The nucleon mass and pion-nucleon sigma term from a chiral analysis of lattice QCD world data

The nucleon mass and pion-nucleon sigma term from a chiral analysis of lattice QCD world data L. Alvarez-Ruso 1, T. Ledwig 1, J. Martin-Camalich, M. J. Vicente-Vacas 1 1 Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC, E-46071 Valencia, Spain Department of Physics and Astronomy, University of Sussex, B1 9QH, Brighton, UK MEU 013, Rome, Italy

Outline 1 Introduction BχPT, Perturbative ucleon Mass, Chiral Extrapolator 3 Results: Fits to LQCD f = Ensembles 4 Results: Fits to LQCD f = +1 Ensembles

Introduction ucleon mass non-perturbative regime of QCD BχPT: Effective theory using the QCD chiral symmetry for m q =0 SU(): u, d quarks pions, nucleons, (13) SU(3): u, d, s quarks p.-scalar mesons, 8/10 baryons LQCD: Lattice simulations using the discretized QCD-action flavors with m u = m d = m +1 flavors with m u = m d = m and m s m s(phys)

Introduction > 11 LQCD collab. BχPT LQCD: fixing LECs χlogs + natural LECs Hellmann-Feynman theorem GOR: m M π m m M (m) = σ π = m (p) uu + dd (p) σ π : π scattering, LQCD, BχPT π scattering: σ π = 43 66 MeV 59(7) MeV Alarcon et al. 01 LQCD: σ π = 37 67 MeV [aive RQM: σ π = 3m = 1 MeV]

Introduction > 11 LQCD collab. BχPT LQCD: fixing LECs χlogs + natural LECs Hellmann-Feynman theorem GOR: m M π m m M (m) = σ π = m (p) uu + dd (p) σ π : π scattering, LQCD, BχPT quark mass contribution (strangeness) of the nucleon mass σ-terms dark matter searches [aive RQM: σ π = 3m = 1 MeV]

Perturbative ucleon Mass ucleon mass pole position of its full propagator at /p = M 1 /p M 0 Σ(/p) M = M 0 +Σ(M ) Expand Σ around bare mass M 0 Rewrite propagator as Σ(/p) = Σ(M 0 )+( /p M 0 )Σ (M 0 )+R( /p) 1 /p M 0 Σ(/p) = 1 /p M 0 Σ(M 0) 1 Σ (M 0 ) ucleon mass M and its residue Z R( /p) 1 Σ (M 0 ) 1 1 Σ (M 0 ) M = M 0 +Z Σ(M 0 )+Z R(M ) Z = 1 1 Σ (M 0 )

BχPT O(p 4 ) Self-Energy: Σ p 4( /p) Covariant baryon chiral perturbation theory up to order O(p 4 ) with explicit (13) degrees of freedom: Σ p 4 ( /p) = Σ () +Σ (3) ( /p)+σ (4) ( /p) 1 1 4 1 1 ΣC Σ3 ΣC4 ΣT4 Σ4 1 1 1 1 1 1 Σ 3 Σ 4 Σ 4a Σ 4b Perturbation in the light scales: p/λ, m π /Λ, (M M )/Λ with Λ = 4πf π Fits: including (13) ( ) and excluding it ( / ) (EOMS) EOMS - Gegelia, Fuchs et al. (1999, 003) consistent (13) couplings - Pascalutsa (1998,001,03) L () - Long, Lensky (011) π

M (4) and Hellmann-Feynman Theorem BχPT ucleon mass M (m π ) BχPT Scalar Form factor σ π = m (p) uu + dd (p) ΣC 1 1 + Σ3 +... σ 1 1 σ3 + +... Chiral structures from BχPT M (4) ( ) m π = M0 +Σ C +Σ C4 +Σ (3)+(4) loops σ (4) π Hellmann-Feynman Theorem ( ) ] m π = Z [σ C +σ C4 +σ (3)+(4) loops q =0 ( ) ( ) σ π m π = m π M mπ m π

Fixed Parameters f π, g A experimental values LECs c and c 3 from BχPT fits to π scattering data Alarcon et al. 01 (13) decay width h A =.87 (13) mass up to O(p 3 ) c 1 0.9±0.4 GeV 1 ucleon fits to LQCD Fits are insensitive to variations in c, c 3 Changes in c 1 compensated by changes in α LQCD data from: PRD64(054506), PRD70(094505), PRL107(141601)

Discretization and Finite Volume Finite Spacing Corrections O(a) - improved LQCD actions M = M a=0 + c a a +O(a 3 ) c a action specific constant Wilson (1974), Sheikholeslami (1985), Luscher et al.(1996) Tiburzi (005) LQCD data given in the dimensionless form (amπ, am ) Finite Volume Corrections d 4 l (4π) 4 = dl0 π d l (π) 3 dl0 1 π L 3 n with l = π L n, n Z 3 Kahn et al. (004)

Ingredients covariant SU() BχPT p 4 EOMS explicit (13) (consistent couplings) 10 parameters, only 3 are fitted finite volume effects finite size effects scaling/correlation of LQCD data points f = +1 LQCD data points for m s m s(phys) strange quark effects integrated out into the LECs

Results for f = Ensembles Fitting to LQCD data from ( deg. light quarks): BGR (010), ETMC (011), Mainz (01), QCDSF (013) Data is given in dimensionless form: (am π, am ) Ratio r 0 /a is known for each collaboration r 0 Sommer-scale (1994) (am π, am ) (r 0 M π, r 0 M ) Condition on points: LM π > 3.8 and r 0 M π < 1.11 χ = i ) M (n) ( M π + (n) Σ ( M π, L) + c a ã d i ( M π, L σ i ) with M(n) = r 0M (n), M π = (r 0 M π ), Σ(n) = r 0Σ (n) Fixing r 0 self-consistently and recursively inside the fit r k 0 = M (n) ( r k 1 0 M π(phys) ) M (phys) until r k 0 r k 1 0 < 0.001 fm

no explicit (13) Results for f = Ensembles with explicit (13) one σ π data point at 90 MeV QCDSF p 4 c 1 GeV 1 χ dof r 0 fm σ π MeV / /σ 1.18(14).5 0.470 6(13) / σ 0.91(4).9 0.507 41(3) /σ 0.77(9).4 0.494 38(10) σ 0.80(1).5 0.489 41()

Results for f = +1 Ensembles Fitting to LQCD data from ( deg. light quarks + 1 heavy quark): BMW (01), HSC (009), LHPC (009), MILC (01), PLQCD (011), PACS (010), RBCUK-QCD (01) SU() BχPT s-quark effects integrated out into the LECs Data is given in dimensionless form (am π, am ) and a a-uncertainty points from same collaboration are correlated χ A = i [ M (n) ( M π ) +Σ (n) ( M π, L ) ( f i d i M π, L ) ] [ fi 1 + f i σ i σ fi ] χ B = T V 1 [ i = M (n) ( ) Mπ,i ( ( )] + c i a i +Σ (n) Mπ,i, L i ) d i Mπ,i, L i Correlation included: (A) through collaboration specific fit constants f i (B) through correlation matrix V

Results for f = +1 Ensembles Condition on points: LM π > 3.8 and M π < 415 MeV p 4 c 1 GeV 1 χ dof σ π MeV / /a 1.15(3) 1.3 58(3) / a 1.10(5) 1. 55(3) /a 0.89(3) 1.4 49() a 0.84(4) 1. 44(3)

Results f = vs f = +1 f = σ π = 41(5)(4) MeV incl. the σ(90) point f = +1 σ π = 5(3)(8) MeV average of and / results Chiral convergence (comparing only fits with ) σ π MeV f p p 3 p 4 full 6 7 6 41 +1 69 6 6 48 M () = M 0 4c 1 M π

Results f = vs f = +1 f = M f = +1 σ π M π [MeV] 139 300 400 100 M [MeV] 1100 1000 900 0.0 0.05 0.1 0.15 0. M π [GeV ] 41(5)(4) vs 5(3)(8): f = : more M data for < 0.075 GeV f = +1: already one σ π data point for < 0.1 GeV

Summary Fitted SU() BχPT O(p 4 ) mass formula to LQCD nucleon mass data for and +1 flavor ensembles Considered scale-setting, finite volume and finite spacing effects Extracted LECs and calculated the σ π Correlation between c 1 and α f = : Either uncertainty of the σ(90) data-point is too small or inclusion of (13) is necessary f = +1: Consistent descriptions of M data by different strategies, small slope variations σ π changes by 10 MeV More f = M data points in the region m π < 0.1 GeV Direct f = +1 σ π LQCD points for mπ < 0.1 GeV

Chiral Extrapolator M (4) ( ) M π = M 0 c 1 4Mπ + 1 αm4 π + c 1 8π fπ Mπ 4 ln M π M 0 ( ) M π, M 0, M 0, f π, g A, h A, c 1, c, c 3, c 1 +Σ (3)+(4) loops Parameters: 10 low energy constants Only 3 are fitted L (n) p 1 p p 3 p 4 fixed f π, g A, h A, M 0 c, c 3, c 1 fitted M 0 c 1 α = 4(8e 38 + e 115 + e 116 ) Pion-mass O(p 4 ): α = α + c 16 1 f π l 3 r ( M 0 ) Mπ ( mπ ) (Λ = mπ + lr 3 ) f π0 mπ 4 + 1 3π f π0 mπ 4 ln m π Λ Shifting: M (mπ ) M (M π ) for σ π l 3 (139 MeV) = 3.(8), Colangelo (011)

Results f = vs f = +1 f = σ π = 41(3)(1) MeV σ π = 45(13)(10) MeV f = +1 σ π = 5(3)(8) MeV Chiral convergence σ π MeV full p p 3 p 4 f = σ 41 6 7 6 f = +1 48 69 6 6