lattice qcd and the search for new physics
TRANSCRIPT
Lattice QCD and the search for new physicsfrom the perspective of the Budapest-Marseille-Wuppertal
collaboration (BMWc)
Laurent Lellouch
CPT Marseille
Excuses to ALPHA, CLS, ETMC, JLQCD, MILC, PACS-CS, QCDSF,RBC/UKQCD, . . . , for not having the time to cover some of their very nice
work
Labex OCEVU
lavinet
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
The place of low-energy searches for New Physics
Standard model: SU(3)c × SU(2)L × U(1)Y is an effective theory
LSM = LG&M(Aµ, ψ, φ) + LH&Y(φ, ψ, v) +1MO(5)
Maj +∑d≥6
∑i
Cd,i
Λd−4i
O(d)i
LG&M: 3 couplings; well tested
LH&Y: ≥ 15 couplings ; less well tested ⊃ 10 quark Yukawas→ flavor physics
Naturalness: δm2H ∼
3GF√2π2 m2
t Λ2 ' (0.3Λ)2
−→ Λ <∼ few TeV
Constraints on flavor conserving ops from precision EW data−→ Λ >∼ 5 TeV
FCNC: e.g. K 0−K 0 mixing−→ Λ ∼ 103 TeV
If there are no RH ν’s−→ M ∼ 1015 GeV
⇒ low-energy processes can be sensitive to very high-energy scales⇒ if quarks are involved, have to deal w/ confinement
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Motivation for lattice QCD
Verify that QCD is theory of strong interaction at low energies↔ verify the validity of the computational framework
→ light hadron masses→ hadron widths→ look for exotics→ . . .
Fix fundamental parameters and help search for new physics
→ mu, md , ms, . . .→ 〈N|mq qq|N〉, q = u,d , s for dark matter→ FK/Fπ ↔ Gq
Gµ
[|Vud |2 + |Vus|2 + |Vub|2
]= 1 ?
→ BK ↔ consistency of CPV in K and B decays ?→ . . .
Make predictions in nuclear physics?Full description of low energy particle physics→ include QED
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Why do we need lattice QCD?
QCD fundamental d.o.f.: q and gQCD observed d.o.f.: p, n, π, K , . . .
q and g are permanently confined w/in hadronshadrons hugely different from d.o.f. present in the Lagrangian
⇒ perturbation in αs has no chance⇒ Need a tool to solve low energy QCD to address important
questions above→ numerical lattice QCD
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
What is lattice QCD?
Lattice gauge theory −→ mathematically sound definition of NP QCD:
UV (and IR) cutoffs and a well defined pathintegral in Euclidean spacetime:
〈O〉 =
∫DUDψDψ e−SG−
∫ψD[M]ψ O[U, ψ, ψ]
=
∫DU e−SG det(D[M]) O[U]Wick
DUe−SG det(D[M]) ≥ 0 and finite # of dof’s→ evaluate numerically using stochasticmethods
rrrrrrrrrr
rrrrrrrrrr
rrrrrrrrrr
rrrrrrrrrr
rrrrrrrrrr
rrrrrrrrrr
rrrrrrrrrr
rrrrrrrrrr
-6�?
6
?
T
-�L
?6a
Uµ(x) = eiagAµ(x) ψ(x)
NOT A MODEL: LQCD is QCD when a→ 0, V →∞ and stats→∞
In practice, limitations . . .
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Ingredients of my dream calculation
Nf = 2 + 1 simulations to include u, d and s sea quark effectsSimulations all the way down to Mπ <∼ 135 MeV to allow smallinterpolation to physical mass pointLarge L >∼ 5 fm to have sub-percent finite V errorsAt least three a <∼ 0.1 fm for controlled continuum limitReliable determination of the scale w/ a well measured physicalobservableUnitary, local gauge and fermion actionsFull nonperturbative renormalization and nonperturbativecontinuum running if necessaryComplete analysis of systematic uncertainties
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Huge challenge
# of d.o.f. ∼ O(109) and large overhead for computing det(D[M]) (∼ 109 × 109 matrix)
+ Capri 1989: LQCD will require “both a 108 increase in computing power AND spectacularalgorithmic advances before a useful interaction with experiments starts taking place”⇒ 1−10 Exaflop/s = 106−107 Tflop/s→ nevertheless quenched LQCD provided useful interaction
+ Berlin wall ca. 2001: beginning of unquenched calculations
cost ∼ Nconf(L3 × T )cL m−cmud a−ca
with cm ∼ 2.5−3, ca ∼ 7, cL ∼ 5/4 and large prefactor (Gottlieb ’02, Ukawa ’02)
⇒ Mπ <∼ 300 MeV and a <∼ 0.1 fm completely out of reach
+ >∼ 2004:
algorithmic breakthroughs: DD-HMC (Lüscher ’03-’04), multiple time scale integrationand mass preconditioning (Sexton et al ’92, Hasenbusch ’01, Urbach et al ’06, BMWc ’07), . . .more effective unitary discretizations of QCD which allow mud ↘ mph
ud (BMWc ’07 using
Morningstar et al ’04 & Hasenfratz et al ’01)
⇒ cm ↘ 1−2, ca ↘ 4−6, cL ↘ 1−5/4 and much reduced prefactor (e.g. Jansen ’08)
+ >∼ 2008: supercomputers capable of 102−103 Tflop/s
⇒ calculations w/ full control over errors are becoming possible
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Not far from dream in 2008
Dürr et al (BMWc) Science 322 ’08, PRD79 ’09
20 large scale Nf = 2 + 1 Wilson fermion simulations
Mπ >∼ 190 MeV a ≈ 0.065,0.085,0.125 fm L→ 4 fm
00 2002
3002
4002
5002
6002
Mπ
2 [MeV
2]
0
0.0502
0.1002
0.1502
a2 [
fm2]
BMWc ’08
1002
2002
3002
4002
5002
6002
Mπ
2 [MeV
2]
2
3
4
5
6
7
L[f
m]
0.1%
BMWc ’08
0.3%
1%
Good enough for ab initio calculation of light hadron masses, . . .
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Postdiction of the light hadron masses
0
500
1000
1500
2000M
[MeV
]
p
K
rK* N
LSXD
S*X*O
experiment
width
input
Budapest-Marseille-Wuppertal collaboration
QCD
(Partial calculations by MILC ’04-’09, RBC-UKQCD ’07, Del Debbio et al ’07, JLQCD ’07, QCDSF ’07-’09, Walker-Loud et al ’08,
PACS-CS ’08-’10, ETM ’09, Gattringer et al ’09, . . .)
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Dream comes true in 2010
Dürr et al (BMWc) PLB 701 (2011); JHEP 1108 (2011)
47 large scale Nf = 2 + 1 Wilson fermion simulations
Mπ >∼ 120 MeV 5a’s ≈ 0.054÷ 0.116 fm L→ 6 fm
00 2002
3002
4002
5002
6002
Mπ
2 [MeV
2]
0
0.0502
0.1002
0.1502
a2 [
fm2]
BMWc ’08BMWc ’10
1002
2002
3002
4002
5002
6002
Mπ
2 [MeV
2]
2
3
4
5
6
7
L[f
m]
0.1%
BMWc ’08BMWc ’10
0.3%
1%
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
State of the art in 2011
Comparison of parameters reached by different LQCD collaborations
00 2002
3002
4002
Mπ
2 [MeV
2]
0
0.0502
0.1002
0.1502
a2 [
fm2]
MILC ’10JLQCD/TW. ’09
QCDSF/UK. ’10
PACS-CS ’09RBC/UK. ’10HSC ’08ETMC ’10BMWc ’08BMWc ’10
1002
2002
3002
4002
Mπ
2 [MeV
2]
2
3
4
5
6
7
L[f
m]
0.1%
MILC ’10JLQCD/TW. ’09
QCDSF/UK. ’10
PACS-CS ’09RBC/UK. ’10HSC ’08ETMC ’10BMWc ’08BMWc ’10
0.3%
1%
(only Nf ≥ 2 + 1 simulations are shown)
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Direct WIMP dark matter detectionχ χ
N N
H
Lqχ =∑
q
λq qqχχ −→ LNχ = λNNNχχ
→: requires nonperturbative tool
Spin-independent WIMP-N cross section (Ellis et al ’08)
σSI =4M2
r
π[Zfp + (A− Z )fn]2 Mr = MχMN/(Mχ + MN)
where
fNMN
=∑
q=[ud ],s
fqNλq
mq+
227
fGN
∑q=c,b,t
λq
mqfGN = 1−
∑q=[ud ],s
fqN
and, in terms of sigma terms,
fudNMN = σπN = mud〈N|uu + dd |N〉 fsNMN = σssN/2 = ms〈N|ss|N〉
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Sigma terms from phenomenology
σπN = mud〈N|uu + dd |N〉 σssN = 2ms〈N|ss|N〉yN = 2〈N|ss|N〉/〈N|uu + dd |N〉
Besides DM, important for: hadron spectrum, ms/mud , πN and KN scattering,counting rates in Higgs search . . .
Not measured directly in experiment⇒ history of phenomenological determinations from relation to Born-subtracted,
I = 0, πN amplitude at Cheng-Dashen point, Σ (Brown et al ’71)
Using Gasser et al ’91 and Bernard et al ’96 to get σπN , then Borasoy et al ’97 for yN
and ms/mud = 24.4(1.5) (Leutwyler ’96) for fsN , find
Canonical result
Σ = 59(2) MeV [3%](Gasser et al ’88)
→ σπN = 44(3) MeV [6%]→ fudN = 0.047(3) [7%]→ yN = 0.18(17) [95%]→ fsN = 0.10(10) [100%]
← differ by > 3σ →
← differ by ×1.5→
← differ by ×4→
More recent
Σ = 81(6) MeV [7%](Hite et al ’05)
→ σπN = 66(6) MeV [10%]→ fudN = 0.070(7) [10%]→ yN = 0.45(12) [26%]→ fsN = 0.39(11) [29%]
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Sigma terms from the latticeσπN = mud〈N|uu + dd |N〉 σssN = 2ms〈N|ss|N〉
Matrix element (ME) method
straightforward challenging
Spectrum method: Feynman-Hellmann (FH) theorem, X = N, Λ, Σ, Ξ
σπX = mud∂MX
∂mud−→ M2
π∂MX
∂M2π
σssX = 2ms∂MX
∂ms−→ 2M2
ss∂MX
∂M2ss
Pros: - Can use hadron spectrum result (e.g. from BMWc ’08)
- Can use whole octet and symmetry constraints
Con: - Model dependence when no lattice results at Mphπ or symmetry
constraints not checked w/ LQCD
−→ fix w/ e.g. BMWc ’10 simulations at Mphπ
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Combined analysis of baryon octet
Dürr et al PRD85 ’12 w/ BMWc ’08 simulations
σπX ' M2π∂MX
∂M2π
σssX ' 2M2ss∂MX
∂M2ss
Example CBχPT fits
MX = M0 − 4cπX M2π − 4cssX M2
ss +∑
α=π,K ,η
gαX
F2α
M30 h
(MαM0
)+ 4dπM4
π + 4dssM4ss
h(x) = −x3
4π2
√
1−( x
2
)2arccos
x
2+
x
2log x
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
MX [G
eV
]
Mπ
2 [GeV
2]
NΛ
Σ
Ξ
Fit 0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
0.45 0.5 0.55 0.6 0.65 0.7
MX [G
eV
]
2MK2 - M
π
2 [GeV
2]
NΛ
Σ
Ξ
Fit
Mπ < 410 MeV, χ2/dof = 39/34
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Error analysis and results864 distinct analyses of octet spectrum corresponding to different choices for:a↘ 0, Mπ ↘ 135 MeV, . . .
Weigh each one by fit quality→ systematic error distribution
repeat for 2000 bootstrap samples
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
-10 0 10 20 30 40 50 60 70 80 90σ
πN
χPTTaylorPade
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
-20 0 20 40 60 80 100 120σ
πN
M < 410.0 MeV
M < 550.0 MeV
σπX [MeV] σssX [MeV] yX fudX fsXN 39(4)(+18
−7 ) 67(27)(+55−47) 0.20(7)(+13
−17) 0.042(5)(+21−4 ) 0.036(14)(+30
−25)
Λ 29(3)(+11−5 ) 180(26)(+48
−77) 0.51(15)(+48−27) 0.027(3)(+5
−10) 0.083(12)(+23−31)
Σ 23(3)(+19−3 ) 245(29)(+50
−72) 0.82(21)(+87−39) 0.019(3)(+17
−3 ) 0.104(12)(+23−31)
Ξ 16(2)(+8−3) 312(32)(+72
−77) 1.7(5)(+1.9−0.7) 0.0116(18)(+59
−22) 0.120(13)(+30−30)
Significant improvements are possible w/ BMWc ’10 simulations at Mphπ
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Comparison and impact
0.02
0.02
0.04
0.04
0.06
0.06
0.08
0.08
fudN
GLS 88
HKJ 05
JLQCD 08, FH
YT 09, XPT+LQCD
BMWc 11, FH
QCDSF 11, FH
Bali et al 11, FH
Pheno.N
f=2
Nf=2+1
0
0
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
fsN
MILC 10, FH
GLS 88
HKJ 05
JLQCD 10, ME
YT 09, XPT+LQCD
BMWc 11, FH
QCDSF 11, FH
ETM 12, ME (Nf=2+1+1)
Bali et al 11, ME
Pheno.N
f=2
Nf=2+1
[caveat: most calculations have incomplete systematic error analysis]
2 4 6 8 10 12 14 16 18 20
mχ
10 (GeV)
1e−43
1e−42
1e−41
1e−40
1e−39
σS
I
P(c
m2) XENON−100
CDMS−II
XENON−10
DAMA
DAMA (with channeling)
CoGeNT
CDMS−09 fit
NMSSM upper limit
Impact of lattice results on DM search
fudN fsN
dashed-red: 0.078 0.63red: 0.059 0.29BMWc ’11: 0.042(5)(+21
−4 ) 0.036(14)(+30−25)
⇒ σSI(lattice) < σSI(red)
e.g. Das et al ’10
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Light quark masses: motivation
Determine mu, md , ms ab initio
Fundamental parameters of the standard modelPrecise values→ stability of matter, N-N scattering lengths,presence or absence of strong CP violation, etc.Information about BSM: theory of fermion masses must reproducethese valuesNonperturbative (NP) computation is requiredWould be needle in a haystack problem if not for χSB
⇒ interesting first “measurement” w/ physical point LQCD
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Light quark masses circa Aug. 2010FLAG→ analysis of unquenched lattice determinations of light quark masses(arXiv:1011.4408v1)
2
2
3
3
4
4
5
5
6
6
MeV
CP-PACS 01-03JLQCD 02
SPQcdR 05QCDSF/UKQCD 04
QCDSF/UKQCD 06ETM 07RBC 07
HPQCD/MILC/UKQCD 04HPQCD 05MILC 07CP-PACS 07RBC/UKQCD 08PACS-CS 08
FLAG 10v1 Nf = 2
Dominguez 09
mud
Nf=
2+
1N
f=2
Narison 06Maltman 01
PDG 09
MILC 09MILC 09AHPQCD 09PACS-CS 09Blum 10
JLQCD/TWQCD 08A
FLAG 10v1 Nf = 2+1
mMSud (2 GeV) =
{3.4(4) MeV [12%] FLAG3.0÷ 4.8 MeV [25%] PDG
60
60
80
80
100
100
120
120
140
140
MeV
CP-PACS 01-03JLQCD 02
ALPHA 05SPQcdR 05
QCDSF/UKQCD 04
QCDSF/UKQCD 06ETM 07RBC 07
HPQCD/MILC/UKQCD 04HPQCD 05MILC 07CP-PACS 07RBC/UKQCD 08PACS-CS 08
FLAG 10v1 Nf = 2
PDG 09
ms
Nf=
2+
1N
f=2
Dominguez 09Chetyrkin 06Jamin 06Narison 06Vainshtein 78
MILC 09MILC 09AHPQCD 09PACS-CS 09Blum 10
FLAG 10v1 Nf = 2+1
mMSs (2 GeV) =
{95.(10) MeV [11%] FLAG80÷ 130 MeV [25%] PDG
Even extensive study by MILC still has:
MRMSπ ≥ 260 MeV ⇒ mMILC,eff
ud ≥ 3.7×mphysud
perturbative renormalization (albeit 2 loops)
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Light quark masses: overall strategy
Dürr et al (BMWc) PLB 701 (2011); JHEP 1108 (2011)
BMWc ’10 set: 47 large scale Nf = 2 + 1 simulations w/ Mπ >∼ 120 MeV,5a’s ≈ 0.054÷ 0.116 fm and L→ 6 fm
improved definition of quark masses
full nonperturbative renormalization in RI/MOM scheme w/ improvements⇒ 21 additional Nf = 3 simulations at same 5a’s
full nonperturbative running up to high µ and four-loop conversion to RGI⇒ masses in other schemes w/ only errors proper to that scheme
renormalized mud (µ) and ms(µ) fitted as fn of (M2π, 2M2
K −M2π, a, L)
→ fully controlled V →∞ extrapolation→ fully controlled interpolation to mph
ud and mphs
→ fully controlled a→ 0 extrapolation
use phenomenology to determine individual mu and md
extensive systematic and statistical error analysis: 288 full analyses on 2000bootstrap samples
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Combined mass interp. and continuum extrap. (1)
Project onto mud axis: chiral interpolation to Mphπ
Illustration of chiral behavior (4/47 simulations)• Fixed a'0.09 fm and Mπ∼130÷ 410 MeV
0.005 0.01
amud
PCAC
2.4
2.5
2.6
2.7
2.8
aM
π2/m
ud
PC
AC
131 MeV
258 MeV368 MeV
409 MeV
• Fit to NLO SU(2) χPT (Gasser et al ’84)
0.005 0.01
amudPCAC
0.04
0.045
0.05
0.055
aFπ
3 Consistent w/ NLO χPT for Mπ <∼ 410 MeV
⇒ 2 safe interpolation ranges: Mπ < 340, 380 MeV
⇒ SU(2) NLO χPT & Taylor interpolations to physical point
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Combined mass interp. and continuum extrap. (2)
Project onto a axis: continuum extrapolation
Leading order is O(αsa)
Allow also domination of sub-leading O(a2)a=
0.0
6 f
m
a=
0.0
8 f
m
a=
0.1
0 f
m
a=
0.1
2 f
m
0
1
2
3
4
mu
d
RI (4
GeV
) [M
eV
]
0 0.01 0.02 0.03 0.04α
sa[fm]
a=
0.0
6 f
m
a=
0.0
8 f
m
a=
0.1
0 f
m
a=
0.1
2 f
m
0
40
80
120
msR
I (4G
eV
) [M
eV
]
0 0.01 0.02 0.03 0.04α
sa[fm]
(continuum extrapolation examples – errors on points are statistical)
⇒ fully controlled continuum limitLaurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Individual mu and md
Calculation performed in isospin limit:
mu = md NO QED⇒ leave ab initio realm
Use dispersive Q from η → πππ
Q2 ≡m2
s −m2ud
m2d −m2
u
* Precise mud and ms/mud ⇒
mu/d = mud
{1∓ 1
4Q2
[(ms
mud
)2
− 1
]}
Use conservative Q = 22.3(8) (Leutwyler ’09)
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Results
RI 4 GeV RGI MS 2 GeVms 96.4(1.1)(1.5) 127.3(1.5)(1.9) 95.5(1.1)(1.5)mud 3.503(48)(49) 4.624(63)(64) 3.469(47)(48)mu 2.17(4)(3)(10) 2.86(5)(4)(13) 2.15(4)(3)(9)md 4.84(7)(7)(10) 6.39(9)(9)(13) 4.79(7)(7)(9)
ms
mud= 27.53(20)(8)
mu
md= 0.449(6)(2)(29)
Additional consistency checks
3 Additional continuum, chiraland FV terms+ all compatible with 0
3 Unweighted final result andsystematic error+ negligible impact
3 Use mPCAC only+ compatible, slightly largererror
3 Full quenched check ofprocedure + cf. referencecomputation (Garden et al ’00)
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Comparison
2
2
3
3
4
4
5
5
6
6
MeV
CP-PACS 01JLQCD 02
SPQcdR 05QCDSF/UKQCD 04
QCDSF/UKQCD 06ETM 07RBC 07
MILC 04, HPQCD/MILC/UKQCD 04HPQCD 05MILC 07CP-PACS/JLQCD 07RBC/UKQCD 08PACS-CS 08
FLAG 10v1 Nf = 2
Dominguez 09
mud
Nf=
2+
1N
f=2
Narison 06Maltman 01
PDG 10
MILC 09MILC 09AHPQCD 09PACS-CS 09Blum 10
JLQCD/TWQCD 08A
FLAG 10v1 Nf = 2+1
HPQCD 10RBC/UKQCD 10A
ETM 10B
MILC 10APACS-CS 10BMWc 10A
70
70
80
80
90
90
100
100
110
110
120
120
MeV
CP-PACS 01JLQCD 02
ALPHA 05SPQcdR 05
QCDSF/UKQCD 04
QCDSF/UKQCD 06ETM 07RBC 07
HPQCD/MILC/UKQCD 04HPQCD 05MILC 07CP-PACS/JLQCD 07RBC/UKQCD 08PACS-CS 08
FLAG 10v1 Nf = 2
PDG 10
ms
Nf=
2+
1N
f=2
Dominguez 09Chetyrkin 06Jamin 06Narison 06Vainshtein 78
MILC 09MILC 09AHPQCD 09PACS-CS 09Blum 10
FLAG 10v1 Nf = 2+1
HPQCD 10RBC/UKQCD 10A
ETM 10B
PACS-CS 10BMWc 10A
mud and ms are now known to 2%, ms/mud to 0.7%. . . mu to 5% and md to 3% w/ help of phenomenology
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Quark flavor mixing constraints in the SM and beyond
Test SM paradigm of quark flavor mixing and CP violation and look for new physics
Unitary CKM matrix
V
u
b W
ub
d s b
→ V =
u
c
t
1− λ
2 λ Aλ3(ρ− iη)
−λ 1− λ2 Aλ2
Aλ3(1− ρ− iη) −Aλ2 1
+O(λ4)
Test CKM unitarity/quark-lepton universality and constrain NP using, e.g.
1st row unitarity:G2
q
G2µ
|Vud |2[1 + |Vus/Vud |2 + |Vub/Vud |2
]= 1 + O
(M2
W
Λ2NP
)
Unitarity triangle:G2
q
G2µ
(Vcd V ∗cb)
[1 +
Vud V ∗ub
Vcd V ∗cb+
Vud V ∗ub
Vcd V ∗cb
]= O
(M2
W
Λ2NP
)
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
CKM matrix and lattice QCD
CKM process LQCD precision (%)|Vus| K → `ν fK ≤ 1.5
K → π`ν f Kπ+ (0) 1.0
|Vus|/|Vud | K → µν/π → µν fK/fπ ≤ 1.5|Vcd | D → `ν fDs/fD 4.0
D → π`ν f Dπ+ (0) ∼ 10
|Vcs| Ds → `ν fDs 2.5D → K `ν f DK
+ (0) ∼ 7|Vub| B → `ν fB 6.0
fBs/fB 4.0B → π`ν f Bπ
+ (q2) ∼ 15|Vcb| B → D(∗)`ν FB→D(∗) (1) 4.0(ρ, η) ε BK ≤ 1.5
ε′ K → ππ
|V ∗tbVtq | ∆md BBs/BBd , ξ ∼ 3.0∆ms BBs ∼ 6.0
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
CPV and the K 0-K 0 system
Two neutral kaon flavor eigenstates: K 0(ds) & K 0(sd)
In experiment, have predominantly:
K 0S → ππ ⇒ K 0
S ∼ K1, the CP even combination
K 0L → πππ ⇒ K 0
L ∼ K2, the CP odd combination
However, CP violation⇒ K 0L → ππ
Experimentally:(PDG ’11)
∆MK ≡ MKL −MKS = 3.483(6)× 10−12 MeV [0.2%]
∆ΓK ≡ ΓKS − ΓKL = 7.339(4)× 10−12 MeV [0.05%]
|ε| = 2.228(11) · 10−3 [0.5%]
Re(ε′/ε) = 1.65(26) · 10−3 [16%]
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
K 0-K 0 mixing in the SM
u, c, t W W
su, c, ts dd d
s
u, c, t
Wd
sdsWs d u, c, t
M12−i2
Γ12 =〈K 0|H∆S=2
eff |K 0〉2MK
− i2MK
∫d4x〈K 0|T {H∆S=1
eff (x)H∆S=1eff (0)}|K 0〉︸ ︷︷ ︸
gives Γ12 and LD contributions to M12
+O(G3F )
Neglecting long-distance (LD) corrections
2MK M∗12SD= 〈K 0|H∆S=2
eff |K 0〉 = CSM1 (µ)〈K 0|O1(µ)|K 0〉
O1 = (sd)V−A(sd)V−A 〈K 0|O1(µ)|K 0〉 =163
M2K F 2
K BK (µ)
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Indirect CPV in K → ππ
Parametrized by
Re ε = ReT [KL → (ππ)0]
T [KS → (ππ)0]= cosφε sinφε
[ImM12
2 ReM12− ImΓ12
2 ReΓ12
]w/ φε = tan−1 (2∆MK/∆ΓK ) = 43.51(5)o, ∆MK ' 2 Re M12, ∆ΓK ' −2 Re Γ12
−→ ε = κεeiφε√
2ImM12
∆MK
w/ κε = 0.94(2) (Buras et al. (2010))
To NLO in αs
|ε| = κεCε Imλt {Reλc [η1Scc
−η3Sct ]− Reλtη2Stt} BK
∝ A2λ6η [cst + cst
×A2λ4(1− ρ)]
BK
w/ λq = Vqd V ∗qs
γ
α
α
dm∆
Kε
Kεsm∆ & dm∆
ubV
βsin 2(excl. at CL > 0.95)
< 0βsol. w/ cos 2
α
βγ
ρ
0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0
η
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
exclu
ded
are
a h
as
CL >
0.9
5
Winter 12
CKMf i t t e r
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Overall strategy for BK
Dürr et al [BMWc], PLB705 ’11
BMWc ’10 set: 47 large scale Nf = 2 + 1 simulations w/ Mπ >∼ 120 MeV,5a’s ≈ 0.054÷ 0.116 fm and L→ 6 fm
bare Q1,...,5 computed on Nf = 2 + 1 ensembles
full nonperturbative renormalization and mixing in RI/MOM scheme w/improvements⇒ 21 additional Nf = 3 simulations at same 5a’s
full nonperturbative running up to high µ and two-loop conversion to RGI⇒ BK in other schemes w/ small conversion errors
renormalized BK (µ) fitted as fn of (M2π, 2M2
K −M2π, a, L)
→ fully controlled V →∞ extrapolation (using Becirevic et al. ’04)
→ fully controlled interpolation to mphud and mph
s
→ fully controlled a→ 0 extrapolation w/ 4 smallest a’s
extensive systematic and statistical error analysis: 5760 full analyses on 2000bootstrap samples
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Combined chiral interp. and continuum extrap. (1)
Project onto M2π and M2
ss axes: interpolation to Mπ = 134.8(3) MeV andMK = 494.2(5) MeV
→ nearly flat mud -dependence near mphud
→ much steeper ms-dependence near mphs
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Combined chiral interp. and continuum extrap. (2)Project onto a axis: continuum extrapolation
→ mild extrapolation for β ≥ 3.5
→ β ≥ 3.31 was found to be outside scaling regime
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Systematic and statistical error2 time-fit ranges for π and K masses
2 time-fit ranges for Q1,...,5
O(αsa) or O(a2) for running
3 intermediate renormalization scales
2 fit fns and 4 ranges in p2 for ∆1i
5 fit fns for mass interpolation
2 pion mass cuts (Mπ < 340, 380 MeV)
BKRI
(3.5 GeV)
0.51 0.52 0.53 0.54 0.55
BKRI
(3.5 GeV)
0.51 0.52 0.53 0.54 0.55
→ 5760 analyses, each of which is a reasonablechoice, weighted by fit quality
→ median and central 68% give central value andsystematic uncertainty
2000 bootstraps of the median give statisticalerror
Many cross checks
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Results
Procedure gives BRIK (3.5 GeV) fully nonperturbatively
Can convert to other schemes w/ perturbation theory (PT)→ perturbative uncertainty
conv. RGI MS-NDR 2 GeV4-loop β, 1-loop γ 1.427 1.0474-loop β, 2-loop γ 1.457 1.062
ratio 1.021 1.01376
Take blanket 1% for 3 loop uncertainty
RI @ 3.5 GeV RGI MS-NDR @ 2 GeVBK 0.5308(56)(23) 0.7727(81)(34)(77) 0.5644(59)(25)(56)
Total error 1.1-1.5%, statistical (and PT) dominated
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
BK comparison
BK
CKM Fitter (LP 2011)
this work (2011, 2HEX-CIW)
SWME (2011, HYP-STAG/MILC)
Laiho et al. (2011, DW/MILC)
Aubin et al. (2010, DW/MILC)
RBC-UKQCD (2010, DW)
ETMC (2009, TM)
0.6 0.8 1 1.2
Dominant error in CKMfitter global fit results is |Vcb|4 ∼ (Aλ2)4
→ our result is an encouragement to reduce uncertainties in otherparts of the calculation of ε
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
∆S = 2 processes beyond the SM
e.g. gluino mediated FCNC in mass insertion approximation (Ciuchini et al ’99)
+ · · · OPE−→
O1 = [sd ]V−A[sd ]V−A
O2,3 = [sd ]S−P [sd ]S−P (unmix,mix)
O4,5 = [sd ]S−P [sd ]S+P (unmix,mix)
〈K 0|H∆S=2eff,BSM|K 0〉 =
5∑i=1
CBSMi (µ)Qi (µ) =
5∑i=1
CBSMi (µ)〈K 0|Oi (µ)|K 0〉
CBSMi (µ) short distance coefficients ⊃ flavor mixing parameters of BSM model
Qi (µ) are chirally enhanced
Qi (µ)
Q1(µ)∼ 〈qq〉(µ)
F 2π(ms + mud )(µ)
∼ 10, i = 2, · · · , 5
⇒ ε and ∆MK impose strong constraints on Im and Re of BSM parameters
only 2 old quenched calculations of Qi (µ), i = 2, · · · , 5 (Donini et al ’99, Babich et al ’06)
⇒ modern calculation required
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Conclusion
After > 30 years we are finally able to perform fully controlled LQCDcomputations all the way down to Mπ ≤ 135 MeV
Presented fully controlled results for sigma terms and for light quark masses andBK w/ σtot < 2%
There Nf = 2 + 1 LQCD calculations of many observables relevant for particlephenomenology, some of which have fully controlled errors, see e.g.http://itpwiki.unibe.ch/flag (Colangelo et al EPJC71 ’11)
http://www.latticeaverages.org (Laiho et al PRD81 ’10)
⇒ LQCD has become an indispensable tool in the search for new physics
2 4 6 8 10 12 14 16 18 20
mχ
10 (GeV)
1e−43
1e−42
1e−41
1e−40
1e−39
σS
I
P(c
m2) XENON−100
CDMS−II
XENON−10
DAMA
DAMA (with channeling)
CoGeNT
CDMS−09 fit
NMSSM upper limit
σSI(lattice) < σSI(red)
γ
α
α
dm∆
Kε
Kεsm∆ & dm∆
ubV
βsin 2(excl. at CL > 0.95)
< 0βsol. w/ cos 2
α
βγ
ρ
0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0
η
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
exclu
ded
are
a h
as
CL >
0.9
5
Winter 12
CKMf i t t e r
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012
Conclusion
Lattice QCD is undergoing a major shift in paradigm
it is now possible to control and reliably quantify all systematic errors with“data” (for processes w/ at most 1 initial and/or final hadron state)
⇒ we are getting QCD NOT LQCD predictionsrequires numerous simulations with Mπ < 200 MeV and preferably↘ 135 MeV, more than 3 a < 0.1 fm and lattice sizes L→ 4÷ 6 fmrequires trying all reasonable analyses of “data” and combining results insensible way to obtain a reliable systematic error
A LQCD result is only as good as its systematic error
Calculations with anything less can yield very interesting results which are not,however, QCD only predictions
Expect many more very interesting nonperturbative QCD predictions in comingyears
Laurent Lellouch RPP 2012, Montpellier, 14-16 May 2012