Extended Glauber Model of Antiproton-Nucleus Annihilation for All Energies and Mass Numbers

12 Pages·2014·0.26 MB·English

Extended Glauber Model of Antiproton-Nucleus Annihilation
for All Energies and Mass Numbers
Teck-Ghee Lee1 and Cheuk-Yin Wong2
1Department of Physics, Auburn University, Auburn, AL 36849, U.S.A. and
2Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A.
Previous analytical formulas in the Glauber model for high-energy nucleus-nucleus collisions de-
velopedbyWongareutilized andextendedtostudyantiproton-nucleusannihilations forbothhigh
andlowenergies, aftertakingintoaccounttheeffectsofCoulombandnuclearinteractions,andthe
4
change of the antiproton momentum inside a nucleus. The extended analytical formulas capture
1
themain features oftheexperimentalantiproton-nucleusannihilation cross sections for allenergies
0
andmass numbers. At high antiproton energies, theyexhibit thegranularproperty for thelightest
2
nuclei and the black-disk limit for the heavy nuclei. At low antiproton energies, they display the

y effectofantiprotonmomentumincreaseduetothenuclearinteractionforlightnuclei,andtheeffect
a of magnification dueto theattractive Coulomb interaction for heavy nuclei.
M
PACSnumbers: 24.10.-i,25.43.+t,25.75.-q,

2
1
I. INTRODUCTION what one would expect just from the geometrical radii

] alone [6].
h
t On the theoretical side, a theoretical optical poten-
- Recently, there has been interest in the interaction of
tial based on the Glauber model [23, 24] has been devel-
l
c antimatter with matter, as it is central to our under- oped by Kuzichev, Lepikhin and Smirnitsky to investi-
u standingofthebasicstructureofmatterandthematter-
gate the antiproton annihilation cross sections on Be, C,
n antimatterasymmetryintheUniverse. Ontheonehand,
Al, Fe, Cu, Cd, and Pb nuclei in the momentum range
[
theland-basedFacilityforAntiprotonsandIonsResearch 0.70-2.50 GeV/c [22]. In this range of relatively high

2 (FAIR) at Darmstadt[1, 2] and the Antiproton Deceler-
antiproton momenta, the Glauber model gives a good
v ator (AD) at CERN [3] have been designed to probe the
agreement with the experimental data, with the excep-
8 interactionofantiprotonswithmatteratvariousenergies
tionofthe deviationsatthemomentumof0.7GeV/cfor
8
and environments. The orbiting Payload for Antimat-
1 heavy nuclei. Their study suggested that the A depen-
ter Matter Exploration and Light-Nuclei Astrophysics
6 dence of the annihilation cross sections is influenced by
(PAMELA) [4] and the Alpha Magnetic Spectrometer
. Coulomb interaction at low momenta. In another anal-
1 (AMS) [5] measure the intensity of antimatter in outer
ysis, Batty, Friedman, and Gal have developed a unified
1
space. They have provided interesting hints on the pres-
3 optical potential approach for low-energy p¯ interactions
ence of extra-terrestrial antimatter sources in the Uni-
1 with protons and with various nuclei [25, 26]. Starting
verse. In support of these facilities for antimatter inves-
: with a simple optical potential v¯determined by compar-
v tigations,itisofinteresttoexamineheretheantiproton-
i isonwith p¯-p experimentaldata, a density-folded optical
X nucleus annihilation cross sections that represent an im- potential V = ρv¯ was formulated for the collision of
opt
portantaspectoftheinteractionbetweenantimatterand
r the p¯-nucleus system. They found that even though the
a matter.
density-folding potential reproduces satisfactorily the p¯
To date, significant experimental and theoretical ef- atomic level shifts and widths across the periodic table
forts have been put forth to understand the process of for A>10 and the few annihilation cross sections mea-
annihilation between p¯and various nuclei across the pe- sured on Ne, it does not work well for He and Li. They
riodic table from low energy to high energies. On the attributed this discrepancy to the spin and the isospin
experimental side, annihilation cross sections for p¯A col- averagingand the approximationsmade inthe construc-
lisions,σp¯A,havebeenmeasuredattheLow-EnergyAn- tions of the optical potentials. An extended black-disk
ann
tiproton Ring (LEAR) at CERN [6]-[21] and compiled strong-absorptionmodel has also been considered to ac-
in Ref. [6], and at the Detector of Annihilations (DOA) count for the Coulomb focusing effect for low-energy p¯
at Dubna [22]. A surprising difference in the behavior interactions with nuclei and a fair agreement with the
at high and low energies has been detected. In light measured annihilation cross sections was achieved [26].
nuclei (H, 2H and 4He), the p¯A annihilation cross sec- There are many puzzling features of the p¯A annihi-
tions at p¯ momenta below 60 MeV/c have comparable lation cross sections that are not yet well understood.
values,whereasatmomentagreaterthan500MeV/cthe With respect to the mass dependence, why at high an-
p¯-nucleus annihilation cross sections increases approxi- tiprotonincident momentado the crosssectionsincrease
mately linearly with the mass number A for the lightest almost linearly with the mass number A for the lightest
nuclei[6–9]. Ontheotherhand,forcollisionswithheavy nuclei,butwithapproximatelyA2/3 asthe massnumber
nuclei at low energies, the annihilation cross section ex- increases? Why in the low antiprotonmomentum region
hibits large enhancements that are much greater than dotheannihilationcrosssectionsnotrisewithAasantic-
2
ipatedbutarecomparableforp,2H,andHe,andbecome thatenhancesgreatlytheannihilationcrosssection. Fur-
subsequentlygreatlyenhancedasAincreasesfurtherinto thermore, the antiproton is subject to the nuclear inter-
the heavy nuclei region? With respect to the energy de- actionthatchangesthe antiprotonmomentum in the in-
pendence, how does one understand the energy depen- terior of the nucleus. The change of antiproton momen-
dence of the p¯A annihilation cross sections and the rela- tumisespeciallyimportantinlowenergyannihilationsof
tionship between the energy dependence of the p¯p cross lightnucleibecauseofthestrongmomentumdependence
sections and the p¯A cross sections? In what roles do the of the basic p¯p annihilation cross section. It is necessary
residualnuclear interactionand the long-rangeCoulomb to modify the analytical formulas to take into account
interactioninterplayinthecrosssectionsasafunctionof these effects sothat they canbe appliedto p¯-nucleus an-
chargenumbersandantiprotonmomenta? Wewouldlike nihilations for all energies and mass numbers. Success
to design a model in which these puzzles can be brought in constructing such an extended model will allow us to
up for a close examination. resolve the puzzles we have just mentioned.
Inordertobeabletodescribethecollisionwithallnu- This paper is organized as follows. In Sec. II, we
clei, including deuteron, it is clear that the model needs review and summarize previous results in the Glauber
to be microscopic, with the target nucleon number A modelforhigh-energynucleon-nucleuscollisions,topave
appearing as an important discrete degree of freedom. the way for its application to p¯-nucleus collisions. An-
Furthermore, with the conservation of the baryon num- alytical formulas are written out for the p¯-nucleus an-
ber, an antiproton projectile can only annihilate with a nihilation cross sections in terms of basic p¯-nucleon an-
single target nucleon. The annihilation process occurs nihilation cross section, σap¯n−nnucleon. We use the quark
between the projectile antiproton and a target nucleon model to relate σap¯nnn to σap¯npn so that it suffices to use
locallywithinashorttransverserangealongtheantipro- only σap¯npn to evaluate the p¯-nucleus crosssection. InSec.
ton trajectory. This process of annihilation occurring in III, we represent the basic p¯p annihilation cross section
a short range along the antiproton trajectory is similar by a 1/v law. In Sec. IV, we extend the Glauber model
in character to the high-energy pA reaction process in to study antiproton-nucleus annihilations at both high
which the incident project p interacts with target nu- and low energies, after taking into account the effects
cleons along its trajectory. In the case of high energy of Coulomb and nuclear interactions, and the change of
collisions,the trajectoryofthe incidentprojectilecanbe the antiproton momentum inside a nucleus. In Sec. V,
assumedtobealongastraightline,andthereactionpro- we compare the results of the analytical formulas in the
cess can be properly described by the Glauber multiple extended Glauber model to experimental data. We find
collision model [22–24, 27, 28]. that these analytical formulas capture the main features
oftheexperimentalantiproton-nucleusannihilationcross
Previously, analytical formulas for high-energy
sections for all energies and mass numbers. Finally, we
nucleon-nucleus and nucleus-nucleus collisions in the
conclude the presentstudy with somediscussions inSec.
Glauber multiple collision model have been developed
VI.
by Wong for the reaction cross section in pA collisions
as a function of the basic nucleon-nucleon cross section
[27, 28]. The analytical formula involves a discrete
sum of probabilities whose number of terms depend on II. GLAUBER MODEL FOR p¯-NUCLEUS
the number of target nucleons as a discrete degree of ANNIHILATION AT HIGH ENERGIES
freedom. They givethe resultthat the pA reactioncross
section is proportional to A for small A, and approaches We shall first briefly review and summarize the ana-
the black-disk limit of A2/3 for large A, similar to the lytical formulas in the Glauber multiple collision model
mass-dependent feature of the p¯A annihilation cross [27, 28] for its application to p¯-nucleus annihilations at
sections at high energies mentioned earlier. Hence, it high energies. The Glauber model assumes that the in-
is reasonable to utilize these analytical formulas and cident antiproton travels along a straight line at a high
concepts in the Glauber multiple collision model [28] to energy and makes multiple collisions with target nucle-
provide a description of the annihilation process in p¯A onsalongits way. The targetnucleuscanbe represented
reactions. by a density distribution. The integral of the density
As the analyticalresults inthe Glauber model[27,28] distribution along the antiproton trajectory gives the
pertain to high-energy nucleon-nucleus collisions with thickness function which, in conjunction with the basic
a straight-line trajectory, the model must be extended antiproton-nucleonannihilationcrosssectionσp¯p ,deter-
ann
and amended to make them applicable to low-energy p¯- mines the probability for an antiproton-nucleon annihi-
nucleusannihilations. The incidentantiprotonissubject lation and consequently the p¯-nucleus annihilation cross
tothe initial-stateCoulombinteraction[22,26]. Thean- section [23, 24, 27, 28].
tiproton trajectory deviates from a straight line in low- To be specific, we consider a target nucleus A with a
energy collisions. Before the antiproton comes into con- thickness function T (b ) and mass number A, and a
A A
tactwiththe nucleus,the antiprotontrajectoryis pulled projectile antiproton with a thickness function T (b )
B B
towardsthetargetnucleus,resultinginamagnifyinglens and a mass number B=1. The integral of all thickness
effect(oralternativelyaCoulombfocusingeffect[22,26]) functionsarenormalizedtounity. Forsimplicity,weshall
3
initiallynotdistinguishbetweentheannihilationofapro- where
ton or a neutron. Refinement to allow for different an-
β2 =β2 +β2 +β2 . (7)
nihilation cross sections will be generalizedat the end of A B p¯p
this section.
For this case with Gaussian thickness functions, Eq. (5)
According to Eq. (12.8) of [28], in general, the thick-
then gives the simple analytical formula [27, 28]
ness function T(b) for the annihilation between the pro-
jectile antiproton B and a nucleon in the target nucleus A 1 (1 f)n
A at high energies along a straight-line trajectory at the σap¯nAn(σap¯npn)=2πβ2 − n− , (8)
transverse coordinate b is given by n=1
X
where
T(b)= db db T (b )T (b )t (b b +b ), (1)
A B A A B B ann − A B σp¯p σp¯p
Z Z f = ann = ann . (9)
wheretann(b bA+bB)istheannihilationthicknessfunc- 2πβ2 2π[βA2 +βB2]+σap¯npn
−
tion, specifying the probability distribution at the rela-
Tocheckourtheory,weapplytheresultsfirsttothecase
tivetransversecoordinateb b +b fortheannihilation
− A B of A=1, and we obtain
of a target nucleon at b with an antiproton at b .
A B
The thickness function t (b) for p¯p annihilationat b 1 (1 f)1
ann σp¯p =2πβ2 − − =σp¯p , (10)
can be represented by a Gaussian with a standard devi- ann 1 ann
(cid:18) (cid:19)
ation β ,
p¯p
asitshouldbe. Next, forp¯2H collisionswhere A=2,we
1 b2 have
t (b)= exp . (2)
ann 2πβp2¯p {−2βp2¯p} σp¯2H =2πβ2 2f f2 . (11)
The cross section for a p¯p annihilation is then given by ann − 2
(cid:20) (cid:21)
σp¯p = db(πb2)t (b)=2πβ2 , (3) The situation depends on the size of βp¯p (or σap¯npn), rela-
ann ann p¯p tive to β andβ . There aretwo different limits ofσp¯p
Z in compaArison wBith 2π[β2 +β2]. If σp¯p 2π[β2 +βan2n]
wheredb=2πbdb. Therefore,thestandarddeviationβp¯p then f 0 and A B ann ≪ A B
inthe p¯p annihilationthicknessfunctionisrelatedtothe →
p¯p annihilation cross section by σp¯2H 2σp¯p , (12)
ann ∼ ann
σp¯p
β2 = ann. (4) whichexhibitsthegranularpropertyofthenucleuswhen
p¯p 2π thebasicantiproton-nucleoncrosssectionismuchsmaller
Inap¯-Acollisionathighenergies,theprobabilityforthe than the radius of the nucleus. On the other hand, if
occurrence of an annihilation is T(b)σp¯p . The probabil- σp¯p 2π[β2 +β2], then f 1 and the cross section
ity for no annihilation is [1 T(b)σp¯pan].n With A target baencnom≫e A B →
− ann
natuchliegohnse,ntehregiaesn,naihsialaftuionnctciroonssofseσcap¯tnpino,nisinthaepr¯eAfocroellision σap¯n2Hn ∼ 23σap¯npn. (13)
σp¯A(σp¯p )= db 1 [1 T(b)σp¯p ]A . (5) Inactualcomparisonwith experimentaldata, we use βB
ann ann − − ann = 0.68 fm, and we parametrize β = r′A1/3/√3. The
Z (cid:26) (cid:27) A 0 ′
standardroot-mean-squaredradiusparameterr isofor-
It should be noted that σp¯p depends on the magni- 0
ann der 1 fm (see Table I below). The p¯p annihilation cross
tude of the antiproton momentum relative to the tar- section σp¯p is about 50 mb at pp¯ =2 GeV/c and about
get nucleons. For example, in our later applications to ann lab
1000 mb at pp¯ =50 MeV/c. Thus, f 1 for pp¯p=2
extend the Glauber model to low energies, the antipro- GeV/c and flab 1 for pp¯ =50 MeV/c.≪If the Glalaubber
ton momentum at the moment of p¯-nucleon annihilation ∼ lab
modelremainsvalidforthewholemomentumrange,then
may be significantly different from the incident antipro-
one expects that
ton momentum, and it becomes necessary to specify the
momentum dependence σp¯p in Eq. (5) explicitly. For
brevity of notation, we shaanlnl not write out the momen- σp¯2H/σp¯p 2 for high p¯momenta,(14)
tum dependence explicitly except when it is needed to ann ann(cid:12)Glauber mod∼el (3/2 for low p¯momenta.
(cid:12)
avoid momentum ambiguities. (cid:12)
Analytical expressions of σp¯A(σp¯p ) can be obtained The experim(cid:12)ental data indicate
ann ann
for simple thickness functions [27, 28]. If the thickness
functionsofT andT areGaussianfunctionswithstan- σp¯2H/σp¯p 2 for high p¯momenta, (15)
darddeviationAβ andBβ respectively,thenEq.(1)gives ann ann(cid:12)experimenta∼l (1 for low p¯momenta.
A B (cid:12)
1 b2 The predic(cid:12)(cid:12)ted ratio of σp¯2H/σp¯p appears correct for
T(b)= exp , (6) ann ann
2πβ2 {−2β2} the high-energy region. However, for low-momentum p¯
4
annihilations,weshallseethatthereareimportantmodi- where β2 = β2 +β2 +σp¯p /2π and β2 = β2 +β2 +
p A B ann n A B
ficationsthatmustbemadetoextendtheGlaubermodel σp¯n /2π.
ann
to the low-momentum region, and these modifications On the other hand, as the value of A increases, the
will alter the σap¯n2Hn /σap¯npn ratio in that region. Tp¯x(b) function becomes a uniform distribution. With
As the nuclear mass number A increases, the density thissharp-cut-offdistribution,Eq.(20)leadstothecross
distribution of the nucleus become uniform. The thick- section given by
ness function for the collision of p¯with a heavy nucleus
can be approximated by using a sharp-cut-off distribu- σp¯A(σp¯−nucleon)=πR2 Z N ′ (−1)1+i+jZ!N!
tion of the form (see Ref. [27, 28]) ann ann c,n (Z i)!(N j)!i!j!
i=0 j=0 (cid:18) − − (cid:19)
X X
T(b)= 3 (2Rπc2R−3 b2)θ(Rc−b), (16) 3σap¯npn i 3σap¯nnn jI(i,j,a), (22)
p c × 2πR2 2πR2
(cid:18) c,p(cid:19) (cid:18) c,n(cid:19)
where the contact radius can be taken to be
R =R +R +R . (17) where R = R +R + (3σp¯x /2π) and x = p,n .
c A B pp¯ c,x A B ann
{ }
With this sharp-cut-offdistribution, Eq. (5) leads to the The function I(i,j,a) is q
cross section given by [27, 28]
Rc,n 2bdb b2 i/2 b2 j/2
σp¯A(σp¯p )=πR2 (18) I(i,j,a)= 1 1
ann ann c R2 − R2 − R2
2 1 (1 F)A+2 1 (1 F)A+1 Z0 c,n (cid:18) c,p(cid:19) (cid:18) c,n(cid:19)
1+ − − − − , 1
× F2 A+2 − A+1 = dy(1 ay)i/2(1 y)j/2, (23)
(cid:18) (cid:20) (cid:21)(cid:19) − −
where F is a dimensionless ratio, Z0
F = σap¯npn . (19) uwahteerdenau=meRric2c,na/llRy.c2,p < 1. The I(i,j,a) function is eval-
2πR2/3
c
The p¯p annihilation radius R in Eq. (17) can be cal-
pp¯
ibrated to be R = (3σp¯p /2π) by using the above III. THE BASIC p¯p ANNIHILATION CROSS
pp¯ ann SECTION σap¯npn
equation (18) for the cqase of p¯p collision as point nucle-
ons.
In the foregoing discussions, we assume that σp¯p and In the previous section, analytical formulas have been
ann
σp¯n are the same. They actually differ by about 20%. written out for the annihilation cross sections in the col-
ann lision of an antiproton in terms of σp¯−nucleon (i) for a
Wecantakeintoaccountdifferentannihilationcrosssec- ann
light target nucleus in Eq. (21) [or Eq. (8) if we assume
tionsandthicknessfunctionsT andT forprotonsand
p¯p p¯n σp¯p σp¯n ], and (ii) for a heavy nucleus in Eq. (22) [or
neutrons. Equation (5) can be generalized to become Eaqn.n(∼18)anifnwe assume σp¯p σp¯n ]. The evaluation of
ann ∼ ann
σp¯A(σp¯−nucleon)= db 1 [1 T (b)σp¯p ]Z the p¯A annihilation cross section will require knowledge
ann ann − − p¯p ann of the basic σp¯−nucleon annihilation cross section.
Z (cid:26) ann
From the quark model, a proton p consists of uud,
×[1−Tp¯n(b)σap¯nnn]N a neutron n consists of udd, and a p¯ is u¯u¯d¯. If only
(cid:27) flavor and antiflavor can annihilate, one would expect
Z N ′ ( 1)1+i+jZ!N! theprobabilityofp¯-nannihilationequals(4/5)timesthe
= −
(Z i)!(N j)!i!j! probability of p¯-p annihilation, that is,
i=0 j=0 (cid:18) − − (cid:19)
X X
σp¯n =(4/5)σp¯p . (24)
(σp¯p )i(σp¯n )j db[T (b)]i[T (b)]j,(20) ann ann
× ann ann Z p¯p p¯n The experimental value for the ratio (σap¯nnn)2H/(σap¯npn)2H
where the argument σp¯−nucleon on the left-hand side for the annihilation inside a deuteron has been de-
stands for σp¯p and σp¯na,nnand the summation ′ allows termined by Kalogeropoulos and Tzanakos [14] to be
ann ann j
for all cases except when i=j =0. 0.745 0.016 and 0.863 0.016 for p¯ at rest and
P ± ±
For small and moderate mass numbers A < 40, the in flight, respectively, giving an average of 0.804 0.022,
±
thickness functions T (b) canbe assumedto be a Gaus- in approximate agreement with the prediction from the
p¯x
sianfunction witha standarddeviationβ , andEq.(20) naive quark model. We shall use the relation Eq. (24)
x
becomes in our subsequent data analysis. By relating σp¯n with
ann
Z N ′ ( 1)1+i+jZ!N! σap¯npn usingEq.(24),theevaluationofthep¯Aannihilation
σap¯nAn(σap¯n−nnucleon)=2π (Z− i)!(N j)!i!j! ccrroosssssseeccttiioonn.willrequireonlythe basicσap¯npn annihilation
i=0 j=0 (cid:18) − − (cid:19)
X X
In our previous theoretical study in connection with
σp¯p i σp¯n j β2β2
ann ann p n ,(21) the annihilationlifetime of matter-antimatter molecules,
×(cid:18)2πβp2(cid:19) (cid:18)2πβn2(cid:19) iβn2 +jβp2! we note that the experimental data of the p¯p total cross
5
section and the p¯p elastic cross section as a function of IV. EXTENDING THE GLAUBER MODEL TO
theantiprotonmomentumforafixedprotontarget,p , LOW ENERGIES
p¯lab
are related by [29]
The results in Sec. II pertains to annihilation of the
σ
σ =σ + 0, (25) antiprotonathighenergieswithstraight-linetrajectories.
tot elastic
v
Toextendthe rangeofthe applicationto lowenergies,it
is necessary to forgo the assumption of straight-line tra-
where v is the velocity of the antiproton
jectories. We need to take into account the modification
pp¯lab of the trajectories due to residual Coulomb and nuclear
v = . (26)
interactions that are additional to those between the in-
p2 +m2
p¯lab p¯ cident antiproton and an annihilated target nucleon.
q The residual Coulomb and nuclear interactions affect
As the difference of the experimental total cross sec-
the annihilation process in different ways. The Coulomb
tion andthe elastic crosssection,the secondtermin Eq. interaction is long-range, and the trajectory of the an-
(25) represents the p¯p inelastic cross section. It is essen-
tiprotonisattractedandpulledtowardthetargetnucleus
tially the annihilation cross section as the latter domi- before the antiproton makes a contact with the nucleus
nates among the inelastic channels. The p¯p annihilation
[Fig. 2(a)]. It leads to a large enhancement of the an-
cross section can therefore be parametrized in the form nihilation cross sections at low energies [22, 26]. The
nuclear interaction is short-range, and it becomes oper-
σ
σp¯p = 0. (27) ative only after coming into contact with the nucleus.
ann v
The interactions modify the antiproton momentum as it
In the present work, we use the experimental p¯p anni- travels in the nuclear interior in which p¯-nucleon annihi-
hilationcrosssectionsdirectlytofine-tunetheparameter lation takes place. As the basic antiproton-nucleon an-
σ0. We find that nihilation cross section σap¯npn as given by (27) is strongly
momentum-dependent, we need to keep track of the an-
σ =43 mb (28) tiproton momentum along the antiproton trajectory.
0
We shall work in the p¯-A center-of-mass system and
gives a good description of the experimental σp¯p (p ) shall measure the antiproton momentum in terms of the
ann p¯lab
data as shown in Fig. 1. For brevity of notation, the relative momentum pp¯A defined as
quantity p will be abbreviated as p in all figures.
p¯lab lab m p m p
Figure 1 indicates that the p¯p annihilation cross sec- p = A p¯− p¯ A, (29)
p¯A m +m
tion has a strong momentum dependence. It decreases p¯ A
by an order of magnitude as the antiproton momentum whereinthecenter-of-masssystemwithp +p =0,we
p¯ A
increases from 30 to 600 MeV/c. have p =p .
p¯ p¯A
It should be pointed out that the 1/v law, Eq. (27),
for the inelastic (or annihilation) cross section of slow
particlesiswellknown. ItwasfirstobtainedbyBethe[30] A. Initial-State Coulomb Interaction
and is discussed in text books [31–33] and other related
work [34]. It arises from multiplying the S-wave partial We consider the collision of an antiproton with a nu-
cross section, π/k2, by the transmission coefficient T0 cleus in the p¯A center-of-mass system with a center-of-
in passing through an attractive potential well, and the massenergyE. Theinitialantiprotonmomentump is
p¯A
transmission coefficient T0 is proportional to k at low related to E by
energies. Whilethe1/vbehaviorisessentiallyanS-wave
p2
result,higher-l partialwaveswillgraduallycontributeas p¯A
E = . (30)
the antiproton momentum reaches the GeV/c region. It 2µ
isnonethelessinterestingtonotethatthe simple1/v law
After the projectile antiproton travels along a Coulomb
ofEq.(27)continuestoprovideareasonableandefficient
trajectory, it makes contact with the nucleus at r = R
description of the experimental data even in the GeV/c with a momentum p′ determined by c
region [29]. p¯A
Recently, an elaborate and model-independent [p′ ]2
p¯A
coupled-channel partial-wave calculation, solving the E = +Vc(Rc), (31)
2µ
problem from first principle, has been used to determine
p¯-p scattering cross sections for momenta below 0.925 where Vc(Rc) is the Coulomb energy for the antiproton
GeV/c. The calculation yields excellent agreement with to be at the nuclear contact radius Rc,
theexperimentalp¯-pannihilationcrosssectionforenergy
(Z 1)α
A
ranging from 0.200 to 0.925 GeV/c [35]. Refinement of V (R )= − , (32)
c c
the presentσp¯p descriptioncan be made, if desired, but − Rc
ann
with additional complications. and Z is the target charge number.
A
6
5 p Ni
10
D Cu
4
He Cd
Be Sn
C Pt
Pt Ne Pb
4 Al
10
)
b Sn
m
Ni
( Ne
3He PbPt

Sn
A n Cd
n 3
p a10 NCiu
σ
Al
Ne
C
Be
4
He
2
10
D
p
100 1000
p (MeV/c)
lab
FIG. 1. (Color online) Antiproton-nucleus annihilation cross sections as a function of the antiproton momentum in the labo-
ratory, plab ≡ pp¯lab, for different nuclear targets. The solid curve for the proton target nucleus is the σ0/v phenomenological
representation of the p¯p annihilation cross section in Eq. (27). The other curves are the results from the extended Glauber
model using the basic p¯p annihilation cross section as input data. The solid curves are for Gaussian density distributions and
thedashedcurvesareforuniformdensitydistributions. Thehigh-momentumdatapointsarefrom[22]. Theotherexperimental
data points are from the compilation of [6], where the individualexperimental sources can be found.
7
The p¯p annihilation cross section arises from the nu- From Eqs. (34) and (35), the above Eq. (37) becomes
clear and Coulomb interaction between the antiproton
[p′ ]2
aemndenattianrgtehtepmroutlotnip.leBcyoellmispiolnoypinrgocσeaps¯nspninasththeeebxatesnicdeeld- Σpa¯nAn(pp¯A)= pp¯2A db′ 1−{1−Tp¯p[b′(b)]σap¯npn(p′p¯A)}Z
p¯A Z (cid:26)
Glauber model of Eq. (5), the parts of the Coulomb and
nuclearinteractionthatareresponsibleforthep¯pannihi- ×{1−Tp¯n[b′(b)]σap¯nnn(p′p¯A)}N
lation have already been included. Therefore in Eq. (31) (cid:27)
for p¯-nucleus annihilation, we are dealing with residual = 1 Vc(Rc) db′ 1 1 T [b′(b)]σp¯p (p′ ) Z
interactionsthat are additionalto those betweenthe an- − E −{ − p¯p ann p¯A }
(cid:26) (cid:27)Z (cid:26)
tiprotonandtheannihilatednucleon. Hence,Eq.(32)for
1 T [b′(b)]σp¯n (p′ ) N (38)
the residual Coulombinteractioncontains the coefficient ×{ − p¯n ann p¯A }
(Z 1). (cid:27)
A
Fr−om angular momentum conservation, we have From the results in Eq. (6), the above equation becomes
pp¯Ab=p′p¯Ab′. (33) Σpa¯nAn(pp¯A)= 1− Vc(ERc) σap¯nAn(σap¯npn(p′p¯A)). (39)
(cid:26) (cid:27)
We obtain
′
p
p¯A ′
b= b. (34)
p
p¯A
From Eq. (31), we have p
pA s
V (R )
′ c c
pp¯A =pp¯A 1− E , (35) b Rc
r b′
which is the relative momentum of the antiprotonin the
p¯A system at the nuclear contact radius R . O
c (a)
′ target A
Starting now fromthe transversecoordinatesb at nu-
clear contact at r = Rc, one can follow the antiproton p-nucleus collision
trajectory in the nuclear interior. This trajectory will Coulomb interaction magnifies b′ to b
be modified by residualinteractions. One canevaluate a
′ ′
thickness function T (b) by integrating the nuclear den-
sityalongthemodifiedtrajectory. Asthethicknessfunc-
′ ′
tionT (b)isgovernedmainlythe geometryofthetarget
′ ′
nucleus,wethereforeexpectthatT (b)willbecharacter-
ized by a length scale that will not be too different from
the length scale in the thickness function T(b′) without s R b′
residualinteractions. Tothelowestorder,itisreasonable b c
′ ′ ′
to approximate T (b) by T(b). With such a simplifying O
(b)
assumption,thep¯Aannihilationcrosssectionataninitial target A
antiproton momentum p is
p¯A heavy-ion collision
Coulomb interaction reduces b′ to b
Σp¯A (p )= db 1 1 T [b′(b)]σp¯p (p′ ) Z
ann p¯A −{ − p¯p ann p¯A }
Z (cid:26)
1 T [b′(b)]σp¯n (p′ ) N , (36) FIG. 2. (Color online) Schematic picture of the bending
×{ − p¯n ann p¯A } of the antiproton trajectories s as the antiproton approaches
(cid:27)
the nucleus: (a) Under the action of the attractive p¯-nucleus
′
where we use a new symbol Σ to indicate that this is Coulomb interaction, the impact parameter b in the nuclear
the result in an extended Glauber model for p¯-nucleus interior is magnified to the initial impact parameter b which
annihilation, modified to take into account the Coulomb determines the annihilation cross section. (b) Under the ac-
tionofarepulsiveCoulombinteractioninheavy-ioncollisions,
interactionthatchangesthe antiprotonmomentumfrom
′ the initial impact parameter b′ in the nuclear interior is re-
initial p to p at the nuclear contact radius R . We
p¯A p¯A c ducedtotheinitialimpactparameterbwhichdeterminesthe
can carry out a change of variable
reaction cross section.
Σp¯A (p )= db′ bdb 1 1 T [b′(b)]σp¯p (p′ ) Z With the aboveformulation,the initial-state Coulomb
ann p¯A b′db′ −{ − p¯p ann p¯A }
Z (cid:26) interaction can be incorporated into the extended
×{1−Tp¯n[b′(b)]σap¯nnn(p′p¯A)}N , (37) Getlearubbetromthoedeimlapsaactmpaaprpaimngetoefrtbh′eaitnintuiaclleimarpcaocnttpaacrta.mIt-
(cid:27)
8
′′
canbe pictoriallydepictedasa lenseffect inFig. 2. The aboveequation,wecanthenapproximatep bythe av-
p¯A
attractive Coulomb interaction between the antiproton erage root-mean-squared momentum (p′′ )2 1/2 in the
and the nucleus acts as a magnifying lens that magni- interior of the nucleus, h p¯A i
′
fies the impact parameter b at the contact radius R to
c
turn it into the initial impact parameter b, with which V (r) + V (r)
′′ c n
the reaction or annihilation cross section is measured pp¯A ∼pp¯A 1− h iEh i. (42)
r
[Fig. 2(a)]. The magnifying lens effect for the attractive
Coulomb interaction with b>b′ leads to a p¯-nucleus an- The p¯-nucleus annihilation crosssection Σp¯A is there-
ann
′ ′′
nihilationcrosssectiongreaterthanthegeometricalcross fore modified by changing p in Eq. (39) to p to be-
p¯A p¯A
section for heavy nuclei in low-energy collisions, behav- come
ingas[1 V (R )/E]σp¯A(σp¯p (p′ ))asgiveninEq.(39),
whereV−(Rc),tcheCoualnonmbaennnergp¯Ayatthenuclearcontact Σp¯A (p )= 1 Vc(Rc) σp¯A(σp¯p (p′′ )). (43)
c c ann p¯A − E ann ann p¯A
radius Rc, is negative. (cid:26) (cid:27)
It is interesting to note in contrast that in heavy-ion The experimentaldataof σp¯p andΣp¯A arepresented
collisions,the repulsive Coulombinitial-state interaction ann ann
as a function of p for a fixedprotonor nucleus target
p¯lab
actsasareducinglensthatreducestheimpactparameter
atrest. Accordingly,weconverttheantiprotonmomenta
′
b at contact to become the initial impact parameter b
inthecenter-of-masssysteminthaboveEq.(43)tothose
′
with b < b as illustrated in Fig. 2(b). The lens effect
in the laboratory system with fixed targets as
for a repulsive Coulomb interaction leads to a reaction
cross section reduced from the geometrical cross section A
πRc2 to[1−Vc(Rc)/E]πRc2,whereVc(Rc)istheCoulomb pp¯A = A+1pp¯lab, (44)
energyattheCoulombbarrierandispositive[31,36,37]. A
′′ ′′
Therefore,oneobtainsthe unifying picture thatforboth pp¯A = A+1pp¯lab, (45)
the p¯-nucleus and heavy-ion collisions, the initial-state
Coulomb interaction act as a lens, leading to the same and Eq. (42) gives
Coulomb modifying factor [1 V (R )/E].
c c
−
V (r) + V (r)
′′ c n
pp¯lab =pp¯lab 1− h iEh i. (46)
r
B. Change of the Antiproton Momentum in the
Therefore, the antiproton-nucleus annihilation cross sec-
Nucleus Interior
tion Σp¯A(p ) in the extended Glauber model for an
in p¯lab
antiproton with an initial momentum p is given in
p¯lab
Because the basic p¯p annihilation cross section is
the following compact form
stronglymomentumdependent,thereis,however,anad-
ditional important amendment we need to make. In the V (R )
Σp¯A (p )= 1 c c σp¯A(σpp¯ (p′′ )), (47)
presence of the nuclear and Coulomb interactions in the ann p¯lab − E in ann p¯lab
nuclear interior, the antiproton momentum p′ at the (cid:26) (cid:27)
p¯A
contact radius is changed to the momentum p′′ in the where σp¯A(σ (p′′ )) is given by Eq. (8) for light nuclei
p¯A in pp¯ p¯lab
interiorofthe nucleus. Thep¯pannihilationoccursinside with a Gaussian thickness distribution, and by Eq. (18)
′′
the nucleus ata momentum pp¯A. It is necessaryto mod- for heavy nuclei with a sharp-cut-off thickness function,
ify Σp¯p in Eq. (39) to take into account this change of withthebasicquantityσp¯p inEqs.(9)or(19)evaluated
ann ann
the antiproton momentum by replacing p′ with p′′ . atp′′ givenintermsofp byEq. (46). TheCoulomb
p¯A p¯A p¯lab p¯lab
′′
The antiproton momentum p is a function of the energy at contact V (R ) is given by Eq. (32), and the
p¯A c c
radialpositionr andis relatedtopp¯A bythe energycon- average Vc(r) intheinteriorofthenucleus(ZA,A)with
h i
dition: a radius R in Eq. (46) is given by
A
E = p22p¯µA = (p2′p¯′Aµ)2 +Vc(r)+Vn(r). (40) hVc(r)i=−3(ZA−1)α 5R1c20−R3RA2 . (48)
(cid:18) c (cid:19)
In order to obtain an analytical formula for general InourapplicationoftheextendedGlaubermodel,con-
purposes for our present approximate treatment, it suf- cepts such as the contact radius R and V (r) are sim-
c c
h i
fices to consider average quantities and use the root- plest for a uniform density distribution. For the evalua-
mean-square average (p′′ )2 1/2 given by tion of these quantities in the case of small nuclei with
h p¯A i
a Gaussian thickness function, we shall approximate the
(p′′ )2 Gaussianas a uniform distribution [only for the purpose
E = h p¯A i + V (r) + V (r) , (41)
2µ h c i h n i ofcalculatingRc and Vc(r) inEqs.(32),(46),and(48)]
h i
with an equivalence between R and β. We note that
c
where V (r) and V (r) are the average interior the dimensionless quantity f in Eq. (9) for a Gaussian
c n
h i h i
Coulombandnuclearinteractions,respectively. Fromthe thicknessfunctionandthe quantityF inEq.(19)forthe
9
sharp-cut-off distribution have the same physical mean-
TABLEI. Fitting parameters
ing. It is reasonable to equate the corresponding quan-
tities 2πβ2 in Eq. (9) with the corresponding quantity Nuclei Gaussian Uniform hVni(MeV)
′
2πR2/3 in Eq. (19), leading to the approximate equiva- r0(fm) r0(fm)
c 2H 1.20 -1.0
lence 4He 1.20 -4.0
R (for Gaussian distribution) √3β, (49) Be 1.20 -10.0
c ∼ C 1.20 -15.0
Ne 1.10 -25.0
and similarly,
Al 1.10 -30.0
R (for Gaussian distribution) √3β . (50) Ni 1.04 -30.0
A ∼ A Cu 1.04 -30.0
Cd 1.04 -35.0
These equivalence relations enable us to obtain the
Sn 1.04 -35.0
Coulomb factor [1 V (R )/E] in (47) and the antipro-
− c c ′′ Pt 1.04 -35.0
tonmomentumchangefromp top inEq.(46),for
p¯lab p¯lab Pb 1.04 -35.0
light nuclei with Gaussian thickness functions.
1500
V. COMPARISON OF THE EXTENDED (a)
GLAUBER MODEL WITH EXPERIMENT mb) 1000 <Vn> = 0 MeV p
ThecentralresultsoftheextendedGlaubermodelcon- A (n <Vn>< V=n ->1 .=0 -M2.e0V MeV D
pan 500 D
sist of Eq. (43) or (47) and their associated supplemen- σ
tary equations. With the basic p¯p annihilationcrosssec- p
tionσap¯npnanditsmomentumdependencewellrepresented 20000
from experimental data by Eq. (27), as discussed in Sec. 3
He
III, it is only necessary to specify the residual nuclear
(b)
interaction VN andthe nucleargeometricalparameters 1500
h i
inEq.(43)toobtainthep¯-nucleusannihilationcrosssec- b) <Vn> = -2.0 MeV
tdinieonEns.qit.Fy(o6dr)i,sltiwgrihibtthuntgiueoconlmewiei(ttwrhiicttahhlepAGara<aums4se0it)ae,nrwstheβicuks=neears′sGAfua1n/u3cs/tsii√oann3 pA (mann1000 <V<nV> n=> - =4. 0-6 M.0e MVeV 34HHee
andβ =0.68. We findthat 1.10<r′ <A1.20f0m givesa σ 4He
B 0 500
gooddescription. Fortheheavynuclei(withA>40),we
use a uniform density distribution with the sharp-cut-off
thickness function in Eq. (16) with geometrical parame- 0
100 200 300 400 500 600
tersR =r A1/3andR =0.8fmTheradiusparameter p (MeV/c)
A 0 B lab
r =1.04 fm fit the data well.
0
InFig.1,thesolidcurvefortheprotontargetnucleusis FIG. 3. (Color online) p¯annihilation cross sections of (a) p
fromthe σ /v phenomenologicalrepresentationofthep¯p andDnucleiand(b)4Henucleus. Theexperimentaldataare
0
annihilation cross section in Eq. (27). The other curves fromRef. [6]. Thep¯pcurveisfromtheσ0/vrepresentationof
are the results from the extended Glauber model using Eq.(27),andtheothercurvesarefromtheextendedGlauber
model.
thebasicp¯pannihilationcrosssectionasinputdata. The
solid curves are for Gaussian density distributions and
the dashed curves are for uniform density distributions.
Thefitting parametersthatgivethetheoreticalp¯Aanni- Coulomb interaction energy Vc(Rc) is also small in com-
hilation cross sections in Fig. 1 are listed in Table I. parison with the incident energy E. As a consequence,
The comparison of the extended Glauber model with the corrections due to the Coulomb and nuclear inter-
the experimental data in Fig. 1 indicates that although actions are small for high-energy collisions. At these
thefitsarenotperfect,theextendedGlaubermodelcap- high energies, the antiproton makes multiple collisions
tures the main features of the annihilation crosssections with target nucleons and probes the granular property
forallenergiesandmassnumbers. Wewillmentionafew of the nucleus when the spacing between the nucleons
ofthenotablefeaturesofthedataandthecorresponding is large compared with the dimension of the p¯ probe,
explanations in the extended Glauber model. similar to the additive quark model in meson-mesoncol-
We examine first p¯A annihilation at high energies. In lisions [38, 39]. The experimental Σp¯2H/σp¯p is consis-
ann ann
these high-energy annihilations, the momentum depen- tently slightly greater than the theoretical prediction of
dence of the basic σp¯p is not sensitive to the antiproton Σp¯2H/σp¯p 2. This indicates that while the Glauber
ann ann ann∼
momentum change arising from the residual nuclear and model for the antiproton annihilation of the deuteron
Coulomb interactions V and V . The initial-state may be approximately valid, future investigations will
c N
h i h i
10
needtoincludeadditionalrefinementstogetabetterde- inFig.3 in a linearplot inboth the lowandhighenergy
scriptionfortheantiproton-deuteronannihilation. There regions. As shown in Figs. 1 and 3(a), the basic cross
are not many high-momentum data points for 4He, and section σp¯p is of order 1000 mb at the low momentum
ann
the onlyhigh-momentumdata point at600MeV/c gives of p 20 MeV/c, corresponding to a effective annihi-
p¯lab
∼
reasonable agreement with the Glauber model results. lation radius between p¯and p of R =(σp¯p /π)1/2=6.8
p¯p ann
Asthenumberofnucleonsincreasesinhigh-energyan- fm. The large cross section and annihilation radius arise
nihilations,theGlaubermodelresultsofEqs.(8)[or(21) from the magnifying lens effect of the attractive initial-
forσp¯n =σp¯p ]and(18)[or(22)forσp¯n =σp¯p ]giveap¯A state Coulomb interaction that magnifies the proton ra-
annihanilna6tionancnross section proportionaanlnt6o βa2nnfor a Gaus- dius as seen by the incoming antiproton, as discussed in
sian thickness distribution and to R2 for a uniform den- Sec. III. In this case, as R R or R ,
c p¯p deuteron antiproton
sity distribution. Both β2 and R2 vary as A2/3. Hence, the quantity f in Eq. (8) is≫close to unity. Accord-
c
thep¯Aannihilationcrossapproachestheblack-disklimit ing to the analysis given in Eq. (14) of Sec. III, the
of A2/3 limit as the nuclear mass number increases. The Glauber model with f 1 would predict a ratio of
comparisoninthehigh-energyregioninFig.1showsthat Σp¯2H/σp¯p 3/2forlow-∼energyannihilationsinthemul-
the experimental data agree with predictions for nuclei tipanlencolalnisnio∼n process.
across the periodic table, indicating that the experimen-
tal data indeed reach this black-disk limit of A2/3 in the 14000
heavy-nucleiregion,However,there arediscrepanciesfor
12000
Pbat700MeV/c,whichmayneedneedtobe re-checked
experimentally, as the other data points for large nuclei 10000 Pt
appear to agree with theoretical predictions. b)
m8000
1200 (a) C pAσ (ann6000 Sn
4000 Ni
900 <Vn>= -10 MeV
<Vn>= -15 MeV 2000
<Vn>= -20 MeV
600 0 60 80 100 120 140 160 180 200 220
A
300 FIG. 5. (Color online) p¯annihilation cross sections on Ni,
0 100 200 300 400 500 600 700 800 900 SnandPtnucleiatpp¯lab =100MeV/c. Thesolidcurvegives
3000 resultsfromtheextendedGlaubermodelandthedatapoints
(b) Ne
are from Ref. [6].
mb) <Vn> = -15 MeV
A (nn2000 <<VVnn>> == --2205 MMeeVV Σp¯E2Hxp/σerp¯ipmeinstaolflyo,rderfournity.lowIn-etnheergeyxtenadnendihGillaatuiobners
pa1000 ann ann
σ model, the reduction of the ratio Σp¯2H/σp¯p arises from
ann ann
the combination of two effects: (i) the increase of the
0
0 100 200 300 400 500 600 700 800 900 antiproton momentum inside the nucleus due to the
2000
attractive residual interactions and (ii) the momentum
(c) Al dependence of the basic σp¯p decreases sensitively as a
ann
1500 <Vn> = -20 MeV function of an increase in antiproton momentum. The
<Vn> = -25 MeV combined effects bring the ratio Σp¯2H/σp¯p from 3/2
<Vn> = -30 MeV ann ann
1000 to about unity. In Fig. 3, we show the variation of the
p¯2H and p¯4He cross sections as a function of p for
p¯lab
different residual nuclear interactions V . There is a
500 h Ni
great sensitivity of the p¯-annihilation cross section on
0 100 200 300 400 500 600 700 800 900
plab (MeV/c) VN in the low energy region for the lightest nuclei for
which the Coulomb interaction is weak. However,as the
FIG. 4. (Color online) p¯annihilation cross sections of (a) C targetchargenumberincreases,theCoulombinteraction
nucleus,(b)Nenucleus,and(c)Alnucleus. Solidcurvesgive becomes stronger and the p¯-annihilation cross section
resultsfromtheextendedGlaubermodelandthedatapoints becomes less sensitive to the strength of the nuclear
are from thecompilation of Ref. [6]. interaction V , as shown in Fig. 4 for p¯C, p¯Ne, and
N
h i
p¯Al collisions.
Thesituationatthe low-energyregionismorecompli- As the target charge number Z increases further in
cated. In addition to the multiple collision process, the the low-energy region, the cross section increases sub-
Coulomb and nuclear interactions also come into play. stantially. For example, the crosssectionreachesa value
We can examine the data for p, 2H, and He more closely ofabout9000mbforthePtnucleus,correspondingtoan