The gradient flow coupling in the Schrödinger Functional.
We study the perturbative behavior of the Yang-Mills gradient flow in the Schrödinger Functional, both in the continuum and on the lattice. The energy density of the flow field is used to define a running coupling at a scale given by the size of the finite volume box. From our perturbative computation we estimate the size of cutoff effects of this coupling to leading order in perturbation theory. On a set of gauge field ensembles in a physical volume of fm we finally demonstrate the suitability of the coupling for a precise continuum limit due to modest cutoff effects and high statistical precision.
Keywords:Lattice Gauge Field Theories, Non-perturbative effects, QCD
Finite-volume renormalization schemes have now a long history in lattice field theory (see Luscher:1991wu (); Luscher:1992an (); deDivitiis:1994yp () or the pedagogical reviews Luscher:1998pe (); Sommer:1997xw ()). Asymptotic freedom tells us that at small distances QCD is well described by perturbation theory, while at large scales QCD is a strongly interacting theory. Instead of trying to accommodate these two scales in a single lattice simulation, the idea of finite-size scaling exploits the size of a finite volume world as renormalization scale. A single lattice simulation can resolve only a limited range of scales, but one can match different lattices and adopt a recursive procedure to cover a large range of scales. In this way one can connect the perturbative and non-perturbative regimes of QCD.
Beside a successful application of the finite-size scaling technique in the case of pure Yang-Mills theory Luscher:1992an (); deDivitiis:1994yp (), the running coupling Luscher:1992zx (); Luscher:1993gh (); DellaMorte:2004bc (); Aoki:2009tf (); Tekin:2010mm () and quark mass Capitani:1998mq (); DellaMorte:2005kg (); Aoki:2010wm () have been computed non-perturbatively in QCD with different flavour content by ALPHA and other collaborations. Since the general idea of finite-size scaling is a very powerful tool to solve scale dependent renormalization problems, it is not surprising that it is broadly used also in other strongly interacting theories, even in effective theories such as HQET Heitger:2003nj (); Sommer:2006sj (); Sommer:2010ic (). There has been a growing interest in other than QCD strongly interacting gauge theories, especially in connection with electroweak symmetry breaking and quasi-conformal behavior (see for example Neil:2012cb () and references therein). Finite-size scaling techniques are also a powerful tool to study these systems.
Basically there are two things that are needed to perform the previously sketched program. First one needs to define exactly what is meant by a finite-volume scheme, i.e., one has to specify the boundary conditions of the fields. Second, one needs a non-perturbative definition of the coupling. In principle there are many valid possibilities, but practical considerations have to be taken into account. Good options should allow for an easy evaluation of the coupling constant both in perturbation theory and in a numerical, non-perturbative (lattice) simulation.
The rest of this section is mainly dedicated to explain why we choose the Schrödinger functional (SF) scheme Luscher:1992an () as our finite-volume setup and the Wilson flow for a non-perturbative definition of the coupling Luscher:2010iy (). To simplify the following discussion we will argue about a pure gauge theory in 4-dimensional Euclidean space-time.
In the Schrödinger functional Luscher:1992an (); Sint:1993un () one embeds the fields in a finite volume box of dimensions . Gauge fields in the SF are periodic in the three spatial directions and have Dirichlet boundary conditions in time direction (i.e. one fixes the value of the gauge fields at ). The value of the gauge fields at the time boundaries are called boundary fields. One can interpret the partition function of the theory as the transition amplitude of the gauge field to propagate from the boundary value at to the boundary value at . Such a setup has nice properties in perturbation theory. In particular with a smart choice of the boundary fields one can guarantee that there is a unique gauge field configuration (up to gauge transformations) that is a global minimum of the action. This avoids some difficulties with perturbation theory Coste:1985mn (); GonzalezArroyo:1981vw (); Luscher:1982ma (). The reader interested in this issue will appreciate the original literature, as well as the nice discussion in Fodor:2012qh ().
Recently, the gradient flow has been used in different contexts Luscher:2009eq (); Narayanan:2006rf (); Lohmayer:2011si (), but it is the proposal made in Luscher:2010iy () to define a renormalized coupling through the gradient flow in non-abelian gauge theories what inspires this work. The gradient flow defines a family of gauge fields parametrized by a continuous flow time . The flow equation brings the gauge field towards the minimum of the Yang-Mills action, and therefore represents a smoothing process. The key point is that correlation functions of the smoothed gauge field defined at are automatically finite Luscher:2011bx (). One can use the expectation value of the energy density,
where is the field strength of the gauge field at flow time , to give a non-perturbative definition of the gauge coupling. This idea was applied to set the scale in lattice simulations Luscher:2010iy (); Borsanyi:2012zs (), to tune anisotropic lattices Borsanyi:2012zr () and more recently in a similar context of this work (finite-size scaling, but using a box with periodic boundary conditions) to compute the step scaling function in with four fermion species Fodor:2012td ().
In this paper we investigate the perturbative behavior of the Wilson flow in the Schrödinger functional. This motivates us to propose a gradient flow coupling
with a normalization factor to be determined later, valid for an arbitrary gauge field coupled (or not) to fermions. Relating and the coupling depends only on one scale, the size of the finite volume box, and therefore can be used for a finite-size scaling procedure in the same way as the traditional SF coupling.
The paper is organized as follows: in the next section we investigate the perturbative behavior of in the SF, both in the continuum and on the lattice. Section 3 uses this information to define the gradient flow coupling in the SF, and to discuss some practical issues: cutoff effects, boundary fields and fermions. In section 4 we investigate this coupling numerically on a set of lattices in a physical volume of fm and finally conclude in section 5. Details needed for the computation have been summarized in form of appendices: a summary with some useful notation A, heat kernels B, propagators in the SF C and finally some practical details on how to integrate the Wilson flow in numerical simulations D.
2 Perturbative behavior of the Wilson flow in the SF
We would like to start this section by recalling the original proposal of using the Wilson flow and the energy density as a definition for a coupling in gauge theories Luscher:2010iy (). Later it will become clear what role the SF setup plays.
By considering the gauge fields to be functions of an extra flow time , not to be confused with Euclidean time, denoted , the Wilson flow is defined by the non-linear equation
is the field strength. Due to gauge fields along the flow become smoother, eventually reaching a local minimum of the Yang Mills action: the flow smooths the fields over a region of radius . The somewhat surprising result of Luscher:2010iy (); Luscher:2011bx () is that correlation functions made of this smoothed field have a well-defined continuum limit. In particular the energy density in Yang-Mills theory in infinite volume has the perturbative behavior
At a scale , is a numerical constant and is the renormalized coupling in the scheme. Therefore one can define a running coupling constant from
These expressions are valid in infinite volume. What about the Schrödinger Functional? The computation is completely analogous, but we have to impose the correct boundary conditions to the gauge fields. As we have mentioned in the SF gauge fields are restricted to a box of dimensions . They are periodic in the three spatial directions and the spatial components have Dirichlet boundary conditions at and . We are going to work exclusively with zero boundary fields, which means
The flow equation (3) has to be solved maintaining these boundary conditions at all flow times . To apply the idea of finite-size scaling, as has previously been done in Fodor:2012qh () in a periodic box, one simply has to run the renormalization scale with the size of the finite volume box given by via
Here is a dimensionless constant that represents the fraction of the smoothing range over the total size of the box. In this way the flow coupling will not depend on any scale other than . The renormalization scheme will depend on the values of , and111Note that in the SF the boundary conditions break the invariance under time translations. Therefore will depend explicitly on .
where will be computed in the next section in order to ensure
Our computation follows the lines of Luscher:2011bx (). First we consider the modified flow equation
One can transform a solution of the last equation into a solution of the canonical flow equation (3) (corresponding to ) by a flow-time dependent gauge transformation. In particular, if is a solution of (12) one can construct a solution of (3) via
as long as obeys the equation
This shows that gauge-invariant quantities are independent of . For instance, setting turns out to be a very convenient choice for perturbative computations. Due to the periodicity in the spatial directions it is natural to expand the gauge fields as
As already mentioned, in the SF the gauge field is periodic in the three spatial directions and its spatial components have Dirichlet boundary conditions in time, eq. (7) and (8) respectively. On the other hand the boundary conditions of the time component of the gauge field are not fixed but naturally emerge through the gauge fixing condition.222The authors want to thank M. Lüscher for helping us to understand this point. To properly derive the boundary conditions for it is convenient to work in the lattice formulation and derive the boundary conditions by taking the continuum limit. We will postpone this derivation to the next section and simply state the result here: obeys Neumann boundary conditions at non-vanishing spatial momentum, while for zero momentum obeys mixed boundary conditions. Thus in the present set-up the full set of boundary conditions reads
The modified Wilson flow equation with is given by
After rescaling the gauge potential with the bare coupling , the flow becomes a function of the coupling
Inserting this expression in the modified flow equation, we find that to leading order in the flow equation is just the heat equation with initial condition :
i.e., to leading order the Wilson flow is the heat flow. We also observe that different momentum modes do not couple to each other at this order. Together with the fact that the zero momentum mode does not contribute to the observable of interest, , we can safely neglect the special treatment that the boundary conditions of the zero momentum mode would otherwise require in the following discussion.
We have to solve the heat equation respecting the boundary conditions (16). This is easily done by using appropriate heat kernels
Since the boundary conditions of the field are inherited from the boundary conditions of the heat kernels, we have to choose them with the correct boundary conditions. Heat kernels with either Dirichlet () or Neumann () boundary conditions can be constructed from the basic periodic () heat kernel in given by
Explicit expressions are given in appendix B.
Our observable, the energy density , has an expansion in powers of . The leading contribution is given by
We are going to split the computation in two parts, one involving only the spatial components of , and the other involving the mixed time-space components of
The final result is obtained inserting the SF gluon propagator Luscher:1996vw (); Weisz:int1996 (). Since our observable is invariant under gauge transformations of the field we will use the Feynman gauge, where the expression for the gluon propagator turns out to be more easy (for additional details see appendix C)333We have checked that the result is independent of the gauge choice..
To shorten notation we use
After some algebraic work one arrives at the expression
and a very similar computation leads to
On the lattice one defines the Wilson flow as
If is an arbitrary function of the link variable , the components of its Lie-algebra valued derivative are defined as
In a neighborhood of the classical vacuum configuration the lattice fields and are parametrized as follows:
2.3.1 Gauge fixing
To simplify our perturbative computations it is useful to study a modified equation with a gauge damping term. It is easy to check that the lattice flow equation (31) is invariant under flow-time independent gauge transformations. On the other hand one can consider the modified equation
with and the forward lattice covariant derivative acting on Lie-algebra valued functions according to
With we denote the forward/backward finite differences respectively as defined in appendix A.
and therefore one can freely choose the function . To fix the gauge the most natural choice is to use the same functional that is used for the conventional gauge fixing. As is detailed in appendix C, we choose
with initial condition
Note does not depend on at and , as a decent gauge transformation should be in the Schrödinger functional according to our conventions (see appendix C for details).
We observe that on the lattice the time component of the gauge field is completely free and does not obey any particular boundary conditions. To understand how the boundary conditions for arise in the continuum theory, one can extend the domain of definition of to and choose to fix the additional variables with the condition
This equation can be interpreted as a boundary condition for the field. In particular has Neumann boundary conditions at , except for its spatial momentum zero mode that has a mixture of Neumann boundary conditions at and Dirichlet boundary conditions at .
2.3.2 Behaviour of in lattice perturbation theory
We again note that the value of any gauge invariant observable is independent of our choice of in equation (41). In particular, with the choice the modified flow equation reads
and to first order in
Using periodicity in the spatial directions, we expand
and the flow equation becomes
where is the usual spatial lattice momentum, see appendix A.
Now we have to solve a special type of heat equation in which the Laplacian is substituted by a discrete version, but the flow time remains a continuous variable. The strategy is very similar: We find the fundamental solutions of this equation, i.e., the discrete heat kernels given in appendix B, and write
Then we have to insert this in our lattice observable . We use the clover definition for that to leading order in reads
where . The computation is completed by using the lattice gluon propagator
For the spatial part of the contribution to the energy density we arrive at
while for the mixed part we obtain
The definitions of the lattice momenta , , and the functions are summarized in appendix A.
There are several tests that can be performed to check the previous computations. Let us first concentrate on the continuum computation. At fixed , the infinite volume limit (with kept constant) is taken through . For this case the continuum expression transforms into an integral via
and we obtain
thus recovering the infinite volume result of Luscher:2011bx ().
Another rather obvious check is that one should recover the continuum result from the lattice expression in the limit . This can be easily checked by noting that the sums (51) and (52) are dominated by terms with small lattice momenta .
Finally we have performed some simulations with the openQCD code Luscher:2012av () at small values of the bare coupling in a pure gauge theory. Using a lattice and varying the value of the bare coupling () we compare the analytical lattice prediction and the numerical results after collecting 10000 measurements of the gradient flow coupling for each value of . We use the clover definition for to compute the value of
The lattice computation of
can be checked in the following way: plotting
versus one expects a linear behavior with zero intercept for all values of and . A couple of typical cases are shown in figure 1, while table 1 shows the results of the of the fits and the intercepts for all values of and .
All intercepts are of the order and compatible with zero within errors. The difference in for different values of , or between the continuum and the lattice result varies between 5% and 10%. We note that this last test is highly non-trivial since it is done for arbitrary at on a small lattice where cutoff effects tend to be larger.
3 Definition of the flow coupling
Using our continuum result
we define the gradient flow coupling for non-abelian gauge theories in the SF by means of
This definition of the coupling is valid if the gauge field is coupled to fermions in arbitrary representations. As the reader may have noticed the scheme that defines the coupling depends not only on the quantities , but also on the value of the fermionic phase angle and the background field. In the simulations of the Schrödinger functional it is customary to include a phase angle in the fermionic spatial boundary conditions. In principle different values of are different schemes, although we have observed in some practical situations that the difference of the gradient flow coupling between and is below the 2%.
Up to now we have worked exclusively with zero background fields, but the generalization to other values is straightforward. It only requires the modification of the heat kernels to preserve the value of the boundary fields and a modified form of the propagator Weisz:int1996 (). Nevertheless common wisdom suggests that cutoff effects are reduced for zero background field, therefore we prefer to work in this scheme. In this case the definition of the coupling is also symmetric about and we choose that value to minimize boundary effects. Also choosing seems reasonable and leaves us with a one-parameter family of couplings, parametrized by the smoothing ratio .
By comparing the lattice and continuum behavior of the energy density as a function of we can compute the leading order size of cutoff effects in the gradient flow coupling. As the reader can see in figure 2, the cutoff effects are large for small values of , reach a minimum around and then grow again. We recall that with the smoothing radius is equal to , and therefore one is effectively smoothing over all the lattice. For cutoff effects are smaller than 10% for a lattice of size , while for even the lattice has cutoff effects of about 10%.
This figure suggests using as a preferred scheme, but later, when lattice simulations enter into the game, we will see that the statistical errors of the coupling also grows with , and therefore in practice it is better to stay with , probably depending on the particular case, but this is the subject of the next section.
We would also like to comment that if one is performing numerical simulations with the Wilson gauge action, one can benefit from smaller cutoff effects by using the lattice prediction to normalize the coupling. Defining
the coupling is given by
Obviously both definitions of the coupling differ only by cutoff effects.
We finally want to mention that it is possible to define analogous couplings by using only the spatial components . In a lattice simulation one stays further away from the boundaries by not including plaquettes with links in the time direction. This may result in smaller cutoff effects, although this point needs further investigations.
4 Non-perturbative tests
In this section we would like to analyze the gradient flow coupling numerically. We want to estimate both the size of cutoff effects and the numerical cost of evaluating the new gradient flow coupling. The main result of this section is that both quantities depend on the particular scheme via the parameter . When is increased cutoff effects decrease, but the numerical cost increases. We find that the window of values allows a very precise determination with a mild continuum extrapolation.
4.1 Line of constant physics
As framework for our tests we choose a set of Schrödinger functional simulations at a line of constant physics as given through