Isospin breaking corrections to leptonic decay rates on the lattice - - PowerPoint PPT Presentation

Isospin breaking corrections to leptonic decay rates on the lattice

James Richings

RBC/UKQCD Internal Seminar 2018

11/12/2018

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 1 / 22

RBC/UKQCD Collaboration

BNL and BNL/RBRC Yasumichi Aoki (KEK) Mattia Bruno Taku Izubuchi Yong-Chull Jang Chulwoo Jung Christoph Lehner Meifeng Lin Aaron Meyer Hiroshi Ohki Shigemi Ohta (KEK) Amarjit Soni UC Boulder Oliver Witzel Columbia University Ziyuan Bai Norman Christ Duo Guo Christopher Kelly Bob Mawhinney Masaaki Tomii Jiqun Tu Bigeng Wang Tianle Wang Evan Wickenden Yidi Zhao University of Connecticut Tom Blum Dan Hoying (BNL) Luchang Jin (RBRC) Cheng Tu Edinburgh University Peter Boyle Guido Cossu Luigi Del Debbio Tadeusz Janowski Richard Kenway Julia Kettle Fionn Ó hÓgáin Brian Pendleton Antonin Portelli Tobias Tsang Azusa Yamaguchi KEK Julien Frison University of Liverpool Nicolas Garron MIT David Murphy Peking University Xu Feng University of Southampton Jonathan Flynn Vera Gülpers James Harrison Andreas Jüttner James Richings Chris Sachrajda Stony Brook University Jun-Sik Yoo Sergey Syritsyn (RBRC) York University (Toronto) Renwick Hudspith

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 2 / 22

Outline

Lattice QCD introduction Isospin breaking effects Isospin breaking corrections to the Pion Isospin breaking corrections to leptonic decay rates Sea quark effects

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 3 / 22

Lattice QCD Introduction

Lattice QCD

Path integral:

O = 1 Z

• D[ ¯

ψ, ψ, U]OeiS

Free fermion action:

SF =

• d4x
• f

¯ ψf (iγµ∂µ − mf )ψf

Lattice naive free fermion action:

SF = a4

n∈Λ

• f

¯ ψf (n)

• γµ

ψ(n + ˆ µ) − ψ(n − ˆ µ) 2a − mψf (n)

• Fermion action:

SF = a4

n∈Λ

• f

¯ ψf (n)

• γµ

ψ(n + ˆ µ)Uµ(n) − U−µ(n)ψ(n − ˆ µ) 2a − mψf (n)

• James Richings

(Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 4 / 22

Lattice QCD Introduction

Lattice QCD: gauge fields and the path integral

Gauge links:

Uµ(x) = exp(iqaAµ(x))

Gauge action:

Uµν(x) = exp(−iqa2Fµν) Sg = 2 g 2

• x
• µ≤ν

Tr

• 1 − 1

2[Uµν(x) + U†

µν(x)]

• U†

ν(x)

U†

µ(x + aˆ

ν) Uµ(x) Uν(x + aˆ µ) x

Euclidean path integral:

< O >= 1 Z

• D[U, ψ, ¯

ψ]e−SLat[U,ψ, ¯

ψ]O[ψ, ¯

ψ] < O >=

• D[U]p(U)O(U)

, p[U] = 1 Z

f

det(Df [U])

• e−Sg

< O >= lim

N→∞

1 N

N

• n=1

O(Un) + O(N−1/2)

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 5 / 22

Lattice QCD Introduction

Example Calculation: Pion mass

Pion correlator:

γ5 γ5 < 0|Oπ+(x)O†

π+(y)|0 >

= 0| ¯ d(x)γ5u(x)[ ¯ d(y)γ5u(y)]† |0 = 0| ¯ d(x)γ5u(x)¯ u(y)γ5d(y) |0 = −tr[γ5Sd(x, y)γ5Su(y, x)] < 0|Oπ(t)Oπ(0)†|0 > = 1 Z

• D[U, ψ, ¯

ψ]

• tr[γ5S(x, 0)γ5S(0, x)]
• e−S[U,ψ, ¯

ψ]

< 0|Oπ(t)Oπ(0)†|0 > =

• n

< 0|Oπ(t)|n >< n|Oπ(0)†|0 > =

• n

| < 0|Oπ(0)|n > |2e−Ent = |Aπ|2e−mπt(1 + O(e−∆Et))

Correlator:

Cπ = |Aπ|2(e−mπt + e−mπ(T−t))

Effective mass:

meff = ln

• Cπ[t]

Cπ[t + 1]

• James Richings

(Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 6 / 22

Lattice QCD Introduction

The propagator

Inverting the Dirac operator:

( / D + m)ψ = η [ / D]size = (483

space × 96time × 12spin/colour

× 2complex × 8double)2

The propagator is the sum over paths of link variables.

Low modes:

Sf = [ / Df + m]−1 = [ / Df + m −

• λ

λ |ψλ ¯ ψλ|]−1 +

• λ

λ−1 |ψλ ¯ ψλ|

Deflation:

Using the low modes of the Dirac

• perator it is possible to form a better

initial guess of the propagator.

100 200 300 400 500 600

Number of low modes

500 1000 1500 2000 2500 3000

Number of CG iterations

m=0.005 m=0.01

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 7 / 22

Lattice QCD Introduction

Lattice QCD: Calculations

Spectral quantities:

u ¯ d γ5 γ5

Leptonic decays:

u ¯ d W + π+ ν l+

1.14 1.18 1.22 1.26 = + + = + =

Semi-leptonic decays:

s ¯ d u k0 ¯ ν π+ l−

0.95 0.97 0.99 1.01 = + + = + = non-lattice

+( )

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 8 / 22

Lattice QCD Introduction

CKM matrix: Precision of Lattice QCD

Unitarity constraints on the CKM ma- trix are an important bound on the BSM theories. Lattice QCD is now at the precision of one percent in the light sector:

0.22 0.23 = + + = + =

| |

0.973 0.975 = + + = + =

• r

• r

| |

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 9 / 22

Isospin breaking corrections

Isospin breaking corrections

Two sources of isospin breaking (IB) corrections:

QED isospin breaking corrections:

• Difference in the electromagnetic charge on the up and down type quarks.

Strong isospin breaking corrections:

• Difference in the up and down quark masses.

How to include these IB effects?

• These effects by power counting are of order 1 %:

α ≈ 1/137 ≈ 1% , (mu − md) ΛQCD ≈ 1%

• This give us an expansion parameter for each contribution.

What makes this difficult?

• Small effect
• QED difficult
• Expensive

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 10 / 22

Isospin breaking corrections

Lattice QFT with Isospin breaking corrections

Both of the IB corrections can be calculated in terms of a pertubative expansion:

QED Isospin breaking:

• Order alpha correction can be calculated by using a perturbative approach:

O = O0 + e2 2 ∂2 ∂e2 O

• e=0

+ O(α2)

• If the operator O is α independent then the correction has the form:

O = O0 − e2qf qf ′ 2 OV c

µ(x)V c ν (y)0 ∆µν(x − y) − (eqf )2

2 OTµ(x)0 ∆µµ

[G.M.de Divitiis et al.Phys.Rev.D87(2013)114505] Strong Isospin breaking at first order:

Omu=md = Omu=md+(md−mu) ∂ ∂m O

• mu=md

= Omu=md+(md−mu) SO

• mu=md

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 11 / 22

Isospin breaking corrections

Example Calculation: IB Corrections to the Mπ

The connected corrections are the following four diagrams: QED IB:

γ5 γ5 V µ

c

V ν

c

γ5 γ5 V µ

c

V ν

c

γ5 γ5 T µ

Strong IB:

γ5 γ5 S

The correlator has the form:

Cπ = (A + δA)(e−(m+δm)t + e−(m+δm)(T−t))

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 12 / 22

Isospin breaking corrections

Example Calculation: IB Corrections to Mπ

Exchange diagram (Preliminary)

Self energy diagram (Preliminary)

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 13 / 22

Isospin breaking corrections

Leptonic decays

u ¯ d W + π+ ν l+

The decay rate is:

Γ(π+ → l+ν) = mπ 8π G 2

F|fπ+|2|Vud|2m2 l

• 1 − m2

l

m2

π

2 0| ¯ dγµγ5u |π+(p) = ipµfπ+

IR finite order α decay rate:

Γα = Γ0 + Γ1

[Bloch and Nordsieck (1937)] [N.Carrasco et al, Phys.Rev.D91(2015)no.7,07450] − Γ0: Order alpha corrections without a final state photon. − Γ1: Order alpha corrections with a final state photon.

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 14 / 22

Isospin breaking corrections

IB corrections to leptonic decay rates

The connected diagrams that contribute to the order α QED correction to

leptonic decays: π, K ν l

Lepton coupling diagram

π, K ν l

Exchange diagram

π, K ν l

Self-energy diagram

π, K ν l

Tadpole diagram

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 15 / 22

Isospin breaking corrections

Initial results: QED IB corrections

π π

−0.001 −0.0005 0.0005 0.001 0.0015 0.002 16 24 32 Tr[pν / Cql(t)(pl / − ml)Γ0

L]/Cπ(t)

t

Preliminary

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 16 / 22

Isospin breaking corrections

Sea quark: IB corrections to Mπ

γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 17 / 22

Isospin breaking corrections

Sea quarks: IB corrections to leptonic deacys

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 18 / 22

Isospin breaking corrections

Conclusion

We are working towards a calculation of leptonic decay rates with isospin

breaking corrections.

Our implementation phase is concluding and a number of tests have been

undertaken on ensembles with unphysical quark masses.

Next phase is move to our physical point ensemble and begin generating data

with a view towards calculating a physical result.

In the future we want to study IB corrections to semi-leptonic decays.

However some theoretical developments are required.

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 19 / 22

Isospin breaking corrections

Back up Slides

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 20 / 22

Isospin breaking corrections

All to All Propagator and the meson field

All to all propagator:

D−1

A2A(x, y) = Nl+Nh

• i=0

vi(x)w †

i (y) = Nl

• l=0

vl(x)w †

l (y) + Nl+Nh

• h=Nl

vh(x)w †

h(y) Low modes (from eigenvectors):

vl(x) = φl(x) wl(x) = φl(x)/λl

High modes (from stochastic solves):

vh(x) = D−1ηh(x) wh(x) = ηh(x)

Two point function:

vi(x) vj(y) w †

i (y)

w †

j (x)

γ5 γ5

Meson Field:

Πji(tx; γ5) =

• x

w †

j (x)γ5vi(x) [J.Foley et al, CPC 172 (2005)0010-4655] [M.Peardon et al,Phys.Rev.D.80.054506(2009)] 3,4,... pt functions can be made contracting the relevant meson fields with

the correct gamma structure.

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 21 / 22

Isospin breaking corrections

All to All: Connected leptonic decay diagrams

Leptonic coupling correlator using meson fields:

Πij(t; γ5) Πik(t; V c

µ)

Πjk(t; γ0γ5)

Similarly the other diagrams that contribute to the decay rate can be split

into meson fields:

James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 22 / 22