B meson potentials in
quenched lattice QCD
William Detmold
National Tsing-Hua University, Oct 11th 2007
WD, K Orginos and M Savage, hep-lat/0703009
• Hadron potentials: NN,YN and BB
• Lattice QCD
• BB potentials from lattice QCD
• Perturbative Coulomb corrections
• Finite volume effects
Hadron potentials
• Not uniquely defined: r?, unitary equivalence
• Encode of information about scattering amplitude
• Successful phenomenology
• Fit NN potential from scattering data
• Add 3N, 4N forces (tuned to more data)
• Accurate description of A<10 spectrum
GFMC caclulations
[Carlson, Pieper, Wiringa,...]
YN scattering
• Λ(Σ)N interactions important in EOS in NS
• Poorly known experimentally
Static heavy quarks
• Static limit for heavy quark: m → ∞
• B meson (bd) mass is infinite
b
Mb = mb + Λ + O(1/mb )
• B and B* degenerate
• Heavy quark spin decouples
Static BB potentials?
• Static hadrons: defined, observable potential
• LDOF quantum numbers ⊃ NN
• EFT: potential has same form for |!r| > Λ
−1
χ
2
−mπ |#
r|
g
e
−1
πN N
VN N (|!r| > Λχ ) −→ # 2
f
|!r|
• Short range very different: 1/|!r| Coulomb
• Shallow bound states, molecular states, ...
Static BB potentials?
• Static hadrons: defined, observable potential
• LDOF quantum numbers ⊃ NN
• EFT: potential has same form for |!r| > Λ
−1
χ
2
−mπ |#
r|
g
e
∗
−1
BB π
VBB (|!r| > Λχ ) −→ #
f2
|!r|
• Short range very different: 1/|!r| Coulomb
• Shallow bound states, molecular states, ...
Lattice QCD
•
•
•
Numerical solution of QCD field equations
QCD partition function
!
−SQCD [A,ψ,ψ]
Z ∼ DAµ Dψ Dψ e
•
Quark functional integral done exactly
Observable
!
1
−Sg [A]
!O" ∼
DAµ det[M[A]]O[A, M]e
Z
Lattice QCD
• Numerical method to solve QCD nonperturbatively [K Wilson 1974]
• Computers are finite but space-time is
infinite
• To make progress
• Discretise space-time: a
• Compactify space-time: L
• Euclidean space
Space-time
a
L
Lattice QCD
• Quarks live on lattice sites, gluons on the links
between them
•
Functional integral is finite dimensional but still too
many integrals (>107 !) to do exactly
•
Estimate using importance sampling (Monte Carlo)
•
•
Configurations {φi } generated with
Boltzmann weight det[M] exp(−SQCD )
N
!
√
1
Observable: !O" →
O[φi ] errors ∼ 1/ N
N i=1
Huge computers
• Calculations use worlds largest computers
• Measured in millions of CPU hours
• Specifically designed processors for LQCD
Ex: energy spectrum
•
•
Measure correlator (χ = source with q# of hadron)
!
ip·x
e
!0|χ(x, t)χ(0, 0)|0"
G2 (p, t) =
x
Long times: only ground state survives
t→∞
−→ e
−E0 (p)t
#0|χ(0, 0)|E0 , p$#E0 , p|χ(0, 0)|0$
0.8
0.6
C!t" 0.4
0.2
0
0
5
10
t
15
20
Extrapolations
• To get real world physics from the lattice
calculations we need to take:
• Lattice spacing to zero
• Lattice volume to infinity
• Quark masses to their physical values
• Physical masses too demanding
• Mostly ignore today...
Heavy-light systems
Lattice parameters
fm
• O(300) quenched DBW2
• Wilson light quark propagators at a single mass
• Light hadron spectrum [MeV]:
3
lattices:16 x32, a=0.1
mπ = 403 mρ = 743 mN = 1140
• Static heavy quarks:!
SQ (x, t; t0 ) =
" #
t
1 + γ4
!
U4 (x, t )
2
!
t =t0
Lattice correlators
• Single particle correlation functions: C(t)
0
B
t
Λ b , Σb
• Energy: plateau in effective mass ratio
!
C(t − 1)
aE(t) = log
C(t)
"
Single particle energies
1.2
"b
1
!b
0.8
bE
B
0.6
0.4
2
4
6
8
10
t!b
12
14
16
Exotic hadrons
• Byproduct of r=(0,0,0) potential
• Binding energy of exotic hadron states:
(3 ⊗ 3)Q ⊗ (3 ⊗ 3)q → 3Q ⊗ 3q + 6Q ⊗ 6q
• Two exotic states: Σ6 , Λ6
long ago: viable BSM
• Considered
[Karl, Wilczek&Zee, Marciano,... 70s]
Exotic particle energies
1.4
1.3
"6
!6
b E 1.2
1.1
1
2
4
6
8
t!b
10
12
LDOF energies: Λ
•
Lattice energies are unphysical ~ a-1
Elatt = δm + Λ
• δm : interaction of static quark with gluons
+...
• Λ : energy of light degrees of freedom
Eq. (11.23) with
c1 " −1:40686 :
(11.25)
There are other proposals in which c2 and c3 are nonzero, which we will not consider here.
The main drawback of all these improved gauge actions is that they have no re!ection positivity
(L"uscher and Weisz, 1984). This means it is not possible to construct a transfer matrix at #nite a.
This also causes problems with numerical simulations. The violation of physical positivity in fact
leads to unphysical poles in the propagators, corresponding to unphysical states that create a sizeable
disturb while extracting physical observables (Necco, 2002b).
Another problem is that perturbation theory is not very manageable. The gluon propagator in a
covariant gauge for generic c1 is given by
!
#
qi
"
1
q̂i = 2(11.26)
sin
G!" (k) =
#k̂ ! k̂ " +
(k̂ $ %!" − k̂ " %!$ )k̂ $ A$" (k)
2
(k̂ 2 )2
$
Lattice PT: improved glue
with
A!" (k) = A"! (k) = (1 − %!" )&(k)
−1 
2 2
2

(k̂ ) − c1 k̂ 2
"
k̂ 4'
+ k̂
2
'
!

#2
"
" "
*
4
2
4
2
2 2
2
k̂ ' + k̂
k̂ '
k̂ ( + (k̂ )
k̂ 2' 
+ c1
'
and

!
2
&(k) = k̂ − c1
− 4c13
"
'
"
'
k̂ 4'
'
k̂ 4'
#-
"
("='
2
k̂ − c1
k̂ 2( :
("=!;"
!
2 2
(k̂ ) +
"
'"=!;"

k̂ 2' 
(11.27)
'"=!;"
"
(
k̂ 4(
#
1
+ c12
2
!
2 3
(k̂ ) + 2
"
(
k̂ 6(
− k̂
2
"
(
k̂ 4(
#.
(11.28)
c = Improvement coeff
The gluon vertices are quite complicated, and1we will not report them here. They can be found in
Weisz and Wohlert (1984). The quark–gluon vertices are of course untouched.
Perturbative calculations using improved gauge actions have been recently presented in Aoki et al.
(2000) for three-quark operators and in DeGrand et al. (2002) for two- and four-quark operators,
[Weisz & Wohlert
and they have even been employed in connection with domain wall calculations (Aoki et al., 2002).
84]
LDOF energies: Λ
•
Lattice energies are unphysical ~ a-1
Elatt = δm + Λ
• δm: interaction of! static quark with gluons
(α)
δmf
α(µ)
=
3π 2 a
d
BZ
3
(f )
q G00 (q̂x , q̂y , q̂z , 0)
• Evaluate numerically
• Λ : energy of light degrees of freedom
qi
q̂i = 2 sin
2
Scale setting
• Scale of one loop contribution not welldefined
• Brodsky-Lepage-Mackenzie procedure
• Sum vacuum polarisation effects
• Subtlety: depends on choice of action
[BLM ‘83/LM ‘93]
Binding energies
• Extracted physical binding energies [MeV]
B
649(31)(10)
Λb
1123(36)(04)
Σb
1250(38)(15)
Λ6
1364(64)(04)
Σ6
1413(65)(10)
B meson potentials
[cf bb potential: Derek Leinweber]
BB potentials
• Lattice energies for BB(I,s ):
l
latt
VI,sl (r, L)
=
latt
EI,sl (r, L)
−
latt
2EB (L)
• Perturbative QCD contribution
latt
EI,s
(r,
L)
=
Λ
(r,
L)
+
2δm
−
δV
(r,
L)
I,s
R
l
l
• “Continuum” potential
VI,sl (r, L) = ΛI,sl (r, L) − 2Λ =
latt
VI,sl (r, L)
+ δVR (r, L)
• Infinite volume continuum potential
Central & tensor potentials
• Separate central and tensor S=1 potentials
V
Ŝ12 =
3
2
(S=1)
!
(!r) = VC (|!r|) + Ŝ12 VT (|!r|)
Ŝ+ (r̂x − ir̂y ) + Ŝ− (r̂x + ir̂y ) + 2Ŝz r̂z
"2
• Tensor potentials: |V | < 40 MeV ∀ !r
• Expect nonzero for NN
• Treat off-axis points as if central
T
− 2Ŝ 2
BB correlators
• Each spin-isospin channel
r
t
• Calculate for separations
!r = n(1, 0, 0), n(0, 0, 1) [n = 0, . . . 8]
!r = (1, 0, 1), (2, 0, 1)
BB correlators
• Each spin-isospin channel
r
t
• Calculate for separations
!r = n(1, 0, 0), n(0, 0, 1) [n = 0, . . . 8]
!r = (1, 0, 1), (2, 0, 1)
512 propagators/lattice!
Effective mass plots
1.15
bE
"r!" " 1
"r!" " 1
1.1
1.25
1.05
1.2
1
b E 1.15
0.95
1.1
0.9
1.05
0.85
2
4
6
8
10
2
12
4
6
1.35
10
12
"r!" " 1
"r!" " 1
1.3
8
t!b
t!b
|!r| = 1
1.1
1.25
1.2
1
bE
bE
1.15
I, sl
0.9
1.1
1.05
0.8
2
4
6
8
t!b
10
12
2
4
6
8
t!b
10
12
14
0,0 1,0
0,1 1,1
Effective mass plots
"r!" " 4
1.2
"r!" " 4
1.2
1.15
1.15
bE
bE
1.1
1.1
1.05
1.05
2
4
6
8
10
2
12
4
6
"r!" " 4
1.2
8
10
12
t!b
t!b
"r!" " 4
|!r| = 4
1.2
1.15
1.15
bE
bE
1.1
I, sl
1.1
1.05
1.05
2
4
6
8
t!b
10
12
2
4
6
8
t!b
10
12
0,0 1,0
0,1 1,1
Effective mass plots
"r!" "#2
1.15
"r!" "#2
1.25
1.2
1.1
1.15
1.05
b E 1.1
bE
1
1.05
0.95
1
0.9
0.95
2
4
6
8
t!b
10
12
14
2
4
6
8
10
12
t!b
"r!" "#2
1.25
0.9
"r!" "#2
1.25
|!r| =
1.2
1.2
1.15
b E 1.15
b E 1.1
I, sl
1.05
1.1
1
1.05
0.95
2
4
6
8
t!b
10
12
0.9
√
2
4
6
8
t!b
10
12
0,0 1,0
0,1 1,1
2
Lattice V = EBB - 2EB
200
200
0
0
V0,0
"MeV#
V1,0
"MeV#
"200
"400
"400
"600
"600
0
2
4
!r!!
6
8
200
0
2
4
!r!!
6
8
0
2
4
!r!!
6
8
200
0
0
V0,1
"MeV#
"200
V1,1
"MeV#
"200
"200
"400
"400
"600
"600
0
2
4
!r!!
6
8
Coulomb potential
• OGE Coulomb potential: modified by a
lattice, add continuum
• Subtract
!"
"
(α)
δVf ;3
α(µ)
=
3π 2 a
iq·r
e
3
d q
−
|q|2
iq·r
e
BZ
(f )
G00 (q̂x , q̂y , q̂z , 0)d3 q
• BLM improve, FV effects
• Shift by 150 MeV at |r|=1 (∞ at r=0)
#
Coulomb potential
• OGE Coulomb potential: modified by a
lattice, add continuum
• Subtract
!"
"
(α)
δVf ;3
α(µ)
=
3π 2 a
iq·r
e
3
d q
−
|q|2
iq·r
e
BZ
(f )
G00 (q̂x , q̂y , q̂z , 0)d3 q
• BLM improve, FV effects
• Shift by 150 MeV at |r|=1 (∞ at r=0)
#
“Continuum” potentials
200
V0,0
"MeV#
200
0
V1,0
"MeV#
"200
0
2
4
!r!!
6
8
200
"MeV#
"200
"400
"400
V0,1
0
0
2
4
!r!!
6
8
0
2
4
!r!!
6
8
200
0
V1,1
"MeV#
"200
"400
0
"200
"400
0
2
4
!r!!
6
8
NB: residual O(a, α2(a)/a) effects
Michael & Pennanen ‘99
a=0.18 fm
163x24
20 cfgs
Figure 7: Results for the binding energy between two B mesons with light
Figure 8: Results for the binding energy between two B mesons with light
quarks in (Iq , Sq )=(0,0) at separation R in units of R0 ≈ 0.5fm. The light
quarks in (Iq , Sq )=(1,0) at separation R in units of R0 ≈ 0.5fm. The light
quark mass used corresponds to strange quarks. Results from variational
quark mass used corresponds to strange quarks. Results from variational
method using basis from t 4:3 and effective mass in that basis from t 6:5.method using basis from t 4:3 and effective mass in that basis from t 6:5.
28
29
Figure 6: Results for the binding energy between two B mesons with light
Figure 9: Results for the binding energy between two B mesons with light
quarks in (Iq , Sq )=(1,1) at separation R in units of R0 ≈ 0.5fm. The light
quarks in (Iq , Sq )=(0,1) at separation R in units of R0 ≈ 0.5fm. The light
quark mass used corresponds to strange quarks. Results from variational
quark mass used corresponds to strange quarks. Results from variational
method using basis from t 4:3 and effective mass in that basis from t 6:5.method using basis from t 4:3 and effective mass in that basis from t 6:5.
[Richards... ‘90, Mihály... ‘97, Stewart... ‘98, Fiebig... ‘02, Takahashi/Doi... ‘06]
Michael & Pennanen ‘99
BBbar pote
BB potential
DF: B=5.2, K=0.1395
DF: B=5.2, K=0.1
Q: B=5.7, K=0.14077
0.5
0.0
0.0
!0.4
!0.8
Iq,Sq=(0,0)
!0.5
R0EBinding
!1.2
0.0
!0.4
R0EBinding
Unquenched
a=0.14 fm
163x24
54 cfgs
Iq,Sq=(1,1)
!0.8
0.4
!1.5
!2.0
Iq,Sq=(0,1)
0.2
!1.0
0.0
!2.5
!0.2
0.4
Iq,Sq=(1,0)
0
1
Figure 3. B B̄ potentials.
a QQ̄ and a pion should
dynamical, 54 confs.
quenched, 20 confs.
0.2
0.0
0
1
R/R0
2
3
a state with particular qu
lowest energy at a particu
tion as a) a hybrid QQ̄ m
b) a ground state QQ̄ me
t-channel potentials
• Potentials with t-channel quantum numbers
VI,sl (|r|) = V1 + σ1 · σ2 Vσ + σ1 · σ2 τ1 · τ2 Vστ + τ1 · τ2 Vτ
1
(V0,0 − V0,1 − V1,0 + V1,1 )
Vστ =
16
1
(−V0,0 − 3V0,1 + V1,0 + 3V1,1 )
Vτ =
16
1
(V0,0 + 3V0,1 + 3V1,0 + 9V1,1 )
V1 =
16
1
(−V0,0 + V0,1 − 3V1,0 + 3V1,1 )
Vσ =
16
t-channel potentials
• Potentials with t-channel quantum numbers
VI,sl (|r|) = V1 + σ1 · σ2 Vσ + σ1 · σ2 τ1 · τ2 Vστ + τ1 · τ2 Vτ
1
(V0,0 − V0,1 − V1,0 + V1,1 )
Vστ =
16
1
(−V0,0 − 3V0,1 + V1,0 + 3V1,1 )
Vτ =
16
1
(V0,0 + 3V0,1 + 3V1,0 + 9V1,1 )
V1 =
16
1
(−V0,0 + V0,1 − 3V1,0 + 3V1,1 )
Vσ =
16
t-channel potentials
Vlatt$"L#
Σ
$MeV%
0
"25
"50
"75
"100
"125
Vlatt$"L#
ΣΤ
$MeV%
0
Vlatt$"L#
Τ
$MeV%
2
4
!r!!
6
8
0
"25
"50
"75
"100
"125
Vlatt$"L#
1
$MeV%
0
2
4
!r!!
6
0
"25
"50
"75
"100
"125
8
0
2
4
!r!!
6
8
0
2
4
!r!!
6
8
0
"50
"100
"150
"200
"250
t-channel potentials
Vlatt$"L#
Σ
$MeV%
0
"25
"50
"75
"100
"125
Vlatt$"L#
ΣΤ
η’
0
Vlatt$"L#
Τ
$MeV%
$MeV%
2
4
!r!!
6
0
"25
"50
"75
"100
"125
$MeV%
ρ
2
4
!r!!
6
8
π
0
8
Vlatt$"L#
1
0
0
"25
"50
"75
"100
"125
2
4
!r!!
6
0
"50
"100
"150
"200
"250
8
σ
0
2
4
!r!!
6
8
t-channel potentials
0
V"L#
Σ
$MeV%
0
V"L#
ΣΤ
"40
$MeV%
η’
"80
0
2
4
!r!!
6
$MeV%
π
"80
0
8
0
V"L#
Τ
"40
2
4
!r!!
6
8
0
"40
V"L#
1
$MeV%
"40
ρ
"80
0
2
4
!r!!
6
σ
"80
"120
8
0
Coulomb corrected
2
4
!r!!
6
8
Periodic lattice
Periodic lattice
Periodic lattice
Periodic lattice
Finite volume effects
• Periodicity of lattice modifies potentials:
V (L) (r) = V (r) +
!
V (r + nL)
n!=0
assuming single particle exchange
• Strictly: V from impossible
• Long range potential from EFT
• Short range: FV effects smallest
(L)
V
Periodic potential
1.2
1
0.8
V!r,L" 0.6
0.4
0.2
0
0
2
4
r
6
8
Periodic potential
1.2
1
0.8
V!r,L" 0.6
0.4
0.2
0
0
2
4
r
6
8
Periodic potential
1.2
1
0.8
V!r,L" 0.6
0.4
0.2
0
0
2
4
r
6
8
Periodic potential
1.2
1
0.8
V!r,L" 0.6
0.4
0.2
0
0
2
4
r
6
8
Infinite volume: Vστ
10
FV
0
|r|≥5
!
FV A
∞V
VΣΤ
"MeV# "10
Fit 1
−mπ |r|
e
|r|
"
Fit 2
"20
0
2.5
5
7.5
!r!!
10
12.5
15
|r|≥2
!
"
e−Λχ |r|
+FV B
|r|
Infinite volume: Vτ
Fit 1
FV
20
|r|>5
!
10
VΤ
"MeV#
"
e−mρ |r|
FV Â
|r|
0
∞V
"10
"20
Fit 2
"30
|r|>3!
0
2.5
5
7.5
!r!!
10
12.5
15
"
−Λχ |r|
e
+FV B̂
|r|
Infinite volume
• BBπ/BBρ couplings extracted
gBB ∗ π = 0.63 ± 0.05 ± 0.06
•g
BB ∗ π
gρ = 2.6 ± 0.1 ± 0.4
consistent with direct calculations
0.42(4)(8), 0.69(18), 0.48(3)(11), 0.517(16)
• Other channels problematic
• Quenching effects
• Noise: vacuum quantum numbers
[‘98, ’02, ’03, ’06]
Summary
• BB potentials from lattice QCD
• t-channel potentials measured cleanly
• Leading lattice spacing artefacts removed
• Infinite volume extraction attempted
• Future improvements
• Unquenched
• Multiple volumes/lattice spacings/m
• Heavy baryon potentials
q