NASA/TP-2002-

211933

String Fragmentation Model in Space Radiation Problems

Alfred Tang and John W. Norbury

University of Wisconsin-Milwaukee, Milwaukee, Wisconsin

R. K. Tripathi Langley Research Center, Hampton, Virginia

September 2002

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NASA/TP-2002-

211933

String Fragmentation Model in Space Radiation Problems

Alfred Tang and

John W. NorburyUniversity of Wisconsin-Milwaukee, Milwaukee, Wisconsin

R. K. Tripathi Langley Research Center, Hampton, Virginia

September 2002

Available from:

NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS)7121 Standard Drive 5285 Port Royal RoadHanover, MD 21076-1320 Springfield, VA 22161-2171(301) 621-0390 (703) 605-6000

Acknowledgments

This work was supported by NASA grants NCC-1-354 and NGT-1-52217. John W. Norbury gratefully acknowl-edges the hospitality of the Physics Department at La Trobe University, Bundoora, Australia, during the summermonths. This paper is dedicated to the memory of Eloise Johnson, a superb technical editor who will be missed by hercolleagues and researchers alike. She went to her eternal abode during the process of publishing this paper.

Nomenclature

Natural units used in this work are such that ! = c = 1.

c speed of lightcm center of massdσdE

spectral distribution cross sectionE total energy of particleEd3σdp3

Lorentz invariant differential cross section

f(y) transitional probability from one vertex to next with rapidity y! Planck constant divided by 2πH(Γ) probability distribution of vertex occurring at Γm massp one-dimensional momentumpmax maximum momentum of produced particlesp± plus or minus light-cone momentum, E ± ps Mandelstam variable representing total energy square in cm frame,

(p1 + p2)2 or

√s = E1cm + E2cm = Ecm

t,τ timeVn nth spacetime position or vertex of breakup point along string,

κ(x+, x−)W± kinetic energy of quark or antiquark along ± light-cone coordinatex coordinate of position or fragmentation variable, pz

pmax

x± plus or minus light-cone conÞguration, t± xy rapidity, 1

2ln E+p

E−pγ Lorentz factor, 1√

1−v2Γ proper time, κx+x− = κ(t2 − x2) = κτ 2κ string constant

iii

Abstract

String fragmentation models such as the Lund Model Þt experimental particleproduction cross sections very well in the high-energy limit. This paper givesan introduction of the massless relativistic string in the Lund Model and showshow it can be modiÞed with a simple assumption to produce formulas for mesonproduction cross sections for space radiation research. The results of the stringmodel are compared with inclusive pion production data from proton-protoncollision experiments.

1. Introduction

The Lund model is a (1 + 1) massless relativistic string fragmentation model which ismodiÞed with a simple assumption in this paper to produce formulas for meson particleproduction cross sections for HZETRN. The idea of modelling hadronic systems withstrings went back to the 1960s (ref. 1). The general skepticism on the extra dimensionspredicted by string theory and the advent of QCD in the 1970s put string theory out ofcommission for the next decade. Interests in string theory rekindled in the 1980s whenGreen and Shwarz showed that string theory is anomaly free and probably Þnite to allorders in perturbation theory. Today many advances in string theory have been madeon the theoretical front. Despite all the excitement string theory has generated in thehigh-energy theory community, no evidence exists that string theory will make any low-energy limit prediction to be tested by experiments in the near future. Supersymmetryand superstring may be testable by experiments in the TeV scale but perhaps not directlyrelevant to the focus of space radiation research in the GeV scale. The interest liesmostly in the low-energy limit of a nonsupersymmetric string theory, with the possibleexception of understanding the Greisen-Zatsepin-Kuzmin (GZK) cutoff in the ultrahigh-energy cosmic ray spectrum using TeV strings (ref. 2).

String fragmentation has been a work horse in analyzing high-energy particle pro-duction. Isgur proposed a ßuxtube model in which the color force Þeld is thought asa string-like object (refs. 3 and 4). B. Andersson implemented the idea of string frag-mentation formally in the Lund Model (refs. 5, 6, and 7), which is implemented in theJETSET and PYTHIA Monte-Carlo programs (ref. 8). If a Monte-Carlo cross-sectionprogram is put into a target code, it will not run in a short time. Therefore we needsimple parameterizations of cross-section formulas, which is the aim of the present work.In this paper, we Þrst review the original Lund Model and expand some of the deriva-tions of reference 5. Later we show how to modify the Lund Model by inserting a simpleassumption to generate formulas for meson production cross sections.

Appendix A deÞnes the basic kinematic notations. Appendix B explains the basicconcepts of the Lund Model and derives the invariant amplitude formulas. The formulas

in the appendixes are taken from reference 5, chapters 6 to 8. This work merely servesto focus on a subsection of reference 5 that is relevant to our discussion and expands thederivations. Section 2 of this paper explains a new idea on how to use a simple ansatz toobtain formulas for production cross sections in the Lund Model.

2. Cross-Section Formulas

The basic result of the Lund Model is the Area Law which is summarized as (ref. 5)

dPext = dsdz

z(1− z)a e−bΓ (1)

dPint =n!j=1

N d2p0j δ+(p20j −m2) δ

"prest −

n#j=1

p0j

$e−bArest (2)

and derived in appendix B. The symbols Γ and Arest are Lorentz invariant kinematicvariables. The variable Γ deÞnes the surface of constant proper time along which the stringis broken. Traditionally the Lund Model results in equations (1) and (2) are incorporatedinto Monte Carlo simulation programs such as JETSET and PYTHIA to compute crosssections. Since the goal of the HZETRN program is to derive simple formulas for the crosssections, numerical calculations are not wanted. In this section, some simple assumptionsare made to derive analytical results.

Equation (2) resembles the quantum mechanical result

dσ = dΩ(s; p01, . . . , p0n) |M|2 (3)

and (ref. 9)

dσ =(2π)4

4%(p1 · p2)2 −m2

1m22

δ4

"p1 + p2 −

n+2#i=3

pi

$n+2!i=3

d3pi(2π)32Ei

|M|2 (4)

By analogy, it is assumed that (ref. 5)

|M|2 = e−bArest (5)

In principle, Atotal = Γ+Arest can also be used in equation (5) in place of Arest. The onlydifference is a proportionality constant exp(−bΓ) in the invariant amplitude |M|2. Hencethe assumption in equation (5) is the simplest. The string breaks up along a surface ofconstant proper time Γ which is determined by the distribution (ref. 5)

H(Γ) = C Γa e−bΓ. (6)

By setting dH/dΓ = 0, we can easily show that the maximum of H(Γ) is (ref. 5)

Γmax =a

b(7)

2

The mean proper time is (ref. 5)

< Γ >=

&∞0ΓH(Γ) dΓ&∞

0H(Γ) dΓ

=a+ 1

b(8)

The fact that a and b are constants in Γ as shown in appendix B does not exclude thepossibility that they are functions of rapidity. Since string fragmentation is a stochasticprocess that occurs over a surface of constant proper time on the average, < Γ > isexpected to be an absolute constant; this means that

a = a0 f(y)− 1 (9)

b = b0 f(y) (10)

for some function of rapidity f(y). The goal of a consistent string theory is to calculatea0, b0, and f(y) analytically. In this paper, the simple assumption is that

f(y) =2mT sinh y√

s$=2pz√s$≈ x, (11)

where mT =%m2 + p2T . The kinematic identities of equation (11) are taken from ref-

erence 9. The ansatz f(y) = x is motivated by a posteriori considerations when Þttingexperimental data. In NASA space radiation research, cosmic particles are highly en-ergetic with production mostly forward. In the center-of-mass frame, the longitudinalmomentum is pz and the total energy of the colliding particles is

√s$. According to the

Fermi Golden Rule,dσ

dx∼ |M|2 (12)

Together with equations (10) and (11), the differential cross section is found to be

dσ

dx= Ae−Bx (13)

for some constants A and B. This prediction is consistent with experimental data fromreferences 10 and 11 as shown in Þgures 1 and 2.

The spectral distribution can be obtained from the cross-section formula in reference 12

dσ

dE= 2πp

' θmax

0

dθEd3σ

dp3sin θ (14)

By using approximations pz ≈ E and x ∼ pz/√s in the high-energy limit and equa-

tion (12) in conjunction with equation (14), the spectral distribution is found to be

dσ

dE∼ 2π(1− cos θmax) pz|M|2

= cE e−kE (15)

3

where c and k are constants. Unfortunately no experimental data exist for dσ/dE forpion production in proton-proton scattering. Blattnig et al. produced a parameterizedspectral distribution by integrating E d3σ/dp3 (ref. 12).

3. Results

The Monte-Carlo programs JETSET and PYTHIA cannot be used in HZETRN dueto excessive running time. Parametrizations of cross-section formulas are needed. Thenew idea is to make simple assumptions within the Lund Model framework to obtainanalytic meson production cross-section formulas. Our purpose was not to produce anyparameterization. To this end, all the Þts in this work have parameters that are simplyhandpicked for the sake of illustrating the correctness of the qualitative aspects of crosssections predicted by string theory. Figures 1 and 2 clearly demonstrate the success of theÞts. A major assumption in this paper is that a and b in equation (6) are not constantsas proposed in the original Lund Model but are functions of rapidity. This assumptionallows us to obtain the correct qualitative features of the cross sections without resortingto a Monte-Carlo simulation. At the present, the functions a and b are adjusted to Þt thedata.

The exact forms of the parameters a and b (eqs. (9) and (10)) for various types ofproduction are calculated from nonsupersymmetric string theory, such as the QCD stringmodel (ref. 13). The concept of conÞnement by a string is quantiÞed in terms of theminimal area law of the Wilson loop. QCD string models also include the gluonic degreesof freedom. The constants A and B in equation (13) and the constants c and k in equation(15) are also calculated from string theory. The (1 + 1) Lund Model cannot give angulardependence which is important for low pz. Future work in string phenomenology hopefullycan include the angular dependence of the cross sections near the threshold by extendingthe model to (3 + 1) or higher dimensions. Production formulas will also be extractedfrom JETSET and PYTHIA to be used in HZETRN.

4

Appendix A

Kinematics

The quark-antiquark pair of a meson is massless in the string model. In this case, theconcept of the mass of a meson is associated with the mass of the string Þeld and not thequarks. Massless quarks move at the speed of light. This result is not surprising becausestring theory predicts that the ends of open strings move at the speed of light either byimposing a Neumann boundary condition for open strings (ref. 14) or by solving theclassical equations of motion (ref. 15). In the highly relativistic problems, the light-conecoordinates are the natural choice. In the (1 + 1) case, Lorentz transformations can bewritten as(

E$

p$

)=

(γ −γv−γv γ

)(Ep

)≡(

cosh y − sinh y− sinh y cosh y

)(Ep

)(16)

(t$

x$

)=

(γ −γv−γv γ

)(tx

)≡(

cosh y − sinh y− sinh y cosh y

)(tx

)(17)

where γ is the Lorentz factor

γ =1√1− v2 (18)

These equations imply

γ = cosh yp (19)

γv = sinh yp (20)

v = tanh yp (21)

The energy and momentum of a particle can now be written as

E = γm = m cosh yp (22)

p = γmv = m sinh yp (23)

The boosted energy Eb and the boosted one-momentum pb are given as

Eb = γ (E − v p)= m (cosh yp cosh y − sinh yp sinh y)= m cosh(yp − y) (24)

pb = γ (p− v E)= m (sinh yp cosh y − cosh yp sinh y)= m sinh(yp − y) (25)

5

The relativistic velocity addition formula is simple in light-cone coordinates (ref. 5) with

v$ = tanh y$ =v − vb1− vbv = tanh(y − yb) (26)

illustrating the additivity of rapidity,

y = y$ + yb (27)

This simplication motivates the deÞnition of rapidity

yp =1

2ln

(1 + v

1− v)=1

2ln

(E + p

E − p)

(28)

Momenta along the light-cone can be deÞned as (ref. 5)

p+ = E + p = m cosh yp +m sinh yp = meyp (29)

p− = E − p = m cosh yp −m sinh yp = me−yp (30)

With these deÞnitions, boosts are simpliÞed along the light cone (ref. 5),

p$± = me±(yp−y) = p±e∓y (31)

Light cone coordinates can also be deÞned in conÞguration space as

t ≡ m

κcosh yq (32)

x ≡ m

κsinh yq (33)

where κ is the string constant and is used here to give t and x the correct dimension.A new subscript is adopted for yq in conÞguration space to distinguish yp in momentumspace. These deÞnitions are consistent with the requirement that v = dx/dt = tanh yq, asin equation (21). Similar results are obtained as in the momentum space case,

x+ = t+ x =m

κcosh yq +

m

κsinh yq =

m

κeyq (34)

x− = t− x = m

κcosh yq − m

κsinh yq =

m

κe−yq (35)

x$± =m

κe±(yq−y) = x±e∓y (36)

6

Appendix B

Lund Model

The focus of this paper is primarily mesons. The force Þeld between a quark-antiquarkpair is presumed to be constant and conÞned to a ßux-tube. This appendix is based onthe work of reference 5, chapters 6 to 8. The equation of motion of any one member ofthe quark-antiquark pair acted upon by a constant force is (ref. 5)

F =dp

dt= −κ (37)

where κ is the string constant. The solution of the equation is

p(t) = p0 − κt = κ(t0 − t) (38)

From E2 = p2 +m2, E dE = p dp or

p

E=dE

dp(39)

is obtained. There is also

p = γmv = γmdx

dt= E

dx

dt(40)

Equations (39) and (40) together yield

dx

dt=p

E=dE

dp(41)

such thatdE

dx=dE

dp

dp

dt

dt

dx=dp

dt= −κ (42)

Its solution gives the expected QCD ßux tube result

E = κ(x0 − x) (43)

Combining equations (38) and (43) and m = 0, the relativistic equation of state is

m2 = E2 − p2 = κ[(x0 − x)2 − (t0 − t)2] = 0 (44)

Let x0 = t0 = 0, the equation of motion |x| = t is obtained. The maximum kinetic energyat x = 0 puts an upper bound on the displacement of the particle such that κxmax =W .The path of the particle is illustrated in Þgure 3. The zig-zag motion of the particle isthe reason for the name yoyo state. The quark-antiquark pair of a meson are assumedto be massless. The mass square of the meson is proportional to the area of the rectangle

7

deÞned by the trajectories of the quark-antiquark pair. The period τ of the yoyo motionis

τ =2E0κ

(45)

where E0 is the maximum kinetic energy of each quark. In a boosted Lorentz frame, theperiod is transformed as

τ $ = τ cosh y (46)

The breakup of a string occurs along a curve of constant proper time such that theprocess is Lorentz invariant. The Lund model assumes that the produced mesons areranked, meaning that the production of the nth rank meson depends on the existence ofrank n−1, n−2, . . . , 1 mesons. A vertex V is a breakup point and the location κ(x+, x−)where a quark-antiquark pair is produced. The breakup of a string is represented inÞgure 4. The produced particle closer to the edges are the faster moving ones correspond-ing to larger rapidities.

Let p±0 and p±j be momentum of the parent quark and the jth of the n-producedquarks moving along the x± light-cone coordinate, respectively. Then

p±0 =n#j=1

p±j (47)

and the momentum fraction at Vj is deÞned as

z±j =p±jp±0

(48)

In order to simplify notations, zj equals z+j unless speciÞed otherwise. The invariantinterval is

ds2 = dt2 − dx2 = dt2(1− v2) (49)

giving

ds = dt√1− v2 = dt

γ= dτ (50)

where τ is the proper time. The proper time is also x+x− = (t− x)(t+ x) = t2 − x2 = τ 2and can be used as a dynamic variable such that

Γ = κ2x+x− (51)

Since Γ is proportional to the proper time square τ2, its Lorentz invariant property makesit a good kinematic variable in the light-cone coordinates. Let W±1 and W±2 be thekinetic energies along x± at vertices 1 and 2, respectively. The following identities can

8

be easily proven geometrically by calculating the areas of the rectangles Γ1 = A1 + A3,Γ2 = A2 +A3, and m

2 as shown in Þgure 5:

Γ1 = (1− z−)W−2W+1 (52)

Γ2 = (1− z+)W−2W+1 (53)

m2 = z−z+W−2W+1 (54)

where m is the mass of the produced meson. Equations (52) and (53) can be expressedwith the help of equation (54) as

Γ1 =m2(1− z−)z+z−

(55)

Γ2 =m2(1− z+)z+z−

(56)

Differentiating these equations gives

∂Γ1∂z−

= − m2

z+z2−(57)

∂Γ2∂z+

= − m2

z2+z−(58)

Let H(Γ1) be a probability distribution such that H(Γ1) dΓ1 dy1 is the probability ofa quark-antiquark pair being produced at the spacetime position V1. From now on, thesymbol Vn also represents the breakup event that leads to the creation of the nth quark-antiquark pair along a surface of constant τ . Let f(z+)dz+ be the transition probabilityof obtaining V2 given that V1 occurs. The transition probability of V2 via V1 is thenH(Γ1) dΓ1 dy1 f(z+)dz+. Similarly the probability of V1 via V2 is H(Γ2) dΓ2 dy2 f(z−)dz−.It is reasonable to assume that the probability of V1 via V2 is equal to that of V2 via V1such that

H(Γ1) dΓ1 dy1 f(z+)dz+ = H(Γ2) dΓ2 dy2 f(z−)dz− (59)

The Jacobian J indΓ1 dz+ = J dΓ2 dz− (60)

is

J =

***** ∂Γ1∂Γ2

∂Γ1∂z−

∂z+∂Γ2

∂z+∂z−

***** = z+z−

(61)

which can be easily computed by using equations (57) and (58) and ∂Γ1/∂Γ2 = ∂z+/∂z− =0. Equation (60) can now be reexpressed as

dΓ1dz+z+

= dΓ2dz−z−

(62)

9

Since rapidity is additive ( i.e., y2 = y1 +∆y)

dy1 = dy2 (63)

With equations (62) and (63), equation (59) is simpliÞed as

H(Γ1)z+f(z+) = H(Γ2)z−f(z−) (64)

New deÞnitions are now made to facilitate the solution of the equation

h(Γ) = logH(Γ) (65)

g(z) = log(zf(z)) (66)

The new deÞnitions transform equation (64) as

h(Γ1) + g(z+) = h(Γ2) + g(z−) (67)

Notice that∂2g(z+)

∂z+∂z−=∂2g(z−)∂z+∂z−

= 0 (68)

Differentiate equation (67) with respect to z+ and z−, eliminate terms with ∂2g/∂z+∂z−using equation (68) and cancel out factors of ∂Γ/∂z on both sides of the equation toobtain

∂h(Γ1)

∂Γ1+ Γ1

∂2h(Γ1)

∂Γ21=∂h(Γ2)

∂Γ2+ Γ2

∂2h(Γ2)

∂Γ22(69)

or equivalentlyd

dΓ

(Γdh

dΓ

)= −b (70)

where b is a constant. The solution is

h(Γ) = −bΓ+ a lnΓ+ lnC (71)

where C is a constant of integration. It yields the distribution

H(Γ) = eh(Γ) = C Γa e−bΓ (72)

Substituting equations (55) and (56) into equation (67) gives

g12(z+) +bm2

z++ a1 ln

m2

z+− a2 ln 1− z+

z++ lnC1

= g21(z−) +bm2

z−+ a2 ln

m2

z−− a1 ln 1− z−

z−+ lnC2 (73)

10

where g12(z+) is g(z+) of V1 changing to V2 and g21(z−) is g(z−) of V2 changing to V1. Ifa = a1 = a2, the transition probability distribution is

f(zj) = N1

zj(1− zj)a e−

bm2

zj (74)

where N is a constant of integration. Otherwise, the transition probability from Vα to Vβis

fαβ(zj) = Nαβ1

zjzaαj

(1− zjzj

)aβe− bm2

zj (75)

where Nαβ is a constant of integration speciÞc to the vertices Vα and Vβ. Let z0j be thejth rank momentum fraction scaled with respect to p+0, then

z01 = z1 (76)

z02 = z2(1− z1) (77)

The probability of two dependent events is the product of the probabilities of the twoindividual events. The existence of the rank-2 hadron depends on that of the rank-1hadron according to the Lund Model so that joint probability of their mutual existence isthe product of the two individual probabilities of V1 and V2. With equations (75) to (77)the combined distribution of the rank-1 and -2 hadrons is

f(z1)dz1 f(z2)dz2 = f(z01)dz01 f

(z02

1− z01

)dz021− z01

=N dz01z01

N dz02z02

(1− z01)a(1− z02

1− z01

)ae− bm2

z1− bm2

z2

=N dz01z01

N dz02z02

(1− z01 − z02)a e−b(A1+A2) (78)

The geometrical identity Aj = bm2/zj is used in the last step. Generalizing the product

of two vertices to that of n vertices, the differential probability for the production of nparticles is easily seen as

dP (1, . . . , n) = (1− z)an!j=1

N dz0jz0j

e−bAj (79)

where z =+n

j=1 z0j; let p0j = z0j p+0 and d2p = dp+ dp−, and use the identity'

dC dB δ(BC −D) = dB

B(80)

to obtaindz01z01

dz02z02

= d2p01 d2p02δ

+(p201 −m2) δ+(p201 −m2) (81)

11

With equation (81), equation (79) can be rewritten as (ref. 5)

dP (p01, . . . , p0n) = (1− z)an!j=1

N d2p0j δ+(p20j −m2) e−bAj (82)

From simple geometry in Þgure 6, the kinetic energies of the quarks along the ± lightcones are shown to be

W+ = z p0+ (83)

W− =n#j=1

m2j

z0jp0+(84)

and the total kinetic energy square at Vn is

s =W+W− =n#j=1

m2z

z0j(85)

The total differential probability of n-particle production is (ref. 5)

dP (z, s; p01, . . . , p0n)

= dz δ

"z −

n#j=1

z0j

$ds δ

"s−

n#j=1

m2z

z0j

$dP (p01, . . . , p0n)

=dz

zδ

"1−

n#j=1

uj

$ds δ

"s−

n#j=1

m2

uj

$dP (p01, . . . , p0n) (86)

where uj ≡ p0+j/Wn+. Use the method

δ

"1−

n#j=1

z0jz

$δ

"s−

n#j=1

m2

uj

$

= δ

"Wn+ −Wn+

n#j=1

uj

$δ

"Wn− −

n#j=1

m2

ujWn+

$

= δ

"Wn+ −

n#j=1

p0j+

$δ

"Wn− −

n#j=1

p0j−

$

≡ δ2

"prest −

n#j=1

p0j

$(87)

12

where prest = (Wn+,Wn−), to reorganize equation (86) as

dP (z, s; p01, . . . , p0n) = dsdz

z(1− z)a e−bΓδ

"prest −

n#j=1

p0j

$

×n!j=1

N d2p0j δ+(p20j −m2) e−bArest (88)

with the deÞnition

Atotal ≡n#j=1

Aj = Γ +Arest (89)

The claim is that Atotal is Lorentz invariant and is called the Area Law. (See ref. 7.)Equation (88) can be separated in the external and internal parts as in reference 5:

dPext = dsdz

z(1− z)a e−bΓ, (90)

dPint =n!j=1

N d2p0j δ+(p20j −m2) δ

"prest −

n#j=1

p0j

$e−bArest (91)

The external part contains kinematic variables s, z, and Γ. The internal part containsdynamic variables p0j . Equations (90) and (91) are the Þnal results.

13

14

References

1. Kaku, Michio: Introduction to Superstrings. Springer-Verlag, 1988.

2. Illana, José I.: TeV Strings and Ultrahigh-Energy Cosmic Rays. Acta Phys. Polon. B, vol. 32,

no. 11, Nov. 2001, pp. 3695–3706. http://xxx.lanl.gov/abs/hep-ph/0110092 Accessed Sept. 9,

2002.

3. Isgur, Nathan: Flux Tube Zero-Point Motion, Hadronic Charge Radii, and Hybrid Meson

Production Cross Sections. Phys. Rev. D, vol. 60, 1999, 114016.

4. Isgur, Nathan; and Thacker, H. B.: Origin of the Okubo-Zweig-Iizuka Rule in QCD. Phys.

Rev. D, vol. 64, 2001, 094507.

5. Andersson, Bo: The Lund Model. Cambridge Univ. Press, 1998.

6. Andersson, B.; and Söderberg, F.: Theoretical Physics: The Diagonalisation of the LundFragmentation Model I. European Phys. J. C, vol. 16, no. 2, 2000, pp. 303–310.

7. Andersson, Bo; Mohanty, Sandipan; and Söderberg, Frederik: Theoretical Physics: The Lund

Fragmentation Process for a Multi-Gluon String According to the Area Law. European Phys.

J. C, vol. 21, no. 4, July 2002, pp. 631–647. http://xxx.lanl.gov/abs/hep-ph/0106185

Accessed Sept. 9, 2002.

8. Sjöstrand, Torbjörn; Lönnblad, Leif; and Mrenna, Stephen: PYTHIA 6.2: Physics and

Manual. LU TP 01-21, Lund Univ., Aug. 2001. http://xxx.lanl.gov/abs/hep-ph/0108264Accessed Sept. 9, 2002.

9. Amsler, C; and Wohl, C. G.: Review of Particle Physics. 13. Quark Model. European Phys.

J. C, vol. 3, no. 1–4, 1998, pp. 109–112.

10. Bailly, J. L.; et al.: Inclusive pi0 Production in 360 GeV pp Interactions Using the EuropeanHybrid Spectrometer. Z. Phys. C, vol. 22, 1984, pp. 119–124.

11. NA22 Collaboration (Ajinenko, I.V., et al.): Inclusive pi0 Production π+p, K+p and pp

Interactions. Z. Phys. C, vol. 35, 1987, pp. 7–14.

12. Blattnig, Steve R.; Swaminathan, Sudha R.; Kruger, Adam T.; Ngom, Moussa; Norbury, John

W.; and Tripathi, R. K.: Parameterized Cross Sections for Pion Production in Proton-Proton

Collisions. NASA/TP-2000-210640, 2000.

13. Allen, Theodore J.; Goebel, Charles; Olsson, M. G.; and Veseli, Sinisa: Analytic

Quantization of the QCD String. Phys. Rev. D, vol. 64, 2001, 094011.

14. Polchinski, Joseph: String Theory. Cambridge Univ. Press, 1998.

15. Hatfield, Brian F.: Quantum Field Theory of Point Particles and Strings. Addison-Wesley,

1992.

15

.1 .2 .3 .4 .5x

0

1000

2000

3000

4000

Reference 10dσ/dx = 3600 exp(−30x)

dσ/d

x, m

b

Figure 1. Experimental dσ/dx and string model results. Constants used in exponential function chosen to fit dataand not parameterizations; p + p → π° + X; Plab = 360 GeV.

.1 .2 .3 .4 .5 .6x

0

5.0

10.0

12.5

2.5

7.5

Reference 11Edσ/dpL = 10 exp(−7x)

Edσ

/dp L

, mb

Figure 2. Experimental E dσ/dpL and string model results. Constants used in exponential function chosen to fit dataand not parameterizations; p + p → π° + X; Plab = 250 GeV.

16

x

t

xmax−xmax

m2

Figure 3. Yoyo motion of quark-antiquark pair confined by linear potential.

Constant τsurface

V1

V2V3V4

Figure 4. Breakup of quark-antiquark pair along surface of constant proper time τ. Bold arrows represent velocitiesof produced mesons; breakup points labeled as vertices V1 to Vn.

17

m2

A3

A2 A1

z+ W+1z− W−2

W+1 = κx+1W−2 = κx−2

V2 V1

Figure 5. Geometry of kinematics of two adjacent vertices V1 and V2, which are vertices or spacetime positions thatalso represent energy carried by string field. Quark moves along positive light cone and antiquark moves alongnegative light cone; antiquark of V1 combines with quark of V2 to produce meson of mass m; W+1 is energy of V1along x+; z+W+1 is fraction of energy used to create quark from V2; W−2 is energy of V2 along x−; z−W−2 is fractionof energy used to create antiquark from V1; A1, A2, A3, and m2 are areas of rectangles. From reference 5.

m2/(z0np+)

m2

z0np+

Arest

S

W−

W+

VnVn−1

Γ

Figure 6. Geometry of kinematics of involving total area of spacetime diagram such that Atotal = Γ + Arest. Totalenergy square s = W+W− is represented by rectangle s; p± is energy of parent quark (antiquark) along ± light-conecoordinates; z0np+ is fraction of energy used to create quark from Vn; m2/(z0np+) is energy used to create antiquarkfrom Vn−1. From reference 5 with minor modifications.

REPORT DOCUMENTATION PAGE Form ApprovedOMB No. 0704-0188

1. REPORT DATE (DD-MM-YYYY)

09-20022. REPORT TYPE

Technical Publication 4. TITLE AND SUBTITLE

String Fragmentation Model in Space Radiation Problems5a. CONTRACT NUMBER

6. AUTHOR(S)

Tang, Alfred; Norbury, John W.; and Tripathi, R. K.

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research CenterHampton, VA 23681-2199

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National Aeronautics and Space AdministrationWashington, DC 20546-0001

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13. SUPPLEMENTARY NOTESTang and Norbury: University of Wisconsin, Milwaukee, Wisconsin. Tripathi: Langley Research Center, Hampton, Virginia.An electronic version can be found at http://techreports.larc.nasa.gov/ltrs/ or http://techreports.larc.nasa.gov/cgi-bin/NTRS

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14. ABSTRACT

String fragmentation models such as the Lund Model fit experimental particle production cross sections very well in the high-energy limit. This paper gives an introduction of the massless relativistic string in the Lund Model and shows how it can be modified with a simple assumption to produce formulas for meson production cross sections for space radiation research. The results of the string model are compared with inclusive pion production data from proton-proton collision experiments.

15. SUBJECT TERMS

String fragmentation model; High energy limit; Lund model; Meson production cross section in proton-proton collisions

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