arX

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v1 [

nucl

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201

3

Extended Glauber Model of Antiproton-Nucleus Annihilation

for All Energies and Mass Numbers

Teck-Ghee Lee1 and Cheuk-Yin Wong21Department 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-veloped by Wong are utilized and extended to study Antiproton-nucleus annihilations for both highand low energies, after taking into account the effects of Coulomb and nuclear interactions, and thechange of the antiproton momentum inside a nucleus. The extended analytical formulas capturethe main features of the experimental antiproton-nucleus annihilation cross sections for all energiesand mass numbers. At high antiproton energies, they exhibit the granular property for the lightestnuclei and the black-disk limit for the heavy nuclei. At low antiproton energies, they display theeffect of the antiproton momentum increase due to the nuclear interaction for the light nuclei, andthe effect of the magnification due to the attractive Coulomb interaction for the heavy nuclei.

PACS numbers: 25.43.+t 25.75.-q 25.90.+k

I. INTRODUCTION

Recently, there is much interest in the interaction ofantimatter with matter, as it is central to our under-standing of the basic structure of matter and the matter-antimatter asymmetry in the Universe. On the one hand,the land-based FAIR (Facility for the Research with An-tiprotons and Ions) at Darmstadt[1, 2] and the AD (An-tiproton Decelerator) at CERN [3] have been designedto probe the interaction of antiprotons with matter atvarious energies and environments. On the other hand,the orbiting PAMELA (Payload for Antimatter MatterExploration and Light-nuclei Astrophysics) [4] and theAMS (Alpha Magnetic Spectrometer) [5] measure the in-tensity of antimatter in outer space. They have providedinteresting hints on the presence of extra-terrestrial an-timatter sources in the Universe. In support of thesefacilities for antimatter investigations, it is of interest toexamine here the antiproton-nucleus annihilation crosssections that represent an important aspect of the inter-action between antimatter and matter.

To date, significant experimental and theoretical ef-forts have been put forth to understand the process ofannihilation between p and various nuclei across the pe-riodic table from low energy to high energies. On theexperimental side, annihilation cross sections for pA col-lisions, σpA

ann, have been measured at LEAR (Low-EnergyAntiproton Ring) at CERN [6]-[22] and compiled in Ref.[6], and at the DOA (Detector of Annihilations) at Dubna[23]. A surprising difference in the behavior at high andlow energies has been detected. In light nuclei (H, Dand 4He), the pA annihilation cross sections at p mo-menta below 60 MeV/c have comparable values, whereasat momenta greater than 500 MeV/c the p-nucleus an-nihilation cross sections increases approximately linearlywith the mass number A for the lightest nuclei [6–9]. Onthe other hand, for collisions with heavy nuclei at lowenergies, the annihilation cross section exhibits large en-hancements that are much greater than what one would

expect just from the geometrical radii alone [6].On the theoretical side, a theoretical optical potential

based on the Glauber model [24, 25] has been developedby Kuzichev, Lepikhin and Smirnitsky to investigate theantiproton annihilation cross sections on Be, C, Al, Fe,Cu, Cd, and Pb nuclei at the momentum range of 0.70-2.50 GeV/c [23]. In this range of relatively high antipro-ton momenta, the Glauber model gives a good agreementwith the experimental data, with the exception of the de-viations at the momentum of 0.7 GeV/c for heavy nuclei.Their study suggested that the A-dependence of the anni-hilation cross sections is influenced by Coulomb interac-tion at low momenta. In another analysis, Batty, Fried-man and Gal have developed a unified optical potentialapproach for low-energy p interactions with proton andwith various nuclei [26, 27]. Starting with a simple opti-cal potential v determined by comparison with p-p exper-imental data, a density-folded optical potential Vopt = ρvwas formulated for the collision of the p-nucleus system.They found that even though the density-folding poten-tial reproduces satisfactorily the p atomic level shifts andwidths across the periodic table for A>10 and the few an-nihilation cross sections measured on Ne, it does not workwell for He and Li. They attributed this discrepancy tothe spin and the isospin averaging and the approxima-tions made in the constructions of the optical potentials.An extended black-disk strong-absorption model has alsobeen considered to account for the Coulomb focusing ef-fect for low-energy p interactions with nuclei and a fairagreement with the measured annihilation cross sectionswas achieved [27].There are many puzzling features of the pA annihi-

lation cross sections that are not yet well understood.With respect to the mass dependence, why at high an-tiproton incident momenta do the cross sections increasealmost linearly with the mass number A for the lightestnuclei, but with approximately A2/3 as the mass numberincreases? Why in the low antiproton momentum regiondo the annihilation cross sections not rise with A as an-ticipated but are comparable for p, D, and He, and they

2

become subsequently greatly enhanced as A increasesfurther into the heavy nuclei region? With respect tothe energy dependence, how does one understand the en-ergy dependence of the pA annihilation cross sections andthe relationship between the energy-dependence of the ppcross sections and the pA cross sections? In what rolesdo the residual nuclear interaction and the long-rangeCoulomb interaction interplay in the cross sections asa function of charge numbers and antiproton momenta?We would like to design a model in which these puzzlescan be brought up for a close examination.

In order to be able to describe the collision with all nu-clei, including deuteron, it is clear that the model needsto be microscopic, with the target nucleon number Aappearing as an important discrete degree of freedom.Furthermore, with the conservation of the baryon num-ber, an antiproton projectile can only annihilate with asingle target nucleon. The annihilation process occursbetween the projectile antiproton and a target nucleonlocally within a short transverse range along the antipro-ton trajectory. This process of annihilation occurring ina short range along the antiproton trajectory is similarin character to the high-energy pA reaction process inwhich the incident project p interacts with target nu-cleons along its trajectory. In the case of high energycollisions, the trajectory of the incident projectile can beassumed to be along a straight line, and the reaction pro-cess can be properly described by the Glauber multiplecollision model [23–25, 28, 29].

Previously, analytical formulas for high-energynucleon-nucleus and nucleus-nucleus collisions in theGlauber multiple collision model have been developedby Wong for the reaction cross section in pA collisionsas a function of the basic nucleon-nucleon cross section[28, 29]. The analytical formula involves a discretesum of probabilities whose number of terms depend onthe number of target nucleons as a discrete degree offreedom. They give the result that the pA reaction crosssection is proportional to A for small A, and approachesthe black-disk limit of A2/3 for large A, similar to themass-dependent feature of the pA annihilation crosssections at high energies mentioned earlier. Hence, itis reasonable to utilize these analytical formulas andconcepts in the Glauber multiple collision model [29] toprovide a description of the annihilation process in pAreactions.

As the analytical results in the Glauber model [28, 29]pertain to high-energy nucleon-nucleus collisions with astraight-line trajectory, the model must be extended tomake them applicable to low-energy p-nucleus annihila-tions. The incident Antiproton is subject to the initialstate Coulomb interaction [23, 27]. The antiproton tra-jectory deviates from a straight line in low energy col-lisions. Before the antiproton comes into contact withthe nucleus, the antiproton trajectory is pulled towardsthe target nucleus, resulting in a magnifying lens effect(or alternatively a Coulomb focusing effect [23, 27]) thatenhances greatly the annihilation cross section. Further-

more, the antiproton is subject to the nuclear interactionthat changes the antiproton momentum in the interiorof the nucleus. The change of antiproton momentum isespecially important in low energy annihilations of lightnuclei because of the strong momentum dependence ofthe basic pp annihilation cross section. It is necessaryto modify the analytical formulas to take into accountthese effects so that they can be applied to p-nucleus an-nihilations for all energies and mass numbers. Successin constructing such an extended model will allow us toresolve the puzzles we have just mentioned.This paper is organized as follows. In Section II, we

review and summarize previous results in the Glaubermodel for high-energy nucleon-nucleus collisions, to pavethe way for its application to p-nucleus collisions. Ana-lytical formulas are written out for the p-nucleus anni-hilation cross sections in terms of basic pp annihilationcross section, σpp

ann. In Section III, we represent the basicpp annihilation cross section by a 1/v law. In SectionIV, we extend the Glauber model to study Antiproton-nucleus annihilations at both high and low energies, aftertaking into account the effects of Coulomb and nuclearinteractions, and the change of the antiproton momen-tum inside a nucleus. In Section V, we compare the re-sults of the analytical formulas in the extended Glaubermodel to experimental data. We find that these ana-lytical formulas capture the main features of the experi-mental antiproton-nucleus annihilation cross sections forall energies and mass numbers. Finally, we conclude thepresent study with some discussions in Section VI.

II. GLAUBER MODEL FOR p-NUCLEUS

ANNIHILATION AT HIGH ENERGIES

We shall first briefly review and summarize the ana-lytical formulas in the Glauber multiple collision model[28, 29] for its application to p-nucleus annihilations athigh energies. The Glauber model assumes that the in-cident antiproton travels along a straight line at a highenergy and makes multiple collisions with target nucle-ons along its way. The target nucleus can be representedby a density distribution. The integral of the densitydistribution along the antiproton trajectory gives thethickness function which, in conjunction with the basicantiproton-nucleon annihilation cross section σpp

ann, deter-mines the probability for an antiproton-nucleon annihi-lation and consequently the p-nucleus annihilation crosssection [24, 25, 28, 29].To be specific, we consider a target nucleus A with a

thickness function TA(bA) and mass number A, and aprojectile antiproton with a thickness function TB(bB)and a mass number B=1. The integral of all thicknessfunctions are normalized to unity. For simplicity, we shallnot distinguish between the annihilation of a proton ora neutron. Refinement to allow for different annihilationcross sections can be easily generalized.According to Eq. (12.8) of [29], in general, the thick-

3

ness function T (b) for the annihilation between the pro-jectile antiproton B and a nucleon in the target nucleusA at high energies along a straight-line trajectory at thetransverse coordinate b is given by

T (b) =

∫

dbA

∫

dbBTA(bA)TB(bB)tann(b− bA+bB), (1)

where tann(b−bA+bB) is the annihilation thickness func-tion, specifying the probability distribution at the rela-tive transverse coordinate b−bA+bB for the annihilationof a target nucleon at bA with an antiproton at bB.The thickness function tann(b) for pp annihilation at b

can be represented by a Gaussian with a standard devi-ation βpp,

tann(b) =1

2πβ2pp

exp{− b2

2β2pp

}. (2)

The cross section for a pp annihilation is then given by

σppann =

∫

db (πb2) tann(b) = 2πβ2pp, (3)

where db = 2πbdb. Therefore, the standard deviation βpp

in the pp annihilation thickness function is related to thepp annihilation cross section by

β2pp =

σppann

2π. (4)

In a p-A collision at high energies, the probability for theoccurrence of an annihilation is T (b)σpp

ann. The probabil-ity for no annihilation is [1 − T (b)σpp

ann]. With A targetnucleons, the annihilation cross section in a pA collisionat high energies, as a function of σpp

ann, is therefore

σpAann(σ

ppann) =

∫

db

{

1− [1− T (b)σppann]

A

}

. (5)

It should be noted that σppann depends on the magnitude

of the antiproton momentum relative to the target nucle-ons. For example, in our later applications to extend theGlauber model to low energies, the antiproton momen-tum at the moment of p-nucleon annihilation may be sig-nificantly different from the incident antiproton momen-tum, and it becomes necessary to specify the momentumdependence σpp

ann in Eq. (5) explicitly. For brevity of no-tation, we shall not write out the momentum dependenceexplicitly except when it is needed to avoid momentumambiguities.Analytical expressions of σpA

ann(σppann) can be obtained

for simple thickness functions [28, 29]. If the thicknessfunctions of TA and TB are Gaussian functions with stan-dard deviation βA and βB, then Eq. (1) gives

T (b) =1

2πβ2exp{− b2

2β2}, (6)

where

β2 = β2A + β2

B + β2pp. (7)

For this case with Gaussian thickness functions, Eq. (5)then gives the simple analytical formula [28, 29]

σpAann(σ

ppann) = 2πβ2

A∑

n=1

1− (1− f)n

n, (8)

where

f =σppann

2πβ2=

σppann

2π[β2A + β2

B ] + σppann

. (9)

To check our theory, we apply the results first to the caseof A = 1, we obtain

σppann = 2πβ2

(

1− (1− f)1

1

)

= σppann, (10)

as it should be. Next, for pD collisions where A = 2, wehave

σpDann = 2πβ2

[

2f − f2

2

]

. (11)

The situation depends on the size of βpp (or σppann), rela-

tive to βA and βB . There are two different limits of σppann

in comparison with 2π[β2A + β2

B]. If σppann ≪ 2π[β2

A + β2B]

then f → 0 and

σpDann ∼ 2σpp

ann, (12)

which exhibits the granular property of the nucleus whenthe basic antiproton-nucleon cross section is much smallerthan the radius of the nucleus. On the other hand, ifσppann ≫ 2π[β2

A + β2B], then f → 1 and the cross section

become

σpDann ∼ 3

2σppann. (13)

In actual comparison with experimental data, we use βB

= 0.68 fm, and we parametrize βA = r′0A1/3/

√3. The

standard root-mean-squared-radius parameter r′0 is of or-der 1 fm (see Table I below). The pp annihilation crosssection σpp

ann is about 50 mb at pplab=2 GeV/c and about

1000 mb at pplab=50 MeV/c. Thus, f ≪ 1 for ppplab=2

GeV/c and f ∼ 1 for pplab=50 MeV/c. If the Glaubermodel remains valid for the whole momentum range, thenone expects that

σpDann/σ

ppann

∣

∣

∣

∣

Glauber model

∼{

2 for high p momenta,

3/2 for low p momenta.(14)

The experimental data indicate

σpDann/σ

ppann

∣

∣

∣

∣

experimental

∼{

2 for high p momenta,

1 for low p momenta.(15)

The predicted ratio of σpDann/σ

ppann appears correct for

the high-energy region. However, for low-momentum pannihilations, we shall see that there are important modi-fications that must be made to extend the Glauber modelto the low-momentum region, and these modificationswill alter the σpD

ann/σppann ratio in that region.

4

As the nuclear mass number A increases, the densitydistribution of the nucleus become uniform. The thick-ness function for the collision of p with a heavy nucleuscan be approximated by using a sharp-cut-off distribu-tion of the form (see Ref. [28, 29])

T (b) =3√

(R2c − b2)

2πR3c

θ(Rc − b), (16)

where the contact radius can be taken to be

Rc = RA +RB +Rpp. (17)

With this sharp-cut-off distribution, Eq. (5) leads to thecross section given by [28, 29]

σpAann(σ

ppann) = πR2

c (18)

×(

1 +2

F 2

[

1− (1− F )A+2

A+ 2− 1− (1− F )A+1

A+ 1

])

,

where F is a dimensionless ratio,

F =σppann

2πR2c/3

. (19)

The pp annihilation radius Rpp in Eq. (17) can be cal-

ibrated to be Rpp =√

(3σppann/2π) by using the above

equation (18) for the case of pp collision as point nucle-ons.

III. THE BASIC pp ANNIHILATION CROSS

SECTION σpp

ann

In the previous section, analytical formulas have beenwritten out for the annihilation cross sections in the colli-sion of an antiproton in terms of σpp

ann (i) for a light targetnucleus in Eq. (8), and (ii) for a heavy nucleus in Eq. (18).The evaluation of the pA annihilation cross section willrequire the knowledge of the basic pp annihilation crosssection, σpp

ann.In our previous theoretical study in connection with

the annihilation lifetime of matter-antimatter molecules,we note that the experimental data of the pp total crosssection and the pp elastic cross section as a function ofthe antiproton momentum for a fixed proton target, pplab,are related by [30]

σtot = σelastic +σ0

v, (20)

where v is the velocity of the antiproton

v =pplab

√

p2plab +m2p

. (21)

As the difference of the experimental total cross sec-tion and the elastic cross section, the second term in Eq.(20) represents the pp inelastic cross section. It is essen-tially the annihilation cross section as the latter domi-nates among the inelastic channels. The pp annihilationcross section can therefore be parametrized in the form

σppann =

σ0

v. (22)

In the present work, we use the experimental pp anni-hilation cross sections directly to fine-tune the parameterσ0. We find that

σ0 = 43 mb (23)

gives a good description of the experimental σppann(pplab)

data as shown in Fig. 1. For brevity of notation, thequantity pplab will be abbreviated as plab in all figures.Fig. 1 indicates that the pp annihilation cross section

has a strong momentum dependence. It decreases byan order of magnitude as the antiproton momentum in-creases from 30 MeV/c to 600 MeV/c.It should be pointed out that the 1/v law, Eq. (22),

for the inelastic (or annihilation) cross section of slowparticles is well known. It was first obtained by Bethe [31]and is discussed in text books [32–34] and other relatedwork [35]. It arises from multiplying the S-wave partialcross section, π/k2, by the transmission coefficient T0

in passing through an attractive potential well, and thetransmission coefficient T0 is proportional to k at lowenergies. While the 1/v behavior is essentially an S-waveresult, higher-l partial waves will gradually contribute asthe antiproton momentum reaches the GeV/c region. Itis nonetheless interesting to note that the simple 1/v lawof Eq. (22) continues to provide a reasonable and efficientdescription of the experimental data even in the GeV/cregion [30].Recently, an elaborate and model-independent

coupled-channel partial-wave calculation, solving theproblem from first principle, has been used to determinep-p scattering cross sections for momenta below 0.925GeV/c. The calculation yields excellent agreement withthe experimental p-p annihilation cross section for energyranged from 0.200 to 0.925 GeV/c [36]. Refinement ofthe present σpp

ann description can be made, if desired, butwith additional complications.

IV. EXTENDING THE GLAUBER MODEL TO

LOW ENERGIES

The results in Section II pertains to annihilation of theantiproton at high energies with straight-line trajectories.To extend the range of the application to low energies, itis necessary to forgo the assumption of straight-line tra-jectories. We need to take into account the modificationof the trajectories due to residual Coulomb and nuclearinteractions that are additional to those between the in-cident antiproton and an annihilated target nucleon.The residual Coulomb and nuclear interactions affect

the annihilation process in different ways. The Coulombinteraction is long range, and the trajectory of the an-tiproton is attracted and pulled toward the target nucleusbefore the antiproton makes a contact with the nucleus(Fig. 2(a)). It leads to a large enhancement of the annihi-lation cross sections at low energies [23, 27]. The nuclearinteraction is short-range, and it becomes operative only

5

100 1000p

lab (MeV/c)

102

103

104

105

σ pA an

n (

mb)

3He

Pt

Sn

Ni

p

D

4He

C

Al

Cu

Ne

Be

CdPb

pD4HeBeCNeAl

NiCuSnCdPtPb

Ne

NiSn

Pt

FIG. 1. (Color online) Antiproton-nucleus annihilation cross sections as a function of the antiproton momentum in the labo-ratory, plab ≡ pplab, for different nuclear targets. The solid curve for the proton target nucleus is the σ0/v phenomenologicalrepresentation of the pp annihilation cross section in Eq. (22). The other curves are the results from the extended Glaubermodel using the basic pp annihilation cross section as input data. The solid curves are for Gaussian density distributions andthe dashed curves are for uniform density distributions. The high-momentum data points are from [23]. The other experimentaldata points are from the compilation of [6], where the individual experimental sources can be found.

6

after the contact with the nucleus. The interactions mod-ify the antiproton momentum as it travels in the nuclearinterior in which p-nucleon annihilation takes place. Asthe basic antiproton-nucleon annihilation cross sectionσppann as given by (22) is strongly momentum-dependent,

we need to keep track of the antiproton momentum alongthe antiproton trajectory.We shall work in the p-A center-of-mass system and

shall measure the antiproton momentum in terms of therelative momentum ppA defined as

ppA =mApp −mppA

mp +mA, (24)

where in the center-of-mass system with pp+pA = 0, wehave pp = ppA.

A. Initial-State Coulomb Interaction

We consider the collision of an antiproton with a nu-cleus in the pA center-of-mass system with a center-of-mass energy E. The initial antiproton momentum ppA isrelated to E by

E =p2pA

2µ. (25)

After the projectile antiproton travels along a Coulombtrajectory, it makes contact with the nucleus at r = Rc

with a momentum p′

pA determined by

E =[p′

pA]2

2µ+ Vc(Rc), (26)

where Vc(Rc) is the Coulomb energy for the antiprotonto be at the nuclear contact radius Rc,

Vc(Rc) = − (ZA − 1)α

Rc, (27)

and ZA is the target charge number.The pp annihilation cross section arises from the nu-

clear and Coulomb interaction between the antiprotonand a target proton. By employing σpp

ann as the basic el-ement in the multiple collision process in the extendedGlauber model of Eq. (5), the parts of the Coulomb andnuclear interaction that are responsible for the pp annihi-lation have already been included. Therefore in Eq. (26)for p-nucleus annihilation, we are dealing with residualinteractions that are additional to those between the an-tiproton and the annihilated nucleon. Hence, Eq. (27) forthe residual Coulomb interaction contains the coefficient(ZA − 1).From angular momentum conservation, we have

ppAb = p′pAb′. (28)

We obtain

b =p′pAppA

b′. (29)

From Eq. (26), we have

p′pA = ppA

√

1− Vc(Rc)

E, (30)

which is the relative momentum of the antiproton in thepA system at the nuclear contact radius Rc.Starting now from the transverse coordinates b′ at nu-

clear contact at r = Rc, one can follow the antiprotontrajectory in the nuclear interior. This trajectory willbe modified by residual interactions. One can evaluate athickness function T ′(b′) by integrating the nuclear den-sity along the modified trajectory. As the thickness func-tion T ′(b′) is governed mainly the geometry of the targetnucleus, we therefore expect that T ′(b′) will be character-ized by a length scale that will not be too different fromthe length scale in the thickness function T (b′) withoutresidual interactions. To the lowest order, it is reasonableto approximate T ′(b′) by T (b′). With such a simplifyingassumption, the pA annihilation cross section at an initialantiproton momentum ppA is

ΣpAann(ppA) =

∫

db

{

1− [1− T [b′(b)]σppann(p

′

pA)]A

}

, (31)

where we use a new symbol Σ to indicate that this isthe result in an extended Glauber model for p-nucleusannihilation, modified to take into account the Coulombinteraction that changes the antiproton momentum frominitial ppA to p′pA at the nuclear contact radius Rc. Wecan carry out a change of variable

ΣpAann(ppA)=

∫

db′bdb

b′db′

{

1−[1− T [b′(b)]σppann(p

′

pA)]A

}

.(32)

From Eqs. (29) and (30), the above Eq. (32) becomes

ΣpAann(ppA) =

[p′pA]2

p2pA

∫

db′{

1− [1− T (b′)σppann(p

′

pA)]A

}

=

{

1− Vc(Rc)

E

}∫

db′{

1− [1− T (b′)σppann(p

′

pA)]A

}

.

From the results in Eq. (5), the above equation becomes

ΣpAann(ppA) =

{

1− Vc(Rc)

E

}

σpAann(σ

ppann(p

′

pA)). (33)

With the above derivations, the initial-state Coulombinteraction can be incorporated into the extendedGlauber model as a mapping of the initial impact param-eter b to the impact parameter b′ at nuclear contact. Itcan be pictorially depicted as a lens effect in Fig. 2. Theattractive Coulomb interaction between the antiprotonand the nucleus acts as a magnifying lens that magni-fies the impact parameter b′ at the contact radius Rc toturn it into the initial impact parameter b, with whichthe reaction or annihilation cross section is measured(Fig. 2(a)). The magnifying lens effect for the attrac-tive Coulomb interaction with b > b′ leads to a p-nucleusannihilation cross section greater than the geometrical

7

b

O

b′

target A

Rc

ppA s

p-nucleus collision

b′b

Rc

target A

heavy-ion collision

(b)O

s

Coulomb interaction magnifies b′ to b

(a)

Coulomb interaction reduces b′ to b

FIG. 2. (Color online) Schematic picture of the bendingof the antiproton trajectories s as the antiproton approachesthe nucleus: (a) Under the action of the attractive p-nucleusCoulomb interaction, the impact parameter b′ in the nuclearinterior is magnified to the initial impact parameter b whichdetermines the annihilation cross section. (b) Under the ac-tion of a repulsive Coulomb interaction in heavy-ion collisions,the initial impact parameter b′ in the nuclear interior is re-duced to the initial impact parameter b which determines thereaction cross section.

cross section for heavy nuclei in low energy collisions, be-having as [1− Vc(Rc)/E]σpA

ann(σppann(p

′

pA)) as given in Eq.

(33), where Vc(Rc), the Coulomb energy at the nuclearcontact radius Rc, is negative.

It is interesting to note in contrast that in heavy-ioncollisions, the repulsive Coulomb initial-state interactionacts as a reducing lens that reduces the impact parameterb′ at contact to become the initial impact parameter bwith b < b′ as illustrated in Fig. 2(b). The lens effectfor a repulsive Coulomb interaction leads to a reactioncross section reduced from the geometrical cross sectionπR2

c to [1−Vc(Rc)/E]πR2c , where Vc(Rc) is the Coulomb

energy at the Coulomb barrier and is positive [32, 37, 38].Therefore, one obtains the unifying picture that for boththe p-nucleus and heavy-ion collisions, the initial-stateCoulomb interaction act as a lens, leading to the sameCoulomb modifying factor [1− Vc(Rc)/E].

B. Change of the Antiproton Momentum in the

Nucleus Interior

Because the basic pp annihilation cross section isstrongly momentum-dependent, there is however an ad-ditional important amendment we need to make. In thepresence of the nuclear and Coulomb interactions in thenuclear interior, the antiproton momentum p′pA at the

contact radius is changed to the momentum p′′pA in theinterior of the nucleus. The pp annihilation occurs insidethe nucleus at a momentum p′′pA. It is necessary to mod-

ify Σppann in Eq. (33) to take into account this change of

the antiproton momentum by replacing p′pA with p′′pA.

The antiproton momentum p′′pA is a function of theradial position r and is related to ppA by the energy con-dition:

E =p2pA

2µ=

(p′′

pA)2

2µ+ Vc(r) + Vn(r). (34)

In order to obtain an analytical formula for generalpurposes for our present approximate treatment, it suf-fices to consider average quantities and use the root-mean-square average 〈(p′′

pA)2〉1/2 given by

E =〈(p′′

pA)2〉

2µ+ 〈Vc(r)〉 + 〈Vn(r)〉, (35)

where 〈Vc(r)〉 and 〈Vn(r)〉 are the average interiorCoulomb and nuclear interactions, respectively. From theabove equation, we can then approximate p′′pA by the av-

erage root-mean-squared momentum 〈(p′′

pA)2〉1/2 in the

interior of the nucleus,

p′′pA ∼ ppA

√

1− 〈Vc(r)〉 + 〈Vn(r)〉E

. (36)

The p-nucleus annihilation cross section ΣpAann is there-

fore modified by changing p′pAin Eq. (33) to p′′pA to be-come

ΣpAann(ppA) =

{

1− Vc(Rc)

E

}

σpAann(σ

ppann(p

′′

pA)). (37)

The experimental data of σppann and ΣpA

ann are presentedas a function of pplab for a fixed proton or nucleus targetat rest. Accordingly, we convert the antiproton momentain the center-of-mass system in above Eq. (37) to thosein the laboratory system with fixed targets as

ppA =A

A+ 1pplab, (38)

p′′pA =A

A+ 1p′′plab, (39)

and Eq. (36) gives

p′′plab = pplab

√

1− 〈Vc(r)〉 + 〈Vn(r)〉E

. (40)

8

Therefore, the antiproton-nucleus annihilation cross sec-

tion ΣpAin (pplab) in the extended Glauber model for an-

tiproton with an initial momentum pplab is given in thefollowing compact form

ΣpAann(pplab) =

{

1− Vc(Rc)

E

}

σpAin (σpp

ann(p′′

plab)), (41)

where σpAin (σpp

ann(p′′

plab)) is given by Eq. (8) for light nuclei

with a Gaussian thickness distribution, and by Eq. (18)for heavy nuclei with a sharp-cut-off thickness function,with the basic quantity σpp

ann in Eqs. (9) or (19) evaluatedat p′′plab given in terms of pplab by Eq. (40). The Coulomb

energy at contact Vc(Rc) is given by Eq. (27), and theaverage 〈Vc(r)〉 in the interior of the nucleus (ZA, A) witha radius RA in Eq. (40) is given by

〈Vc(r)〉 = −3(ZA − 1)α

(

5R2c −R2

A

10R3c

)

. (42)

In our application of the extended Glauber model, con-cepts such as the contact radius Rc and 〈Vc(r)〉 are sim-plest for a uniform density distribution. For the evalua-tion of these quantities in the case of small nuclei witha Gaussian thickness function, we shall approximate theGaussian as a uniform distribution [only for the purposeof calculating Rc and 〈Vc(r)〉 in Eqs. (27), (40), and (42)]with an equivalence between Rc and β. We note thatthe dimensionless quantity f in Eq. (9) for a Gaussianthickness function and the quantity F in Eq. (19) for thesharp-cut-off distribution have the same physical mean-ing. It is reasonable to equate the corresponding quan-tities 2πβ2 in Eq. (9) with the corresponding quantity2πR2

c/3 in Eq. (19), leading to the approximate equiva-lence

Rc(for Gaussian distribution) ∼√3β, (43)

and similarly,

RA(for Gaussian distribution) ∼√3βA. (44)

These equivalence relations enable us to obtain theCoulomb factor (1− Vc(Rc)/E) in (41) and the antipro-ton momentum change from pplab to p′′plab in Eq. (40), forlight nuclei with Gaussian thickness functions.

V. COMPARISON OF THE EXTENDED

GLAUBER MODEL WITH EXPERIMENT

The central results of the extended Glauber model con-sist of Eq. (37) or (41) and their associated supplemen-tary equations. With the basic pp annihilation cross sec-tion σpp

ann and its momentum dependence well representedfrom experimental data by Eq. (22), as discussed in Sec-tion III, it is only necessary to specify the residual nuclearinteraction 〈VN 〉 and the nuclear geometrical parametersin Eq. (37) to obtain the p-nucleus annihilation cross sec-tion. For light nuclei (with A < 40), we use a Gaussian

density distribution with the Gaussian thickness functionin Eq. (6), with geometrical parameters βA = r′0A

1/3/√3

and βB = 0.68. We find that r′0 = 1.05 fm gives a gooddescription. For the heavy nuclei (with A > 40), weuse a uniform density distribution with the sharp-cut-offthickness function in Eq. (22) with geometrical parame-ters RA = r0A

1/3 and RB = 0.8 fm The radius parameterr0 = 1.00 fm fit the data well.In Fig. 1, the solid curve for the proton target nucleus is

from the σ0/v phenomenological representation of the ppannihilation cross section in Eq. (22). The other curvesare the results from the extended Glauber model usingthe basic pp annihilation cross section as input data. Thesolid curves are for Gaussian density distributions andthe dashed curves are for uniform density distributions.The fitting parameters that give the theoretical pA anni-hilation cross sections in Fig. 1 are listed in Table I.

TABLE I. Fitting parameters

Nuclei Gaussian Uniform 〈Vn〉(MeV)r′0(fm) r0(fm)

D 1.05 -1.04He 1.05 -4.0Be 1.05 -10.0C 1.05 -15.0Ne 1.05 -25.0Al 1.05 -30.0Ni 1.00 -30.0Cu 1.00 -30.0Cd 1.00 -35.0Sn 1.00 -35.0Pt 1.00 -35.0Pb 1.00 -35.0

The comparison of the extended Glauber model withthe experimental data in Fig. 1 indicates that althoughthe fits are not perfect, the extended Glauber model cap-tures the main features of the annihilation cross sectionsfor all energies and mass numbers. We may mention a fewof the notable features of the data and the correspondingexplanations in the extended Glauber model.We examine first pA annihilation at high energies. In

these high energy annihilations, the momentum depen-dence of the basic σpp

ann is not sensitive to the antiprotonmomentum change arising from the residual nuclear andCoulomb interactions 〈Vc〉 and 〈VN 〉. The initial-stateCoulomb interaction energy Vc(Rc) is also small in com-parison with the incident energy E. As a consequence,the corrections due to the Coulomb and nuclear inter-actions are small for high-energy collisions. At thesehigh energies, the antiproton makes multiple collisionswith target nucleons and probes the granular propertyof the nucleus when the spacing between the nucleonsis large compared with the dimension of the p probe,similar to the additive quark model in meson-meson col-lisions [39, 40]. The theoretical prediction in Eq. (14)give ΣpD

ann/σppann ∼ 2 in agreement with the experimen-

tal ΣpDann/σ

ppann. There are not many high momentum

data points for 4He, and the only data high momentum

9

0

500

1000

1500σ

pA ann

(m

b)

D

p

<Vn> = 0 MeV

<Vn> = -1.0 MeV

<Vn> = -2.0 MeV

p D

100 200 300 400 500 600 p

lab (MeV/c)

0

500

1000

1500

2000

σ pA an

n (

mb)

4He

<Vn> = -2.0 MeV

<Vn> = -4.0 MeV

<Vn> = -6.0 MeV

3He

4He

(a)

3He

(b)

FIG. 3. (Color online) p annihilation cross sections of (a) pand D nuclei and (b) 4He nucleus. The experimental data arefrom Ref. [6]. The pp curve is from the σ0/v representation ofEq. (22), and the other curves are from the extended Glaubermodel.

point at 600 MeV/c gives reasonable agreement with theGlauber model results.

As the number of nucleon increases in high energy an-nihilations, the Glauber model results of Eqs. (8) and(18) give a pA annihilation cross section proportional toβ2 for a Gaussian thickness distribution and to R2

c fora uniform density distribution. Both β2 and R2

c vary asA2/3. Hence, the pA annihilation cross approaches theblack-disk limit of A2/3 limit as the nuclear mass numberincreases. The comparison in the high-energy region inFig. 1 shows that the experimental data agree with pre-dictions for nuclei across the periodic table, indicatingthat the experimental data indeed reach this black-disklimit of A2/3 in the heavy-nuclei region, However, thereis a discrepancy for Pb at 700 MeV/c, which may needneed to be re-checked experimentally, as the other datapoints appear to agree with the theoretical predictions.

The situation at the low energy region is more com-plicated. In addition to the multiple collision process,the Coulomb and nuclear interactions also come intoplay. We can examine the data for p, D, and He moreclosely in Fig. 3 in a linear plot in both the low andhigh energy regions. As shown in Figs. 1 and 3(a),the basic cross section σpp

ann is of order 1000 mb atthe low momentum of pplab∼20 MeV/c, correspondingto a effective annihilation radius between p and p ofRpp = (σpp

ann/π)1/2=6.8 fm. The large cross section and

annihilation radius arise from the magnifying lens ef-fect of the attractive initial-state Coulomb interactionthat magnifies the proton radius as seen by the incom-

0 100 200 300 400 500 600 700 800 900300

600

900

1200

<Vn>= -10 MeV

<Vn>= -15 MeV

<Vn>= -20 MeV

0 100 200 300 400 500 600 700 800 9000

1000

2000

3000

σ pA an

n (

mb) <V

n>= -15 MeV

<Vn>= -20 MeV

<Vn>= -25 MeV

0 100 200 300 400 500 600 700 800 900p

lab (MeV/c)

500

1000

1500

2000

<Vn>= -20 MeV

<Vn>= -25 MeV

<Vn>= -30 MeV

(a) C

(b) Ne

(c) Al

FIG. 4. (Color online) p annihilation cross sections of (a) Cnucleus, (b) Ne nucleus, and (c) Al nucleus. Solid curves giveresults from the extended Glauber model and the data pointsare from the compilation of Ref.[6].

ing antiproton, as discussed in Section III. In this case,as Rpp ≫ Rdeuteron or Rantiproton, the quantity f in Eq.(8) is close to unity. According to the analysis given inEq. (14) of Section III, the Glauber model with f ∼ 1would predict a ratio of ΣpD

ann/σppann ∼ 3/2 for low-energy

annihilations in the multiple collision process.

40 60 80 100 120 140 160 180 200 220A

0

2000

4000

6000

8000

10000

12000

14000

σ pA an

n (

mb)

NiSn

Pt

FIG. 5. p annihilation cross sections on Ni, Sn and Pt nucleiat pplab = 100 MeV/c. The solid curve gives results from theextended Glauber model and the data points are from Ref.[6].

Experimentally, for low-energy annihilationsΣpD

ann/σppann is of order unity. In the extended Glauber

10

model, the reduction of the ratio ΣpDann/σ

ppann arises from

the combination of two effects: (i) the increase of theantiproton momentum inside the nucleus due to theattractive residual interactions and (ii) the momentumdependence of the basic σpp

ann decreases sensitively as afunction of an increase in antiproton momentum. Thecombined effects bring the ratio ΣpD

ann/σppann from 3/2 to

about unity. In Fig. 3, we show the variation of thepD and p4He cross sections as a function of pplab fordifferent residual nuclear interactions 〈VN 〉. There is agreat sensitivity of the p-annihilation cross section onVN in the low energy region for the lightest nuclei forwhich the Coulomb interaction is weak. However, as thetarget charge number increases, the Coulomb interactionbecomes stronger and the p-annihilation cross sectionbecomes less sensitive to the strength of the nuclearinteraction 〈VN 〉, as shown in Fig. 4 for pC, pNe, andpAl collisions.As the target charge number Z increases further in

the low-energy region, the cross section increases sub-stantially. For example, the cross section reaches a valueof about 9000 mb for the Pt nucleus, corresponding to anannihilation radius of Rann = (σpA

ann/π)1/2=17 fm. Again,

such a large annihilation radius arises from the Coulombmagnifying lens effect that magnifies the nuclear radiusof Pt as seen by the incoming antiproton. The attrac-tive Coulomb interaction as well as the nuclear interac-tion also changes the momentum of the antiproton insidethe nucleus. The combined effect of the Coulomb initial-state interaction, the change of the antiproton momen-tum inside a nucleus, together with the Glauber multiplecollision process of individual antiproton-nucleon annihi-lation, give a good description of the cross sections forthe heaviest nuclei at low energies, as shown in Figs. 1and 5.There are only a few cases where the experimental data

points deviate from the general trend and the theoreticalpredictions. In particular, there are discrepancies for pDat plab∼280 MeV/c, p(3He) at 55 MeV/c, pPb at 700MeV/c, and pBe at high momenta. It will be necessary tofind out the origins for the discrepancies by theoretical re-examination or experimental re-measurements for thesecases in the future.

VI. SUMMARY AND DISCUSSIONS

We are motivated to examine the antiproton-nucleusannihilations because it is an important process in theinteraction between antimatter and matter. The experi-mental data reveal interesting features whose high-energybehavior coincides with what one would expect from theGlauber multiple collision model formulas of [28, 29].However the peculiarities in the low-energy region are be-yond the application of the high-energy Glauber model.It is an interesting question to extend the model to thelow-energy region, to provide a simple and systematic un-derstanding of the antiproton-nucleus annihilation pro-

cess for all energies and mass numbers.

For such an extension to low energies, it is necessaryto take into account the Coulomb and nuclear interac-tions and forgo the assumption of straight-line trajecto-ries. The Coulomb interaction is long range. Before theantiproton makes a contact with the nucleus, the tra-jectories of the antiproton are pulled toward the targetnucleus, resulting in a magnifying lens effect that mag-nifies the target radius as it appears to the incident an-tiproton. The end result is to make the nucleus appearlarger than what it really is. It can also alternativelydescribed as a Coulomb focusing effect [23, 27]. The de-scription of attractive Coulomb interaction as a magni-fying lens effect unifies it with the reducing lens effectin repulsive Coulomb interactions that one encountersin heavy-ion collisions, with the same Coulomb factor of(1− Vc(Rc)/E).

Starting from the nuclear contact radius at an im-pact parameter b′, the antiproton is further subject tothe attractive residual nuclear and Coulomb interactions.These residual interactions change the thickness functionfrom T (b′) (without residual interactions) to T ′(b′) (withresidual interactions) because they modify the trajecto-ries of the antiproton inside the nucleus. The change inthe thickness function is expected to lead to only minorcorrections to the length scales in the thickness function,as both T ′(b′) and T (b′) depend first and foremost on thelength scale of the target nucleus. The use of the samethickness function T (b′) for T ′(b′) suffices for the low-est order calculations. Clearly, future refinements canfollow these non-parallel trajectories of the antiprotonfrom the contact radius and evaluate a corrected thick-ness function, for a refinement of the thickness functionin annihilation studies.

There is an additional important effect in low-energyannihilations. The attractive residual nuclear andCoulomb interactions modify the antiproton momentumin the interior of the nucleus in which the pp annihila-tion occurs. The change of the antiproton momentumhas a significant effect on the antiproton-nucleus annihi-lation cross sections because of the sensitive momentum-dependence of the basic pp annihilation cross section atlow energies. The basic pp annihilation cross section atlow energies changes by an order of magnitude as theantiproton momentum increases from 30 MeV/c to 600MeV/c.

We have extended the high-energy Glauber model tolow energy annihilation processes after taken into ac-count the residual interactions. The results is a com-pact equation (41) (or the equivalent (37)) with supple-mentary equations that capture the main features of theannihilation process and provide a simple analytical wayto analyze antiproton-nucleus annihilation cross sections.At high antiproton energies, they exhibit the granularproperty for the lightest nuclei and the approach to ablack-disk for heavy nuclei. At low antiproton energies,they display the modification of the antiproton momen-tum due to the nuclear interaction for the lightest nuclei,

11

and the large attractive magnifying lens effect due to theCoulomb interaction for heavy nuclei.

We can properly respond to the specific questions weraised in the Introduction. With respect to the mass de-pendence at high antiproton incident momenta, the crosssections increase almost linearly with the mass numberA for the lightest nuclei, but with approximately A2/3

as the mass number increases because the basic processis a Glauber multiple collision process of the antiprotonpassing through a target of individual nucleons. In thelow antiproton momentum region, the annihilation crosssections do not rise with A as anticipated but are com-parable for p, D, and He, because the residual nuclearinteraction causes an increase in the antiproton momen-tum inside the nucleus and the increases in antiprotonmomentum leads to a decrease in the basic p-nucleonannihilation cross section. As the charge number Z in-creases, the initial-state Coulomb interaction magnifiesthe target nucleus and the pA annihilation cross sectionat low energies become subsequently greatly enhancedin the heavy nuclei region. With respect to the energydependence, the energy dependence of the pA annihila-tion cross sections is intimately related to the energy-

dependence of the pp cross sections.The simple picture we have presented can be refined,

and individual nuclear properties can be revealed, withnew data that fill in the gaps in Fig. 1. The deviations ofexperimental data with theoretical predictions for a fewcases also call for a re-examination of both the experi-mental measurements as well as theoretical refinements.The annihilation cross sections for pp and pn are dif-

ferent. A proton p consists of uud, a neutron n consistsof udd, p is uud. In the quark model, we expect (theprobability of p-p annihilation)=(5/4)×(the probabilityof p-n annihilation), if only flavor and antiflavor can an-nihilate. One can refine the theory to separate out theneutrons and protons and work out a modification factorthat depends on Z/A ratio.The extended Glauber model may find future applica-

tions in similar problems such as in the collision of mesonsor baryons with nuclei at both high energies as well aslow energies.

Acknowledgment

This research was supported in part by the Division ofNuclear Physics, U.S. Department of Energy.

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