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Effective Field Theory Effective Field Theory Iain Stewart MIT Nuclear Physics 2007 Long Range Plan Joint Town Meeting on QCD Jan, 2007 Effective Field Theories of QCD ChPT HTL NRQCD cc states pions large T large HDET unstable density particle top quark QCD energetic EFT hadrons SCET jets nuclear SCET single nucleon forces bd states NNEFT Heavy Hadron HQET ChPT Arrows: Effective theories describe infrared limits of QCD m ChPT π 1 π Λ ! talks by D. Kaplan, D. Phillips, p H. Griesshammer 1 Λ ! Effective Field Theories of QCD ChPT HTL NRQCD cc states pions large T large HDET unstable density particle top quark QCD energetic EFT hadrons SCET jets nuclear SCET single nucleon forces bd states NNEFT Heavy Hadron HQET ChPT Arrows: Effective theories describe infrared limits of QCD Q Soft-Collinear Effective hard probes Theory (SCET) Λ 1 • Q ! Effective Field Theories of QCD ChPT HTL NRQCD cc states pions large T large HDET unstable density particle top quark QCD energetic EFT hadrons SCET jets nuclear SCET single nucleon forces bd states NNEFT Heavy Hadron HQET ChPT Lattice QCD a Chiral EFT ’s g L QCD, π Nuclei 1 mπ− q a, L, QCD Q perturbative 1 F (Q2) = dxdy H(x, y) φ(x) φ(y) Q2 SCET Λ ! QCD non-perturbative 2 ChPT mπ f µ (4) = tr ∂ Σ∂ Σ† + v tr m†Σ+m Σ† + (L ) L 8 µ q q L i ! " ! " Lattice & EFT - extrapolations Symanzik’s Effective Field Theory Symanzik’s EffeχctiSBvewFieithldWTilsheonoQryuarks ! −1 Near continuum a ΛQCD, lattice action described by a continuum effective fieldFtehremorioy n Discretization ! W−1ilson solves the doubling radically with second Near continuum a Λ2 QCD, lattice a1 ction describ1ed by a 2 2 0 1 2 Sy−manzik’s Effective Field Theory Symanzik Action: SSymanzik = S + a S + a Sder+iv.a.t.ive Go (m = 0) = iγµ a sin(pµa) + a sin (pµa/2) FecormniotinnuDumLisactetricefefetizdctiscriavtioeetizafientilodnt(hsheoortrryange) in χPT? Operators in EFT respect symmetries of laCttihceir−a1al ctsymionmetry1 recovered only in the continuum limit Naive discretization G (m = 0) = iγµ sin(p2µa) Symanzik Action: SSymanziko = S0 + a S1a + a S2 + . e.g. gauge invariance, hypercubic invarianLceeading Symanzik action contains!the−P1auli term Lattice discrSoelutitzaiontisonto(shferomrtiornandgoeu)bilninχgPgTe?neraNlley aleracod tonchtiniruaul m a ΛQCD, lattice action described by a OperatorssyminmEetFryTbreeaspkinegctat finsymite am, weitthriems=o0f lattice action S0 is the QCD action: S0 = ψ(D/ + m)ψ −1 S1 = a cS1W ψσµνcoGµnνtψinuum effective field theory Naive discretization Go (m = 0) = iγµ a sin(pµa) e.g. gaugBeuiildnvχaPTriafonr cethe,Shymypaenzircuk abctiiconinvariance 2 Discretization effects enter as higher dimenNsioonn-perturbative O(a) improvement renders cSW = 0 Solutions Shtoarfpee ramnd iSionngledtoonu’9b8,lLineegangdeShnaerprea’9ll9y lead toSchymiraalnzik Actireonvie: wSSedym byan zik = S0 + a S1 + a S2 + . operators, e.g., O S40sym=isc mth4eeψtryDQbCrDeDakiDnagctγaitoψfinn∈:iteS(aP20,ew=rithuψrm(bD/a=+t0ivme )imψprovement schemes deplete size of ci’s) H H Gµoldstµ onµe µmode propagation sensitOivepteoreaxptolircist iχnSEB FTTiburziresp ate ct symmetries of lattice action Build χPT for the Sym2 anzikFaoctriounnimproved actions mChiralust de Dynamicsal with O’06(a) errors, Hidden log 1/a-depeDniscrdenecetiza: Ectaiioch=nlcaeit(tfgicefe)ctfesrmeionnthearsadsiffehringghχeEreF.dTgi.mgeanusigoeninvariance, hypercubic invariance Sharpe and SinWgletiolnso’98n, ,LeTewanidstSheadrpem’i9an9 ssstrWuctilsoivne, Stotagcogenrestdr,uct χPT. opGeorldasttoornse,meo.dge.,pOropHa4ga=tiocnHse4 ψnsiDtivµeDtoµDexpµγliciµψt χ∈SBS2 Ginsparg-Wilson (Domain Wall), HybSr0idis(MthixedQACctiDon)action: S0 = ψ(D/ + m)ψ Each lattice fermion has difχfeSrinBg fχrEoFmT Pauli2term has the same form as the quark Hidden log 1/a-dependence: ci = ci(g ) Wilson, Twisted mass Wilsomn,aSsstagtgeerrmend., Discretization effects enter as higher dimension Partially O(a ) in χPT – p.3 n Ginsparg-Wilson (Domain Wall), Hybrid (MixeodpAectriaotno)rs, eO(.ag) i.n χ,PTO– p.4 H4 = cH4 ψDµDµDµγµψ ∈ S2 Quenched n O(a 2) in χPT – p.5 Hidden log 1/a-dependence: ci = ci(g ) O(an) in χPT – p.3 n valence O(a ) in χPT – p.4 sea n O(a ) in χPT – p.3 fits F and G which are the same as fit E except the argument of the log2 contribution is chosen to be M = mπ and M = mK , respectively. These choices lead to a larger variation in L˜5 and consequently in fK /fπ. C. Discussion To determine fK /fπ at the physical point and its associated uncertainty we synthesize the results of the NLO and NNLO fits. Fitting the lowest two mass points at NLO gives fK/fπ = 1.215 0.002, while fitting the three data points with pion masses below m 500 MeV ± π ∼ gives fK/fπ = 1.218 0.002. The difference between them is within statistical errors (1.5σ) but there appears to±be a systematic trend in the data which can be attributed to higher orders in the chiral expansion. As we are unable to fit the full NNLO expression to our small data set we can estimate the systematic uncertainty in this calculation by looking at the range of values of fK /fπ that result from the two types of NNLO extrapolation, both with and without the log2 contribution, including variation in the argument of the NNLO logarithm and including statistical errors. The range of variation in the NNLO estimate is an order of magnitude larger than the statistical error found at NLO. We take this NNLO +0.011 uncertainty, ∆(fK/fπ) =−0.022, to be an estimate of the systematic error in our calculation due to the truncation of the chiral expansion. We also assign a systematic error due to fitting procedures, obtained by varying the fitting ranges displayed in fig. 1, which gives +0.000 ∆(fK /fπ) =−0.010. Therefore, our final number is: fK +0.011 = 1.218 0.002 −0.024 , (17) fπ ± wfihtesreFthaendfirGst werhriocrhisarsetatthiseticsaml aendasthfiet sEeceoxncdepetrrothreisasrygsutmemenatico,fwthiteh ltohge2 ecxotnratrpiobluattiioonn is error and fitting error added in quadrature. The error in this lattice QCD determination of chosen to be M = mπ and M = mK , respectively. These choices lead to a larger variation f /f˜ is clearly dominated by the systematics. iKn Lπ5 and consequently in fK /fπ. Decay constants fK , fπ Using a similar procedure, we arrive at a value for L5: Calculation of fπ, fK phy +0.18 −3 L (f ) = 5.65 0.02 − 10 , (18) PrCe.limDisicnusasiroyn Res5uStalπtgsgered: De cFaermions±y Co0n.54st×ants MILC from C.Bernard where the first error is statistical and the second is an estimate of the systematic error To determine fK /fπ at the physical point and its Casosmopcaiarteewdituhncertainty we synthepshiyze the due to omitted higher ordersunquenched,in the chiral expfaanst,s•ionligh. Thtismthπen scales to give L5(mη ) = results of th+e0.N18LO and−3NNLO fits. Fitting the lowepehsyxpt tewroimmenats.s points a+t0.N18LO giv−e3s fK/fπ = 2.22 0f.π02=−128.6 100.4 at 3th.0e Mη-emVass an[129d L.5(m 0.)9 = 31.452MeV0.]02 − 10 at the 1.215 0.0020.,54whi±le fittin±g the three data po5int±sρwith±pion masses be0l.o54w m 500 MeV ρ-ma±ss. As stated×previouslybut, th 4is “tais stanese” ffforect eachive L flaavors it, includ±es the higher×orπder strange gives f±K/f= 155= 1..2318 0.04.0023..1ThMeedVifferen[c156e b.e6twe5e1n.0them3.1i6s/4MwietVhi]n statistical e∼rrors (1.5σ) quark coKntriπbution. ± ±Pissue:relimdet(iD/n+amry) Rd±etes(D/u+l±tms): Decay Constants but there appears to±be a syste+ma7tic trend i→n the data which can be attributed to higher The resultfsKf/forπf=/f1.208(have2)a(n−1a4d)dition[1a.l210(syste4)m(a13)tic]error due to the non-zero lattice orders in the chiraKl exπpanslotsion. ofA paramets we arerse u nina bChPTle to fit the full NNLO expression to our spacing which we expect to be O((m m )b2). In principle one can reduce this error small data set we can estimate the sysst−emautic uncertainty in this calculation by looking at bOy lfidttriensug tltostihneraepdpraorperifartoemχPMTIfLπfCo=r,mP128uRlaDs.6t7h0at(02i.n40c0lu4d3)e.10t1hM4e5eOV0(1g.2b2)[129effe.c5ts d0u.e9to fl3a.5voMr- eV] ± ± ± ± sythmemreatnrgyeborefavkailnugesinotfhfeKs/efaπ-qtuhaarkt rsescutoltr [f2ro4m]. Hthoewetwveor,toyupresdaotfaNfiNt LwOelletxotrtahpeocloanttioinnu,ubmoth with and without the log2 nuclearcontfriKbu beta=tio155n,-decayin.3clud 0igivesn.4g var3i.a1tiMoneVin the [a156rgu.m6 ent1o.0f the3.N6NMLOeV] χPT fEorxpmuelarismaendnthaelnrcaetewefodroπnot eµxνpe+ctVtuhdatf±ruosme o(fsu±thpeeexratellnodwededχ)PT formu±las of±Ref. [24] log•arithm and including statist→ical errors. The range of v+ari7ation in the NNLO estimate is wouldnsuigclneifiacranbtelytaimdpercaovyse our refsπul=tsf130aKt /fth.7πis =st0a1g.1.e208(. O0u2).r4fi(M−na1elV4)r.esult[1is.210(consi4)st(e13)nt w]ith the an order of magnitude large⇒r than the statis±tical e±rror found at NLO. We take this NNLO MILC number [1] +0.011 uncerRtaeisuntyl,t f∆o(rffKK/f/fπ)π=, p−lus V, utod baendanKestimµaνte(eoxpf tth)e, gsyivsetesmaVtuics e.rror in our calculation O0.f0ld2K2 results in red are from MILC, PRD 70 (2004) 114501. du•e to the truncation of the chiral =ex1p.a2n1→0sion0. .0W04e al0s.o01a3ss,ign| a s|ystematic error d(u19e)to Following Marciano, fPR!L 93 (2004)±231803±, get fitting procedures, obtained πb!yMvILaCrying the fitting ranges displayed in fig. 1, which gives Expe+ri2m6 ental rate for π µν + V from (superallowed) where the firs+t0.e00r0ror Vis sta=tis0t.i2223(c!al and th)e sec[o0n.2219(d is th26)e t]otal systeumdatic error estimated ∆(fK /fπ) =− . Thuesrefore•, o!ur fin−a1l4number is: → Chiral Dynamics, Sept. 19, 2006 – p.16 0.010 | | Domainnucle War albel tvalenca decaeys NPLQCDfπ = 130.7 (‘060.1) 0.4 MeV. by MIPLDCG. S(i2n0ce06o)u,rfrvoamlenKce quπaµrkνs (aerxpe dt)omanadin-(wnaolnl -f⇒leartmticeion)s,thienocroy,n±trast w±ith the KS f • →ReKsult for fK /fπ, pl+u0s.01V1ud and K µν (expt), gives Vus .
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