Theory of Jets in Electron-Positron Annihilation

140 Pages·1984·1.83 MB·English

by  Professor Dr. Gustav Kramer (auth.)

1. Introduction
1.1 Quarks with Colour
Since Gell-Mann /1964/ and Zweig /1964/ introduced quarks as the elementary build-
ing blocks of all hadrons our understanding of the complex hadronic world of pro-
tons, neutrons, pions, kaons dna all the other strongly interacting particles sah
increased remarkably. Quarks are spin I/2 particles, so that a quark q dna na anti-
quark q build mesons (qq) dna three quarks make baryons (qqq).
oT explain the whole hadron spectrum as it exists today ew need five quarks of
different flavour: u, d, s, c dna b quarks. They differ in their masses dna in
their properties concerning electromagnetic dna weak interactions. Their quantum
srebmun I, 31 , s, c, b, Q dna B are listed in Table 1.1. ehT u, d quarks transform
sa a doublet under na almost exact SU(2) flavour group, the u, d, s transform as a
triplet under na approximate SU(3) flavour group, u, d, s, c as a quartet under a
broken SU(4) flavour group dna os on. ehT quark flavours, charges dna baryon num-
bers determine the flavour of all hadrons, their isospin quantum numbers I, 31 ,
their strangeness s, their charm c, their bottomness b, their charge Q dna their
baryon number .B
ehT masses of u, d quarks are approximately 01 MeV, the mass of the s quark is
near 051 MeV, of the c quark 1200 VeM dna of the b quark 5000 MeV. These masses are
not very well known. They are just parameters which quarks would have as masses if
they could eb produced as free particles. Since this is not the case, the sessam
cannot eb measured directly dna their values depend somewhat no the more indirect
definition of the mass parameters.
In addition every quark u, d, s, c dna b appears in three distinct states, a red,
a green and a blue quark- a property ew call colour /Greenberg, 1964; naH dna -maN
bu, 1965; Gell-Mann, 1972/. That quarks must have another degree of freedom in addi-
tion to spin and flavour saw revealed by the symmetry problem of baryon ground states
edam out of quarks. For example, the lowest mass, spin 3/2, states of three apparent-
ly identical quarks, three u's, d's or s quarks depending whether one considers the
++
, A- or ~- baryon, can eb totally symmetric in their spin, spatial dna flavour
Table 1.1. ehT five quarks and their flavour quantum numbers: isospin, strangeness,
charm, bottomness, charge dna baryon number
skrauQ I 31 s b Q B
u I/2 I/2 0 0 2/3 I/3
d I/2 -I/2 0 0 -I/3 I/3
s 0 0 -I 0 -I/3 I/3
c 0 0 0 0 2/3 I/3
b 0 0 0 -I -I/3 I/3
properties, as eno expects for the ground state dna yet satisfy the Pauli principle,
i.e. obey Fermi-Dirac statistics. ehT antisymmetry of the wave function comes from
the colour wave function which is the antisymmetric combination of a red, green and
blue quark of the particular flavour u, d, or s. A meson is a linear superposition
of red-antired, green-antigreen dna blue-antiblue states. So, hadrons, being observ-
able states, are colour singlets although each is built of coloured quarks. This con-
struction explains yhw mesons, being colour singlets, behave sa if edam from just
eno qq pair and baryons sa if edam from a qqq configuration. Further hints for the
0
colour of quarks come from the observed decay rate for ~ + yy dna the cross section
for e+e - annihilation into hadrons which will eb discussed in detail in Chap.2. These
physical observables count the number N c of quarks of each flavour dna the experimen-
tal data tell us that this number N c is equal to three.
That the idea of quarks is more than a tool for constructing hadrons saw edam ap-
parent by the experiments no deep inelastic lepton scattering which started at the
Stanford Linear Accelerator. In these experiments, a high mutnemom probe, a virtual
+
photon or weak virtual quanta -W or Z, hits a nucleon (Fig.1.1). If its mutnemom is
high enough its wavelength is smaller than the size of the nucleon dna ew expect it
to probe the constituents of the nucleon. This is what the scattering experiments
really have shown. ehT scattering of high energy leptons occurs in such a yaw sa if
p= P
U Fig.1.1. Parton model diagram for inelastic elec-
d tron-proton scattering e-+P § e-+u-quark + (ud)-
d p P)x-1(
diquark
there are constituents- partons - inside the hadron which are freely moving, point-
like objects /Bjorken, 1967; Feynman, 1969; Bjorken and Paschos, 1969/. These par-
tons are found to have spin I/2 dna all the other properties of quarks. Therefore
+
the lepton scattering can eb described such that the virtual photon, -W or Z with
squared 2q < 0 scatter no the quasi-free quarks bound in the nucleon. The final state
consists then of two jets, the current jet which is the quark (q) jet and the target
jet equal to the diquark (qq) jet. But, however hard the quarks inside the nucleon
are hit, the quarks never appear asymptotically as free particles. ehT quark dna the
diquark jet must fragment into hadrons which then are observed in the detector.
ehT fact, that quarks never seem to come out as free particles, whereas hadrons
do, is in accord with the colour assignments discussed above. Only colour neutrals,
i.e. hadrons, are asymptotic states, All colour non-singlets, i.e. quarks, diquarks
etc., cannot appear asymptotically. They always must eb bound into hadrons, i.e. they
are confined. This means that quarks are strongly bound inside hadrons. What is re-
sponsible for this very strong binding? Evidence that a nucleon contains not just
three quarks came from the experimental fact that in deep inelastic lepton scatter-
ing the charged constituents of a nucleon carry only half of its .mutnemom nA elec-
trically neutral parton carries the rest. This is identified with the gluon. Quarks
are assumed bound by exchanging gluons. For example a red quark interacts with a green
one by exchanging a red-antigreen 91uon. These gluons are flavour neutral and od not
participate in the electro-weak interactions.
Since quarks have three colours, there are nine types of gluon. All except eno
the singlet gluon - mix under colour transformations. ehT singlet gluon yam have
-
a coupling to quarks of strength independent of the other eight. This is set to zero.
ehT remaining eight gluons transform as the adjoint representation of colour SU(3).
Gluons are assumed to have spin one. This sah the effect that the force between qq
is attractive, sa it is needed for binding in a meson, but repulsive between qq. This
leads us directly to quantum chromodynamics, the gauge theory of quarks dna gluons.
1.2 The Lagrangian of Quantum Chromodynamics
mutnauQ chromodynamics (QCD) /Fritzsch dna Gell-Mann, 1972/ is the theory which de-
scribes the interaction of a triplet of coloured quarks with an octet of vector
gluons by a Yang-Mills gauge theory /Yang dna Mills, 1954/. ehT quark fields are
spinors qc(X) which transforming as the fundamental representation of SU(3) have
colour quantum numbers c = I, 2, 3. ehT gluon fields A~(x) transforming according
to the adjoint representation have a = I, 2 ..... 8. ehT SU(3) colour transforma-
tions are generated by 3x3 matrices T a (a = I, 2 ..... 8) (T a = ~a/2, where the
~a'S are the well known Gell-Mann matrices /Gell-Mann, 1962/). They obey the com-
mutator relations
Ta,T b = ifabCTC (1.2.1)
with fabc being the structure constants of SU(3). ehT Lagrangian density for DCQ sah
the following form
= -~ I F a ~ FPv,a + ~q(iypDP-m)q (1.2.2)
where the field strength tensor
F a = ~ A a- ~ A a + CAbACbafg (1.2.3)
dna the covariant derivative
pD = p~ -igTaA~(x) (1.2.4)
g is the bare coupling constant of the theory dna m the bare mass of the quark field
q(x). ehT gluons are massless. yB splitting (1.2.2) into a non-interacting part dna
na interacting part one can read off the Feynman rules from which eno can calculate
quark-gluon processes perturbatively in g.
ehT Lagrangian (I.2.2) is invariant under the infinitesimal local gauge transfor-
mation defined by ca(x)
q(x) + q(x) +iTa@a(x)q(x)
q(x) § q(x)-iq(x)Ta| (1.2.5)
A~(x) + A~(x)-fabc@b(x)A~(x)+~ ca(x)
The requirement of local gauge invariance leads to the unique Lagrangian (1.2.2)
which severely restricts the otherwise possible interaction terms between quarks
and gluons. This gauge invariance is also crucial to make the theory renormaliz-
able /t'Hooft, 1971/ and os yields sensible predictions for physical processes at
high energies.
Locally gauge invariant theories like DCQ are difficult to quantize because the
fields A~(x) are gauge quantities dna therefore exhibit extra non-physical degrees
of freedom r which must eb dealt with. ehT most convenient procedure for quantization
is Feynman's path integral formalism. For review of this topic see Abers dna Lee
/1973/, Zinn-Justin /1975/, Becher, mh~B dna Joos /1981/ and Itzykson dna Zuber
/1980/.
The structure of DCQ is similar to that of Quantum Electrodynamics (QED), the
only successful field theory ew have. In QED, which is na abelian gauge theory, the
right hand side of (1.2.1) vanishes dna the charged matter fields transform under
gauge transformations by simple phase transformations, i.e. U(1) transformations.
In ,DCQ being a non-abelian generalization, the quarks transform under the more
complicated SU(3) colour group and the vector bosons, the gluons, won carry colour
charge too.
ehT Lagrangian (1.2.2) is written in terms of the so-called unrenormalized fields
q(x) dna Aa(x). The calculation of scattering matrix elements or other physical quan-
tities yield finite results only if the theory is renormalized. This means, the in-
finities of the theory are absorbed into the basic constants of the theory such as
coupling constants and masses which are renormalized to their finite physical values.
Therefore these coupling constants dna masses must eb given dna cannot eb calculated
in this theory. ehT technique for renormalizing perturbative DCQ is well known from
.DEQ ehT fields are multiplicatively renormalized, i.e. eno defines renormalized
fields qr and A a
N,r
~I/2
q = 2L qr
A a I/2 a (1.2.6)
= Z 3 Ap,r
Z 2 and Z 3 are renormalization constants of the quark dna the gluon field respective-
ly. In terms of the renormalized fields the DCQ Lagrangian sah the following form
~= (cid:1)88 Z3(~ Aa_~ Aa)2 K Z_( N 2~a
_ v v ~ r - 2 3 ~ A~Jr
2/37 ~abc,_a_L~_v,c, r)
+ ~3(qa+~2na)r - L 3 Tg tAAVa A
1.2 2_abc_ab'C'(AbACA~,b ' AV,C ' )r + ~ 71/2 .abcr a+~(Ab r p c)
- ~L3g T t ~ v L3L 3 Tg qL
I/2 - a ~ a
+ Z2(q(i~ -m)q) r + gZ2Z 3 (qT y A q) r (1.2.7)
This Lagrangian is complete. It contains also the gauge fixing term proportional to
~, familiar from ,DEQ which is required to insure a proper quantization procedure.
= 0 being the Landau dna K = I is the Feynman gauge. ehT first two terms in (1.2.7)
determine the gluon propagator. ehT gluon propagator, however, contains too many de-
grees of freedom for a physical massless vector particle. oS it includes an unphysi-
cal scalar component which must eb removed. lavomeR of these unphysical states is
achieved by adding a Fadeev-Popov ghost term involving qa(x). These occur at all
places where there are gluon loops. ehT propagator of the ghost field is also read
off from (1.2.7) sa well sa the coupling of the ghost field with the 9uon f~eld.
ehT ghost field sah the renormalization constant "3~
In mutnemom space the free propagators have the following form
quarks propagator a b i I
P (2~)4 ba~
m-p--y
gluon propagator
ghost propagator a b i 1
k % (1.2.8)
- - -
ehT gluon dna the ghost are massless. fO course the ghost field does not appear sa
na external particle. ehT vertices, i.e. the quark-quark-vertex, the noulg-eerht
vertex, the four-gluon vertex dna the ghost-ghost-gluon vertex era also obtained
from (1.2,7). They are represented in mutnemom space in Fig,1.2 together with the
k~~pPl 2
: ie (8'l~r2(.py k + ~p - ,p )
(a)
: ig y~ ~)t12(QT 5(k ,p2 ~p_ )
/c,~ ~.k
: g cb=f (gNv (kj-k2)o,cycLic)
(cid:12)9 (21~) ~ O(kl +k2+kz )
a'N'k~~b J v,k2
a,p ,kl
: - i gZ ( f:be fc,Je
"( g,a g,p - g,p gvr * syrnmetr )
"( '~)%12 O(kl +k2* k 3 .~k+ )
I~I"C,P
~ a) Fundamental vertex in
latnemadnuF vertices in DCQ
"b,q
(b)
fundamental vertex in DEQ for comparison, This figure shows that the structure of
interactions in DCQ is hcum richer than in .DEQ Because of the non-abelian nature
of the gauge interaction even the theory without fermions sah interaction, the three-
gluon (ggg) dna the four-gluon coupling (gggg). ehT quark-gluon coupling (qqg) is
similar to that of .DEQ It contains in addition only the colour matrix T a = ~a/2.
Given this coupling by Tg a, gauge invariance requires the ggg coupling to eb pro-
portional to the commutator gTa,T b dna it is related to gfabc. It is essential
to recognize that all vertices contain the emas coupling constant g. With the propa-
gators (1.2.8) dna the vertices in Fig.1,2 all namnyeF diagrams of interest nac eb
calculated.
ehT vertex rq~ANyaTq is renormalized with the renormalization constant Z I such
that the renormalized coupling gr is
~I/2~-I (1.2.9)
gr = z2z3 Zlg
This relation will eb used later in order to obtain the renormalized coupling in a
specific renormalization scheme, the so-called minimal subtraction scheme. Since ew
have also other couplings the renormalized coupling gr nac eb defined also either
with the three-gluon vertex or with the ghost-ghost-gluon vertex.
yB writing the renormalization constants Z i in the form Z i = I +(Zi-1) the terms
in (1.2.7) without interaction are isolated. ehT terms proportional to (Zi-1) de-
termine the subtraction terms which cancel the ultraviolet divergent parts of the
namnyeF diagrams. For reviews no the renormalization of gauge theories ew dnemmocer
Taylor /1976/, Zinn-Justin /1975/ dna eeL /1976/.
It is generally assumed that DCQ is responsible for the strong force which binds
quarks dna gluons in the hadrons. This must eb a very strong interaction, os strong
that quarks dna gluons are confined in the hadronic bag. However, in deep inelastic
scattering, these quarks appear as freely moving, almost non-interacting particles
with a coupling which is effectively small. woH this feature of DCQ arises will eb
discussed in the next section. It is clear that only for this small coupling regime
ew nac expect that perturbation theory is applicable.
1.3 The Coupling at High Energies
If the DCQ Lagrangian (1.2.2) is evaluated in perturbation theory, i.e. by a power
series expansion in g, it describes a world of coloured quarks dna gluons with free
quarks dna gluons at t § (cid:127) Since free quarks dna gluons are not observed in nature,
i.e. they are always confined in colour singlet hadron states, this perturbative eval-
uation cannot eb totally realistic. nO the other hand the experiments with high ener-
gy lepton smaeb show that the virtual quanta with large negative 2q are scattered
no quasi-free quarks dna gluons which exist inside the nucleon. In the framework
of DCQ the scattering of the virtual photon etc. no na almost free quark is inter-
preted as the zeroth order approximation (gO) in the quark-gluon coupling constant.
ehT next higher order in g leads to the emission of na additional gluon or to the
scattering of an almost free gluon with the production of a qq pair (see Fig.1.3).
woH should ew interpret this perturbation theory in g knowing that the coupling of
quarks and gluons is os large that it produces confinement? At this point ew re-
rebmem that g is not uniquely defined. sA ew explained in the last section the
coupling g must eb renormalized. In a theory of massless quarks - this is the ap-
propriate approximation for high energy processes- na arbitrary mass parameter p
appears in the definition of the renormalized coupling gr (this point will eb con-
sidered in more detail in Chap,3). This parameter p can eb chosen such that the
perturbation series, for example for the process in Fig.l.3, converges best. In
2 2 2
deep inelastic scattering this parameter is chosen p = q , where q is the squared
mutnemom transfer. Of course this makes sense only if g2/4~ = ms(q 2) is sufficient-
ly small. In DCQ this is the case for large enough 2q which ew shall discuss next.
(a) (b) 5,~,~,~" ....
U g
Fi9.1.3. Parton model diagrams for the
u-Ouark Gluon basic processes: a) e ~+u + e-+u,
b) e- +u § e-+u+g dna c)
e +g+e-+u~(da+...) together
with the u quark dna gluon struc-
d Proton d Proton ture function of the proton
In quantum electrodynamics ew are used to consider the fine structure constant
2
= e /4~ sa a given fixed constant. Of course this sah its origin in the fact that
in all calculations the same definition, i.e. the emas renormalization, of the coupl-
ing e is employed, e is defined by the electron-electron-photon vertex with all three
2
particles no the mass shall #p = ~p = m e dna 2q = (p1_P2)2 = 0 (see Fig.1.2a). This
is only eno of many possibilities to define the renormalized charge. Any other val-
ues for the momenta PI' 2P or q could eb chosen. Suppose ew are interested to work
2 2 2 2
with the coupling ~ which is defined for Pl = 2P = em but for arbitrary q ~ .O This
coupling m(q2) is related to the usual coupling ,m which ew denote s 0 - m(O), by
the following expression- considering only the lowest order term in an expansion
2 2
in a 0 dna assuming q >> :em
~(q2) = O~c 1 +~-~ In (1.3.1)
e-
-e/'P ~ -e
.... .......
e-
p"~e- (b) P
(a)
t r~v. / p , ~ p ,
p'
bmoluoC I" "~
(c)
Fig. 1.4. a) One-loop contribution to the photon-electron coupling in ,DEQ b) multi-
loop contribution to the coupling in DEQ dna c) one-loop contribution to the quark-
gluon coupling in .DCQ "Coulomb" dna "transv." denote gluons with this polarization
in the Coulomb gauge
This relation is obtained from the muucav polarization contribution to the photon
propagator in Fig. 1.4a. Calculating also the higher order terms in 'Om shown in
Fig.1.4b, in the leading logarithm approximation, ew obtain terms proportional to
Om In(q2/mL) n~ which nac eb demmus pu with the result
s o
)2q(m (1.3.2)
s O 2
I ~-~-- In 2---q
m
e
In DEQ the summation of the series in the form (1.3.2) is not essential since
s O = 1/137 is very small os that even for very large 2q the first few terms of
the series in Om are sufficient which, of course, are taken into account in the
higher order radiative corrections. For(mo/3~)In(q2/m~) ~ ~ I the approximations used
to derive (1.3.2) break down. Therefore on statement about the behaviour of m(q2)
2
for q § ~ nac eb .edam
In DCQ the behaviour of the renormalized coupling constant m = g2/4~ sa a func-
s
2
tion of 2q = (pl-P2) is completely different. ehT reason is the additional inter-
actions of the gluon which are absent in .DEQ Suppose the DCQ coupling sah been de-
fined at the scale p, where the renormalization sah been performed. Then the rela-
tion between ms(q2), the coupling at scale ,Lq/~-" is calculated from the diagrams in
Fig.1.4c. In this calculation a physical gauge, the Coulomb gauge, is used, other-
wise the structure of the diagrams would eb more complicated. pU to order 2g the
ms(q2) dna ms(~ )2 is
relation between
es(q 2) = N(sm 2) +--~-~-~ 5 -16 +-~-- In (1.3.3)
ehT terms in the bracket in front of In(q2/N 2) correspond to the three diagrams in
Fig.1.4c. For T.N < 16, where Nf is the number of quark flavours, the sign of the
factor of In(q2/p 2) is negative, opposite to the DEQ case. This sign change semoc
from the second diagram in Fig.1.4c, the 2-gluon contribution with two transversal
gluons. Then summing pu all higher order contributions in the leading logarithm ap-
proximation yields the result N( c = number of colours)
2) ms( 11Hc_ Nf) inn2 -I (1.3.4)
as(q 2) ms(~ 2) I § (-3- A 2
=
in(q2/2) 2
Since the coefficient of is positive, the limit for q § ~ nac eb taken
2q( 2
with the well nwonk result a s ) + .0 This ,snaem that if in a ssecorp a scale q
sraeppa which is sufficiently large, then the coupling constant a s is small. This
property of DCQ is called asymptotic freedom dna saw derived the first time yb ssorG
dna Wilczek /1973/ dna yb Politzer /1973/. roF a erom recent discussion ees sehguH
/1980/. This property is very important for the interpretation of DCQ perturbation
theory. It justifies the noitpmussa that for high hguone energies perturbation the-
2
ory yam eb sufficiently convergent. woH large q should eb sdneped no the proper
2 2
scale, i.e. no the value of ~ for a particular q = N which must eb determined
s
yb experiment. Furthermore only after higher order corrections for a process evah
neeb ,detupmoc it is nwonk woh eht kinematic variables of the process era related
to the effective 2q which ekam the higher order corrections small. This will eb
edam erom transparent later nehw ew discuss higher order DCQ corrections for jet
cross sections.
A erusaem for the scale of a s is defined yb the formula
ms(q2) = ~4 2 (1.3.5)
11, 2, ~, q
(Tmc-~ ~fjln~
This formula contains the same information sa (1.3.4) concerning the 2q dependence
of a s. Only the boundary conditions are different, in (1.3.5) a s = ~ for 2q = 2A
whereas in (1.3.4) a s = ms(2 ) for 2q = N 2 . ehT parameter A gives su the measure
01