Improving the precision of light quark masses Christian Sturm - PowerPoint PPT Presentation

Improving the precision of light quark masses Christian Sturm Brookhaven National Laboratory Physics Department High Energy Theory Group Upton, NY I. Introduction & Motivation II. Concepts & Framework III. Results & Discussion IV. Summary & Conclusion Based on arXiv:1004.4613 [hep-ph] , Phys. Rev. D 80, 014501 (2009) in collaboration with: L. G. Almeida, Y. Aoki, N.H. Christ, T. Izubuchi, C.T. Sachrajda and A. Soni C. Sturm, Improving the precision of light quark masses Pheno 2010, Madison, May 10th, 2010 0/10

Introduction Light quark masses (up-, down- and strange-quark masses) can be determined non-perturbatively with lattice simulations in QCD Result from RBC/UKQCD Coll., domain-wall fermions: m MS ud ( 2 GeV ) = 3 . 72 ( 0 . 16 ) stat ( 0 . 18 ) syst ( 0 . 33 ) ren MeV m MS s ( 2 GeV ) = 107 . 3 ( 4 . 4 ) stat ( 4 . 9 ) syst ( 9 . 7 ) ren MeV C. Allton et al. Error (11%) from renormalization dominates (>60% of tot.) Pert. calculations are performed in dim. reg. � not directly amenable to lattice calculations Direct calculation of bare quantity with lattice spacing acting as ultra-violet cutoff in some particular discretization of QCD instead of space-time dimension d � = 4 Minimal subtraction(MS) à la dim. reg. not directly possible C. Sturm, Improving the precision of light quark masses Pheno 2010, Madison, May 10th, 2010 1/10

Introduction Regularization invariant momentum subtraction schemes Use for renormalization regularization invariant(RI) scheme, which removes ultraviolet divergences at a certain momentum point(subtraction point) � RI/MOM-scheme Martinelli et al. ’93-’95 Determine QCD parameters: m R = Z m m B , Ψ R = Z 1 / 2 Ψ B ,.. q � fix renormalization constants, define scheme in PT: + k + . . . S − 1 = Z − 1 S − 1 ∝� p Σ V R ( p 2 ) − m R Σ S R ( p 2 ) ⇔ q R B p p k + p p " γ µ ∂ S − 1 #˛ ˛ R ( p ) 1 1 ˛ Tr [ S − 1 ˛ lim R ( p )] = 1 lim 48 Tr = − 1 RI/MOM ˛ ˛ 12 m R ∂ p µ ˛ m R → 0 m R → 0 ˛ p 2 = − µ 2 ˛ p 2 = − µ 2 ˛ ˛ 1 1 Tr [ S − 1 12 p 2 Tr [ S − 1 ˛ ˛ lim R ( p )] = 1 lim R ( p ) � p ] = − 1 RI’/MOM ˛ ˛ 12 m R m R → 0 m R → 0 ˛ p 2 = − µ 2 ˛ p 2 →− µ 2 C. Sturm, Improving the precision of light quark masses Pheno 2010, Madison, May 10th, 2010 2/10

Introduction Ward-Takahashi identities Ward-Takahashi identities(WI) q µ Λ µ V , B ( p 1 , p 2 ) = S − 1 B ( p 2 ) − S − 1 B ( p 1 ) − iq µ Λ µ A , B ( p 1 , p 2 ) = 2 m B Λ P , B ( p 1 , p 2 ) − i γ 5 S − 1 B ( p 1 ) − S − 1 B ( p 2 ) i γ 5 WI valid for renorm. quantities: O R = Z O O , Λ O , R = Z O Z q Λ O , B Renormalization condition on S ⇔ condition on Λ O 1 �� q = p 1 −p 2 � � N Tr Λ O , R ( p 1 , p 2 ) P O = 1 � � mom . conf . p 1 p 2 � study quark bilinear operators with vector( γ µ ), axial- vector( γ 5 γ µ ), pseudo-scalar( γ 5 ) and scalar(1 1) operators Renormalization constants related: Z A = 1 = Z V , Z P = Z S , Z P = 1 / Z m C. Sturm, Improving the precision of light quark masses Pheno 2010, Madison, May 10th, 2010 3/10

Motivation RI/MOM 1 i˛ 1 i˛ h h Λ µ ˛ Λ µ ˛ lim 48 Tr V , R ( p 1 , p 2 ) γ µ = 1 , lim 48 Tr A , R ( p 1 , p 2 ) γ 5 γ µ = 1 ˛ ˛ m R → 0 m R → 0 ˛ ˛ asym asym 1 ˜˛ 1 ˜˛ ˆ ˛ ˆ ˛ lim 12Tr Λ S , R ( p 1 , p 2 ) 1 1 = 1 , lim 12 i Tr Λ P , R ( p 1 , p 2 ) γ 5 = 1 ˛ ˛ m R → 0 m R → 0 ˛ ˛ asym asym Asymmetric/exceptional momentum config.(MOM): µ 2 > 0 , p 2 1 = p 2 2 = − µ 2 , p 1 = p 2 , q = 0 q = p 1 −p 2 Symmetric/nonexceptional momentum config(SMOM): 2 = q 2 = − µ 2 , µ 2 > 0 , p 2 1 = p 2 q = p 1 − p 2 p 1 p 2 Renormalization constants need to be determined through simulation Need to introduce renormalization scale µ typically µ ∼ 2 GeV for this problem C. Sturm, Improving the precision of light quark masses Pheno 2010, Madison, May 10th, 2010 4/10

Motivation Symmetric subtraction point implies a lattice simulation with suppressed contamination from infrared effects For asymmetric subtraction point effects of chiral symmetry breaking vanish slowly like 1 / p 2 for large ext. momenta 0 −0.02 Τ[Λ ] V MOM − SMOM Τ[Λ ] A −0.04 −0.06 −0.08 0 0.5 1 1.5 2 2.5 2 (pa) Y. Aoki For SMOM infrared effects better behaved, vanishing with larger powers of p N.H. Christ, et al. C. Sturm, Improving the precision of light quark masses Pheno 2010, Madison, May 10th, 2010 5/10

Motivation R I-scheme = ⇒ MS-scheme Conversion/Matching factor: ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ m MS R = C RI/MOM m RI/MOM ( C m in general gauge dependent) m R RI/MOM scheme intermediate scheme before conversion to the MS scheme C m can be computed in cont. PT, e.g. RI/MOM, RI’/MOM: known up to 3-loop G. Martinelli et al.; Franco, Lubicz; Chetyrkin, Retey; Gracey C RI/MOM = 1 . 0 − 0 . 1333 − 0 . 0759 − 0 . 0557 α s ( 2 GeV ) /π ∼ 0 . 1 m n f = 3 C RI’/MOM = 1 . 0 − 0 . 1333 − 0 . 0816 − 0 . 0603 m Observation: ✿✿✿✿✿✿✿✿✿✿✿✿✿ Size of NLO, N 2 LO, N 3 LO contr. amount ∼ 13 % , ∼ 8 % , ∼ 6 % � poor convergence � big error in renormalization Matching to pert. theo.: reduce truncation error: large µ � window problem C. Sturm, Improving the precision of light quark masses Pheno 2010, Madison, May 10th, 2010 6/10

Concepts & Framework RI/SMOM Task: Find a RI/MOM type scheme which is "close" to ✿✿✿✿ MS scheme, e.g. which has a matching factor with small corrections � Small expansion coefficients Idea: Use subtraction point with symmetric momenta ✿✿✿✿ RI/SMOM conditions: 1 ˜˛ 1 ˜˛ ˆ ˛ ˆ ˛ lim 12 Tr Λ S , R ( p 1 , p 2 ) 1 1 = 1 , lim 12 i Tr Λ P , R ( p 1 , p 2 ) γ 5 = 1 ˛ ˛ m R → 0 m R → 0 ˛ ˛ sym sym 1 i˛ 1 i˛ h h q µ Λ µ ˛ q µ Λ µ ˛ lim 12 q 2 Tr V , R ( p 1 , p 2 ) � q = 1 , lim 12 q 2 Tr A , R ( p 1 , p 2 ) γ 5 � q = 1 ˛ ˛ m R → 0 m R → 0 ˛ ˛ sym sym Alternative scheme (RI/SMOM γ µ ) using different projectors i˛ i˛ 1 1 h h ˛ ˛ Λ µ V , R ( p 1 , p 2 ) γ µ = 1 , Λ µ A , R ( p 1 , p 2 ) γ 5 γ µ = 1 lim 48 Tr lim 48 Tr ˛ ˛ m R → 0 m R → 0 ˛ ˛ sym sym C. Sturm, Improving the precision of light quark masses Pheno 2010, Madison, May 10th, 2010 7/10

The NLO, NNLO calculation (order α s , α 2 s ) RI/SMOM ( γ µ ) scheme Need to compute the matching factor C m in the new RI/SMOM schemes 1 diagram at LO, 1 at NLO and 11 at NNLO: q = p 1 −p 2 q = p 1 −p 2 q = p 1 −p 2 q = p 1 −p 2 q = p 1 −p 2 q = p 1 −p 2 k + p 1 k + p 2 k p 1 p 2 p 1 p 2 p 1 p 2 p 1 p 2 p 1 p 2 p 1 p 2 q = p 1 −p 2 q = p 1 −p 2 q = p 1 −p 2 q = p 1 −p 2 q = p 1 −p 2 q = p 1 −p 2 q = p 1 −p 2 p 1 p 2 p 1 p 2 p 1 p 2 p 1 p 2 p 1 p 2 p 1 p 2 p 1 p 2 Calculation: ✿✿✿✿✿✿✿✿✿✿✿✿ Two steps: A) IBP/Laporta’s algorithm K.G. Chetyrkin, F .V. Tkachov / S. Laporta, E. Remiddi � reduction to small(7) set of MI B) Solve MI, here known Davydychev et al. C. Sturm, Improving the precision of light quark masses Pheno 2010, Madison, May 10th, 2010 8/10

Results at NNLO Results & Comparison of the conversion factors for n f = 3, α s /π ≃ 0 . 1, scale of ∼ 2 GeV RI ′ /MOM ✿✿✿✿✿ ⇐ ⇒ ✿✿✿✿✿✿✿✿✿✿✿ RI/SMOM: ✿✿✿✿✿✿✿✿✿✿ C RI ′ /MOM = 1 − 0 . 1333333 ... − 0 . 07585848 ... − 0 . 0556959 ... m , L C RI/SMOM = 1 − 0 . 0161380 ... − 0 . 00660442 ... ← New m , L ֒ → in agreement with Jaeger, Gorbahn RI/MOM ✿✿✿✿✿ RI/SMOM γ µ : ⇐ ⇒ ✿✿✿✿✿✿✿✿✿✿✿✿✿✿ ✿✿✿✿✿✿✿✿✿ C RI/MOM = 1 − 0 . 1333333 ... − 0 . 0815876 ... − 0 . 0602759 ... m , L RI/SMOM γµ C = 1 − 0 . 0494713 ... − 0 . 0228421 ... ← New m , L Matching factors for schemes with symmetric subtraction point show better convergence behavior � significant reduction of syst. error on light quark masses Informative to have multiple schemes � better assessment of syst. errors C. Sturm, Improving the precision of light quark masses Pheno 2010, Madison, May 10th, 2010 9/10

Summary & Conclusion Framework + concepts of renorm. of quark bilinear operators in the RI/SMOM schemes has been discussed Results can be used to convert light quark masses in this scheme to the MS scheme The conversion factors in the RI/SMOM schemes are now available up to NNLO + show a better convergence behavior than in the traditional RI/MOM schemes The pert. truncation error is smaller than in RI/MOM RI/SMOM schemes are less sensitive to infrared effects in latt. simulation � The use of the RI/SMOM schemes will reduce the systematic error and improve precision of light quark mass determinations from lattice simulations obtained in this approach C. Sturm, Improving the precision of light quark masses Pheno 2010, Madison, May 10th, 2010 10/10