Table Of ContentSpectrum and mass anomalous dimension of
SU(2) gauge theories with fermions in the adjoint
representation: from N = 1/2 to N = 2
f f
Georg Bergner
∗
7
Friedrich-Schiller-UniversityJena,InstituteofTheoreticalPhysics,
1
0 Max-Wien-Platz1,D-07743Jena,Germany
2 E-mail: [email protected]
n
a Pietro Giudice, Gernot Münster
J UniversitätMünster,InstitutfürTheoretischePhysik,
1
Wilhelm-Klemm-Str. 9,D-48149Münster,Germany
3
E-mail: [email protected], [email protected]
]
t Istvan Montvay
a
l DeutschesElektronen-SynchrotronDESY,
-
p Notkestr. 85,D-22603Hamburg,Germany
e
E-mail: [email protected]
h
[
Stefano Piemonte
1 UniversitätRegensburg,InstituteforTheoreticalPhysics,
v
D-93040Regensburg,Germany
2
9 E-mail: [email protected]
9
8
We summarize our results concerning the spectrum and mass anomalous dimension of SU(2)
0
. gaugetheorieswithvariousnumbersoffermionsintheadjointrepresentation,whereeachMajo-
1
0 ranafermioncorrespondseffectivelytohalfaDiracflavourN . Themostrelevantexamplesfor
f
7
extensionsofthestandardmodelaresupersymmetricYang-Millstheory(N =1/2)andMinimal
1 f
: WalkingTechnicolour(N =2). Inadditiontothesetheorieswewillalsoconsiderthecasesof
v f
i N =1 and N =3/2. The results comprise the particle spectrum of glueballs, triplet and sin-
X f f
gletmesons,andpossiblefractionallychargedspinhalfparticles. Inadditionwewilldiscussour
r
a recentresultsforthemassanomalousdimension.
The34rdInternationalSymposiumonLatticeFieldTheory
24-30July2016
UniversityofSouthampton,UK
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
LatticesimulationsofadjointQCD GeorgBergner
1. TechnicolourandadjointQCD
Gauge theories with fermions in higher representations of the gauge group offer interesting
alternativesforextensionsofthestandardmodelofparticlephysicsandforamoregeneralviewon
theoreticalproblemsliketheunderstandingoftheconfinementmechanism. Oneofthemostinter-
estingexamplesinthiscontextistheadjointrepresentation,sinceitisrelatedtoseveralapproaches
to extensions of the standard model, in particular supersymmetry and Technicolour models. They
representtheoriesthatareinsomerespectsquitedifferentfromQCD-likeexampleswithfermions
inthefundamentalrepresentation. Fromthatperspectivetheyareinterestingfromthemoregeneral
theoreticalpointofview.
InthisworkwepresentourinvestigationsrelatedtotheWalkingTechnicolourapproachtoex-
tensionsofthestandardmodel,see[1,2]forareview. Inthisapproachamorenaturalelectroweak
sector is obtained from a new strongly coupled dynamics. The Higgs particle appears as a bound
state in this strongly coupled theory. There are severe constraints on Technicolour candidates,
since they have to be consistent with the electroweak precision data and still potentially explain
thefermionmassesbasedonanextendedTechnicolourtheory. Furthermorealargemasshierarchy
between the Higgs particle and so far undiscovered additional bound states has to be explained.
One possible explanation is a dynamics that is quite different from QCD due to a near conformal
walkingbehaviour. Thismeansthatsuchkindoftheorieshavetobeclosetotheconformalwindow.
Theconformalwindowisdefinedbytheappearanceofaninfraredfixedpointoftherunning
gaugecoupling. Intheperturbativecalculationstherunningofthegaugecouplingoftheconfining
gauge theory gets weaker with an increasing number of fermions N . Above a certain (N ) the
f f c
runningstopsataninfraredfixedpoint. Increasingthenumberoffermionsevenfurther,asymptotic
freedom is lost. The conformal window is defined as the region between (N ) and the loss of
f c
asymptoticfreedom.
Thedeterminationoftheconformalwindowisofgeneraltheoreticalinterestsinceitmarksthe
parameter regions for a completely different behaviour of the theory in the infrared. Irrespective
of the considered application, it is important to know this behaviour. While the upper end of the
conformal window is in the perturbative regime, the lower end can only be determined by non-
perturbativemethods.
Therehavebeenseveralinvestigationsoftheconformalwindowforfermionsinthefundamen-
talrepresentationinascanofseveraltheorieswithdifferentN ,see[3,4,5]forareviewoflattice
f
results. Inourinvestigationsweconsiderthesameapproachforadeterminationoftheconformal
windowinthecaseoffermionsintheadjointrepresentation. Higherfermionrepresentationsallow
toapproachthelowerendoftheconformalwindowwithasmallernumberoffermions(N ) . This
f c
allowstoconstructtheoriesthatmighthavelesstensionswiththeelectroweakprecisiondata.
In our numerical investigations on the lattice, a finite mass term is included in the theory,
which leads to mass deformed versions of the conformal candidates. In this case the mass is the
onlyrelevantparameteratthefixedpoint, whichimpliesthatforsmallmallparticlemassesscale
like M m1/(1+γm) with the mass anomalous dimension γm at the fixed point. It also implies that
∼
a signal for conformality are the constant ratios of different particle masses. The value of γ can
m
be obtained from the scaling of the mass ratios. A more precise measurement is provided by the
modenumber,theintegratedeigenvaluedensityoftheDiracoperator[6,7]. Fromtheperspective
2
LatticesimulationsofadjointQCD GeorgBergner
of the Walking Technicolour approach, a large anomalous dimension is desirable. It is therefore
interestingtostudytheN dependenceofthisquantity.
f
N = 2 SU(2) adjoint QCD, also called Minimal Walking Technicolour (MWT), has been
f
studiedinseveralinvestigations[8,9,10,11,12,13]. Itshowsthesignaturesofaconformaltheory
and a clear separation between the lightest scalar bound state and the rest of the spectrum. The
case of N =1/2, corresponding to SU(2) N =1 supersymmetric Yang-Mills theory, has been
f
investigated by us in several studies. This theory clearly shows the signals of a theory below the
conformalwindow. InmorerecentinvestigationstheN =1casehasbeenconsidered. Inthisshort
f
reportweaddourresultsforN =2andN =3/2andprovideasummaryofthepropertiesofthe
f f
differenttheorieswithfermionsintheadjointrepresentation.
InourstudiesofsupersymmetricYang-Millstheorytheinvestigationofdifferentinversegauge
couplings β turned out to be important. In a QCD-like theory this corresponds to a study of the
approachtowardsthecontinuumlimit. Intheconformalcasethisparameterisirrelevantinthenear
vicinity of the fixed point. Since we want to investigate all the theories on the same ground, we
includeastudyoftheβ-dependenceinourinvestigations.
2. TheexpectedmassspectrumforadjointQCD
AdjointQCDinthecontinuumisdefinedbythefollowingaction
(cid:34) (cid:35)
1 Nf
L =Tr F Fµν+∑ψ¯ (D/+m)ψ , (2.1)
µν i i
−4
i=1
withthecovariantderivativeforfermionsintheadjointrepresentation
D ψ =∂ ψ+ig[A ,ψ]. (2.2)
µ µ µ
N counts the numberof Dirac fermions, which correspondsto 2N Majorana fermions, sincethe
f f
adjointrepresentationiscompatiblewiththeMajoranaconditionψ =Cψ¯T.
The chiral symmetry breaking pattern, which is different from QCD, is implied by the repre-
sentation in terms of Majorana flavours. In adjoint QCD there is a larger chiral symmetry group
withthespontaneousbreakingpattern
SU(2N ) SO(2N ) (2.3)
f f
→
in the presence of a fermion condensate. The chiral symmetry breaking leads to the appearance
of light pseudo-Nambu-Goldstone particle states. A signal for these states can be obtained by the
pseudoscalartripletmeson,similartothepioninQCD.Inourcurrentworkthisadjointpionmass
is represented by the pseudoscalar mass m . In our simulations we employ a discretisation of
PS
the theory with a Wilson-Dirac operator including stout-smeared link variables and a tree-level
Symanzik-improvedgaugeaction.
We study various particle states in the theory. One class are the flavour triplet mesons: the
pseudoscalar m , scalarm , vectorm , and pseudovectorm mesons. We have also considered
PS S V PV
the flavour singlet scalar and pseudoscalar mesons. In addition we study the scalar 0++ glueball,
3
LatticesimulationsofadjointQCD GeorgBergner
1.6 1.4
√σ/mPS √σ/mPS
11..24 msmpinm0−F+V+π21////mmmmPPPPSSSS 1.21 msmpinm0−F+V+π21////mmmmPPPPSSSS
mS/mPS mS/mPS
1 mPV/mPS mPV/mPS
/mPS 0.8 346284333××666444 /mPS 0.8
M × M 0.6
0.6
0.4
0.4
0.2 0.2
0 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26
amPCAC amPCAC
(a) MassesinunitsofmPSatβ =1.5 (b) MassesinunitsofmPSatβ =1.7
Figure1: ThemassesofvariousboundstatesinMWTinunitsofthepseudoscalarmassm . Thetriplet
PS
mesonsinthescalar(m ),vector(m ),andpseudovector(m )channelareshowntogetherwiththescalar
S V PV
glueball(m )andthespin1/2particle(m ). Thefigurealsoshowsthepseudoscalardecayconstant
0++ spin1/2
F andthestringtensionσ.
π
State β =1.5 β =1.7
m 1.0825(58) 1.051(12)
V
m 1.285(24) 1.190(14)
S
m 0.620(35) 0.398(48)
0++
m 0.948(24) 0.86394(52)
1/2
Table 1: The mass ratios averaged over a range of PCAC masses for two different β values. The values
correspondtothemassesofthetripletvector(m )andtripletscalar(m )mesons,theglueball(m ),and
V S 0++
thespin-1/2particle(m )dividedbythetripletpseudoscalarmesonmass(m ).
1/2 PS
providing a signal for a possible light Higgs-like state. As an example for more exotic fermion-
gluonoperatorsthatarepossibleinadjointQCD,wemeasurethemassobtainedfromthecorrelator
ofthespin-1/2operator
∑σ tr[Fµνψ]. (2.4)
µν
µ,ν
3. ResultsforMinimalWalkingTechnicolourincludingtheβ dependence
Our investigation of Minimal Walking Technicolour has now been finalised. We have mea-
sured the spectrum and determined the mass anomalous dimension from the mode number. The
generalpictureofourresultsisconsistentwithearlierinvestigations[10,13]: theparticlespectrum
contains a light scalar particle, namely the 0++ glueball state, while the triplet mesons, including
the pseudoscalar meson, are heavier. Hence the spectrum in not consistent with chiral symmetry
breaking, which predicts the appearance of light pseudo-Goldstone particles separated from the
restofthespectrum. ThemassratiosareapproximatelyconstantasafunctionofthePCACmass,
seeFigure1(a)and1(b).
4
LatticesimulationsofadjointQCD GeorgBergner
N N β κ γ
s t
× ∗
24 64 1.5 0.1325 0.39(3)
×
32 64 1.5 0.1335 0.38(1)
×
48 64 1.5 0.1344 0.380(10)
×
32 64 1.5 0.1350 0.375(4)
×
average 1.5 0.376(3)
32 64 1.7 0.1285 0.270(15)
×
32 64 1.7 0.1290 0.260(20)
×
32 64 1.7 0.1300 0.285(15)
×
average 1.7 0.274(10)
Table2: ThisTablecontainstheestimatesforthemassanomalousdimensionγ atthefixedpoint,thatwe
haveobtainedfromafitofthemodenumber. Theresultsareobtainedattwodiff∗erentvaluesofβ.
In our more detailed investigations we have also added some new states that have not been
consideredinearlierinvestiations: thefermionicspin-1/2particle, andtheflavoursingletmesons.
The fermion-gluon particle is lighter than the pseudoscalar meson, but considerable heavier than
thescalarglueball. Thisobservationisrelevantfromaphenomenologicalpiontoffew,sincethese
statesmightleadtofractionallychargedparticles.
Weobtainedresultsattwodifferentβ-valuesthataresuperficiallyconsistentwitheachother,
asshowninFigure1(a)and1(b). Thesameorderingofthestateswithconstantmassratiosisob-
served. Amorecarefulinvestigationofthemassratiosshows,however,asignificantβ dependence
of the results. We have averaged the mass ratios over a certain range of the PCAC mass for both
valuesofβ. AsshowninTable1,theresultsforbothβ valuesarenotconsistentwitheachother. In
particularthescalarglueballatthelargervalueofβ islighterincomparisontom thanatβ =1.5.
PS
Thisindicatesaremnantβ-dependenceandscalingcorrectionsfortheconformaltheory.
We have estimated the mass anomalous dimension γ at the fixed point from our measure-
∗
mentofthemodenumber. Themodenumbercorrespondstotheintegratedspectraldensityofthe
Wilson-Dirac operator. Our results at β =1.5 are consistent with [13] (γ =0.371(20)), but for
∗
β =1.7 a considerable smaller γ is obtained, see Table 2. The remnant scaling corrections that
∗
are indicated by these results might also explain the discrepancies between different results in the
literature, for example the small γ =0.20(3) obtained with a clover improved Dirac operator in
∗
[12].
4. NewresultsforN =3/2
f
In our most recent simulations we have started an investigation of the theory with three Ma-
jorana flavours, corresponding to N =3/2 Dirac flavours. When integrating out an odd number
f
of Majorana fermions, one obtains, in addition to the usual fermion determinant to the power of
N , also the sign of the Pfaffian of the Dirac operator. For our investigations of supersymmetric
f
Yang-Mills theory we have developed methods to calculate the sign of the Pfaffian, but for our
currentparametersthiscontributioncanbeneglected.
Wehaveinvestigatedthemassspectrumofthistheoryandfoundthatitisquitesimilartothe
N =2case. Thescalarglueballisagainthelighteststateinthetheory,andalsothemasshierarchy
f
5
LatticesimulationsofadjointQCD GeorgBergner
1.8
a√σ
M 111...2461 a1m6a3saampma×amianm0m−FP+3PVV+π221SS mPS 2.235 mspinmm1−F6VSπ123////×mmmmPPPP32SSSS
a 0.8 M/ 1.5
0.6
1
0.4
0.2 0.5
0
0.000000 0.050000 0.100000 0.150000 0.200000 0.250000 0
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
amPCAC amPCAC
(a) Massesinlatticeunits (b) Massesinunitsofm
PS
Figure 2: Mass spectrum of adjoint QCD with N =3/2 Dirac flavours at β =1.5 with the same lattice
f
actionasfortheN =2investigations,seeFigure1.
f
Theory scalarparticle γ smallβ γ largerβ
∗ ∗
N =1/2adjQCD(SYM) partofmultiplet – –
f
N =1adjQCD light 0.92(1) 0.75(4)
f ∗
N =3/2adjQCD light 0.40(5) 0.32(5)
f ∗ ∗
N =2adjQCD(MWT) light 0.376(3) 0.274(10)
f
Table 3: The comparison of adjoint QCD with a different number of Dirac fermions N . The table states
f
the relation of the scalar glueball to the rest of the bound state spectrum and shows the mass anomalous
dimension. Theresultsindicatedbya symbolarestillinapreliminarystate.
∗
isthesameinbothcases,seeFigure2(a)and2(b). Themainexceptionthatwehavediscoveredso
faristhespin-1/2state: itchangesfromm <m intheN =2casetom <m forN =3/2.
1/2 PS f PS 1/2 f
We have done a first preliminary investigation of the mass anomalous dimension from the
mode number. For the two different coupling constants that we have currently considered, we
obtainavalueofγ =0.40(5)atβ =1.5,andγ =0.32(5)atβ =1.7. Thevaluesarehencelarger
∗ ∗
thanintheN =2case,andapparentlythereareagainsomescalingcorrectionsthatwouldhaveto
f
beincludedinordertoobtainthefinaluniversalresult.
5. Conclusions: comparisonofN =1/2toN =2
f f
Our results can be combined with other findings in order to obtain a more complete general
picture for adjoint QCD with various numbers of fermions, see Table 3. The smallest possible
numberoffermions,N =1/2correspondstosupersymmetricYang-Millstheory. Thistheoryhas
f
been considered in our earlier investigations [14]. It is a confining theory with supersymmetric
multiplets of bound states. The lightest scalar is necessarily a component of the lightest multiplet
andcanhencenotbeseparatedfromtherestofthespectrum.
One-flavouradjointQCDhasbeenconsideredin[15]. Evenwiththisrathersmallnumberof
fermions, the theory seems to be in the conformal window. The lightest scalar particle is a scalar
boundstateandthereisnoclearsignatureofchiralsymmetrybreaking. Ourresultsshowhowthe
6
LatticesimulationsofadjointQCD GeorgBergner
mass anomalous dimension consistently decreases with the number of fermions. In addition also
thegapbetweenthelightestscalarboundstateandtherestofthespectrumincreases. Thesebounds
of the conformal behaviour can also be generalised to different SU(N ) and other representations,
c
asdetailedin[16].
Acknowledgments
ThisprojectissupportedbytheJohnvonNeumannInstituteforComputing(NIC)withgrants
ofcomputingtime. WealsogratefullyacknowledgetheGaussCentreforSupercomputinge.V.for
fundingthisprojectbyprovidingcomputingtimeontheGCSSupercomputerSuperMUCatLeib-
niz Supercomputing Centre. Further computing time has been provided by the computer cluster
PALMAoftheUniversityofMünster.
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