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

LATTICE GAUGE THEORY FOR BEGINNERS

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.

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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.

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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).

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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