Renormalization

Renormalization of the Action

Starting from a perturbative expansion in the coupling g we are now able to dimensionally regularize all occurring UV divergences. Especially we can calculate radiative corrections to the terms that occur in the action of quantum chromodynamics. This gives rise to corrections of the quark-gluon vertex, the three-gluon vertex, the four-gluon and ghost-gluon vertex, as well as of the self-energies for the various propagators. As we have already mentioned the regularized divergences must be cured by a redefinition of the bare quantities, the renormal-ization. In the following we will demonstrate this explicitly for the quark propagator. We will furthermore show all radiative one-loop corrections to the variables in the action, whereby we follow a systematic approach and sort the diagrams according to their external legs.

We begin with the Feynman diagrams that have one incoming and one outgoing quark leg, Figure 2.2. To first order we find two of them, the tree-level contribution consisting of the free quark propagator and a first order correction which manifests itself in a gluon loop attached to the free propagator. We have already come across the latter diagram when discussing the need for regularization. It contributes to the so-called self-energy of the quark field and diverges in the limit →0. On the other hand the expectation value should be finite in the end, because the physical quark field always comes with this contribution as every quark is dressed in a gluon cloud. Let us therefore have a closer look at how renormalization absorbs the unphysical divergence into a redefinition of the bare variables of the action and thus solves this dilemma.

We focus again on the Feynman graphs for the quark self-energy shown in Figure 2.2. The left diagram in this figure is the tree-level contribution and evaluates to the free quark propagator:

S(p)^free= 1

ip/+m1. (2.39)

Then one calculates the radiative correction and adds it to the tree-level result. The result can be cast in the following form, which resembles the structure of the free propagator:

S(p)^reg= 1

Σ1(p²)ip/+ Σ2(p²)1. (2.40) Both functions Σ1 and Σ2 are perturbative series of the coupling constant and depend on the cutoff parameter. In our first order expansion they can be decomposed up to vanishing higher powers of into

Σ1(p²) = 1 +g_R² Σ⁽⁰⁾₁ +g_R² 1

¯

Σ⁽¹⁾₁ , (2.41)

Σ2(p²) =m+g_R² Σ⁽⁰⁾₂ +g_R² 1

¯

Σ⁽¹⁾₂ . (2.42)

Figure 2.2: Diagrams contributing to the quark self-energy in first order.

2.5. RENORMALIZATION 33 Note that the fieldsψ, that appear in the action, have no direct physical meaning. Hence we are free to modify them. Especially we could have written the action in terms of the fields ψ^reninstead ofψfrom the very beginning. With a proper redefinition at the scaleµ² =p² we get rid of the divergence in Σ1 (we will come back to the implications of the renormalization scaleµlater):

ψ→ψ^ren= q

Zq(µ)ψ, (2.43)

Z_q⁻¹= 1−g²_R1

¯

Σ⁽¹⁾₁ . (2.44)

Since the propagator is a bilinear of the quark fields, hψψi, it gets multiplied by¯ Zq upon redefinition of the quark fields. Doing so amounts to the following expression:

S(p) =˜ 1

Σ₁(p²)^renip/+ ˜Σ₂(p²)1,

Σ^ren₁ = 1 +g²_RΣ⁽⁰⁾₁ , (2.45)

Σ˜2=m+g_R² Σ⁽⁰⁾₂ +g²_R1

¯

(Σ⁽¹⁾₂ −mΣ⁽¹⁾₁ ).

The remaining divergence in ˜Σ2 can now be eliminated by a redefinition of the quark mass in the action:

m→m^ren=Z_m(µ)m, (2.46)

Zm = 1−g_R² 1

¯

(Σ⁽¹⁾₂ /m−Σ⁽¹⁾₁ ). (2.47) This alters also the ˜Σ₂ term to a finite expression,

Σ2(p²)^ren=m+g²_RΣ⁽⁰⁾₂ , (2.48) and finally results in the finite, renormalized quark propagator:

S(p)^ren= 1

Σ1(p²)^renip/+ Σ2(p²)^ren1. (2.49) Absorbing the divergences in a well-defined manner, like in the above example, is called renormalization. The resulting matrix elements are renormalized matrix elements as opposed to their so-called bare counterparts that are merely regularized but still contain divergent parts in the limit →0. Renormalized quantities can be calculated from bare quantities by multiplication with a renormalization coefficient Z. For the propagator we have, e.g.,

S(p)^ren=Zq

1

Σ1ip/+ZmΣ21. (2.50)

Once all divergences are absorbed into a redefinition of the fields, the masses and the coupling constant in a consistent way, all matrix elements should be finite. So we summarize in the following all remaining one-loop diagrams that contribute to the renormalization of the action.

Just as for the quarks there are also self-energy contributions for the gluons, see Figure 2.3.

In a first order expansion we find four of them. One is produced by a gluon loop that is

Figure 2.3: Diagrams contributing to the gluon self-energy in first order.

Figure 2.4: Diagrams contributing to the ghost self-energy in first order.

connected to two three-gluon vertices. The second stems from the four-gluon vertex and the third from a quark loop. The last contribution finally is due to the ghost fields that cancel the unphysical degrees of freedom introduced by the gluon loop.

Also the ghost propagator itself picks up radiative corrections, cf. Figure 2.4. They look similar to those of the quark propagator. However, ghost propagators are restricted to loops within gluon lines and therefore will not show up at all in the first order perturbative expansion for the three-quark operators considered in this work.

Apart from the already presented diagrams with two external legs there are three dia-grams with three external legs. They originate from the different vertices and their radiative corrections. The first one under investigation is the quark-gluon vertex in Figure 2.5. Its first one-loop diagram looks similar to the correction of the electron-photon vertex in QED, however, the second diagram would be absent in the abelian case. It is genuinely non-abelian since it originates from the self-interaction of the gauge fields in a three-gluon vertex.

Two one-loop diagrams contribute to the three-gluon vertex itself, compare Figure 2.6.

While the first Feynman graph has a gluon-loop attached by two further three-gluon vertices, the second correction is indebted to the four-gluon vertex. For reasons of completeness we give in Figure 2.7 also the one-loop correction to the ghost-gluon vertex.

The last class of diagrams is shown in Figure 2.8. These diagrams summarize the Feynman

Figure 2.5: Diagrams contributing to the quark-gluon vertex in first order.

2.5. RENORMALIZATION 35

Figure 2.6: Diagrams contributing to the three-gluon vertex in first order.

Figure 2.7: Diagrams contributing to the ghost-gluon vertex in first order.

graphs with four external legs that belong to the four-gluon vertex. The first correction originates from attaching a gluon loop with two three-gluon vertices to the four-gluon vertex, the second from inserting another four-gluon vertex.

All these diagrams contain divergences that manifest themselves after dimensional regular-ization in poles at isolated values of the space-time dimension. As quantum chromodynamics is a renormalizable quantum field theory these divergences can be absorbed order by order into a redefinition of the fields, the masses, the gauge parameter and the coupling constant in a consistent way. This was shown by induction in the early 1970s by ’t Hooft and Veltman [39, 40]. In the minimal subtraction scheme they introduced, this redefinition amounted to the subtraction of the terms proportional to 1/ for graphs without further subdivergences.

However, as we have already mentioned, each factor 1/always comes together with ln 4π−γ_E which is almost of order 2. It has proven advantageous to also get rid of this large coefficient in the perturbative expansion by absorbing it into the definition of the renormalization constant Z. This subtraction of the 1/¯ term instead of the mere 1/ pole is known as the modified minimal subtraction scheme, MS [41]. Today it can be said to be the standard scheme for any perturbative calculation and hence will also be used in this thesis.

Figure 2.8: Diagrams contributing to the four-gluon vertex in first order.

Renormalization of Composite Operators

It is important to note that introducing renormalized fields and a renormalized coupling con-stant is not the whole story, when trying to extract physics from quantum chromodynamics.

In many situations we will meet further divergences due to so-called composite operators.

These composite operators are a product of fields at the same space-time coordinatex. They typically occur in the operator product expansion, where non-local operators at large mo-mentum transfer q are rewritten as a 1/q²-series of local operators. A generic application is deep inelastic scattering, but also the electromagnetic currents Jµ in the process of electron-positron annihilation can be expanded in such a way:

J_µ(x)J_ν(0)∼C_µν¹ (x) 1 +C_µν^qq¯ q(0)q(0) +¯ C_µν^F2(F_αβ^a )²(0) +. . . , (2.51) where the Wilson coefficient C_µν¹ (x) is ∼ (q²)⁰. The two other Wilson coefficients,Cµν^qq¯ and C_µν^F2, are already suppressed by two powers of the inverse photon momentum square,∼(q²)⁻². Let us have a closer look at such a composite operator, e.g., at ¯q(0)q(0). A small distance between two objects in coordinate space amounts to large values in momentum space. Hence it is comprehensible that reducing the distance between two objects to zero amounts to an ultraviolet divergence in momentum space. Going beyond this hand-waving argument, one will find the mentioned UV divergences in the first order perturbative expansion of the associated matrix elements. If one demands the Wilson coefficients to be finite, then one has to renormalize the composite operators in a way consistent with the renormalization of the action presented in the previous section. This implies that both the Wilson coefficients and the composite operators also have to be renormalized in the MS scheme.

A typical bare, i.e., regularized but not yet renormalized composite operatorO will look in one-loop order as follows:

O^bare(p²) =O^tree(p²) + g_R²(µ) 16π² γ

1

¯

+ ln p² µ²

O^tree(p²) +. . . . (2.52) Its sought renormalized counterpart is given in the MS scheme by subtraction of the pole part at some renormalization scaleµ² =p²:

O^ren(µ²) =O^tree(p²) +g_R²(µ) 16π² γ

lnp²

µ²

O^tree(p²) +. . .|_µ2=p². (2.53) And the relationship between the bare and renormalized operators is established by a renor-malization constantZ, as usual,

O^ren(µ²) =Z(µ²)O^bare(p²)|_µ2=p². (2.54) More general, the renormalized operator can even be a linear combination of different bare operators with the same quantum numbers. This behavior is known as operator mixing under renormalization:

O^ren_i =ZijO_j^bare. (2.55)

In practice one tries to avoid mixing as much as possible, because it introduces further sys-tematic and – in lattice QCD – statistic uncertainties. In this thesis we will focus on the renormalization of local three-quark operators,

O(x) =q(x)q(x)q(x). (2.56) Also here a first and important step will be the controlling of the mixing behavior.

2.5. RENORMALIZATION 37 Renormalization Group Equation and Running Coupling

Let us come back to the renormalization scale µ, which we have used without commenting on its implications, e.g., in eq. (2.53). The value of µis determined by the typical momenta involved in the process under investigation and it sets the scale at which we subtract the divergences and thereby define our renormalized operators. Hence both Z and the renor-malized operator O^ren depend on the renormalization scale. This dependence is quantified by the renormalization group equation (RGE). This equation is a direct consequence of the observation that the regularized, but not yet renormalized operator cannot depend on the renormalization scale. Upon reexpressing the bare operator in terms of its renormalized counterpart and the inverse of the renormalization matrix, one arrives at the condition

0=^! µ² d

dµ² O^bare =µ² d

dµ² Z⁻¹(µ)O^ren(µ)

. (2.57)

After some trivial mathematics this relation becomes the RGE in the accustomed notation.

It was derived independently by Callan and Symanzik [42, 43] in 1970 and since then is also referred to as the Callan-Symanzik equation:

0=^!

µ² ∂

∂µ² +β(α_s) ∂

∂α_s +γ

O^ren(µ²). (2.58)

Here we have introduced the beta and gamma functions with the conventions β =µ² d

dµ² αs(µ), (2.59)

γ =−Z⁻¹(µ)µ² d

dµ² Z(µ). (2.60)

The beta function describes the running of the strong coupling constant when the renormal-ization scale µ is changed. We will come back to this universal quantity at the end of this section. Right now it is sufficient to note that the inverse of the momentumµcan be under-stood as the resolution with that a given process is observed.

The gamma function is specific to the operator under investigation and amounts to its scaling behavior under a variation of the renormalization scale. For perturbative calculations it is reasonable to also expandβ and γ in powers of the strong couplingαs(µ) = ^gR_4π^(µ)2:

β(αs) 4π =−

∞

X

i=0

βi

αs(µ) 4π

i+2

, (2.61)

γ(α_s) =−

∞

X

i=0

γ_i 2

α_s(µ) 4π

i+1

. (2.62)

In the MS schemeαs as well as the beta function are known to high order in perturbative QCD. Throughout this thesis we will use the following expression:

α^MS(q²)

4π = 1

β₀ ln(q²) − β₁ ln lnq² β₀³(lnq²)²

+ 1

β₀⁵(lnq²)³

β₁²(ln lnq²)²−β₁² ln lnq²+β^MS₂ β₀−β₁²

, (2.63) q² = µ²

Λ⁽ⁿ_QCD^f) 2. (2.64)

0.2 0.25 0.3 0.35 0.4 0.45 0.5

1 2 3 4 5 6 7 8 9 10

αs(µ2 )

µ²/GeV²

Figure 2.9: The running couplingαs as a function of the renormalization scaleµ.

And the required coefficients of the beta function read for an arbitrary number of colors N_c and flavors nf [44]:

β0 =11

3 Nc−2 3nf, β1 =34

3 N_c²− 10

3 Ncnf −N_c²−1 Nc

nf, β₂^MS=2857

54 N_c³+(N_c²−1)²

4N_c² n_f −205

36 (N_c²−1)n_f

− 1415

54 N_c²n_f +11 18

N_c²−1

N_c n²_f+ 79

54Ncn²_f. (2.65) Since we are interested in quantum chromodynamics and will work with two flavors, we set Nc = 3 andnf = 2. The parameter ΛQCD is the position of the Landau pole in the running couplingα_s. We take a two flavor lattice result matched to the MS scheme [45]:

Λ⁽²⁾_QCD = 261 MeV. (2.66)

It was first observed by Gross, Wilczek [46, 47] and independently by Politzer [48] that the beta function is negative in QCD. According to eq. (2.59) this implies that the running coupling decreases when increasing the momentum scale µ, compare Figure 2.9. Now large momenta correspond to small distances in coordinate space and vice versa. Hence the coupling becomes weak for small separations between the quarks and gluons, whereas it becomes strong for large separations. Quantum chromodynamics is therefore said to be asymptotically free, i.e., the quarks behave like quasi-free particles when probed at very high momentum. On the other hand the quarks are confined within color-neutral hadrons, as the coupling between two quarks gets stronger and stronger if one increases their separation. If one forces the separation, e.g., a quark-antiquark pair is created from the QCD vacuum that restores the color-neutrality and confinement of the fragments.

The running of the coupling also reminds us that a perturbative expansion can only be valid in regimes, where the coupling g_Ris small. Due to the property of asymptotic freedom this condition is only fulfilled for large momentum scales µ.

Chapter 3

Lattice QCD

In this chapter we want to give a short introduction to lattice QCD. We have seen in the previous chapter that the path integral approach leads to an infinitely dimensional integral over an infinite series represented by the exponential of the action, eq. (2.15). Since it is unfeasible to treat both expressions exactly we have truncated the infinite series at a fixed low order of the strong coupling in perturbative (continuum) QCD. Lattice QCD takes the complementary approach and keeps the full interaction part, but approximates the infinitely dimensional integration. This is accomplished on a twofold basis. The first mainstay is to carry out the integration over the gluon fields by Monte-Carlo techniques. This facilitates to evaluate the fermionic integration analytically by means of Wick contractions on each of the sampled gauge configurations. The second mainstay is a reduction of the continuous four-dimensional space-time to a discrete and finite hypercubic lattice. Note that this discretization guarantees in turn the feasibility of the Monte-Carlo integration.

Let a denote the spacing of our four-dimensional lattice and V = a³N_s³ ×aN_t its physical volume, i.e., the lattice extendsNs slices in each spatial direction and Nt slices in the time-like direction. Then the continuous space-time coordinate x is replaced by its discretized counterpart:

x→a n≡a(n1, n2, n3, n4) withni ∈ {0,1,2, . . . , Ni−1}. (3.1) The first advantage of this method is that the regulators for the IR and UV regimes come for free, as we will explain in the following. The discretization and finiteness of the space-time results in the existence of a discretized Brillouin-zone and the allowed lattice momenta are limited to the thereby defined values. Before we summarize them, we must specify the boundary conditions at the borders of the lattice. These are defined periodic in the spatial directions and (anti-)periodic in the time-direction for the gauge (quark) fields. For evenNs

andNt the allowed momenta are then given by the following parametrizations:

pi = 2π a Ns

mi, mi=−Ns

2 ,−Ns

2 + 1, . . . ,Ns

2 , i= 1,2,3, p4 = 2π

a Nt

m4, m4=−Nt

2 ,−Nt

2 + 1, . . . ,Nt

2 (for bosons), p4 = 2π

a N_t

m4+1 2

, m4 =−Nt

2 ,−Nt

2 + 1, . . . ,Nt

2 (for fermions). (3.2) These equations unveil how the lattice acts as a regulator: While the finite lattice spacing a results in an UV cutoff located at ^π_a, the finite volume, i.e., the finite numberN of lattice sites

39

in each direction, results in a discretization of the momenta and thus provides a valuable IR regulator. This ensures that any matrix element is regulated by the lattice and yields a finite result without additional effort. To allow contact with experiments we nevertheless still have to perform a finite renormalization that translates the regularized results into a well-defined continuum renormalization scheme.

The second advantage of a discretized space-time is the already mentioned applicability of the Monte-Carlo approach for the integration of the gauge fields. Consequently the whole setup of lattice QCD can be implemented in a computer system and used to perform fully non-perturbative calculations. This facilitates detailed studies not only of the hadron spectrum, but also, e.g., of the internal structure of hadrons like the nucleon distribution amplitude. In the following we will briefly summarize some more theoretical details underlying lattice QCD.

For a general introduction we refer to [49, 50].

3.1 Naive Discretization of the Free Action

Lattice QCD can be best understood from the path integral approach, and the most important step towards the path integral formulation of lattice quantum chromodynamics is to adapt the action of continuum QCD to the discretized space-time. In eq. (3.1) we have introduced a set of coordinates, also referred to as sites, that define our lattice. The fermionic degrees of freedom “live” on these sites, i.e., the spinor-valued quark fields are placed on the lattice sites only:

ψ(x),ψ(x)¯ →ψ(n),ψ(n).¯ (3.3)

In continuum QCD the Lagrangian density of free fermions is described by the following massive Dirac operator, compare eq. (2.6):

L_F,free= ¯ψ(x)(γ_µ∂_µ+m)ψ(x). (3.4)

Here we have suppressed the color and spinor indices. Moreover, analogous to the continuum case, an implicit sum over the different flavors is implied.

Since on the lattice the quark fields are not anymore a function of a continuous space-time, but are defined on the discrete sites nonly, one has to find a suitable solution for the imple-mentation of the space-time derivative ∂_µ. Apart from the so-called forward and backward versions one possible choice is the symmetric discretization of the derivative,

∂µψ(x)→ 1

2a(ψ(n+ ˆµ)−ψ(n−µ))ˆ . (3.5) Here ˆµstands for a unit-vector inµ-direction. Thereby it is clear how a lattice version of the free fermionic Lagrangian density may be chosen. In the continuum the action is defined by a space-time integral over the Lagrangian density. On the lattice this integral is replaced by a sum over all sites nof the lattice, so that one arrives at

S_F,free=a⁴X

n

ψ(n)¯

γ_µ 1

2a ψ(n+ ˆµ)−ψ(n−µ)ˆ

+mψ(n)

. (3.6)

It is evident that one introduces unwanted discretization errors when replacing the derivatives and integrals by finite differences and sums. The above expression reproduces the continuum

3.2. INTRODUCING GAUGE INVARIANCE 41

Im Dokument Renormalization of Three-Quark Operators for the Nucleon Distribution Amplitude Dissertation (Seite 36-45)