Renormalization, asymptotic freedom and connement

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Renormalization, asymptotic freedom and con nement

The functional integral formulation Eq. (2.18) has no proper meaning in this form, as the functional measures are not well de ned. This can be overcome by discretizing space-time { the lattice QCD approach, which we describe comprehensively in Sec. 2.2, thereby introducing a nite lattice spacing a, which we shall eventually take to 0. This approach naturally introduces a regularization of the theory, in the sense that it imposes a momentum cut-o at a , and makes momentum integrals nite. Eq. (2.18) is then de ned as the limit as a ! 0 of its discrete counterpart.
Consider now some operator O(a; g(a)) written in terms of the bare elds appearing in the action (2.20). To get its vacuum expectation value using our functional integral formalism, we must compute (2.18) at non-zero a (i.e. nite momentum cut-o ) and then take the continuum limit a ! 0. Following Wilson’s approach [8], we must then de ne a renormalized action and a renormalized operator in order to obtain a smooth limit as a ! 0. These renormalized quantities are de ned to smoothly incorporate and account for contributions coming from momenta higher than a . The renormalized action has the same form as the original one, but with rescaled elds, and parameters which now depend on a. Assuming that O does not mix with other operators in the presence of radiative corrections, the renormalized operator OR is related to the bare one by a multiplicative constant, called the renormalization constant, and should be independent of a: OR Z(a; g) O(a; g). Z is a dimensionless quantity, so that a new scale must be introduced in the de nition of OR to compensate for the dimension of a and form the dimensionless a quantity. has therefore the dimension of an energy, and is called the renormalization scale as it is in practice the scale at which we shall specify renormalization conditions to properly de ne the integral. This arti cially introduces a dependence of OR on the renormalization scale , i.e. OR = OR( ; g( )).
However, if the theory is to give robust predictions, any physical observable involving OR( ; g( )) must be independent of the scale at which we imposed the conditions to completely de ne the Lagrangian. This independence can be expressed through the so-called Callan-Symanzik equation [9] [10].

Symanzik improvement program

The lattice gauge and fermion actions built above are discretized versions of their contin-uum counterparts. As such, they give rise to discretization e ects, which go like powers of the lattice spacing a. More speci cally, using the expansion of gauge links into gauge elds U (n) = 1 + iaA (an) + O(a2) and the de nition of the Wilson term, one can show that the discretization e ects for the Wilson action are of O(a2) for the gauge part and of O(a) for the fermion part, i.e.
SWilson;G[U ] = SG[A] + O(a2) (2.42).
SWilson;F[q; q; U ] = SF[q; qA] + O(a) (2.43).
Practical calculations are performed at nite a, and taking the continuum limit with the raw Wilson action often proves to be a costly task as one needs to go to very small lattice spacings in order to control the extrapolation. However, the discretization errors can be reduced by adding to the Wilson action irrel-evant terms which vanish in the continuum and compensate the leading nite a e ects. A way to systematically cancel discretization e ects in lattice theories was proposed by Symanzik [28, 29] and is known as the Symanzik improvement program. The idea is to consider the lattice action as an e ective action Slat which matches the continuum one as a ! 0, i.e.

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Gauge-link smearing

The study of hadronic correlation functions physically involves the long-range behaviour of QCD, but violent UV uctuations can hinder our calculations, and give rise to strong discretization e ects. As mentioned above, the Symanzik improvement program provides a well-de ned approach to control these nite a contaminations, but the determination of the improvement coe cients is di cult, and the iteration of the procedure to higher orders in a turns out to be unworkable. Therefore, we use the low order tree-level improvement described in Sec. 2.2.2, and combine it with a smearing of the gauge links coupled to the fermions 2. The smearing consists in some kind of averaging of the links over some neighbourhood, and helps smoothing strong local uctuations of the gauge con guration. This is equivalent to a rede nition of the coupling between quarks and gluons with a di erent discretization of the action. Several smearing procedures have been devised, the most popular being studied in [33]. These are covariant procedures, which do not require xing the gauge, but need projection of the resulting averaged fat link onto the gauge group SU(3) as it is not stable under matrix addition. The smearing methods can be iterated, but keeping in mind that every iteration increases the averaging range and hence a ects longer-range behaviours.

Table of contents :

Aknowledgements
1. Introduction
2. QCD on the lattice 
2.1. QCD basics
2.1.1. Quarks, gluons and chromodynamics
2.1.2. Quantization
2.1.3. Renormalization, asymptotic freedom and connement
2.1.4. Chiral symmetry
2.2. QCD on the lattice
2.2.1. QCD lattice action
2.2.2. Symanzik improvement program
2.2.3. Gauge-link smearing
2.3. Computing vacuum expectation values
2.3.1. Path integrals on the lattice
2.3.2. Hybrid Monte-Carlo method
2.4. Hadron spectrum and scale setting
2.4.1. Gauge conguration ensembles
2.4.2. Hadron spectrum
2.4.3. Scale setting
3. scattering: the resonance 
3.1. Introduction
3.2. scattering in nite volume
3.2.1. Luscher’s method
3.2.2. Symmetry considerations
3.2.3. Finite volume formula
3.3. Methodology
3.3.1. Extraction of the energies: the generalized eigenvalue method
3.3.2. Kinematics and interpolating operators
3.3.3. Parametrization of the resonance
3.3.4. Pion mass dependence of the resonance
3.4. Lattice calculation details
3.4.1. Ensembles
3.4.2. Contractions and stochastic propagators
3.5. Results
3.5.1. GEVP energies
3.5.2. Results for the \GEVP » ensembles
3.5.3. Global t and systematic error analysis
3.5.4. Resonance parameters
3.5.5. Conclusion
4. Nucleon form factors 
4.1. Form factors
4.1.1. Electromagnetic form factors
4.1.2. Axial form factors
4.2. Methodology
4.2.1. Lattice currents, renormalization and Ward identities
4.2.2. Spectral decomposition
4.2.3. Form factor extraction
4.2.4. Excited states
4.3. Lattice calculation details
4.3.1. Ensembles, kinematics and parameters
4.3.2. Contractions and smearing
4.4. Results
4.4.1. Form factors
4.4.2. Electric charge radius
4.4.3. Magnetic moment
4.4.4. Axial charge
4.5. Conclusion
5. Conclusion and outlooks 
Appendices

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