The topological susceptibility from grand canonical simulations in the interacting instanton liquid model: chiral phase transition and axion mass

0.91 MB·English

by  Olivier Wantz

The topological susceptibility from grand canonical
simulations in the interacting instanton liquid model:
chiral phase transition and axion mass.
Olivier Wantza, E.P.S. Shellarda
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1 aDepartment of Applied Mathematics and Theoretical Physics, Centre for Mathematical
0
Sciences,
2
Universityof Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

n
a
J

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1 Abstract

] This is the last in a series of papers on the topological susceptibility in the
h
interacting instanton liquid model (IILM). We will derive improved finite tem-
p
- perature interactions to study the thermodynamic limit of grand canonical
p
Monte Carlo simulations in the quenched and unquenched case with light,
e
physical quark masses. In particular, we will be interested in chiral sym-
h
[ metry breaking. The paper culminates by giving, for the first time, a well-

motivated temperature-dependent axion mass. Especially, this work finally
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provides a computation of the axion mass in the low temperature regime,
v
4 m2f2 = 1.46 10−3Λ4 1+0.50T/Λ . It connects smoothly to the high tem-
a a 1+(3.53T/Λ)7.48
2 perature dilute gas approximation; the latter is improved by including quark
3
threshold effects. To compare with earlier studies, we also provide the usual
0
8. power-law m2a = fa2α(Ta/ΛΛ4)n, where Λ=400MeV, n=6.68 and α=1.6810−7.
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0 1. Introduction
:
v
i The purpose ofthis paper is to continue the study of the interactinginstan-
X
ton liquid model (IILM) at finite temperature, with an ancillary goal being to
r improve our understanding of the temperature dependence of the axion mass.
a
For the first time, we will be able to give a well-motivated axion mass that
covers all temperatures down to T =0.
In[44]wesetuptheformalismunderlyingtheIILManddevelopedanumer-
ical framework to compute the interactions given by an arbitrary background
ansatz. Simulationsatzerotemperaturewereperformedwiththeso-calledratio
ansatz to determine the parameters that enter the model: the lambda parame-
ter, Λ, and the quark masses, m .
q
Email addresses: [email protected] (OlivierWantz),
[email protected] (E.P.S.Shellard)
Preprint submitted toElsevier January 14, 2010
1600
1400
1200
1000
>
N
<
800
600
400 RE (Quenched)
R (Quenched)
200 H
0 2000 4000 6000 8000 10000 12000
V (L -4)
Figure1: TheRH ansatzdoesnotleadtoawell-definedthermodynamiclimitatfinitetemper-
ature,whereastheRE ansatzdoesexhibitthecorrectlinearscalingofanextensivequantity.
Here we display results from quenched simulations, but the same problems persist with dy-
namical quarks. The instanton number N shows the strongest response to the unphysical
interactions.
InthispaperwewillinvestigatetheIILMatfinitetemperaturebasedonthe
caloron solution of Harrington and Shepard [14]. Using as input the physical
parameters we determined at zero temperature, we will study chiral symmetry
restoration, based on the ideas of instanton–anti-instanton molecule formation
[15], [16], [35], and determine the topological susceptibility.
In order to deal with light, physical quark masses, we study the thermo-
dynamic limit. As mentioned in the previous paper [44], we found that the
interactions derived previously in [39] are deficient in this respect. Although
the old and new interactions agree rather well at zero temperature, this is no
longer the case at T =0 because of an unphysical behaviour for the instanton–
6
instantoninteractionthatonlydecaysasO(1/R ),withR thespatialinstanton
s s
separation, see Eq. (3.11) in [39]. This long-range interaction prohibits a ther-
modynamic limit as it is not integrable, see Fig. 1. In their paper, the authors
dodiscussthislong-rangeinteractionandreporttheyfoundtheO(1/R )dyon–
s
dyon behaviour for a wide range of intermediate separations. It might well be
that for the simulation boxes used in a subsequent numerical investigation(see
[34]) the interactions are still well described by this ansatz, i.e. that the spatial
extent of the box is bounded by these intermediate separations. However, for
studying the large volume behaviour this ansatz is not appropriate. Removing
this particular part of the interactions, we were able to retrieve a well-defined
thermodynamic limit.
Apartfromthisdeficiency,itseemsobviousthatatfinitetemperatureitwill
2
bemuchhardertofindagoodparametrisationfortheactionofthebackground
ansatz because the underlying O(4) symmetry is broken and the constituent
gauge fields are more complicated. This fact was already pointed out in [39].
Weimprovetheexistinginteractionsbyextendingtheformalismsetupin[44]to
finite temperature; as we will see, the only difficulties are of a technical nature.
Withthispaperwewillachieveourinitialaimofcomputingthetemperature
dependent axion mass. So far the mass was computed within a dilute gas ap-
proximation which breaks down at low temperatures. The connection between
the hightemperatureregime,wherethe dilute gasbecomes evermoreaccurate,
and the zero temperature result, obtained through chiral perturbation theory,
hasbeenperformedinarathercrudemanneruptodate: eitherbyunsmoothed
matching[41,4]orbyanadhocinterpolationprescription[22]. Ourdetermina-
tion ofthe axionmass will for the first time give a well-motivatedinterpolation
betweenthe zeroandfinite temperature regimes. Comparisonwith lattice data
will allow for a critical evaluation of the systematic uncertainties. In partic-
ular, considerations regarding the anthropic axion with large decay constant
[27, 42, 40] are potentially very sensitive to the non-perturbative effects of the
QCD phase transition, when their mass becomes sizable.
In section 2 we will re-derive the finite temperature interactions for the ra-
tio ansatz. We will then discuss, in section 3, the new elements that finite
temperature introduces in the numerical framework. After these technical pre-
liminaries we will have a short investigation of the topological susceptibility in
the quenched sector in section 4, before we discuss the main numerical results
of the unquenched IILM regarding chiral symmetry restoration, section 5, and
the topological susceptibility and axion mass, section 6.
2. Interactions in the IILM at finite temperature
Intermsofthe’tHooftpotential1+Π,theHarrington–Shepardcaloron[14],
an infinite sequence of singular gauge BPST instantons [5] along the Euclidean
time direction, is given by
∂ Π(x, y,ρ )
Aa = Oabζb ν { } , (1)
µ − i µν1+Π(x, y,ρ )
{ }
where O is the colour matrix in the adjoint representation of the embedding
SU(2) SU(3), and ζb = η¯b (ζb = ηb ) for instantons (anti-instantons); η
→ µν µν µν µν
are the ’t Hooft symbols. The ’t Hooft potential has the following form
πρ2 sinh2πr
Π(x, y,ρ )= β , (2)
{ } βr cosh2πr cos2πt
β − β
withr2 =(~x ~y)2 andt=x4 y4;thecollectivecoordinatesare: y thecentre,ρ
− −
the size and O the colour orientation. At finite temperature bosonic quantities
such as Π are periodic in the Euclidean time direction with period β = 1/T.
NotethatΠapproachesthezerotemperatureinstantonpotentialinthesingular
gauge for β .
→∞
3
We willuse as backgroundthe ratioansatzand, asfor the zerotemperature
case, only consider two-body interactions. The gauge field is then given by
Oabζb ∂ Π (x, y ,ρ )
Aa = i i µν ν i { i i} , (3)
µ − 1+ Π (x, y ,ρ )
P i i { i i}
with O = OtO . This pair interactPion has been derived in [39], and we will
1 2
refertoitasR ; the correspondingforcesderivedinthis paperwillbe denoted
H
by R . However, in studying the volume dependence of various quantities,
E
we noticed that these interactions did not allow for a thermodynamic limit.
The problem can be traced back to the log term in the instanton–instanton
interaction, (3.11) in [39], only decaying like O(1/R ) for large R , where R
s s s
is the spatial separation of the pair. Note that this term is attributed to the
dyon–dyon interaction for intermediate separations, β R ρ2/β, in the
s
≪ ≪
high temperature limit.
The ratio ansatz has the same functional form in terms of Π as for zero
temperature, and so we can use our previous result [44] to write
Fa Fa =I+(TrOtO+(η¯Oη) )J +(η¯Oη) I
µν µν µνµν ρµρν µν
+(η¯Oη) I +(ηOtOη) J +(η¯Oη) (η¯Oη) K . (4)
µρνσ µρνσ µρνσ µρνσ αµαρ βνβσ µρνσ
The different terms are given in appendix Appendix A. Due to charge renor-
malisation the action, S[A] = 1 Fa Fa , acquires a quantum contribution
4g2 µν µν
in the form of the running coupling constant. The classical interaction is given
R
by
Sg /S V (S[A]/S 2), (5)
12 0 ≡ 12 ≡ 0−
whereS =8π/g2isthesingleinstantonaction. Thequantumeffectssubstitute
0
g in S for the running coupling constant; the RG scale is estimated by the
0
geometric mean √ρ1ρ2, as proposed in [38, 34].
First,wewilllookatthedependenceoftheinteractionsonR ,theinstanton
t
separationinthe(imaginary)timedirection. Comparedtothezerotemperature
case, the differences are substantial even for pairs with equal sizes. The major
difference comes from the fact that the R interactions are not periodic. For
H
unequal sizes, the difference is even more pronounced as was the case at zero
temperature, see Fig. 2. The reason is again that the functional form on the
instanton sizes in R is not general enough.
H
The interaction between instantons and anti-instantons does not lead to as
close a match between R and R as in the T = 0 case, but is still qualita-
E H
tively similar for not too large separations. In the instanton–instanton case,
however,we have found significant differences, see Fig. 3: we do find the dyon–
dyon behaviour for intermediate distances but for separations R ρ2/β the
s
≫
functional dependence changes into an integrable O(R−4) for R whereas the
s E
non-integrabledyon–dyoninteractionpersistsinthe R ansatz. Thatthelarge
H
separation fall-off should be different from the dyonic regime is clear from the
discussionin[12]wheretheauthorsshowthatthecaloronfielddevelopsadipole-
likecharacterinthe far-fieldregion. Notethatthis leadstoathree-dimensional
4
R : r /r =1
E 1 2
10 R : r /r =1
H 1 2
R : r /r =3
8 RE: r 1/r 2=3
H 1 2
6
gS12 4
2
0
-2
0 0.2 0.4 0.6 0.8 1
R / b
t
0
-0.5
-1
g12
S
-1.5
R : r /r =1
E 1 2
R : r /r =1
-2 H 1 2
R : r /r =3
E 1 2
R : r /r =3
H 1 2
0 0.5 1 1.5 2 2.5 3
R / r
s
Figure2: ThemajordifferencefollowsfromthefactthatRH interactionsarenotperiodicin
Rt. Thisisclearlyadeficiencyoftheanalytical formulasbecauseitfollowsdirectlyfrom(4)
thattheinteractionsshouldhaveperiodβ. Forspatialseparations,themaindifferencesoccur
for unequal size parameters, e.g. ρ1/ρ2 =3 in this case. The reason is that the dependence
onthesizesismorecomplicatedthanthefunctionalform,√ρ1ρ2,usedinRH. (Wehaveset
ρ¯=qρ21+ρ22.)
5
R
E
R
10-1 O(R1s) dyonic region H
g1210-2
S
10-3 O(R14) large separation region
s
1 2 3 4 5 6 7 8 9
R / r
s
Figure 3: The instanton–instanton interaction behaves very differently for RH and RE. In
the high temperature, large size limit we do see the dyon–dyon behaviour but for larger
separations,Rs ρ2/β,theinteractiondecays muchfaster,O(R−s4). Thisfall-offbehaviour
≫
is integrable and we can study the thermodynamic limit, in contrast to the purely dyonic
interaction. (Wehavesetρ¯=qρ21+ρ22.)
dipole–dipole interactionfor aninstanton–anti-instantonpair, but that the R
H
ansatz retains the zero temperature (four-dimensional) dipole–dipole interac-
tion1.
The fermionic interaction follows from the quark wave function overlaps
(D/ +m) = ξ D/ +mξ =D/ +mδ . (6)
ij i j ij ij
h | | i
Even though the set of eigenfunctions ξ is generally not an orthonormal
i
{ }
basis, the extra contributions are neglected; effectively, we treat ξ as being
i
{ }
orthonormal which leads to the diagonal mass term. The fermionic zero mode
at finite temperature is given by [13], [12], [39],
ξ = 1 1+Π ∂/ χI UIϕ , (7)
I 2πρ I 1+Π 0
I I (cid:18) (cid:19)
p
1 χ 0
A
ξ = 1+Π ∂/ , (8)
A 2πρA A 1+ΠA (cid:18) UAϕ (cid:19)
cospπt
β
χ = Π , (9)
coshπr
β
1instanton–instantonpairsdecayslightlyfaster,atO(1/R4),however,andtheirinteraction
s
is not of dipole type; note that this happens at T = 0 too, where instanton–anti-instanton
pairsexhibitadipole–dipoleinteractionbutinstanton–instanton pairsdon’t[9].
6
with ϕ = ǫ , normalised according to ǫ = 1, and U the collective coordi-
αa αa 12 i
nates for the colour embedding in the fundamental representation.
The overlaps T = ξ†iD/ξ have a slightly more complicated form than
IA I A
their T =0 counterparts and are given by
R
1 1 i
T = Tr(Uτ+)I Tr(Uτ+τ )η¯a J
IA ZR3×S1 4π2ρIρA (cid:18)2 β β − 2 β a µα βµα
i
+ Tr(Uτ τ+)ηa K . (10)
2 a β µα βµα
(cid:19)
The different contributions can be found in appendix Appendix B.
The difference between the R and R interactions is quite large for tem-
E H
poral and spatial separations, see Fig. 4. However, it is not straightforward
to compare both ansa¨tze because the R overlaps are computed on a different
H
background [39], the sum ansatz, whereas we use the full ratio ansatz in R .
E
As in the gluonic case, the quark overlaps are not periodic for the R ansatz
H
as opposed to those of R . As always, unequal sizes increase the differences
E
between the R and R ansa¨tze even more. For large separations, when the
H E
ratioansatzbecomesindistinguishablefromthesumansatz,andforequalsizes,
we find very good agreement between R and R .
H E
The total interaction, after normalising to the dilute gas approximation, is
given by
Nf lndet(I+ TT†) ,Q<0
Sint = S0(√ρiρj)Vij − ( lndet(I+ Tm†2nT) ,Q>0 , (11)
paiXrs(i,j) nX=1 m2n
where Q = N N is the topological charge and N the number of quark
I A f
−
flavours. For details see [44].
Even though the analytic formulas of the R ansatz are not periodic, this
H
is notreallya majorshortcomingbecause periodicity canbe realisedby folding
back the instantons into the fundamental interaction region R [ β/2,β/2].
t
∈ −
We demonstrate this for the total interactionin Fig. 5. We can clearly see that
the R ansatz is intrinsically periodic.
E
3. Numerical Implementation
3.1. Interpolation and asymptotic matching
As for zero temperature, we will need to set up a grid for the numerical
evaluationofthetwo-bodyinteractions. Thelook-uptableswilldependonfour
variables: the spatial separation R , the temporal separation R and the two
s t
sizes ρ and ρ . The colour degrees of freedom O, or equivalently U, have been
1 2
completely factored out and can be treated exactly.
As we have seen in section 1, the dyon–dyon regime is characterised by a
fairlyslowfall-off. However,notallbosonicinteractionsinappendixAppendix A.1
7
R : r /r =1
E 1 2
3 R : r /r =1
H 1 2
R : r /r =3
2.5 RE: r 1/r 2=3
H 1 2
2
|A
TI
| 1.5
1
0.5
0
0 0.2 0.4 0.6 0.8 1
R / b
t
R : r /r =1
E 1 2
2.5 R : r /r =1
H 1 2
R : r /r =3
E 1 2
2 R : r /r =3
H 1 2
1.5
|A
TI
|
1
0.5
0
0 0.5 1 1.5 2 2.5 3
R / r
s
Figure4: ThelargediscrepancybetweenREandRHforbothtemporalandspatialseparations
isdue tothe factthat RE uses the fullratioansatz intheDiracoperator whereas forRH a
sumansatzisused. Periodicity,whichfollowsdirectlyfrom(6),islackingforoverlaps inthe
RH ansatzbutisrealisedintheRE ansatz. (Wehavesetρ¯=qρ21+ρ22.)
8
15
R
E
10 RH
R (folded)
E
5
R (folded)
H
0
-5
2
1
S
-10
-15
-20
-25
-30
0 0.2 0.4 0.6 0.8 1
R / b
t
Figure 5: Folding back the temporal separation into the fundamental interaction region,
Rt [ β/2,β/2], we explicitly retrieve periodic interactions for RH. The RE ansatz is
∈ −
intrinsicallyperiodic.
decay this slowly; rather they can be grouped according to polynomial and ex-
ponentialdecay. The fermionic overlapsfall-offexponentially, butslightly more
slowly than their bosonic counterparts. Therefore,we found it advantageousto
define three different grids:
For the polynomial grid, the maximal separation Rmax will depend on
• s
ρ˜ πρ2/β, the natural size parameter for separations beyond β. Note
≡
that for very small sizes Rmax < β the prescription for Rmax should
s s
switch over to the T = 0 case; this is implemented by setting Rmax =
s
max(αpρ,αpπρ2/β).
0 T
The exponential decay sets in at R β. To accommodate very small
s
• instantons we set again Rmax =min(α≈egρ,αegβ).
s 0 T
The grid for quark overlaps is set in an analogous fashion, i.e. Rmax =
• min(αeqρ,αeqβ). s
0 T
The constants αj are fine-tuned so as to achieve fast and stable numerical in-
i
tegrations and a good matching to the analytic expressions used for separa-
tions beyond Rmax. In the temporal direction the grid is bounded by R =
s | t|
β/2; care needs to be taken again for small instantons and we set Rmax =
| t |
min(Rmax,β/2).
s
Thesizedistributionissupportedon30gridpointsandforeachpair(ρ ,ρ )
i j
theR R planeconsistsof30 29nodes;thisleadstoatotalofroughly1million
s t
− ·
interpolation points. The grid is ‘logarithmic’ in the size and R direction. A
s
typical R R plane is shown in Fig. 6.
s t
−
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Interpolation among four points
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
Rt
R
s
Figure6: It turns out that the interactions aremore or less identical on the Rs Rt plane,
when expressed in units of ρ21+ρ22. We therefore choose the grid to be irregu−lar, i.e. the
points on the Rs Rt plane are different for different values of the sizes. Since instantons
−
are localised field configurations, the separation between grid points grows geometrically in
the Rs direction. In the ρ directions the mesh is regular, but still ‘logarithmic’. The inner
andouterregionsarematchedtoasymptoticexpansions: forsmallseparationsthematching
isperformedradiallyfromtheoriginbecauseinthisregiontheinteractions areroughlyO(4)
symmetric; for large separations we match along the Rs direction with constant Rt because
the interactions become t-independent. (Note that for graphical reasons not all 870 grid
points are displayed and that we actually display logRs. Also the actual gridonly contains
theRs>0points.)
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