Parton showers with quantum interference

TLDR: In this article, the authors proposed recursive equations that could be used to generate a lowest order parton shower for hard scattering in hadron-hadron collisions, based on factorization soft and collinear interactions from relatively harder interactions in QCD amplitudes.

Abstract: We specify recursive equations that could be used to generate a lowest order parton shower for hard scattering in hadron-hadron collisions. The formalism is based on the factorization soft and collinear interactions from relatively harder interactions in QCD amplitudes. It incorporates quantum interference between different amplitudes in those cases in which the interference diagrams have leading soft or collinear singularities. It incorporates the color and spin information carried by partons emerging from a hard interaction. One motivation for this work is to have a method that can naturally cooperate with next-to-leading order calculations.

Figures

Figure 1: Illustration of eq. (2.18). The wide ovals represent the evolution operator U while the narrow ovals represent the no change operator N , which provides the Sudakov exponentials, and the small circle is a parton splitting. Shower time t runs from left to right.

Figure 10: The starting diagram from figure 9 in the leading color approximation, in which gluons are emitted from color dipoles. The i-j color dipole is a unit that emits a gluon and absorbs it on the other side of the diagram. At the next stage there will be an i-(m + 1) color dipole and an (m + 1)-j color dipole.

Figure 9: Illustration of eq. (10.2). The first diagram shows the color configuration corresponding to the operator G(i, j; {f̂}m+1): the gluon’s color is emitted from parton i and absorbed on gluon j. Using eq. (7.24) to write this in terms of color basis states, one contribution is shown in the middle diagram. This is the leading color contribution. Another contribution is shown in the right-hand diagram. This contribution is subleading. If i and j had not been color connected to each other, there would have been no contribution that survives in the leading color approximation.

Figure 3: Momentum space mapping for a final state splitting. Coordinate axes in the two lightlike directions in the plane of pl and Q are shown. Points pl and Q are shown in the case that pl is nearly lightlike. Then Pl lies on the hyperbola (Q− Pl)2 = const. that passes through pl.

Figure 5: Illustration of factorization at the amplitude level, leading to the definition of the splitting function in eq. (6.19). When the final state gluon becomes collinear with the initial state quark, the full amplitude is approximately the amplitude for one fewer parton, with the incoming quark after the gluon radiation approximated as being on shell, times a splitting function simply made from the QCD qqg vertex and the singular quark propagator. If the final state gluon is soft, then this diagram is one of the possibilities. Then, the soft gluon could have been emitted from any of the external legs of the diagram and we must sum over all of the possible emissions. In this case, we can, however, use a simpler splitting function.

Figure 7: Inner products for color basis states. The left hand picture illustrates the inner product of the [1, 2, 3, 4] with itself. The color diagram shown gives C2FNc, which is just canceled by the normalization factor 1/n(S) from eq. (7.7). The right hand picture illustrates the inner product of the [1, 2, 3, 4] with [1, 3, 2, 4]. The color diagram gives −CF/2. Multiplying by 1/n(S) gives −1/(N2c − 1).

Full text

JHEP09(2007)114
Published by Institute of Physics Publishing for SISSA
Received: June 1, 2007
Revised: August 8, 2007
Accepted: August 27, 2007
Published: September 25, 2007
Parton showers with quantum interference
Zolt´an Nagy
Theory Division, CERN,
CH-1211 Geneva 23, Switzerland
E-mail: Zoltan.Nagy@cern.ch
Davison E. Soper
Institute of Theoretical Science, University of Oregon,
Eugene, OR 97403-5203, U.S.A.
E-mail: soper@uoregon.edu
Abstract: We specify recursive equations that could be used to generate a lowest order
parton shower for hard scattering in hadron-hadron collisions. The formalism is based
on the factorization soft and collinear interactions from relatively harder interactions in
QCD amplitudes. It incorporates quantum interference between different amplitudes in
those cases in which the interference diagrams have leading soft or collinear singularities.
It incorporates the color and spin information carried by partons emerging from a hard
interaction. One motivation for this work is to have a method that can naturally cooperate
with next-to-leading order calculations.
Keywords: Hadronic Colliders, QCD, Jets.
c
° SISSA 2007 http://jhep.sissa.it/archive/papers/jhep092007114/jhep092007114.pdf

JHEP09(2007)114
Contents
1. Introduction 2
2. A notation for parton showers 7
3. Structure of the calculation 11
3.1 The space of quantum parton states 11
3.2 The density matrix 14
3.3 Statistical states 17
3.4 The resolution scale 18
3.5 Parton shower evolution 20
4. Momentum and flavor mapping 24
4.1 Splitting a final state parton 25
4.2 Combining two final state partons 29
4.3 The integration measure for final state splitting 30
4.4 Splitting an initial state parton 30
4.5 Combining an initial state parton with a final state parton 34
4.6 The integration measure for initial state splitting 35
5. Spin states 35
6. Splitting functions for the quantum states 37
6.1 Definition of the splitting functions v
l
38
6.2 Initial state q → q + g splitting, quark scatters 39
6.3 Initial state q → q + g splitting, gluon scatters 41
6.4 Other qqg splittings 43
6.5 Splitting with a ggg vertex 43
6.6 Soft splitting function 45
7. Description of color 45
7.1 Color basis 46
7.2 Parton insertion operators 48
7.3 Color evolution for the quantum states 50
8. Evolution for the statistical states 52
9. The operator H
I
(t) 58
10. Color evolution of the statistical states 59
11. Soft gluon coherence 61
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JHEP09(2007)114
12. Inclusive evolution 64
13. End of the shower 69
14. Conclusions 71
A. Limit on momentum fraction after splitting 75
B. Counting factors for the density matrix 76
1. Introduction
Parton shower Monte Carlo event generators, such as Herwig [1] and Pythia [2], have
proven to be enormously useful since the development of the main ideas in the 1980s [3 – 5].
These computer programs perform calculations of cross sections according to an approx-
imation to the standard model or some of its possible extensions. Because of the great
success of these programs, it is worthwhile to investigate possible improvements. In this
paper, we propose a theoretical structure for event generators that generalizes the structure
of current programs and allows the elimination of certain approximations used currently.
Parton showers are mostly reflections of QCD interactions. In order to present a
reasonably complete discussion of the QCD issues in a parton shower while keeping the
length of this paper within reasonable bounds, we limit the presentation to QCD and omit
any discussion of how electroweak and beyond-the-standard-model interactions are to be
added to the QCD interactions to make a useful event generator.
What is a parton shower Monte Carlo event generator? Let us consider hadron-hadron
collisions, which is the case relevant for the Tevatron and the Large Hadron Collider. An
experiment will produce a large number of events f , where one can characterize an event as
a list of the momenta and flavors of the final state particles produced. The experiment can
measure a cross section σ[F ] corresponding to an observable
1
that assigns to each event f
a number F (f). The relation of the cross section and the function F is
σ[F ] ≈
1
L
N
X
n=1
F (f
n
) , (1.1)
where L is the integrated luminosity for an experimental run and the f
n
are the observed
events. For example, the cross section to produce a Higgs boson and two jets having certain
characteristics is specified by setting F (f) = 1 if f contains a Higgs boson and two jets
having these characteristics and F (f ) = 0 otherwise.
2
1
In order to be subject to reliable calculation in QCD perturbation theory, the function F should have the
property known as infrared safety. However, a parton shower event generator is also useful for observables
that are not infrared safe.
2
The case in which F (f) takes values 0 or 1 is the most common, but other possibilities are allowed.
For instance, the energy-energy correlation function in electron-positron annihilation is of the more general
variety.
– 2 –

JHEP09(2007)114
A parton shower Monte Carlo event generator calculates this cross section by producing
a large number N of simulated events f
n
, each with an accompanying weight w
n
. The
calculated cross section is then
σ[F ] ≈
1
N
N
X
n=1
w
n
F (f
n
) . (1.2)
Most typically, the weights are all equal, so that 1/w
n
is the simulated luminosity per point
L/N. Our definition of the category of parton shower Monte Carlo event generator includes
the possibility that the weights are complex numbers produced for each event. It is always
possible to throw away the imaginary parts of the w
n
since we know in advance that the
imaginary part of the sum in eq. (1.2) vanishes, so having complex weights is equivalent to
having real weights that can be positive or negative. This situation occurs in typical event
generators [6, 7] that are based on next-to-leading order perturbation theory.
3
In a typical parton shower event generator, the physics is modeled as a process in
classical statistical mechanics. Some number of partons are produced in a hard interac-
tion. Then each parton has a chance to split into two partons, with the probability to
split determined from an approximation to the theory. Parton splitting continues in this
probabilistic style until a complete parton shower has developed.
The parton splitting probability is biggest when the two daughter partons are almost
massless with nearly collinear momenta or when one of their momenta is soft (near p = 0),
or both. There is a simple underlying approximation used: the amplitude for producing
m + 1 partons when two of the momenta p
i
and p
j
are nearly collinear or one is soft factors
into a splitting function times the matrix element for producing m partons.
The underlying approximation is the factorization of amplitudes in the soft or collinear
limits. However, further approximations are usually added:
1. The interference between a diagram in which a soft gluon is emitted from one hard
parton and a diagram in which the same soft gluon is emitted from another hard
parton is treated in an approximate way, with the “angular ordering” approximation.
2. Color is treated in an approximate way, valid when 1/N
2
c
→ 0 where N
c
= 3 is the
number of colors.
3. Parton spin is treated in an approximate way. According to the full quantum ampli-
tudes, when a parton splits, the angular distribution of the daughter partons depends
on the mother parton spin and even on the interference between different mother-
parton spin states. This dependence is typically ignored.
With the use of these further approximations, one can get to a formalism in which the
shower develops according to classical statistical mechanics with a certain evolution oper-
ator.
Our purpose in this paper is to investigate whether one can have a formulation of
parton showers based on the factorization of amplitudes in the soft or collinear limits in
3
The recent paper [8] provides an exception to this rule.
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JHEP09(2007)114
which one does not make the additional approximations enumerated above. For this, one
would have to use quantum statistical mechanics instead of classical statistical mechanics.
It might seem that doing the problem in quantum mechanics is hopelessly complicated.
However, within the soft/collinear factorization approximation, the problem is fairly simple
because it is almost classical. In fact, if partons did not have color or spin, the problem
would be classical (as we discuss in section 2). Thus what we need is a fully quantum
treatment of color and spin. We arrange for this by making use of the quantum density
operator in color ⊗ spin space.
In the subsequent sections, we define evolution equations for the quantum density ma-
trix within the soft/collinear factorization approximation. The matrix evolves in “shower
time” from harder splittings to softer splittings. The iterative solution of these equations
gives σ[F ] in the form of a sum of integrals. To give some idea of the structure, we omit
any mention of hadronization and write the result in a notation that is quite abbreviated
compared to the notation in the body of the paper,
σ[F ] =
Z
dP
0
f
0
∞
X
N=1
N
Y
j=1
µ
Z
dζ
j
f
j
¶
F . (1.3)
There is, first of all, an integration (including sums, for discrete variables) over momenta,
flavors, spins, and colors for initial partons that emerge from the hard matrix element
and its complex conjugate. Here we call all of these variables collectively P
0
, the initial
partonic variables. There is a function f
0
that depends on P
0
and represents the hard
matrix element at the start of the shower times its complex conjugate. Then there is a
sum over how many splittings, N, there are.
4
Next there is an integration over splitting
variables ζ
j
for the jth splitting. The splitting variables include the label telling which
parton split and momentum variables, for which a dimensionless virtuality y, a momentum
fraction z, and an azimuthal angle φ might be used. There are also discrete flavor, color,
and spin variables. At each splitting, there is a set of starting partonic variables, P
j− 1
and a set of new partonic variables P
j
that are determined by P
j− 1
and the splitting
parameters ζ
j
. For each splitting, there is a function f
j
that depends on P
j− 1
and ζ
j
. We
have integrations over the splitting parameters for splittings 1 through N. At the end,
there is the measurement function F that depends on the partonic variables P
N
reached
after all of the splittings.
The structure of this representation is similar to that in conventional parton showers,
with the functions f
j
made from splitting functions and Sudakov exponentials that express
the probability for not splitting. There are, however, some important structural differences
that result from not making the approximations 1, 2, and 3 above. Chief among them is
the use of the spin and color variables.
What we develop in this paper is an evolution equation that results in a representation
of σ[F ] as integrals of known functions. Of course, one will want to turn the integrals into
4
We have formally iterated the evolution equation an infinite number of times, allowing any number of
splittings. However, we imagine that there is a cutoff on splitting hardness, so that very large values of N
are seldom encountered. Some of our splittings are 1 → 1 self interactions rather than 1 → 2 splittings.
– 4 –