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

Aug-21-2007

Charm Charm and and bottom bottom Heavy baryon Heavy baryon mass spectrum mass spectrum from from Lattice QCD Lattice QCD with with 2+1 flavors 2+1 flavors

Heechang Na Heechang Na with with Steven Gottlieb Steven Gottlieb

Indiana University Indiana University

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 :

• Lattice QCD with

2+1 flavors

* * * * *

, , , , , , ' , , , ,

HH HH HH HH H H H H H H H H

• PDG, J. Phys. G 33, 1 (2006)

• New measurements

• Lattices and Propagators
• MILC coarse lattices

– 203×64, a ≈ 0.12 fm – 3 ensembles with four different time sources

• ml = 0.007 ms = 0.05
• ml = 0.01 ms = 0.05
• ml = 0.02 ms = 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

• Operators (K.C. Bowler et al., PRD 54, 3619 (1996))

c H b aT abc c H b aT abc

C O C O

• =
• =

) ( , ) (

2 1 2 5 1 5

• µ

µ

Ω*

hh

s h h s h h Ξ*

hh

l h h l h h Ω*

h

s s h s s h Ξ*

h

l s h l s h Σ*

h

l l h l l h 3/2+ Ωhh s h h s h h Ξhh l h h l h h Ωh s s h s s h Ξ’h l s h l s h Σh l l h l l h 1+ Oµ Ξh l s h l s h Λh l l h l l h 0+ 1/2+ O5 Baryon Content sπ Jp

• Formalism

• Two point function with a Staggered light quark and a Wilson heavy quark
• Conversion between a Naive propagator and a Staggered propagator!
• Now, we can write the Heavy-Light correlator
• (M. Wingate et al. PRD67, 054505 (2003))

) , ( ˆ ) ( ) ( ) ; (

4

y x G I G y G x y x G

• =
• =
• +
• eipx

x

• < Wsc

+ (x)Wsk (0) > =

eipx

x

• Tr scG(0;x)sk

+GH (x;0)

[ ]

= eipx

x

• tr{sc+(x)sk

+GH c'c(x;0)}G cc'(0;x)

[ ]

c,c'

• where W =

H (x)(x)

• =
• =

+ µ µ

• µ
• a

x

x y x G y x y x G

/

) ( ) ( where ) , ( ) ( ) ( ) , (

• Two point function for the heavy baryon

) , ( ) , ( ) , (

5 5 5

r r r

r r r

O t x O e t p C

x p i x

• =

) , ( ] ) )( , ( ) , ( [ tr

' 5 ' 2 5 ' 1 ' ' '

x G C x G C x G e

cc H bb T aa c b a abc x p i x +

• =
• r

r r

) , ( ) , ( ) , ( ] ) )( ( ) ( [ tr ) , (

' ' 2 ' 1 5 5 ' ' ' 5

x G x G x G C x C x e t p C

cc H bb aa T c b a abc x p i x

• =

+

• r

r r

r

4 ] ) 1 ( ) 1 [( tr ] ) )( ( ) ( [ tr

3 1 3 1

5 5

=

• =
• +

+ + x x x x T

C x C x

• )

, ( ) , ( ) , ( 4 ) , (

' ' 2 ' 1 ' ' ' 5

x G x G x G e t p C

cc H bb aa c b a abc x p i x

• r

r r

r

• =

Finally,

This is for O5, what about Oµ?

tr[T (x)C i(x)(C j)+] = 4(1)xi ij

) , ( ) , ( ) , ( ] ) )( ( ) ( [ tr ) , (

' ' 2 ' 1 ' ' '

x G x G x G C x C x e t p C

cc H bb aa j i T c b a abc x p i x ij

• =

+

• r

r r

r

3 2 1

3 2 1

) (

x x x t

x

• =
• - Surprisingly, Cij is a diagonal matrix for i and j indices

t m t m j i j i ij ij ij ij

e t C e t C t C t C t C P t C P t C

2 / 1 2 / 3

) ( , ) ( ) ( 3 1 ) ( ) 3 1 ( ) ( ) ( ) (

2 / 1 2 / 3 2 / 1 2 / 3 2 / 1 2 / 1 2 / 3 2 / 3

• +
• =

+ =

• Cij(r

p ,t) =

r x

eir

p r x 4(1)xi ijabca'b'c'G1 aa'(x,0)G2 bb'(x,0)GH cc'(x,0)

• Taste mixing?

Oµ = abc(1

aTC µ2 b)H c

• (x) =

a(x) a(x)

q i,a(y) = 1 8 i() a(y + )

• x = y +

a(y + ) = 2+ i()q i,a(y)

• : Naive quark

a : 4 copies of staggered quark q i,a : Staggered quark in taste basis (x) = 0

x01 x1 2 x2 3 x3

a : Copy index : Staggered spin index

• : Naive spin index

i : Taste index

• (x) =

a() a(y + ) = a()2+ i()q i,a(y)

Dµ = (1

T (x)C µ2(x))

Dµ

conti(y) =

• (1

T (x)C µ2(x))

Dµ = (1

T (x)C µ2(x))

Dµ

conti(y) =

• 2+ i()q i,a(y)Ta

()(C µ) b()2+ j ()q j,b(y)

=

• 4+ i()q i,a(y)(1)

µ (C µ)a b+ j ()q j,b(y)

• + i()(1)

µ + j () = 4(C µ) ( µC1)i j

Dµ

conti(y) =16q i,a(y)(C µ) ( µC1)i jq j,b(y)(C µ)a b

D5

conti(y) =16q i,a(y)(C 5) ( 5C1)i jq j,b(y)(C 5)a b

a, b : Copy index i, j : Taste index , : Staggered spin index , : Naive spin index

• Di-quark operator

Overlap with 1+ and 0+ spin state with single taste

• K. Nagata et al., arXiv:0707.3537

• Two-point function of the di-quark operator

Cµ

conti(y;0) = < Dµ conti(y)D

• conti(0) >

=162Tr[G1(y,0)(C µ) (C µ)+G2(y,0)(C )+ (C )] (C µ)ab (C µ)

b a + b b a a

=162Tr[G1(y,0)(C µ) (C µ)+G2(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

) ( ~ * ~ * ) ( * * ) ( ~ ~ ) (

* * * *

~ ) 1 ( ~ ) 1 ( ~ ) 1 ( ~ ) 1 ( ) (

t T m t t m t t T m t m t T m t t m t t T m mt

e A e A e A e A e A e A Ae Ae t P

• +
• +

+ +

• +
• +

+ =

• Correlated least squares fit
• Error estimation

– 1000 bootstrap samples

• Linear chiral extrapolation

• Results
• 1/2+ singly charmed heavy baryons

243×48 : 60 confs (a≈0.068)

• 1/2+ singly charmed heavy baryons : Other groups (Quenched calculations)

123×32 : 720 confs (as≈0.22) 143×38 : 442 confs (as≈0.18) 183×46 : 325 confs (as≈0.15)

K.C. Bowler et al., PRD 54,3619 (1996)

• R. Lewis et al.,

PRD 64,094509 (2001)

• 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, ml=0.2ms, ml=0.4ms

• 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.

Mphy = Mcal+ Δ Constant Mass Shift = Average (Mexp - Mcal)

Mkin = | r p |2 [Mcal(r p ) Mcal(0)]2 2[Mcal(r p ) Mcal(0)]

• 1/2+ singly bottom heavy baryons conti.

Confidence level ~ 40% in averag e

• Extrapolation of light valence quark mass

0.00148 Real quark masses are quotations from MILC. PRD 70, 114501 (2004)

• Interpolation of Strange quark mass and

extrapolation of Light sea quark mass

0.00148 0.039

• Full QCD extrapolation