:105) (cid:104) cc(cid:105)

1 c¯c=S wave, octet
 −
Uˆ((cid:126)r )= 2S(rc¯c a2S) S wave, singlet , (46)
cc N(cid:16) − (cid:17) −
 x√c¯c±iyc¯c P wave
2(cid:104)rcc(cid:105) −
where x , y are the components of the vector r , vµ¯ and uµ are spinors which correspond to the c¯and c-quark 3,
c¯c c¯c c¯c
and f is given by (43). For the color singlet S-wave we assume that the wave function has a node 4, i.e. it is a
5
bound state of ψ(2S) and a proton, as suggested in [18]. The node position a = r √π/ 2 and the value of a
2S cc
(cid:0) (cid:1) (cid:104) (cid:105)
normalization constant = 2/ r √4 π may be extracted from orthonormality with the 1S state and are in
2S cc
N (cid:104) (cid:105) −
reasonable agreement with numerical solutions from potential models of charmonium [67],
For a singlet c¯c, the relevant cross-section is controlled by an overlap (4), which in this case takes the form
(cid:16) (cid:17) Uˆ((cid:126)r ) (cid:18) r2 R2 (cid:19)
µ¯µ α , ξ, (cid:126)r , R(cid:126) = cc vµ¯(p )uµ(p )f (ξ, α )exp c¯c c¯c , (47)
H c cc c¯c π r R c¯ c 3,5 c −2 r2 − 2 R2
(cid:104) cc(cid:105)(cid:104) cc(cid:105) (cid:104) cc(cid:105) (cid:104) cc(cid:105)
and we introduced a shorthand notation
ˆ 3 (cid:32)(cid:26) (cid:27) (cid:33) (cid:32) (cid:33)
f (ξ, α ) (cid:89)dα f αi , ξαc , ξα¯c f (α )δ (cid:88)α =1 . (48)
3,5 c i 5 3 i i
≡ 1+ξ 1+ξ 1+ξ
i=1 light i
3 Theparametrization(45)assumesthatthec¯cpairinapentaquarkcanhaveanyspinwithequalprobability. Inthecaseofmodelswith
internalstructureadditionalspinprojectorsshouldbeadded. Weexpecttheuncertaintyduetospinfactorstoresultinafactoroftwo
uncertaintyinthecross-section.
4 Ifweassumethatthec¯cpairinsideaPc+ isinacolorsinglet1S stateanddoesnothaveanynodes,itwouldbechallengingtoexplain
arelativelynarrowwidthofitsdecayintoJ/ψ andp.