Isospin-breaking corrections in Giusti Kaon decays on the lattice - - PowerPoint PPT Presentation

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Isospin-breaking corrections in Giusti Kaon decays on the lattice - - PowerPoint PPT Presentation

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Davide Isospin-breaking corrections in Giusti Kaon decays on the lattice OUTLINE Motivations Current and future QCD + QED on the lattice: RM123 method status of the first-row CKM unitarity Leptonic decays of hadrons Amherst, MA,

slide-1
SLIDE 1

Isospin-breaking corrections in Kaon decays on the lattice

In collaboration with:

  • V. Lubicz, G. Martinelli, C.T. Sachrajda, F. Sanfilippo,
  • S. Simula and N. Tantalo

Current and future status of the first-row CKM unitarity

Amherst, MA, USA

17th May 2019

Davide Giusti OUTLINE

▪ Motivations ▪ QCD + QED on the lattice: RM123 method ▪ Leptonic decays of hadrons ▪ Future perspectives K + → µ+νµ γ

( ) π + → µ+νµ γ ( )

Vus Vud

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SLIDE 2

Qu ≠ Qd : O(αem) ≈ 1/100 mu ≠ md : O[(md-mu)/ΛQCD] ≈ 1/100

“Electromagnetic” “Strong”

Isospin-breaking effects are induced by:

Since electromagnetic interactions renormalize quark masses the two corrections are intrinsically related

Isospin symmetry is an almost exact property

  • f the strong interactions

Though small, IB effects play often a very important role (quark masses, Mn - Mp, leptonic decay constants, vector form factor)

ISOSPIN-BREAKING EFFECTS

2

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SLIDE 3

Phenomenological motivations

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SLIDE 4

The determination of some hadronic

  • bservables in flavor physics has reached such

an accurate degree of experimental and theoretical precision that electromagnetic and strong isospin-breaking effects cannot be neglected anymore

Phenomenological motivations

4

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SLIDE 5

The relevant processes are leptonic and semileptonic K and π decays

The determination of Vus and Vud

5

K/π

Vus/Vud

K

π

Vus

Γ K + → ℓ+νℓ γ

( )

( )

Γ π + → ℓ+νℓ γ

( )

( )

= Vus Vud fK fπ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2 M K + 1− mℓ 2 M K + 2

( )

2

Mπ + 1− mℓ

2 Mπ + 2

( )

2 1+δ EM +δ SU 2

( )

( )

Γ K +,0 → π 0,−ℓ+νℓ γ

( )

( ) = GF

2M K +,0 5

192π 3 CK +,0

2

Vus f+

K 0π − 0

( )

2

IKℓ

( )SEW 1+δ EM

K +,0ℓ +δ SU 2

( )

K +,0π

( )

slide-6
SLIDE 6

Vus and Vud: experimental results

K/π K

π

V

u s

V

ud

fK fπ = 0.2760(4) V

u s f+ K 0π − (0) = 0.21654(41)

PDG

  • M. Moulson, arXiv:1704.04104

6

Γ K + → ℓ+νℓ γ

( )

( )

Γ π + → ℓ+νℓ γ

( )

( )

= Vus Vud fK fπ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2 M K + 1− mℓ 2 M K + 2

( )

2

Mπ + 1− mℓ

2 Mπ + 2

( )

2 1+δ EM +δ SU 2

( )

( )

Γ K +,0 → π 0,−ℓ+νℓ γ

( )

( ) = GF

2M K +,0 5

192π 3 CK +,0

2

Vus f+

K 0π − 0

( )

2

IKℓ

( )SEW 1+δ EM

K +,0ℓ +δ SU 2

( )

K +,0π

( )

Vus Vud fK fπ = 0.27599 38

( )

< 0.2%

Vus f+ 0

( ) = 0.21654 41 ( )

slide-7
SLIDE 7

Vus and Vud: results from lattice QCD

fK± / fπ± = 1.1932(19) Nf=2+1+1 fK± / fπ± = 1.1917(37) Nf=2+1

0.2%

f+(0) = 0.9706(27) Nf=2+1+1 f+(0) = 0.9677(27) Nf=2+1

0.3%

7

fK ± fπ ± = fK fπ 1+δ SU 2

( )

9

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SLIDE 8

Given the present exper. and theor. (LQCD) accuracy, an important source of uncertainty are long distance electromagnetic and SU(2)-breaking corrections ChPT is not applicable to D and B decays

M.Knecht et al., 2000; V.Cirigliano and H.Neufeld, 2011

δ EM = − 0.0069 (17)

At leading order in ChPT both δEM and δSU(2) can be expressed in terms of physical quantities (e.m. pion mass splitting, fK/fπ, …) 25% of error due to higher orders 0.2% on ΓKl2/Γπl2 For ΓKl2/Γπl2

J.Gasser and H.Leutwyler, 1985; V.Cirigliano and H.Neufeld, 2011

25% of error due to higher orders

0.1% on ΓKl2/Γπl2

Electromagnetic and isospin-breaking effects

K/π K

π

8

Γ K + → ℓ+νℓ γ

( )

( )

Γ π + → ℓ+νℓ γ

( )

( )

= Vus Vud fK fπ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2 M K + 1− mℓ 2 M K + 2

( )

2

Mπ + 1− mℓ

2 Mπ + 2

( )

2 1+δ EM +δ SU 2

( )

( )

Γ K +,0 → π 0,−ℓ+νℓ γ

( )

( ) = GF

2M K +,0 5

192π 3 CK +,0

2

Vus f+

K 0π − 0

( )

2

IKℓ

( )SEW 1+δ EM

K +,0ℓ +δ SU 2

( )

K +,0π

( )

δ SU 2

( ) =

fK + fπ + fK fπ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

−1= −0.0044 12

( )

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SLIDE 9

Given the present exper. and theor. (LQCD) accuracy, an important source of uncertainty are long distance electromagnetic and SU(2)-breaking corrections

Electromagnetic and isospin-breaking effects

K/π K

π

Γ K + → ℓ+νℓ γ

( )

( )

Γ π + → ℓ+νℓ γ

( )

( )

= Vus Vud fK fπ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2 M K + 1− mℓ 2 M K + 2

( )

2

Mπ + 1− mℓ

2 Mπ + 2

( )

2 1+δ EM +δ SU 2

( )

( )

Γ K +,0 → π 0,−ℓ+νℓ γ

( )

( ) = GF

2M K +,0 5

192π 3 CK +,0

2

Vus f+

K 0π − 0

( )

2

IKℓ

( )SEW 1+δ EM

K +,0ℓ +δ SU 2

( )

K +,0π

( )

  • M. Moulson, arXiv:1704.04104

15

For ΓKl3

9

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SLIDE 10

Unitarity of the CKM first-row

10

Vus Vud fK fπ = 0.27599 38

( )

Vus f+ 0

( ) = 0.21654 41 ( )

9

f+ 0

( )

fK ± fπ ±

Vu

2 = 0.99884 53

( )

Vu

2 = 0.99986 46

( )

≈ 2.2σ ≈ 0.4σ

  • J. Hardy and
  • I. S. Towner, 2016

Vud from

Vu

2 = 0.99778 44

( )

≈ 5σ Vu

2 = 0.99875 37

( ) ≈ 3.4σ

C.-Y. Seng et al., 2018

Vu

2 ≡ Vud 2 + Vus 2 + Vub 2

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SLIDE 11

Isospin-breaking effects on the lattice

RM123 method

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SLIDE 12

A strategy for Lattice QCD: The isospin-breaking part of the Lagrangian is treated as a perturbation Expand in:

arXiv:1110.6294

+

arXiv:1303.4896

RM123 Collaboration

αem md – mu

12

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SLIDE 13
  • Identify the isospin-breaking term in the QCD action

S

m =

m

uuu + m ddd

⎡ ⎣ ⎤ ⎦

x

∑

= 1 2 m

u + m d

( ) uu + dd

( )− 1

2 m

d − m u

( ) uu − dd

( )

⎡ ⎣ ⎢ ⎤ ⎦ ⎥

x

∑

= = m

ud uu + dd

( )− Δm uu − dd ( )

⎡ ⎣ ⎤ ⎦

x

∑

= S

0 − Δm

ˆ

S

  • Expand the functional integral in powers of Δm

O = Dφ O e

−S

0+Δm ˆ

S

∫

Dφ e

−S

0+Δm ˆ

S

∫

1st

!

Dφ O e

−S

0 1+ Δm ˆ

S

( )

∫

Dφ e

−S

0 1+ Δm ˆ

S

( )

∫

! O

0 + Δm O ˆ

S 1+ Δm ˆ S = O

0 + Δm O ˆ

S

  • At leading order in Δm the corrections only appear in the

valence-quark propagators:

(disconnected contractions of ūu and ƌd vanish due to isospin symmetry)

1 The (md-mu) expansion

Advantage: factorized out

13

Ŝ = Σx(ūu-ƌd)

for isospin symmetry

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SLIDE 14

SQED = 1 2 A

ν(x) −∇µ −∇µ +

( ) A

ν(x) x;µν

∑

=

( p.b.c.) 1

2  A

ν *(k) 2sin(kµ / 2)

( )

2 

A

ν(k) k;µν

∑

  • Non-compact QED: the dynamical variable is the gauge potential A(x)

in a fixed covariant gauge ( )

∇µ

− Aµ(x) = 0

  • The photon propagator is IR divergent subtract the zero momentum mode

2 The QED expansion!

+

  • Full covariant derivatives are defined introducing QED and QCD links:

Aµ(x) → Eµ(x) = e

−iaeAµ (x)

Dµ

+q f (x) = Eµ(x)

⎡ ⎣ ⎤ ⎦

ef Uµ(x) q f (x + ˆ

µ)− q f (x)

QED QCD

  • Since the expansion leads to:

Eµ(x) = e

−i e Aµ (x) = 1− i e Aµ(x) −1/ 2 e 2

Aµ

2(x) +…

+ counterterms 24

2

14

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SLIDE 15

The QED expansion for the quark propagator

In the electro-quenched approximation:

15

slide-16
SLIDE 16

All masses in MSbar at 2 GeV

The down- and up-quark mass difference

RM123 Collaboration, arXiv:1704.06561

md − mu = 2.38(18) MeV

0.01 0.02 0.03 0.04 0.05

ml (GeV)

2.4 2.6 2.8 3 3.2

[M

2 K

0 - M

2 K

+]

QCD/(md-mu) (GeV)

β=1.90, L/a=20 β=1.90, L/a=24 β=1.90, L/a=32 β=1.95, L/a=24 β=1.95, L/a=32 β=2.10, L/a=48 physical point

M2

K 0 - M2 K +

⎡ ⎣ ⎤ ⎦

QCD

md − mu = 2.51(18) GeV

QED QCD

Δmud

QED

MK+ − MK0 ⎡ ⎣ ⎤ ⎦

QCD

= −6.00(15) MeV

MK+ − MK0 ⎡ ⎣ ⎤ ⎦

QED

= 2.07(15) MeV and from the experimental value

mu = 2.50(17) MeV md = 4.88(20) MeV

electro-quenched approximation

16

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SLIDE 17

QED corrections to hadronic decays

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SLIDE 18

QED corrections to hadronic decays

In general the amplitudes are infrared divergent.

On the lattice, a natural infrared cutoff is provided by the finite volume. But a delicate procedure to remove it is needed. A method to solve this problem is presented

We consider the leptonic decay of a charged pseudoscalar meson, but the method is general

(it can be used for semileptonic decays)

  • N. Carrasco,
  • V. Lubicz, G. Martinelli, C.T. Sachrajda, N. Tantalo, C. Tarantino, M. Testa

PRD 91 (2015) 074506, arXiv:1502.00257

18

slide-19
SLIDE 19

The rate is:

In the absence of electromagnetism, the non-perturbative QCD effects are contained in a single number, the pseudoscalar decay constant

AP

( ) ≡ 0 q2γ 4γ 5q1 P ( ) = fP ( )M P ( )

K+ s u ℓ+ νℓ

K+ s u ℓ+ νℓ

Leptonic decays at tree level

Since the masses of the pion and kaon are much smaller than MW we use the effective Hamiltonian

This replacement is necessary in a lattice calculation, since

1/ a ≪ MW

q1 q2 ℓ+ νℓ W q1 q2 ℓ+ νℓ

Heff = GF 2 Vq1q2

*

q2γ µ 1−γ 5

( )q1

( ) νℓγ µ 1−γ 5

( )ℓ

( )

19

ΓP±

tree

( ) P± → ℓ±νℓ

( ) = GF

2

8π Vq1q2

2

fP

( )

⎡ ⎣ ⎤ ⎦

2 M P±mℓ 2 1− mℓ 2

M P±

2

⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

In the presence of electromagnetism it is not even possible to give a physical definition of fP

  • J. Gasser and G.R.S. Zarnauskas, PLB 693 (2010) 122
slide-20
SLIDE 20

!The!Fermi!constant!GF!is!conven3onally!defined!from!the!muon!life3me!

using!

!+!!diagrams!with!

the!real!photon!

S.M.Berman,!PR!112!(1958)!267;!!T.Kinoshita!and!A.Sirlin,!PR!113!(1959)!1652!

The!explicit!O(α)!correc3ons!are!those!obtained!in!the!effec3ve!theory!(UV!finite! for!muon!decay).!All!other!EW!correc3ons!are!absorbed!into!the!defini3on!of!GF! 1 τ µ = GF

2mµ 5

192π 3

1− 8me

2

mµ

2

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1+ α 2π 25 4 −π 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

Leptonic decays at O(α)!

When!including!the!O(α)!correc3ons,!the!UV!contribu3ons!in!the! effec3ve!theory!are!different!from!those!in!the!Standard!Model:!!!!!!! !!!!!!!!!A!MATCHING!BETWEEN!THE!TWO!THEORIES!IS!REQUIRED!

34

20

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SLIDE 21

The W-regularization!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!UV!finite.!Not!included!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!in!GF.!Equal!to!the! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!corresponding!diagram!in!!! the!effec3ve!theory!with!the!“Wjregulariza3on”!! (up!to!negligible!terms!of!O(q2/MW

2))!!

!A!convenient!way!to!separate!out!in!the!Standard!Model!the!tradi3onal!photonic! correc3ons!is!to!write!the!(Feynman!gauge)!photon!propagator!as:!

A.!Sirlin,!! RMP!50!(1978)!573,!! PRD!22!(1980)!971!

UV!divergent.!Absorbed!! in!the!defini3on!of!GF!together! with!the!other!EW!correc3ons!

1 k2 = 1 k2 − MW

2 +

MW

2

MW

2 − k2

1 k2

Also!the!γ-W!box!diagram,!which!is!finite!in!the!SM,!is!equal!to!the!corresponding! diagram!of!the!effec3ve!theory!with!Wjregulariza3on!(up!to!O(q2/MW

2))!

[!UV!finite!]!

Effec3ve!theory! !!!!!with!the!! Wjregulariza3on! 1 k2 → MW

2

MW

2 − k2

1 k2 Standard!Model!

=!

35

21

slide-22
SLIDE 22

The effective Hamiltonian at O(α)!

!!!The!Wjregulariza3on!provides!a!convenient!way!to!separate!out!the!tradi3onal! photonic!correc3ons.!For!muon!decay,!all!other!EW!correc3ons!are!absorbed!in!GF!! muon! decay! pion! decay!

Same!weak! isospin!charges!

Most!of!the!terms!which!are!absorbed!into!the!defini3on!of!GF!are!common! also!to!other!processes,!including!the!leptonic!decays!of!pseudoscalar!mesons! Some!shortjdistance!contribu3ons,!however,!do!depend!on!the!electric!charges!

  • f!the!fields!in!the!4jfermion!operator.!These!lead!to!the!effec3ve!Hamiltonian!

Heff = GF 2 Vud

* 1+ α

π

log M Z

MW ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ dγ µ(1−γ 5)u

( ) νℓγ µ(1−γ 5)ℓ

( )

A.Sirlin,!NP!B196!(1982)!83;!E.Braaten,!C.S.Li,!PRD!42!(1990)!3888!

This!factor!provides!the!matching!between!the!SM!and!the!local!Fermi!theory! Wjregulariza3on!

36

  • A. Sirlin, NPB 196 (1982) 83; E. Braaten and C.S. Li, PRD 42 (1990) 3888

22

slide-23
SLIDE 23

Matching the W- and lattice regularizations

23

The lattice 4-fermion operator is renormalized in RI’-MOM and then matched to the one in the W-regularization in perturbation theory

µ2 ∂ ∂µ2 + β α s,α

( ) ∂

∂α s ⎡ ⎣ ⎢ ⎤ ⎦ ⎥U RI' = γ α s,α

( )U RI'

O1

W-reg MW

( ) = 1+ α

4π 2 1− α s µ

( )

4π ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ln MW

2

µ2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + C W-RI' ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ O1

RI' µ

( )

The W-regularization cannot be implemented directly in lattice simulations since:

1 a ≪ MW

O1

W-reg MW

( ) = Z W-RI' α s MW ( ),α

( )U RI' MW ,µ,α

( )O1

RI' µ

( )

p p p p k

…

The evolution operator is solution of the renormalization group equation

C W-RI' = −5.7825 +1.2373ξ

At first order in and up to terms of , the result is

O αα s MW

( )

( )

α

slide-24
SLIDE 24

O1

RI' µ

( ) =

ZO

( )1i aµ

( )Oi

bare a

( )

i=1 5

∑

For Wilson and Twisted-Mass fermions

RI’-MOM in QCD+QED

24

ZΓO aµ

( )ΓO ap ( )

p2=µ2 = ˆ

1

δZO = −ZΓO

( )δΓO + 1

2 δZq1 +δZq2 +δZℓ

( )

+ [mf m0

f]

⌥ [mcr

f mcr 0 ]

The weak 4-fermion operator is renormalized non-perturbatively on the lattice to all orders in 𝛽s and up to first order in 𝛽 by imposing the RI’-MOM condition:

ZO = ˆ 1+ α 4π δ ZO ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ZO

( )

By decomposing the RCs as , it follows:

ΓO = Tr ΛOP

O

[ ]

ZΓO = ZO Z f

−1 2 f

∏

δS

slide-25
SLIDE 25

Both Γ0 and Γ1(ΔE) can be evaluated in a fully non-perturbative way in lattice simulations.

However, as a first approach to the the problem, we have considered a different strategy, applicable to both pion and kaon decays

Leptonic decays at O(α): the IR problem

At O(α), Γ0 contains infrared divergences. One has to consider:

F . Bloch and A. Nordsieck, PR 52 (1937) 54

with 0 ≤ Eγ ≤ ΔE . The sum is infrared finite

... + ...

Γ0

Γ1 ΔE

( )

25

K+ s u ℓ+ νℓ

K+ s u ℓ+ νℓ K+ s u ℓ+ νℓ

Γ P

ℓ2 ±

( ) = Γ P± → ℓ±νℓ ( )+ Γ P± → ℓ±νℓγ ΔE

( )

( ) ≡ Γ0 + Γ1 ΔE

( )

slide-26
SLIDE 26

The strategy

We propose to consider sufficiently soft photons (i.e. they do not resolve the internal structure of the pion (kaon)), so that the pointlike approximation can be used to compute Γ1(ΔE) in perturbation theory, but hard with respect to the experimental resolution

ΔE ∼ O(20 MeV)

F . Ambrosino et al., KLOE Collaboration, PLB 632 (2006) 76; EPJC 64 (2009) 627; 65 (2010) 703(E)

A cut-off appears to be appropriate, both experimentally and theoretically

ΔE ≪ ΛQCD

  • J. Bijnens et al., NPB 396 (1993) 81; V.Cirigliano

and I.Rosell, JHEP 0710 (2007) 005

26

K+ s u ℓ+ νℓ

P+

slide-27
SLIDE 27

Γ P

ℓ2 ±

( ) = Γ0 − Γ0

pt

( )+ Γ0

pt + Γ1 pt ΔE

( )

( )

In order to ensure the cancellation of IR divergences with good numerical precision, we rewrite:

ΔE ∼ O(20 MeV)

Γ ΔE

( )= Γ0 + Γ

1

pt ΔE

( )

Montecarlo simulation Lattice QCD Perturbation theory with pointlike pion

Γ0

pt = Γ π + → +ν

( )

pt

The strategy

27

Γ P± → ℓ±νℓ

( )

Γ P± → ℓ±νℓγ ΔE

( )

( )

Γ P

ℓ2 ±

( )

is an unphysical quantity

Γ P

ℓ2 ±

( ) = Γ0 − Γ0

pt

( )+ Γ0

pt + Γ1 pt ΔE

( )

( )

slide-28
SLIDE 28

The strategy

The contributions from soft virtual photon to and in the first term are exactly the same and the IR divergence cancels in the difference .

Γ0 − Γ0

pt

Γ0 Γ0

pt

The sum in the second term is also IR finite since it is a physically well defined quantity. This term can be thus calculated in perturbation theory with a different IR cutoff.

Γ0

pt + Γ1 pt ΔE

( )

The two terms are also separately gauge invariant.

28

Γ P

ℓ2 ±

( ) = Γ0 − Γ0

pt

( )+ Γ0

pt + Γ1 pt ΔE

( )

( )

ΔΓ0 L

( ) = Γ0 L ( )− Γ0

pt L

( )

Γ pt ΔE

( ) = lim

mγ →0 Γ0 pt mγ

( )+ Γ1

pt ΔE,mγ

( )

⎡ ⎣ ⎤ ⎦

slide-29
SLIDE 29

1 k2 → MW

2

MW

2 − k2

1 k2

+!!QED!!

for!π!and!l+!

Calculation of Γpt ΔE ( ) = Γ0

pt + Γ1 pt ΔE

( )

is!calculated!in!perturba3on!theory!with!a!pointlike!pion!

Γpt ΔE

( )

UV!divergences!in!!!!!!!!!! are!regularized!with!! the!Wjregulariza3on!

Γ0

pt

IR!divergences!are!regularized!with!the!a!photon!mass!

42

29

Calculation of Γ pt ΔE

( ) = lim

mγ →0 Γ0 pt mγ

( )+ Γ1

pt ΔE,mγ

( )

⎡ ⎣ ⎤ ⎦

slide-30
SLIDE 30

Γ0

pt mγ

( )

Γ1

pt ΔE,mγ

( )

Γ pt ΔE

( ) = lim

mγ →0 Γ0 pt mγ

( )+ Γ1

pt ΔE,mγ

( )

⎡ ⎣ ⎤ ⎦

30

slide-31
SLIDE 31

The result is:

IMPORTANT CHECK: For ΔE=ΔEMAX the well known result for the total rate as in S. M. Berman, PRL 1 (1958) 468 and T . Kinoshita, PRL 2 (1959) 477 is reproduced

31

Γ pt ΔE

( ) = lim

mγ →0 Γ0 pt mγ

( )+ Γ1

pt ΔE,mγ

( )

⎡ ⎣ ⎤ ⎦

NEW

slide-32
SLIDE 32

For π and K decays, the size of the neglected structure-dependent contributions can be estimated, as a function of ΔE, using ChPT at O(p4) For B decays, due to the presence of the small scale, , the radiation of a soft photon may still induce sizeable SD effects and a full non-perturbative calculation of real emission is likely necessary

  • D. Becirevic, B. Haas, E. Kou, PLB 681 (2009) 257

32

Estimates of SD contributions to

H ν k, pπ

( ) = εµ

* k

( ) d 4x eikxT 0

∫

j µ x

( )JW

ν 0

( ) π pπ

( )

k2 = 0, ε * ⋅k = 0

Γ1 ΔE

( )

expressed in terms of two hadronic form-factors FV,A

5 10 15 20 25 30 EMeV 4.107 2.107 2.107 R1Π ΜΝΓ INT SD

π → µν γ

( )

sef

50 100 150 200 250 EMeV 0.0015 0.0010 0.0005 0.0000 R1K ΜΝΓ INT SD

SD negligible

R1

A ΔE

( ) =

Γ1

A ΔE

( )

Γ0

α ,pt ΔE

( )+ Γ1

pt ΔE

( ), A= SD, INT { }

K → µν γ

( )

slide-33
SLIDE 33

is the first term in the master formula IR divergences (Log(L)) cancel in the difference. Also 1/L corrections are universal and cancel in the difference

Montecarlo simulation Lattice QCD

1

Perturbation theory with pointlike pion in finite volume

2

Γ P

ℓ2 ±

( ) = lim

L→∞ Γ0 L

( )− Γ0

pt L

( )

⎡ ⎣ ⎤ ⎦ + lim

mγ →0 Γ0 pt mγ

( )+ Γ1

pt ΔE,mγ

( )

⎡ ⎣ ⎤ ⎦

arXiv:1711.06537

33

ΔΓ0 L

( ) = Γ0 L ( )− Γ0

pt L

( )

ΔΓ0 L

( )

First Lattice Calculation of the QED Corrections to Leptonic Decay Rates

PHYSICAL REVIEW LETTERS 120, 072001 (2018)

  • D. Giusti,1 V. Lubicz,1 G. Martinelli,2 C.T. Sachrajda,3
  • F. Sanfilippo,4 S. Simula,4 N. Tantalo,5 and C. Tarantino1

Light-meson leptonic decay rates in lattice QCD+QED

  • M. Di Carlo,1 D. Giusti,2 V. Lubicz,2 G. Martinelli,1

C.T. Sachrajda,3 F. Sanfilippo,4 S. Simula,4 and N. Tantalo5

arXiv:1904.08731

NEW

slide-34
SLIDE 34

Lattice calculation of Γ0(L) at O(α)

The Feynman diagrams at O(α) can be divided in 3 classes

Connected Disconnected 1 The photon

is attached to quark lines

2 The photon

connects one quark and one charged lepton line

3 Leptonic wave

function renormalization. It cancels in

Γ0(L)− Γ0

pt L

( )

34

NEGLECTED [QUENCHED QED]

K+ s u ℓ+ νℓ

K+ s u ℓ+ νℓ

K+ s u ℓ+ νℓ

+ …

K+ s u ℓ+ νℓ

K+ s u ℓ+ νℓ

K+ s u ℓ+ νℓ

K+ s u ℓ+ νℓ

+ …

δ SU 2

( )

K+ s u ℓ+ νℓ

slide-35
SLIDE 35

Lattice calculation of Γ0(L) at O(α)

K+ s u ℓ+ νℓ

K+ s u ℓ+ νℓ

δC

qq

( ) t

( )

C

( ) t

( )

t>>a

⎯ → ⎯⎯ δ ZPAP

qq

( )

⎡ ⎣ ⎤ ⎦ ZP

( )AP ( )

− δ M P M P

( ) 1+ M P ( )t

( )

35

K+ s u ℓ+ νℓ

δC

qℓ

( ) t

( )

C

( ) t

( )

t>>a

⎯ → ⎯⎯ δ AP

qℓ

( )

AP

( )

0.000 0.003 0.006 0.009 0.012 10 20 30 40 50

δR

P µ(t)

t / a

M

π ~ 310 MeV

M

K ~ 550 MeV

D30.48 K

+

π

+

δC

qℓ

( ) t

( ) C

( ) t

( )

  • 0.4
  • 0.3
  • 0.2
  • 0.1

0.0 0.1 5 10 15 20 25 30 35 self energy (1a)+(1b) exchange (1c) (1a)+(1b)+(1c)

δ C

qq(t) / C (0)(t)

t / a

B35.32 M

π ~ 300 MeV

M

K ~ 550 MeV

δC

qq

( ) t

( ) C

( ) t

( )

total

slide-36
SLIDE 36

a

C1(t)αβ = − d 3

∫

! x d 4x1

d 4x2 0 T JW

ν (0) jµ(x1) φ†(!

x,−t)

{ } 0

× Δ(x1,x2) γ ν(1−γ 5) S(0,x2) γ µ

( )αβ eEℓt2−i "

pℓ⋅" x2

ω ℓ = " kℓ

2 + mℓ 2

ωγ = ! kγ

2 + mγ 2

We!need!to!ensure!that!the!t2!integra3on!converges!as!!t2!→ ∞ . The!large!t2!! behavior!is!given!by!the!factor!

exp Eℓ −ω ℓ −ωγ

( ) t2

⎡ ⎣ ⎤ ⎦

Eℓ = " pℓ

2 + mℓ 2

! kℓ + ! kγ = ! pℓ

ω ℓ +ωγ

( )min =

mℓ

2 + mγ 2

( )+ "

pℓ

2 > Eℓ

The!integral!is!convergent!and!the!con3nua3on!from!Minkowski!to!Euclidean!space! can!be!performed!(same!if!we!set!mγ=0!but!remove!the!photon!zero!mode!in!FV).! A!technical!but!important!point:! j!mass!gap!between!the!decaying!par3cle!and!the!intermediate!states! j!absence!of!lighter!intermediate!states! CONDITIONS:!

51

Lattice calculation of Γ0(L) at O(α) [2]!

36

K+ s u ℓ+ νℓ

slide-37
SLIDE 37

Calculation of

Γ ΔE

( ) = lim

V→∞ Γ0 L

( )− Γ0

pt L

( )

( ) + lim

mγ →0 Γ0 pt + Γ1 pt ΔE

( )

( )

Calculation of Γ0

pt(L)

is!calculated!in!perturba3on!theory!with!a!pointlike!pion!

Γ0

pt L

( )

IR!divergences!are!regularized!by!the!finite!volume!(same!of!!!!!!!!!!!!!!)!

Γ0 L

( )

! q = 2π L nx,ny,nz

( )

with!

d 3q… →

∫

2π L ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

3

…

! q

∑

! q ≠ 0,0,0

( )

53

The!result!has!the!form:! r

ℓ = mℓ / mP

Γ0

pt L

( ) = !

C0(r

ℓ)log mPL

( )+ C0(r

ℓ)+ C1(r ℓ)

mPL +O 1 L2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

37

slide-38
SLIDE 38

Calculation of Γ0

pt(L)

Both the leading [log(mPL)] and next-to-leading [O(1/L)] volume dependence cancel in . UNIVERSAL

Γ0 L

( )− Γ0

pt L

( )

The remaining, structure dependent, O(1/L2) finite volume effects are milder and can be fitted from the lattice data evaluated at different volumes.

PRD 95 (2017) 034504

  • 0.015
  • 0.010
  • 0.005

0.000 0.005 0.010 0.015 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030

δR

P

a

2 / L 2

δR

Kπ

δR

K

δR

π

β = 1.90 M

π ~ 320 MeV

M

K ~ 580 MeV

A40.40 A40.32 A40.24 A40.20

QEDL FV formulation

38

slide-39
SLIDE 39

Γ K ± → ℓ±νℓ γ

( )

( ) Γ π ± → ℓ±νℓ γ

( )

( )

39

δ RK ±π ±

Reduction of systematic uncertainties

Uncertainty due to qQED approximation is expected to be smaller for . Within SU(3) ChPT the effects of the sea-quark electric charges depend on unknown LECs at NNLO.

The residual SD FVEs start at O(1/L2). In are found to be much milder.

δ RK ±π ±

In the effects related to the mixing of operators of different chiralities cancel out (Twisted Mass Wilson fermions).

δ RK ±π ±

  • J. Bijnens and N. Danielsson, PRD 75 (2007) 014505

Γ Kℓ2

±

( )

Γ π ℓ2

±

( )

= Vus Vud fK fπ

2 Mπ ± 3

M K ±

3

M K ±

2 − mℓ 2

Mπ ±

2 − mℓ 2

⎛ ⎝ ⎜ ⎞ ⎠ ⎟

2

1+δ RK ± −δ Rπ ±

( )

δ RK ±π ±

slide-40
SLIDE 40

Details of the lattice simulation

We have used the gauge field configurations generated by ETMC, European Twisted Mass Collaboration, in the pure isosymmetric QCD theory with Nf=2+1+1 dynamical quarks

  • Gluon action: Iwasaki
  • Quark action: twisted mass at maximal twist

(automatically O(a) improved) OS for s and c valence quarks

  • Scale setting:

40

ensemble

  • V/a4

aµud aµ aµ Ncf aµs M⇡ MK (MeV) (MeV) A40.40 1.90 403 · 80 0.0040 0.15 0.19 100 0.02363 317(12) 576(22) A30.32 323 · 64 0.0030 150 275(10) 568(22) A40.32 0.0040 100 316(12) 578(22) A50.32 0.0050 150 350(13) 586(22) A40.24 243 · 48 0.0040 150 322(13) 582(23) A60.24 0.0060 150 386(15) 599(23) A80.24 0.0080 150 442(17) 618(14) A100.24 0.0100 150 495(19) 639(24) A40.20 203 · 48 0.0040 150 330(13) 586(23) B25.32 1.95 323 · 64 0.0025 0.135 0.170 150 0.02094 259 (9) 546(19) B35.32 0.0035 150 302(10) 555(19) B55.32 0.0055 150 375(13) 578(20) B75.32 0.0075 80 436(15) 599(21) B85.24 243 · 48 0.0085 150 468(16) 613(21) D15.48 2.10 483 · 96 0.0015 0.1200 0.1385 100 0.01612 223 (6) 529(14) D20.48 0.0020 100 256 (7) 535(14) D30.48 0.0030 100 312 (8) 550(14)

fπ

( ) = 130.41 20

( ) MeV

slide-41
SLIDE 41

41

Leptonic decays at O(α): RESULTS

  • 0.015
  • 0.010
  • 0.005

0.000 0.00 0.01 0.02 0.03 0.04 0.05

physical point β = 1.90, L/a = 20 (FVE corr.) β = 1.90, L/a = 24 (FVE corr.) β = 1.90, L/a = 32 (FVE corr.) β = 1.90, L/a = 40 (FVE corr.) β = 1.95, L/a = 24 (FVE corr.) β = 1.95, L/a = 32 (FVE corr.) β = 2.10, L/a = 48 (FVE corr.) continuum limit fit at β = 1.90 fit at β = 1.95 fit at β = 2.10

δ R

Kπ

m

ud (GeV)

m

s = m s phys

PDG

δ RKπ = C0 + Cχ log mud

( )+ C1mud + C2mud

2 + Da2

+ K2 L2 1 M K

2 − 1

Mπ

2

⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + K2

µ

L2 1 Eµ

K

( )

2 −

1 Eµ

π

( )

2

⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ +δΓ pt ΔEγ

max,K

( )−δΓ pt ΔEγ

max,π

( )

δ RK ±π ± = −0.0126 10

( )stat 2 ( )input 5 ( )chir 5 ( )FVE 4 ( )disc 6 ( )qQED

= − 0.0126 14

( )

LATTICE RESULT

V.Cirigliano and H.Neufeld, PLB 700 (2011) 7

δR K − δR π = − 0.0112 (21)

ChPT

Vus Vud fK

( )

fπ

( ) = 0.27683 29

( )exp 20 ( )th

fK

( )

fπ

( ) = 1.1966 13

( )

Vus Vud = 0.23134 24

( )exp 30 ( )th

FLAG(2019) Nf=2+1+1

Vus = 0.22538 30

( )

Vus = 0.22526 38

( )

H & T, 2016

Vud from

S et al., 2018

slide-42
SLIDE 42

42

Leptonic decays at O(α): RESULTS

0.00 0.01 0.02 0.03 0.04 0.00 0.01 0.02 0.03 0.04 0.05

β = 1.90, L/a = 20 β = 1.90, L/a = 24 β = 1.90, L/a = 32 β = 1.90, L/a = 40 β = 1.95, L/a = 24 β = 1.95, L/a = 32 β = 2.10, L/a = 48 physical point β = 1.90, L/a = 20 (FVE corr.) β = 1.90, L/a = 24 (FVE corr.) β = 1.90, L/a = 32 (FVE corr.) β = 1.90, L/a = 40 (FVE corr.) β = 1.95, L/a = 24 (FVE corr.) β = 1.95, L/a = 32 (FVE corr.) β = 2.10, L/a = 48 (FVE corr.) continuum limit fit at β = 1.90 fit at β = 1.95 fit at β = 2.10

δ R

π

m

ud (GeV)

PDG

π

+ -> µ +ν[γ]

  • 0.01

0.00 0.01 0.02 0.03 0.00 0.01 0.02 0.03 0.04 0.05

β = 1.90, L/a = 20 β = 1.90, L/a = 24 β = 1.90, L/a = 32 β = 1.90, L/a = 40 β = 1.95, L/a = 24 β = 1.95, L/a = 32 β = 2.10, L/a = 48 physical point β = 1.90, L/a = 20 (FVE corr.) β = 1.90, L/a = 24 (FVE corr.) β = 1.90, L/a = 32 (FVE corr.) β = 1.90, L/a = 40 (FVE corr.) β = 1.95, L/a = 24 (FVE corr.) β = 1.95, L/a = 32 (FVE corr.) β = 2.10, L/a = 48 (FVE corr.) continuum limit fit at β = 1.90 fit at β = 1.95 fit at β = 2.10

δ R

K

m

ud (GeV)

PDG

K

+ -> µ +ν[γ]

fπ

( ) Vud = 127.28 2

( )exp 12 ( )th MeV

fK

( ) Vus = 35.23 4

( )exp 2 ( )th MeV Γ P± → µ±νµ γ

[ ]

( ) = Γ

tree

( ) 1+δ RP±

⎡ ⎣ ⎤ ⎦

δ RK ± = 0.0024 6

( )stat 8 ( )syst

= 0.0024 10

( )

ChPT/PDG ChPT/PDG δRπ = 0.0176 21

( )

δRK = 0.0064 24

( )

V.Cirigliano, H.Neufeld; PLB 700 (2011) 7

δ Rπ ± = 0.0153 16

( )stat 10 ( )syst

= 0.0153 19

( )

Vus = 0.22567 42

( )

fK

( ) = 156.11 21

( ) MeV

FLAG(2019) Nf=2+1+1

cannot be predicted

Vud

Vus = 0.2253 7

( )

PDG

because is used to set the scale

fπ

( )

slide-43
SLIDE 43

43

Semileptonic decay amplitudes

K0 π+

d s u

ℓ− νℓ

K 0 → π +ℓ−νℓ

sπℓ = pπ + pℓ

( )

2

q = pK − pπ = pℓ + pν

K0 π+ ℓ− νℓ

π pπ

( ) sγ µu K pK ( ) = f0 q2

( ) M K

2 − Mπ 2

q2 qµ + f+ q2

( )

pπ + pK

( )µ − M K

2 − Mπ 2

q2 qµ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

Without QED

d 2Γ dq2dsπℓ = GF

2 Vus 2 a+ q2,sπℓ

( ) f+ q2 ( )

2 + a0 q2,sπℓ

( ) f0 q2 ( )

2 + a0+ q2,sπℓ

( ) f0 q2 ( ) f+ q2 ( )

⎡ ⎣ ⎢ ⎤ ⎦ ⎥

FV corrections due to em rescattering To be addressed: IR divergences cancel out 1/L corrections depend on df± dq2

d 2Γ dq2dsπℓ = lim

L→∞

d 2Γ0 L

( )

dq2dsπℓ − d 2Γ0

pt L

( )

dq2dsπℓ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + lim

mγ →0

d 2Γ0

pt mγ

( )

dq2dsπℓ + d 2Γ1

pt ΔE,mγ

( )

dq2dsπℓ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

lighter intermediate states πℓ γ

( )

0.02 0.04 0.06 0.08 0.1 0.12 q 2 (GeV 2) 0.22 0.23 0.24 0.25 0.26 Vusf Vusf+(q 2) Vusf0(q 2) exp data fit
  • N. Carrasco et al., 2016

Vus f+ q2

( )

Vus f0 q2

( )

  • D. Giusti et al., arXiv:1811.06364
slide-44
SLIDE 44

Conclusions and future perspectives

We have performed the FIRST lattice calculation (arXiv:1711.06537, arXiv: 1904.08731) of isospin-breaking corrections to light-meson leptonic decay rates The inclusion of disconnected diagrams is mandatory for removing the qQED (quenched-QED) approximation Extensions to leptonic heavy-light meson decays and semileptonic Kl3 decays are being targeted Inclusion of a non-perturbative calculation of the real photon emission Setting the lattice scale of our simulations with an hadron mass (e.g. ) allows to predict Vud

M Ω

slide-45
SLIDE 45

Supplementary slides

slide-46
SLIDE 46

▪︎ Isospin-breaking effects due to the up-down mass difference in lattice QCD

RM123 Collaboration: G.M. de Divitiis, P . Dimopoulos, R. Frezzotti, V. Lubicz, G. Martinelli,

  • R. Petronzio, G.C. Rossi, F. Sanfilippo, S. Simula, N. Tantalo, C. Tarantino

JOURNAL OF HIGH ENERGY PHYSICS 04 (2012) 124

▪︎ Leading isospin-breaking effects on the lattice

RM123 Collaboration: G.M. de Divitiis, R. Frezzotti, V. Lubicz, G. Martinelli, R. Petronzio, G.C. Rossi, F. Sanfilippo, S. Simula, N. Tantalo PHYSICAL REVIEW D 87, 114505 (2013)

▪︎ Leading isospin-breaking corrections to pion, kaon and charmed-meson masses with Twisted-Mass fermions

RM123 Collaboration: D. Giusti, V. Lubicz, G. Martinelli, F. Sanfilippo, S. Simula, N. Tantalo,

  • C. Tarantino

PHYSICAL REVIEW D 95, 114504 (2017)

QED and isospin corrections to quark and hadron masses The RM123 method

slide-47
SLIDE 47

▪︎ QED corrections to hadronic processes in lattice QCD

  • N. Carrasco, V. Lubicz, G. Martinelli, C. T. Sachrajda, N. Tantalo, C. Tarantino, M. Testa

PHYSICAL REVIEW D 91, 074506 (2015)

QED and isospin corrections to hadronic processes

▪︎ Finite-volume QED corrections to decay amplitudes in lattice QCD

  • V. Lubicz, G. Martinelli, C. T. Sachrajda, F. Sanfilippo, S. Simula, N. Tantalo

PHYSICAL REVIEW D 95, 034504 (2017)

▪︎ First lattice calculation of the QED corrections to leptonic decay rates

  • D. Giusti, V. Lubicz, G. Martinelli, C. T. Sachrajda, F. Sanfilippo, S. Simula, N. Tantalo, C. Tarantino

PHYSICAL REVIEW LETTERS 120, 072001 (2018)

▪︎ Light-meson leptonic decay rates in lattice QCD+QED

  • M. Di Carlo, D. Giusti, V. Lubicz, G. Martinelli, C. T. Sachrajda, F. Sanfilippo, S. Simula, N. Tantalo

ArXiv:1904.08731

▪︎

NEW

slide-48
SLIDE 48

LATTICE QED

  • Non-compact QED: the dynamical variable is the gauge potential Aµ(x)

in a fixed (covariant) gauge

  • The covariant derivatives are defined by introducing the QED links:

Aµ(x) → Eµ(x) = e

−iaeAµ (x)

a Dµ

+q f (x) = Eµ(x)

⎡ ⎣ ⎤ ⎦

ef q f (x + a ˆ

µ)− q f (x)

  • Gauge transformations for the quarks and photon fields are:

Aµ(x) → Aµ(x) + ∇µ

+λ(x) =

= Aµ(x) + λ(x + a ˆ µ) − λ(x)

( ) / a

q f (x) → e

ieef λ(x)q f (x)

q f (x) → e

−ieef λ(x)q f (x)

  • The QED link then

transforms as:

Eµ(x) → e

−iae Aµ (x)+∇µλ(x) ⎡ ⎣ ⎤ ⎦ = e ieλ(x)Eµ(x) e −ieλ(x+a ˆ µ)

and is manifestly covariant.

Dµ

+q f (x) = ∂µ− ieef Aµ(x)

( )q f (x)+O(a)

slide-49
SLIDE 49

LATTICE QED

  • For non-compact QED the pure gauge action is:

SQED = 1 4 Fµν(x)Fµν(x)

x

∑

= 1 4 ∇µ

+ A ν(x) − ∇ν + Aµ(x)

( )

2

=

x;µν

∑

= − 1 4 A

ν(x)∇µ − ∇µ + A ν(x) − ∇ν + Aµ(x)

( )− Aµ(x)∇ν

− ∇µ + A ν(x) − ∇ν + Aµ(x)

( )

⎡ ⎣ ⎤ ⎦

x;µν

∑

∇µ

− Aµ(x) = 0

  • By using a covariant gauge fixing one gets:

SQED = 1 2 A

ν(x) −∇µ −∇µ +

( ) A

ν(x) x;µν

∑

  • Imposing periodic b.c. and looking at the action in momentum space

reveals a problem with the zero momentum mode:

SQED = 1 2  A

ν *(k) 2sin(kµ / 2)

( )

2 

A

ν(k) k;µν

∑

The photon propagator is infrared divergent

slide-50
SLIDE 50

LATTICE QED

  • The infrared problem is not specific of the lattice regularization but

it is general for QED in a finite volume with periodic b.c. Already at the classical level, the Gauss’ law for a charged particle is inconsistent for the zero mode:

∇µ

−Fµν(x) = jν(x)

∇i

−Ei(x) = ρ(x)

0 = ∇i

−Ei(x)  x

∑

= e δ 3(t,x) = e

 x

∑

  • A solution to the infrared problem

consists in removing the zero mode:

Dµν

⊥ (x − y) =

δ µν eik(x−y) 2sin(kρ / 2) ⎡ ⎣ ⎤ ⎦

2

k≠0

∑

  • We subtracted the zero

mode in x-space and applied a stochastic technique

P⊥φ(x) ≡ φ(x) − 1 V φ(y)

y

∑

−∇ρ

−∇ρ +

⎡ ⎣ ⎤ ⎦φµ(x) = P⊥ηµ(x) φµ(x) = δ µν −∇ρ

−∇ρ + P⊥

⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ην(x) = Dµν

⊥ (x − y)ην(y) y

∑

Real Z2 noise

slide-51
SLIDE 51

Tuning the critical mass

The Dashen theorem: in the massless theory, the neutral pion and kaon are Goldstone bosons even in the presence of electromagnetic interactions: 1

lim

mf →0 Mπ 0 = lim mf →0 M K0 = 0

With twisted mass fermions, one can extend the method used also in the isosymmetric QCD case, based on a specific Ward-Takahashi identity: 2

∇µ Vµ

1(x)P 5 2(0) = 0

More precise: it does not require a chiral extrapolation

slide-52
SLIDE 52

Estimates of structure dependent! contributions to Γ1(ΔE)

We!es3mate!the!size!of!the!neglected!structurejdependent!contribu3ons!to! the!decay!!!!!!!!!!!!!!!!!!!!!!!!!!!using!chiral!perturba3on!theory!at!O(p4)!

K + /π + → ℓνℓγ

J.!Bijnens,!G.!Ecker,!J.!Gasser,!NPB!396!(1993)!81;!!V.Cirigliano,!I.Rosell,!JHEP!0710!(2007)!005!!

Start!with!the!decomposi3on!in!terms!of!Lorenz!invariant!form!factors!of!the! hadronic!matrix!element! and!separate!the!contribu3on!corresponding!to!the!approxima3on!of!a! pointlike!pion!!!!!!!!!!!from!the!structure!dependent!part!

Hpt

µν

HSD

µν

!!!!!!!!!is!simply!given!by:!

Hpt

µν

Hpt

µν = fπ gµν − (2pπ − k)µ(pπ − k)ν

(pπ − k)2 − mπ

2

⎡ ⎣ ⎢ ⎤ ⎦ ⎥ kµHpt

µν = fπ pπ ν

( )

H µν = Hpt

µν + HSD µν

H µν(k, pπ ) = d 4x

∫

eikx 0 T j µ(x) JW

ν (0)

( ) π(pπ )

61

  • J. Bijnens et al., NPB 396 (1993) 81; V. Cirigliano and I. Rosell, JHEP 0710 (2007) 005
slide-53
SLIDE 53

The!structure!dependent!component!!!!!!!!!!!can!be!parametrized!by!four! independent!invariant!form!factors!which!we!define!as!

HSD

µν

HSD

µν = H1 k2gµν − k µkν

⎡ ⎣ ⎤ ⎦ + H2 k ⋅ pπ

( )k µ − k2pπ

µ

⎡ ⎣ ⎤ ⎦ pπ − k

( )

ν

kµHSD

µν = 0

( )

−i F

V

mπ ε µναβkα pπβ + F

A

mπ k ⋅ pπ − k2

( )gµν − pπ − k

( )

µ kν

⎡ ⎣ ⎤ ⎦ For!the!decay!into!a!real!photon,!only!FV!and!FA!contribute! At!O(p4)!in!chiral!perturba3on!! theory!FV!and!FA!are!constant:! For!our!es3mates!we!use:! ,! F

V =

mP 4π 2 fπ F

A = 8mP

fπ L9

r + L10 r

( )

F

V (π ) = 0.0254

F

A (π ) = 0.0119

F

V (K ) = 0.096

F

A (K ) = 0.042

J.!Bijnens,!G.!Ecker,!J.!Gasser,!NPB!396!(1993)!81!

Direct!measurement!

PDG!2014!

ChPT!

62

Estimates of structure dependent! contributions to Γ1(ΔE)

  • J. Bijnens et al., NPB 396 (1993) 81

PDG

slide-54
SLIDE 54

Structure dependent contributions ! to decays of D and B mesons!

For!the!studies!of!D!and!B!mesons!decays!we!cannot!apply!ChPT! For!B!mesons!in!par3cular!we!have!another!small!scale,! !!!!!!!!the!radia3on!of!a!sos!photon!may!s3ll!induce!sizeable!SD!effects!

mB* − mB ! 45 MeV

A!phenomenological!analysis!based!on!a!simple!pole!model!for!FV!and!FA! confirms!this!picture!!

D.!Becirevic,!B.!Haas,!E.!Kou,!PLB!681!(2009)!257!

F

V !

" CV 1− pB − k

( )

2 / mB* 2

F

A !

" CA 1− pB − k

( )

2 / mB1 2

Under!this!assump3on!the!SD!contribu3ons!to!!!!!!!!!!!!!!!!!!!!!! for!Eγ!!20!MeV!can!be!very!large,!but!are!small!for!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!and! B → eν(γ ) B → µν(γ ) B → τν(γ ) A!laWce!calcula3on!of!FV!and!FA!would!be!very!useful! B → eν(γ ) B → µν(γ ) B → τν(γ ) SD large SD small

64

A lattice calculation of FV and FA would be very useful

  • D. Becirevic et al., PLB 681 (2009) 257
slide-55
SLIDE 55

R1

A(ΔE) =

Γ1

A(ΔE)

Γ0

α ,pt + Γ1 pt(ΔE) , A = { SD, INT }

SD!=!structure!dependent! INT!=!interference!

π → µν(γ ) K → eν(γ ) K → µν(γ ) π → eν(γ )

ΔE = 20 MeV

Interference!contribu3ons!are!negligible!in!all!the!decays! Structurejdependent!contribu3ons!can!be!sizable!for!!!!!!!!!!!!!!!!!!!!!!!!!but!they!! are!negligible!for!!!!!!!!!!!!!!!!!!!!!!!!!!!(which!is!experimentally!accessible)!

K → eν(γ )

ΔE < 20 MeV

63