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.