Running coupling in SU(2) with adjoint fermions

0.12 MB·English

by  Jarno Rantaharju

January14,2013 2:21 WSPC-ProceedingsTrimSize:9.75inx6.5in proceedings
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Running coupling in SU(2) with adjoint fermions
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JarnoRantaharju∗
n
a RikenAdvanced Institute for Computational Science,
J 7-1-26, Minatojima-minami-machi, Chuo-ku, Kobe, Hyogo, Japan

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1 KariRummukainen†

Department of Physics and Helsinki Institute of Physics,
]
t P.O.Box 64, FI-00014 University of Helsinki, Finland
a
l
- KimmoTuominen‡
p
e Department of Physics, P.O.Box 35 (YFL),
h FI-40014 Universityof Jyv¨askyla¨, Finland,
[ and
Helsinki Institute of Physics, P.O. Box 64,
1 FI-00014 Universityof Helsinki, Finland.
v
3
7 WepresentameasurementoftheSchro¨dingerFunctionalrunningcouplinginSU(2)lat-
3 ticegaugetheorywithadjointfermions.WeuseHEXsmearingandcloverimprovement
2 to reduce the discretization effects. We obtain a robust continuum limit for the step
. scaling,whichconfirmstheexistenceofanon-trivialfixedpoint.
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0 Keywords: Latticefieldtheory,Conformalfieldtheory
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1. The Model
:
v
We study the runningof the Schro¨dinger Functional coupling inthe SU(2) lattice field
i
X theorywith2fermionsintheadjointrepresentation.Themodel,dubbedMinimalWalk-
ing Technicolor, has been studied recently as a possible candidate for a walking tech-
r
a nicolor theory and as a part of the ongoing mapping of the conformal window on the
lattice.1–5 Wedefinethelatticemodelbytheaction
S=SG+SF, (1)
whereSF isthecloverimprovedWilsonfermionactionwithsmearedgaugelinksandSG
is a partiallysmeared version of the Wilson plaquette action. We use hypercubic stout
smearing, or HEX smearing,6 to reduce the discretization errors and allow simulations
atlargercoupling.Thegaugeactionisdefinedas
SG=βL (1−cg)Lx,µ(U)+cgLx,µ(V) (2)
xX;µ<ν
1
Lx,µ(U)=(cid:18)1− 2Tr[Uµ(x)Uν(x+aµˆ)Uµ†(x+aνˆ)Uν†(x)](cid:19),
∗[email protected]
†kari.rummukainen@helsinki.fi
‡kimmo.i.tuominen@jyu.fi
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whereV isthesmearedgaugefieldandwechoosecg=0.5.
Thefermionactionisgivenby
i
SF =a4 (cid:20)ψ¯(x)(iDW +m0)ψ(x)+acswψ¯(x)4σµνFµν(x)ψ(x)(cid:21), (3)
Xx
Weexpectthesmearingtobringthediscretizationerrorsclosetothetree-levelvaluesand
choose csw = 1. We have performed a few short measurements of the clover coefficient
andfindthemconsistentwiththetree-levelvalue,evenatsmallvaluesofβ.
Thesmearedlinksarecalculatedinthreesequentialstoutsmearingsteps,summing
overthe directionsthat areorthogonal tothoseintheprevious steps.6 Wehavechosen
thesmearingcoefficients foreachsteptobeα1=0.78,α2=0.61andα1=0.35.
WemeasuretherunningcouplingusingtheShc¨odingerfunctionalmethod.7,8Weuse
boundary conditions to the temporal direction to induce a chromoelectric background
fieldandmeasurethecouplingbytheresponsetoachangeinthebackgroundfield.The
boundaryconditions are
Uµ(x¯,t=0)=e−iησ3a/L, Uµ(x¯,t=L)=e−i(π−η)σ3a/L (4)
withσ3 thethirdPaulimatrixandwechooseη=π/4.Thespatialboundaryconditions
areperiodicforthegaugefield.Thefermionfieldissettozeroatthetemporalboundaries
and have twisted periodic boundary conditions to the spatial directions: ψ(x+Lˆi) =
exp(iπ/5)ψ(x).
Attheclassicallevelthederivativeoftheactionwithrespecttoη is
∂Scl. k
= ,
∂η g2
0
wherek isafunctionofN andη. Wedefine thefullrenormalizedcouplingatquantum
levelby
∂S k
h i= ,
∂η g2
Therunningofthecouplingisquantifiedbythestepscalingfunctionσ.Itdescribes
thechangeofthemeasuredcouplingwhenthelinearsizeofthesystemischangedfrom
LtosLkeepingthebarecouplingg2 constant.
0
Σ(u,s,L/a)=g2(g2,sL/a)|g2(g2,L/a)=u (5)
0 0
Σ(u,s,L/a)=σ(u,s)+c(u,s)a2 (6)
We choose s = 2 and obtain the continuum limit from measurements at L/a = 6 and
L/a = 8. The measured values of the coupling squared are shown in figure 2. We also
show the lattice step scaling function Σ(g2,2,L/a)/g2 = g2(g2,2L/a)/g2(g2,L/a) on
0 0
theleftsideinfigure1.
The Schro¨dinger functional boundary conditions reduce the amount of zero modes
inthesystemandallowsimulationsatzeroquarkmass.SincetheWilsonfermionaction
breaks chiral symmetry and allows additive renormalization of the quark mass, we use
the PCAC relation to find the value of κ where the renormalized mass vanishes. The
quarkmassM isdefinedby
aM(x0)= 1(∂0∗+∂0)fA(x0). (7)
4 fP(x0)
Wedefineκc asthevalueoftheparameterκwherethemassaM(L/2)vanishes.Tofind
κcwemeasurethemassat3to7valuesofκonlatticesofsizeL/a=16andinterpolate
tofindwherethemassiszero.InpracticeweachieveaM <0.003.
Weusereweightingtocorrectfortheresidualeffectofthenonzeromass.Thediffer-
ence can beseen infigure1,wherewe show the lattice step scalingfunction calculated
fromboththeoriginalandthereweightedmeasurements.
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Table1. Thevaluesofχ2 anddegreesoffreedomintheinterpolationineq.8.
L/a=4 L/a=6 L/a=8 L/a=12 L/a=16 combined
χ2 32.77 38.02 23.71 19.27 11.48 125.2
d.o.f 4 5 5 5 5 24
The continuum limitσ(g2,2) is taken keeping the coupling g2 equal, but the mea-
surementshavebeencalculatedatconstantg2,anddonotcorrespondtothesamevalue
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ofg2atdifferentlatticesizes.Themeasurements,therefore,needtobeshiftedtomatch-
ingvaluesofg2.Themosteconomicalandconvenientwaytoachievethisistointerpolate
the measurements ateach latticesizeL/abyfitting toafunction ofg2.Thisresults in
0
avalueofg2(g2,L/a)overacontinuous rangeofg2.
0 0
Weusetheinterpolatingfunction
1 = 1 1+ ni=1aig02 , (8)
g2(g02,L/a) g02 (cid:20)1+Pmi=1big02(cid:21)
withn=4andm=2.ThesevalueswerechosenPtomaximizethecombinedP valuefor
thefit,calculatedfromthesumofχ2anddegreesoffreedomforeachfit.Theinterpolating
functions are then used to calculate Σ(u,2,L/a) at a continuous range of u, and the
continuum limit is calculated by fitting to the quadratic function in equation 6. The
resultisshowninfigure3.
1.1 1.1
L=4 L=4
L=6 L=6
L=8 L=8
1 1
22σ(g,2)/g0.9 22σ(g,2)/g0.9
0.8 0.8
0.7 0.7
0 2 4 6 8 0 2 4 6 8
g2 g2
Fig. 1. The scaled lattice step scaling function Σ(g2,2,L/a)/g2 = g2(g2,2L/a)/g2(g2,L/a).
0 0
The plot on the left shows the result using the original measurements and the plot on the right
usingreweightedvalues.Theblackdashedlinegivesthecontinuum 2-loopperturbativeresultfor
σ(g2,2)/g2.
2. Conclusions
Both the lattice results and the estimated continuum limit show a non-trivial infrared
fixed point between g2 = 2 and 4. This is in agreement with previous studies. Fur-
thermore, a large part of the discretization errors in the interesting region seems to be
removed by the clover improvement and the HEX smearing.Smearing alsoreduces the
computation time required to generate lattices and enables simulations at larger cou-
pling. We plan to verify this further by repeating the measurement with an increased
latticesize.
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8 β=0.98
β=1 1.05
β=1.05
β=1.1
6 β=1.2
β=1.3
2g4 ββββ====1248.5 22σ(g,2)/g0.98
0.91
8-16
2 extrapolation
2-loop
0.84
0 0 1 2 3 4 5 6
0 4 8 12L 16 20 24 g2
Fig.2. Ontheleft,themeasuredvaluesofg2(g2,L/a)againsta/L.Theblackdashedlinegives
0
an exampleof the runningin2-loopperturbation theory at modestcoupling, normalizedsothat
it matches the measurement at L/a=6. The measurements have been reweighted to zero mass.
Ontheright,thescaledstepscalingfunctionσ(g2,2)/g2 usingreweightedmeasurements.Thered
line with the hashed band correspond to the continuum extrapolation using lattice sizes L = 6
and8andtheblacklinewiththegreenbandcorrespondtothelatticeresultatthelargestlattice
sizeL=8.Theblackdashedlinegivesthe2-loopperturbativeresult.
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