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1994
Glueball wave functions and production cross sections
Anthony B. Wakely College of William & Mary - Arts & Sciences
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Recommended Citation Wakely, Anthony B., "Glueball wave functions and production cross sections" (1994). Dissertations, Theses, and Masters Projects. Paper 1539623854. https://dx.doi.org/doi:10.21220/s2-n4e0-hq80
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Glueball wave functions and production cross sections
Wakely, Anthony Bruce, Ph.D.
The College of William and Mary, 1994
UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with with permission permission of the of copyright the copyright owner. owner.Further reproduction Further reproduction prohibited without prohibited permission. without permission. GLUEBALL WAVE FUNCTIONS AND
PRODUCTION CROSS SECTIONS
A Dissertation
Presented to
The Faculty of the Department of Physics
The College of William and Mary in Virginia
In Partial Fulfillment
Of the Requirements for the Degree of
Doctor of Philosophy
by
Anthony B. Wakely
1994
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPROVAL SHEET
This dissertation is submitted in partial fulfillment of
the requirements for the degree of
D octor o f Philosophy
Anthony B. Wakely
Approved, January 1994
Carl E. Carlson
Marc Sh
Robert T. Siegel / 1
( q m J. Dirk W alecka
<>t Richard H. Prosl Department of Computer Science
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TO
MY MOTHER AND FATHER
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS...... vi
LIST OF T A B L E S ...... vii
LIST OF FIGURES ...... viii
ABSTRACT ...... x
CHAPTER 1. INTRODUCTION ...... 2 I. HISTORY OF Q C D ...... 2 II. QCD INSPIRED M ODELS ...... 4 III. GLUEBALLS ...... 5 IV. QCD SUM R U LES ...... 7 V. OUTLINE OF THE T H E SIS ...... 8
CHAPTER 2. PSEUDOSCALAR GLUEBALL WAVE FUNCTIONS FROM QCD SUM RULES ...... 9 I. INTRODUCTION ...... 9 II. DEFINITIONS AND HADRONIC EVALUATION OF THE PSEUDOSCALAR GLUEBALL CORRELATOR ...... 10 III. THE GLUEBALL DISTRIBUTION AMPLITUDE FROM A QCD SUM R U LE ...... 17 IV. THE DISTRIBUTION AM PLITUDE ...... 42
CHAPTER 3. TWO-PHOTON PRODUCTION OF PSEUDOSCALAR GLUEBALLS ...... 45 I. INTRODUCTION ...... 45 II. CALCULATION OF THE SCATTERING AM PLITUDE ...... 45 III. CALCULATION OF THE CROSS SECTION ...... 54 IV. CONCLUSIONS ...... 60
CHAPTER 4. SEMI-INCLUSIVE PION PHOTOPRODUCTION AT LARGE TRANSVERSE MOMENTA ...... 65 I. IN T R O D U C T IO N ...... 65 II. THE DIRECT SUBPROCESS ...... 67
iv
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. III. THE FRAGMENTATION SUBPROCESS ...... 73 IV. RESULTS ...... 76 V. CONCLUDING REM ARKS ...... 80 APPENDIX A. COMMENTS ON FRAGMENTATION FUNCTIONS 82 '
REFERENCES ...... 88
VITA ...... 94
v
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS
First and foremost, I would like to thank Dr. Carl Carlson for his patient guidance, support, and assistance during the completion of my graduate studies including this work. His insightful direction, encouragement, and enthusiasm enabled me to greatly increase my understanding of hadronic physics, especially glueballs,
I would also like to express my appreciation to:
Dr. Marc Sher for serving on my thesis and review commitees and for providing useful advice concerning particle physics and physics in general,
Dr. Dirk Walecka for serving on my thesis committee and for teaching his courses in quantum mechanics and nuclear physics,
Drs. Robert Siegel and Richard Prosl for serving on my thesis committee,
and Drs. Nathan Isgur and Franz Gross for serving on my annual review committee.
I would like to thank SURA and CEBAF for support during the completion of this work.
I would like to thank Paula Perry, Sylvia Stout, and Dianne Fannin for helping to deal with the maze of college paperwork.
I would also like to thank the rest of the Physics Department for the encouragement and support of my efforts.
And, last but not least, I would like to thank my fellow graduate students and friends, especially Doug Baker, Pete Markowitz, and Matt Johnson, for making my life here at William and Mary an enjoyable experience.
vi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES
TABLE I. Moments of the distribution amplitude and the decay constant of the pseudoscalar glueball evaluated as a function of the mass of the glueball ...... 39
vii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES
Fig. 2.1. The lowest order purely perturbative and a purely vacuum contribution to the correlator...... 19
Fig. 2.2. Non-perturbative contributions to the correlator ...... 24
Fig. 2.3. The gluon propagator modified by the interaction with one gluon which connects to the vacuum ...... 26
Fig. 2.4. The gluon propagator modified by the interaction with two gluons which connect to the vacuum and combine to form the gluon condensate ...... 30
Fig. 2.5. The gluon propagator modified by the interaction with a quark-antiquark pair; one of the pair connects to the vacuum to form the quark condensate ...... 31
Fig. 2.6. Other possible non-perturbative contributions to the correlator ...... 33
Fig. 2.7. The sum rule for /J for Mc = 2.0 G eV ...... 38
Fig. 2.8. The pseudoscalar glueball distribution amplitude for MG = 2.0 GeV. . . . 43
Fig. 3.1. Typical Feynman diagrams contributing to the process y+ y —> G + 48
Fig. 3.2. One of the Feynman diagrams contributing to the process y + y —» G+ ft0...... 49
Fig. 3.3. Predictions for the production cross section for y+ y G + nP and 7 + y -» 7i° + tP...... 59
viii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 3.4. Experimental cross section for yy —» + K+K~...... 61
Fig. 3.5. Predictions of the cross section for y+ y -» G + from Ref. [3.1]. . . 64 '
Fig. 4.1. A typical Feynman diagram for the fragmentation subprocess for y p - > 7 t X ...... 66
Fig. 4.2. A typical Feynman diagram for the direct subprocess for the process y p - ^ n X ...... 66
Fig. 4.3. Typical Feynman diagrams for the process yq —»(qq) + q...... 69
Fig. 4.4. A typical Feynman diagram contributing to the pion electromagnetic form
factor...... 72
Fig. 4.5. Typical partonic Feynman diagrams for the fragmentation subprocess for the process y p —> n X ...... 74
Fig. 4.6. Predictions for the cross section, da/dkj, for yp -*n X for C0 y= 100 GeV and kL > 40 G eV ...... 77
Fig. 4.7. Predictions for the cross section, da/dkj, for yp -*n X for C0 y = 20 GeV and all allowed kL...... 78
Fig. 4.8. Predictions for the invariant cross section, coK da/d^k for y p ^ n X for 4 values of coy, 20, 30, 70, and 100 GeV and xp= 0.5...... 79
Fig. A.I. The fragmentation functions of [A.2] compared to experimental data. . . . 83
Fig. A.2. Primary and secondary meson production from quark fragmentation. . . . 84
Fig. A.3. Our fragmentation functions com pared to experimental data ...... 85
ix
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT
Three topics are studied in this dissertation.
Using QCD sum rules, I first calculate the first few moments of the distribution amplitude of the pseudoscalar glueball, a bound state of gluons. The distribution amplitude is the momentum wave function integrated over the transverse momentum. QCD sum rules are a method of finding hadronic parameters using perturbative QCD. I use an approximation in which the glueball is treated as a narrow resonance. The moments of the distribution amplitude then give the corresponding first few coefficients of the distribution amplitude for the glueball expanded in Gegenbauer polynomials. The distribution amplitude is rather close to its asymptotic form, and the glueball's “decay constant” is about 105 MeV for a pseudoscalar glueball mass of 2.0 GeV.
Using the previously calculated distribution amplitude, Icalculate the normalized differential cross section for two-photon pseudoscalar glueball plus meson production, y + y —» G + tt°. I compare to a previous calculation and to y+ y —> tt° + jP.
Lastly, I show that very high momentum transfer semi-inclusive photoproduction of pions from nucleons, y p —>7tX, is dominated by a subprocess where the pions are produced directly. Although the “direct subprocess” may be thought of as a limiting case of a quark fragmentation subprocess, it is an exclusive process at the quark level, fully calculable using perturbative QCD, and thus distinct from the usual thinking about fragmentation processes. The numerical results, concentrating on neutral pions, indicate that the direct subprocess is dominant at pion transverse momenta slightly higher than those already measured. Incidentally, the integral over the pion distribution amplitude in the present case is the same as that for the pion electromagnetic form factor, but the momentum transfers involved correspond to measuring the pion form factor at momentum transfers up to several hundred GeV2. These calculations can now be extended to glueball production at high transverse momentum.
x
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. GLUEBALL WAVE FUNCTIONS
AND PRODUCTION CROSS SECTIONS
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. INTRODUCTION
I. HISTORY OF QCD
Once upon a time, in the 1960’s, particle physics was drowning in a sea of new
hadrons. One of the first indications that there might be order among these hadrons was
the discovery by Gell-Mann and Ne’eman in 1961 that the known mesons and baryons
could be placed into SU(3) multiplets. In 1964, Gell-Mann and Zweig expanded the idea
of SU(3) symmetry to the quark model of hadrons. After the concept of color was
introduced to the quark model to explain the spin-symmetry properties of bound states of
quarks and after partons were discovered in hadrons, quantum chromodynamics (QCD)
was developed as a gauge theory of quarks and gluons based on color charge. Several
models based on QCD have been been developed to explain various properties of hadrons.
As a calculational technique, QCD sum rules were developed to calculate masses, wave
functions, and other properties of hadrons.
The original SU(3) symmetry (now known as flavor symmetry) introduced by Gell-
Mann and Ne’eman was known as the “Eightfold Way” because one of the SU(3)
multiplets is the octet [1.1]. The Q", discovered in 1964, was predicted in 1962 [1.2] on
the basis of its position in an SU(3) multiplet. The SU(3) multiplets, octets and decuplets,
used for the hadrons, however, are not the fundamental multiplet, the triplet, or the
adjoint multiplet but are product multiplets of the fundamental multiplets. The pseudoscalar
mesons can be placed into an octet and a singlet which can be obtained by multiplying the
triplet with the adjoint multiplet, 3 ® 3*. The spin-1/2 baryons, which includes the
2
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nucleons, and the spin-3/2 baryons, which includes the delta resonance, can be placed
into an octet and a decuplet, respectively; these multiplets can be obtained by multiplying
the triplet by itself twice, 3 0 3 0 3.
In 1964, Gell-Mann and Zweig identified the fundamental representation of SU(3),
the triplet, with quarks of charge 2/3, the up quark, and-1/3, the down and strange
quarks, and spin 1/2; however, since no particles with these unusual charges had been
identified, this scheme was at first mostly regarded as a mathematical model of hadrons
without the quarks being identified as actual particles [1.3]. Baryons are then composed of
three quarks and mesons composed of a quark and an anti-quark. There was a problem
with this scheme, though, because the delta resonance, a baryon, which is a spin-3/2
particle requires an overall antisymmetric wave funcdon but three spin-1/2, to form spin-
3/2, and three isospin-1/2, to form isospin-3/2, quarks in the ground state combine to
form a symmetric wave function. This was solved by adding a hidden quantum number,
color [1.4]. Each quark carries one of three colors which forms a triplet under SU(3). The
wave funcdon of all baryons are then composed of an antisymmetric color wave funcdon
and a symmetric space-spin-flavor (or isospin) wave funcdon; the color wave funcdons are
colorless. No free quarks of fractional charge, however, were ever seen1.
In the late 1960’s, electron scattering experiments on nucleons were performed which
seemed to indicate that nucleons are composed of point-like particles with weak interactions
where the coupling constant becomes small at short distances [1.6]. In 1973, Gross,
Wilczek, Politzer, and Coleman showed that non-Abelian or Yang-Mills gauge field
theory and only Yang-Mills theories have just such properties [1.7,1.8,1.9], and so QCD,
which is a non-Abelian theory of quarks and gluons based on the SU(3) group of color
charge, was postulated [1.10]. This property of weak forces at short distances is known
1 Fractionally charged particles were found on small niobium balls using a sophisticated Millikan oil-
drop experiment by LaRue et al [1.5], but these results were never confirmed by other experiments.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4
as asymptotic freedom. Gluons, the vector bosons which mediate the chromodynamic
force, also carry color charge and so, in addition to interacting with quarks, interact with
themselves. This theory explained why bound states with color fractional charges are not
seen by postulating that colored quarks and gluons are confined within hadrons by strong
forces [1.8,1.11], Therefore the color wave function is a singlet under SU(3) as postulated
earlier. The lightest and possibly only bound states of quarks that are colorless are
quark-antiquark states, mesons, and three quark states, baryons. QCD also predicts
colorless bound states of gluons, glueballs; however, no glueball candidates have been
universally accepted.
II. QCD INSPIRED MODELS
As we have seen, QCD predicts large or confining forces at large distances and weak
forces at short distances, and this fact is exploited in several models of hadrons based on
QCD. Potential models approximate the forces between quarks using non-relativistic
potentials. There are usually two components to the potential, a long distance confining
piece which is usually given by a linear potential and a short distance one-gluon exchange
piece which produces a 1/r potential. Using these models, the spectrum of hadrons
consisting of non-relativistic quarks such as the 7/ i/r and T can be explained quite well
[ 1. 12].
The M.I.T. bag model is based on the fact that the quarks within a hadron are confined
to a particular volume and that at short distances the forces between the quarks are weak
[1.13,1.14], The bag model carries this to an extreme by assuming that the quarks are
confined by an infinite potential at the bag surface and that the forces between the quarks
are zero. In this model, the bag creates a bubble in the vacuum and it takes an amount of
positive internal energy to create this bubble. Using, for example, the nucleon mass as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. input, excited states of nucleons, static and transition magnetic moments, and various
other properties can be predicted fairly well. As an improvement, a one-gluon exchange
potential can then be added to the model as a perturbation [1.15]. The bag model can also
be extended by adding meson fields to model the long distance behavior of the wave
functions which then exhibit a meson tail at long distance; this is the basis for the Cloudy
[1.16], Little [1.17], and Chiral [1.18] Bag models [1.14],
Experiments in the physical world indicate that the actual number of colors (Nc) is
three. QCD can be solved exactly in the limit where the number of colors is infinite. This
may not seem like such a good idea, but the fact that naive expectations may not always be
bome out is shown by examining the electromagnetic fine structure constant:
^2/4^1/137 and therefore £=0.30 as pointed out by Ed Witten [1.19]. Of course, in
QED perturbative calculations using e give accurate results. Some results of the infinite
Nc limit include:
1) The baryon spectrum starts at an energy proportional to 0(N C), i.e., they are
infinitely heavy, but then the spacing between excited baryons continues
independent o f Nc.
2) The meson spectrum is independent of Nc.
3) Meson-meson scattering goes as 0(1/NC).
4) Baryon-meson scattering is independent of Nc.
and
5) Baryon-baryon scattering is also independent of Nc.
These results indicate that mesons and baryons can be regarded as the low-energy effective
degrees of freedom of QCD [1.19,1.20,1.14],
III. GLUEBALLS
As mentioned previously, gluons are self-interacting and so QCD predicts that bound
states of gluons (“glueballs” or “gluonia”) should form [1.21]. Gluons, being of spin
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one, will combine to form bound states with integral spin, and since some glueballs will
have the same quantum numbers as quarkic mesons, it may be difficult to distinguish one
from the other. Another difficulty in discovering glueballs is that when the quantum
numbers are the same between gluonic and quarkic states there may be quantum mechanical
mixing between pure glueball and pure quarkic states. However, there are several possible
glueball states that have quantum numbers that cannot be formed from qq states; these
have JPC (spin, parity, and charge conjugation) of 0", 0+‘, 1 '+, 2+" If one of
these states, known as oddballs [1.22], were discovered, this would be the best evidence
for glueballs or at least for something not qq. But since evidence for these states is
lacking, the search for glueball candidates continues in other ways as oudined in the
Review of Particle Properties [1.23]. One search method is to examine the meson spectra,
and if there are more discovered states than can be placed in qq muldplets, then it may be
surmised that a non -qq candidate, possibly a glueball, has been found. Another method
is to examine the production and decay channels of particular glueball candidates. Naively
one might expect a glueball to exhibit flavor-singlet couplings, reduced photon couplings,
and enhanced production in gluon-rich decays such as 7/f'decays. However, even
though glueballs are composed of uncharged components, our results (see Chap. 3) seem
to indicate that the photon-glueball coupling is not necessarily reduced relative to neutral
quarkic meson-photon coupling.
Lattice gauge theoretic calculations indicate that the mass of the pseudoscalar glueball
(jPC=o- +) is about 2.8(3) GeV, the mass of the scalar glueball (J ^ = 0++) js about
1.5(1) GeV, and the tensor glueball (J^ = 2 ++) is about 2.3(3) GeV [1.24]. QCD sum
rule calculations indicate that the mass of the pseudoscalar glueball is about 2.3(3) GeV
[1.25] or 1.7(1) GeV [1.26], the mass of the scalar glueball is 1.6(1) GeV [1.26], and the
mass of the tensor glueball is about 1.5 GeV [1.25] or 1.7(1) GeV [1.26]. See the review
by Close [1.27] for a more complete review of the status of theoretical calculations of
glueball properties.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7
Some of the best experimental glueball candidates include the/0( 1590) and/0(1710)
(jPC=o++), and the 77(1440) (jPC-q- +)_ The 77(1440) situation is complicated because
there are likely two resonances in this region. For a more complete review of the status of
non-<7 <7 candidates see the Review of Particle Properties [1.23] or the review by Close
[1.27]. No glueball candidates have been indisputably discovered or confirmed.
Another problem in trying to establish whether glueball candidates are actually glueballs
is that glueball production rates have not been accurately calculated. In this thesis, I hope
to contribute to the glueball search by first calculating the glueball wave function which can
then be used to calculate the production rate for glueballs in various processes; one process
that we chose to study was glueball production in two-photon processes, yy-> Gtfi (see
Chap. 3).
IV. QCD SUM RULES
QCD sum rules, based on the operator product expansion in QCD and developed by
Shifman, Vainshtein, Zakharov, and Novikov, analyzes hadrons, bound states of
quarks and gluons, by computing a sum rule. These sum rules can be used to determine
masses, wave functions, decay and coupling constants, and other hadronic properties
[1.28,1.25].
All QCD sum rules are calculated by saturating (i.e., inserting a complete set of states
and then summing over these states) the vacuum expectation value of the time-ordered
product of two operators each of which have some overlap with the hadron. One side of
this sum rule is calculated using the partons (quarks and gluons) which make up the hadron
and the other side is calculated using the hadron. The first term of the partonic sum rule
includes the perturbative contributions to the bound state; succeeding terms include the
non-perturbative corrections which are calculated using quarkic and gluonic condensates
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8
which describe the effects of confinement These quarkic and gluonic condensates are the
number density of quarks and the energy density of gluons in the physical vacuum; values
of these are obtained elsewhere. The quark and gluon condensates measure the intensity of
the quark and gluon fields in the non-perturbative QCD vacuum. The other side of the sum
rule is then calculated by saturating it with a complete set of hadronic states. Both sides of
the sum rule are then smoothed, typically using the Borel transform which introduces the
Borel parameter.
The smoothed partonic sum should then be equal to the smoothed hadronic sum;
however, approximations have been made in both sums. The sum rule can then be
rearranged to give a function of the desired hadronic property on the LHS of the sum rule
and a function of the Borel transform parameter on the RHS. The sum rule is then
evaluated over a suitable range of the Borel parameter and the RHS should be
approximately constant over a broad range of the Borel parameter. The desired hadronic
property can then be obtained from this evaluation [1.29]. See Chap. 2 for a more detailed
description of this method as it is applied to the determination of the moments of the
glueball distribution amplitude.
V. OUTLINE OF THE THESIS
In Chapter 2, I describe in detail how the distribution amplitude of a glueball can be
obtained from QCD sum rules [1.30]. In Chapter 3, I calculate two-photon production of
glueballs using perturbative QCD and the glueball distribution amplitude obtained in
Chapter 2 [1.31]. In Chapter 4, I show that semi-inclusive pion photoproduction at large
transverse momenta is dominated by a subprocess that can be calculated using perturbative
QCD [1.32].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. PSEUDOSCALAR GLUEBALL WAVE FUNCTIONS FROM QCD SUM RULES
I. INTRODUCTION
To calculate the production rate for glueballs using perturbative QCD, we first need the
wave function of the glueball or, more accurately, the distribution amplitude, which is the
integral over the transverse momentum of the momentum wave function of the glueball
[2.1 ]. Although other people have previously tried to calculate the glueball wave function,
their calculations were usually plagued by relatively large non-perturbative contributions to
the partonic evaluation of the correlator which came from non-loop diagrams exemplified
by Figs. 2.1(b), 2.2(e), and 2.6(a) below [2.2], One therefore had no confidence that the
calculation was correct. We were able to find an operator such that the non-loop
contributions are zero, and thus we find that the overall non-perturbative contribution is
less than a few tens of percent of the perturbative contribution.
We concentrate on the pseudoscalar glueball (1^= 0" +). In section II, the definitions
of the pseudoscalar glueball distribution amplitude, its expansion in terms of Gegenbauer
polynomials, and the correlator and the operators used in the QCD sum rule calculation are
given. The hadronic evaluation of the correlator is also calculated in section II. In section
IK, the partonic evaluation of the correlator is calculated and the matching of the hadronic
and partonic evaluations is done. In section IV, the distribution amplitude is given.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10
II. DEFINITIONS AND HADRONIC EVALUATION OF THE
PSEUDOSCALAR GLUEBALL CORRELATOR
We start with the lowest Fock component of the pseudoscalar glueball which is given
by [2.3]
| G ) - ^[d.x][d2kT] ~ j= ~J2~ V2 \Sa^x\^\T ^\)S b ^x2 ->k2 T '^ 2 ))
(2.II.1)
where Ay = ± l is the helicity, x,- is the momentum fraction, kiTis the transverse
momentum of a gluon, a and b are color indices, ng = 8 is the number of gluon colors,
and £i ^ 2 is antisymmetric in its arguments with £+_-1. The abbreviated symbols [dx]
and [cPkT} are given by
<2-"-2>
and
The momentum wave function y/g is required to be antisymmetric under the interchange of
its arguments because Bose statistics require that the wave function of a boson be
symmetric and the L%=0 part of the wave function is the leading contribution to the
scattering amplitude. Given relativistic normalization for the states
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t§l(*l>*17-’* l)M * 2 ’*27-^2)) = 16ff35 (^ - X 2)8 2(klT -*2r)^Aj,A2 ’ (2-11.4)
satisfies the following normalization
d 2k 2 \ dx^ ^ xl x2 \ v g(x,kT) f = P2g (2.II.5) 16 it
where P^g represents the probability of finding the two gluon Fock state.
High momentum transfer reactions, for which the transverse momentum of the bound
state is negligible, depend on the distribution amplitude
where Q2, the momentum transfer, is some scale relevant to the process. If we examine
the evolution equation, or Q1 behavior, of the glueball wave function, we see that the
eigenfunctions of this equation are the Gegenbauer polynomials of 5/2 order, C„/2.
Therefore, the pseudoscalar glueball distribution amplitude can be expanded at high Q in
the odd Gegenbauer polynomials of 5/2 order as [2.4,2.5]
where bn depends on In Q2 to a known power which grows monotonically more negative
with increasing n. The dependence of bn on In Q2 can be written as
^ln(<22 /A 2) ^(ln (a2)) = ^(ln(eb2)) (2.II.8) ln(2 b2/A 2)
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where A is the QCD running coupling constant parameter, typically about 200 MeV, y"
is [2.3,2.4]
n+1 12 ^ y I L-HL (2.II.9) 11 — n t (n + l)(n + 2) £ 2j 12 18
and n f is the number of quark flavors at a given energy scale. Here we will neglect the
weak Q2 dependence in bn so that bn is evaluated at some fixed Q2. The first few odd
Gegenbauer polynomials of 5/2 order are [2.6]:
C?/2(jc) = 5x
C35/2« = y (3 * 3-*)
d /2(x) = — (l4 3 x 5 - 1 10x3 + 15jc) (2.0 .10)
The Gegenbauer polynomials, C„/2, satisfy an orthonormality relation
j lo dx,(xxx2 )2 d ,2(Xl- x 2 ) d /2(Xl- x 2 ) = ± K n 8mn (2.0 . 11)
where
2 (n + 4)! K = 9(2n + 5) n!
First, some coordinate definitions are in order. We will often use light-cone
coordinates for four-vectors and tensors. This coordinate system is defined as follows; x
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and y are four-vectors:
xM =(ar+,.t",xT)
x+ = x ° + j:3 x ~ = x ° - x 3 xT = |x 1,x 2j
;c'>’ = ~(;c+3’_ + jr_)'+)- XT,yT • (2.II.12)
For the QCD sum rule calculation, we choose operators
Mn(y) = - i e i} Fa+i(y) (,D +) ^ Fb+J(y) (2.II.13)
where / and j represent transverse components (i, j = 1, 2), e^-is totally antisymmetric
with£12= l, F is the gluonic field tensor, F will be the dual field tensor,
F^v - £^vaPFap, and D is the covariant derivative, [p+)^ = d+8ab-ig{Tc)abA^. The
operators Mn are components of the gauge and Lorentz invariant operators
- / F™ in ( D “ ‘ ...D a" U F™ gp7 . (2.II.14)
The color indices are summed over. The operator Mn{y) is obtained by setting all of the
free indices in (2.II.14) equal to +. If we use light-cone gauge for the dynamical gauge
fields AF{x),
A+(v) = 0 ,
then M n(y) becomes
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14
Mn(y) = -i£'><5^ +A‘(y)(/3 +)V /^'(?) . (2.II.15)
We will see that with this choice of operator the non-loop contributions to the partonic
evaluation of the correlator are zero, and therefore the overall non-perturbative
contribution remains small compared to the perturbative contribution. Note that the twist,
defined as the naive dimension of Mn minus the number of free dimensions [2.7], of Mn
is two. The operators of lowest twist (i.e., twist-two) dominate in the light-cone
expansion. This operator was first used by Tsokos to study glueball wave functions [2.3].
We now want to calculate the matrix element of Mn between a one glueball state and
the vacuum. We use
where k is the momentum and A is the helicity of the gluon. For the matrix element, we
then get
(0|M„(0)|G) = J [dx][d2kj] *jL ? te-y,g(x,kT)-
><9X(0)(;a+) d+AJb( o) gc(kM s A k2^2)
= -ieij^ j m [ d 2kT\ ¥g(x,kT)kt^(kt-k^)n
{e + (*1 )£- {k2 ) “ £- (*1 )£+ (^2 )]
= 2 ^ ( k +f +2 j[dx][d2k t ] xlx2(xl - x2)n ¥g(x,kT)
= 2^ ( k + )"+2 £ dxx xix2(xx - x2 )" (2.II.17) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 We define the “moments” of the distribution amplitude as I " = J ^ i x\x2{xx - x2) where £ = ^ - x2 so that (0 |M „(0 )|g ) = 2 ^ ( * +)n+2i ” . (2.II.19) The “decay constant” of the glueball, fQ, which is related to £, is defined [2.3] as (0|Ml(0)|G) = (*+)3/c = 2^(*+)3I . (2.II.20) By substituting the expansion of the distribution amplitude in Gegenbauer polynomials into (2.II. 18) and using the orthogonality of the Gegenbauer polynomials, we can obtain the coefficients in the expansion, bn, in terms of the moments. We get the first coefficient, b[, from the first moment, we get b% from £ and | 3, etc. For example, ^ = 4 2 1 = - ^ ^ V"« ^ = 33(3^3-^ ) fc5= ^(l4 3 £ 5-110£310£3 ++ 115S) 5 ^ (2.II.21) The QCD sum rule calculation of the moments proceeds by twice evaluating the correlator Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 /„(p) = ij d*yeipy(0\TMy{y)Mn(0)\0) , (2.11.22) once by saturating it with glueball intermediate states and once by saturating it with gluonic and quark intermediate states. Here the hadronic evaluation is outlined. First, we evaluate the correlator for a single glueball of massmg. Saturating the correlator, or in other words, inserting a summation of a complete set of glueball states 1 = f (2n?2EklG(k)){Gm = J(0 5+(*2 ' mG2)|GW>(G(*)| where the + means ko>0 and using Mn(y) = e ^ M n( 0 ) ^ , we get for the correlator In(p) = ijd* y - ^ f SA kl - ^)[^ (> 'oy (- i)^{o|A/1(0)|G(()){G(()!A/n(0)|o) + 0 (-y ° y (P+<:»'(O|Mn(O)|C(/:)){G(/:)|Ml(O)|O>] (2.II.23) W e now use JQdy°e^P - - iT +7iS(p° + k°) , pv Tkv +ie Pu + r f^ 3 e -'W y' = 53(pTk) , J (2tt) v 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 and the fact that (o[Mn(0)|G^ is real to obtain Im In(p) = njd*k 5+(k2 -i»^)(0|W 1(0)|C(ft)){G|Afll(0)|0) x £(5 (p — k) + 8 (/> + £)j ( 2 I I 24) Substituting (2.II.19) and (2.II.20) into (2.II.24), we get finally Im In(p) = K(p+)n+5S+(p 2- m * ) f £ l r / l . (2-11-25) Thus, the correlator is related to the moments of the distribution amplitude through the hadronic evaluation. We should sum over all glueballs and add contributions from the continuum. Here we keep only the lightest glueball, approximating the rest of the glueball spectrum and the continuum as described in the next section. We use the narrow width approximation by supposing that the lightest glueball is the lowest mass state that gives a substantial contribution to the correlator. The details of the partonic evaluation of the correlator using the gluon and quark condensates are described in section ID. III. THE GLUEBALL DISTRIBUTION AMPLITUDE FROM A QCD SUM RULE We now turn to the partonic evaluation of the correlator using gluon intermediate states and the gluon and quark condensates. First, we calculate the perturbative contribution to the correlator, Eq. (2.II.22). The Feynman diagram for this contribution is given in Fig. 2.1(a). The operator Mn{y) can be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 used in the form of (2.II. 15) to get Ij-Hp) = -ijd*yeipy£ijekll 0 rj (2.DI.1) We use the propagator (0|7>1‘ M A '(y )|o ) = - i j (dk) e - ^ x~y) Sat, (2.HI.2) d*k where (dk) is Wick's theorem, which allows us to write the time-ordered product (2tt) of operators as a sum of products of time ordered products, to match the two pairs of gluon fields together to form two propagators, and to make two sets of contractions, and do the integral over y to write the correlator as k f k f I^\p) = -32i\{dkl){dk2)(2iz)H \p-kl-k2),y ^ r (kt-kt)n+X . (2.ffl.3) k\ k2 Here k{ and k2 represent Iq +ie and k2 +ie. Using the Feynman parametrization (n + m+ 1)! ri xn( \ - x f [ dx (2.m .4) n\m\ Jo n+m+2 doing the integral over k2, and changing the integration variable to q = k{ - (1 - x)p, we get Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 (a) (b) Fig. 2.1. (a) The lowest order purely perturbative contribution to the correlator, (b) A purely vacuum contribution to the correlator. Wiggly lines are propagating gluons and dashed lines represent background gluons in the physical vacuum. / x {z • [~2q + (x - (1 - x ))]}n+1 + x(l-x)p2 + ie| After we do the integral j(dq)(z-q)n f[q2)= 0 which is zero because of the angular part of the integral, perform a Wick rotation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 qE =iq° J — ©O d?°= - / ]J — '. loo dq*E = i\ J — o©dq4E which transforms the Minkowski momentum, q, into the Euclidean momentum, qg, calculate the resulting four-dimensional angular integral using the following four- dimensional integrals, JQ 4 = sin2 6{ ddi sin d2 dd2 d(p JjQ4=2;r2 | d4q£ /( where Q4 is the four-dimensional solid angle, and do the integral over dqE, the correlator can be written as ' P f +5 \\d x (1 - x )2 x2[x - (1 - x)]+l In - 1 7T x(\-x)P2-ie (2.m.7) where P2 = - p 2 and A is an infinitely large momentum cutoff. As is customary, to eliminate the dependence on A, we take the imaginary part of the correlator by using Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 2 Im In ^ - 3 ------= - I m ( ln |p 2| - indl-P 1)) (/>2 < o) x(l-x)Px(l-x)P2-ie ' 1 1 \ I) \ > = n We then apply the binomial expansion to the polynomial in x to get o i n+1 f h j_ | \ Im/^^-Cz-p)^5} ^(l-x)2x2X . (-l)' (l-x)' x' * Jo 1 ; /i+l n + 1 ,-( n - i + 3)!(i + 2)! = - ( « - p r 5 I (-D * /=o7~i (n + 6 )! . (2.HL8) The integral in x is done using the standard Beta integral. The sum in (2.III.8) can be sim plified to 2 = !------2 (n + 2)(rt + 4)(n + 6) and so we finally get for the perturbative contribution to the correlator 1 Im I{nl)(p) = -(.z-p)n+5 n (n + 2)(n + 4)(n + 6 ) c(l) = ^ ( Z.Pr 5 n (2.DI.9) We now turn to the non-perturbative contributions to the correlator. These contributions form an expansion in the QCD coupling constant g (g2 = 4nas) and the momentum of the glueball squared p 2. We will keep terms up to order g2/p4 relative to the leading, purely perturbadve term. For the gluon fields that are connected to the vacuum via the gluon condensate, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 background fields, as in Figs. 2.1(b) and 2.2(a-e), we use the fixed point gauge xmA ^ M = 0 . (2.IIL10) This gauge enables us to write the gluon field A% (x) in terms of the gluon field tensor F£v(0) as shown in the text by Pascual and Tarrach [1.29]; this enables us to write a gauge and Lorentz invariant gluon condensate. To show this, we begin with the identity r1 • <2'nu» When we use the gauge condition (2.III. 10) and the gluon field tensor, Eq. (2.III. 11) can be rewritten as A a^y ) + yp^ f - = yp Fp^iy) . (2.HI.12) Substituting y= ax and noticing that the LHS of Eq. (2.III. 12) is a total derivative, we write j-[ccA°(ax)] = axpF^{ax) (2.III.13) which can be integrated from 0 to 1 to get A“(ax) = ^ d a ax? Fp^(ax) . (2.DI.14) Substituting the series Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 fa {a x ) = (0) + a ^ f a (0 ) + ^ a 2* ^ x " 2 f a (0 )+... (2.III.15) ' into (2.m. 14) and performing the integration, we obtain L xpxa>l...x0)n da ... da f a (0) . (2.HI.16) ,^o(n + 2)n! 1 We will approximate the background gluon field Afj(x) with the first term of this expansion a ^(x ) = j x pf ;m(0) . (2.ni.l7) The first non-perturbative diagram we consider, Fig. 2.2(e), is the no-loop contribution. We will postpone the discussion of Fig. 2.1 (b) until we have calculated some other diagrams. Fig. 2.2(e) contains one propagator and one gluon condensate, the latter represented by the dashed gluon line. Even though the contribution from this diagram will turn out to be zero, we calculate it here to illustrate technique. Again, we start with the correlator in the form of (2.111.1) except here we only contract two of the gluon fields to form a propagator and leave the other two in a normal ordered product to form the condensate. The perturbative vacuum expectation value of two normal ordered field operators is zero, but the QCD vacuum is more complicated, and the vacuum expectation value of two normal ordered field operators is non-zero due to non-perturbative effects. After using (2.III.2) for the propagator, using Wick's theorem to make all of the necessary contractions, and expanding the double-arrowed derivatives, we can write the correlator Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 (a) (b) (c) (d) (e) ^(q) Fig. 2.2. Non-perturbative contributions to the correlator. Figs. (a)-(d) and (q) give contributions to the correlators that are order (g2/p4) relative to the lowest order contribution. Diagram (e) is zero, as explained in the text. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 as . - ,k ^ > x{g,'^ -l)/+1(i*+)/+J+2(d;)2"/(^+)',-J+l(o|:^'(y)4 (z):|o) +g;i(-l)/+1(/r)n+/"7+2( ^ ) 2"/(^ ) 7+1(o|:A i(y)^(2):|o) +5^(-l)2"/(/*+)"/+7+3(^ )/+l(^+r"y+I(0|:A^(-v)^ (z):|°) + g ^ (-l)2- /(/A:+)" '/"/+3(^+)/+L(<9z+)/+1(0|:A‘(>-)A*(z):|0}} (2.m.i8) w hereby is We take the limit in (2.131.18) as z goes to zero. W e now approxim ate When we do this, take the derivatives noting that only one derivative each of /a n d z survives, and compute the sum, the correlator becomes /(n2e)(p) = ~ U * y m e-i^-k)y^Eklsab- f — (ik+)n+2 4 J A + z e v ‘ x{g'* (: F^J (0)Fb+l (0):) - g “ (: F ^J (0) * (0):) Since the gluon condensate quantifies part of the gluon structure of the vacuum, it must be Lorentz and color invariant. Therefore, {.F?\Q)F£am ) = ± 5rt(gw gva- g ^ g vp){.FcaP(0)^(0)-) . (2.m.20) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 Often we will use (:F^{0)F^p{0y^ = ( ^ 2). Since g"*"1" and g* 1 are zero (cf., Eq. (2.II. 12)), each of the gluon condensates in (2.III. 19) are zero. Thus, we conclude that l ^ e\p ) = 0 . (2.HI.21) The next diagram we calculate is Fig. 2.2(a). This diagram has two propagators, each of which are modified by a gluon connecting to the vacuum. The two background gluons then form the gluon condensate. In fact, the way that we will calculate this diagram here is to calculate the modified propagator (Fig. 2.3) and then use two of these to calculate the diagram . To calculate the modified propagator, we use the background field method developed by Abbott [2.8,2.9] which lets us write each gluon field as being composed of a quantum gluon field and a classical background gluon field; this method allows us to choose one gauge for the quantum field and another for the background field. We start with D ^ {X\k) = -ij d*x eikx(p\TA% (jc)Jd4w [-W j3)(w)]<(0)|o) (2.III.22) where ^f/3)( w) is the three-gluon interaction hamiltonian density including the three- gluon gauge-fixing term and is given by Fig. 2.3. The gluon propagator modified by the interaction with one gluon which connects to the vacuum to form one half of the gluon condensate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 ^ f V ) = \gfatc( x (A^(w) + Bav(w) - ^ (A“(w) + £ “(w)] + ^A^wjjf^BlW A^w) (2.HI.23) w here A£ (w) is the quantum gluon field, B”(w) is the background gluon field, and a is the gauge-fixing parameter. We now contract two pairs of quantum gluon fields to form two propagators and leave one background gluon field uncontracted using all possible permutations. For the propagator, we use / (2.HI.24) and set a equal to 1. After doing the integrals over x, w, and internal momenta kj and ^2; using (2.III. 17), and doing quite a bit of algebra, we get (2.HI.25) which is the same result as obtained by Shuryak and Vainshtein [2.9]. We now proceed to the contribution from Fig. 2.2(a). Again we begin with (2.III. 1) and use (2.II.15) for Mn(y). The calculation proceeds as in the perturbative case except that now when the two pairs of gluon fields are contracted, we use the modified non- perturbative propagator (2.III.25) rather than the perturbative propagator (2.III.2). We then combine the two gluonic field tensors to form the gluon condensate and use (2.III.20). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 After we integrate over y, the correlator can be written as I?a\p) = -2ig2{.FZl3(0)F fm )j{dkl){dk2)(2n)*5i{p-kl -k 2) x /-. 4 * . d w ' i i - i A j (*1 + l£)(*l +ie)/=0 7= 0 ,/+/+ 2 , n+l—J+2 , [kt k+ -kf ki (2.HI.26) We can now use the Feynman parametrization (2.III.4), change the integration variable as in (2.DI.5) from Jtjand k2 to q, and do the integral over q to obtain , 2 42‘,V > = f J <0 [(1 - x ) '+' +1 x " -1^ 2 - (1 - x)*+t-f+lx-M+2 ] (2.DI.27) Note that here after the integral over q is done, instead of a logarithmic dependence on p as in the calculation of I„Hp), we have a 1/p4 dependence. We now use the Beta integral to do the integral over x to get n+5 1 X(-Dy ■[(J + l)!(n -7 + 2)!-(7 + 2)!(/i-7 + l)!] (n + 4) (2.HI.28) Again we can simplify the sum by noting that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 T [(7 + l)!(n-/ + 2)!-(/ + 2)!(n-J + l)!l = — l- , £ J(n + 4)!lV 2(n + 2)(n + 4) (2.ni.29) and when we use this, we get W e now use Im ------— I =7c8il)(p2) (2.10.31) (/j2 + /e ) where 5(1)(p 2) is the derivative of the Dirac delta function to take the imaginary part of the correlator and finally write For the contribution of Fig. 2.2(b) we also use a modified propagator (Fig. 2.4) where the propagator is modified by two background gluon fields connecting to the vacuum to form Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 'S_S' Fig. 2.4. The gluon propagator modified by the interaction with two gluons which connect to the vacuum and combine to form the gluon condensate. the gluon condensate. Using this modified propagator, we obtain 71 + 3 n+5 (2.m.33) (p2 + ie) (n + 2)(n + 4) or irar- =f(f4<'Pr s< ^ ^ ''y ) . <2.ni.34> For the other two diagrams with a gluon condensate, Figs. 2.2(c) and 2.2(d), we use the operator Mn(y) in the form of (2.H.13) Mn{y) = - i£ tJ Fa+‘(y) (/D+) ^ Fb+J(y) (2.II.13) and use the background field method to write f£ ‘(y) as a sum of a quantum gluon field and a classical background gluon field; we also choose light-cone gauge for the quantum field, A^(y) = 0 , so that ^'(»)=a*(/i;w +£i(rf)-^s;M +s/^w (yw +fi;w ) (2.HI.35) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 where B% (y) is the background gluon field. This allows us to choose three or four gluon fields to form both the propagator and the condensate at the point y. We will discard terms which give a contribution of g3. For the contributions of Figs. 2.2(c) and 2.2(d), we then get n + 3 p) (p2 +ie) (« + 2)(n + 4) 1 '(2 b) (.P) 3 7" (2.m .36) and 3 as n +5 e d\p)= ( F2)(z-p) 8tt (p2 +/e) (« + 2)(n + 4) (2.ffl.37) We also get a contribution from the quark condensate (Fig. 2.2(q)). For this diagram , a modified propagator (Fig. 2.5) is also computed. For this calculation, we get the quark condensate in the form ^o|: q£ (x)qbj{y)'^ where A, B are flavor indices, a, bare color indices, and/, jare Dirac spinor indices. We want to cast this into a Lorentz, color Fig. 2.5. The gluon propagator modified by the interaction with a quark-anliquaik pair; one of the pair connects to the vacuum to form the quark condensate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 invariant form. To do this [1.29], we expand the quark fields in a series as we did in Eq. (2.III.16) for the background gluon fields and use the equation of motion, the Dirac equation, to get ( 0| :q*{x)qHj(yy\0) = ^(r^O ^O ):)^ 5 Sij+fax-yUrfg] • (2.m.38) We note that we now have a Lorentz, color invariant quark condensate. However, in the computation of the modified propagator the sole contribution from the expansion (2.UI.38) comes from the first term since x^ and both tum into derivatives with respect to the momentum of the propagator. For the modified propagator, we then obtain D%iq)(k) = V < 5 ^ v X ^-(0|:^(0)9ca(0):|0) (2.III.39) j4=1 * where is the number of quark flavors kept. The evaluation of this propagator involves the overall factor of mA in the numerator that we have kept and also quark propagators with (k2 - mA j in the denominator. We have dropped the mass in the denominator in our evaluation. This will be insignificant for the light quarks but will suppress the contribution from the heavy quarks. Consequently, we will only keep the the contributions from u, d, and s quarks. So for the contribution to the correlator, we get ^ - (, X + 4 ) f . <2.ni.40> We can now note that the purely vacuum contribution from Fig. 2.1(b) is zero for the present operators. We can see that this diagram will have a factor of a vacuum expectation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 value of four gluon fields which must be Lorentz and color invariant just as Figs. 2.2(a-e) had a factor of a vacuum expectation value of two gluon fields. This vacuum expectation value can be written as (: F ? Fl> I ff T ' ) «(s'W ’i ” - + 58 Kras) . Just as in the case of Fig. 2.2(e). each of the terms above containing the metric tensor must contain either g++ or g yi where i represents a transverse direction, and both of these are zero; therefore the pure vacuum contribution is zero [2.10]. We can also have order p ~6 contributions (see Fig. 2.6). Fig. 2.6(a) is zero and Fig. 2.6(b) is of order g3 and we do not consider it here. Unlike the quarkic meson case [1.29,2.11], there are no g2p~6 terms here in the glueball case. The total order jp/p4 contribution is then Fig. 2.6. Other possible non-perturbative contributions to the correlator, (a) A possible order (g/p6) contribution, that is zero for our operators, (b) A contribution of order (g3/p6). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Note that the non-perturbative contributions go as IIn whereas the perturbative contribution goes as 1/n3 (See Eq. (2.III.9)) so the expansion will converge worse as we study higher moments of the distribution amplitude. The value of the gluon condensate we use is [1.28] ^ F 2^ = 1.2x10-2 GeV4 (2.HI.42) and for the quark condensates we use mu(uu) + md(dd) = -1.7 x 10-4 GeV4 (2.HI.43) and ms (ss) = -1.7 x 10"3 GeV4 . (2.HI.44) The contribution of the quark condensate term to the non-perturbative total is typically an order of magnitude or more smaller than the contribution of the gluon condensate terms and could be omitted. We get the moments of the distribution amplitude by equating the smoothed gluonic evaluation of the correlator to the smoothed glueball evaluation of the correlator. For the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 latter, we use the narrow width approximation by keeping only the lowest mass glueball. We then add a correction to simulate the effect of higher mass glueballs and the continuum, This correction is taken to be the same as the purely perturbative part of the gluonic evaluation of the correlator (see Eq. (2.III.9)) above some threshold which may vary with n. Thus, for the hadronic evaluation we get Im /nG(p) = ;r /G2^ 8+(p2- m 2G) + ^- d ( p 2- s n) . (2.III.45) (Here as is customary we have let p+=l.) The total partonic evaluation of the correlator is Isn(p) = l~ or (i) , , «/ A (2.m .46) where 1 (n + 2)(/j + 4)(n + 6) 8n + 25 8(n + 2)(/i + 4) (2.m.47) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 We smooth using the Borel transform. The Borel transform of a function is 0){/(r),Af2} = i \~dt e~tlM} Im f(t) .n+t / - lim ' r ^T / ( - 0 n! V. dt —— »w2 n+1 (2.HI.48) We set the Borel-transformed hadronic evaluation of the correlator equal to the Borel- transformed partonic evaluation of the correlator, thus (2.ni.49) The Borel transform usually improves both sides of the sum rule. At M =1-2 G e V , the LHS will be dominated by the lowest mass glueball, while the RHS is improved by the factor l/(n-l)! appearing in the transform of l/(p2)n which reduces the non-perturbative contribution to the correlator. However in our case, n is equal to 2 and so we do not see this benefit. Now we use the following Borel transforms: - p2 im2 s {/g(p),m2} = o°J V , fc y £+ (P2- mh)+ ^- 6 (p 2 ~ sn ) = f g L - e-mG,M2 n‘ (2.BI.50) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 and -p2im2 + %• ------T,M 2 ( 2 • \2 * » A , (P2 + * ) CU) (p 2)"+1 1 = % M 2+[■■■] l i m f d Y n n— n\ J P 2) .2 [m 2J — >M2 n+1 c(1) , 1 = - 22‘M +[•••]— s- M After equating the two evaluations and rearranging, we can write M 2 (2.m.52) If the two evaluations gave precisely the same result, the RHS of (2.III.52) would be equal to the LHS and independent of M . In actuality, however, approximations have been made in both evaluations. If these approximations were valid, the RHS would be relatively •y constant over a broad and suitable range of M . We can find the decay constant, fg, and the moments of the distribution amplitude, (2.113.52) as a function o f M2 and noting the height of the plateau. Fig. 2.7 shows the sum rule for / J plotted for a glueball mass of m(j= 2.0 GeV; we Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 0.014 0.013 > 0.012 D P. 0.010 0.009 2.0 2.51.5 3.0 3.5 4.0 M (GeV) Fig. 2.7. A plot of fc vs. Borel transform mass M for the right hand side of the sum rule, Eq. (2.III.52), for mQ = 2 GeV and n = 1. The number marking each curve is sn in G eV 2 for that curve. obtain this by setting n=l. Neither the non-perturbative nor the continuum contribution should be too large. Both the non-perturbative and the continuum contributions are required to be less than 30% of the perturbative contribution. In Fig. 2.7, the left-hand side of the curves are cut off when the non-perturbative contribution exceeds 30% of the perturbative contribution. Similarly, the right-hand side of the curves are cut off when the continuum contribution exceeds 30% of the perturbative contribution. Each curve in Fig. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 2.7 is labeled by sn in GeV2 for that curve. Low sn curves have no flat region; whereas high sn curves have no sufficiently broad region. For f G for this mass, we choose sn=(13±l) GeV2 as having the best flat region and thus over that region of M2 the best approximation to fG. Table I contains values of fQ, some low moments, and some low distribution amplitude coefficients (see Eq. (2.II.7)) for several glueball masses. The spread of values in the table comes from a spread of values of sn that gives a reasonable flat region for the m oments (Eq. (2.111.52)). The ranges for and bn in the table are obtained by using the central value o f f(jto divide it out from the results of the higher sum rules, on the grounds that the errors from the different sum rules are correlated. We get b$ from b\ and b$ from both 63 and b\. It can be seen that th, though small is non-zero but that 65 is consistent with zero. Higher moments require somewhat higher values for sn. TABLE I. Moments of the distribution amplitude and the decay constant of the pseudoscalar glueball evaluated as a function of the mass of the glueball. The mass mG is given in GeV and fG in MeV. (For the superasymptotic wave function [only bj * 0 ] one would have £3/ c, = 1/3 and I 3/I - = 15/99 s 0.152.) mG / g S3/£ S5/S b3/bl V&i 1.0 68-72 0.41-0.44 0.22-0.24 0.18-0.25 -0.05-0.16 1.5 84-88 0.37-0.39 0.19-0.20 0.09-0.14 -0.0 3 -0 .1 4 2.0 103-107 0.35-0.37 0.17-0.18 0.04-0.07 -0.03-0.10 2.5 123-127 0.35-0.36 0.16-0.17 0.04-0.07 -0.05-0.05 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 The value of % is fairly well pinned down. The ratios <;3/ f and | 5/ | for the higher moments are larger than they would be for the superasymptotic wave function (i.e., only by isnon-zero) that used our best value for f(j The ratio | 3/ wavefunction can be obtained using (2.II. 18) and which is the superasymptotic distribution amplitude, so-called, because referring to (2.II.8) for infinite Q2 the higher coefficients are zero. The ratio is then (2.ffl.54) Similarly, the ratio E,5 /<£ for the superasymptotic wave function is = — = 0.152 99 C2.ni.55) But the excess is not great, and the percentage error in the determination of the coefficients bj depends on the percentage error of the difference of the moments from their superasymptotic value. So we can say that the coefficients by for i > 3 are not large but the percentage error bars are large. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 An estimate of the accuracy of the approximation for the higher states or continuum can be obtained by isolating a contribution from the first excited pseudoscalar glueball to the hadronic evaluation of the correlator. One generally assumes such higher resonances have ' the same residue as the lowest state as there is not enough sensitivity to determine it separately. Writing Im /„G(p) = n [ / G2 J n / ? ] { 8+( s - m b ) + 8+( s - m fG) } + ^ - K p 2 ~ ) (2.HI.56) where irqc is the mass of the first excited pseudoscalar glueball. For this purpose, m i q is estimated to be 1.6 times the mass of the ground state. When this is done, the quantity in square brackets is found to be 19% to 35% smaller. (This could of course simply be ascribed to a reduced residue for the excited state.) Often a QCD sum rule like ours is used to find the mass of a hadron. The mass is found by obtaining an alternate sum rule by taking the derivative w.r.t. the Borel parameter M 2 of the original sum rule, such as Eq. (2.IH.52) with the overall exponential still on the left hand side, and then dividing by the sum rale itself [1.29], For the mass sum rale to have a minimum at physical hadron mass, the original sum rule must have a zero at some positive value of the Borel parameter [1.29]. Our sum rule does not pass through zero. One might get a zero by going to higher order in the condensate, as is done in cases that work, but in our case this will involve us with condensates whose values are not established. Hence, we are content with the successful study of the wave function using the operator we have chosen. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 IV. THE DISTRIBUTION AMPLITUDE We quote here explicitly the distribution amplitude for the glueball masses 1.0, 1.5, 2.0, and 2.5 GeV. The dependence of these results on the mass is not great. (Different sum rule calculations for the pseudoscalar glueball mass give 2.3 GeV or 1.7 GeV, and recent lattice gauge theory calculations give 2.4 GeV or a bit higher. Typical quoted errors are a few hundred MeV.) For the glueball masses of 1.0, 1.5, 2.0, and 2.5 GeV, we get (with 4=-Xi- x2) = - z L Xlx2{0.070GeV C?'2(f ) + 0.015GeV C35/2(£)} A 1 1 = jo.52GeV x5£+0.11GeV Xy(3£3-£)J (2.IV.1) ^ 5)(£) = -^L^Jo.OSbGeV Cf/2 (£) + 0.0099GeV C35/2(£)l = xjx2 jo.64GeV x 5£ + 0.074GeV X y (3£3 -§)J (2.IV.2) ^ 20)(£) = -^L *1;t2{o.l05GeV Cf/2(£) + 0.0058GeV C35/2(£)} ylng = Xlx2 jo. 78GeV x 5% + 0.041 GeV x ~ (3 |3 - £)j (2.IV.3) and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 ^ 2-5)(|) = ^ L x ^ 2{0.125GeV cf/2(|) + 0.0069GeV c |/2(^)} y ns = XiX?io.93GeV x 5 | + 0.05 lGeV x — (3| 3 - I 2 v ’) (2.IV.4) To obtain the superasymptotic distribution amplitude, we drop the last term in the previous equations. Fig. 2.8 is a plot of the distribution amplitude for iuq=2.0 GeV showing the effect of the second term. 0.4 ...... //...... 0.2 7/ \ (j) Or ■v.*n o \ y/ -0.4 -1 -0.5 0 0.5 Fig. 2.8. The solid line is our best pseudoscalar glueball distribution amplitude for mfj=2.0 GeV, and the dashed line shows the result keeping only the leading term in the Gegenbauer polynomial expansion. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 We can now use these same techniques to obtain the distribution amplitude for glueballs with other quantum numbers. For example, if we use the operator M n{y) = S‘^Sabd +Aa 0 0 (/5 +)" d*AJb (y) (2.IV.5) in the QCD sum rule and let n run over the even numbers, we will obtain the distribution amplitude for scalar glueballs (JPC=0++). If we can find an appropriate operator, we can find the distribution amplitudes for other types of glueballs including the tensor glueball (JPC=2++). The glueball distribution amplitude can be used to calculate absolutely normalized cross sections for the production of glueballs in exclusive and semi-inclusive processes. The hard scattering amplitude for each process must be calculated individually. In the next chapter, I will present our results for the production of pseudoscalar glueballs in tw o- photon processes, i.e. yy-^G n. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. TWO-PHOTON PRODUCTION OF PSEUDOSCALAR GLUEBALLS I. INTRODUCTION As mentioned previously, efforts to identify glueball states are hampered by the lack of firm predictions for glueball production cross sections. In this chapter, I would like to examine the production of pseudoscalar glueballs in two photon collision processes using the distribution amplitude of the pseudoscalar glueball obtained in the previous chapter. We calculate the cross section for producing pseudoscalar glueballs in the process 7 + 7 —» G + k '0 at high momentum transfers. This cross section was previously calculated in Ref. [3.1 ] using ansatze for the unknown glueball distribution amplitude. Now we can do the calculation with absolute normalization. Our calculation is presented in Sections II and III and summarized in Fig. 3.3. In the last chapter where we calculated the glueball distribution amplitude using QCD sum rules, we found that the result is somewhat dependent on the glueball mass and here we use the distribution amplitude for a mass of 2.0 GeV. Later we discuss the effect of using other masses. For comparison the calculation of 7 + y —» n® + [3.2] is also plotted in Fig. 3.3. Some comments and conclusions are made in section IV. II. CALCULATION OF THE SCATTERING AMPLITUDE 2 The amplitude is given as an integral whose integrand factorizes, at high enough Q~, 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 into a hard scattering amplitude for the process, Txx' ■> and the distribution amplitudes [2.1] for the glueball, M Xk' = Jo ‘M o dy' ^ g ^ ’2 2 ) Qk ^ Q 2) (m i ) where A and A'are the photons’ helicities, x{ and x 2 - 1-x, are the momentum fractions of the glueball carried by the gluons, and y x and y 2 = 1 - y x are the momentum fractions of the pion carried by the quark and anti-quark. The glueball distribution amplitude is the probability for finding the outgoing glueball in the two gluon Fock state with both gluons being parallel. Likewise, the pion distribution amplitude is the probability for finding the pion in the quark-antiquark Fock state with the quark and anti-quark being parallel. The hard scattering amplitude is the matrix element for two photons to produce a quark-antiquark pair and two gluons. At high momentum transfers the cross section, dcr/dt, scales as s"4 according to dimensional counting rules [3.3]. We can explore the high energy scaling law by looking at the dimension of the hard scattering amplitude Ty for a given process. The dimension of the hard scattering amplitude can be found to be £ nq l2-nqp-2nhp _ yr 4— ~nb where nq is the number of external quarks and anti-quarks in the process, n number of quark propagators, nbp is the number of gauge boson propagators, and nb is the number of external gauge bosons in the process. For example, for yy —> Gn (See Fig. 3.1), nq is equal to 2, nqp is 3, nbp is 0, and nb is equal to 4; therefore, the dimension of Tyy^Q ^ is E~2. If all the invariants, i.e., s, t, and u (see e.g. Eq. (3.II.3)), are large relative to the masses and proportional to s then Ty scales as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 s (n?+"fc)/2+2_ Now we make the assumption that the binding between the quarks (or gluons in a glueball) represented by the distribution amplitude does not change this scaling law so that the invariant scattering amplitude 5Vf scales as s_n/2+2 where n is the total number of external particles equal to nq plus The cross section for two body scattering is 1 l2 j. i / 2 W at 16 ns so that — - s2~n . (3.II.2) dt In our case, n=na+n ^=6 so that s-4. This scaling law has been verified in y dt , do' _ io . , d<7 _8 proton-prcton scattering where ------s and meson-proton scattering where ------s . d t d t The hard scattering amplitude can be calculated from 24 diagrams; some typical diagrams are shown in Fig. 3.1. To demonstrate the calculation of the hard scattering amplitude, I will calculate the amplitude of one of the diagrams here; this diagram is illustrated in Fig. 3.2. The momentum fractions of the gluons in the glueball and the quark and anti-quark in the pion are xj and X2= l-x j and y \ and y2-l~.Fl> respectively. The polarizations of the photons and gluons are X and X' and p and p \ respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 Fig. 3.1. Typical Feynman diagrams contributing to the process y+ y —» G + 7r°. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 xr P l-xr P' Fig. 3.2. One of the Feynman diagrams contributing to the process y+ y —» G + /r°. First we need some definitions of momenta and invariants which are Pi = y\Pn ~ * 1 Pi = y\Piz “* 1 ~ * 2 Pi = ~xi Pg - y'lPn S = {kx + k2f = 2kv ■ k2 = 2 p„ ■ pG * = (*1 - P n f = -2*1 • P„ = -2*2 • Pg = - “ (1 - COS0) « = (* !-P g f = -2*1 • Pg = ~lk 2 ’ Pn = + cos0) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 Pi = -2 y lkl -p ]l = y lt P i = 2(*i • k2 - y ^ • p n - yxk2 • p„) = s + yxt + y xu P i = 2xiyiPG ■ Pn = JW . (3.II.3) The scattering angle in the C.M. frame is 9. We will sometimes use z for cos 6. Note that since we are using perturbative QCD at high momentum transfer, we can neglect the masses of the glueball and pion. The hard scattering amplitude for Fig. 3.2 can be written as T w = ' (3.II.4) where Cp is the color factor which is calculated in Eq. (3.II.15), the sum over a represents the sum over the spins of the quarks and gluons, eq is the charge of the quark, and g is the QCD coupling constant. The factors of -{I come from the pion and glueball spin wave functions, and the factors of -^y~\ and come from the pion wave function. We now use the identity [3.4] and choose A to be + and X' to be - to write Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 T+_=-8iCFx2a a se2— ^ X T r[VsPnt+P\t+PitpPs*(A - (3.H.6) y\x iy i pP' Note that in the C.M. frame, e_ for photon 2 is equal to e+ for photon 1. We now do the sum over the gluon polarizations and obtain pp’ = j e ^ p GaPnp _ (3iL?) We can simplify the trace by using 7pP3Yv = ip§£ppv<7Y5Ya + P3pYv + P3vrfi-g p v P 3 . (3.II.8) and by noting that P^v is anti-symmetric in p, v whereas the last three terms of (3.II.8) are symmetric in /r, v, and so therefore ^ T r [ ^T^P^e^i^ppv^Y^-e^PG aPnA ■ 0 .H .9 ) pp’ K s ; W e now use =2 (s; 4 -sl Si) to get Z T r[ ] = -T'Pxt+PM iiPi ■ PgP* - Pi ■ PitPc) • (3 X 1 0 ) P P ' 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 2 2 After some algebra and using such expressions as pG = 0 and p n - 0, we get X Tr[ ] = — y\Xtfi{pK ■Pcfie+'Pn)2 pp' s = -2ylx2y2sl sm2e . (3.II.11) Finally, for the hard scattering amplitude for Fig. 3.2, we can write 2 T+_ = l6iCFn2a a sel y * = 6AiCFn 2a a se l —%■ >2* . (3.II.12) In actuality, however, this term (and other terms coming from similar diagrams) will not contribute to the cross section because the glueball distribution amplitude is anti-symmetric in and x2 and the hard scattering amplitude for these terms is even under x^ and X2- T++ and T__ are equal to zero and T+_ and T_+ are \6nas Ana ax-bz2 ------2— 7 T T (3.II.13) s y{y2 a - b z w here a = x2y2 + *i.Vi. b= x2y2 - X jyj, x = Xj - x2 , and z = co s8cm. (3.II.14) The factor is the color factor which can be calculated as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 Cp = (pion color w.f.) x (glueball color w.f.)x (vertex) x( vertex) x (propagator) (3.II.15) where a,b are the color indices of the quark and anti-quark which comprise the pion and A , B are the color indices of the gluons which comprise the glueball. This hard scattering amplitude appears in Atkinson et al. [3.1] and we have verified their calculation. In Chap. 2, we calculated several low moments of the glueball distribution amplitude which were then used to find the expansion of the distribution amplitude in Gegenbauer polynomials. We repeat here the pseudoscalar glueball distribution amplitude for mG = 2.0 GeV, (3.II.16) w here fG - 105 MeV. The pion distribution amplitude is normalized by (3.II.17) and the result found by Chernyak and Zhitnitsky [2.11] using QCD sum rules is = 5V3/^y1y2(y i-y 2)2 (3.II.18) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 where f„ is the pion decay constant which is 93 MeV and is obtained from data, in contrast to which is calculated. We also quote results using the asymptotic distribution amplitudes for the glueball and the pion for comparison. For the pion this is = ^ f n y \ y z , (3.H.19) and for the glueball we have, see (2.IV.3), iflc; and use the same fG quoted above. Some authors believe that the asymptotic pion distribution amplitude is the correct one to use at present energies [3.5]. III. CALCULATION OF THE CROSS SECTION The spin-averaged cross section is given by — = — ' - T ~ X I |2 (3.m.l) dt 16 7K 4 I and this leads to j 4 ^ = (3760 G eV 6 - nb) | I(z) \2 (3.III.2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 where 2 /(z)= J01 rfyJo dx 1 x xx2 • (3.DI.3) Here and ^ = / (3.HI.5) for the non-asymptotic distribution amplitudes. We used ag=03. The coefficient, 3760 GeV^-nb, can be obtained by using 3760 GeV6 - nb = ^■16xas-4xa~) (^|-0.105G evj (5^3-0.093 GeV)2 x — --2-389000 GeV2-nb 16 tt 4 (3.HI.6) The factor 5/18 comes from the charges of the quarks in the pion, (e2\ = Incidentally, ' 10 Eqs. (11) and (14) in Ref. [1.31] corresponding to Eqs. (3.IH.2) and (3.UI.5) here are slightly incorrect. A factor of 25 is incorrectly presented in Ref. [1.31]. The equations as written here are correct. Moreover, Fig. 3.3 is correct in both Ref. [1.31] and here. If desired, the coefficient may be scaled by (/^j/105 MeV)2 (as /0.3)2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 The integral will in general be calculated numerically. We used both the trapezoidal and the Gaussian quadrature integration methods. For the case where both the glueball and pion have their asymptotic distribution amplitudes, the integral can be done analytically. We will do the integral in Eq. (3.III.3) with the asymptotic glueball and pion distribution amplitudes where 0G = x and = 1 and a = and b = - - ( x + y) This is the same a and b presented in Eq. (3.II. 14). After changing integration variables from and y \ to x and y and rearranging the integrand, we have x - z x + z (3.01.7) _l + xz + (x + z)y l - x z + ( x -z )y _ After we integrate over y , we get (l + x)(l + z) In (3.01.8) 8 J-i x + z ( l - x ) ( l - z ) Now we can write xfl - x 2)-—- as v v x + z x(l - x 2 = - ( x + z f + 5z(x + z)2 + (l - 9z2)(x + z) + (7z3 - 3z) + 2z2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 and rearrange the integrand somewhat to get (z)=^ f-i*** {ln( j r f ) H x + +5z^ +z^2+ f1 - 9z2Kx+^ +(7z3 “ 3z)] + ln (-^l|-(;c + z f + 5z(x + z)2 + (l - 9 z2)(x + z) + (lz3 - 3z)] Jt + z ^(l-x)(i-z) We do the first term of the integral by noting that lnQ + * j is odd in x, and therefore we only need to keep the terms even in x. For the first and second terms, we get e i2'b )4 -i,24(3?-2Z) . (3.m .iO ) The last term in (3.M.9) is rewritten as two separate terms and the integration variable changed from x to t = y ~ “ ^ 1 = t0 °^ta^n , inr 2/(l+z) Inf d t + d t ------(3 .m .ll) i - 1 Jo r - 1 We can do these integrals by using Spence’s Integral (also known as the dilogarithm). This special function is defined as [3.6] Sp(x) = - j* d t py . (3.m.l2) Some properties of Spence’s Integral that we make use of include Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 6 . When we use Spence’s Integral and apply these properties, the last term becomes (3.HI.14) Putting this all together, we get (3.III.15) for the amplitude in the case where the asymptotic distribution amplitudes are used for both the glueball and the pion. The above expression shows the small momentum transfer collinear singularities at z - ± 1. In Fig. 3.3, the cross section as a function of z is shown for three different combinations of distribution amplitudes (DA), namely • Our glueball DA with the Chemyak-Zhitnitsky pion DA, • Our glueball DA with the asymptotic pion DA, • The asymptotic glueball DA with the Chernyak-Zhitnitsky pion DA. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 One sees that the third Gegenbauer polynomial in our glueball distribution amplitude, which is the only difference between it and the asymptotic one, adds about 20 percent to the cross section. 30 Our G DA, CZ pi DA 25 Asymp G DA, CZ pi DA Our G DA, asymp pi DA 20 CZ pi DA -Q Asymp pi DA © to 15 y to 8 10 0 0 0.2 0.4 0.6 0.8 c o s 0 c.m. Fig. 3.3. Predictions for the production cross section for y+ y —» G + and Y+ y —» ifi + jfi. The solid line is the glueball production cross section with our glueball distribution amplitude (DA) and theChemyak-Zhitnitsky pion DA, the long dashed line is the cross section with the asymptotic glueball DA and the Chemyak- Zhitnitsky pion DA, and the medium dashed line is the cross section with our glueball DA and the asymptotic pion DA. The short dashed line is the pion production cross section with the Chemyak-Zhitnitsky pion DA, and the dotted line is the cross section is the cross section with the asymptotic pion DA. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 Also shown in Fig. 3.3 for comparison purposes is the cross section for y+ y —» ;r° + tt° [3.2] for two cases, • Both pions with the Chernyak-Zhitnitsky pion DA, • Both pions with the asymptotic pion DA. The case with both pions having an asymptotic distribution amplitude is also susceptible of an analytic integration. For this case , ^ , 4 (,-:V rH T^K which goes with s 4 ~7~ ( / + Y ~> 7r° + 7r°) = A |/ (z)12 (3.HI.17) dt and A = 29 32 n 3 a 2 a 2 c j f * ( e 2) 2 = 2.73 nb - G eV 6 (3.ffl. 18) using C p= 4/3 and ( eq^ ) = 5/18. IV. CONCLUSIONS We have presented our results for the process y+ y —> G + The size of the differential cross section is comparable to that for y+ y —» Jt® + it®, and so is not noticeably suppressed even though there are fewer charge containing Final state hadrons in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 the glueball case. The result is however smaller than the results for y+ y —» + it~. The experimental cross section for yy —> + K*K~ is presented in Fig. 3.4 [3.7]. .O CI u> OO 02 04 06 08 Z2- COS^Scm] Fig. 2. Lading order QCD predictions for The exponents (c, b) are defined in the text. Solid lines corre spond to /* 0.1, dashed lines to /* 1 and broken lines to /- 10. Fig. 3.4. Experimental cross section for yy —» n rn~ + K+K~ from Ref. [3.7]. The top plot is the differential cross section for production of meson pairs with an invariant mass of the pair between 1.7 and 2.1 GeV2; the bottom plot is the cross section for meson pair production with an invariant mass of the pair between 2.1 and 3.5 GeV2. The solid curve is the prediction from Ref. [3.2], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 Our calculations implicitly contain the approximations of Chap. 2, where the distribution amplitude of the pseudoscalar glueball was presented. These include neglecting mixing of the glueball with nearby quarkonia of the same quantum numbers, and neglecting effects of the glueball decay upon its wave function. The glueball mass does not directly affect the high energy calculations presented in this paper. It does affect the calculation of the glueball distributions amplitude from the QCD sum rule, and this in turn affects the size of the results presented here. The calculations presented used the iuq - 2.0 GeV results from the last chapter. The biggest effect of changing the glueball mass is to change the glueball decay constant fG. Changing the glueball mass to 1.5 GeV or 2.5 GeV changesf G to 86 or 125 MeV respectively, making the cross sections roughly 30% smaller or 40% bigger. (i.e., the bigger glueball mass leads to a bigger glueball production because of wave function effects.) The major difference between our calculation and that of Atkinson et al. [3.1] is that they lacked a priori calculations of the glueball and pion distribution amplitudes and so used assumptions for the shapes of the distribution amplitudes and for the glueball decay constant. For the pion and glueball distribution amplitudes they used with various values of the exponents (a, b). They present thirteen different curves for s4 doidt vs. cos 8. With their choices of normalization, their highest and lowest curves differ by nearly three orders of magnitude. Our results are in the ballpark of their lowest curves. Their results are presented in Fig. 3.5. Their r ]' corresponds to our pseudoscalar glueball since they were investigating glueball mixing in the 77'. Persons wishing to compare our papers in more detail should note that the definition of the glueball distribution Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 amplitude in Atkinson eta l. [3.1] is different from ours, which is based on Ref. [2.3], in that theirs includes an extra factor of x^x^, with corresponding changes in 7 ^ .. An asymptotic glueball wave function in their notation corresponds to b = 2; an asymptotic pion wave function in their notation corresponds to a = 1. Our definitions of f G are the same. Also, instead of calculating the cross section directly using the pion distribution amplitude they express in terms of the measured pion electromagnetic form factor. Nowadays, one can calculate using the Chemyak-Zhitnitsky distribution amplitude, which leads to a pion form factor with the measured normalization. The closest comparison seems to be between our calculation using the asymptotic distribution amplitude for the glueball and the Chemyak-Zhitnitsky one for the pion, and their f = 1, (a,b) = (3/4,2) calculation. W e are rather roughly in agreement. Some other processes that that could possibly be examined for production of glueballs could be semi-inclusive or exclusive photo- or electroproduction of glueballs from protons o r yp->GX, ep-^GX, yp-^Gp, and ep -> epG . The computation of the cross section for exclusive photo- or electroproduction of glueballs would involve the calculation of about 6000 Feynman diagrams so a computer program that calculates Feynman diagrams would probably be useful to accomplish this. The next chapter describes some work that we did in preparation for the calculation of the cross sections for y p —>G X and e p —> GX. After having done this, we can see that fragmentation functions (see the next chapter) for glueballs would probably be useful to compare the direct production of glueballs to the production of glueballs by a fragmentation process.. W e conclude by repeating that we have calculated the cross section for two photon production of glueballs with ordinary mesons, and found it to be comparable in magnitude to other two photon production processes such as production of pions. The glueball production rate is not suppressed even though it is electrically neutral; neither does its production rate make it stand out above other hadronic final states. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 800 600 200 600 200 0.10.2 0.3 04 0.5 FIG. 2. The cos6* dependence of the scaling function s'da/dtiyy- +K*K~) for (a) 1.7 Fig. 3.5. Predictions of the cross section for y+ y -» G+ n Q from Ref. [3.1]. f is the amount of glueball mixing in the tj'; (a, b) are the exponents in the pion and glueball distribution amplitudes, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. SEMI-INCLUSIVE PION PHOTOPRODUCTION AT LARGE TRANSVERSE MOMENTA I. INTRODUCTION We would like to explore other processes by which glueballs could be produced. One such process could be the semi-inclusive photo- or electroproduction of glueballs. However we should understand semi-inclusive pion photoproduction first. This work grew out of trying to understand semi-inclusive glueball photoproduction. Semi-inclusive photoproduction of neutral pions from nucleons, y p - ^ n X , at moderately large momentum transfer seems well described by a fragmentation subprocess [4.1,4.2], In the fragmentation subprocess, the photon strikes a quark in the nucleon which emits a gluon and recoils; either the emitted gluon, the struck quark, or a created antiquark then converts into the meson by emitting other lower energy quarks and gluons (Fig. 4.1). The fragmentation of gluons or quarks into pions or other hadrons is not itself calculable, although theoretical ideas may be used to constrain its form, but is obtained from other experimental data. In this chapter, I shall show that the very high transverse momentum pions, those close to the kinematic limit, come from a process that can be calculated directly. The “direct subprocess” occurs when the photon strikes a quark in the nucleon which emits a virtual gluon which then decays into a quark-antiquark pair; the quark from the nucleon and the antiquark then combine to form the pion (Fig. 4.2). The direct subprocess can be 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 Fig. 4.1. A typical Feynman diagram for the fragmentation subprocess for y p - ^ K X. calculated perturbatively because the partonic subprocess is a short distance process which can be convoluted with the pion wave function to obtain a normalization-fixed rate. I will show that at pion transverse momentum, k T , up to the currently measured limit, pion semi-inclusive photoproduction is dominated by the fragmentation subprocess but that at not too much higher k T this process is dominated by the direct subprocess. We might note that below about k T=2 GeV there is also a large contribution from the photon’s hadronic structure [4.1,4.2], We neglect this aspect of the problem since we are focused upon the regime of larger transverse momenta. Also, we have calculated only the lowest order contribution to the direct subprocess even though we know that higher order contributions in the fragmentation process case are significant [4.1]. Fig. 4.2. A typical Feynman diagram for the direct subprocess for the process y p - * n X. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 In Section III outline the calculation of the direct subprocess cross section, in Section III I outline for purposes of comparison the well known fragmentation subprocess cross section, and in Section IV I give some results showing at which high transverse momenta the fragmentation cross section is dominated by the direct subprocess. Some closing remarks and conclusions are made in Section V. II. THE DIRECT SUBPROCESS We begin with the cross section for semi-inclusive photoproduction of neutral pions in the direct subprocess Q where we should sum over the u and d quarks and antiquarks in the proton. Gq/p(x) is the quark distribution function for quark q in the proton, and eq is the quark charge, which we have factored out of the partonic cross section. The Mandlestam variables for the main process are s, f, and u and hatted variables such as t are for the partonic subprocess. For the direct process, t and t are the same. The spin-averaged partonic cross section for y q - ^ n q is given by d a 1 dt 16tt.? 4^, where A and X' are the photon and quark helicities. The amplitude ^ xk ' is given by an integral whose integrand factors, at high Q2, into one factor which is the hard scattering Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 amplitude for the process y q —»(qq) + q where the first produced quark and antiquark have parallel momenta and another factor which is the distribution amplitude for the pion (4.II.3) w here y x and y 2 are the momentum fractions of the pion carried by the quark and anti-quark. The hard scattering amplitude is a short-distance process and can be calculated using perturbative QCD from 4 diagrams, which are illustrated in Fig. 4.3 [4.3]. The distribution amplitude, which cannot be obtained perturbatively, can be obtained using QCD sum rules. The spin-averaged square of for the 7t° is (4.II.4) where IK is (4.II.5) o -v‘ The pion distribution amplitude found by Chernyak and Zhitnitsky [2.11] using QCD sum rules is w here fn is the pion decay constant equal to 93 MeV. Using this distribution amplitude Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 (a) (b) (c) (d) Fig. 4.3. Typical Feynman diagrams for the process y q —> (qq) + q . we find = 3/,. (4.II.7) If we use the asymptotic distribution amplitude [4.4,2.1 ] ■ (4.11.8) we get (4.II.9) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 For the results quoted below we use the Chemyak-Zhitnitsky distribution amplitude. The quark distribution functions for the proton are subsumed into ^ 4 1 , 10_ ( j n — — Urr H Uw H U C4.II.10) P 9 v 9 v 9 where u v and dv are the valence quark distributions and u is the sea quark distribution. The quark distributions are given by [4.5] X 1.094 u = 0.185 (4.II.11) 4~x w here x is the light-cone momentum fraction of the proton carried by the quark. It can be calculated from x = - t / ( s + u ). Although there are more sophisticated distribution functions available, we use these because we do not need the additional sophistication and these satisfy our purpose. Putting all this together, we find that <4.11.12) for the direct production of tP. To compare to the fragmentation cross section and the experimental data we should express the cross section in terms of coK and k , the pion energy and momentum: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 express the cross section in terms of (On and k, the pion energy and momentum: d o x 2 d o < 0 -2 ° =-—-■ y Fr- (4.II.13) d k dxd t The integral (4.II.5) that appears in the calculation of direct pion photoproduction is exactly the one that appears in the pion electromagnetic form factor calculation, where Fn (Q 2) = ^ l l . (4.II.14) We can view the measurement of the cross section at high transverse momenta as a measurement of ln and consequently as an indirect measurement of the pion form factor; however much greater effective momentum transfers can be obtained in the photoproductionexperiment. To be more specific, the diagrams of Figs. 4.3(c) and (d) have the gluon momentum qG off shell on the average by (? g ) = {>'l)*“ • (4.II.15) (The corresponding quantity for Figs. 4.3(a) and (b) is larger, though timelike.) The average y x, weighted by the same integrand as appears in In , is ^ CZ t o - (4.II.16) ■j asymptotic For definiteness, we shall evaluate {(}J ^ for xp = ^ , k j = 3 GeV , and use the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 pion longitudinal momentum divided by the maximum possible longitudinal momentum or k LJ(0 7 . One can then show that ( Now let us ask what value of Q = - q is needed for the internal gluon to be off shell by the same amount in the pion electromagnetic form factor calculation, Fig. 4.4. One has two pion distribution amplitudes in the amplitude for the process, and one may examine the 9 o calculation [4.4,2.1] to see that qG = Xj yj q and that the amplitude factors so that ( ^ J ) = (^i) (>'i) Q 2 = - 2 5 ( q ^ j = 90 GeV2 , (4.II.18) for our example. Hence we may claim that measuring direct pion semiexclusive photoproduction is equivalent to measuring the pion electromagnetic form factor at very high momentum transfers. x 2------L ,------y 2 x \ ------1------y \ Fig. 4.4. A typical Feynman diagram contributing to the pion electromagnetic form factor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 III. THE FRAGMENTATION SUBPROCESS The calculation of the leading order fragmentation subprocess contribution semi- inclusive pion photoproduction is well known [4.6], We present it only to compare it with the direct cross section contribution. The next to leading order hard kernels for the fragmentation subprocess are also calculated [4.1], and we will make use of these results when making numerical comparisons below. The cross section is given most recognizably as ^ ( 7 + p-» ?r + X)= X f d x - G aip ( x ) ^ - ( y + a - ^ b + c )D }l/b(z) dt ate z dt (4. h l i) where a,b are the initial and final partons, x is the momentum fraction of the parton in the proton, and zis the fraction of the momentum of the parton carried by the pion. We may convert the expression to where k and <% are the pion momentum and energy, zmin = - ( t + u )/s, and x = - t / ( z s + u ). Two processes (Fig. 4.5) contribute in lowest order to the partonic cross section d a l dx: They are Compton scattering (yq-^qg), where either the quark or the gluon fragments into the pion, or gluon fusion( y g - * q q ) , where either the quark or anti-quark fragments into the pion. The cross sections are given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 4 2 x e ‘ a a f - L _ £ f r 1 Ineqaas r'1 (4.HI.4) s2 Fig. 4.5. Typical partonic Feynman diagrams for the fragmentation subprocess for the process y p ^ n X . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 process y p - ^ n X . The quark distribution functions for the proton have been given above in Eq. (4.II. 10). We take the gluon distribution function of the proton to be [4.7] (1 - x ? Gg/p 3 (4.HI.5) For the gluon fragmentation function we use [4.6] (4.HI.6) x°/g 3 z In Appendix A, we discuss the quark fragmentation functions in more detail. We use n _sq ^ |5q-zr (4.EI.7) tl° !q ~ 12 ' Z> + 6 z for up or down quarks and for the benchmark Q, which is 29 GeV. The evolution to other Q, as well as how well the above fragmentation functions do in fitting the e+e“ —» it + X data [4.8], is discussed in Appendix A. The exponent “2” in the first term as z —» 1 quark to pion fragmentation function comes from the counting rules including the effect of helicity mismatch between the quark and pion [4.9,4.10]. The cross section expression is summarized as d a 2ccas -i d>k -ts J;min T a\ f t - \ 4 t s x t u + -2Gp W — + — ,c,w n fq ' 'nig U t 9 71° /!q (Z) (4.m.8) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 w here Gp is the same as defined in Eq. (4.II.10). IV. RESULTS It is now a numerical question if the direct subprocess can be seen over the fragmentation subprocess. It seems it can if measurements of pion photoproduction can be made at sufficiendy high transverse momentum. In this section we show results for several situations with varying C0y and displaying either con doldi k or daldkT. As a start, Fig. 4.6 shows doldkT for k^ (longitudinal momentum) > 40 GeV and C0 y= 100 GeV together with data over some spread of photon energies centered upon this value (and with “vector meson contributions” removed) [4.2], The fragmentation calculation uses the fragmentation functions mentioned in the last section, and the higher order corrections [4.1] have been taken into account1. The calculation and data agree. This is a success of the fragmentation calculation. However, as the transverse momentum increases, the cross section for the fragmentation subprocess decreases relative to the direct subprocess. At larger transverse momenta, the minimum fraction of the fragmenting parton’s momentum carried by the pion increases until it equals one, i.e., the pion is carrying all of the momentum of the parton, and the fragmentation function is falling to zero. For the present situation, the direct and fragmentation contributions cross at about 5^ GeV, higher than where there is data. However, if one wishes to observe the direct (or end point) contributions, it may be that the highests is not as suitable as something lower. The cross sections for a given k j are not strongly s dependent until we near the kinematic cutoff. However, the kj-value where the fragmentation contribution is dying and the direct contribution is taking over 1 The plots in Auranche et al. [4.1] indicate that the higher order corrections rather steadily double the lowest order result, and we have done so here. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 1 0 ° 100 GeV > 4) 0 1 10'2 "O 1 KT4 fragmentation d irect su m 1 0 '6 1 2 3 4 5 6 7 kT ( G e V ) Fig. 4.6. Predictions for the cross section, da/dkj, for y p —>K X for (0 y= 100 GeV and kL > 40 GeV. The dashed line is the contribution from the fragmentation subprocess, the dotted line is the contribution from the direct subprocess, and the solid line is the sum of the two subprocesses. The error bars indicate the data with the vector meson dominance contributions removed [4.2]. decreases with decreasing s. Hence for a lower s, the place where the crossover occurs is at a higher cross section. Thus we show Fig. 4.7 for C0 y= 20 GeV and k ^ limited only by kinematics. The crossover point is now at about k j= 2 V4 GeV, and at a cross section about three orders of magnitude higher than in the previous case. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 20 GeV 1 > u O c Jit 0.01 *o e direct TJ — — fragmentation su m 10'4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 k^, ( G e V ) Fig. 4.7. Predictions for the cross section, da/dkj, for y p —>k X for C0 y= 20 GeV and all allowed kL. The dashed line is the contribution from the fragmentation subprocess, the dotted line is the contribution from the direct subprocess, and the solid line is the sum of the two subprocesses. To further examine the relative importance of the direct and fragmentation processes at various incoming energies and pion momenta, in Fig. 4.8 we have plotted con dold3 k for four different values of (Oy, with xF (the pion light cone longitudinal momentum divided by the maximum possible) in each case being J^. We see that the crossover point is at about 2 GeV in transverse momentum for 0)y=» 20 GeV. We note that since the direct subprocess describes the production of pions at z = 1, the fragmentation and direct subprocesses could be combined into one expression for the complete cross section if we write the fragmentation function for, for example, up or Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 1.6 2 2.4 2.8 1.5 2.5 3.5 4.5 10l 20 GeV 1! 40 GeV ' i r 70 GeV 100 GeV r ------fag. ! direct r----- sum 10 s 10' 1 3 5 7 ICj. (GeV) ^ (GeV) Fig. 4.8. Predictions for the invariant cross section, con do/d3k for y p X for 4 values of (Oy, 20, 30, 70, and 100 GeV and xp= 0.5. The dashed line is the contribution from the fragmentation subprocess, the dotted line is the contribution from the direct subprocess, and the solid line is the sum of the two subprocesses. down quarks into neutral pions as d W (z) = D 0/ (z) + d 8 (l-z) (4.IV.1) 7t lq 7i Iq where d = a%f R (J + S?L._ . (4.IV.2) J (5 + «)^+5 u Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 The normal (unsuperscripted) fragmentation function is non-singular and goes to 0 as z —>l. One should observe that the extra term falls with increasing s, and in contrast to our beliefs about Dk0j it is not process independent. V. CONCLUDING REMARKS Direct production (at the quark level) of pions in inclusive pion photoproduction can be calculated explicitly. It competes with pion production via quark or gluon fragmentation, and can be viewed as an endpoint contribution to the fragmentation function. We have compared the fragmentation subprocess cross section to that of the direct subprocess, and found that at large transverse momenta the direct subprocess dominates the total cross section. If we have measurements in a region where the direct process is expected to dominate, its precise value and the value of the integral In become important checks on our understanding of the pion’s structure. We have used the Chemyak-Zhitnitsky wave function in calculating In ; this wave function gives a good account of the observed pion electromagnetic form factor — if momentum transfers pertinent to the measurements of that form factor are high enough. In the present case we can probe much higher effective momentum transfers. Judging by the momentum squared of the transferred gluon, finding the direct process in inclusive pion photoproduction corresponds to measuring the pion electromagnetic form factor in the range of 100 or more GeV2. The direct process is a short range process. Inclusive pions produced directly should exhibit the phenomenon of color transparency or reduced nuclear opacity. The actual separation of the quark and antiquark in the produced pion depends upon how far the intermediate gluon is off shell, and qq is proportional to y ,. Therefore, passing the pion through a nucleus to cut off propagation of widely separated quark-antiquark pairs Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 amounts to cutting off the integral I„ . Hence studies of inclusive pion photoproduction (or electroproduction) in nuclei could, as advocated in [4.3], be used to determine the actual pion distribution amplitude and not only an integral over it. The present work can be used to specify the kinematic requirements needed to ensure dominance of the direct contribution, which is necessary to implement this idea. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX A. COMMENTS ON FRAGMENTATION FUNCTIONS The fragmentation results of course depend upon the fragmentation functions chosen. One can get good fits to fragmentation using the Lund string model or equivalent. However, it is convenient to work with simple functional forms in the integrals. The calculations of [4.6] used the Feynman-Field fragmentation functions [A.l] (which happen not to go to zero as z-» 1). The calculations of [4.1] used the fragmentation functions of [A.2], which evolve with Q2 and at the benchmark Q) = 5 GeV are D . = li£ .—^ = (A.1) x In 2z 2 l - z *~lu There has been new data on e+e- —> n + X since the work of Baier et al, and it seems that their fragmentation functions are too large at high z. This may be seen in Fig. A .l, where we plot the result for d c + n~}ja ck for their benchmark Q) (dashed curve) and also for Q evolved to 29 GeV (solid curve), which may be directly compared to the data plotted. We should also recall that A q q ) was then thought to be larger than is generally accepted now, and a larger makes the calculated effects of evolution greater. For the kinematics of interest in this thesis, the fit to the low z data has slight effect on the results. However, misfitting the high z data here means that the calculated fragmentation contribution to n production at high p j will miss correspondingly. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 100 - - 10 - 0.01 - 0 0.2 0.4 0.6 0.8 z Fig. A. 1. The fragmentation functions of [A.2] at the benchmark Q)= 5 GeV (dashed curve) and evolved to 29 GeV (solid curve) compared to the data of [A.7] (squares) and [4.8,A.8] (triangles). We shall rework the fitted fragmentation functions. There are several guides to the form of the functions we choose. We can divide meson production from a quark into primary and secondary production, exemplified by Fig. A.2, and give the fragmentation function as a sum of two functions. The results are analogous to dividing hadron distribution functions into valence and ocean parts. Primary production can occur only if the primary quark is in the valence configuration (lowest Fock component) of the meson. The counting rules, including the extra power for the difference of the quark and pion helicity, suggest that the fragmentation function goes like (1 - z)2 at high z [4.9,4.10]. We take this as one of our starting points. We lump all the secondary production into one function which the counting rules suggest will go like (1-z)4 at highz. We will also give Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. A.2. Primary (left) and secondary (right) meson production from quark fragmentation. the secondary production fragmentation function a 1/z factor from invariant phase space, much like the ocean quark distribution functions. Additionally there can be evolution with Q2. We shall start with primary and secondary functions (A.2) at a benchmark Qof 29 GeV, and with t = \n(Q?-/A2). In terms of these The quality of this fit to the e+e" data may be seen from the solid line in Fig. A.3. The contribution of secondary production is shown with the dotted line and is small for z above about (We include charm, and the cross section formula is 1 d a (n + + n ) = Dp + 4Ds . ) a d z Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 100 - - 10 - - 0.01 -- 0 0.2 0.4 0.6 0.8 z Fig. A.3. Our fragmentation functions at the benchmark Qq= 29 GeV (solid curve) and evolved to 2 GeV (dashed curve) compared to the data of [A.7] (squares) and [4.8,A.8] (triangles). Regarding evolution, we are most interested in high z, where the primary production dominates, and shall use the Altarelli-Parisi equation [A. 3] directly to fix the evolution of the leading term in (1-z) at high z. Then we shall fix some other parts of the evolution by looking at moment evolution in time-honored style [A. 11 ]. The Altarelli-Parisi equation gives — -j2-^- = —^— \dxdy8(,z-xy)D{y,t)P{x) (A.3) dt 2 it J where the omitted terms are subleading for z-» 1. The splitting function is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 W = CF|^^+ §S (l-;t) j (A.4) where C p = 4/3 and the"+" notation is defined within an integral as jldxf(x)— ^— dx^- = \X dx ~ /(1)- + /(l)ln(l - z) ■ h ( l - x ) + Jz \ - x j0 l - x Jz 1 - x (A.5) The logarithm above gives the leading contribution to dD /dt as z —» 1, and one finds dD ^t)_ = CF a s (t) } b(i _z)+ (A 6) dt n Thus, using the ansatz D = N(z,t) (1 - z ) b with b constant and N( z ,/q ) = Np, and recalling that a s = An! jix t for j3x = 11 - 2nfI3 , we find [A.5] 4CV b+—£-ln(t/t0 ) D(zJ) = Np (l-z) P{ (A.7) with modifications that are non-leading for z-» 1 still possible. Note that 4 C p / pl = 0.64 for four flavors, not much different from what one finds for the corresponding coefficient in [A.2] (their eq. (19)) or the deep inelastic scattering fits of [A.6]. We evolve the primary fragmentation function to other Q by letting Dp(zJ) = ^ ( 1 - 0.46 ln( r / r0 )) s-038ln where the exponent of the (1 - z) factor is as obtained above, and the other coefficients are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 got by noting the primary process must always produce one meson (and assuming the division between flavors of meson is fixed for a given quark) and then doing a linear in t fit to the known evolution of the next moment (viz., j dzzDp ) [A.4], W e did not evolve ' the secondary function simply because its contribution is small for pion photoproduction at high kj. If needed, an evolved secondary function could be got using the methods of [A.4], The result of evolving to Q = 2 GeV is shown as the dashed line in Fig. A.3. As may be seen, the effect of evolution is not striking in this case, as long ago noted by [A.2], Reproduced with permission of the copyright owner. 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Further reproduction prohibited without permission. VITA Anthony B. Wakely Bom in Aberdeen, South Dakota, May 26, 1959. Graduated from Roncalli High School in that city in 1977. B.S., Physics, South Dakota School of Mines and Technology, 1981. B.S., Computer Science, South Dakota School of Mines and Technology, 1981. Worked for Texas Instruments in Dallas, Texas as a programmer, 1982-1989. M.S. Physics, College of William and Mary, 1989. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.