2.4 Renormalization in a Nutshell
2.4.2 Renormalization Group Equations
2.35has a pole at µ= ΛQCD. This argument, however, is flawed.
Eq.
2.35was obtained in perturbation theory and therefore, cannot hold in a region where perturbative QCD breaks down. Still, the growth of the coupling at low energies or, equivalently, large distances indicates a peculiar behavior exhibited by QCD: quarks appear only in bound states, the hadrons, and no free quarks have been (experimentally) observed. This phenomenon is called confinement.
2.4.2 Renormalization Group Equations
Equation
2.27is one example of a so-called renormalization group equation (RGE). This class of equations describes the dependence of renormalized quan-tities on the renormalization scaleµ. For example, the RGE for the mass will take the form [40]
µdm(µ)
dµ =−γm(g)m(µ) with γm(g) = µ Zm
dZm
dµ .
2.36 The renormalization group functionγmis called quark massanomalous dimen-sion. Using the same line of argumentation as previously for the β-function, one easily finds thatγm can be obtained from the residue of the corresponding Zmfactor via
γm=−2g2dRes (Zm)
dg2 =−2αs
dRes (Zm)
dαs .
2.37 This formula is true not only for masses but for all renormalized quantities in theM S scheme. The anomalous dimensions can generically be expanded in a series inαs
γm(αs) =γm(0)αs
2π +γm(1)αs
2π 2
+. . . . In leading order inαsthe RGE
2.36can be solved by separation of variables.
One obtains
m(µ) =
αs(µ) αs(µi)
γb(0)m0
m(µi),
2.38
14
2.4. RENORMALIZATION IN A NUTSHELL
where µi, m(µi) are the initial conditions. It should be noted that the RGE given in
2.36can trivially be rewritten in the “standard” form
µ ∂
∂µ+β(g) ∂
∂g +γO
Oren(µ) = 0
2.39 for a generic renormalized local operator Oren(µ, αs(µ)) with anomalous di-mension γO. An operator that follows such a RGE is called multiplicatively renormalizable, as its dependence on the scaleµ can be expressed via a single multiplicative factor, see
2.38.
In general, the situation is more complicated. Due to renormalization an operator can be affected by admixtures of operators with the same quantum numbers. If a set of operators{Oi},i= 1, . . . , nis closed under renormalization, that is each operator in the set only receives admixtures due to operators also in{Oi}, the RGEs have the form
µ ∂
∂µ+β(g) ∂
∂g+γij
O~jren(µ) = 0,
2.40 whereγis then×nmatrix of anomalous dimensions andO~ a vector consisting of the operatorsOi. By diagonalizing the matrixγ, Eq.
2.40can be reduced tondecoupled differential equations of the form
2.39. What to keep in mind
The subtleties and challenges of the renormalization procedure are numerous and we refer the reader to standard textbooks, such as [50, 51], for further read-ing. For our purposes this small detour will be sufficient. Apart from Eqs.
2.39,
2.38and
2.40it is useful to keep this convenient property of our renormal-ization scheme in mind:
In theM S scheme the leading order anomalous dimension (of an oper-ator) is equal to the negative of the double residue of the corresponding renormalization constant.
“Shut up and calculate!”
N. David Mermin
’What’s Wrong with this Pillow?’
3
Technical Background
As mentioned briefly in the Introduction, two more advanced concepts are to be presented in this chapter; the first one being the so-called spinor formalism.
It is based on the early observation by Weyl [52] and van der Waerden [53]
that the Dirac equation [54] for 4-spinors of massless fermions can be rewritten in terms of two separate differential equations for two-component spinors, the Weyl spinors. Over the course of Sect. 3.1 we will show that working with Weyl spinors as basic objects of the theory allows for a simple classification of the transformation properties of generic tensors with respect to the Lorentz group.
While this formalism is utilized frequently in supersymmetric theories, where the spinor nature of the supersymmetric generators makes this a natural choice, it rarely sees use in QCD or QED, although its merits have been pointed out frequently [55, 56].
We begin in Sect. 3.1 with van der Waerden’s idea that the transformation of chiral and antichiral Weyl spinors can conveniently be indicated by employing dotted and undotted indices. After explaining how to include Lorentz indices in this spinor notation, we give a short summary of translation rules and relations useful for working in this formalism. The following section deals with gauge fields and we show that, analogously to the 4-spinor, the field strength tensor can be decomposed into two components transforming according to irreducible representations of the Lorentz group. In Sect. 3.1.3 we discuss how to project arbitrary tensor operators onto definite spin, which turns out to be straightfor-ward in spinor notation and is, in fact, the main reason why we work with it.
Note that only recently a detailed review [56] on these two-component spinor techniques was published1.
The second part of Chap. 3 is dedicated to the concept of conformal sym-metry and its application to multi-particle light-ray operators (the study of the
1[56] uses different sign conventions in its definitions and therefore one has to be careful not to miss a sign. The authors of [56] provide alternative versions with different conventions on their website [57] .
CHAPTER 3. TECHNICAL BACKGROUND
renormalization properties of such operators is the aim of Chapter 4). The sym-metry itself has been known for more than a century and its early applications included complicated problems in electrostatics. Moreover, the study of the conformal properties of two-dimensional field theories is a large area of research due to connections with string theory. In 4-dimensional field theories, conformal symmetry has generally been treated rather stepmotherly, as an existing con-formal symmetry is usually broken at quantum level. We explain this in more detail in Section 3.2. A notable exception are the so-called super-conformal N = 4 Yang-Mills theories, which feature a vanishingβ-function. They are the basis of the famous AdS/CFT conjecture [12] – one the most active fields in mathematical physics [58].
As conformal symmetry is not among the standard tools of QCD, see [59]
for the current state-of-the-art, we give a short introduction to the structure of the conformal algebra in Sect. 3.2.1 and show how the generators look like in spinor notation. After restricting ourselves to the so-calledSL(2,R) subgroup, which corresponds to the projective Moebius transformations on a light-ray, we construct a basis of one-particle operators with “good” conformal properties2 in Sect. 3.2.3. While our explicit derivation does not take into account issues related to the fact that QCD is a gauge theory, this basis can be generalized to full QCD with the tools introduced in this chapter. This construction strategy, along with the new, complete one-particle basis, see Eq.
3.63, represents the main result of this chapter and one of the central novelties of this thesis.
3.1 Spinor Formalism
In the case of free massless spin-1/2 fermion fields q the Dirac equation [54]
assumes the form
pµ(γµ)ijqj(p) = 0,
3.1 wherepis the momentum of the fermion,i, jare spinor indices andµis a Lorentz index. It is possible to decouple the equation for the upper two components of the bispinor from the equation for the lower two components by a specific choice for theγ matrix basis. This is the so-called Weyl representation, see e.g. [33], which, of course, respects the usual commutation relations
{γµ, γν}= 2gµν for µ, ν= 0,1,2,3.
3.2 The fact that such a separation is possible is equivalent to the statement that the Dirac equation preserves the chirality of massless fermions.
2Each operator has well-defined collinear twist, helicity and conformal spin.
18
3.1. SPINOR FORMALISM
Following [53], it is then convenient to introduce the following notation for the four-dimensional Dirac bispinor:
q= ψα
¯ χβ˙
!
, q¯= (χβ,ψ¯α˙).
3.3 ψ corresponds to the chiral, ¯χ to the anti-chiral Weyl spinor. The somewhat peculiar notation with dotted and undotted indices has a distinct advantage [53]:
The irreducible representations of the Lorentz group are labeled by two spins (s,s). The chiral spinor¯ ψα transforms according to (1/2,0), the anti-chiral spinor ¯χα˙ according to (0,1/2). Hence, it is very simple to read off the trans-formation properties in this notation. It is now obvious that the standard Dirac spinor does not transform according to an irreducible representation of the Lorentz group; it rather transforms as (1/2,0)⊕(0,1/2).
However, in order for the separation of dotted and undotted indices, i.e.
chiral and anti-chiral fields, to be useful for the classification of the transforma-tion properties of a generic operator, it is necessary to convert also all Lorentz indices into spinor indices. This can be achieved in the following way, see also [60]:
• take an arbitrary covariant four-vectorxµ
• then define
xαα˙ :=xµ(σµ)αα˙
3.4 where
σµ= (1, ~σ)
3.5 and~σare the usual Pauli matrices
• the 2×2 matrix xαα˙ then contains the full information on the vector xµ
and has the transformation properties under Lorentz transformations as indicated by its indices.
One can see that this procedure actually does what it claims by observing that each covariant four-vectorxµ can be mapped to a hermitian 2×2 matrixx
x= x0+x3 x1−ix2
x1+ix2 x0−x3
!
≡xµσµ.
3.6 It can easily be checked that detx = xµxµ = x2 and that a Lorentz trans-formation x′µ = Λµνxν corresponds to a rotation of the form x′ = M xM†, where M ∈ SL(2,C). The homomorphism Λ → M must then define a two-dimensional (spinor) representation of the Lorentz groupu′ =M u[60]. At first
CHAPTER 3. TECHNICAL BACKGROUND
glance, it is actually possible to find four homomorphisms Λ→M, M∗, M−1,T and M−1†, each of which could define a different representation and the cor-responding spinors are usually denoted asuα,u¯α˙, uα and ¯uα˙, respectively, i.e.
u′α = Mαβ
uβ, ¯u′α˙ = Mα∗˙β˙u¯β˙ etc. However, not all these representations are independent, as the Lorentz group has only two non-equivalent spinor represen-tations (1/2,0) and (0,1/2). One finds that
iσ2M =M−1,Tiσ2 and iσ2M∗=M−1,†iσ2.
3.7 The transformation properties indicated by the notation of Eq.
3.4are there-fore indeed realized.