3.2 Conformal Symmetry
3.2.3 Construction of the Conformal Basis
3.44
3.2.3 Construction of the Conformal Basis
For the renormalization group equations to be manifestly conformally invariant, we have two fundamental prerequisites:
• We need an operator basis with “good” conformal properties. That is, a complete set of one-particle light-ray operators which transform according to an irreducible representation of the SL(2,R) group. These operators will serve as building blocks for multi-parton operators (operators for DIS, baryon operators, etc.).
• In order to find and classify these “good” fields, it is necessary to have a set of good quantum numbers that uniquely determines each one-particle operator. Hence, additional quantum operators commuting with the gen-eratorsS± and S0are required.
While the first point is non-trivial, there are indeed two operators to be found that commute with theSL(2,R) generators6:
E=i Eis exactly the operator measuring thecollineartwist. This can be seen in the following way: the dilatation operatoriDin
3.45gives the scaling dimension (cf.
3.35) of a field, whereas iM21 and iM˙1˙2 count the difference between dotted or undotted 1 and 2 spinor indices, which gives exactlyminus the spin projection on the light-ray. SoEdetermines the difference of dimension and spin
6This can be verified by explicit calculation of the commutator.
CHAPTER 3. TECHNICAL BACKGROUND
ψ+ ψ− χ¯+ χ¯− f++ f+− f−−
j 1 1/2 1 1/2 3/2 1 1/2
E 1 2 1 2 1 2 3
H 1/2 −1/2 −1/2 1/2 1 0 −1
Table 3.1: The SL(2,R) spin, twist and helicity of the fundamental fields. The table is taken from [62].
projection of a quantum field; this is by definition justcollinear twist. At this point there is no straightforward interpretation ofH which we will suggestively call “helicity operator”. This nomenclature will become clear shortly.
A light-ray operator with definite collinear twist Etransforms according to an irreducible representation of theSL(2,R) group withconformal spin [59]
j=lcan−E/2.
3.47 For such fields theSL(2,R) generators acquire their simple canonical form [59]
S+ =z2∂z+ 2jz , S0=z∂z+j , S− =−∂z.
3.48 Here all derivatives have lost their spinor indices and act on the light-ray coor-dinatez.
In particular the upper and lower component of the chiral quark field, ψ1 and ψ2, have conformal spin j = 1 andj= 1/2. Since the two fields correspond to the projection on the ‘plus’ and ‘minus’ light-cone coordinate, they are usually labeled byψ+andψ−. This coincidently agrees with the sign of their light-cone projected spin, providing a useful mnemonic. Thus
ψ(z) =λ ψ−(z)−µ ψ+(z),
3.49 where
ψ+(z) =λαψα(z)≡ψ1(z), [Eψ+](z) = ψ+(z), [Hψ+](z) = 1 2ψ+(z), ψ−(z) =µαψα(z)≡ψ2(z), [Eψ−](z) = 2ψ−(z), [Hψ−](z) =−1
2ψ−(z).
3.50 Note that the eigenvalues±12ofHcorrespond to the helicity of the fieldsψ+and ψ−. This is not true for fields with bad transformation properties with respect to the collinear subgroup. A decomposition following
3.49can be performed for each field. We define
¯
χ+ = ¯χα˙¯λα˙ , χ¯− = ¯χα˙µ¯α˙
3.51 28
3.2. CONFORMAL SYMMETRY
for antichiral quark fields7 and
f++(z) =λαλβfαβ(z), f+−(z) =λαµβfαβ(z), f−−(z) =µαµβfαβ(z).
3.52 for self-dual gluon fields. The projections of the anti self-dual gluon fields ¯f are given by ¯f±± = (f±±)∗. Table 3.1 summarizes the quantum numbers of the
“good” fields. This set of fields is our starting point for the construction of a one-particle light-ray operator basis.
In what follows, we restrict ourselves to the case of massless fermions alone, i.e. we ignore all issues concerning gauge invariance. After we finish our con-struction for the fermions, we sketch the strategy, how our result can be gener-alized to full QCD.
One of the problems that has to be addressed is the appearance of so-called non-quasipartonicoperators. A nonlocal operator
O= Φ(z1)Φ(z2). . .Φ(zN)
3.53 is calledquasipartonic if the number of fieldsN is equal to the light-cone twist E. One can easily read off, see Table 3.1, that each “plus” field adds exactly one unit of twist. So every quasipartonic operator consists only of “plus” fields.
Every “minus” field increases the twist of the operator further, making it non-quasipartonic.
The quasipartonic operators have been studied in great detail in the liter-ature and it was understood that the renormalization of such operators only requires so-called 2-to-2 kernels (see Chapter 4). This simplifies the treatment of such operators. One of the reasons for this is that operators of given twist cannot mix with operators of different twist and operators withN fields will not mix with operators with less thanN fields8. As quasipartonic operators have by definition the minimal possible twist for any given number of fields, the set is closed under renormalization.
This is obviously no longer true for non-quasipartonic operators. They can not only mix with operators corresponding to higher Fock states, but also with operators containing the derivatives
∂−−˙ =∂1 ˙1, ∂+ ˙−=∂2 ˙1 and ∂−+˙ =∂1 ˙2.
3.54 Let us now consider the action of the generators
3.35on the chiral quark operator with a derivative, i.e.
[∂1 ˙1ψ±](z), [∂1 ˙2ψ±](z), [∂2 ˙1ψ±](z) and [∂2 ˙2ψ±](z).
7For simplicity we neglect the dots on the + and−symbols, as they can unambiguously be restored from the corresponding field.
8The reverse is not true, operatorscanmix with operators with a higher number of fields.
CHAPTER 3. TECHNICAL BACKGROUND
IfS±andS0do not assume their canonical form, the operators do not have defi-nite conformal spin and will, if included in our operator basis, veil the conformal invariance of the RGEs.
The generator S− trivially commutes with all derivatives, so there is no source for complications. Let us have a closer look at the action ofS0. Here one must again distinguish between the quantum operatorS0 and the differential operatorS0. It follows that
h
S0,[∂1 ˙1ψ+](z)i
=
∂1 ˙1[S0ψ+](x)
x=zn= (z∂z+ 1) [∂1 ˙1ψ+](z), h
S0,[∂1 ˙1ψ−](z)i
=
∂1 ˙1[S0ψ−](x)
x=zn=
z∂z+1 2
[∂1 ˙1ψ−](z),
3.55 where one has to take into account that∂αα˙xββ˙ = 2δαβδβα˙˙ andx2 ˙2 =z. Analo-gously one gets:
S0[∂1 ˙2ψ+](z) =
z∂z+3 2
[∂1 ˙2ψ+](z) S0[∂2 ˙1ψ+](z) =
z∂z+3 2
[∂2 ˙1ψ+](z) S0[∂1 ˙2ψ−](z) = (z∂z+ 1) [∂1 ˙2ψ−](z) S0[∂2 ˙1ψ−](z) = (z∂z+ 1) [∂2 ˙1ψ−](z) S0[∂2 ˙2ψ+](z) = (z∂z+ 2) [∂2 ˙2ψ+](z) S0[∂2 ˙2ψ−](z) =
z∂z+3
2
[∂2 ˙2ψ−](z).
3.56 One sees that the canonical form is always acquired. However, this is not the case forS+. Repeating the steps above we obtain:
S+[∂2 ˙2ψ−](z) = (z2∂z+ 4z)[∂2 ˙2ψ−](z), S+[∂2 ˙1ψ+](z) = (z2∂z+ 3z)[∂2 ˙1ψ+](z), S+[∂1 ˙1ψ+](z) = (z2∂z+ 2z)[∂1 ˙1ψ+](z),
S+[∂1 ˙2ψ+](z) = (z2∂z+ 3z)[∂1 ˙2ψ+](z)−2ψ−(z)
3.57 and
S+[∂2 ˙2ψ−](z) = (z2∂z+ 3z)[∂2 ˙1ψ−](z), S+[∂2 ˙1ψ−](z) = (z2∂z+ 2z)[∂2 ˙1ψ−](z), S+[∂1 ˙1ψ−](z) = (z2∂z+z)[∂1 ˙1ψ−](z),
S+[∂1 ˙2ψ−](z) = (z2∂z+ 2z)[∂1 ˙2ψ−](z).
3.58 Obviously S+ deviates from the standard form only for [∂1 ˙2ψ+](z). One can use the equations of motion to circumvent this. The Dirac equation, Eq.
3.24, connects transversal, plus and minus derivatives:
∂1 ˙2ψ+=−∂2 ˙2ψ−, ∂2 ˙1ψ−=−∂1 ˙1ψ+.
3.59 The “bad” term [∂1 ˙2ψ+](z) can be removed from each expression by replacing it with [−∂2 ˙2ψ−](z), which has good transformation properties. The second
30
3.2. CONFORMAL SYMMETRY
equation, ∂2 ˙1ψ− =−∂1 ˙1ψ+, relates two objects with favorable transformation properties. One is free to keep any one of the two in the operator basis. Elimi-nating∂2 ˙1ψ−in favor of∂1 ˙1ψ+ is slightly more advantageous, as the basis will be more symmetric with respect to the appearance ofψ+andψ−. As∂2 ˙2∼∂z∂ , it cannot give rise to new operators and must be removed from the basis. We then end up with four independent operators with one derivative:
∂1 ˙1ψ+, ∂1 ˙1ψ−, ∂2 ˙1ψ+ and ∂1 ˙2ψ−.
3.60 Since there are also operators with more than just one derivative, it might seem necessary to repeat the analysis presented above ad infinitum. However, as the four operators in
3.60transform according to irreducible representations of the collinear subgroup, they can take the place ofψ+ andψ− in
3.55-
3.59 if one adjusts the expressions for the changed conformal spin. This allows us, for an arbitrary number of derivatives, to find a set of fields which belong to a conformal spin-j representation of theSL(2,R) group
ψ(j,m)+ (z) =[(∂2 ˙1)2j−2(∂1 ˙1)2mψ+](z),
ψ(j,m)− (z) =[(∂1 ˙2)2j−1(∂1 ˙1)2mψ−](z)
3.61 by removing all unwanted combinations of derivatives with help of
3.55. All previous observations do not take the (self) interacting gauge fields of QCD into account. In order to move to a true gauge theory, the following adjustments are necessary:
• All derivatives must be replaced by covariant ones
∂→D=∂−igA .
Thus, the conformal transformation properties of the gauge field A are needed. The commutation of two covariant derivatives gives an additional field, a field strength tensor. However, we can drop terms proportional to commutators and treat the derivatives as commuting ones, since we are only interested in a one-particle basis.
• The (anti-)self-dual gluon fields ¯f andf have to be included
• Each light-ray field must be connected to a gauge link Φ(z)→[0, z]Φ(z),
where
[0, z] = Pexp
−1 2igz
Z 1 0
du A2 ˙2(uz)
3.62 is the path-ordered exponent along the light-ray in the appropriate rep-resentation of the gauge group (adjoint for the gluon fields, fundamental for quark fields).
CHAPTER 3. TECHNICAL BACKGROUND
While the calculations are more extensive, everything turns out to work analo-gously. Therefore, we just quote the final results in the next section.