genie september 1, 2016...lattice gauge theory k. wilson, prd 10 (1974) 2445 • invented to...
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CKM Matrix from Lattice QCD and the z Expansion
Andreas S. KronfeldFermilab & IAS TU München
GENIE z-expansion TutorialSeptember 1, 2016Fermilab
Outline
• Basics of Lattice QCD
• iz Expansions Used in B Physics
• iz Fits: Lattice QCD, Experiment, Combined
• Compare and Contrast ν and B Physics
2
The QCD Lagrangian
• SU(3) gauge symmetry and 1 + nf + 1 parameters:
• Observable CP violation ∝ θ = ϑ – arg det yf (if all Yukawas nonvanishing):
• neutron electric-dipole moment sets limit θ ≲ 10–11;
• bafflingly implausible cancellation called the strong CP problem.
LQCD =1g2
0tr[FµnFµn]
Âf
y f (/D+m f )y f
+iq
32p2 eµnrs tr[FµnFrs]
mΩ, Υ(2S-1S), or r1, w0,....
mπ, mK, mJ/ψ, mY, ....
θ = 0.
4
Lattice Gauge Theory K. Wilson, PRD 10 (1974) 2445
• Invented to understand asymptotic freedom without the need for gauge-fixing and ghosts [Wilson, hep-lat/0412043].
• Gauge symmetry on a spacetime lattice:
• mathematically rigorous definition of QCD functional integrals;
• enables theoretical tools of statistical mechanics in quantum field theory and provides a basis for constructive field theory.
• Discrete space (a la Heisenberg & Pauli) and discrete time (a la Feynman).
h•i = 1
Z
ZDUD D ¯ exp (S) [•]
5
Numerical Lattice QCD
• Nowadays “lattice QCD” usually implies a numerical technique, in which the functional integral is integrated numerically on a computer.
• Big computers:
• Some compromises:
• finite human lifetime ⇒ Wick rotate to Euclidean time: x4 = ix0;
• finite memory ⇒ finite space volume & finite time extent;
• finite CPU power ⇒ light quarks until recently heavier than up and down.
6
MC hand
Lattice Gauge Theory
•Infinite continuum: uncountably many d.o.f. (⇒ UV divergences);
•Infinite lattice: countably many; used to define QFT;
•Finite lattice: finite dimension ~ 108, so compute integrals numerically.
a
L = NSa
L 4 = N
4a
h•i = 1
Z
ZDUD D ¯ exp (S) [•]
=
1
Z
ZDU Det(D/+m) exp (Sgauge) [•0]
• Two-point functions for masses, , :
• Two-point functions for decay constants:
• Three-point functions for form factors, mixing:
• LHS needs supercomputers; RHS needs students, postdocs, junior faculty.
Correlators Yield Masses & Matrix Elements
h(t)†(0)i =
X
n
|h0||ni|2 exp(mnt)
hJ(t)†(0)i =
X
n
h0| ˆJ |nihn|†|0i exp(mnt)
h(t)J(u)B†(0)i =
X
mn
h0||mihn| ˆJ |BmihBm| ˆB†|0i
exp[mn(t u)mBmu]
8
p(t) = yug5Syd(t) N(t) = cyug5ydyd(t)
QCD Hadron Spectrumπ…Ω: BMW, MILC, PACS-CS, QCDSF; ETM (2+1+1);
η-ηʹ: RBC, UKQCD, Hadron Spectrum (ω); D, B: Fermilab, HPQCD, Mohler&Woloshyn
numerous quarkonium
omittedρ K K∗ η φ N Λ Σ Ξ ∆ Σ
∗Ξ∗ Ωπ η′ ω0
500
1000
1500
2000
2500
(MeV
)
D, B D*, B
*
D s,B s D s
* ,B s*
Bc Bc*
© 2012−2014 Andreas Kronfeld/Fermi Natl Accelerator Lab.
B mesons offset by −4000 MeV
9
Neutron-Proton Mass Difference BMW Collab., arXiv:1406.4088 (see also Horsley et al. arXiv:1508.06401)
0 1 2α/αphys
0
1
2
(md-m
u)/(m
d-mu) ph
ys
physical point
1 MeV
2 MeV
3 MeV
4 MeV
Inverse β decay region
Mn = Mp
: Mn < Mp + me
1H → n+νe
Form-Factor Calculations I
• Matrix elements decomposed into Lorentz-covariant forms, multiplied by “form factors”.
• Compute them from three-point functions, with several
• daughter momenta;
• lattice spacing;
• quark masses, physical volumes, etc.
• Fit lattice data to EFT formulas to get continuum limit.11
Form-Factor Calculations II
• Output of these fits in continuum and at physical quark masses is a set of fit parameters, errors, and their correlations for the EFT formula.
• Fit this information to the z expansion:
• “synthetic data”: evaluate fit function at # of points less than # of fit parameters;
• functional z expansion (here): finds best fit of second polynomial, akzk, to an initial one (EFT expression).
12
z Expansions Used in B Physics
Three Channels
• scattering νn → pl, νB → πl, …: q2 < 0.
• decay n → plν, B → πlν, …: 0 = q2 ≤ (Mparent – Mdaughter)2.
• s-channel, lν annihilation into pn, Bπ, etc.: s = q2 ≥ tcut, but in general also subthreshold poles:
• flavor-tag in Bπ implies tcut = (MB + Mπ)2, but note also pole at s = M2B* when J = 1;
• radius of convergence of q2 expansion is tpole or tcut.
14
Polology
• Physics looks exciting for annihilation kinematics,
• but dull for decay and scattering:
• all connected by analyticity.
-1500 -1000 -500 0 500 1000 1500 2000 2500-0.02
-0.01
0.
0.01
0.02
0.03
-1500 -1000 -500 0 500 1000 1500 2000 2500-0.02
-0.01
0.
0.01
0.02
0.03
JP = 1–
JP = 0+
Ep = (M2B +M2
p q2)/2MB
General z Expansion
• In general, write
• Unitarity constrains |f(t)|2:
f (t) =1
P(t)f(t)
•
Âk=0
akz(t, t0)k
poles expressed as “inner function” aka “Blaschke factor”
unitarity constraints managed with
“outer function”
1 •
Âk=0
|ak|2
f = mess
1 •
Âk=0
B jkajak, B jk = mess
f = 1
BCL:
BGL:
16
References
• Meiman, Sov. Phys. JETP 17 (1963) 830.
• Okubo and Shih, PRD 4 (1971) 2020.
• Singh and Raina, Fort. Phys. 27 (1979) 561.
• Boyd, Grinstein, and Lebed (BGL), hep-ph/9412324.
• Boyd and Savage, hep-ph/9702300.
• Bourrely, Caprini, and Lellouch (BCL), arXiv:0807.2722.
17
z Fits
Combining Lattice QCD with Experiment
• B → πlν, RBC/UKQCD, arXiv:1501.05373.
0.2
0.4
0.6
0.8
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
χ2/dof = 1.32, p = 7%
(1 -
q2 /mB*2
) f+
z
BABAR 2012 (untagged)BABAR 2010 (untagged)BELLE 2013 B0 (tagged)BELLE 2013 B- (tagged)BELLE 2010 (untagged)This work
103|Vub| = 3.61(32)
Combining Lattice QCD with Experiment
• B → πlν, Fermilab/MILC, arXiv:1503.07839.
0
0.2
0.4
0.6
0.8
1
1.2
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
(1-q
2 /MB*
2 )f +(
z)
z
All expt. Nz=3 tLattice Nz=4 t
BaBar untagged 6 bins (2011)Belle untagged 13 bins (2011)
BaBar untagged 12 bins (2012)Belle tagged B0 13 bins (2013)
Belle tagged B- 7 bins (2013)103|Vub| = 3.72(16)
Combining Lattice QCD with Experiment
• B → Dlν, Fermilab/MILC, arXiv:1503.07237.
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0 0.02 0.04 0.06
form
factor
z
p = 0.40
f+f0
BaBar 2009
103|Vcb| = 39.6(1.8)
Combining Lattice QCD with Experiment
• B → Dlν, HPQCD, arXiv:1505.03925.
103|Vcb| = 40.2(2.1)
Combining Lattice QCD with Experiment
• Λb → plν/Λb → Λclν, Detmold, Lehner, Meinel, arXiv:1503.01421.
• LHCb measures these rates (over some range):
|Vub|/|Vcb| = 0.083(4)(4)
arXiv:1504.01568.
• Quiz: what’s tcut here?
0 2 4 6 8 10
q2 (GeV2)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
d/d
q2
|Vcb|2
(ps
1G
eV
2)
b ! c µ µ
0 5 10 15 20
q2 (GeV2)
0.0
0.5
1.0
1.5
2.0
2.5
d/d
q2
|Vub|2
(ps
1G
eV
2)
b ! p µ µ
Synthesis of |Vub| & |Vcb| Calculations
24
• Experimental errors for B → D(*) will shrink with Belle 2.
• Other errors bars: QCD and expt comparable.
• Refs in backup.35 36 37 38 39 40 41 42 43 44 45
103|Vcb|
3.0
3.5
4.0
4.5
103 |V
ub|
© 2015 Andreas Kronfeld, Fermi National Accelerator Laboratory
|Vub|/|Vcb| (latQCD + LHCb)|Vub| (latQCD + BaBar + Belle)|Vcb| (latQCD + BaBar + Belle)|Vcb| (latQCD + HFAG, w = 1)p = 0.27∆χ
2 = 1∆χ
2 = 2inclusive |Vxb|
How Are Lattice QCD Results Reported?
• Curve and error band, described by
• the z formulas (t0, tcut, inner and outer functions);
• coefficients ak, their errors, and their correlation matrix;
• cross correlations, e.g., between FV and FA.
• Lattice-QCD community working on ways to easily and robustly combine calculations with each other, and with experimental data.
25
ν vs B
Similarities
• Same theoretical underpinnings:
• analyticity constraints on shape ⇒ z expansion with (more than) adequate radius of convergence;
• lattice gauge theory = first-principles tool for QCD.
• Similar interval in z (for optimal t0) for D decay and ν scattering—
• larger (smaller) range for B decay (K decay).
27
Differences
• Kinematic region—unimportant in practice (I think).
• Experiments measure
• form factor shape (essentially) directly in B physics;
• complicated nuclear process in ν physics.
• Meson form factors may have poles, a mild complication;
• nucleons don’t have this (thresholds at 3mπ, 2mπ).
28
Outlook
Further Calculations of Interestim
porta
nce
(to n
eutri
no p
hysic
s)
difficulty in lattice QCD
f K!p+ FNC
A
FNCPFCC
P
nN ! N0,D,Np, . . .
NN forces
pN forces
ss in N
FCCAf B!p
+
hNN|J|NNi
30
The Advocated Paradigm
• Replace Ansätze for nucleon-level physics with ab-initio QCD (i.e., continuum limit of lattice QCD).
• Use success of lattice QCD for meson form factors to bolster confidence in nucleon form factors.
• (With further checks from FV and gA, of course.)
31
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