4.5 Anomalous Dimensions
4.5.1 Spectrum of Anomalous Dimensions I: Chiral Case
4.33.
4.5.1 Spectrum of Anomalous Dimensions I: Chiral Case
Additional Symmetries in the Chiral Sector
The chiral basis, at first glance, seems to be more “symmetric” than its mixed chirality counterpart. To put this, somewhat purely esthetical statement, on more solid ground, let us introduce the generator of cyclic permutations,
P =Pa⊗Pz,
4.124 wherePapermutes the quantum numbers of an operator andPzthe coordinates, e.g.
PaΨiN,q(z1, z2, z3) =Ψi+1N,q(z1, z2, z3)
4.125 PzΨiN,q(z1, z2, z3) =ΨiN,q(z3, z1, z2).
4.126 The chiral Hamiltonian Hψψψq commutes with the generatorP and we can, in principle, find simultaneous eigenfunctions of both operators. Since obviously
P3=1,
4.127 the eigenvaluesεofP are the third roots of 1, i.e. ε∈ {1, ei23π, e−i23π}. Thus, the eigenfunctions10ofHψψψq can be chosen to have a definite parity with respect toP:
PΨ(ε)N,q=εΨ(ε)N,q.
4.128 Each eigenfunction Ψ(ε)N,q then depends only on a single (scalar) functionψN,q(ε)
ΨεN,q(z1, z2, z3) =
ε0 ψεN,q(z1, z2, z3) ε1 ψεN,q(z2, z3, z1) ε2 ψεN,q(z3, z1, z2)
.
4.129
This is obvious, sinceP simultaneously permutes the coordinates as well as the rows of the vector; each eigenfunction has to be of the form given in
4.129. The chiral quark Hamiltonian features an additional symmetry. It commutes with the operatorP12 permuting the first two vector entries and the first two
10Recall thatHψψψq is a 3×3 matrix, its eigenfunctions are three dimensional vectors.
70
4.5. ANOMALOUS DIMENSIONS
coordinates. This can be checked by direct calculation. As
ε0 ψN,qε (z1, z2, z3) ε1 ψN,qε (z2, z3, z1) ε2 ψN,qε (z3, z1, z2)
P→12
ε1 ψεN,q(z3, z2, z1) ε0 ψεN,q(z2, z1, z3) ε2 ψεN,q(z1, z3, z2)
→P
ε0 ψN,qε (z3, z2, z1) ε2 ψN,qε (z2, z1, z3) ε1 ψN,qε (z1, z3, z2)
,
4.130 [P12ΨεN,q] is an eigenfunction ofP to the eigenvalueε2. It then follows due to
[Hψψψq ,P12] = 0 and ei23π2
=e−i23π
that the spectra of anomalous dimensions for ε=e−i23π andε =ei23π are the same.
For the mixed chirality HamiltonianHψψq χ¯ no such permutation symmetry can be found and the spectrum cannot be decomposed into different sectors.
Integrability
It has been known for some ten years that the chiral twist-3 Hamiltonian pos-sesses an integral of motion [61]. That is, there exists an operator Q that commutes with the Hamiltonian and its eigenvalues are conserved charges of the system. The Hamiltonian corresponds to a one-dimensional three-body problem and the total conformal spin as well as its projection on the light-cone already are good quantum numbers [61]. The existence of the third conserved chargeQthen implies that the system is completely integrable.
It turns out that the quark part of the chiral twist-4 HamiltonianHψψψalso possesses such a hidden integral of motion. To find this charge some amount of sophisticated guessing is necessary.
Let us introduce an operator [62]
Sik =∂k(zk−zi)≡(∂/∂zk)(zk−zi).
4.131 It can be checked that this operator connects theSL(2,R) representationsTj1⊗ Tj2 andTj2⊗Tj1:
SikTjk=1/2⊗Tji=1=Tjk=1⊗Tji=1/2Sik.
4.132 It is referred to as intertwining operator [62]. The conformal two-particle Casimir operatorJ12 for conformal spins (1,1) [61] and (1/2,1) then takes the form
J122 =S21(S12+ 1) and J122 =S12S21+1
4 ,
4.133 respectively. The HamiltonianHψψψq is a three by three matrix and so must be the conserved charge. The matrix Casimir operator can be defined as follows
CHAPTER 4. BARYON DISTRIBUTION AMPLITUDES
Figure 4.2: The spectrum of the conserved charge for twist-3 and twist-4 chiral quark operators. The figures are taken from [62]
[86] whereJik depends on the conformal spins of the representation.
To write the operators ˆJik2 in a compact way, one can introduce 3×3 matrices Q+ik andQ−ik,i < k, which are defined by forjdifferent fromiandk, with all other matrix elements equal to zero. Then Jˆik2 can be written as The conserved charge should be constructed from the operators ˆJik2 and indeed one finds that
4.5. ANOMALOUS DIMENSIONS
commutes withHψψψq :
[Qb3,Hψψψq ] = 0.
4.140 This can be shown by calculating the commutator in the conformal basis, cp. [61, 72]. Due to Eq.
4.140we can label any eigenfunction Ψ of Hψψψq by its conserved chargeq,Qb3Ψ =qΨ. Further, one can show that
[Qb3,P] ={Qb3,P12}= 0.
4.141 Therefore, eigenstates with q 6= 0 must be degenerate, as the eigenfunctions with chargeqand−qcorrespond to the same eigenvalue. The spectrum for the operatorQb3 can be found in the right panel of Fig. 4.2. The left panel shows the spectrum of the twist-3 conserved charge for comparison [61].
The Chiral Quark Spectrum
It is known that an operator with a total derivative has the same anomalous dimensions as the corresponding operator without the total derivative. So, as far as the spectrum is concerned, these operators can be omitted. To do this, the coefficient functions ΨN,q of these operators have to identified in the expansion of the non-local operators
O(~z) =X
N,q
ΨN,qON,q over the complete set of local operatorsON,q.
This is possible using the following observation: Let us consider an expansion of a fictitious non-local operator
Ofree(z1, z2, z3) =X
N,q
ΨfreeN,qOf reeN,q ,
that does not involve any operators with total derivatives. The action of the generator of translation along the light-ray P2 ˙2 = P++ on Ofree(z1, z2, z3) is given by
i
P2 ˙2,X
N,q
ΨfreeN,qOf reeN,q
=iX
N,q
ΨfreeN,qh
P2 ˙2,Of reeN,q
i
=iX
N,q
h
P2 ˙2,ΨfreeN,qi Of reeN,q ,
4.142 where the boldface generator acts on quantum fields and the generator in normal font acts on the coordinates, cp.
3.35. Inserting the explicit expression for P++, one obtains
X
N,q
h(∂1+∂2+∂3)ΨfreeN,q(z1, z2, z3)i
ON,q=X
N,q
ΨN,q(z1, z2, z3) [∂+ON,qfree].
4.143
CHAPTER 4. BARYON DISTRIBUTION AMPLITUDES
The left-hand side contains by definition only operators free of total derivatives, whereas the right-hand side is a sum over operators which explicitly contain at least one total derivative. So both sides must vanish identically, otherwise they cannot be equal to each other. Since the operators on the l.h.s. are indepen-dent, each coefficient function must be equal to zero. This provides us with a criterion to single out coefficient functions corresponding to operators without total derivatives, sometimes referred to as conformal operators:
(∂1+∂2+∂3)ΨN,q(~z) = 0.
4.144 It is possible to derive a second constraint by considering the generator of translations perpendicular to the light-cone P1 ˙2. As it moves the fields away from the light-ray, this generator can – just as the generator of the Lorentz rotationM21considered in Sect. 4.4 – increase the twist of an operator. In fact
i[P1 ˙2, ψ+] (z) =−2∂zψ−. Thus, applyingP1 ˙2 to the chiral twist-3 operator
Otw−3(~z) =ǫijkψ+a,i(z1)ψb,j+ (z2)ψ+c,k(z3) leads to
i[P1 ˙2,Otw−3(~z)] =−2 X3 k=1
∂
∂zk
Qk(~z) =−2X
N,q
X3 k=1
∂
∂zk
Ψ(k)N,q(~z)
! QN,q,
4.145 whereQk(~z) is defined in Eq.
4.11. The l.h.s. only contains operators with a total derivative, whereas the r.h.s. involves both conformal and non-conformal operators. For the equation to be fulfilled, the coefficients in front of conformal operators must vanish:
∂1Ψ(1)N,q(~z) +∂2Ψ(2)N,q(~z) +∂3Ψ(3)N,q(~z) = 0.
4.146 The set of shift-invariant homogeneous polynomials
eN,k(z1, z2, z3) = (z1−z2)k(z1−z3)N−k k!(N−k)!
4.147 automatically fulfills the condition
4.143and it is possible to calculate the HamiltonianHψψψq in this basis, i.e.
Hψψψ
q eN,k= XN k′=0
(Hψψψ
q )k′keN,k′.
4.148 The resulting (N+ 1)×(N + 1) matrix11 Hψψψ
q can be diagonalized numeri-cally for each choice of ε [62]. This “brute-force” ansatz is the actually most
11Originally we had to deal with a 3(N+1)×3(N+1) matrix, but the permutation symmetry reduced the size of the matrix by a factor three, cf.
4.129.
74
4.5. ANOMALOUS DIMENSIONS
N EN,0 EN,1 EN,2 EN,3
0 −2∗ - -
-1 −23
4 3
∗ -
-2 43∗ 4 -
-3 2 29−6√57∗ 29+6√57∗ -4 167−303√481∗ 173 167+330√481∗
-5 349 7715∗ 679∗ 13715
6 3.633418∗ 311/60 6.687457∗ 7.724361∗
Table 4.1: Anomalous dimensions of local twist-4 chiral-quark operators withN co-variant derivatives in units ofαs/(2π). The entries marked with an asterisk correspond to the operators withP-parityε=e±i2π/3and the remaining ones toε= 1. The table is taken from [62].
effective option, since analytic methods such as the algebraic Bethe-Ansatz are too sophisticated for such a simple problem.
The final result for the spectrum of the chiral quark Hamiltonian for the eigenvalues withN <7 is presented in Table 4.1.
All eigenvalues except the lowest one for each N are doubly degenerate.
Therefore, the lowest eigenvalue must correspond to a conserved chargeQb3= 0.
The eigenvalue spectum for a larger range in N is displayed in Fig. 4.3. The upper panels show the chiral twist-4 and, for comparison, the chiral twist-3 spectrum. The smoothness of the spectra is a manifestation of the integrability of the chiral kernels. The lower two panels represent the different sectors with ε= 1 andε=ei2π3 .
The Chiral Quark-Gluon Spectrum The chiral four-particle HamiltonianHψψψf¯
g exhibits the same permutation sym-metries as the three-particle Hamiltonian, and the eigenfunctions can be classi-fied in the same way, that is by their eigenvaluesε = 1, e±i2π3 with respect to P. An additional constraint comes from Eq.
4.13, namely X3
i=1
Ψ(i)N,q(z1, z2, z3, z4) = 0.
4.149
The shift invariant polynomials
eN,k,m(z1, z2, z3, z4) = (z1−z4)k(z2−z4)m(z3−z4)N−2−k−m k!m! (N−2−k−m)!
4.150
CHAPTER 4. BARYON DISTRIBUTION AMPLITUDES
Figure 4.3: Upper panels: spectra of the chiral Hamiltonians for twist 3 and twist 4. The lower panels show the two sectors with ε = 1 and ε = ei2π3 separately. The figures are taken from [62].
76
4.5. ANOMALOUS DIMENSIONS
Figure 4.4: Spectra of the chiral twist-4 four-particle (open circles) and three-particle (crosses) Hamiltonians. The left panel shows the full spectrum, whereas the right panel depicts only theε =e±i2π3 sector. The figures are taken from [62].
again provide us with a suitable basis for evaluating the Hamiltonian. Note that the degree of the polynomials in
4.150is N−2. This choice is useful for the calculation of the multiplicatively renormalizable operators, as operators of different dimension do not mix and gluon operators have canonical dimension lqqqcanf¯=lcanqqq+ 2.
CalculatingHψψψf¯in the basis
4.150, the resultingN(N−1)×N(N−1) matrix can be diagonalized numerically. The result is shown in Fig. 4.4. In the left panel the spectrum of anomalous dimensions for the chiral four-particle operator is indicated by the open circles. The crosses denote the twist-4 three-particle spectrum for comparison. The right panel corresponds to theε=e±i2π3 sector alone. The two spectra start to overlap forN >7 and the operator mixing between three and four particle operators is expected to become very strong at this point. For N = 2 and N = 3 the gap between the two spectra suggests a rather weak mixing, which supports the claim that the four-particle Fock states do not play a prominent role in actual applications. We can understand this qualitative claim in terms of our Schr¨odinger equation-like renormalization group equation picture. The anomalous dimensions depend only on the diagonal blocks, Hq and Hg, but are independent of the off-diagonal block Hqg, which determines the mixing of three- and four-particle operators. One can split the Hamiltonian in two pieces:
H→ Hq 0 0 Hg
!
+ 0 Hgq
0 0
!
.
4.151 The first summand gives the “energy eigenvalues” (anomalous dimensions) of
CHAPTER 4. BARYON DISTRIBUTION AMPLITUDES
N EN,0 EN,1 EN,2 EN,3 EN,4
0 −2 - - -
-1 2/9 2 - -
-2 2(14−
√43)
9 32/9 2(14+
√43)
9 -
-3 197−45√5089 49−9√73 49+9√73 197+45√5089 -4 3.706620 589−√9011161 6.634936 589+√9011161 7.858442
Table 4.2: Anomalous dimensions of twist-4 quark operators of mixed chirality in units of αs/(2π); N is the total number of covariant derivatives. The table is taken from [62].
the Hamiltonian, the second term can be viewed as a perturbation. Classical Rayleigh-Schr¨odinger perturbation theory tells us that the correction to a (non-degenerate) state|nidue to a small perturbationV is given by [87]
|nperti=|ni+X
p6=n
hp|V|ni
En−Ep|pi,
4.152 whereEp is an (unperturbed) energy eigenvalue andplabels the different eigen-functions. Therefore, even if Hgq qualifies as small, once the spectra overlap and the distance between the eigenvalues becomes tiny the mixing gets strong nonetheless12.