η and η mesons from N f = flavour lattice QCD

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1 η and η mesons from N f = 2++ flavour lattice QCD for the ETM collaboration C. Michael, K. Ottnad 2, C. Urbach 2, F. Zimmermann 2 arxiv: Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool 2 Helmholtz - Institut für Strahlen- und Kernphysik (Theorie), Bethe Center for Theoretical Physics, Universität Bonn INT Workshop 2-2b

2 Flavour Singlet Pseudo-Scalar Mesons nine lightest pseudo-scalar mesons show a peculiar spectrum: 3 very light pions (4 MeV) kaons and the η around 6 MeV η around GeV The large mass of the η meson is thought to be caused by the QCD vacuum structure and the U A () anomaly η meson is not a (would be) Goldstone Boson massive even in the SU(3) chiral limit

3 Lattice Status disconnected contributions significant hard problem only a limited amount of lattice results available no control of systematics usually only one lattice spacing and/or only one pion mass no clear picture in particular at light pion masses Mη,η [GeV].5. experimental values HSC RBC/UKQCD UKQCD.2.3 M 2 PS [GeV2 ].4 filled symbols: η open: η [HSC, J. J. Dudek et al., Phys. Rev. D83 (2)] [RBC/UKQCD, N. Christ et al., Phys. Rev. Lett. 5 (2)] [UKQCD, E. B. Gregory et al., Phys.Rev. D86 (22)]

4 N f = 2++ Wilson Twisted Mass Fermions with twisted mass formulation of LQCD only doublets of quarks can be considered light doublet, mass-degenerate: D l = D W + m crit + iµ l γ 5 τ 3, χ l = heavy doublet, mass-split, flavour non-diagonal: ( χu D h = D W + m crit + iµ σ γ 5 τ +µ δ τ 3, χ h = χ d ) ( χc χ s ) [Frezzotti, Rossi (24)] rotation from χ- to standard ψ-basis (at maximal twist): ψ l = e iπγ5τ 3 /4 χ l, ψ h = e iπγ5τ /4 χ h

5 N f = 2++ Wilson Twisted Mass Fermions Pros: O(a) improvement at maximal twist [Frezzotti, Rossi; JHEP 48 (24)] by tuning only one parameter excellent scaling behaviour observed mixing patterns under renormalisation can be simplified note: could easily introduce u-d mass splitting as well Cons: flavour and parity symmetries broken at finite values of the lattice spacing technical complication unphysical splittings, mostly in between m π ± and m π [Dimopoulus, Frezzotti, Michael, Rossi, CU; Phys.Rev. D8 (2)]

6 The + Doublet heavy doublet: D h = D W + m crit + iµ σ γ 5 τ +µ δ τ 3, χ h = think of µ σ as the mean s/c mass and µ δ the splitting splitting µ δ orthogonal to twist µ σ (τ 3 versus τ ) obtain renormalised quark masses of the doublet ˆm s = ZP µ σ Z S ˆm c = ZP µ σ + Z S µ δ µ δ ( χc χ s ) fermion determinant positive and O(a) improvement remains valid [Frezzotti, Rossi (24)]

7 Flavour Singlet Pseudo-Scalar Mesons in the SU(3) symmetric case (sloppy notation) η 8 : η : SU(3) symmetry broken ūiγ 5 u + diγ 5 d 2 siγ 5 s ūiγ 5 u + diγ 5 d + siγ 5 s mixing SU(2) plus strange: ( ) η η = ( ) cosφ sinφ sinφ cosφ ( ) ηl η s N f = 2++ possible charm contribution [Feldmann, Int. J. Mod. Phys. A5, 59 (2)]

8 Flavour Singlet Pseudo-Scalar Mesons need to estimate correlator matrix C = η ll η ls η lc η sl η ss η sc η cl η cs η cc η XY correlator of appropriate interpolating fields, e.g. η ss (t) siγ 5 s(t) siγ 5 s() projected to zero momentum η: lowest state, η : first state, η c...

9 A Little Twisted Mass Algebra rotation to twisted basis 2 ( ψ u iγ 5 ψ u + ψ d iγ 5 ψ d ) 2 ( χ d χ d χ u χ u ) O l, and in the heavy sector ) T ( ψc ±τ 3 ( ) ψc iγ ψ 5 s 2 ψ s therefore ) T ( χc τ ± iγ 5 τ 3 ( ) χc χ s 2 χ s O c,s. O c Z( χ c iγ 5 χ c χ s iγ 5 χ s )/2 ( χ s χ c + χ c χ s )/2, O s Z( χ s iγ 5 χ s χ c iγ 5 χ c )/2 ( χ s χ c + χ c χ s )/2. with ratio of non-singlet renormalisation constants Z Z P Z S

10 Estimating Disconnected Diagrams fermionic disconnected contributions noisy need as many as possible observations use R stochastic volume sources ξ r lim R [ξ i ξ j ] R = δ ij, lim [ξ iξ j ] R = R then we get with [ξi r φ j ] = (D ) ji + noise φ j = (D ) jk ξ r k noise V s /R while signal O()

11 A Very Efficient Variance Reduction Method [Jansen, Michael, CU, Eur.Phys.J. C58 (28)] relation in between up and down Dirac operator D u D d = 2µ l γ 5 multiply with /D u from left, /D d from right = 2µ l γ 5 D d D u D u D d lhs is what we want, rhs has an extra volume loop can be estimated from noise Vs 2 /R, but signal V [φ φ] R + noise only applicable in the light sector involving τ 3

12 A Very Efficient Variance Reduction Method: Example R(t = 6) dr(t = 6) Hopping parameter expansion twisted mass noise reduction 5σ limit σ limit N example for strangeness content of the nucleon [ETMC, Dinter et al., JHEP 28 (22)] easily a factor four improvement

13 Ensemble-Details gauge configurations from ETM Collaboration [ETMC, R. Baron et. al., JHEP 6 (2)] Iwasaki Gauge action [Iwasaki, Nucl. Phys. B258, 4] three lattice spacings: a A =.86 fm, a B =.78 fm and a D =.6 fm charged pion masses range from 23 MeV to 5 MeV L 3 fm and M π L 3.5 for most ensembles 6 up to 25 gauge configuration per ensemble µ σ, µ δ fixed for each β use r =.45(2) fm (from f π ) throughout the talk

14 Analysis Procedure 24 to 32 volume sources per gauge for disconnecteds have tested one ensemble with 64 sources local and smeared operators two γ-combinations iγ 5, iγ γ 5 two independent fitting methods (up to 2 2 matrix) solving the GEVP [Michael, Teasdale, Nucl.Phys. B25, 433 (983); Lüscher, Wolff, Nucl.Phys. B339, 222 (99)] using a factorising fit errors computed by bootstrapping and blocking bootstrap samples significant autocorrelation for η : 2 trajectories

15 Effective Masses B matrix 6 6 matrix 2 η η 2 η η.5.5 am am 5 t/a 5 5 t/a 5 ground state η well determined next state hardly plateaus

16 Flavour Content B25 η state η state flavour content light content strange content charm content flavour content light content strange content charm content 5 t/a 5 5 t/a flavour content qualitatively as expected no charm contribution to η and η third state (not shown) is charm only (almost)

17 Summary Masses η mass quite precise 4 η rather noisy large systematic uncertainties mild pion mass dependence in M η 2.5 physical strange quark mass not fixed for different β values! what can we say about lattice artifacts? rmη,η A-Ensembles A8.24s, A.24s B-Ensembles D-Ensembles D45.32sc how to perform the chiral extrapolation? (r M PS) 2

18 Strange Quark Mass Dependence 2 µ σ and µ δ fixed for each β m s unfortunately not perfectly tuned to its physical value.5 we have two re-tuned ensembles for a A (β =.9) can estimate m s dependence need to come up with a strategy to correct for this effect! rmk.2.4 physical value A-Ensembles A8.24s, A.24s B-Ensembles D-Ensembles D45.32sc (r M PS) 2

19 Scaling Test for M η () fix r M PS, r M K and V/r we don t see a volume dependence in M η ignore it 2.5 estimate rmη D η d(am η) 2 d(am K )2 =.6(2) from two A-ensembles now assume: D η independent of β,µ l,µ σ,µ δ.2.4 physical value A-Ensembles A8.24s, A.24s.6.8 (r M PS) 2.2.4

20 Scaling Test for M η (2) use ensembles A6, B55, D45 with r M PS.9 correct M η using D η linearly in M 2 K r M K =.34 fixed rmη 2.5 compatible with both, constant and linear continuum extrapolation assign conservative 8% error from difference to all our results..2 a 2 /r 2 average linear fit A6.24 B55.32 D45.32sc.3.4.5

21 Chiral Extrapolation of M η more ambitious: shift all M η to physical strange mass fit g K = a+b(r M PS ) 2 to data for (r M K ) 2 from A ensembles rmk 2.5 adjust a to match physical M K for M PS = M π g K compute δ K [(r M PS ) 2 ] = (r M K ) 2 g K [(r M PS ) 2 ].2.4 g K g K physical value A-Ensembles A8.24s, A.24s.6.8 (r M PS) for all ensembles

22 Chiral Extrapolation of M η now correct all (r M η ) 2 by corresponding 2 D η δ K [(r M PS ) 2 ] (r Mη ) 2 [(r M PS ) 2 ].5 all β-values fall on the same curve! extrapolate (r Mη ) 2 linearly in (r M PS ) 2 to M PS = M π result rmη.2.4 r M η,exp r M η,phys A-Ensembles A8.24s, A.24s B-Ensembles D-Ensembles D45.32sc M η = 549(33) stat (44) sys MeV (r M PS) 2

23 Chiral Extrapolation of M η alternatives to avoid D η use the ratio (M η /M K ) 2.5 all β-values on a single curve in particular: m s dependence seems to cancel Mη/MK extrapolate linearly in (r M PS ) 2 2 to physical point result.2.4 (M η/m K) exp (M η/m K) phys A-Ensembles A8.24s, A.24s B-Ensembles D-Ensembles D45.32sc M η = 558(3) stat (45) sys MeV (r M PS) 2

24 Chiral Extrapolation of M η or use GMO relation 3M 2 η = 4M2 K M2 π.5 valid for SU(3) experimentally.925 result 3M 2 η 4M 2 K M2 π M η = 559(4) stat (45) sys MeV weighted average over three methods.2.4 experimental value physical value A-Ensembles A8.24s, A.24s B-Ensembles D-Ensembles D45.32sc.6.8 (r M PS) M η = 557(5) stat (45) sys MeV

25 Comparing to Other Lattice Results overall mutual agreement apart from maybe UKQCD at smallest M PS.5 we have pushed significantly more chiral Mη,η [GeV] ETMCs η is much noisier despite similar number of independent gauges (RBC, HSC, ETMC) similar number of inversions (RBC, ETMC). experimental values ETMC HSC RBC/UKQCD UKQCD.2.3 M 2 PS [GeV2 ].4

26 η and η Mixing write correlator matrix C qq (t) = A q,n A [ ] q,n exp( m (n) t)+exp( m (n) (T t)) 2m (n) n with amplitudes A q,n corresponding to qq n (n η,η,... and q = l, s, c) define mixing angles via (ignoring charm) ( ) ( ) Al,η A s,η fl cosφ = l f s sinφ s A l,η A s,η f l sinφ l f s cosφ s for φ s φ l tan 2 φ = A l,η A s,η A l,η A s,η

27 η and η Mixing φ l and φ s are too noisy separately 9 single mixing angle φ can be determined linear fit in (r M PS ) 2 φ = 44(5) φ L [deg] or in singlet/octet basis θ = (5) good agreement with other determinations A-Ensembles A8.24s, A.24s B-Ensembles D-Ensembles D45.32sc.75 (r M PS) note: statistical error only

28 Mixed Action Approach idea: use different valence action for s and c to avoid s/c flavour mixing to vary strange and charm quark masses up/down stay unitary add two valence strange quarks s and s s s(+) : D W + m crit +µ s iγ 5 s s( ) : D W + m crit µ s iγ 5 can proof that continuum limit is correct [Frezzotti, Rossi, JHEP 8, 7 (24)] the different signs allow to use the variance reduction trick ψ s ψ s = 2 ( ψ s ψ s + ψ s ψ s ) = 2 ( χ sχ s χ s χ s )

29 Matching Unitary and Mixed Actions there are different matching quantities possible match µ s with µ σ Zµ δ unitary with mixed s(+)d( ) kaon (denote M K +) smallest lattice artifacts for f PS and M PS [Sharpe, Wu,Phys. Rev. D 7 (25), Frezzotti et al., JHEP 64 (26)] unitary with mixed s( )d( ) kaon usually larger lattice artifacts (denote M K OS) and of course many other quantities tried first to match with M K +

30 Matching Unitary and Mixed Actions there are different matching quantities possible match µ s with µ σ Zµ δ unitary with mixed s(+)d( ) kaon (denote M K +) smallest lattice artifacts for f PS and M PS [Sharpe, Wu,Phys. Rev. D 7 (25), Frezzotti et al., JHEP 64 (26)] unitary with mixed s( )d( ) kaon usually larger lattice artifacts (denote M K OS) and of course many other quantities tried first to match with M K +... and got pathological results so, why is that?

31 Matching Unitary and Mixed Actions η,η have significant disconnected contributions which don t know about µ s!.2. M ηss M K OS M K + connected and disconnected have to match to produce the correct correlation matrix (am) am unit K am unit ηss match connected only M ηss.4 due to M PS ± - M PS splitting: significantly smaller matching µ s -value.2.6 aµ s = aµ l =.85, B aµ s.22

32 µ s -Dependence B85.24 ensemble η unitary η η unitary η meff r µ s r at M ηss matching point we find reasonable agreement in between unitary and mixed approach currently exploring this further

33 Summary η and η for three lattice spacings and various quark mass values.5 η can be extracted precisely M η = 557(5) stat (45) sys MeV η noisy, significant systematics single mixing angle φ = 44(5) ( or θ = (5) ) small lattice artifacts in M η Mη,η [GeV]. experimental values ETMC HSC RBC/UKQCD UKQCD.2.3 M 2 PS [GeV2 ].4

34 Outlook.5 noise reduction techniques for η larger operator basis Mη,η [GeV] ηπ scattering length η γγ. experimental values ETMC HSC RBC/UKQCD UKQCD.2.3 M 2 PS [GeV2 ].4

35 Acknowledgements computing time from FZ Jülich on JUGENE and JUDGE on local GPU cluster founded by DFG financial support from DFG in CRC 6 supported by the Bonn-Cologne Graduate School (BCGS) thanks to all members of ETMC

36 φ l and φ s φ L l [deg] 5 4 φ L s [deg] A-Ensembles A8.24s, A.24s B-Ensembles D-Ensembles D45.32sc A-Ensembles A8.24s, A.24s B-Ensembles D-Ensembles D45.32sc (r M PS) 2 (r M PS) 2