Numerical simulations with two flavours of twisted-mass Wilson quarks and DBW2 gauge action

Abstract

Discretisation errors in two-flavour lattice QCD with Wilson quarks and DBW2 gauge action are investigated by comparing numerical
simulation data at two values of the bare gauge coupling. Both non-zero- and zero-twisted-mass values are considered. The
results, including also data from simulations using the Wilson plaquette gauge action, are compared to next-to-leading order
chiral perturbation theory formulas.

Full text

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arXiv:hep-lat/0512017v2 12 Apr 2006
DESY 05-250
MS-TP-05-33
SFB/CPP-05-83
Numerical simulations with two flavours of
twisted-mass Wilson quarks and DBW2 gauge action
F. Farchionia, P. Hofmanna, K. Jansenb, I. Montvayc,
G. M¨ unstera, E.E. Scholzc∗, L. Scorzatod, A. Shindlerb,
N. Ukitac†, C. Urbachb,e‡, U. Wengerb§, I. Wetzorkeb
aUniversit¨ at M¨ unster, Institut f¨ ur Theoretische Physik,
Wilhelm-Klemm-Strasse 9, D-48149 M¨ unster, Germany
bNIC/DESY Zeuthen, Platanenallee 6, D-15738 Zeuthen, Germany
cDeutsches Elektronen-Synchrotron DESY, Notkestr.85, D-22603 Hamburg, Germany
dInstitut f¨ ur Physik, Humboldt Universit¨ at zu Berlin, D-12489 Berlin, Germany
eFreie Universit¨ at Berlin, Institut f¨ ur Theoretische Physik,
Arnimallee 14, D-14196 Berlin, Germany
Abstract
Discretisation errors in two-flavour lattice QCD with Wilson-quarks and DBW2
gauge action are investigated by comparing numerical simulation data at two values of
the bare gauge coupling. Both non-zero and zero twisted mass values are considered.
The results, including also data from simulations using the Wilson plaquette gauge
action, are compared to next-to-leading order chiral perturbation theory formulas.
∗Present address: Physics Department, Brookhaven National Laboratory, Upton, NY 11973 USA
†Present address: Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan and
Department of Physics, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
‡Present address: Theoretical Physics Division, Dept. of Mathematical Sciences, University of Liverpool,
Liverpool L69 3BX, UK
§Present address: Institute for Theoretical Physics, ETH Z¨ urich, CH-8093 Z¨ urich, Switzerland
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1Introduction
The singular point of QCD at vanishing quark masses is distorted in Wilson-type lattice
formulations: as a result of lattice artefacts, in the region of small quark masses an
extended phase structure is developed. This phase structure can be predicted and
classified in Chiral Perturbation Theory (ChPT) [1] if lattice artefacts are taken into
account [2]. If, in addition to the usual quark mass parameter, a twisted quark mass
is introduced [3, 4] then in the plane of untwisted and twisted quark mass a first order
phase transition line with second order endpoints appears. Depending on the sign of
the leading term representing lattice artefacts, the first order phase transition line is
either on the untwisted quark mass axis (“Aoki-phase scenario” [3]) or perpendicular
to it (“normal scenario”) [5, 6, 7].
In numerical simulations it pays off to try to reduce lattice artefacts at fixed (non-
vanishing) lattice spacing by an appropriate choice of the lattice action. An important
issue in this respect is to bring the phase structure at small quark masses as close
as possible to the point-like singularity appearing in the continuum limit. In fact,
the strong first order phase transition observed earlier in numerical simulations with
Wilson-type quarks [8, 9, 10] presents a serious obstacle for QCD simulations with light
quarks.
In previous work we systematically investigated the phase structure of lattice QCD
with twisted-mass Wilson-type quarks (for a recent review see [11]). In Ref. [12] we have
shown that at lattice spacings near a ≃ 0.2fm the phase structure with Wilson-quarks
and Wilson-plaquette gauge action is consistent with the “normal scenario” of ChPT.
This differs from the situation in the strong coupling regime, where the “Aoki-phase
scenario” has been previously observed [13].
A consequence of the “normal scenario” is that for fixed gauge coupling (β) the
mass of charged pions have a positive lower bound (mmin
data in Ref. [12] have shown that this lower bound is at a ≃ 0.2fm quite high, namely
about 600MeV. Such a high lower bound would prohibit the study of light quarks.
Therefore, an important question is the behaviour of this lower bound as a function of
the gauge coupling (or lattice spacing) towards the continuum limit. In a subsequent
paper it has been shown [14] that, as expected, the lower bound becomes clearly smaller
for decreasing lattice spacing. Its decrease in the range 0.20fm ≥ a ≥ 0.14fm is roughly
consistent with the prediction of next-to-leading-order (NLO) ChPT [2, 5, 6, 7, 15],
namely mmin
π
∝ a (at aµ = 0). A minimal pion mass of mmin
to occur near a ≈ 0.07 − 0.10fm, but this estimate is rather uncertain and has to be
checked in future simulations if the Wilson gauge action ought to be used. The question
arises whether one could lower mmin
π
by a suitable change of the lattice action.
An early observation by the JLQCD Collaboration has been [9] that the strength
π
). The numerical simulation
π
≃ 300MeV is estimated
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of the first order phase transition near zero quark mass is sensitive to a change of
the gauge action. Following this hint, we have shown in a previous paper [16] that
combining two flavours (Nf= 2) of Wilson-quarks with the DBW2 gauge action [17]
leads to a phase structure near zero quark mass with substantially weaker first order
phase transition. As a consequence, the minimal pion mass is at least by a factor of
two lower compared to the plaquette gauge action at similar lattice spacings.
This implies that numerical simulations with light quarks become possible on
coarser lattices and hence with much less computational costs if the DBW2 gauge
action is used. Of course, for the choice of the gauge action also other criteria may be
relevant. For instance, it has been reported in quenched studies [18, 19] that in some
quantities strong scale breaking effects appear if the DBW2 action is used. Another
problem could be the late convergence of lattice perturbation theory, implied by the
results of the QCDSF Collaboration [20].
In general, the question of the scaling behaviour of the results obtained by a given
lattice action is very important. In case of the Wilson twisted-mass formulation of
lattice QCD it has been shown [21] that the leading lattice artefacts are of O(a2) if
the bare quark masses are appropriately tuned. Detailed investigations have shown
[22, 23, 24] that in the quenched approximation excellent scaling behaviour can be
achieved, indeed, also at light quark masses. The same question in the full theory with
dynamical quarks is obviously very important.
In the present paper we perform first exploratory scaling tests for the combination
of Wilson-fermion lattice action with the DBW2 gauge action by comparing numerical
simulation data at two values of the gauge coupling, namely β = 0.67 and β = 0.74.
We consider data points with both vanishing and non-vanishing value of the twisted
mass. Moreover, since one can extract useful information on multiplicative renormali-
sation factors from the dependence of matrix elements on the twist angle in the plane
of untwisted and twisted quark mass, we exploit this method and derive from our
simulation data the values of ZV, ZAand ZP/ZS. In addition, we compare the NLO-
ChPT formulas of Refs. [5, 6, 7, 15, 25] to the results of the numerical simulations. For
comparison, ChPT fits of the data obtained by the Wilson plaquette gauge action [14]
are also considered.
The outline of the paper is as follows: in the next section, after specifying the lattice
action and the simulation algorithms, the numerical simulation runs are discussed and
some scaling tests are presented. Section 3 is devoted to a detailed description of the
results on the twist angle in the plane of untwisted and twisted quark mass together
with an explanation how the aforementioned multiplicative renormalisation Z-factors
can be determined. The knowledge of the twist angle and Z-factors makes it possible
to obtain results on physical quantities, such as the quark mass and the pion decay
constant. In Section 4 the ChPT fits of the data with DBW2 gauge action are presented.
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Section 5 contains a discussion and a summary. In an Appendix alternative chiral fits
of the DBW2 data are shown and compared to similar ChPT fits of Wilson plaquette
data.
2Numerical simulations
The lattice action and simulation algorithms are defined here for the reader’s conve-
nience. The notations are similar to those in Ref. [16].
2.1Lattice action and simulation algorithms
We apply for quarks the lattice action of Wilson fermions, which can be written as

Sq=
?
x

(χx[µκ+ iγ5τ3aµ]χx) −1
2
±4
?
µ=±1
?
χx+ˆ µUxµ[r + γµ]χx
?

.

(1)
Here the (“untwisted”) bare quark mass in lattice units is denoted by
µκ≡ am0+ 4r =
1
2κ,
(2)
r is the Wilson-parameter, set in our simulations to r = 1, am0is another convention for
the bare quark mass in lattice units and κ is the conventional hopping parameter. The
twisted mass in lattice units is denoted here by aµ. (This differs from the notation in
[16] where µ has been defined without the lattice spacing factor a in front.) Uxµ∈ SU(3)
is the gauge link variable and we also defined Ux,−µ= U†
For the SU(3) Yang-Mills gauge field we apply the DBW2 lattice action [17] which
belongs to a one-parameter family of actions obtained by renormalisation group con-
siderations. Those actions also include, besides the usual (1×1) Wilson loop plaquette
term, planar rectangular (1 × 2) Wilson loops:

c0
with the condition c0= 1 − 8c1. For the DBW2 action we have c1= −1.4088.
For preparing the sequences of gauge configurations two different updating algo-
rithms were used: the Hybrid Monte Carlo (HMC) algorithm [26] with multiple time
scale integration and mass preconditioning as described in [27] and the two-step multi-
boson (TSMB) algorithm [28] which has been tuned for QCD applications following
[29, 12].
x−ˆ µ,µand γ−µ= −γµ.
Sg= β
?
x
4
?
µ<ν;µ,ν=1
?
1 −1
3ReU1×1
xµν
?
+ c1
4
?
µ?=ν;µ,ν=1
?
1 −1
3ReU1×2
xµν
?
,(3)
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2.2Simulation parameters and a first scaling test
In our numerical simulations we considered two values of the gauge coupling, namely
β = 0.67 and β = 0.74. The simulations at the lower β-value have been performed
on a 123· 24 lattice as in [16]. The higher β-value (β = 0.74) was chosen in such a
way that the physical volume of the 163· 32 lattice remains approximately the same,
that is a(β = 0.74) ≃3
extrapolating the Sommer scale parameter in lattice units r0/a [30] to zero quark mass
and assuming r0≡ 0.5fm. The simulation parameters and the amount of statistics are
specified in Table 1.
As Table 1 shows, both zero and non-zero twisted mass points were simulated. The
non-zero values of the twisted mass were also chosen according to the assumed scale
ratio, that is aµ(β = 0.74) =
4aµ(β = 0.67) = 0.0075. In other words, the bare
twisted mass µ is kept (approximately) constant.
In several points of the parameter space simulation runs have been performed with
both the HMC and the TSMB updating algorithms. Having run the two algorithms in
the same points allowed to compare their performance. It turned out that the optimised
HMC algorithm of Ref. [27] is substantially faster than TSMB. For instance, in long
runs at the simulation point (A) (163· 32 lattice, β = 0.74, κ = 0.1580, aµ = 0) HMC
with multiple time scale integration and mass preconditioning is almost by a factor
of 10 faster. Therefore, in the majority of simulation points the final data analysis is
based on HMC runs. Results from TSMB updating were only used in the runs of the
first part of Table 1 (those at β = 0.67 and aµ = 0). Even if results with both updating
algorithms were available in several other points, in the final analysis we never mixed
results from different updating procedures.
The results for some basic quantities are collected in Tables 2 and 3. The pseu-
doscalar meson (“pion”) mass amπ is obtained from the correlator of the charged
pseudoscalar density
x= ¯ χxτ±
4a(β = 0.67). The value of the lattice spacing was defined by
3
P±
2γ5χx
(4)
where τ±≡ τ1± iτ2. In case of the vector meson (“ρ-meson”) mass amρ, for generic
values of the bare untwisted and twisted quark mass, the correlators of both vector
(Va
xµ) bilinears of the χ-fields can be used:
xµ) and axialvector (Aa
Va
xµ≡ χx
1
2τaγµχx,Aa
xµ≡ χx
1
2τaγµγ5χx
(a = 1,2) .(5)
The reason is that the physical vector current is, in general, a linear combination of
Va
xµ(see Section 3). In a given simulation point we determined amρfrom the
correlator possessing the better signal.
In Table 3 the values of the bare (untwisted) PCAC quark mass amPCAC
given. It is defined by the PCAC-relation containing the axialvector current Aa
xµand Aa
χ
are also
xµin
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(5) and the pseudoscalar density P±
x:
amPCAC
χ
≡?∂∗
µA+
2?P+
xµP−
x P−
y?
y ?
.(6)
Here ∂∗
Besides amPCAC
doscalar decay constant afχπdefined by
µdenotes, as usual, the backward lattice derivative.
, Table 3 also contains the values of the bare “untwisted” pseu-
χ
afχπ≡ (amπ)−1?0|A+
x=0,0|π−? .(7)
The relation of the bare (untwisted) quantities amPCAC
physical quantities will be discussed in the following section.
The squared ratio of the pion mass to the ρ-meson mass is plotted in Figure 1 as
a function of (r0mπ)2, both of which are expected to be approximately proportional
to the quark mass for small quark masses. (This holds if the effect of the “chiral
logarithms” is negligible in the quark mass depedence of m2
constant near zero as a function of the quark mass.) The straight line in the figure
connects the origin and the point with the physical values mπ = 140MeV, mρ =
770MeV and r0 = 0.5fm.As the figure shows, in this plot there are observable
scale breaking effects between β = 0.67 and β = 0.74, but the β = 0.74 points are
already close to the continuum expectation. Within the (large) statistical errors there
is no noticeable difference between the points with vanishing and non-vanishing twisted
mass. (According to Table 9 the twisted mass values are given by r0µ = 0.02845(68)
and r0µ = 0.0283(15) for β = 0.67 and β = 0.74, respectively.)
χ
and afχπto the corresponding
πand if r0is approximately
3Twist angle and renormalisation factors
3.1Twist angle
In this section we discuss the determination of the twist angle ω. For given (µκ,aµ) this
is defined as the rotation angle relating twisted-mass QCD (TMQCD) to the physical
theory QCD. An important point is that the connection can be made only after (lattice)
renormalisation of the theory. The renormalisation of the local bilinears in the Wilson
twisted-mass formulation is therefore involved. Some of the arguments of this section
were already discussed in previous publications of this collaboration [31, 16].
Following [32] we operationally define [31, 16] the twist angle ω as the chiral rota-
tion angle between the renormalised (physical) chiral currents and the corresponding
bilinears of the twisted formulation. We denote withˆVa
and axialvector currents, while Va
xµare the bilinears of the χ-fields defined in
Eq. (5). In order to establish the correspondence with the physical currents, the bilin-
ears of the χ-fields have to be properly renormalised. This is obtained, as in QCD, by
xµandˆAa
xµthe physical vector
xµand Aa
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multiplying them by the respective renormalisation constants ZV and ZA. In a mass
independent scheme these are functions of β alone and coincide with the analogous
quantities in Wilson lattice QCD for the same value of β. So the relation reads:
ˆVa
xµ
=ZVVa
xµcosω + ǫabZAAb
xµsinω ,(8)
ˆAa
xµ
=ZAAa
xµcosω + ǫabZVVb
xµsinω(9)
where only charged currents are considered (a=1,2) and ǫabis the antisymmetric unit
tensor.
The conserved vector current of the χ-fields
˜Va
xµ≡1
4
?
χx+µτaUxµ(γµ+ r)χx+ χxτaU†
xµ(γµ− r)χx+µ
?
(10)
satisfies by construction the correct Ward-Takahashi identity of the continuum. In this
case the Formulas (8), (9) apply with ZV replaced by 1, in particular
ˆAa
xµ= ZAAa
xµcosω + ǫab˜Vb
xµsinω .(11)
In practical applications it is useful to define two further angles ωV and ωA:
ωV = arctan(ZAZ−1
Vtanω) ,ωA= arctan(ZVZ−1
Atanω) .(12)
In terms of ωV, ωAEqs. (8) and (9) read
ˆVa
xµ
ˆAa
xµ
=
NV(cosωVVa
NA(cosωAAa
xµ+ ǫabsinωVAb
xµ) , (13)
=
xµ+ ǫabsinωAVb
xµ).(14)
The unknown multiplicative renormalisations are now contained in an overall factor
(X = V,A):
NX=
ZX
cosωX
√1 + tanωVtanωA
.(15)
From the definition (12) it follows
ω = arctan
?√tanωVtanωA
tanωV/tanωA.
?
(16)
ZA
ZV
=
?
(17)
As already proposed in [31, 16], we determine the twist angle ω by imposing parity-
restoration (up to O(a) precision) for matrix elements of the physical currents. Due
to the presence of unknown lattice renormalisations, two conditions are required. The
most suitable choice in the case of the vector current is
?
? x
?ˆV+
x0P−
y? = 0 .(18)
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Indeed, for asymptotic times, the pion state dominates the matrix element1and the
condition reads
?0|ˆV+
In case of the axialvector current we choose the condition2
x0|π−? = 0 . (19)
?
? x,i
?ˆA+
xiˆV−
xi? = 0(20)
or asymptotically
?0|ˆ A+
xi|ρ−? = 0 .(21)
In terms of (13), (14) Eqs. (18), (20) admit the solution
tanωV
=
−i
?
?
? x?V+
? x?A+
x0P−
x0P−
y?
y ?, (22)
tanωA
=
−i?
?
? x,i?A+
? x,i?V+
xiV−
xiV−
yi?+tanωV
yi?+itanωV
?
?
? x,i?A+
? x,i?V+
xiA−
xiA−
yi?
yi?
. (23)
Eqs. (16), (17), (22) and (23) allow the numerical determination of ω and of the ratio
ZA/ZV.
It is obvious that the definition of the twist angle in the lattice theory is subject
to O(a) ambiguities. Different choices of the parity-restoration conditions, including
also the form of the lattice currents, result in different definitions of the twist angle
differing by O(a) terms. The situation of full twist corresponds to ω = ωV = ωA= π/2.
Numerically it is most convenient to use ωV = π/2 as a criterion. The reason is that
a safe determination of the twist angle is obtained in the asymptotic regime where the
lightest particle dominates as intermediate state. This is the pseudoscalar state in the
case of ωV which, as one would expect, delivers a better signal than the vector meson
in case of ωA. Therefore we impose [31, 16]
ωV =π
2
⇐⇒
?
? x
?A+
x0P−
y? = 0(24)
or asymptotically
?0|A+
x0|π−? = 0(25)
and denote with µκcrthe corresponding value of µκfor the given µ.
1At small time-separations, due the O(a) breaking of parity, intermediate states with “wrong” parity may
still play a role.
2In [31, 16] the use of the temporal component for the currents was proposed. This choice is however
not optimal: a scalar state with positive parity dominates in this case the matrix element in the continuum
limit, but at finite lattice spacing the O(a) breaking of parity introduces contamination by pion intermediate
states which eventually dominate for light quark masses.
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Another possible determination of ωV is obtained by replacing in (22) the currents
with their divergences. For simplicity, we consider the case of the conserved vector
current which avoids the introduction of a renormalisation constant:
?
?
Here in the last step [7, 15] the Ward identity for the conserved vector current
cot ˜ ωV = i
? x?∂∗
? x?∂∗
µA+
µ˜V+
xµP−
xµP−
y?
y ?
=
mPCAC
χ
µ
.(26)
∂∗
µ˜V+
xµ= 2iµP+
x
(27)
and the definition (6) of the “untwisted” PCAC quark mPCAC
local vector current defined in Eq. (5) is used for the determination of ωV instead of
the conserved one, in Eq. (26) the introduction of the renormalisation constant ZV is
required. In this case one has
?
?
where ZV is determined as explained in the next subsection. Using the definition (12)
for ωV one arrives at the following relation involving this time the twist angle ω:
χ
have been used. If the
cotωV = i
? x?∂∗
? x?∂∗
µA+
µV+
xµP−
xµP−
y?
y ?
= ZV
mPCAC
χ
µ
, (28)
cotω = ZA
mPCAC
χ
µ
. (29)
Notice that the factor ZV cancels in this relation which is, therefore, independent of
the choice for the vector current employed for the determination of the twist angle ω.
One can simply show that the two determinations of ωV given by Eqs. (22) and
(28) coincide under the assumption that the ratio of the correlators is independent
of the time separation; this is in particular true for asymptotic times where the pion
dominates.
To have an effective automatic O(a) improvement, meaning without large O(a2)
effects, the critical line (µκcr(a,µ),µ) has to be fixed in such a way that the lattice
definition of the untwisted quark mass (e.g. mPCAC
from mass independent O(a) errors. For a definition of the critical line where this
condition is not necessarily satisfied, one has to make sure that µ > aΛ2.
The issue of the choice of the critical untwisted mass has been raised by the work of
Aoki and B¨ ar [33] and by the numerical results obtained in [34]. This problem has been
further analyzed in several aspects [15, 35, 36]. In [33, 15, 36] the theoretical framework
is twisted mass chiral perturbation theory (tmChPT) [25] where the cutoff effects are
included in the chiral lagrangian along the lines of [2, 46]. The works [33, 15] agree on
the fact that choosing the critical mass by imposing mPCAC
have automatic O(a) improvement down to quark masses that fulfill µ ≃ a2Λ3. In [35] a
Symanzik expansion was performed (in an approach different from that of refs. [33, 15],
χ
defined above) is free, on that line,
χ
= 0 (or ωV = π/2) allows to
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cf. [15] for a discussion) confirming the results of [33, 15]. For a discussion of these
issues in numerical studies within the quenched approximation see [22, 23, 24] and the
review [11].
3.2Determination of ZV
We adopt here the procedure well known in QCD which relies on the non-renormalisa-
tion property of the conserved current˜Vxµ[38]. A possible determination of ZV in
TMQCD is given by
=?0|˜V+
?0|V+
Note that in TMQCD the time component of the vector current couples the vacuum
to the pseudoscalar particle: in the most interesting region near full twist this coupling
is maximal. (Note that at aµ = 0 the analogous procedure has to rely on the noisier
matrix element with the vector particle or on three point functions.) Alternatively
ZV can be determined without direct use of the conserved current by exploiting the
(exact) Ward identity for the vector current. This implies [39]
Z(1)
V
x=0,0|π−?
x=0,0|π−?.(30)
?0|˜V+
x=0,0|π−? =−2iµ
mπ
?0|P+
x=0|π−? .(31)
Inserting the above relation in (30) a second determination of ZV is obtained:
Z(2)
V
=−2iµ?0|P+
mπ?0|V+
x=0|π−?
x=0,0|π−?
.(32)
Z(1)
V
obtain a mass independent determination of ZV by extrapolating Z(i)
(mPCAC
χ
= 0). In this situation the theory is O(a) improved and the Z(i)
estimate of ZV with O(a2) error (also including O((µa)2) terms).
and Z(2)
V
(differing by O(a) terms) are mass dependent renormalisations. We
V
to full twist
deliver an
V
3.3Physical quantities
The knowledge of the twist angle ω allows the derivation of physical quantities of
interest in QCD for a generic choice of (µκ,aµ).
quark mass and the pion decay constant. It is convenient [39, 40, 22] here to use the
conserved vector current since it possesses already the right continuum normalisation.
The physical PCAC quark mass mPCAC
q
can be obtained from the Ward identity for the
physical axialvector current:
Let us consider the case of the
?∂∗
µˆA+
xµP−
y? = 2amPCAC
q
?P+
xP−
y? . (33)
We use Eq. (8) in order to eliminate Aa
xµin (11) for ω ?= 0
xµcotω + ǫab˜Vb
ˆAa
xµ= −ǫabˆVb
xµ(sinω)−1
(34)
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and insert the result in the Ward identity (33) using isospin invariance forˆVa
result we obtain:
−i
2sinω
xµ. As a
amPCAC
q
=
?∂∗
?P+
µ˜V+
xP−
xµP−
y ?
y?
=
µ
sinω,
(35)
where in the last step we used once again the Ward identity (27). Inserting Eq. (29)
into the last expression in the above equation, we arrive at the following relation for
the untwisted quark mass
mPCAC
χ
= mPCAC
q
Z−1
Acosω .(36)
In the remainder we shall also make use of a definition of the untwisted quark mass
which already incorporates the renormalisation factor of the axial current:
¯ mPCAC
χ
= mPCAC
q
cosω = ZAmPCAC
χ
.(37)
Analogously, for the physical pion decay constant fπwe use
afπ= (amπ)−1?0|ˆ A+
x=0,0|π−? = −i(amπsinω)−1?0|˜V+
x=0,0|π−? . (38)
Also here the matrix element on the right hand side can be replaced by the matrix
element of the pseudoscalar density as in (31) giving
afπ=
−2aµ
(amπ)2sinω?0|P+
x=0|π−? . (39)
Let us note that here the normalisation of fπcorresponds to a phenomenological value
≈ 130MeV. If the local vector current is used in (38) instead of the conserved one, a
factor ZV is missing:
afvπ= −i(amπsinω)−1?0|V+
x=0,0|π−? ,fvπ= Z−1
Vfπ.(40)
3.4Results
In Fig. 2 the local determination of ωV and ωA is shown as a function of the time
separation for a specific simulation point at positive untwisted quark mass. The nu-
merical values of the twist angles ωV, ωAand ω are reported in Table 4. Notice that
the simulation point at β = 0.74 and κ = 0.159 is almost at full twist.
Figs. 3 and 4 show the determinations of µκcrby extrapolating mPCAC
to zero. The theoretical dependence of the twist angle upon the untwisted bare quark
mass µκcan be obtained [16] by starting from the equation [37]
χ
and cotωV
cotω =mχR
µR
+ O(a)(41)
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where µRand mχRare the renormalised twisted and untwisted quark masses in the
continuum limit
µR
=Z−1
Pµ
a−1Z−1
(42)
mχR
=
S(µκ− µκcr) . (43)
Observe that the relation (41) holds up to O(a) terms because the right hand side of
the relation corresponds to a different definition of the twist angle compared to the one
given in Section 3.1. The two definitions only coincide in the continuum limit.
By using the first of Eqs. (12) one obtains for ωV [16]
cotωV = (ZoVµ)−1(µκ− µκcr) + O(a)
ZoV = ZSZAZ−1
V
(44)
PZ−1
.(45)
Note that the angular coefficient of the linear fit gives the finite combination of renor-
malisation factors ZoV. Using as an input the determination of ZA/ZV in Eq. (17) one
can obtain from this the combination ZP/ZS.
We use Eq. (44) for a linear fit to µκcrand ZoV, see Table 5 for the results. As
expected from the discussion in Sec. 3.1, the condition mPCAC
close to those from the parity-restoration condition cotωV = 0. We conclude that
the two methods are essentially equivalent also from the numerical point of view. A
discrepancy is observed between the extrapolation from positive and negative quark
masses for the simulation point β = 0.67: we interpret this as a residual effect of the
first order phase transition at the given value of the lattice spacing. (Whether first
order phase transition or “cross-over” can only be decided in a study of the infinite
volume limit.) Observe also that the ZoV comes out different for the two different
signs of the quark mass: this is due to the breaking of symmetry under reflection of
the untwisted quark mass induced by O(a) terms [36]. The numerical discrepancy
shows that these O(a) corrections are relevant. An O(a)-improved estimate of ZoV
is simply obtained by averaging the determinations for negative and positive quark
masses, corresponding to a Wilson average for the quantity under study. An analogous
observation can be done for other combinations of renormalisation constants (see the
following).
Table 6 reports the determination of the renormalisation constants of the vector
and axialvector currents ZV and ZA. The ratio ZA/ZV comes from the analysis of the
the twist angles, Eq. (17). Using the direct estimate of ZV by Eq. (30) we can also
determine ZA. Observe that the full twist extrapolations of ZA/ZV from the two quark
mass signs present large discrepancies, which in this case cannot be attributed to O(a)
effects (these should disappear at full twist). A possible explanation of the discrepancy
could reside in the relatively bad quality of the data in the negative mass region. The
χ
= 0 gives results very
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discrepancies in ZAand ZP/ZSare a consequence of that for ZA/ZV. In the light of
these considerations we rely on the determinations for positive quark masses.
The full twist extrapolations of ZV are shown in Figs. 5 and 6: the values from
the two signs of the quark mass are rather close, compatible with each other within
statistical uncertainty. For the case β = 0.74 the extrapolation is very short, see Table 7
for the numerical values with comparison with one-loop perturbative estimates [41].
Table 7 also includes the determinations of the ratio ZP/ZSfrom ZoV (see Eqs. (44),
(45)). This quantity is of particular interest for simulations [42] of the theory with an
additional mass-split doublet describing the strange and charm quarks [43]. Defining
rcsas the mass-ratio mc/ms, the positivity of the fermionic measure in the strange-
charm sector imposes
ZP
ZS
>rcs− 1
rcs+ 1. (46)
The most stringent condition considering the experimental bounds [44] for msand mc
is
ZP
ZS
> 0.89 .(47)
Our results and the tadpole improved perturbative determinations for ZP/ZS (for
Nf= 2) seem to indicate that already at our values of β this condition is satisfied.
The results for the physical PCAC quark mass and pion decay constant fπobtained
from Eqs. (35) and (38) are listed in Table 8. In Figs. 7 and 8 the pion decay constant
is plotted as a function of the quark mass. The simulation points for negative quark
masses are not taken into account in the present discussion. The figures also include
the determination of fπby the axialvector current Aa
applies in this case where, however, the factor 1/sinω is replaced by 1/cosω. In the
interesting region near full twist this introduces large fluctuations in the estimate of
fπ, as one can see from the figures. Moreover in the case of the axialvector current,
the decay constant has not yet the right normalisation of the continuum: a ZAfactor
is still missing. On the contrary, in the case of the conserved vector current fπhas
automatically the physical normalisation [39, 40, 22]. If we exclude the lightest point
at β = 0.67, which is likely to be under the influence of residual metastabilities, fπ
seems to be characterised by a linear dependence upon the quark mass. On the basis
of this observation we try a simple linear extrapolation to the chiral limit mPCAC
see Table 9 for the numerical results. Of course, deviations from this linear behaviour
could be present for lighter quark masses where chiral logarithms play a role.
In order to check the scaling between the two β values we need to fix the lattice
spacing. This can be accomplished by extrapolating the value of r0to mPCAC
in this case we obtain two different values for the two different signs of the untwisted
quark mass, again due to O(a) effects. As for ZoV we take the average of the two
values, which delivers an O(a)-improved estimate of r0in the chiral limit. The results
xµ: a formula similar to Eq. (38)
q
= 0,
q
= 0. Also
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are reported in Table 9. We obtain for the lattice spacing (assuming r0= 0.5fm):
a(0.67) = 0.1757(41)fm, a(0.74) = 0.1326(70)fm. Denoting the zero quark mass limit
of the pion decay constant by
f0≡
lim
mPCAC
q
=0fπ, (48)
we obtain for f0r0: f0r0(0.67) = 0.333(10), f0r0(0.74) = 0.274(20). These values are
not far from the phenomenological value (f0r0)phen= 0.308. (The errors here are only
statistical. Systematic errors of the chiral extrapolation are not included.)
4Fits to chiral perturbation theory
Chiral perturbation theory (ChPT) is an expansion around the limit of massless quarks
in QCD [1]. It describes the dependency of physical quantities on the quark masses
in terms of expansions in powers of quark masses, modified by logarithms. In nature,
however, quark masses have fixed values. The question of how observables depend on
them functionally is experimentally unaccessible. Lattice gauge theory, on the other
hand, offers the possibility to vary quark masses. Therefore it represents the ideal
field of application of chiral perturbation theory. On the one hand, chiral perturbation
theory allows to extrapolate results from numerical simulations of QCD into the region
of small physical values for the up- and down-quark masses. On the other hand, lattice
QCD can provide values for the low-energy constants of chiral perturbation theory.
In chiral perturbation theory the effects of the non-zero lattice spacing a can be
taken into account in form of an expansion in powers of a[2, 45, 46, 47, 48]. For the
case of the Wilson twisted-mass formulation of lattice QCD this has been worked out
in next-to-leading order in [25, 49, 6, 15].
The major purpose of the present paragraph is to provide a set of formulas derived
from lattice chiral perturbation theory that can be used to analyze physical quantities
such as the pion mass, decay constants and amplitudes. The novelty here is that these
quantities have to be described across or nearby a phase transition.
The ChPT formulas are expected to be applicable at sufficiently small values of
the lattice spacing and quark mass. It is thus far from obvious whether the data
obtained with the DBW2 action in this work can be described by them, hence it is
interesting to confront the simulation data at our quark masses and lattice spacings
with these formulas. Let us emphasize that we consider this investigation mainly as a
methodological study that does not aim to extract physical values of the low energy
constants in the first place.
Properly determined parameters of the ChPT formulas in the continuum limit are
independent of the lattice action. The parameters describing the dependence on the
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lattice spacing do, however, depend on it. Therefore, in an Appendix we also present
ChPT fits of some simulation data obtained previously with the Wilson plaquette gauge
action [14].
The quark masses in chiral perturbation theory always appear multiplied by 2B0,
where B0 is a low-energy constant.A connection to lattice regularisation can be
established by considering the renormalised quark masses defined in Eqs. (42), (43)
and
mPCAC
χR
ZPmPCAC
A common renormalisation factor 1/ZP in mPCAC
However, since the multiplicative renormalisation of mPCAC
ZA, this has to be taken into account when fitting lattice data (see below).
The lattice spacing enters chiral perturbation theory in the combination
=ZA
χ
.(49)
χR
and µRcan be absorbed into B0.
and µ differs by a factor
χ
ρ = 2W0a, (50)
where W0is another low-energy constant.
For the low-energy constants of lattice QCD in next to leading order [46, 48] with
two quark flavours we use the notation
L54= 2L4+ L5,L86= 2L6+ L8,W54= 2W4+ W5,W86= 2W6+ W8,(51)
W =1
2(W86− 2L86),
Experience in untwisted lattice QCD shows [50] that lattice artefacts are consider-
ably reduced when observables are considered as functions of the PCAC quark mass
instead of the renormalised lattice quark mass. (A possible reason is that the PCAC
quark mass reabsorbs leading order O(a) effects.) Therefore, in our case, instead of
using mχRas a variable, we re-expand the physical quantities in terms of the PCAC
quark mass in the twisted basis mPCAC
χR . Including the relevant prefactor we define
W′=1
2(W′
86− W86+ L86),
?
W =1
2(W54− L54). (52)
χ′
PCAC= 2B0mPCAC
χR
.(53)
For the purpose of fitting data at constant µ it is convenient to define the combination
?
¯ χ = 2B0
(mPCAC
χR
)2+ µ2
R.(54)
(The attentive reader is certainly realising that we use the symbols χ for different quan-
tities. Nevertheless, both the notation for the fermion field of twisted-mass fermions
and the mass parameters in ChPT are standard in the literature and we do not want
to change neither of them in this paper.) Then, for the charged pion masses, chiral
perturbation theory at next-to-leading order including lattice terms of order a gives
m2
π±= ¯ χ +
1
32π2F2
0
¯ χ2ln
¯ χ
Λ2+
8
F2
0
{(−L54+ 2L86)¯ χ2+ 2(W −?
15
W)ρχ′
PCAC}. (55)