Wednesday, December 21, 2005

The Ginsparg-Wilson relation

Time for another post in our series about lattice fermions.

In the previous post in this series we had a look at the Nielsen-Ninomiya theorem, which stated that any acceptable lattice fermion action for which the Dirac operator anticommuted with $$\gamma^5$$ had to have doubler fermions. On the face of it that seems to imply a stark choice between chiral symmetry and freedom from doublers.

There is, however, an interesting way around this apparent dilemma. This was discovered by Ginsparg and Wilson in a 1982 paper, were they studied the result of performing a spin-blocking step on a chirally symmetric continuum fermion action. What they discovered was that the Dirac operator of the blocked theory obeyed the anticommutation relation
$$\{\gamma^5,D\} = 2 a D \gamma^5 D$$
now known as the Ginsparg-Wilson relation.

This relation has a number of interesting consequences: Firstly, it implies that the propagator $$\tilde{S}(p)=\tilde{D}(p)^{-1}$$ obeys the anticommutation relation
$$\{\gamma^5,\tilde{S}(p)\} = 2 a \gamma^5$$
and hence in coordinate space
$$\{\gamma^5,S(x-y)\} = 2 a \gamma^5\delta(x-y)$$
i.e. the propagator is chirally invariant at all non-zero distances. Secondly, Lüscher discovered in 1998 that the Ginsparg-Wilson relation leads to a non-standard realization of chiral symmetry in the theory, which is invariant under the infinitesimal transformations
$$\psi & \mapsto & \psi+\epsilon\gamma^5\left(1-aD\right)\psi\\bar{\psi} & \mapsto & \bar{\psi} +<br />\epsilon\bar{\psi}\left(1-aD\right)\gamma^5$$
The fermion measure, however, transforms anomalously under this symmetry, and a little calculation shows that this gives precisely the correct chiral anomaly.

On the other hand, since the Wilson operator no longer anticommutes with $$\gamma^5$$, the conditions of the Nielsen-Ninomiya theorem no longer apply, and there is hence no reason to expect the existence of any doubler fermions.

What all this means is that the correct chiral physics can be obtained from a lattice theory, provided one is able to find a solution to the Ginsparg-Wilson relation. The next post in this series will look at some of the fermion actions that arise from this.

Monday, December 12, 2005

Analytical results for the glueball spectrum

In a recent paper, Leigh, Minic and Yelnikov present an analytical result for the glueball spectrum in (2+1) dimensions. They employ a Hamiltonian formalism pioneeered in a series of papers by Karabali, Kim and Nair. The main result is that the glueball spectrum of (2+1)-dimensional pure Yang-Mills theory can be expressed in terms of the zeros of the Bessel function $$J_2(z)$$. In particular, the masses of 0++ states can be written as the sum of two Bessel zeros:
$$m(0^{++*^r}) = (j_{2,n_1}+j_{2,n_2})\frac{g^2 N}{4\pi}$$
where n1 and n2 can be determined from r, and it is to be noted that the gauge coupling in (2+1) dimensions has the dimension of $$\sqrt{Mass}$$. Similarly, the masses of 0-- states can be written as the sum of three Bessel zeros:
$$m(0^{--*^r}) = (j_{2,n_1}+j_{2,n_2}+j_{2,n_3})\frac{g^2 N}{4\pi}$$
Their results agree reasonably well with lattice simulations of (2+1)-dimensional pure Yang-Mills theory.

There are some interesting implications of their results which are not discussed in their paper (they say they are going to publish another, more detailed, one). In particular, since for large m the Bessel zeros go like
$$j_{m,n}\simeq\left(n+\frac{m}{2}+\frac{1}{4}\right)\pi$$
for large excitation numbers, there will be almost degenerate states separated by gaps of $$g^2N/4$$, with the (almost) degeneracy of the r-th state given by the number of ways to partition (r+1), or (r+2), into two or three integers, respectively.

Another interesting implication of their results is that the mass difference between successive states of even parity and that of successive states of odd parity should be the same. This does not quite agree with what is found on the lattice, where the mass difference for the ++ states is about 1.6 times that for the -- states (which is similar to the difference in results obtained for the gluonic mass in (2+1) dimensions using self-consistent resummation methods with parity-even and parity-odd mass terms, respectively). From the analytical results this parity-dependence of the mass gap would appear to be some sort of artifact.

It will be interesting to see what is in Leigh, Minic and Yelnikov's detailed paper, in particular how the higher-spin glueballs turn out.

Thursday, December 08, 2005

Lattice QCD in the News

Lattice QCD made the news again. The AIP's Top Physics Stories in 2005 include the Most Precise Mass Calculation For Lattice QCD, an unquenched determination of the B_c mass by members of the HPQCD, Fermilab lattice and UKQCD collaborations published in Physical Review Letters this May.

Monday, December 05, 2005

The Nielsen-Ninomiya theorem

In recent posts in this series, we have been looking at naive, Wilson and staggered fermions. One of the things we have seen is how difficult it is to get rid of the doubler fermions; staggering did a good job at this, but still retained some of the doublers with all the problems they bring, while the Wilson term got rid of the doublers, but only at the expense of spoiling chiral symmetry, which brought on even worse problems. Why should the discretisation of fermions be so hard?

The answer lies in a theorem about lattice fermions, the celebrated Nielsen-Ninomiya no-go theorem, which states that it is impossible to have a chirally invariant, doubler-free, local, translation invariant, real bilinear fermion action on the lattice. The theorem comes from topological arguments: A real bilinear fermion action can be written as
$$S = \sum_{x,y} \bar{\psi}(x)M(x,y)\psi(y)$$
with hermitean $$M$$. Translation invariance means that
$$M(x,y)=D(x-y)$$
and locality requires that the Fourier transform $$\tilde{D}(p)$$ of $$D(z)$$ be a regular function of p throughout the Brillouin zone. Chiral symmetry
$$\{\tilde{D}(p),\gamma^5\}=0$$
requires that
$$\tilde{D}(p)=\sum_\mu \gamma_\mu d_\mu(p)$$
Since the Brillouin zone has the topology of a 4-torus, we thus have a vector field $$d_\mu$$ on the torus. Now it is possible to assign an "index" of +1 or -1 to every zero of this vector field, and the Hopf-Poincare index theorem states that the sum over the indices of the zeros of a vector field on a manifold is equal to the Euler characteristic of the manifold. The Euler characteristic of any n-torus is zero, and therefore the zeros of $$d_\mu$$ must come in pairs of opposite index, which is precisely the origin of the doublers.

OK, so what does all this mathematics mean? Well, prima facie it seems to leave us with the choice between chiral symmetry and freedom from doublers (since locality, translation invariance and hermiticity are too important to abandon). There is, however, a clever way around this, which will be the topic of our next post.

Friday, December 02, 2005

More about staggered quarks

A while back, Matthew was running a number of pedagogical articles on fermions on the lattice. Since I think that those articles were a good idea, I will endeavour to continue them. Obviously there may be some differences in outlook and style, but that is the beauty of diversity.

Matthew's last post in the series was about staggered quarks. To remind ourselves, when we put fermions on the lattice naively, we find that the fermion propagator has extra poles at momenta of order $$\pi/a$$, leading to the emergence of 16 degenerate quark flavours, or "doublers", from a single quark action. Staggering gets rid of some of those doublers by redistributing the fermionic degrees of freedom across different lattice sites. In the end, one is left with 4 degenerate quark flavours, usually referred to as "tastes" to distinguish them from physical quark flavours, with the added bonus of retaining a remnant of chiral symmetry that forbids the generation of an additive mass renormalisation.

There is a downside to all this, however. Since the different components of the staggered quark field live on different lattice sites, they experience a slightly different gauge field, which leads to a breaking of their naive degeneracy. This becomes even clearer when looking at it from a momentum space point of view: A pair of quarks with momenta close to 0 can exchange a gluon with momentum around $$\pi/a$$ to change into a pair of quarks with opposite momenta of order $$\pm\pi/a$$, and these correspond to quarks of a different taste from the original pair. The interaction has changed the taste of the quarks!

These taste-changing interactions are the source of a number of problems: naively, we would expect a theory of four degenerate quark flavours to have 16 degenerate pions. These pions, however, are mixed by the taste-changing interactions, and their degeneracy is therefore lifted. Only one of the 16 pions will be the (pseudo-)Goldstone boson whose mass goes to zero with the quark mass; the others will remain massive in the chiral limit. This also adversely affects the discretisation errors from the finite lattice spacing $$a$$.

The influence of the taste-changing interactions can be suppressed by adding additional terms to the lattice action. This leads to improved staggered quarks, and we will hear more about those in a future post on improved actions.

Another potentially problematic feature of staggered quarks is that they come always in four tastes. Nature, however, has not been so generous as to provide us with four degenerate, or even nearly degenerate, quark flavours. So how do we simulate a single flavour with staggered quarks?

Remember that the fermionic path integral could be done analytically:
$$
\int DU D\bar{\psi} D\psi exp(-S_{G}-S_{F}) = \int DU det(M[U]) exp(-S_{G})
$$
The fermionic determinant can be put back into the exponent as
$$
\int DU det(M[U]) exp(-S_{G}) = \int DU exp(-S_{G}-S_{GF})
$$
where
$$
S_{GF} = - log( det(M[U]) )
$$
incorporates the fermionic contributions to the action. This is additive in the number of quark flavours, so we can get from four staggered tastes to one physical flavour by dividing $$S_{GF}$$ by four, which is equivalent to taking the fourth root of the fermion determinant.

Taking the fourth root of the determinant introduces a nonlocality, and currently nobody kows with certainty whether that nonlocality will go away in the continuum limit $$a\to 0$$, but empirical evidence suggesting that it does is accumulating.

Thursday, December 01, 2005

Two-loop Lamb shift in U^{89+}

Researchers at Lawrence Livermore National Laboratory have used U^{89+} ions (that's Uranium stripped of all but three electrons) to measure the two-loop Lamb shift in Lithium-like ions at large Z, providing one of the most stringent tests of QED in strong fields so far. There are no theoretical predictions for Lithium-like ions, but when extrapolated to the Hydrogen-like case of U^{91+}, their results are in excellent agreement with theoretical predictions. It is always nice to see experiment agree with theory.