Charm and and bottom bottom Heavy baryon Heavy baryon Charm - PowerPoint PPT Presentation

Charm and and bottom bottom Heavy baryon Heavy baryon Charm mass spectrum from from mass spectrum Lattice QCD with with 2+1 flavors 2+1 flavors Lattice QCD Heechang Na Heechang Na with Steven Gottlieb with Steven Gottlieb Indiana University Indiana University Aug-21-2007 INT Summer School 2007 , U of Washington, Seattle

Outline • Introduction • Lattices and propagators • Formalism – Operators – Two point function – Taste mixing? • Data analysis • Results – Charmed heavy baryons – Bottom heavy baryons – Doubly heavy baryons • Future study

● Introduction • Singly and doubly charmed heavy baryons • Singly and doubly bottom heavy baryons : * * * , , , , ' , , , � � � � � � � � H H H H H H H H * * , , , � � � � HH HH HH HH • Lattice QCD with 2+1 flavors PDG, J. Phys. G 33, 1 (2006)

• New measurements

● Lattices and Propagators • MILC coarse lattices – 20 3 × 64, a ≈ 0.12 fm – 3 ensembles with four different time sources • m l = 0.007 m s = 0.05 • m l = 0.01 m s = 0.05 • m l = 0.02 m s = 0.05 • Propagators – 9 different staggered light valence quarks • 0.005 ~ 0.02 – 3 different staggered strange valence quarks • 0.024, 0.03, 0.0415 – One valence clover heavy quark • k = 0.122 (Tuned for charm quark) – 007 : 545 confs 010 : 591 confs 020 : 459 confs • k = 0.086 (for bottom) – 007 : 554 confs 010 : 590 confs 020 : 452 confs

● Formalism • Operators (K.C. Bowler et al ., PRD 54, 3619 (1996)) aT b c aT b c O ( C ) , O ( C ) = � � � � � = � � � � � 5 abc 1 5 2 H abc 1 2 H µ µ Jp s π Content Baryon Λ h l l h l l h O 5 0 + Ξ h l s h l s h l l h l l h Σ h 1/2+ l s h l s h Ξ’ h Ω h s s h s s h Ξ hh l h h l h h s h h s h h Ω hh O µ 1 + l l h l l h Σ * h l s h l s h Ξ * h 3/2+ Ω * s s h s s h h l h h l h h Ξ * hh s h h s h h Ω * hh

• Two point function with a Staggered light quark and a Wilson heavy quark • Conversion between a Naive propagator and a Staggered propagator! G ( x ; y ) ( x ) G ( y ) + = � � � � ˆ G I G ( x , y ) = 4 � � G ( x , y ) ( x ) ( y ) G ( x , y ) + = � � � � x / a where ( x ) ( ) � � = � µ µ µ • Now, we can write the Heavy-Light correlator + ( x ) W � sk (0) > = [ ] � e ip � x � e ip � x + G H ( x ;0) < W � sc Tr � sc G � (0; x ) � sk x x [ ] � e ip � x � c ' c ( x ;0)} G � cc ' (0; x ) tr{ � sc � + ( x ) � sk + G H = x c , c ' where W � = � H ( x ) �� ( x ) • (M. Wingate et al. PRD67, 054505 (2003))

• Two point function for the heavy baryon r r r r r i p x C ( p , t ) e � � O ( x , t ) O ( 0 , 0 ) � = 5 5 5 r x r r i p x aa ' T bb ' cc ' e � � tr [ G ( x , 0 ) C G ( x , 0 )( C ) + ] G ( x , 0 ) � = � � � � abc a ' b ' c ' 1 5 2 5 H r x r r r i p x T C ( p , t ) e � � tr [ ( x ) C ( x )( C ) + ] � = � � � � � � 5 abc a ' b ' c ' 5 5 r x aa ' bb ' cc ' G ( x , 0 ) G ( x , 0 ) G ( x , 0 ) � 1 2 H � � T x x x x tr [ ( x ) C ( x )( C ) ] tr [( 1 ) + ( 1 ) + ] + � � � � = � 1 3 � 1 3 5 5 4 = Finally, r r r i p x aa ' bb ' cc ' C ( p , t ) e 4 G ( x , 0 ) G ( x , 0 ) G ( x , 0 ) � � � = � � 5 abc a ' b ' c ' 1 2 H � � r x This is for O 5 , what about O µ ?

-- Surprisingly, C ij is a diagonal matrix for i and j indices r r r i p x T C ( p , t ) e tr [ ( x ) C ( x )( C ) ] � � + � = � � � � � � ij abc a ' b ' c ' i j r x aa ' bb ' cc ' G ( x , 0 ) G ( x , 0 ) G ( x , 0 ) t x x x ( x ) � � = � � 1 � 2 � 3 1 2 H � � 0 1 2 3 tr[ � T ( x ) C � i � ( x )( C � j ) + ] = 4( � 1) x i � ij C ij ( r � e � i r � r x 4( � 1) x i � ij � abc � a ' b ' c ' G 1 � p aa ' ( x ,0) G 2 � bb ' ( x ,0) G H cc ' ( x ,0) p , t ) = r x 3 / 2 1 / 2 C ( t ) P C ( t ) P C ( t ) = + ij ij 3 / 2 ij 1 / 2 1 1 ( ) C ( t ) C ( t ) = � � � � + � � ij i j 3 / 2 i j 1 / 2 3 3 m t m t C ( t ) e � , C ( t ) e � � 3 / 2 � 1 / 2 3 / 2 1 / 2

• Taste mixing? T ( x ) C � µ � 2 ( x )) aT C � µ � 2 b ) � H c O µ = � abc ( � 1 D µ = ( � 1 � ( x ) = � � � � a ( x ) � a ( x ) � : Naive quark � � � � a : 4 copies of staggered quark q � i , a ( y ) = 1 � � � i ( � ) � a ( y + � ) q � i , a : Staggered quark in taste basis 8 x 0 � 1 x 1 � 2 x 2 � 3 x 3 � � ( x ) = � 0 x = y + � a : Copy index � : Staggered spin index � a ( y + � ) = 2 � + i � ( � ) q � i , a ( y ) � : Naive spin index � i : Taste index � ( x ) = � � � � a ( � ) � a ( y + � ) = � � � a ( � )2 � + i � ( � ) q � i , a ( y ) �

• Di-quark operator T ( x ) C � µ � 2 ( x )) conti ( y ) = � ( � 1 T ( x ) C � µ � 2 ( x )) D µ D µ = ( � 1 � � b ( � )2 � + j � ( � ) q � j , b ( y ) conti ( y ) = � 2 � + i � ( � ) q � i , a ( y ) � Ta � � ( � )( C � µ ) � � � � � � D µ � � µ ( C � µ ) a b � + j � ( � ) q � j , b ( y ) � 4 � + i � ( � ) q � i , a ( y )( � 1) = � � µ � + j � ( � ) = 4( C � µ ) �� � ( � µ C � 1 ) i j � � + i � ( � )( � 1) � conti ( y ) = 16 q � i , a ( y )( C � µ ) �� � ( � µ C � 1 ) i j q � j , b ( y )( C � µ ) a b D µ conti ( y ) = 16 q � i , a ( y )( C � 5 ) �� � ( � 5 C � 1 ) i j q � j , b ( y )( C � 5 ) a b D 5 Overlap with 1 + and 0 + spin state with single taste K. Nagata et al., arXiv:0707.3537 a , b : Copy index i , j : Taste index � , � : Staggered spin index � � , � � : Naive spin index

• Two-point function of the di-quark operator conti ( y ;0) = < D µ conti ( y ) D conti (0) > C µ � � = 16 2 Tr[ G 1 ( y ,0)( C � µ ) � ( C � µ ) + G 2 ( y ,0)( C � � ) + � ( C � � )] + � b � � ( C � µ ) ab � ( C � µ ) � b � a � b � a a = 16 2 Tr[ G 1 ( y ,0)( C � µ ) � ( C � µ ) + G 2 ( y ,0)( C � � ) + � ( C � � )] � Tr[( C � µ )( C � � ) + ] Tr[( C � µ )( C � � ) + ] = 0 � µ � � � 4 µ = � � The delta function appears, because the cancellations between copy indices.

● Data analysis • Fit model function ~ ~ ~ ~ mt m ( T t ) t m t t m ( T t ) P ( t ) Ae Ae ( 1 ) A e ( 1 ) A e � � � � � � = + + � + � ~ ~ ~ ~ * * * * * m t * m ( T t ) t * m t t * m ( T t ) A e A e ( 1 ) A e ( 1 ) A e � � � � � � + + + � + � • Correlated least squares fit • Error estimation – 1000 bootstrap samples • Linear chiral extrapolation

● Results • 1/2 + singly charmed heavy baryons

• 1/2 + singly charmed heavy baryons : Other groups (Quenched calculations) 12 3 × 32 : 720 confs (a s ≈ 0.22) 24 3 × 48 : 60 confs 14 3 × 38 : 442 confs (a s ≈ 0.18) (a ≈ 0.068) 18 3 × 46 : 325 confs (a s ≈ 0.15) R. Lewis et al., K.C. Bowler et al., PRD 64,094509 (2001) PRD 54,3619 (1996)

• 1/2 + singly bottom heavy baryons Recent measurements from CDF and D0

• 1/2 + singly bottom heavy baryons : Other groups (Quenched calculations) K.C. Bowler et al.,PRD 54,3619 (1996) A. Ali Khan et al., PRD 62,054505 (2000) N. Mathur et al., PRD 66,014502 (2002)

• Doubly charmed heavy baryons (Preliminary)

• Doubly bottom heavy baryons (Preliminary)

Future study • Fine lattice – a ≈ 0.09, m l =0.2m s , m l =0.4m s • Increase statistics • More about error analysis • Finite size effect • Discretization errors • Excited states (3/2 + ,1/2 - ,3/2 - )

• Mass differences between bottom and charm hadrons

• 1/2 + singly charmed heavy baryons conti. M phy = M cal + Δ M kin = | r p | 2 � [M cal ( r ) � M cal (0)] 2 p 2[M cal ( r p ) � M cal (0)] Constant Mass Shift = Average (M exp - M cal )

• 1/2 + singly bottom heavy baryons conti.

• Extrapolation of light valence quark mass 0.00148 Confidence level ~ 40% in averag e Real quark masses are quotations from MILC. PRD 70, 114501 (2004)

• Interpolation of Strange quark mass and extrapolation of Light sea quark mass 0.039 0.00148

• Full QCD extrapolation