Probing the mixed regime of ChiPT with mixed actions

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1 Probing the mixed regime of ChiPT with mixed actions I. Introduction and spectral observables Carlos Pena In collaboration with: F Bernardoni E Endreß N Garron P Hernández S Necco G Vulvert Chiral Dynamics with Wilson Fermions, ECT*, October 011

2 Themes Mixed actions: use different regularisations for sea and valence quarks to optimal effect. chiral symmetry / renormalisation Mixed chiral regimes: explore low-energy QCD with quark masses in different kinematical regimes. systematic uncertainties in LECs

3 Outline Mixed-action approach to Lattice QCD. (continuum) Chiral Perturbation Theory. ChiPT and finite volume regimes. Random Matrix Theory. ChiPT in a mixed regime. Results for spectral observables in Nf= QCD. Topology. Low-energy couplings: chiral condensate and L s.

4 Mixed action Best of two worlds? Cheap sea sector with broken chiral symmetry, valence quarks with exact chiral symmetry. Bär, Rupak, Shoresh 0-03 Valence: Neuberger fermions. Sea: non-perturbatively O(a) improved Wilson fermions (Nf = CLS configurations). sophisticated Neuberger fermion setup: Giusti et al

5 Mixed action pros and cons Direct access to chiral limit in valence sector. (optimistic:) Greater possibilities than other regularisations. (pragmatic:) Handle on systematics. finite volume/mixed regimes, saturation of correlators with topological zero modes,... Wilson sea samples all topological sectors, Neuberger valence leads to welldefined topological charge. Simplified renormalisation phenomenology.

6 Mixed action pros and cons Direct access to chiral limit in valence sector. (optimistic:) Greater possibilities than other regularisations. (pragmatic:) Handle on systematics. finite volume/mixed regimes, saturation of correlators with topological zero modes,... Wilson sea samples all topological sectors, Neuberger valence leads to welldefined topological charge. Simplified renormalisation phenomenology. - Neuberger fermions significantly more expensive than e.g. Wilson. - Different cutoff dependence in valence and sea + partial quenching potentially large lattice artifacts. Golterman, Izubuchi, Shamir PRD 71 (005) Dürr et al./aubin, Laiho, van de Water PoS LAT007 Cichy, Herdoíza, Jansen 011

7 Mixed action pros and cons Direct access to chiral limit in valence sector. (optimistic:) Greater possibilities than other regularisations. (pragmatic:) Handle on systematics. finite volume/mixed regimes, saturation of correlators with topological zero modes,... Wilson sea samples all topological sectors, Neuberger valence leads to welldefined topological charge. Simplified renormalisation phenomenology. this talk: use Neuberger-Dirac operator as probe for chiral symmetry breaking (topology, spectral quantities).

8 Chiral Perturbation Theory Subtle dichotomy: Guide mass extrapolations, circumvent difficult problems (weak decays). Test ChiPT: first-principles LECs, check predictions for m,v dependence.

9 Chiral Perturbation Theory Subtle dichotomy: Guide mass extrapolations, circumvent difficult problems (weak decays). Test ChiPT: first-principles LECs, check predictions for m,v dependence ! 1/ FLAG check systematic uncertainties! N f = N f =+1 MILC 10A [SU() fit] JLQCD/TWQCD 10 RBC/UKQCD 10A JLQCD 09 MILC 09A [SU(3) fit] MILC 09A [SU() fit] MILC 09 TWQCD 08 JLQCD/TWQCD 08B PACS-CS 08 [SU(3) fit] PACS-CS 08 [SU() fit] RBC/UKQCD 08 JLQCD/TWQCD 10 ETM 09C ETM 08 CERN 08 JLQCD/TWQCD 08A JLQCD/TWQCD 07A Bernardoni 10 ETM 09B HHS 08 JLQCD/TWQCD MeV 350

10 Finite volume scaling and chiral regimes p-regime: lim [m πl 1, 4πF L 1 ] m 0 standard χpt in finite V: m p mass, volume effects relevant L 1, T 1 p Gasser, Leutwyler 1987; Hansen 1990; Hansen, Leutwyler 1991 "1 m! L ε-regime: lim [mσv O(1), 4πF m 0 L 1 ] reordering of the χ expansion: m p 4 ɛ 4 volume effects relevant, mass effects suppressed L 1, T 1 ɛ

11 Finite volume scaling and chiral regimes p-regime: lim [m πl 1, 4πF L 1 ] m 0 standard χpt in finite V: m p mass, volume effects relevant L 1, T 1 p Gasser, Leutwyler 1987; Hansen 1990; Hansen, Leutwyler 1991 "1 m! L ε-regime: lim [mσv O(1), 4πF m 0 L 1 ] reordering of the χ expansion: m p 4 ɛ 4 volume effects relevant, mass effects suppressed L 1, T 1 ɛ Low energy constants universal: check systematics of matching to QCD

12 Finite volume scaling and chiral regimes Gasser, Leutwyler 1987; Hansen 1990; Hansen, Leutwyler 1991 NLO: χ MU Lµ i µuu, Wµν =( µlν + νlµ); ( ij ) ab = δ ai δ bj p-regime ɛ-regime L Gasser, Leytwyler Leutwyler H QCD L 4 DµU D µ U U χ + χ U L 5 DµU D µ ( U U χ + χ ) U L 6 U χ + χ U L 8 χ Uχ U + U χu χ Kambor,Missimer,Wyler Kambor, Missimer, Wyler H SU(4) weak D ± t± ij,kl ij (χ χ ) kl (χ χ ) D 4 ± t± ij,kl ij L µ kl {L µ, (χ + χ } ) D 7 ± t± ij,kl ij L µ kl Lµ (χ + χ ) D 0 ± t± ij,kl ij L µ kl νwµν D 4 ± t± ij,kl ij W µν kl Wµν

13 ChiPT in the ε-regime Pion zero-momentum modes become non-perturbative. U = U 0 e iξ(x)/f d 4 xξ(x) =0 Z = du 0 d 4 xj(ξ)e S(U 0,ξ) SU(N f ) Gasser, Leutwyler 1987; Hansen 1990; Hansen, Leutwyler 1991 Exact factorisation at LO: S LO (U 0, ξ) = d 4 x Tr[ µ ξ µ ξ] ΣV Tr MU 0 + U 0 M Integrals over zero-mode manifold can be done exactly via master integral. ΣV Z(N f,m,θ) = du 0 exp SU(N f ) Tr e iθ/n f MU 0 +h.c. dθ Z ν (N f,µ)= π e iνθ Z (0) = du 0 det(u 0 ) ν e µ Tr[U 0+U 0 ] =det ij U(N f ) I i j+ν (µ) Brower, Rossi, Tan 1981

14 ChiPT in the ε-regime: the role of topology Correlation functions in the ε-regime depend on topology. Leutwyler, Smilga 199 D 1 = φ k (x) φ k (y) mv k 1/m poles in quark propagator ρ ν λ ( ν +N f )+1 zero modes are repelled Consider averages in fixed topological sectors: topological charge becomes scaling variable. p-regime: match dependence on ε-regime: match dependence on m, L L, ν

15 V = p-regime vs. ε-regime M π L O(1) mσv O(1) V 3 V V V 1 V 1 m a m a m a C NLO (m, Σ,F }{{}, }{{} L i LO NLO,...) Cp NLO (m, V, Σ,F }{{}, }{{} L i LO NLO,...) Cɛ NLO (m, V, Σ,F }{{},...) LO ε-regime cleanest for LO LECs, p-regime probes NLO LECs (mass dependence).

16 p-regime vs. ε-regime p-regime: check mass dependence M π = M 1+ M F π = F M 3π F ln Λ 3 M 1 M 16π F ln Λ 4 + O(p 4 ) + O(p 4 ) M Σm F lk ln Λ k M M=139.6 MeV ε-regime: check volume, topology dependence C P (t) = 1 L 3 d 3 xp (x)p (0) = Σ a P + C A (t) = 1 L 3 d 3 xa 0 (x)a 0 (0)= F V a A + T F L 3 b P h 1 (t/t )+O(ε 4 ) T F L 3 b Ah 1 (t/t )+O(ε 4 ) h 1 (x) = 1 (x 1 ) 1 1 a A,P,b A,P functions of L, T, ν

Zero mode domination or, Random Matrix Theory ε-regime: gap between

"# '()*$ possible to obtain an effective description of ChiPT by

17 Zero mode domination or, Random Matrix Theory ε-regime: gap between zero (E~M) and non-zero (E~π/L) momentum modes. +$! "# '()*$ possible to obtain an effective description of ChiPT by integrating out non-zero modes within ChiPT. Shuryak, Verbaarschot, Zahed 93-94! "#$ %&'()*$ [Z ν ZMChPT] LO = k U(N f ) du 0 det(u 0 ) ν e µ Tr[U 0+U 0 ] = Z RMT (N f,µ,ν) N.B.: ZMChPT matches same RMT at NLO, all corrections absorbed in μeff.

18 Zero mode domination or, Random Matrix Theory ε-regime: gap between zero (E~M) and non-zero (E~π/L) momentum modes. +$! "# '()*$ possible to obtain an effective description of ChiPT by integrating out non-zero modes within ChiPT. Shuryak, Verbaarschot, Zahed 93-94! "#$ %&'()*$ [ZZMChPT] ν LO = k [Z ν ZMChPT] NLO = k U(N f ) U(N f ) du 0 det(u 0 ) ν e µ Tr[U 0+U 0 ] = Z RMT (N f,µ,ν) du 0 det(u 0 ) ν e µ eff Tr[U 0+U 0 ] = Z RMT (N f,µ eff, ν)

19 Zero mode domination or, Random Matrix Theory Corresponding RMT is a Gaussian chiral unitary model. Computations easy. Dirac spectral properties in QCD (in appropriate finite volume chiral regime). Z RMT = ˆD = 0 W W 0 dwe N Tr[W W ] N f i=1 det( ˆD +ˆm), W (N + ν ) N exact at large N

20 Zero mode domination or, Random Matrix Theory Corresponding RMT is a Gaussian chiral unitary model. Computations easy. Dirac spectral properties in QCD (in appropriate finite volume chiral regime). Z RMT = ˆD = 0 W W 0 dwe N Tr[W W ] N f i=1 det( ˆD +ˆm), W (N + ν ) N exact at large N Celebrated example: microscopic spectral density of Dirac operator. ζ k ν RMT = ΣV λ k ν QCD

21 Zero mode domination or, Random Matrix Theory Corresponding RMT is a Gaussian chiral unitary model. Computations easy. Dirac spectral properties in QCD (in appropriate finite volume chiral regime). Z RMT = ˆD = 0 W W 0 dwe N Tr[W W ] N f i=1 det( ˆD +ˆm), W (N + ν ) N exact at large N 7 6 4/1!=0!=1!= 10 8 N f =0, Q=0 N f =, Q=0 N f =0, Q= (m=3mev) 5 4 3/1 6 4/ /1 1 /1 3/ 4/ 4/3 0 /1 4/ 3/ 4/3 Giusti, Lüscher, Weisz, Wittig 003 JLQCD+TWQCD, 007

22 Mixed chiral regimes Bernardoni, Hernández 07; Daamgard, Fukaya 07; BHDF 08; Bernardoni, Hernández, Necco 09 m l ΣV 1 m s ΣV 1 Our specific setup: valence sector in the ε-regime, sea sector in the p-regime. ( partial quenching). Factorisation of perturbative and non-perturbative modes modified as: U = ( ) U0 0 0 I s ( ) iξ exp, F d 4 x tr [ξt a ] =0,T a Algebra(SU(N v )) Power counting rules: m l m s p 4 V 1. All p functions computed to NLO.

23 Mixed chiral regimes Example: two-point function of left-handed current: Tr[T a T b ] C(x 0 )= Bernardoni, Hernández 07; Daamgard, Fukaya 07; BHDF 08; Bernardoni, Hernández, Necco 09 d 3 x J0 a (x)j0 b (0) ν For ε-regime valence quarks, scales order as M vv,l M ss,m sv (4πF ) Heavier (sea) quarks behave as decoupling particles, sea meson mass dependence appears as renormalisation of Nf= LECs in ε-regime quenched expressions. C(x 0 )= F 4T mσ x0 ΣνTH ν 1 T F = F 1 N s M ss F 16π log Mss µ 8L 4 M ss

24 Mixed chiral regimes: RMT ZMChiPT can be derived along same lines. E Λ χ ε mixed QChPT Goldstones with heavy flavours dstones treated with like non-zero momentum modes ( M ss L 1 ) and integrated out. 1/L ChPT ZMChT N l RMT with flavours, LECs of theory with N s + N l flavours. N l + N s N l 1/FL Λ M χ ss

25 Mixed chiral regimes: RMT ZMChiPT can be derived along same lines. E Λ χ ε mixed QChPT Goldstones with heavy flavours dstones treated with like non-zero momentum modes ( M ss L 1 ) and integrated out. 1/L ChPT ZMChT N l RMT with flavours, LECs of theory with N s + N l flavours. N l + N s N l 1/FL Λ M χ ss λ k QCD,Nf = Σ eff (M ss )V = ζ k RMT,Nf =0

26 Mixed chiral regimes: RMT Topological susceptibility and chiral condensate at NLO can be worked out in standard fashion: ν NLO = m sσv N s 1 N s 1 M ss M N s 16π F log ss µ + g 1 (M ss,l,t) + 16M ss F (L r 8(µ)+N s L r 6(µ)+N s L r 7(µ)) Mao et al 09; Aoki et al 09; Bernardoni et al 10 lim Σ eff(m ss )=Σ 1+ M ss N s 0 F l β N s + log(µv 1/4 ) 8π N s + 16N s L r 6(µ) N s 4π log Mss µ β 1 N s F V N s F + g 1(M ss /,L,T) Bernardoni et al.

27 Nf= QCD in partially quenched mixed regime Bernardoni, Garron, Hernández, Necco, CP PRD 83 (011) Mixed action: non-perturbatively O(a) improved Wilson fermions for sea, Neuberger fermions for valence. β =5.3, c sw =1.9095, V/a 4 = label κ am ss N cfg D (14) 156 D (15) 169 D (37) 46 (D 6a :159,D 6b :87) a fm (K, K ), fm (Ω), r0), (F K )

28 Nf= QCD in partially quenched mixed regime Bernardoni, Garron, Hernández, Necco, CP PRD 83 (011) Mixed action: non-perturbatively O(a) improved Wilson fermions for sea, Neuberger fermions for valence. β =5.3, c sw =1.9095, V/a 4 = label κ am ss N cfg D (14) 156 D (15) 169 D (37) 46 (D 6a :159,D 6b :87) a fm (K, K ), fm (Ω), r0), (F K ) M ss 477, 4, 333 MeV Observables: topological charge, low-lying eigenvalues of Dirac operator. Aim (this talk): Study scaling with sea quark mass for topological susceptibility and chiral condensate, test matching to quenched RMT.

29 Topological susceptibility ν ν # conf # conf D 6a D ν ν D 6 D 5

30 Topological susceptibility ν NLO = m sσv N s 1 N s 1 M ss M N s 16π F log ss µ + g 1 (M ss,l,t) + 16M ss F (L r 8(µ)+N s L r 6(µ)+N s L r 7(µ)) Ν a m s

31 Topological susceptibility ν NLO = m sσv N s 1 N s 1 M ss M N s 16π F log ss µ + g 1 (M ss,l,t) + 16M ss F (L r 8(µ)+N s L r 6(µ)+N s L r 7(µ)) Ν At am ref = am ss : ( Z MS S a m s ( M ss = m sσ ) 1 m overlap Z MS S Mπ ref F = m MS ( GeV) ) 1 m overlap M ref π Z MS S ( GeV) = 1.84(10) Wilson Mπ ref = m MS ( GeV) cf. talk by S. Necco Wilson Mπ ref

32 Topological susceptibility ν NLO = m sσv N s 1 N s 1 M ss M N s 16π F log ss µ + g 1 (M ss,l,t) + 16M ss F (L r 8(µ)+N s L r 6(µ)+N s L r 7(µ)) Ν M ss = m sσ F a =0.070 fm F = 90 ± 10 MeV a m s NLO: Σ MS ( GeV) = 87 (35)(5) (36)(7) MeV 3 [L r 8 + (L r 6 + L r 7)] (M ρ )=0.0018(30) cf. LO: Σ MS ( GeV) = [344(10) MeV] 3

33 Topological susceptibility ν NLO = m sσv N s 1 N s 1 M ss M N s 16π F log ss µ + g 1 (M ss,l,t) + 16M ss F (L r 8(µ)+N s L r 6(µ)+N s L r 7(µ)) Ν a m s 3 NLO: Σ MS ( GeV) = 87 (35)(5) (36)(7) MeV Σ MS ( GeV) = [69(18) MeV] 3 FLAG

34 Topological susceptibility ν NLO = m sσv N s 1 N s 1 M ss M N s 16π F log ss µ + g 1 (M ss,l,t) + 16M ss F (L r 8(µ)+N s L r 6(µ)+N s L r 7(µ)) Ν 1 ()((,, , :/33; ,< 10 ()((( #=>?:@A(B,(),,0(' 67893:/33;1 34-< ! / #$%& * ' ()(((. ()((( NLO: Σ MS ( GeV) = a m s 87 (35)(5) (36)(7) MeV 3 ()(((( ()(( ()(* ()(+ (),- (),.! " #$%&' Σ MS ( GeV) = [49(4)() MeV] 3 Chiu et al. 08

35 Eigenvalue ratios ν= / /1 4/ / 0.4 /3 3/ / /1 1/ D 6 D 5 D 4 D 6 D 5 D 4 D 6 D 5 D 4 D 6 D 5 D 4 ν = /4 /4 1/4 λ k QCD,Nf = Σ eff (M ss )V = ζ k RMT,Nf =0 ν = / /1 4/ 4/ / / 1 3/ / / / 0. /1 1/ / 0.8 4/3 3/ D 6 D 5 D 4 D 6 D 5 D 4 D 6 D 5 D 4 D 6 D 5 D / / /4.5 3/1 1 / / / /3 0. 1/ D 6 D 5 D 4 D 6 D 5 D 4 D 6 D 5 D 4 D 6 D 5 D 4

36 Eigenvalue ratios λ k QCD,Nf = Σ eff (M ss )V = ζ k RMT,Nf =0 1.5 ν=0 (<λ k > ν /<λ l > ν ) (<ζ l > ν /<ζ k > ν ν =1 1.5 ν = 1 D 4 D 5 D /1 3/1 4/1 3/ 4/ 4/3

37 lim Σ eff(m ss )=Σ 1+ M ss N s 0 F l Chiral condensate from spectrum β N s + log(µv 1/4 ) 8π N s + 16N s L r 6(µ) N s 4π log Mss µ β 1 N s F V N s F + g 1(M ss /,L,T) a 3 Σ eff D 4, ν=0 D 4, ν =1 D 4, ν = D 5, ν=0 D 5, ν =1 D 5, ν = D 6, ν=0 D 6, ν =1 D 6, ν = ν=0 ν =1 ν = ν=0 ν =1 ν = D 5 /D 4 D 6 /D k ν=0 ν =1 ν = k D 6 /D 5

38 lim Σ eff(m ss )=Σ 1+ M ss N s 0 F l Chiral condensate from spectrum β N s + log(µv 1/4 ) 8π N s + 16N s L r 6(µ) N s 4π log Mss µ β 1 N s F V N s F + g 1(M ss /,L,T) a m s a

39 lim Σ eff(m ss )=Σ 1+ M ss N s 0 F l Chiral condensate from spectrum β N s + log(µv 1/4 ) 8π N s + 16N s L r 6(µ) N s 4π log Mss µ β 1 N s F V N s F + g 1(M ss /,L,T) a M ss = m sσ F a =0.070 fm F = 90 ± 10 MeV m s a NLO: Σ MS ( GeV) = 80 (14)(4) (16)(5) MeV 3 L r 6(M ρ )=0.0010(7)

40 lim Σ eff(m ss )=Σ 1+ M ss N s 0 F l Chiral condensate from spectrum β N s + log(µv 1/4 ) 8π N s + 16N s L r 6(µ) N s 4π log Mss µ β 1 N s F V N s F + g 1(M ss /,L,T) a m s a NLO: Σ MS ( GeV) = 80 (14)(4) (16)(5) MeV 3 Σ MS ( GeV) = [69(18) MeV] 3 FLAG

41 lim Σ eff(m ss )=Σ 1+ M ss N s 0 F l Chiral condensate from spectrum β N s + log(µv 1/4 ) 8π N s + 16N s L r 6(µ) N s 4π log Mss µ β 1 N s F V N s F + g 1(M ss /,L,T) a m s a NLO: Σ MS ( GeV) = 80 (14)(4) (16)(5) MeV 3! +,,!"#""(!"#""'!"#""&!"#""%!"#""$!"!"#"$!"#"&!"#"(!"#")!"#*!"#*$! "# $ %!-!$!./001+ %30!,10 &30!,10 '30!,10 (30!,10 430!, /.! Σ MS ( GeV) = [4(05)(0) MeV] 3 Fukaya et al. 11

42 Summary and Outlook First results seem to validate mixed-regime approach, results for LECs in comparable precision ballpark as other setups. Check finite size scaling, cutoff effects. Get F, other L s from analysis of two-point functions (talk by S. Necco). Provide solid estimation of valence-sea relative cutoff effects. Move to phenomenology: weak LECs for K ππ decays, understanding the role of the charm quark in ΔI=1/ rule. Giusti, Hernández, Laine, CP, Torró, Weisz, Wennekers, Wittig work in progress

43 BACKUP

44 Scale setting uncertainty Different results for scale setting in our lattices: K, K a fm r 0, Ω a fm F K a fm Del Debbio et al. 06 Donnellan et al. 11; Brandt et al. Lat10 Marinkovic Lat11 (cf. R. Sommer s talk) Impact on LECs: topology RMT 1/3 a Σ MS ( GeV) [L r 8 + (L r 6 + L r 7)](M ρ ) (33)(4) (34)(5) MeV 0.003(43) (35)(5) (36)(7) MeV 0.003(43) n/a n/a a Σ MS ( GeV) L r 6(M ρ ) (1)(1) (13)(4) MeV (10) (13)(4) (14)(5) (1)(7) (1)(9) 1/3 MeV (6) MeV (4)

T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.1/34. The Topology in QCD. Ting-Wai Chiu Physics Department, National Taiwan University

T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.1/34 The Topology in QCD Ting-Wai Chiu Physics Department, National Taiwan University The vacuum of QCD has a non-trivial topological structure. T.W.