It is obvious that the lattice action does not possess the full symmetry of the Lorentz group.
What remains is merely its restriction to the symmetry group of the (spinorial) hypercubic lattice. We will exploit this observation in the next chapter to construct a basis of leading-twist three-quark operators that allows to control the mixing under renormalization.
The focus of this section is on an additional symmetry of the massless continuum action with respect to the following global transformations of the quark fields (this discussion closely follows [50]):
ψ→ψ0=eiαTiψ, SU(Nf)V, (3.41)
ψ→ψ0=eiα1ψ, U(1)V, (3.42)
ψ→ψ0=eiαγ5Tiψ, SU(Nf)A, (3.43)
ψ→ψ0=eiαγ51ψ, U(1)A. (3.44)
The transformations for ¯ψ are defined by analogous expressions. Here the quarks ψ are interpreted as vectors in flavor space with Ti denoting the generators of the flavor group SU(Nf). The implications of these symmetries become evident when introducing right- and left-handed quark fields by the projection
ψR/L= (1±γ5)ψ. (3.45)
Then the vector transformations SU(Nf)V and U(1)V leave the handedness of the quarks unchanged, while the axial transformations SU(Nf)A and U(1)A mix right- and left-handed quarks with each other. Moreover the U(1) transformations leave the flavor unchanged, whereas the SU(Nf) transformations express a symmetry between the different quark flavors.
The axial anomaly leads to a breaking of U(1)Aalready in the massless quantized theory.
Introducing degenerate quark masses breaks both U(1)A and SU(Nf)A even on the level of the action. Finally, since SU(Nf)V relates to a symmetry between the different flavors, this invariance can be broken by lifting the degeneracy of the quark masses. This leaves quantum chromodynamics only with Nf factors of the vector symmetry U(1)V.
Let us concentrate on the axial symmetry. Since it is only present in the massless action, it is also referred to as chiral symmetry. The invariance of the massless action under chiral
3.9. CHIRAL SYMMETRY BREAKING AND CHIRAL ACTIONS 49 transformations directly relates to the fact that the massless Dirac operator, D = M|m=0, anti-commutes withγ5:
Dγ5+γ5D= 0. (3.46)
Generally speaking, the axial symmetry is broken both explicitly and spontaneously in the continuum theory. Introducing quark masses in the action (which are believed to be generated in turn by a spontaneous symmetry breaking of the Higgs field) breaks the chiral symmetry explicitly. However, due to the smallness of the up and down quark masses, this explicit breaking of the chiral symmetry is not sufficient to explain phenomenological observations like the large mass difference between the nucleonN and its parity partner N∗. This puzzle can be solved by introducing a spontaneous breaking of the chiral symmetry, which manifests itself in a non-vanishing expectation value of the so-called chiral condensate:
hψψi 6= 0.¯ (3.47)
Associated to the spontaneous breaking of the (approximate) axial symmetry of the up and down quarks are three (pseudo-) Goldstone bosons, the (almost) massless pions in the meson octet of Figure 1.3. The small but finite value of their mass originates from the explicit breaking of the chiral symmetry due to the non-zero quark masses in the action.
Given its large phenomenological implication, it would be favorable to have chiral sym-metry also realized for the lattice action. However, it turns out that the massless Wilson action explicitly breaks chiral symmetry at finite lattice spacingsa. This is due the Wilson term, which was introduced in eq. (3.15) in order to remove the doublers and which leads to a violation of the condition in eq. (3.46).
The Nielsen-Ninomiya no-go-theorem formulates this statement in a more general way [71, 72].
It states that it is impossible to have a chirally invariant, doubler-free lattice action whose Hamiltonian is
• local in the sense that its Fourier transform is a smooth function,
• translational invariant on the lattice by an integer number of lattice constants,
• hermitian N ×N matrix.
A way out of this dilemma is to construct improved actions that do not fulfill eq. (3.46), but rather a lattice version of it, namely the so-called Ginsparg-Wilson relation [73],
Dγ5+γ5D=a Dγ5D. (3.48)
Any lattice Dirac operator that satisfies this relation, restores the invariance under the chiral transformation
ψ→ψ0 =eiαγ5(1−a2D)ψ, (3.49) with an analogous expression for ¯ψ. This facilitates lattice studies of observables that do crucially depend on the chiral properties of quantum chromodynamics. Today, several lattice actions are known and used that implement exact chiral symmetry via the Ginsparg-Wilson relation. These include domain wall fermions [74], when the fifth dimension is taken to infinity, and Neuberger’s overlap operator [75, 76]. However, since simulations with these actions are still rather expensive and chirality is not expected to be of major importance for our approach to the nucleon distribution amplitude, we will stick with the much cheaper orderaimproved Wilson fermions.
Chapter 4
Irreducible Multiplets of Three-Quark Operators
Now, let us turn to the main objective of this thesis. In the phenomenological introduction of Chapter 1 we have investigated the nucleon structure on the basis of inclusive and exclu-sive scattering processes. While the incluexclu-sive process of deep inelastic scattering could be related to inelastic structure functions, exclusive processes gave rise to the elastic form fac-tors of the nucleon. In this context we have outlined the important relation of hard exclusive electron-nucleon scattering to the nucleon wavefunction, which was first worked out in the late 1970s [21, 25]. Upon approximating the nucleon by its leading Fock state and integrating out the transverse degrees of freedom, we ended up with the nucleon distribution amplitude ϕN(x1, x2, x3, Q2). It parametrizes the nucleon in the valence state and defines the ampli-tude to find the three valence quarks of given spin and flavor carrying certain longitudinal momentum fractions xi. The great advantage of the nucleon distribution amplitude is its universality. Because of this process-independence it must be determined only once and can then be used to describe different hard exclusive processes on a quantitative basis.
The nucleon distribution amplitude originates from a factorization of the scattering pro-cess into hard and soft subpropro-cesses. Whereas the hard scattering kernel can be evaluated perturbatively, the distribution amplitude cannot be accessed so easily since it contains all soft contributions. To date only phenomenological models and estimates from sum rule cal-culations were available. As already mentioned, lattice QCD is well suited to provide better insights into this non-perturbative quantity. Here, moments of the nucleon distribution am-plitude, eq. (1.39), can be evaluated from first principles:
ϕlmnN (Q2) = Z
[dx]xl1xm2 xn3ϕN(x1, x2, x3, Q2).
By operator product expansion these moments are related to matrix elements of local three-quark operators O sandwiched between the vacuum and a nucleon state|N(p)i. In order to access the nucleon distribution amplitude these matrix elements are computed on the lattice,
h0|Olmn|N(p)i ∝fNϕlmnN N(p).
On the right-hand side of this equation N(p) denotes the nucleon spinor. We will omit the details of these relations until they are finally needed in Section 7.4. Here it is of central importance to note that the matrix elements computed on the lattice must be renormalized
51
so that the momentsϕlmnN can be used for quantitative predictions of physical processes. To this end the main object of interest are the composite three-quark operators within the matrix elements, which contain three quark fields at a common same space-time coordinate x and have to be renormalized non-perturbatively. Apart from symmetrization to isospin 1/2 and color antisymmetrization, these three-quark operators typically look like
O(x)∝Dλ1. . . Dλlfα(x)·Dµ1. . . Dµmgβ(x)·Dν1. . . Dνnhγ(x). (4.1) Here the three quark fields are denoted by f, g and h with spinor indices α, β and γ, respectively, and the covariant derivatives Dκ act on the adjacent quark field only.
The following sections and chapters of this thesis focus on the non-perturbative renormal-ization of the three-quark operators, since this will provide the key for quantitative statements on the distribution amplitude. In this chapter we will concentrate on the fact that three-quark operators are in general subject to operator mixing under renormalization,
O(i),ren=X
j
ZijO(j),bare. (4.2)
This equation implies that the renormalized three-quark operator is a linear combination of several bare, i.e., regularized but not yet renormalized three-quark operators O(j),bare. Mixing shows up in non-vanishing off-diagonal elements of the renormalization matrixZ. In order to renormalize a matrix element containing the three-quark operator O(i), one must therefore also compute the same matrix element for all mixing operators O(j). Since even without covariant derivatives there exist 43 = 64 independent and hence potentially mixing three-quark operators, compare eq. (4.1), it is of central importance to obtain a detailed understanding of the real mixing pattern before starting any lattice calculation.
In the Euclidean continuum theory mixing between operators is restricted by their trans-formation properties under the symmetry group O4. Operators that transform according to inequivalent irreducible representations of the symmetry group cannot mix under renormal-ization. However, on the lattice this symmetry group is reduced to its discretized counterpart H(4), which means that in general more operators will participate in the mixing process. In the mid-1990s a generic study for quark-antiquark operators was performed by examining their transformation behavior with respect to the symmetry transformations of the lattice [77]. We will adapt this approach in order to deal with the half-integer spin assigned to our three-quark operators. As published in [78, 79], this results in the construction of irreducibly transforming multiplets of three-quark operators with respect to the spinorial hypercubic group H(4). These multiplets allow to control the mixing under renormalization in a well-defined manner.