A LPHA Collaboration Z S / Z P from three-flavour lattice QCD - PowerPoint PPT Presentation

Z S / Z P from three-flavour lattice QCD Fabian Joswig in collaboration with Jochen Heitger and Anastassios Vladikas A LPHA Collaboration

Z S / Z P from three-flavour lattice QCD Outline 1. Motivation: Why is Z S / Z P important in the context of quark mass calculations 2. A new method to determine Z S / Z P based on Ward identities in the Schrödinger functional framework 3. Preliminary results and crosschecks 1 Fabian Joswig

Z S / Z P from three-flavour lattice QCD Calculation of quark masses Why are we interested in heavy quark masses? ◮ Fundamental parameters of the Standard Model ◮ Main source of uncertainty in Higgs partial widths comes from m c , m b and α s (e.g. arXiv:1404.0319) ◮ Matching parameters for Heavy Quark Effective Theory Challenges ◮ Large discretization effects due to the high mass ⇒ Systematic uncertainties have to be treated carefully 2 Fabian Joswig

Z S / Z P from three-flavour lattice QCD How can we calculate quark masses from LQCD We work on N f = 2 + 1 ensembles with Wilson-clover fermions 1. Tune the hopping parameter κ such that a particle containing the desired quark has it’s physical mass. 2. Use an appropriate renormalization pattern to relate this to the renormalized quark mass. 3 Fabian Joswig

Z S / Z P from three-flavour lattice QCD How is the quark mass renormalized in our setup A LPHA The standard Collaboration method uses the PCAC mass �� ˜ ∂ 0 A 12 0 ∂ 0 P 12 ( x 0 ) P 21 ( 0 ) � � 0 ( x 0 ) + ac A ∂ ∗ m 12 = � P 12 ( x 0 ) P 21 ( 0 ) � 2 and it’s renormalization and improvement pattern to calculate the renormalized quark mass M Z A � � 1 + a ( b A − b P ) m q12 + a (¯ b A − ¯ + O ( a 2 ) b P ) tr ( M ) m 12R = Z P ( L ) m 12 ¯ m ( L ) 4 Fabian Joswig

Z S / Z P from three-flavour lattice QCD How is the quark mass renormalized in our setup A LPHA The standard Collaboration method uses the PCAC mass �� ˜ ∂ 0 A 12 0 ∂ 0 P 12 ( x 0 ) P 21 ( 0 ) � � 0 ( x 0 ) + ac A ∂ ∗ m 12 = � P 12 ( x 0 ) P 21 ( 0 ) � 2 and it’s renormalization and improvement pattern to calculate the renormalized quark mass M Z A � � 1 + a ( b A − b P ) m q12 + a (¯ b A − ¯ + O ( a 2 ) b P ) tr ( M ) m 12R = Z P ( L ) m 12 ¯ m ( L ) m 3R = 2 m 13 � 1 + ( b A − b P )( am q3 − am q2 ) � − 1 + O ( a 2 ) m 1R m 12 2 4 Fabian Joswig

Z S / Z P from three-flavour lattice QCD How is the quark mass renormalized in our setup ◮ Method used in N f = 2, arXiv:1205.5380 ◮ Light and strange quark masses for N f = 2 + 1 simulations with Wilson fermions Jonna Koponen, tomorrow 2:40 pm ◮ We want to use a complementary method as a cross check for the future computation of the charm quark’s mass ( , partly joint with the Regensburg group) 5 Fabian Joswig

Z S / Z P from three-flavour lattice QCD How is the quark mass renormalized in our setup ◮ Method used in N f = 2, arXiv:1205.5380 ◮ Light and strange quark masses for N f = 2 + 1 simulations with Wilson fermions Jonna Koponen, tomorrow 2:40 pm ◮ We want to use a complementary method as a cross check for the future computation of the charm quark’s mass ( , partly joint with the Regensburg group) Rolf, Sint, N f = 0 5 Fabian Joswig

Z S / Z P from three-flavour lattice QCD How is the quark mass renormalized in our setup 1. Subtractive renormalization 1 1 am q , f = − 2 κ f 2 κ crit 6 Fabian Joswig

Z S / Z P from three-flavour lattice QCD How is the quark mass renormalized in our setup 1. Subtractive renormalization 1 1 am q , f = − 2 κ f 2 κ crit 2. Multiplicative renormalization (plus O ( a ) improvement) 1 �� m q , f + ( r m − 1 ) tr ( M ) � m fR = Z S ( L ) N f b m m q , f tr ( M ) + ( r m d m − b m ) tr ( M 2 ) b m ) tr ( M ) 2 �� � q , f + ¯ + ( r m ¯ d m − ¯ b m m 2 + O ( a 2 ) + a N f N f 6 Fabian Joswig

Z S / Z P from three-flavour lattice QCD How is the quark mass renormalized in our setup 1. Subtractive renormalization 1 1 am q , f = − 2 κ f 2 κ crit 2. Multiplicative renormalization (plus O ( a ) improvement) 1 �� m q , f + ( r m − 1 ) tr ( M ) � m fR = Z S ( L ) N f b m m q , f tr ( M ) + ( r m d m − b m ) tr ( M 2 ) b m ) tr ( M ) 2 �� � q , f + ¯ + ( r m ¯ d m − ¯ b m m 2 + O ( a 2 ) + a N f N f 6 Fabian Joswig

Z S / Z P from three-flavour lattice QCD How is the quark mass renormalized in our setup We have two different methods to arrive at the renormalized quark mass : � 1 + ( b A − b P )( am q3 − am q2 ) � m 3R = 2 m 13 − 1 + O ( a 2 ) 2 m 1R m 12 1 � �� � 1 + ab m ( m q1 + m q2 ) + a ¯ + O ( a 2 ) m 1R − m 2R = m q1 − m q2 b m tr ( M ) Z S ( L ) 7 Fabian Joswig

Z S / Z P from three-flavour lattice QCD Progress on non-perturbative determination of renormalization constants and parameters ◮ Renormalization and improvement factors for the axial current were previously determined in our group A LPHA [ Collaboration , Bulava, Della Morte, Heitger, Wittemeier; arXiv:1502.04999, arXiv:1604.05827] (There is also an approach to Z A in the chirally rotated Schrödinger functional) M ◮ Results for Z P and the RGI factor m ( L ) were published recently by our collaboration ¯ A LPHA [ Collaboration , Campos, Fritzsch, Pena, Preti, Ramos, Vladikas; arXiv:1802.05243] ◮ Determination of b m and b A − b P is almost finished [arXiv:1710.07020 & in progress: De Divitiis, Fritzsch, Heitger, Köster, Kuberski, Vladikas] ◮ Z S is undetermined so far 8 Fabian Joswig

Z S / Z P from three-flavour lattice QCD Our method for the determination of Z S / Z P We start with the general axial Ward identity : � � � � � � A a µ ( x ) O b int ( y ) O c d 4 x P a ( x ) O b int ( y ) O c d σ µ ( x ) ext ( z ) − 2 m ext ( z ) ∂ R R � � [ δ a A O b int ( y )] O c = − ext ( z ) 9 Fabian Joswig

Z S / Z P from three-flavour lattice QCD Our method for the determination of Z S / Z P We start with the general axial Ward identity : � � � � � � A a µ ( x ) O b int ( y ) O c d 4 x P a ( x ) O b int ( y ) O c d σ µ ( x ) ext ( z ) − 2 m ext ( z ) ∂ R R � � [ δ a A O b int ( y )] O c = − ext ( z ) And use the transformation property of the pseudoscalar density under small chiral rotations: A P b ( x ) = d abc S c ( x ) + δ ab δ a ¯ ψ ( x ) ψ ( x ) N f 9 Fabian Joswig

Z S / Z P from three-flavour lattice QCD Our method for the determination of Z S / Z P We start with the general axial Ward identity : � � � � � � A a µ ( x ) O b int ( y ) O c d 4 x P a ( x ) O b int ( y ) O c d σ µ ( x ) ext ( z ) − 2 m ext ( z ) ∂ R R � � [ δ a A O b int ( y )] O c = − ext ( z ) And use the transformation property of the pseudoscalar density under small chiral rotations: A P b ( x ) = d abc S c ( x ) + δ ab δ a ¯ ψ ( x ) ψ ( x ) N f ◮ d abc � = 0, for SU ( N f ) with N f ≥ 3 9 Fabian Joswig

Z S / Z P from three-flavour lattice QCD Our method for the determination of Z S / Z P Inserting a pseudoscalar density into the chiral Ward identity leads to: � � �� � d 3 y d 3 x A a 0 ( y 0 + t , x ) − A a P b ( y 0 , y ) O ext � 0 ( y 0 − t , x ) � y 0 + t � � � � d 3 y d 3 x P a ( x 0 , x ) P b ( y 0 , y ) O ext − 2 m d x 0 y 0 − t � � � = − d abc d 3 y S c ( y ) O ext 10 Fabian Joswig

Z S / Z P from three-flavour lattice QCD Our method for the determination of Z S / Z P When the Ward identity is evaluated on a lattice with Schrödinger functional boundary conditions we end up with: 1 + ab A m q + a ¯ 1 + ab P m q + a ¯ � �� � b A tr ( M ) b P tr ( M ) Z A Z P × f I , abcd ( y 0 + t , y 0 ) − f I , abcd ( y 0 − t , y 0 ) − 2 m ˜ f abcd � � ( y 0 + t , y 0 − t ) AP AP PP 1 + ab S m q + a ¯ f abcd ( y 0 ) + O ( a 2 ) + O ( am ) � � = − Z S b S tr ( M ) S 11 Fabian Joswig

Z S / Z P from three-flavour lattice QCD Our method for the determination of Z S / Z P When the Ward identity is evaluated on a lattice with Schrödinger functional boundary conditions we end up with: Z A Z P × f I , abcd ( y 0 + t , y 0 ) − f I , abcd ( y 0 − t , y 0 ) − 2 m ˜ f abcd � � ( y 0 + t , y 0 − t ) AP AP PP f abcd ( y 0 ) + O ( a 2 ) = − Z S S 11 Fabian Joswig

Z S / Z P from three-flavour lattice QCD Our method for the determination of Z S / Z P When the Ward identity is evaluated on a lattice with Schrödinger functional boundary conditions we end up with: Z A Z P × f I , abcd ( y 0 + t , y 0 ) − f I , abcd ( y 0 − t , y 0 ) − 2 m ˜ f abcd � � ( y 0 + t , y 0 − t ) AP AP PP f abcd ( y 0 ) + O ( a 2 ) = − Z S S 11 Fabian Joswig

Z S / Z P from three-flavour lattice QCD Figure: Graphical representation of the Wick contractions contributing to f Γ˜ Γ (arXiv:hep-lat/9611015). 12 Fabian Joswig