FERMILAB-Conf-01/387-T

On the Large Nc Expansion in Quantum Chromodynamics


William A. Bardeen
Theoretical Physics Department
Fermilab, MS 106, P.O. Box 500
Batavia, Illinois 60510
bardeen@fnal.gov


Abstract

I discuss methods based on the large Nc expansion to study nonperturbative
aspects of quantum chromodynamics, the theory of the strong force. I apply these
methods to the analysis of weak decay processes and the nonperturbative
computation of the weak matrix elements needed for a complete evaluation of these
decays in the Standard Model of elementary particle physics.


 Introduction.

Field theories are frequently studied through a perturbative expansion in the interaction strength
or coupling constant. In many cases, a nonperturbative analysis is required to apply these theories
in physical situations. The large N expansion is a nonperturbative method of analysis that makes
use of particular limits for parameters unrelated to the coupling constant. For example, O(N) spin
systems where N is the number of spin components can be studied by mean field methods which
become exact in the large N limit. Applications of perturbative QCD, such as the cross-sections
for high p_t jets, are greatly simplified using color-ordered amplitudes and a large Nc expansion
where Nc is the number of colors, the quantum number associated with the QCD gauge dynamics.
The large Nc expansion can also be used to study nonperturbative aspects of quantum
chromodynamics with applications to the structure of chiral symmetry breaking and hadronic string
theory. More recently, it has been established that there exists a duality between the large N limit
of SU(N) supersymmetric Yang-Mills theory and classical supergravity in a higher dimensional
arXiv: 17 Dec 2001 space-time. In this talk, I will discuss the nature of the large Nc expansion in QCD and its
application to the computation of weak decay amplitudes.


 Large Nc expansion in QCD.

The large Nc expansion defines a nonperturbative reordering of the QCD coupling constant
expansion. Each Feynman diagram can be classified by its dependence on the strong coupling
constant, strong, and the number of colors Nc. In 1974, `t Hooft [1] argued that the structure of
the theory simplified in the limit, Nc ,
strong 0 , strong Nc fixed . The theory is
summed to all orders in the rescaled coupling constant,
strong Nc , with corrections being
formally suppressed by powers of 1/Nc. At leading order in this expansion, the gluons are
constrained to be in planar Feynman diagrams. Quarks lines form boundaries of the planar
surfaces formed by the gluons. In this sense, the structure of large Nc QCD is similar to an open
string theory with quarks attached to the ends of the string.


If we assume that large Nc QCD is confining, then the physical states are expected to consist of
stringlike glueballs and mesons formed as quarkantiquark boundstates. The physical states are
towers of meson resonances with increasing mass and spin. The lowest order meson scattering
amplitude has the structure of a gluonic disk with meson bound-states attached to the edge of the
disk. Since the color wavefunction of a meson bound-state is O(1/Nc), the leading order meson
amplitudes with more than two mesons bound-states attached to the edge of the disk are suppressed
by a factor of (1/Nc) for each additional meson. This simple power counting implies that all
mesons are stable at leading order and that the meson S-matrix is that of a trivial free meson theory.
We may still consider the leading contribution to processes at any order in the large Nc expansion.
A meson scattering amplitude at leading nontrivial order involves only meson tree amplitudes with
simple poles at positions of the meson bound-states. Higher order diagrams involve either
additional quark loops which are suppressed by a power of (1/Nc) or nonplanar diagrams which are
suppressed by powers of (1/Nc2). The 1/Nc expansion is a topological diagram expansion with
the complexity increasing at higher order. In two space-time dimensions, large Nc QCD has an
exact solution while only its structure can be inferred in higher dimensions.

 Weak decay amplitudes.

Nonleptonic weak decays are mediated by the virtual exchange of W and Z bosons. These
processes occur at a short distance scale, and their effects can be expressed in terms of effective
local interactions between quark currents and densities at low energy [2]. The weak decay
amplitudes are written as an expansion in terms of short distance Wilson coefficients, C 
i ( ), and
sets of local quark operators, Q 
i ( ),

G
A(K F
) = V C () K Q 
CKM i i ( ) (1)
2 i

where a normalization scale,  , is introduced to separate the short and long distance physics
contributions. Details of the short distance processes and perturbative QCD dynamics are used to
compute the Wilson coefficients. Operator matrix elements are sensitive to long distance physics
including the mechanisms of quark confinement and the formation of hadronic bound-states that
can not be computed using perturbative QCD. The systematic computation of relevant Wilson
coefficient functions for weak decays at two loops, (NLO), have been made by several groups [3].
The coefficient function may be expressed as

C () = Z () + i * Y 
i i i ( ) , (2)

where the coefficient, Y 
i ( ) , incorporates the short distance effects of CP violation.

Operator matrix elements are more difficult to analyze as perturbative methods can not be used.
Several strategies have been employed to compute the matrix elements of the quark operators
appearing in the expansion of the weak Hamiltonian. All of the operators being studied have the
form of products of quark currents and densities which themselves are color-singlet quark bilinears.
The simplest approximation invokes factorization where it is assumed the quark currents and
densities independently couple to the hadrons in the initial and final states. The factorization
approximation has inherent limitations as it fails to reproduce the scale dependence of the Wilson
coefficents. A second method employs numerical computations in the lattice formulation of QCD.
The lattice method is constrained by the size of the numerical effort required and by problems
associated with the chiral structure of lattice QCD. There has be recent progress in the direct
computation of weak matrix elements using the domain wall formulation of chiral fermions on the
lattice [4]. Another nonperturbative method for calculating the weak operator matrix elements
invokes the large Nc expansion of QCD as discussed in the following section.


 Large Nc computation of weak matrix elements.

The large Nc analysis focuses on the computation of weak matrix elements of the four-quark
operators, Qi, generated by the renormalization group expansion.

Q =
i ( i )( i ) (3)

The analysis of the large Nc expansion of QCD relies on the toplogical structure of the quark
diagrams and large Nc power counting to determine the various contributions to the weak matrix
elements. At the leading order of the large Nc expansion, the matrix elements are factorized

Q =
i ( ) ( ) . (4)
F i i

We recall that the leading order quark amplitudes have the structure of a disk with the quark line
forming the boundary of the disk with planar gluons filling in the surface of the disk. Since the
quarks have Nc colors, the factorized quark densities must each be associated with separate disks
and, therefore, two powers of Nc are generated, one for each disk. This factorized contribution is
the leading order contribution (LO). Gluonic interactions between the two disks are nonplanar and
therefore suppressed by at least two powers of Nc.

At next-leading-order (NLO) in the large Nc expansion, O(1/Nc), there are two possible
contributions. An internal quark loop can be added to either disk of the leading order calculation
giving a suppression of at least 1/Nc. Since this insertion does not involve the second disk it
contributes only to the factorized matrix element. This correction represents a meson loop
correction to the matrix element of the quark current or density. It will be assumed that we are able
to measure these current matrix elements which include all order corrections phenomenologically.

The second contribution at NLO is nonfactorized. It arises when both quark bilinear operators
are associated with a single disk. Because only one quark loop is involved instead of two, the
amplitude is suppressed by a power of 1/Nc relative to the LO factorized contribution. At this
order in the large Nc expansion, the two quark currents and the various meson bound-states are
associated with the single quark line forming the boundary of the single disk. Hence, this
amplitude is represented by a tree-level meson amplitude having only poles at the positions of the
infinite tower of meson bound-states.

We can write this amplitude as the momentum integral of a meson amplitude with two external
bilinear currents,

Q = dk A k
( ,
-k, p ...p
i ). (5)
NF 1 N
i i


Knowledge of meson tree amplitudes may be used to compute integrand. At low momentum,
chiral Lagrangian description is an exact representation of QCD dynamics for matrix elements
involving low energy meson states. At high momentum, we can use large Nc version of the QCD
renormalization group equation to compute integrand in terms of perturbative coefficient functions
and factorized matrix elements. The entire integral is obtained by interpolating between the
nonperturbative low energy approximation and the perturbative high energy amplitude.
Calculations based on this duality have achieved some success in explaining the structure of weak
decay amplitudes including the octet enhancement observed in K 2 decays and the CP


violation seen the ' / measurement [5]. The precision of these methods depend on three
ingredients:

 the phenomenological determination of the long distance meson amplitude,
 the order of the perturbative short distance calculation, the strong expansion,
 the accuracy of the interpolation between the long and short distance approximations.

The amplitude calculation requires the evaluation of the momentum integral in Eq.(5). At high
momentum (short distance), k , p ...p
1 - small
N , the operator product expansion (PQCD)
takes the form,

A (k, -k; p ,..., p )  
1 C (k, ) Q
N Q ( )
; (6)
LO

where the coefficient function, C, is O(1/Nc) and the operator matrix element is factorized. Note
that all weak mixing processes are O(1/Nc). Using the renormalization group and the anomalous
dimension matrices of perturbative QCD, we compute evolution of the NLO weak matrix elements,

 Q () = -
( ()) Q
i () (7)
NLO ij j LO

where the LO matrix element is factorized. The integrated form at NLO is

Q = - d2 / 2
2 (()) Q 
i ( ) . (8)
NLO ij j LO

The long distance contributions are described by tree-level meson amplitudes which involve
nonperturbative parameters which are determined phenomenologically A number of
approximations to the low energy meson amplitudes has been used in the study of weak decay
amplitudes. They include chiral Lagrangians, chiral Lagrangians + vector mesons, chiral quark
models, and extended NambuJona-Lasinio models. The chiral Lagrangian is an exact description
of QCD in the light quark limit and for low momenta. The various approximations given above are
different attempts to parameterize the physical low energy dynamics. The models require the input
of masses and coupling constants from data other than the weak decay processes. Further efforts
are needed to improve extrapolation to higher momentum scales to more precisely match the
perturbative QCD calculations of the high momentum part of the two-current correlation function.
These efforts may involve adding additional mesons (vector, axial-vector, scalar, tensor, ...) or
higher dimension operators in the effective field theory (O(p4), O(p6), ...).

In addition to the precise computation of the above tree-level meson amplitudes, matching
conditions are also required to connect the momentum integral of the two-current correlation
function with the standard perturbative analysis of the weak matrix elements. At NLO in the large
Nc expansion, the weak decay amplitudes are divergent at high energy. The standard short distance
analysis uses dimensional regularization, (NDR, HV), to regularize the quark amplitudes and define
normalization scales for the coefficient functions and operator matrix elements. Both the chiral
Lagrangian calculation and the short distance quark calculation may be regularized by a cutoff on
momentum flowing through color singlet currents.

A consistent treatment of the short distance contributions matches the different regularization
schemes by computing quark matrix elements of weak operators using both dimensional
regularization and momentum cutoff methods. To isolate the purely short distance effects, an
infrared regularization is introduced,


dk dk k2 /(k2 - M2 ) , M = IR cutoff
. (9)

We may now compute the perturbative matrix elements of the various quark operators using the
different UV cutoff schemes. The operator matrix elements will have a finite correction factor at
one loop,


QNDR,HV = Qmom -
w NDR,HV Qmom
i i ij j
(10)
4

where wij is rotation matrix between dimensional regularization basis and momentum cutoff basis.

This rotation matrix has been computed for all ten four-fermion quark operators used to expand
the weak Hamiltonian for both NDR and HV reqularization schemes [6]. The effective coefficient
functions to use with momentum subtracted matrix elements can now constructed,

, , ,
Cmom C NDR HV C NDR HV w NDR HV
= -
i i (11)
k ki
4

An example of effect of this shift on coefficient functions can be evaluated. Our numbers are
based on the perturbative QCD analysis of Bosch et al (199) [7]. The normalization scale is
 = 1 3
. Gev and = 0.
QCD 340. The result for certain coefficient functions are given below

CF NDR HV mom mom
NDR HV


Z1 -0.425 -0.521 -0.669 -0.687
Z2 1.244 1.320 1.371 1.394
Y3 0.030 0.034 0.041 0.041
Y4 -0.059 -0.061 -0.063 -0.064
Y5 0.005 0.016 0.013 0.011
Y6 -0.092 -0.083 -0.091 -0.090


There is an enhancement of the Z1 and Z2 coefficients in the momentum basis over the values in
either the NDR or HV schemes. The larger values of Z1 and Z2 will tend imply a larger octet
enhancement and 27 supression factors than the conventional treatment. Of course, the important
point of this aspect of the analysis is that we now have all the elements to make a consistent use of
the 1/Nc expansion for computing the weak matrix elements:

 the standard renormalization group analysis of the weak coefficient functions is used to
compute the short distance contributions and evolve the operators to low energy scales,
 a consistent matching between the dimensional regularization schemes and the momentum
cutoff is used in the NLO analysis of the nonfactorized weak matrix elements,
 chiral Lagrangians or other effective field theories are used to describe the long distance
contributions to the weak matrix elements and consistently matched to the short distance
contributions.

The complete calculation has yet to be fully integrated. There are still some issues regarding
matrix elements involving scalar and pseudoscalar densities that can affect the evaluation of the Q_6
and Q_8 operator matrix elements needed for ' / analysis.


 Conclusions.

Precision tests of the Standard Model require knowledge of nonperturbative aspects of quantum
chromodynamics, the strong dynamics. The Large Nc expansion combined with phenomenological
knowledge of certain meson amplitudes provides one avenue for systematic estimates. The
analysis outlined in this talk provides the elements for this systematic analysis of the weak decay
matrix elements through next-leading-order However it is fundamentally limited by the ability to
compute yet higher order terms in the large Nc expansion.

Direct numerical computation of weak matrix elements using lattice formulations of QCD
provides another avenue. Recent developments related to the chiral symmetry structure of lattice
QCD may lead to more realistic calculations of the necessary matrix elements.

The final picture remains unclear. As yet there is no clear violation of the Standard Model to be
inferred by weak decay processes but the large value of ' / and the large value of the I = 1/ 2
amplitude for K 2 may yet challenge its validity.


 Acknowledgements.

I thank the Alexander von Humboldt Foundation for its hospitality and support. This research
is supported by the Department of Energy under contract DE-AC02-76CHO3000.


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