The Lagrangian of QCD

The theory of strong interactions is based on the Lagrangian density

LQCD =Lclassical+Lgauge−f ixing+Lghost . (2.1) The “classical” Lagrangian is given by

Lclassical =−1

4F^{a, µν}F_µν^a +

Nf

X

f=1

ψ¯_f,i(iγ_µD^µ_ij−m_fδ_ij)ψ_f,j . (2.2)

2.1 The Lagrangian of QCD 9

Here and in the following summation over repeated indices is implicitly understood.

Lclassical describes the interaction of the gauge bosons of the theory, the massless spin-1 gluons, with the fermionic quark fields ψ_f,i of flavor f, mass m_f, and color i. All fields depend on the four-dimensional space-time,x, but we do not indicate the argument explic-itly. Since the quarks belong to the fundamental representation of an SU(N = 3) gauge theory, iruns from 1 toN = 3. In QCD,N denotes the number of colors. We adopt the convention of Bjorken and Drell [39] withg^µν = diag(1,−1,−1,−1) and set c=~= 1.

In four dimensions, the Dirac matricesγ_µ satisfy the anti-commutation relation

{γ_µ, γ_ν}= 2g_µν. (2.3)

We will often use the symbolic notation a/=a_µγ^µ. The D_ij^µ in Eq. (2.2) is the covariant derivative,

D_ij^µ =∂^µδij −igsT_ij^aA^a,µ, (2.4) with the strong coupling gs. The gluonic fields A^µa have color indicesarunning from 1 to (N²−1) = 8. The T^a are the generators of the gauge group and obey

[T^a, T^b] =if^abcT^c . (2.5)

The f^abc are the structure constants characterizing the algebra of the group. F_µν^a in Eq. (2.2) is the field strength tensor built up from the gauge fields A^a_µ,

F_µν^a =∂_µA^a_ν−∂_νA^a_µ+g_sf^abcA^b_µA^c_ν . (2.6) The striking difference between QCD and an Abelian gauge field theory such as Quan-tum Electrodynamics (QED) resides in the non-Abelian term of the field strength tensor, g_sf^abcA^b_µA^c_ν. It accounts for self-interactions amongst the “color-charged” gluons, in con-trast to the neutral gauge bosons of QED, the photons, which couple only to the electrically charged fermions of the theory.

The quantization of QCD requires an additional gauge-fixing condition for the gluon fields. For our purposes it is useful to adopt a manifestly covariant gauge and choose

∂^µA^a_µ = 0. Implementing this condition in the QCD Lagrangian yields an additional term,

Lgauge−f ixing =− 1

2η(∂^µA^a_µ)² . (2.7)

Since all physical observables which can be derived fromLQCD must be independent of the choice of gauge, η can in principle take any arbitrary value. The choiceη = 1 (Feynman gauge) is frequently used and will be adopted throughout this work. In covariant gauges, Lgauge−f ixing must be supplemented by a ghost Lagrangian [40],

Lghost= (∂^µχ^a?)D^ab_µχ^b, (2.8)

with the scalar, anti-commuting Faddeev-Popov ghost fields χ^a and the covariant deriva-tive in the adjoint representation,

D_µ^ab=∂_µδ^ab−g_sf^abcA^c_µ. (2.9)

The ghost term in the Lagrangian is necessary to remove unphysical polarization degrees of freedom in the gluon fields, emerging in covariant gauges. Alternatively, the condition (2.7) could be replaced by a non-covariant gauge, e.g., an axial gauge withA^a₃ = 0, which a priori excludes unphysical gluon polarizations and therefore does not require the introduction of ghost fields.

In a covariant gauge the complete Lagrangian of QCD then takes the form LQCD = −1

4F^{a, µν}F_µν^a +

Nf

X

f=1

ψ¯_f,i(iγ_µD^µ_ij−m_fδ_ij)ψ_f,j

− 1

2η(∂^µA^a_µ)²+ (∂^µχ^a?)D^ab_µχ^b , (2.10) which is invariant under local gauge transformations. The actual calculation of physical observables in the framework of QCD is cumbersome due to the non-Abelian character of LQCD. Depending on the energy-range relevant for the calculation different methods have been proposed to cope with the complexity of the strong interaction.

In the low-energy regime, on the one hand numerical methods are used, which rely on a discretization of the continuous four-dimensional space-time on a lattice. This approach turned out to successfully describe aspects of hadronic structure, such as the baryon mass spectrum or hadronic corrections to weak matrix elements, and the strong running cou-pling [41], and has the advantage of being conceptually consistent. However, it suffers from a high numerical intricateness, requiring very time-consuming computations. Above these technical difficulties lattice calculations are so far restricted to rather small volumes in space-time, and the extrapolation from the lattice to the continuum is not unproblematic.

Another method applied at low energies is chiral perturbation theory [42], constructed as an expansion in the momenta and masses of the physical particles, which are considered to be small. The parameters of this effective field theory have to be determined from experiment. More heuristic approaches rely on phenomenologically inspired models. A large variety of methods has been proposed, starting from bag models [43] via Goldstone boson exchange mechanisms [44], diquark potentials [45], chiral quark solitons [46], and instanton models [47], up to all kinds of attempts to simply fit data from current ex-periments. Although such models are indispensable for a first qualitative description of measurements, they certainly cannot account for a thorough test of QCD itself and fall behind other methods, if precision calculations are required.

One therefore resorts to yet another approach, perturbative Quantum Chromodynam-ics, which is, however, applicable solely in the high momentum-transfer regime of the strong interaction. The basic observation underlying pQCD is the decrease of the strong running coupling α_s, related to the coupling constantg_s enteringLQCD via α_s =g²_s/4π.

This feature of the strong interaction, referred to as “asymptotic freedom” [1], allows treating hard interactions between quasi-free quarks and gluons at high energies as only small perturbations, which can be accounted for by a series expansion in the coupling constant α_s. This approach has the advantage that the Feynman rules, which are used to calculate physical observables describing the interaction of quarks and gluons, can be de-rived fromLQCD directly without any model assumptions or restrictions on space-time as

2.1 The Lagrangian of QCD 11

in lattice gauge theories. We have listed the Feynman rules of QED and QCD in App. A.

The coefficients of a perturbation series in QCD exhibit factorial growth, i.e., they diverge [48]. It is one of the basic assumptions of pQCD that such an expansion, despite being divergent, is asymptotic [49]. The perturbative description of a physical, i.e., ex-perimentally observable quantity Γ by a seriesP_∞

n=0αⁿ_sΓ⁽ⁿ⁾ in the limitαs→0 does not necessarily uniquely define Γ, even if summed to all orders. However, if

¯¯

for all positive integer N, the series is said to be asymptotic to Γ and may reproduce the observable to a good approximation, even though the coefficientsC_N do not converge. The divergent behavior of the perturbation series often indicates non-perturbative effects [3]

and will not concern us in this thesis, as it has been shown in a multitude of reactions that perturbation theory indeed works well in matters of practical relevance.

The qualitative behavior of a physical observable can often be estimated by a leading or-der (LO) analysis where the perturbative expansion is truncated at the first non-vanishing order in α_s. A reliable quantitative understanding, however, requires the inclusion of higher order corrections although the computation of such contributions can be cumber-some in practice. Higher order corrections to an observable can be large, as was found, e.g., for prompt photon production [50]. Above that, theoretical predictions exhibit a de-pendence on unphysical scales, if the perturbation series is truncated at some finite order in αs. For instance, a physical quantity Γ which is a priori independent of the arbitrary scaleµ,

acquires in an N-th order perturbative calculation via renormalization and factorization procedures a residual scale dependence of orderα^N_s ⁺¹:

µ d

Taking into account as many orders as possible in the perturbative expansion therefore reduces the artificial scale dependence and thus the uncertainty of a theoretical prediction to a minimum amount, provided pQCD is applicable. The reduction of scale dependence when extending a perturbative calculation to higher orders is then a “measure” for the reliability of the perturbative expansion.

The calculation of higher order QCD contributions is furthermore called for by the ongoing hunt for signatures of “new physics”, i.e., physics beyond the Standard Model. The large systematic uncertainties of a lowest order pQCD analysis do not allow to disentangle presumably tiny effects of so far unobserved mechanisms from the omni-present QCD background. Only a thorough understanding of QCD opens up ways to a search for such phenomena.