Discretization .1 Scalar fields

We start by considering a simple field theory involving a scalar fieldφ. Post-poning until section 16.3 the question of exactly how we shall use it, we assume

/

so that a typical integral (in one dimension) becomes / dx (∂φ)2 → ae 1 2

[φ(n₁ + 1) − φ(n1)] . (16.6)

∂x a²

n1

As in all our previous work, we can alternatively consider a formulation in momentum space, which will also be discretized. It is convenient to impose

16.2. Discretization 155 a form suitable for numerical simulation, which we defer until section 16.3.

There is, however, a quite separate problem which arises when we try to repeat for the Dirac field the discretization used for the scalar field.

First note that the Euclidean Dirac matricesγμ^E are related to the usual Minkowski ones γ_μ^M by γ_1,2,3^E ≡ −iγ_1,2,3^M , γ₄^E ≡ −iγ₄^M ≡ γ₀^M. They satisfy {γ^E_μ, γ^E_ν} = 2δμν for μ = 1,2,3,4. The Euclidean Dirac Lagrangian is then ψ(x)¯ [

γ_μ^E∂μ+m]

ψ(x), which should be written now in Hermitean form mψ(x)ψ(x) +¯ 1

But here there is a problem: in addition to the correct continuum limit (a→0) found at kν1 → 0, an alternative finite a → 0 limit is found at kν1 → π/a (consider expandinga⁻¹sin [(π/a−δ)a] for smallδ). Thus two modes survive as a→0, a phenomenon known as the ‘fermion doubling problem’. Actually in four dimensions there aresixteensuch corners of the hypercube, so we have far too many degenerate lattice copies (which are called different ‘tastes’, to distinguish them from the real quark flavours).

Various solutions to this problem have been proposed. Wilson (1975), for example, suggested adding the extra term

− 1 2are

n1

ψ(n¯ 1)[ψ(n1+ 1) +ψ(n1−1)−2ψ(n1)] (16.21) 154 16. Lattice Field Theory, and the Renormalization Group Revisited

since (problem 16.1)

Equation (16.9) is a discrete version of the δ-function relation given in (E.25) of volume 1. A one-dimensional version of the mass term in (16.4) then becomes (problem 16.2)

Thus a one-dimensional version of the free action (16.4) is 1 φ˜(kν₁ )

In the continuum case, (16.13) would be replaced by 1 dkφ(k)˜ [

k² + m ²]

φ˜(−k) (16.14)

2 2π

as usual, which implies that the propagator in the discrete case is proportional

to [4 sin²(kν1 a/2) ]−1

+ m 2 (16.15)

a² rather than to [

k² + m²]−1

(remember we are in one-dimensional Euclidean space). The two expressions do coincide in the continuum limit a → 0. The manipulations we have been going through will be easily recognized by readers familiar with the theory of lattice vibrations and phonons, and lead to a satisfactory discretization of scalar fields. For Dirac fields the matter is not so straightforward.

16.2.2 Dirac fields

The first obvious problem has already been mentioned: how are we to rep-resent such entirely non-classical objects, which obey anticommutation rela-tions? This is part of the wider problem of representing field operators in

154 16. Lattice Field Theory, and the Renormalization Group Revisited Equation (16.9) is a discrete version of theδ-function relation given in (E.25) of volume 1. A one-dimensional version of the mass term in (16.4) then becomes (problem 16.2) Thus a one-dimensional version of the free action (16.4) is

1

In the continuum case, (16.13) would be replaced by 1

2 dk 2πφ(k)˜ [

k²+m²]φ(˜ −k) (16.14) as usual, which implies that the propagator in the discrete case is proportional

to [4 sin²(kν1a/2) space). The two expressions do coincide in the continuum limita→0. The manipulations we have been going through will be easily recognized by readers familiar with the theory of lattice vibrations and phonons, and lead to a satisfactory discretization of scalar fields. For Dirac fields the matter is not so straightforward.

16.2.2 Dirac fields

The first obvious problem has already been mentioned: how are we to rep-resent such entirely non-classical objects, which obey anticommutation rela-tions? This is part of the wider problem of representing field operators in

16.2. Discretization 155

a form suitable for numerical simulation, which we defer until section 16.3.

There is, however, a quite separate problem which arises when we try to repeat for the Dirac field the discretization used for the scalar field.

The corresponding ‘one-dimensional’ discretized action is then

a e ¯ a { In momentum space this becomes (problem 16.3)

¯ | sin(kν1 a) |

But here there is a problem: in addition to the correct continuum limit (a →0) found at kν₁ → 0, an alternative finite a → 0 limit is found at kν₁ → π/a

Various solutions to this problem have been proposed. Wilson (1975), for example, suggested adding the extra term

16.2. Discretization 157 where /D is the SU(3)c-covariant Dirac derivative. Any addition toD/ which is proportional to the unit 4×4 matrix will violate (16.24), and hence break chiral symmetry. The Lagrangian massmitself is of this form, and it breaks chiral symmetry, but ‘softly’ – i.e. in a way that disappears asmgoes to zero (thereby preserving the symmetry in this limit). The Wilson addition (16.21) also breaks chiral symmetry, but it remains there even asm→0: it is a ‘hard’

breaking.

This means that in the theory with the Wilson modification (i.e. with

‘Wilson fermions’) fermion mass renormalization will not be protected by the chiral symmetry, so that large additive renormalizations are possible. This will require repeated fine-tunings of the bare mass parameters, to bring them down to the desired small values. And it turns out that this seriously lengthens the computing time.

Another approach (‘staggered fermions’) was suggested by Kogut and Susskind (1975), Bankset al. (1976), and Susskind (1977). This essentially involves distributing the 4 spin degrees of freedom of the Dirac field across different lattice sites (we shall not need the details). At each site there is now a one-component fermion, with the colour degrees of freedom, which speeds the calculations. The 16-fold ‘doubling’ degeneracy can be re-arranged as four different tastes of 4-component fermions, while retaining enough chiral symmetry to forbid additive mass renormalizations.

Since the different components of the staggered Dirac field now live on different sites, they will experience slightly different gauge field interactions.

(These are of course local in the continuum limit, but the point remains true after discretization, as we shall see in the following section.) These interactions will mix fields of different tastes, causing new problems, but they can be suppressed by adding further terms to the action. There is still the 4-fold degeneracy to get rid of, but a trick is available for that, as we shall explain in section 16.3.

One might wonder if a lattice theory with fermions could be formulated such that it both avoids doublers and preserves chiral symmetry. For quite a long time it was believed that this was not possible – a conclusion which was essentially the content of the Nielsen–Ninomaya theorem (Nielsen and Ninomaya 1981a, b, c). But more recently a way was found to formulate chiral gauge theories with fermions satisfactorily on the lattice at finite spacing a.

The key is to replace the condition (16.24) by the Ginsparg–Wilson (1982) relation

γ5D+/ Dγ/ 5=aDγ/ 5D ./ (16.25) This relation implies (L¨uscher 1998) that the associated action has an exact symmetry, with infinitesimal variations proportional to

156 16. Lattice Field Theory, and the Renormalization Group Revisited to the fermion action in this one-dimensional case, where r is dimensionless. how renormalization group ideas provide a different perspective on such non-renormalizable interactions, classifying them as ‘irrelevant’).

How does the extra term (16.21) help the doubling problem? One easily finds that it changes the (one-dimensional) inverse propagator to

|

Unfortunately there is a price to pay. The problem is that, as we learned in section 12.3.2, the QCD lagrangian has an exact chiral symmetry for massless quarks. To the extent that mu and md (and ms, but less so) are small on a hadronic scale such as Λ_MS, we expect chiral symmetry to have important physical consequences. These will indeed be explored in chapter 18. For the moment, we note merely that it is important for lattice-based QCD calcu-lations to be able to deal correctly with the light quarks. Now we cannot simply choose the bare Lagrangian mass parameters to be small, and leave it at that. In any interacting theory, renormalization effects will cause shifts in these masses. In a chirally symmetric theory, or one which is chirally sym-metric as a fermion mass goes to zero, such a mass shift is proportional to the fermion mass itself; in particular it does not simply add to the mass. We drew attention to this fact in the case of the electron mass renormalization in QED, in section 11.2. So in chirally symmetric theories, mass renormalizations are

‘protected’, in this sense. But the modification (16.21), while avoiding phys-ical fermion doublers, breaks chiral symmetry badly. This can easily be seen by noting (see (12.154) for example) that the crucial property required for chiral symmetry to hold is

γ₅ D+ / Dγ/ ₅ = 0, (16.24)

156 16. Lattice Field Theory, and the Renormalization Group Revisited to the fermion action in this one-dimensional case, wherer is dimensionless.

Evidently this is aseconddifference, and it would correspond to the term

−1 2ra

/

d³xdτ ψ(x)(∂¯ _τ²+∇²)ψ(x) (16.22) in the four-dimensional continuum action. Note the presence of the lattice spacing ‘a’ in (16.22), which ensures its disappearance asa→0. The higher-derivative term ¯ψ(∂_τ²+∇²)ψhas mass dimension 5, and therefore requires a coupling constant with mass dimension -1, i.e. a length in unitsh =c = 1;

it is, in fact, a non-renormalizable term. However, if we recall the discus-sion of section 11.8 in volume 1, we would expect it to be suppressed at low momenta much less than the cut-off π/a. Hence it is natural to see a cou-pling proportional toa appearing in (16.22). (We shall see in section 16.5.3 how renormalization group ideas provide a different perspective on such non-renormalizable interactions, classifying them as ‘irrelevant’).

How does the extra term (16.21) help the doubling problem? One easily finds that it changes the (one-dimensional) inverse propagator to

| By considering the expansion of the cosine nearkν1 ≈0 it can be seen that the second term disappears in the continuum limit, as expected. However, for kν1 ≈ π/a it gives a large term of order ¹_a which adds to the mass m, effectively banishing the ‘doubled’ state to a very high mass, far from the physical spectrum.

Unfortunately there is a price to pay. The problem is that, as we learned in section 12.3.2, the QCD lagrangian has an exact chiral symmetry for massless quarks. To the extent that mu and md (and ms, but less so) are small on a hadronic scale such as Λ_MS, we expect chiral symmetry to have important physical consequences. These will indeed be explored in chapter 18. For the moment, we note merely that it is important for lattice-based QCD calcu-lations to be able to deal correctly with the light quarks. Now we cannot simply choose the bare Lagrangian mass parameters to be small, and leave it at that. In any interacting theory, renormalization effects will cause shifts in these masses. In a chirally symmetric theory, or one which is chirally sym-metric as a fermion mass goes to zero, such a mass shift is proportional to the fermion mass itself; in particular it does not simply add to the mass. We drew attention to this fact in the case of the electron mass renormalization in QED, in section 11.2. So in chirally symmetric theories, mass renormalizations are

‘protected’, in this sense. But the modification (16.21), while avoiding phys-ical fermion doublers, breaks chiral symmetry badly. This can easily be seen by noting (see (12.154) for example) that the crucial property required for chiral symmetry to hold is

‘Wilson fermions’) fermion mass renormalization will not be protected by the chiral symmetry, so that large additive renormalizations are possible. This will require repeated fine-tunings of the bare mass parameters, to bring them down to the desired small values. And it turns out that this seriously lengthens the computing time.

Another approach (‘staggered fermions’) was suggested by Kogut and Susskind (1975), Banks et al. (1976), and Susskind (1977). This essentially involves distributing the 4 spin degrees of freedom of the Dirac field across different lattice sites (we shall not need the details). At each site there is now a one-component fermion, with the colour degrees of freedom, which speeds the calculations. The 16-fold ‘doubling’ degeneracy can be re-arranged as four different tastes of 4-component fermions, while retaining enough chiral symmetry to forbid additive mass renormalizations.

Since the different components of the staggered Dirac field now live on different sites, they will experience slightly different gauge field interactions.

(These are of course local in the continuum limit, but the point remains true after discretization, as we shall see in the following section.) These interactions will mix fields of different tastes, causing new problems, but they can be was essentially the content of the Nielsen–Ninomaya theorem (Nielsen and Ninomaya 1981a, b, c). But more recently a way was found to formulate chiral gauge theories with fermions satisfactorily on the lattice at finite spacing a.

The key is to replace the condition (16.24) by the Ginsparg–Wilson (1982) relation

γ₅ D+ / Dγ/ ₅ = a /Dγ₅ D ./ (16.25) This relation implies (L¨uscher 1998) that the associated action has an exact symmetry, with infinitesimal variations proportional to

( 1 )

δψ = γ5 1− a D/ ψ (16.26)

( 1 2 )

δψ = ψ 1− a D/ γ₅ . (16.27)

2

16.2. Discretization 159 Under the gauge transformation (16.29),O(x, y) transforms by

O(x, y)→φ^†(x)e^−ieθ(x)e^{{iefyxAdx'+ie[θ(x)−θ(y)]}}e^ieθ(y)φ(y) =O(x, y), (16.32) and it is therefore gauge invariant. The familiar ‘covariant derivative’ rule can be recovered by lettingy=x+ dxfor infinitesimal dx, and by considering the gauge-invariant quantity

Evaluating (16.33) one finds (problem 16.4) the result φ^†(x)

with the usual definition of the covariant derivative. In the discrete case, we merely keep the finite version of (16.31), and replaceφ^†(n1)φ(n1+1) in (16.28) by the gauge invariant quantity

φ^†(n1)U(n1, n1+ 1)φ(n1+ 1), (16.36) where thelink variable U is defined by

U(n1, n1+ 1) = exp The generalization to more dimensions is straightforward. In the non-AbelianSU(2) orSU(3) case, ‘eA’ in (16.38) is replaced bygt^aA^a(n1) where thet’s are the appropriate matrices, as in the continuum form of the covariant derivative. A link variable U(n2, n1) may be drawn as in figure 16.1. Note that the order of the arguments is significant: U(n2, n1) = U⁻¹(n1, n2) = U^†(n1, n2) from (16.38), which is why the link carries an arrow.

Thus gauge invariant discretized derivatives of charged fields can be con-structed. What about the Maxwell action for the U(1) gauge field? This does not exist in only one dimension (∂μAν−∂νAμ cannot be formed), so let us move into two. Again, our discussion of the geometrical significance ofFμν as [

158 16. Lattice Field Theory, and the Renormalization Group Revisited The symmetry under (16.26)–(16.27), which is proportional to the infinitesi-mal version of (12.152) as a → 0, provides a lattice theory with all the funda-mental symmetry properties of continuum chiral gauge theories (Hasenfratz et al. 1998). Finding an operator which satisfies (16.25) is, however, not so easy – but that problem has now been solved, indeed in three different ways:

Kaplan’s ‘domain wall’ fermions (Kaplan 1992); ‘classically perfect fermions’

(Hasenfratz and Niedermayer 1994); and overlap fermions (Narayanan and Neuberger 1993a, b, 1994, 1995). Unfortunately all these proposals are com-putationally more expensive than the Wilson or staggered fermion alterna-tives.

16.2.3 Gauge fields

Having explored the discretization of actions for free scalars and Dirac fermions, we must now think about how to implement gauge invariance on the lattice. In the usual (continuum) case, we saw in chapter 13 how this was implemented by replacing ordinary derivatives by covariant derivatives, the geometrical signif-icance of which (in terms of parallel transport) is discussed in appendix N. It is very instructive to see how the same ideas arise naturally in the lattice case. to parallel transport one field to the same point as the other, before they can be properly compared. The solution (N.18) shows us how to do this. Consider the quantity

x

O(x, y) = φ^†(x)exp[ie Adx ^']φ(y). (16.31)

y

158 16. Lattice Field Theory, and the Renormalization Group Revisited The symmetry under (16.26)–(16.27), which is proportional to the infinitesi-mal version of (12.152) asa→0, provides a lattice theory with all the funda-mental symmetry properties of continuum chiral gauge theories (Hasenfratz et al. 1998). Finding an operator which satisfies (16.25) is, however, not so easy – but that problem has now been solved, indeed in three different ways:

Kaplan’s ‘domain wall’ fermions (Kaplan 1992); ‘classically perfect fermions’

(Hasenfratz and Niedermayer 1994); and overlap fermions (Narayanan and Neuberger 1993a, b, 1994, 1995). Unfortunately all these proposals are com-putationally more expensive than the Wilson or staggered fermion alterna-tives.

16.2.3 Gauge fields

Having explored the discretization of actions for free scalars and Dirac fermions, we must now think about how to implement gauge invariance on the lattice. In the usual (continuum) case, we saw in chapter 13 how this was implemented by replacing ordinary derivatives bycovariant derivatives, the geometrical signif-icance of which (in terms of parallel transport) is discussed in appendix N. It is very instructive to see how the same ideas arise naturally in the lattice case.

We illustrate the idea in the simple case of the Abelian U(1) theory, QED.

Consider, for example, a charged scalar fieldφ(x), with chargee. To construct a gauge-invariant current, for example, we replacedφ^†∂μφbyφ^†(∂μ+ ieAμ)φ, so we ask: what is the discrete analogue of this? The term φ^†(x)_∂x^∂ φ(x) becomes, as we have seen,

φ^†(n1)1

a[φ(n1+ 1)−φ(n1)a] (16.28) in one dimension. We do notexpect (16.28) by itself to be gauge invariant, and it is easy to check that it is not. Under a gauge transformation for the continuous case, we have

φ(x)→e^ieθ(x)φ(x), A(x)→A(x) + dθ(x)

dx ; (16.29)

thenφ^†(x)φ(y) transforms by

φ^†(x)φ(y)→e−ie[θ(x)−θ(y)]φ^†(x)φ(y), (16.30) and is clearly not invariant. The essential reason is that this operator involves the fields at twodifferentpoints, and so the termφ^†(n1)φ(n1+ 1) in (16.28) will not be gauge invariant either. The discussion in appendix N prepares us for this: we are trying to compare two ‘vectors’ (here, fields) at two different points, when the ‘coordinate axes’ are changing as we move about. We need to parallel transport one field to the same point as the other, before they can be properly compared. The solution (N.18) shows us how to do this. Consider the quantity

O(x, y) =φ^†(x)exp[ie [ x

y

Adx^']φ(y). (16.31)

| [ |

'

16.2. Discretization 159

Under the gauge transformation (16.29), O(x, y) transforms by

{ie f _x

Adx +ie[θ(x)−θ(y)]} ieθ(y)φ(y)

O(x, y)→φ^†(x)e^−ieθ(x)e y ^' e = O(x, y), (16.32) and it is therefore gauge invariant. The familiar ‘covariant derivative’ rule can be recovered by letting y = x+ dx for infinitesimal dx, and by considering the gauge-invariant quantity

|O(x, x + dx)− O(x, x)|

lim . (16.33)

dx→0 dx

Evaluating (16.33) one finds (problem 16.4) the result ( d )

φ^†(x) −ieA φ(x) (16.34)

dx

≡φ^†(x)Dxφ(x) (16.35) with the usual definition of the covariant derivative. In the discrete case, we merely keep the finite version of (16.31), and replace φ^†(n1)φ(n₁ +1) in (16.28) by the gauge invariant quantity

φ^†(n1)U(n1, n1 + 1)φ(n1 + 1), (16.36) where the link variable U is defined by

n1a

U(n1, n1 + 1) = exp ie Adx . (16.37)

(n1+1)a

Note that

U(n1, n1 + 1) →exp[−ieA(n1)a] (16.38) in the small a limit.

Similarly, the free Dirac term ψ¯(n1)γ₁^Eψ(n₁ + 1) −ψ¯(n₁ + 1)γ₁^Eψ(n1) in (16.18) is replaced by the gauge-invariant term

ψ(n¯ 1)γ₁^EU(n1, n₁ + 1)ψ(n₁ + 1)−ψ¯(n₁ + 1)γ₁^EU(n₁ + 1, n1)ψ(n1). (16.39) The generalization to more dimensions is straightforward. In the non-Abelian SU(2) orSU(3) case, ‘eA’ in (16.38) is replaced by gt^aA^a(n1) where the t’s are the appropriate matrices, as in the continuum form of the covariant derivative. A link variable U(n2, n1) may be drawn as in figure 16.1. Note that the order of the arguments is significant: U(n2, n1) = U⁻¹(n1, n2) = U^†(n1, n2) from (16.38), which is why the link carries an arrow.

Thus gauge invariant discretized derivatives of charged fields can be con-structed. What about the Maxwell action for the U(1) gauge field? This does not exist in only one dimension (∂μA_ν −∂_ν A_μ cannot be formed), so let us move into two. Again, our discussion of the geometrical significance of F_μν as

Im Dokument Non-Abelian Gauge Theories (Seite 168-175)