Aug-21-2007

CharmCharm andand bottombottom Heavy baryon Heavy baryonmass spectrummass spectrum from from

Lattice QCDLattice QCD with with 2+1 flavors2+1 flavors

Heechang NaHeechang Nawithwith Steven Gottlieb Steven Gottlieb

Indiana UniversityIndiana 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 doublycharmedheavy baryons

• Singly and doublybottomheavy baryons :

• Lattice QCD with2+1 flavors

**

***

,,,

,,,',,,,

HHHHHHHH

HHHHHHHH

!!""

!!"""##$

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

baT

abc

c

H

baT

abcCOCO !=! = )(,)( 212515 "#"$"#"$ µµ

Ω*hhs h h

Ξ*hhl h h

Ω*hs s h

Ξ*hl s h

Σ*hl l h

3/2+

Ωhhs h h

Ξhhl h h

Ωhs s h

Ξ’hl s h

Σhl l h

1+Oµ

Ξhl s h

Λhl l h0+

1/2+

O5

BaryonContentsπJp

● 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 yxGIG

yGxyxG

!=

""=

#

+

#$

!

eip"x

x

# <W$sc

+(x)W$sk

(0) > = eip"x

x

# Tr $scG% (0;x)$sk+GH (x;0)[ ]

= eip"x

x

# tr{$sc&+(x)$sk

+GH

c 'c(x;0)}G'

cc'(0;x)[ ]

c,c'

#

where W$ = % H (x)$%(x)

!="

""= +

µ

µ

#$

µ%ax

x

yxGyxyxG

/)()( where

),()()(),(

• Two point function for the heavy baryon)0,0(),(),( 555

rrr rr

r

OtxOetpCxpi

x

!"#=

)0,(]))(0,()0,([tr '

5

'

25

'

1''' xGCxGCxGe cc

H

bbTaa

cbaabc

xpi

x

+!"#= $$%%rr

r

)0,()0,()0,(

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

''

2

'

1

55'''5

xGxGxG

CxCxetpC

cc

H

bbaa

T

cbaabc

xpi

x

!!

""##

$

%%=+&'(

rr

r

r

4

])1()1[(tr]))(()([tr 3131

55

=

!!=""+++ xxxxT

CxCx ##

)0,()0,()0,(4),( ''

2

'

1'''5 xGxGxGetpC cc

H

bbaa

cbaabc

xpi

x

!!""rr

r

r #$%=

Finally,

This is for O5, what about Oµ?

!

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

)0,()0,()0,(

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

''

2

'

1

'''

xGxGxG

CxCxetpC

cc

H

bbaa

ji

T

cbaabc

xpi

x

ij

!!

""##

$

%%=+&'(

rr

r

r

321

3210)(xxxt

x !!!!="

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

tmtm

jijiij

ijijij

etCetC

tCtC

tCPtCPtC

2/12/3 )(,)(

)(3

1)()

3

1(

)()()(

2/12/3

2/12/3

2/1

2/1

2/3

2/3

!!""

+!=

+=

####$

!

Cij (r p ,t) =

r x

" e# i

r p $

r x 4(#1)xi %ij&abc&a 'b'c 'G1'

aa'(x,0)G

2'bb'(x,0)GH

cc'(x,0)

• Taste mixing?

!

Oµ = "abc(#

1

aTC$µ#2

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

x0'1

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#µ"2(x))

!

Dµ

conti(y) =

"

# ($1

T(x)C%µ$2

(x))

!

Dµ = ("1

T

(x)C#µ"2(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 )%' ) ((µC

&1)i j

!

Dµ

conti(y) =16q" i,a (y)(C#µ )"$ % (#µC

&1)i j q

$ j,b(y)(C#µ )a b

D5

conti(y) =16q" i,a (y)(C# 5)"$ % (# 5C

&1)i j q

$ 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 cancellationsbetween copy indices.

● Data analysis

• Fit model function

)(~*~*)(**

)(~~)(

**** ~)1(

~)1(

~)1(

~)1()(

tTmttmttTmtm

tTmttmttTmmt

eAeAeAeA

eAeAAeAetP

!!!!!!

!!!!!!

!+!+++

!+!++=

• 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 average

• 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

0.001480.039

• Full QCD extrapolation