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Correlation functions with Karsten-Wilczek fermions 6 1 Johannes Heinrich Weber ∗ 0 TechnischeUniversitätMünchen,PhysikDepartmentT30f 2 E-mail: [email protected] n a J TheKarsten-Wilczekactiondescribestwochiralfermionsbutbreaksthesymmetriesunderboth 5 2 chargeconjugation(C(cid:98))andtimereflection(Θ(cid:98))explicitly,thoughinvarianceunderC(cid:99)Θandamir- rorfermionsymmetryT(cid:99)Θaremaintained.Theseproceedingsoutlinehowtheaction’ssymmetries ] andthepresenceofthesecondfermionemergeinmesoniccorrelationfunctions. t a The residual symmetries explain the non-observation of broken time-reflection symmetry in a l - classofmesoniccorrelationfunctions.Time-reflectionsymmetryisenforcedforcorrelationfunc- p e tionsthataremanifestlyinvariantunderC(cid:98)orT(cid:98). Duetocontributionsfromthesecondfermion, h oscillating contributions arise in some mesonic correlation functions. A second condition for [ non-perturbativetuningoftherelevantcountertermisobtainedfromtheseoscillations. Bothnon- 1 perturbativetuningconditionsareindependentandagreewithinerrors.Duetocontributionsfrom v thesecondfermion,additionalpseudoscalarstatesareobservedinnon-standardchannels. Mass 9 6 splittingsbetweentheseadditionalstatesandtheGoldstonebosonvanishasO(a2). 6 6 0 . 1 0 6 1 : v i X r a The32ndInternationalSymposiumonLatticeFieldTheory, 23-28June,2014 ColumbiaUniversityNewYork,NY Speaker. ∗ (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ CorrelationfunctionswithKarsten-Wilczekfermions JohannesHeinrichWeber 1. Introduction Minimallydoubledfermionsareacategoryofultralocal,chirallatticefermionsthathaveonly twodegenerateflavoursofquarks. Karsten-Wilczekfermions[1]areatypeofminimallydoubled fermions that retains two original poles of the naïve Dirac operator on the Euclidean time axis. SincethetwopolesoftheKarsten-WilczekoperatorlieonalineintheBrillouinzone, thehyper- cubicsymmetryisinevitablybrokenandfurtheranisotropictermsaregenerateddynamically. Thetree-levelKarsten-Wilczekfermionactionreads(withWilczekparameterζ,4ζ2>1) 3 1 (cid:16) (cid:17) Sf [ψ,ψ¯,U]=a4 ∑ ∑ ψ¯ γµ Uµψ Uµ† ψ +m ψ¯ ψ +ζ n Λ µ=0 n2a n n+µˆ − n−µˆ n−µˆ 0 n n ∈ 3 iζ (cid:16) (cid:17) +∑ψ¯ γ0 2ψ Ujψ Uj† ψ (1.1) n2a n− n n+jˆ− n jˆ n jˆ j=1 − − and the gauge action is assumed to be Wilson’s plaquette action. The standard discrete symmetries of this action were reviewed first in [2] and include γ5 hermiticity, cubic symmetry under spatial rotations, symmetry under spatial reflections and symmetey under the product of charge conjugation (C(cid:98)) and time reflection (Θ(cid:98)). Eq. (1.1) varies under charge conjugation (C(cid:98)) or time reflection (Θ(cid:98)), since each transformation flips the sign of the Karsten-Wilczek term [lower line in eq.(1.1)]relativetothenaïveterm. Thischangeoftheactioncanbeparameterisedasachangeof f f theWilczekparameterasS S . However,C(cid:99)Θtogetherisasymmetrytransform. Moreover, +ζ → ζ theactionisinvariant[3]underloc−alvectorandaxialtransforms(thelatteronlyform =0) 0 ψn e+iαnVψn, ψ¯n ψ¯ne−iαnV, ψn eiαnAγ5ψn, ψ¯n ψ¯ne+iαnAγ5. (1.2) → → → → ThereisonefurtherunitarytransformT(cid:98)thatinterchangesthepolesoftheDiracoperator, ψn T(cid:98)ψn=Tnψn, ψ¯n (T(cid:98)ψ¯n)=ψ¯nTn, Tn iγ0γ5( 1)n0, (1.3) → → ≡ − whichleavesthenaïvetermintheupperlineofeq.(1.1)invariantbutflipsthesignoftheKarsten- Wilczek term. It is related to the remnant of the discrete subgroup of theU(4)-symmetry of the naïve Dirac action (cf. [4]) and permits two further symmetry transforms T(cid:99)C and T(cid:100)Θ. The latter has been pointed out as “mirror fermion symmetry” in [5]. Hence, the action has one further independent non-standard discrete symmetry. The action in eq. (1.1) requires inclusion of one relevantandtwomarginalcountertermsthatwerecalculatedperturbativelyin[3]atone-looplevel, i S3 [ψ,ψ¯]=sign(ζ)c(g2)a4 ∑ψ¯ γ0ψ (1.4) +ζ 0 na n n Λ ∈ 1 (cid:16) (cid:17) S4f[ψ,ψ¯,U]=d(g2)a4 ∑ψ¯ γ0 U0ψ U0† ψ , (1.5) 0 n2a n n+0ˆ− n 0ˆ n 0ˆ n Λ − − ∈ β 3 S4g[U]=d (g2) ∑ ∑ReTr(1 Uj0), (1.6) p 0 3 − n n Λj=1 ∈ whereUnj0isthetemporalplaquetteatsiten. Inparticular,thougheq.(1.4)flipsitssignunderC(cid:98),Θ(cid:98) andT(cid:98),botheqs.(1.5)and(1.6)areinvariantunderC(cid:98),Θ(cid:98) andT(cid:98). Theone-loopcoefficients[3]read c1L(g2)= 29.5320C G, d1L(g2)= 0.125540C G, d1L(g2)= 12.69766C G, (1.7) 0 F 0 f P 0 2 − − − 2 CorrelationfunctionswithKarsten-Wilczekfermions JohannesHeinrichWeber whereG=g2/(16π2). Anon-perturbativetuningschemethatfixesc(g2)asthevaluec thathas 0 0 M minimalanisotropyofthepseudoscalarmassatfinitelatticespacingwaspresentedin[6]. Because no method for non-perturbative tuning of d(g2) is known at present due to lack of sensitivity, its 0 perturbativeestimateisused. 2. Higherorderoperators HigherorderoperatorsarerestrictedbyC(cid:99)Θ andchiralsymmetriesinparticular. Allpossible γ5hermitiancontinuumoperatorsofdimensionfivethatrespectchiralsymmetryunlesstheyvanish in the chiral limit are collected in table 1. They are referred to as O in the ensuing discussion, RC whereRandCarerowandcolumnindices. DerivativesaresymmetrisedasDµ = 1(Dµ Dµ). 2 R− L ψ¯ m2ψ ψ¯ m D/ψ ψ¯ m γ0D0ψ m FµνFµν m Fi0Fi0 0 0 0 0 0 3 3 ψ¯ im ∑ Σ0jDjψ ψ¯ i ∑ γjF0jψ ψ¯ i D0,D/ ψ ψ¯ im D0ψ ψ¯ im2γ0ψ 0 { } 0 0 j=1 j=1 3 3 ψ¯ iγ0 ∑ DjDjψ ψ¯ iγ0D0D0ψ ψ¯ iγ0 ∑ ΣjkFjkψ ψ¯ i(γ0D/D/+D/D/γ0)ψ j=1 j<k=1 Fkj(DkF0j) F0j(D0F0j) Fj0(D0Fj0) Fjk(D0Fjk) ∗ ∗ Table1:EighteencontinuumoperatorsofdimensionfivecanbeconstructedifoddnumbersoftheEuclidean index‘0’areallowedandγ5hermiticityisrequired. Chiralsymmetryisrequiredonlyinthechirallimit. a) Chiralactionsareinvariantundertheaxialtransformofeq.(1.2)andasimultaneousflipofthe signofm . ThispropertyprohibitsallO(am )correctionsasinO ,...,O . 0 0 11 15 3 b) Sums are completed by adding zero as ∑ Σ0jDj = i[γ0,D/] and D/,[γ0,D/] =2(cid:126)γ [D0,(cid:126)D]. 2 { } · j=1 Then2O =O +O(a)andO =0+O(a)arederivedfromfieldequations. 21 22 21 c) Completion of sums and field equations yield O = 2O +O(a) and O = O +O(a). 23 25 24 25 − O andO areeliminatedandO isconsideredasO(a2m2)chiralcorrectiontocofeq.(1.4). 23 24 25 0 d) O is the Karsten-Wilczek term’s continuum form. Though 2(O +O )+O =O is an 31 31 32 33 34 exactrelation,nooperatoriseliminatedasthecoefficientζ ofO isfixed(Wilczekparameter). 31 e) IterativeapplicationofthefieldequationyieldsO =2O +O(a)andeliminatesO . 34 25 34 f) AsO41 andO42 lackinvarianceunderΘ(cid:98) butareC(cid:98)andT(cid:98)invariantandO43 andO44 lackinvari- anceunderparity,theyareprohibited. ThesumO +O isatotalderivative. 43 44 Hence, O and O form a complete set of independent additional operators. A lattice operator 32 33 for O33, which couples an axial current to the colour magnetic field, can be constructed using F(cid:98)nkl as in the clover term for Wilson fermions. Invariance under T(cid:99)C requires next-to-next neighbour terms in O . As a next-neighbour operator, it would vanish at only one of the two poles of the 32 Diracoperator. Thetwoadditionalindependentdimensionfivelatticeoperatorsarechosenas: i (cid:16) (cid:17) O : S5t =ca4 ∑ψ¯ γ0 2ψ U0U0 ψ U0† U0† ψ , (2.1) 32 t n4a n− n n+0ˆ n+20ˆ− n 0ˆ n 20ˆ n 20ˆ n Λ − − − ∈ 3 O33: S5B=cBa4 ∑ψ¯n ∑ εjklγ5γjF(cid:98)nklψn. (2.2) n Λ j,k,l=1 ∈ 3 CorrelationfunctionswithKarsten-Wilczekfermions JohannesHeinrichWeber The gauge action is anisotropic due to eq. (1.6) but does not include O(a) operators and retains separate invariance underC(cid:98), Θ(cid:98) and T(cid:98). This invariance holds for any observables without valence quarks even in full QCD, because the fermion determinant is invariant under each of the symmetries. This statement is proved in section 3. Each dimension five operator varies underC(cid:98), Θ(cid:98) and T(cid:98), but is still invariant underC(cid:99)Θ, T(cid:99)C and T(cid:100)Θ. The anisotropy at tree level is deferred to higher orders[atleastO(a2)]ifc ,c andthecoefficientofO arechosentobeequaltoζ. Lastly,though t B 25 O(am ) chiral corrections to bare parameters are prohibited by chiral symmetry, O(a2m2) chiral 0 0 correctionsarerequiredforthecoefficientoftherelevantcounterterm. 3. Symmetryofthefermiondeterminant Invarianceofthefermiondeterminantisinferredfrominvarianceofthepartitionfunction, (cid:90) Z+ζ =N Dψ¯DψDUe−S+fζ[ψ,ψ¯,U]−Sg[U], (3.1) where Sf [ψ,ψ¯,U] and Sg[U] are the fermion action of eq. (1.1) and Wilson’s plaquette action +ζ including the three anisotropic counterterms and N is a normalisation constant. The fermions are formallyintegratedoutandyieldthefermiondeterminantandchangetheconstanttoN , (cid:48) (cid:90) Z =N DUdet(Df [U])e Sg[U]. (3.2) +ζ (cid:48) +ζ − Next,thefermiondeterminantsatisfies(C=iγ0γ2 isthechargeconjugationmatrix) det(Df [U])=det(Df [U]C 1C)=det(CDf [U]C 1)=det(Df [Uc])T, (3.3) +ζ +ζ − +ζ − ζ − wheregaugefieldsU arereplacedbychargeconjugatedfieldsUc,theWilczekparameterζ changes itssignandtheDiracoperatoristransposed. Sincethegaugeactionandthemeasureareinvariant underchargeconjugationandthedeterminantisinvariantundertransposition,Z satisfies (cid:90) (cid:90) Z =N DUdet(Df [Uc])T e Sg[U]=N DUcdet(Df [Uc])e Sg[Uc]=Z . (3.4) +ζ (cid:48) ζ − (cid:48) ζ − ζ − − − AfterrelabelingUc toU,itfollowsthatthepartitionfunctionisanevenfunctionofζ. Sinceeach ofthetransformsC(cid:98),Θ(cid:98) orT(cid:98)flipsthesignofζ,thepartitionfunctionisinvariantundereither. Dueto theinvarianceofthegaugeactionandthemeasureundereach,thefermiondeterminantisinvariant as well. An even simpler proof uses the Tn of the unitary transform T(cid:98) [cf. eq. (1.3)] instead ofC. Thus,thefullQCDvacuumisinvariantunderC(cid:98),Θ(cid:98) andT(cid:98)andhasmoresymmetrythantheaction. 4. Timereflectionofcorrelationfunctions InvarianceoftheKarsten-WilczekactionundereitherC(cid:99)Θ-orT(cid:100)Θ-transformationsisreflected in invariance of the Hamiltonian H(cid:98). A generic correlation function C(tf ti) between initial and − finalstatesO(cid:98)†i|Ω(cid:105)andO(cid:98)†f|Ω(cid:105)usingthetransfermatrixU(cid:98)(t)=e−H(cid:98)t reads C(tf−ti)=(cid:104)Ω|O(cid:98)fe−H(cid:98)(tf−ti)O(cid:98)†i|Ω(cid:105), (4.1) 4 CorrelationfunctionswithKarsten-Wilczekfermions JohannesHeinrichWeber where Ω istheinvariantvacuum. InvarianceofthevacuumforfullQCDisprovedinsection3. In | (cid:105) particular,C(cid:98)Ω = Ω =Θ(cid:98) Ω . DuetoinvarianceofH(cid:98)underC(cid:99)Θ,onehasU(cid:98)(t)=C(cid:98)†Θ(cid:98)†U(cid:98)( t)C(cid:98)Θ(cid:98): | (cid:105) | (cid:105) | (cid:105) − C(tf−ti)=(cid:104)Ω|O(cid:98)fC(cid:98)†Θ(cid:98)†e−H(cid:98)(ti−tf)Θ(cid:98)C(cid:98)O(cid:98)†i|Ω(cid:105). (4.2) IfO(cid:98)f andO(cid:98)i botheithercommuteoranticommutewithC(cid:98),theoperatorC(cid:98)ismovedpastO(cid:98)f andO(cid:98)i, C(tf−ti)=(cid:104)Ω|C(cid:98)†O(cid:98)fΘ(cid:98)†e−H(cid:98)(ti−tf)Θ(cid:98)O(cid:98)†iC(cid:98)|Ω(cid:105), (4.3) andleavesthevacuuminvariant. Insertionof1=Θ(cid:98)Θ(cid:98)† betweenvacuumstatesandoperatorsyields C(tf−ti)=(cid:104)Ω|Θ(cid:98)†Θ(cid:98)O(cid:98)fΘ(cid:98)†e−H(cid:98)(ti−tf)Θ(cid:98)O(cid:98)†iΘ(cid:98)†Θ(cid:98)|Ω(cid:105) (4.4) =(cid:104)Ω|(Θ(cid:98)O(cid:98)fΘ(cid:98)†)e−H(cid:98)(ti−tf)(Θ(cid:98)O(cid:98)†iΘ(cid:98)†)|Ω(cid:105)=Θ(cid:98)C(tf−ti). (4.5) ThisismanifestΘ(cid:98) invarianceofC(cid:98)invariantcorrelationfunctions. Sincealldimensionfiveopera- torsexplicitlybreakeachofC(cid:98),Θ(cid:98) andT(cid:98)symmetries,oddpowersoftheseoperatorsmustcancelif thecorrelationfunctionhassuchsymmetry. Therefore,leadingdiscretisationeffectsaresuppressed toO(a2)inC(cid:98)invariantcorrelationfunctions. AnalogousresultsrelyonT(cid:98)insteadofC(cid:98)andrequire O(cid:98)i,f thatbotheithercommuteoranticommutewithT(cid:98). Aspurelygluonicoperatorsobviouslycom- mute with T(cid:98) (which affects only fermions), Θ(cid:98) invariance and suppression of O(a) corrections are apparent for gluonic observables. Caveat: Generic composite operators of fermion fields that lackadditionalsymmetryunderC(cid:98)orT(cid:98)mayhaveO(a)corrections. 5. Decompositionofinteractingfields ThefreefieldhastwocomponentsrelatedbytheunitarytransformT(cid:98)ofeq.(1.3), ψ =∑gφ φ +T gχ χ , ψ¯ =∑φ¯ gφ† +χ¯ gχ† T , (5.1) n n,m m n n,m m n m m,n m m,n n m m where restriction of the components’ four-momenta is realised through support on multiple sites. Two kernels gφ , gχ (an example is found in [7]), which satisfy gχ =T gφ T =δ +O(a) m,n m,n m,n m m,n n m,n implement extended support. Contraction of unlike components generates terms with alternating sign in correlation functions. The same mechanism [8] generates parity partner contributions for naïveorstaggeredfermions. Withinteractions,therelevantoperatorofeq.(1.4)cancelsdivergent interaction effects that would spoil the continuum limit. In the following, approximate tuning is assumed–amismatchδc=c c(g2)oftheparameterissupposedtobesmall. Thismismatched − 0 relevanttermwouldspoilthecontinuumlimitofkernelslikegφ ,gχ . Alocalfieldtransform m,n m,n δc ψ ψc=ξ ψ , ψ¯ ψ¯c=ψ¯ ξ , ξ eiϕn0, ϕ (5.2) n→ n n n n→ n n n∗ n≡ ≡ 1+d thatmodifiesthetemporalboundarycondition(cf.[2])shiftsthemismatchδcuptoorderO(a,ϕ2) intothedefinitionofthefieldsψc,ψ¯c. Thelatticeproductrulethatisappliedrequireslinearisation in ϕ and a. The field transform’s parameter ϕ is constrained by the given parameters c (or rather δc) and d. As the relevant operator of eq. (1.4) anticommutes with T(cid:98), absorption of δc into two interactingcomponentsrequiresoppositephases[cf.eq.(5.1)], ψ = ∑ξ gφ [U]φc +ξ T gχ [U]χc, ψ¯ = ∑φ¯cgφ† [U]ξ +χ¯cgχ† [U]T ξ , (5.3) n n n,m m n∗ n n,m m n m m,n n∗ m m,n n n m m 5 CorrelationfunctionswithKarsten-Wilczekfermions JohannesHeinrichWeber where kernels gφ [U], gχ [U] are outfitted with Wilson lines. This definition decomposes local m,n m,n fermionicbilinearsatleadingorderintocontributionswithdifferentspace-timequantumnumbers, ψ¯ Mψ =ξ2χ¯ gχ†[U]T Mgφ [U]φ +ξ 2φ¯ gφ†[U]MT gχ [U]χ n n n k k,n n n,l l n∗ k k,n n n,l l +φ¯ gφ†[U]Mgφ [U]φ +χ¯ gχ†[U]T MT gχ [U]χ . (5.4) k k,n n,l l k k,n 0 0 n,l l Dirac structures of bilinears with unlike (e.g. φ¯MT χ) or like (e.g. φ¯Mφ) components differ by 0 T0. Phasefactors( e±2iϕ)n0 =( 1)n0ξn(∗)2forunlikecomponentsineq.(5.4)generateoscillations − − in correlation functions with a frequency that depends on the mismatch δc. Except for this δc- dependence, this is well-known for naïve and staggered fermions [8]. If linearisation in ϕ and a anddecompositionwithkernelsgφ,χ[U]areapplicableinthenon-perturbativeregime,theobserved n,m frequency is shifted by ω =2 ϕ unless c is tuned properly. This prediction yields a new tuning c | | conditionthatisputtothetestinnumericalsimulations. 6. Pseudoscalarcorrelationfunctions 1 R05(n0) 0.3 ωc β =6.2 0.25 0.5 0.2 0 0.15 β 5.8 0.1 6.0 -0.5 6.2 0.05 c: 0.35 0.38 0.40 0.42 0.45 − n c 0 -1 0 0 16 32 48 64 80 96 112 128 -0.6 -0.55 -0.5 -0.45 -0.4 -0.35 -0.3 Figure 1: Left: The frequency of oscillations in the ratio R (n )=C (n )/C (n ) differs from π by a 05 0 00 0 55 0 shiftω thatdependsonc. Right: Thefrequencyshiftω islinearlyinterpolatedasω ∝ c c . c c c 0 | − | Pseudoscalarsarethegroundstatesofγ0 andγ5 channelsduetoeq.(5.4). Oscillationsareap- parentintheratioR (n )=C (n )/C (n )[cf.leftplotinfigure1]. Goodfrequencyresolution 05 0 00 0 55 0 requires a long time direction. The frequency shift is studied on ten quenched configurations of 128 243latticeswithpseudoscalarmassesr M 450MeVandperturbativelytunedd. Splitting 0 55 × (cid:39) betweenpseudoscalarmassesinbothchannels, ∆ =r2(M2 M2 ), (6.1) 05 0 00 55 − causesasmallexponentialdecaythatbroadensthepeaksinfrequencyspectraandcausesasystem- aticalerrorthatdominatesoverthestatisticalerror. Thefrequencyisshiftedasω ∝ c c = δc c 0 | − | | | and agrees with the prediction of linearity [cf. right plot in figure 1]. The error of c is seen to 0 diminish for finer lattices. The mass splitting ∆ is nearly constant over the range 250MeV (cid:46) 05 r M (cid:46)650MeVandapproachesthecontinuumlimitasO(a2)[cf.leftplotinfigure2]. 200(100) 0 55 quenchedconfigurationson48 243(323)-latticeswithβ =5.8,6.0(6.2)areused. Theoscillating × groundstateofC (n )isdegeneratewithaGoldstonebosoninthecontinuumlimit. Theground 00 0 state of C (n ) behaves as a Goldstone boson with quenched chiral logarithms at finite lattice 55 0 spacing. Theseobservationsattesttotheapplicabilityofthedecompositionofinteractingfields. 6 CorrelationfunctionswithKarsten-Wilczekfermions JohannesHeinrichWeber 0.14 0.6 ∆ [GeV2] -c 0.12 05 0.55 0.1 0.5 0.08 0.45 0.06 c0+cc22aa22 0.4 -c1L 0.04 0.35 -cBPT -cM 0.02 0.3 -c0 0 0.25 a2[fm2] g2 -0.02 0.2 0 0 0.005 0.01 0.015 0.02 0.94 0.96 0.98 1 1.02 1.04 Figure 2: Left: The splitting between pseudoscalar masses in C (n ) and C (n ) vanishes as O(a2). 00 0 55 0 Right: c atoneloopfallsshortnon-perturbativeresults(c ,c ). Thefigureisexplainedinthetext. 1L M 0 7. Non-perturbativetuning Various methods for non-perturbative tuning of c are compared in the left plot in figure 2. InterpolationsuseaPadéfituptoO(g4),whereO(g2)isfixedastheone-loopresultc ofeq.(1.7): 0 0 1L c(g2)= c1L(g20)+Ng40. (7.1) 0 1+Dg2 0 ThetwofitparametersN andDforc definedbyω =0read 0 c N = 0.2163(8), D = 0.926(2), (7.2) 0 0 − − and agree within errors with c defined by minimal mass anisotropy [6], though uncertainties of M c are much larger. For fine lattices (a 0.06fm), c from boosted perturbation theory [9] is M BPT ≤ consistentwithnon-perturbativeresults. Thus,useofd appearsjustifiable. BPT Acknowledgements: The speaker thanks C.Seiwerth for technical support and P.Petreczky for invaluable discussions. Simulations were performed on the cluster “Lily” at the Institute for Nuclear Physics, Univ. of Mainz. This work was supported by the “Excellence Cluster Universe” andbytheResearchcenter“ElementaryForces&MathematicalFoundations”(EMG). 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