The spinor productµλ= 1 has not been carried out, as this guarantees that the number of spinors on the right- and left-hand side of the equation is the same.
All three DAs arise as matrix elements of non-quasipartonic operators.
The remaining three four-particle distribution amplitudes have not been studied before and were defined in our work [62] for the first time. Each involves an additional gluon field compared to
4.8: h0|igǫijkψ+u,i(z1) ¯χu,j+ (z2)[ ¯f++(z4)ψd+(z3)]k|Pi=
=1
4mN(pn)2N+↑ Z
Dx e−i(pn)PxiziΦg4(x), h0|igǫijkχ¯u,i+ (z1) [ ¯f++(z4)ψ+u(z2)]jψ+d,k(z3)|Pi=
=1
4mN(pn)2N+↑ Z
Dx e−i(pn)PxiziΨg4(x), h0|igǫijk[ ¯f++(z4)ψ+u(z1)]iψu,j+ (z2)ψd,k+ (z3)|Pi
=1
4mN(pn)2N+↓ Z
Dx e−i(pn)PxiziΞg4(x).
4.9 Note that the integration measure now has to ensure that the sum of the mo-mentum fractions of all four partons is equal to one, hence
Z Dx=
Z 1 0
dx1dx2dx3dx4δ(1− X4 i=1
xi).
4.10 If even higher twist DAs are considered the number of independent amplitudes grows dramatically. Already at twist five the three known three-quark DAs [65]
will be complemented by matrix elements involving the Fock states: |qqqFi,
|qqqF Fi and |qqq¯qqi. At this level the mere classification of all independent distribution amplitudes would be a non-trivial task.
4.2 The Complete Twist-4 Operator Basis
As we have seen in the previous section, there are six twist-4 nucleon distribution amplitudes. Two of them, Ξ4 and Ξg4, involve only chiral quark fields, whereas Φ4, Ψ4, Φg4 and Ψg4feature of both chiral and antichiral quarks. It is well known that chirality is preserved in QCD perturbation theory [56]; therefore, the two sets of distribution amplitudes cannot mix under renormalization and we can treat them separately right from the start. In the following we construct an operator basis for each case; pure chiral and mixed chirality operators that is.
The operators are built from the good one-particle light-ray operators found in Eq.
3.63.
4.2.1 Chiral Operators
The two chiral nucleon distribution amplitudes Ξ4 and Ξg4 have the quantum numbers E = 4 and H = +1/2. All operators of the basis are, therefore,
CHAPTER 4. BARYON DISTRIBUTION AMPLITUDES
required to share these quantum numbers, otherwise operator mixing is not possible.
The chiral three-quark distribution amplitude Ξ4 is related to matrix ele-ments of the operators
Q1(z1, z2, z3) =ǫijkψ−a,i(z1)ψ+b,j(z2)ψ+c,k(z3), Q2(z1, z2, z3) =ǫijkψ+a,i(z1)ψ−b,j(z2)ψ+c,k(z3),
Q3(z1, z2, z3) =ǫijkψ+a,i(z1)ψ+b,j(z2)ψ−c,k(z3).
4.11 i, j, k are color and a, b, c are flavor indices1. For the nucleon one would have to set two flavor indices to up and one to down type flavor. It is, however, convenient to consider the general case, as the nucleon flavor structure can always be restored. The corresponding chiral quasipartonic operators are
G1(z1, z2, z3, z4) =igǫijk(µλ) [ ¯f++(z4)ψa+(z1)]iψ+b,j(z2)ψc,k+ (z3), G2(z1, z2, z3, z4) =igǫijk(µλ)ψa,i+ (z1) [ ¯f++(z4)ψb+(z2)]jψc,k+ (z3), G3(z1, z2, z3, z4) =igǫijk(µλ)ψa,i+ (z1)ψ+b,j(z2) [ ¯f++(z4)ψc+(z3)]k,
4.12 where the factorµλis useful for translating the expressions back to the normal Dirac notation sinceF+,µλ¯=−(µλ) ¯f++.
It turns out that the three operators in
4.12are not independent because G1(z1, z2, z3, z4) +G2(z1, z2, z3, z4) +G3(z1, z2, z3, z4) = 0.
4.13
This identity is a direct consequence of gauge invariance. Consider the operator O:=ǫijk(µλ)ψa,i+ (z1)ψb,j+ (z2)ψc,k+ (z3).
Performing an infinitesimal global gauge transformation
ψ+h →[eigTAǫψ+]h=ψ+h+igTA,hlψl+·ǫ+O(ǫ2), one gets
O→O+ig
TA,ilψl+(z1)ψj+(z2)ψk+(z3) +ψi+(z1)TA,jlψ+l(z2)ψl+(z3) +ψ+i(z1)ψ+j(z2)TA,klψ+l(z3)
·ǫ+O(ǫ2).
4.14 Multiplying the sum in the brackets with ¯f++A gives the left-hand side of
4.13. However, Ois gauge invariant; any term in Eq.
4.14proportional to ǫ must vanish identically, which proves
4.13.
1For simplicity we usually do not display the flavor indices explicitly and assume that the first quark carries flavora, the second flavorband so on.
40
4.2. THE COMPLETE TWIST-4 OPERATOR BASIS
4.2.2 Operators of Mixed Chirality
Analogously, one can define the operator basis for the mixed chirality operators.
The distribution amplitudes Φ4, Ψ4, Φg4 and Ψg4also have collinear twistE= 4, but helicityH =−1/2 as opposed to +1/2 for the chiral amplitudes. Again one can find three independent operators matching these quantum numbers:
Q1(z1, z2, z3) =ǫijkψa,i− (z1)ψb,j+ (z2) ¯χc,k+ (z3), Q2(z1, z2, z3) =ǫijkψa,i+ (z1)ψb,j− (z2) ¯χc,k+ (z3), Q3(z1, z2, z3) =1
2ǫijkψa,i+ (z1)ψb,j+ (z2) [ ¯χ3/2+ ]c,k(z3),
4.15 where ¯χ3/2+ ≡χ¯(3/2,0)+ =−(µDλ) ¯¯ χ+≡ −Dµ¯λχ¯+, cp. Eq.
3.63. Note that the naive choice for the third operator
Qˆ3(z1, z2, z3) =ǫijkψ+a,i(z1)ψ+b,j(z2) ¯χc,k− (z3)
has the wrong helicity (H = 3/2). This can be read off Table 3.1, since the helicity of ˆQ3is the sum of the helicities of the involved one-particle operators.
For the four particle case there again exist three operators
G1(z1, z2, z3, z4) =igǫijk(µλ) [ ¯f++(z4)ψ+a(z1)]iψ+b,j(z2) ¯χc,k+ (z3), G2(z1, z2, z3, z4) =igǫijk(µλ)ψ+a,i(z1) [ ¯f++(z4)ψ+b(z2)]jχ¯c,k+ (z3), G3(z1, z2, z3, z4) =igǫijk(µλ)ψ+a,i(z1)ψb,j+ (z2) [ ¯f++(z4) ¯χc+(z3)]k.
4.16 The same argument as before guarantees the identity
G1(z1, z2, z3, z4) +G2(z1, z2, z3, z4) +G3(z1, z2, z3, z4) = 0,
4.17 so that the operatorsGi are not independent. Therefore, there are only two independent chiral distribution amplitudes with one gluon field, Ψg4 and Φg4, instead of three.
4.2.3 Nucleon Matrix Elements
The matrix elements of the operators between vacuum and nucleon state exhibit additional symmetries. The nucleon has isospin 1/2 and this property is reflected in the matrix elements. Furthermore, the identity of quark flavors, twouquarks in case of the proton and twodquarks for the neutron, generates an additional symmetry.
The matrix elements of the chiral operatorsQiandGi between vacuum and proton can be defined, see also [62], as
φi(z1, z2, z3) =h0|Qi(z1, z2, z3)|Pi,
φgi(z1, z2, z3, z4) =h0|Gi(z1, z2, z3, z4)|Pi,
4.18
CHAPTER 4. BARYON DISTRIBUTION AMPLITUDES
where the first two quarks are of u-type flavor, that is a = u and b = u in Eqs.
4.11and
4.12. The identity of theuquarks leads to
φ1(z1, z2, z3) =φ2(z2, z1, z3),
4.19 whereas the isospin condition enforces the relation2
φ3(z2, z3, z1) =−φ1(z1, z2, z3)−φ1(z1, z3, z2).
4.20 For the four particle operators one obtains:
φg2(z1, z2, z3, z4) =φg1(z2, z1, z3, z4),
φg3(z2, z3, z1, z4) =−φg1(z1, z2, z3, z4)−φg1(z1, z3, z2, z4).
4.21 The matrix elements of the mixed chirality operators feature similar rela-tions, if the quarks of same chirality also have the same flavor. However, the two distribution amplitudes Φ4and Ψ4are directly related to the operatorsQ1 andQ2 with flavorsa=c=uandb=d; theuquarks have different chirality.
The matrix element corresponding to the operatorQ3, 1
2h0|ǫijkψu,i+ (z1)[ ¯χ3/2+ ]u,j(z2)ψ+d,k(z3)|Pi=
= i
4(µλ)(pn)mNN+↑ Z
Dx e−i(pn)PxiziD4(x),
4.22 did not appear in the set of distributions amplitudes
4.8. The reason for this is that the distribution amplitudeD4 is not independent and can be expressed in terms of matrix elements of other twist-4 operators. To show this, let us define the matrix elements of mixed chirality as
ϕk(z1, z2, z3) =h0|Qk(z1, z2, z3)|Pi,
ϕgk(z1, z2, z3, z4) =h0|Gk(z1, z2, z3, z4)|Pi.
4.23
One can show that the relation3 ϕ3(z1, z2, z3) = ∂
∂z1
ϕ1(z1, z2, z3) + ∂
∂z2
ϕ2(z1, z2, z3)
−1 2
Z 1 0
dτ
z13ϕg1(z1, z2, z3, z13τ ) +z23ϕg2(z1, z2, z3, z23τ )
4.24
holds. Here we used the notation
zik =zi−zk, τ¯= 1−τ , zτik= ¯τ zi+τ zk.
4.25 The proof of
4.24is straightforward but lengthy. It can be found in App. B.2.
2Isospin relations for nucleon matrix elements were studied in great detail in [104].
3Note that in [62] an incorrect momentum space representation of this relation is presented.
42