SLIDE 1
Determination of B0
s and B0 d mixing
parameters using lattice QCD
Elvira G´ amiz
In collaboration with: Christine T.H. Davies, G. Peter Lepage, Junko Shigemitsu, Howard Trottier and Matthew Wingate HPQCD Collaboration Madison, 11 May 2009
SLIDE 2
- 1. New Physics effects on B0 − ¯
B0 mixing
B0
q
¯ B0
q
W W u, c, t u, c, t q b b q B0
q
¯ B0
q
u, c, t u, c, t W W q b b q
- B0 mixing parameters determined by the off diagonal elements of the
mixing matrix i d dt |Bs/d(t) | ¯ Bs/d(t) =
2 Γs/d |Bs/d(t) | ¯ Bs/d(t) ∆Ms/d ∝ |Ms/d
12 |
∆Γs/d ∝ |Γs/d
12 |
SLIDE 3
- 1. New Physics effects on B0 − ¯
B0 mixing
B0
q
¯ B0
q
W W u, c, t u, c, t q b b q B0
q
¯ B0
q
u, c, t u, c, t W W q b b q
- B0 mixing parameters determined by the off diagonal elements of the
mixing matrix i d dt |Bs/d(t) | ¯ Bs/d(t) =
2 Γs/d |Bs/d(t) | ¯ Bs/d(t) ∆Ms/d ∝ |Ms/d
12 |
∆Γs/d ∝ |Γs/d
12 |
New physics can significantly affect Ms/d
12
∝ ∆Ms/d * Γ12 dominated by CKM-favoured b → c¯ cs tree-level decays.
SLIDE 4
# Hints of discrepancies between SM expectations and some flavour observables (see, for example, E. Lunghi, talk at BEACH08) * sin(2β) E. Lunghi and A. Soni, PLB 666 (2008) 162 * B0
s mixing phase Nierste and Lenz, JHEP 0706, UTfit coll., arXiv:0803.0659
* CP violating effects B0
d − ¯
B0
d, Buras and Guadagnoli, PRD78(2008)033005
SLIDE 5 # Hints of discrepancies between SM expectations and some flavour observables (see, for example, E. Lunghi, talk at BEACH08) * sin(2β) E. Lunghi and A. Soni, PLB 666 (2008) 162 * B0
s mixing phase Nierste and Lenz, JHEP 0706, UTfit coll., arXiv:0803.0659
* CP violating effects B0
d − ¯
B0
d, Buras and Guadagnoli, PRD78(2008)033005
# These analyses depend on several theoretical inputs: Vcb, Vub, ˆ BK and the SU(3) breaking mixing parameter ξ:
Vts
∆MsMBd * Comparison of ∆M and ∆Γ with experiment also provides bounds for NP effects
SLIDE 6 # Hints of discrepancies between SM expectations and some flavour observables (see, for example, E. Lunghi, talk at BEACH08) * sin(2β) E. Lunghi and A. Soni, PLB 666 (2008) 162 * B0
s mixing phase Nierste and Lenz, JHEP 0706, UTfit coll., arXiv:0803.0659
* CP violating effects B0
d − ¯
B0
d, Buras and Guadagnoli, PRD78(2008)033005
# These analyses depend on several theoretical inputs: Vcb, Vub, ˆ BK and the SU(3) breaking mixing parameter ξ:
Vts
∆MsMBd * Comparison of ∆M and ∆Γ with experiment also provides bounds for NP effects Improvement in B0 − ¯ B0 mixing parameters which enter on those analyses is crucial.
SLIDE 7
- 2. Mixing parameters in the Standard Model
# In the Standard Model
B0 ¯ B0 W W H∆B=2
eff
∆Mq|theor. = G2
F M2 W
6π2
|V ∗
tqVtb|2ηB 2 S0(xt)MBsf2 Bq ˆ
BBq
SLIDE 8
- 2. Mixing parameters in the Standard Model
# In the Standard Model
B0 ¯ B0 W W H∆B=2
eff
∆Mq|theor. = G2
F M2 W
6π2
|V ∗
tqVtb|2ηB 2 S0(xt)MBsf2 Bq ˆ
BBq * Non-perturbative input
8 3 f2 BqBBq(µ)M2 Bq = ¯
B0
q|OL|B0 q(µ) with
OL ≡ [bi qi]V −A[bj qj]V −A In terms of decay constants and bag parameters ξ = fBs
fBd BBd * Many uncertainties in the theoretical (lattice) determination cancel totally or partially in the ratio = ⇒ very accurate calculation
SLIDE 9
- 3. Some details of the lattice formulations
and simulations
Unquenched: Fully incorporate vacuum polarization effects MILC Nsea
f
= 2 + 1 u,d,s Asqtad action: improved staggered quarks = ⇒ errors O(a2αs), O(a4) * good chiral properties * accessible dynamical simulations b NRQCD: Non-relativistic QCD improved through O(1/M2), O(a2) and leading relativistic O(1/M3) * Simpler and faster algorithms to calculate b propagator Improved gluon action * For further reduction of discretization errors
SLIDE 10
Parameters of the simulation
# Lattice spacing: Two different values a ≃ 0.12 fm, 0.09 fm. Extracted from Υ 2S-1S splitting. # Bottom mass: Fixed to its physical value from Υ mass.
SLIDE 11
Parameters of the simulation
# Lattice spacing: Two different values a ≃ 0.12 fm, 0.09 fm. Extracted from Υ 2S-1S splitting. # Bottom mass: Fixed to its physical value from Υ mass. # Light masses: We work with full QCD points (mvalence = msea). * Strange mass: Very close to its physical value (from Kaon masses). * up, down masses: six different values (mmin.
π
≃ 230MeV) → chiral regime
SLIDE 12 Parameters of the simulation
# Lattice spacing: Two different values a ≃ 0.12 fm, 0.09 fm. Extracted from Υ 2S-1S splitting. # Bottom mass: Fixed to its physical value from Υ mass. # Light masses: We work with full QCD points (mvalence = msea). * Strange mass: Very close to its physical value (from Kaon masses). * up, down masses: six different values (mmin.
π
≃ 230MeV) → chiral regime # Renormalization and matching to the continuum: One-loop. < OL >MS∝ (1 + ρLLαs) < OL >latt. +ρLLαs < OS >latt with OS = ¯ b(1 − γ5)q ¯ b(1 − γ5)q
SLIDE 13 # Need 3-point (for any ˆ Q = QX, Q1j
X ) and 2-point correlators
t = 0
ˆ Q ¯ B0 B0
C(4f)(t1, t2) =
x2
0|Φ ¯
Bq(
x1, t1)
Q
¯ Bq(
x2, −t2)|0 C(B)(t) =
0|Φ ¯
Bq(
x, t)Φ†
¯ Bq(
0, 0)|0
Bq(
x, t) = ¯ b( x, t)γ5q( x, t) is an interpolating operator for the B0
q meson.
We carried out simultaneous fits of the 3-point and 2-point correlators using bayesian statistics to the forms → extract < OX > and fBs(d).
SLIDE 14
Results for fBq MBqBBq
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
r1 mf
1.1 1.2 1.3 1.4 1.5 1.6
r1
(3/2) fBs (MBs BBs) (1/2)
Coarse Lattice Fine Lattice Physical point
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
r1 mq
0.9 1 1.1 1.2 1.3 1.4 1.5
r1
(3/2) fBd (MBd BBd) (1/2)
Coarse Lattice Fine Lattice Physical point
fBs
BBs = 266(6)(17)MeV fBd
BBd = 216(9)(12)MeV Chiral+continuum extrapolations: NLO Staggered CHPT. * accounts for NLO quark mass dependence. * accounts for light quark discretization effects through O
sa2Λ2 QCD
- → remove the dominant light discretization errors
SLIDE 15 Results for ξ MBs
MBd
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
r1 mq
0.9 1 1.1 1.2 1.3 1.4 1.5
(MBs / MBd)
1/2 ξ
Coarse Lattice Fine Lattice Continuum Physical point
ξ =
fBs
√
BBs fBd
√BBd = 1.258(25)(21) = ⇒
Vts
* Previous value used in UT fits and another analyses (HPQCD/JLQCD): ξ = 1.20 ± 0.06
SLIDE 16 Results for fBq MBq
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
r1 mf
1.1 1.2 1.3 1.4 1.5 1.6
r1
(3/2) fBs (MBs BBs) (1/2)
Coarse Lattice Fine Lattice Physical point
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
r1 mq
0.9 1 1.1 1.2 1.3 1.4 1.5
r1
(3/2) fBd (MBd BBd) (1/2)
Coarse Lattice Fine Lattice Physical point
fBs = 231(15)MeV fBd = 190(13)MeV
fBs fBd = 1.226(26)
* HPQCD previous numbers are fBs = 260(29)MeV , fBd = 216(22)MeV and
fBs/fBd = 1.20(3)(1).
** chiral extrapolation based only on coarse lattice, no continuum extrapolation.
SLIDE 17 Bag parameters: Calculation of Br(B → µ+µ−)
# Very interesting place to look for the effect of new operators in the effective Hamiltonian.
Hurth et al, NPB 808 (2009); Buras, arXiv:0904.4917.
* Scalar operators can enhance branching ratios to current experimental bounds.
SLIDE 18 Bag parameters: Calculation of Br(B → µ+µ−)
# Very interesting place to look for the effect of new operators in the effective Hamiltonian.
Hurth et al, NPB 808 (2009); Buras, arXiv:0904.4917.
* Scalar operators can enhance branching ratios to current experimental bounds. # The most precise way of extracting this branching ratio is from Br(Bq → µ+µ−) ∆Mq = τ(Bq) 6π ηY ηB
4πMW sin2θW 2 m2
µ
Y 2(xt) S(xt) 1 ˆ Bq * Using our value of ˆ Bs → Br(Bs → µ+µ−) = (3.2 ± 0.3) × 10−9 to be compared with
CDF bound Br(Bs → µ+µ−) ≤ 6 × 10−8
SLIDE 19 Bag parameters: Calculation of Br(B → µ+µ−)
# Very interesting place to look for the effect of new operators in the effective Hamiltonian.
Hurth et al, NPB 808 (2009); Buras, arXiv:0904.4917.
* Scalar operators can enhance branching ratios to current experimental bounds. # The most precise way of extracting this branching ratio is from Br(Bq → µ+µ−) ∆Mq = τ(Bq) 6π ηY ηB
4πMW sin2θW 2 m2
µ
Y 2(xt) S(xt) 1 ˆ Bq * Using our value of ˆ Bs → Br(Bs → µ+µ−) = (3.2 ± 0.3) × 10−9 to be compared with
CDF bound Br(Bs → µ+µ−) ≤ 6 × 10−8
* Similarly for ˆ Bd → Br(Bd → µ+µ−) = (0.98 ± 0.12) × 10−10 to be compared with
CDF bound Br(Bd → µ+µ−) ≤ 2 × 10−8
SLIDE 20
- 5. B0 mixing beyond the SM
# Comparison of experimental measurements and theoretical predictions can constraint some BSM parameters and help to understand BSM physics.
SLIDE 21
- 5. B0 mixing beyond the SM
# Comparison of experimental measurements and theoretical predictions can constraint some BSM parameters and help to understand BSM physics. # Effects of heavy new particles seen in the form of effective operators built with SM degrees of freedom
SLIDE 22
- 5. B0 mixing beyond the SM
# Comparison of experimental measurements and theoretical predictions can constraint some BSM parameters and help to understand BSM physics. # Effects of heavy new particles seen in the form of effective operators built with SM degrees of freedom # The most general Effective Hamiltonian describing ∆B = 2 processes is H∆B=2
eff
=
5
CiQi +
3
Qi with SM
Qq
1 =
ψi
bγν(I − γ5)ψi q
¯ ψj
bγν(I − γ5)ψj q
2 =
ψi
b(I − γ5)ψi q
¯ ψj
b(I − γ5)ψj q
3 =
ψi
b(I − γ5)ψj q
¯ ψj
b(I − γ5)ψi q
4 =
ψi
b(I − γ5)ψi q
¯ ψj
b(I + γ5)ψj q
5 =
ψi
b(I − γ5)ψj q
¯ ψj
b(I + γ5)ψi q
Qq
1,2,3 = Qq 1,2,3 with the replacement (I ± γ5)→(I ∓ γ5)
where ψb is a heavy b-fermion field and ψq a light (q = u, d) fermion field.
SLIDE 23
- 5. B0 mixing beyond the SM
# Comparison of experimental measurements and theoretical predictions can constraint some BSM parameters and help to understand BSM physics. # Effects of heavy new particles seen in the form of effective operators built with SM degrees of freedom # The most general Effective Hamiltonian describing ∆B = 2 processes is H∆B=2
eff
=
5
CiQi +
3
Qi with SM
Qq
1 =
ψi
bγν(I − γ5)ψi q
¯ ψj
bγν(I − γ5)ψj q
2 =
ψi
b(I − γ5)ψi q
¯ ψj
b(I − γ5)ψj q
3 =
ψi
b(I − γ5)ψj q
¯ ψj
b(I − γ5)ψi q
4 =
ψi
b(I − γ5)ψi q
¯ ψj
b(I + γ5)ψj q
5 =
ψi
b(I − γ5)ψj q
¯ ψj
b(I + γ5)ψi q
Qq
1,2,3 = Qq 1,2,3 with the replacement (I ± γ5)→(I ∓ γ5)
where ψb is a heavy b-fermion field and ψq a light (q = u, d) fermion field.
Ci Wilson coeff. calculated for a particular BSM theory
B0|Qi|B0 calculated on the lattice
SLIDE 24 # Some examples:
- F. Gabbiani et al, Nucl.Phys.B477 (1996), D. Be´
cirevi´ c et al, Nucl.Phys.B634 (2002); general SUSY models
- P. Ball and R. Fleischer, Eur.Phys.J. C48(2006); extra Z’ boson; SUSY
Help to constrain the soft SUSY breaking terms and the mechanism of SUSY breaking.
- M. Ciuchini and L. Silvestrini, PRL 97 (2006) 021803; SUSY
Constraints on the mass insertions (|Re(δd
23)RR| < 0.4, |(δd 23)LL| < 0.1,...)
- M. Blanke et al, JHEP 12(2006) 003; Little Higgs model with T-parity
∆Mq can be used to test viability of the model. To constrain and test the model in detail ∆Ms/∆Md and ∆Γq. Lunghi and Soni, 0707.0212; Top Two Higgs Doublet Model Constraints on βH (ratio of vev’s of the two Higgs) and mH+
- M. Blanke et al, 0809.1073; Warped Extra Dimensional Models
Constraints on the KK mass scale (it can be as low as MKK ≃ 3T eV )
SLIDE 25
- 6. Conclusions and outlook
# SM results for the B0
s and B0 d mixing parameters (∆M and ∆Γ)
* fB √BB with 7% error and ξ with 2.6% error. # SM results for decay constants, fBs with 6% error and fBd with 7%. Important tests of the SM are possible # Improvements of the analysis: More statistics, more (and smaller) lattice spacings, improved renormalization techniques, direct extraction
- f bag parameters from fits . . .
# Same accuracy can be achieved for the matrix elements in the general ∆B = 2 effective hamiltonian BSM. * The value of those matrix elements, together with experimental data, will help to constrain the parameter space in BSM theories
SLIDE 26
- 6. Conclusions and outlook
# SM results for the B0
s and B0 d mixing parameters (∆M and ∆Γ)
* fB √BB with 7% error and ξ with 2.6% error. # SM results for decay constants, fBs with 6% error and fBd with 7%. Important tests of the SM are possible # Improvements of the analysis: More statistics, more (and smaller) lattice spacings, improved renormalization techniques, direct extraction
- f bag parameters from fits . . .
# Same accuracy can be achieved for the matrix elements in the general ∆B = 2 effective hamiltonian BSM. * The value of those matrix elements, together with experimental data, will help to constrain the parameter space in BSM theories
SLIDE 27
- 6. Conclusions and outlook
# SM results for the B0
s and B0 d mixing parameters (∆M and ∆Γ)
* fB √BB with 7% error and ξ with 2.6% error. # SM results for decay constants, fBs with 6% error and fBd with 7%. Important tests of the SM are possible # Improvements of the analysis: More statistics, more (and smaller) lattice spacings, improved renormalization techniques, direct extraction
- f bag parameters from fits . . .
# Same accuracy can be achieved for the matrix elements in the general ∆B = 2 effective hamiltonian BSM. * The value of those matrix elements, together with experimental data, will help to constrain the parameter space in BSM theories
SLIDE 28
- 6. Conclusions and outlook
# SM results for the B0
s and B0 d mixing parameters (∆M and ∆Γ)
* fB √BB with 7% error and ξ with 2.6% error. # SM results for decay constants, fBs with 6% error and fBd with 7%. Important tests of the SM are possible # Improvements of the analysis: More statistics, more (and smaller) lattice spacings, improved renormalization techniques, direct extraction
- f bag parameters from fits . . .
# Same accuracy can be achieved for the matrix elements in the general ∆B = 2 effective hamiltonian BSM. * The value of those matrix elements, together with experimental data, will help to constrain the parameter space in BSM theories # Similar analysis from the FNAL/MILC will be completed soon.
SLIDE 29
×
SLIDE 30 Error budget: B0 mixing (in %) fBs
BBs fBd
BBd ξ Statistical + chiral extrapolation 2.3 4.1 2.0 Residual a2 3.0 2.0 0.3 r3/2
1
uncertainty 2.3 2.3
1.0 1.0 1.0 ms and mb tuning 1.5 1.0 1.0
4.0 4.0 0.7 relativistic corrections 2.5 2.5 0.4 Total 6.7 7.1 2.6
SLIDE 31 Error budget: Decay constants (in %) fBs fBd fBs/fBd Statistical + chiral extrapolation 2.2 3.5 1.6 Residual a2 3.0 3.0 0.5 r3/2
1
uncertainty 2.3 2.3
1.0 1.0 0.3 ms and mb tuning 1.5 1.0 1.0
4.0 4.0 0.7 relativistic corrections 1.0 1.0 0.2 Total 6.3 6.7 2.1
