Isospin Breaking Effects Vittorio from Lattice QCD Lubicz · 2011. 4. 19. · Isospin Breaking...

Isospin Breaking Effectsfrom Lattice QCD

Vittorio Lubicz

Outline● Introduction: motivations and strategy● md-mu from MK0-MK+ ● Decay constants: IB effects in FK/Fπ

- The kaon semileptonic f.f.: f K+π0/f

K0π-

- The neutron-proton mass splitting

LPT - Orsay14 avril

20111

lunedì 18 aprile 2011

Isospin Breaking Effectsfrom Lattice QCD

Vittorio Lubicz

The collaboration:P. Dimopoulos, G. de Divitiis, R. Frezzotti,

V. Lubicz, G. Martinelli, R. Petronzio, G. Rossi, F. Sanfilippo, S. Simula, N.Tantalo, C. Tarantino

Roma1/SISSA Roma2 Roma3

Lattice gauge configurations provided by:

LPT - Orsay14 avril

20112


Motivations

Let us discuss as an example an important case: the determination of Vus

Isospin breaking effects are induced by:

Qu≠Qd: O(αem) ≈ 1/100

mu≠md: O((md-mu)/ΛQCD) ≈ 1/100

Though small, these effects are important at the current level of precision in flavour physics

(“Electro-magnetic”)

(“Strong”)

These corrections are the subject of this talk

3


K

Vus

[Marciano 04]

K !

Vus

|Vud|2+ |Vus|2+ |Vub|2 = 1

The 1st row unitarity test

The most stringent unitarity test

4


arXiv:1005.2323 [hep-ph]

arXiv:1011.4408 [hep-lat]

F L A G

THE

OR

Y: L

ATTI

CE

QC

DE

XP

ER

IME

NTS

5


The lattice determinations are obtained in the limit of exact

Isospin Symmetry: mu=md , qu=qd=0

Isospin breaking effects are estimated using Chiral Perturbation Theory:

0.8 % 0.4 %

f+K +π 0 (t)f+K 0π−

(t)

⎛

⎝⎜⎞

⎠⎟QCD−1= 0.029(4)

Kastner, Neufeld 0805.2222

f+K +π 0 (t)f+K 0π−

(t)

⎛

⎝⎜⎞

⎠⎟QCD= 1+ 3

4md − mu

ms − mud

1+ ΔM + Δ f( ) +O(p4 ,(mu − md )2 )

Gasser,Leutwyler (GL) 1985

FK + / Fπ +

FK / Fπ

⎛⎝⎜

⎞⎠⎟QCD

= 1− 12md − mu

ms − mud

FKFπ

−1− MK2 − Mπ

2 − Mπ2 ln(MK

2 /Mπ2 )

64π 2F02

⎧⎨⎩

⎫⎬⎭+O(m2 )

FK + / F

π +

FK / Fπ

⎛⎝⎜

⎞⎠⎟QCD

−1= − 0.0022(6)

Cirigliano, Neufeld 1102.05636


IB effects cannot be neglected at the current level of precision.For the form factor, even the size of the uncertainty is becoming relevant

0.8 % 0.4 %

f+K +π 0 (t)f+K 0π−

(t)

⎛

⎝⎜⎞

⎠⎟QCD−1= 0.029(4)

FK + / F

π +

FK / Fπ

⎛⎝⎜

⎞⎠⎟QCD

−1= − 0.0022(6)

Can IB effects be computed on the lattice?Not easy, a priory, because of of their smallness

The previous example illustrates a relatively favourable situation in which IB effects can be estimated using ChPT. But the uncertainties of ChPT applied to kaons are not small. Moreover, in many other cases, we do not have such a tool. E.g.: the neutron-proton mass splitting

7


Lm = muuu + mddd =12mu + md( ) uu + dd( )− 1

2md − mu( ) uu − dd( ) =

= mud uu + dd( )− Δm uu − dd( )

Δm =12md − mu( )S = S0 − Δm S S = uu − dd( )

x∑with ,

The functional integral can be then expanded in powers of Δm:

O =Dφ O e−S0 +Δm S∫Dφ e−S0 +Δm S∫

Dφ O e−S0 1+ Δm S( )∫Dφ e−S0 1+ Δm S( )∫

O 0 + Δm O S

0

1+ Δm S0

for isospin symmetry

A promising strategy for lattice QCD:the (mu-md) expansion

8


Δ OO 0

≡O − O 0

O 0

ΔmO S

0

O 0

Very precise on the lattice because of the statistical correlation

When applying the Wick theorem to〈OŜ〉the disconnected contractions of Ŝ cancel out because of the isospin symmetry

at LO the corrections only appear in the valence quark propagators:

- = 0u

x

d

x

==

+-

+ ...x + ...x

d

u

x = Δm qqx∑

S = uu − dd( )x∑

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Determination of md-mu: the neutral and charged

kaon masses

Note: because of the symmetry with respect to u↔d, the neutral and charged pion mass splitting is quadratic in (mu-md)

10


CKK (p, t) = K(x, t)K + (0)

x∑ ⋅ei

p⋅x

=t→∞

ZK exp(−EKt)

δCKK (p, t) ≡ ΔCKK (

p, t)CKK (

p, t)= δZK −δEKEKt

us

!"

su/d

su/d

The 2-point functions

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E(p)2 = M 2 + p2 E(p) δE(p) = MδM

The energy-momentum relations

12


The neutral and charged kaon mass splitting

MK 02 − M

K +2 = md − mu( )ΔMK

2

For PGB it is convenient to look at the square mass splitting:

ΔMK2 = 2MkΔMK

ΔMK2 ≡

MK 02 − M

K +2

md − mu

= B0 1+ 132π 2F0

223Mη

2 logMη

2

µ2 +MK

2

MK2 − Mπ

2 Mη2 log

Mη2

µ2 − Mπ2 log Mπ

2

µ2

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥+

⎧⎨⎪

⎩⎪

+16B0

F02 ms + mud( ) 2L8

r − L5r( ) + ms + 2mud( ) 2L6

r − L4r( )⎡⎣ ⎤⎦

⎫⎬⎭+O(m3 )

In Chpt, at NLO:

13


The neutral and charged kaon mass splitting

MK 02 − M

K +2 = md − mu( )ΔMK

2

For PGB it is convenient to look at the square mass splitting:

ΔMK2 = 2MkΔMK

In Chpt, at NLO:

ΔχMK2 =

MK 02 − M

K +2

md − mu

⎛

⎝⎜⎞

⎠⎟=

=MK

2 − Mπ2

ms − mud

⎛⎝⎜

⎞⎠⎟

MK2

ms + mud

⎛⎝⎜

⎞⎠⎟/ Mπ

2

2mud

⎛⎝⎜

⎞⎠⎟

MK 02 − M

K +2

MK2 − Mπ

2

Mπ2

MK2 =

md2 − mu

2

ms2 − mud

2 =1Q2

In order to extract md-mu we need to discuss the electromagnetic effects14


What do we know about the electromagnetic

self energies of kaons ?

15


MP

MP

ΔPγ

= physical mass= QCD mass= e.m. self energy

Define

MP2 = MP

2 + ΔPγ

At LO in the chiral expansion, i.e. in the chiral limit:

Δπ +γ = Δ

K +γ ≠ 0Δ

π 0γ = Δ

K 0γ = 0

ΔK +γ − Δ

K 0γ( ) = Δ

π +γ − Δ

π 0γ( )

,

ΔPγ = A eq + eq( )2 +O(mq )

At the LO:

The Dashen theorem

16


The corrections to the Dashen theorem, appearing at NLO in ChPT, can be parameterized in terms of “small” parameters:

ΔK +γ − Δ

K 0γ( )− Δ

π +γ − Δ

π 0γ( ) = ε Δπ

Δπ 0γ = ε

π 0 Δπ ΔK 0γ = ε

K 0 Δπ

Mπ +2 − M

π 02 = εm Δπ

Because of the u↔d interchange symmetry, the mass difference

in QCD is proportional to . Thus, we also define:Mπ +2 − M

π 02 mu − md( )2

Δπ = Mπ +2 − M

π 02 = 1261 MeV2

The typical size of the charged meson Δ’s is

(and )Mπ + −Mπ 0 = 4.6 MeV

The corrections to the Dashen theoremFLAG, 1011.4408

17


Δπ +γ − Δ

π 0γ = 1− εm( ) Δπ

All pion and kaon e.m. self energies can be expressed in terms of the ε parameters:

ΔK +γ − Δ

K 0γ = 1+ ε − εm( ) Δπ

Δπ 0γ = ε

π 0 Δπ

ΔK 0γ = ε

K 0 Δπ

Δπ +γ , Δ

K +γFor charged mesons are of O(1) in unit of .Δπ

In the chiral expansion:

επ 0

=O(mu ,d )εm =O((mu − md )2 )

εK 0

=O(ms )ε =O(ms )

,

,

,

Thus, and are expected to be smallerεm επ 0 18


Mπ +2 − M

π 02 = mu − md( )2 2B2

F2 l7 1+O(mud )[ ] =

= 14mu − md( )2

ms − mud

B0 1+ 83µη + 2µK +

32F0

2 MK2 −

12L4r + L6

r − 9L7r − 3L8

r +18νK

⎛⎝⎜

⎞⎠⎟ +O(mud ,ms

2 )⎡

⎣⎢

⎤

⎦⎥ =

=M

K +2 − M

K 02( )2

3 Mη2 − Mπ

2( )2 1+ 83ΔGMO +

MK2

8π 2F02 1+ 6 log MK

2

Mη2

⎛

⎝⎜

⎞

⎠⎟ +O(mud ,ms

2 )⎡

⎣⎢⎢

⎤

⎦⎥⎥

εm 0.04Mπ +2 − M

π 02 = εm ΔπΔGMO =

4MK2 − Mπ

2 − 3Mη2

Mη2 − Mπ

2with From:

ε = 0.70(27)

Q2 =ms2 − mud

2

md2 − mu

2 =MK

2

Mπ2

MK2 − Mπ

2

MK 02 − M

K +2 1+O(m2 ){ }

Using Q=22.3(8) from η→3π and taking επ0,K0,m from FLAG

Q 24.3− 3.0ε + 0.9επ 0 − 0.1εK 0 + 2.6εm

NLO corrections cancel in the ratio

Two predictions from ChPT GL, 1985

19


The first lattice QCD calculation of the e.m. self energies of hadrons has been performed by Duncan, Eichten, Thacker, PRL 76 (1996) 3894, hep-lat/9602005.

QED is treated in the quenched approximation. The electromagnetic field is introduced dynamically, using a noncompact formulation:

Sem =14e2

∇µAν (x)−∇νAµ (x)( )xµν∑ 2

∇i Ai (x) = 0 UµQED (x) = eiQqAµ (x )

The action is gaussian distributed. The fields are generated in p-space.

, ,

Coulomb gauge The e.m. link

The presence of massless, unconfined photons implies that the finite volume effects may be much larger than for pure QCD.

Basically, all subsequent lattice calculations (10 years later) followed the same approach.

Lattice calculations of the e.m self energies

20


Fixing Mπ + : ε

π 0 ≈ 0.10Duncan, Eichten Thacker, 1996 0.42(4) 0.034(4) 0.23(3)

Blum et al., 2007 0.24(6)

Blum et al., 2010 0.49(9) 0.02(5)

MILC, Lat 2008 1.0(4)

BMW, Lat 2010 0.66(14) -0.09(4)Phenomenology (model indep.) 0.70(27) 0.04

FLAG 0.7(5) 0.07(7) 0.3(3) 0.04(2)

ε επ 0 ε

K 0εm

✝ ✝✝(✝)

(✝✝)Q from η→3π

From octet masses

Lattice errors are statistical only. LAT 2008 and LAT 2010 results are preliminary

Nf=2+1, NLO chiral fits, 2 volumes

Estimates of the epsilon parameters

21


π 139.57 134.98 4.59 4.4(1) 0.2(1)K 493.68 497.61 -3.94 2.1(6) -6.0(6)

MP+ −MP0( )QCDM

P+ −MP0( )e.m.MP+ −MP0M

P+ MP0

Using the FLAG estimates

Experiments

ε = 0.7(5), επ 0 = 0.07(7), ε

K 0 = 0.3(3), εm = 0.04(2) :

one obtains (all masses in MeV):

Summary of pion and kaon masses

22 10% of uncertaintylunedì 18 aprile 2011

mu /md = 0.46(5)

md − mu = 2.66(29) MeV

The up-down quark mass difference

From the lattice result, using

we obtain:

mu /md =

MK +2 − M

K 02 + M

π +2

MK 02 − M

K +2 + M

π +2 1+O(m){ } 0.56 − 0.08ε − 0.02επ 0 + 0.11εm = 0.50(4)

In ChPT, at the LO:

[preliminary]

The main uncertainty is due to the electromagnetic self energy

MK 02 − M

K +2( )QCD = md − mu( )ΔMK

2

23


IB effects in thekaon decay constants

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CKK (t) =t→∞

ZK exp(−MKt) =

= FK2Mk

3

2 ms + mu( )2 exp(−MKt)

δCKK (t) = δZK −δMKMKt

!"

su/d

su/d

For the kaon and pion decay constants IB effects due to (mu-md) are:

δFK + =O(mu − md ) δF

π + =O((mu − md )2 )

δ (FK + / Fπ + ) = δFK + −δFπ + = δFK + +O((mu − md )

2 )Therefore:

The 2-point function

p=0

25


Extrapolating to the physical quark mass and using the previous determination of md-mu we obtain:

δ (FK + / Fπ + ) = −0.0029(4)

~ 0.1% correction !

δ (FK + / Fπ + ) = −0.0022(6) Cirigliano, Neufeld 1102.0563

[preliminary]

26

The lattice result is in good agreement with the ChPT estimate


δ FK + / Fπ +( ) = 12 md − mu( )B0 ln(MK

2 / µ2 )+164π 2F0

2 +µη − µπ

Mη2 − Mπ

2 −4L5

r

F02

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪+O(m2 )

= −12md − mu

ms − mud

FKFπ

−1− MK2 − Mπ

2 − Mπ2 ln(MK

2 /Mπ2 )

64π 2F02

⎧⎨⎩

⎫⎬⎭+O(m2 )

GL,1985

Extrapolating to the physical quark mass and using the previous determination of md-mu we obtain:

δ (FK + / Fπ + ) = −0.0022(6) Cirigliano, Neufeld 1102.0563

Compare with the ChPT prediction (using lattice data):

27

δ (FK + / Fπ + ) = −0.0029(4)

~ 0.1% correction !

[preliminary]

The lattice result is in good agreement with the ChPT estimate


The neutral and charged kaon semileptonic

form factors

28


The 3-point functions

s

u

u

K+ π0

s u

d

K0 π+

!"

s

u/d

u/d

K πδ =

π (pπ ) |V0 |K(pK ) δ π (pπ ) |V0 |K(pK )

29


The form factor

and its correction[preliminary]

- The numerical analysis is still in progress

30

- The K+→π0 channel also receives a contribution from disconnected diagrams. They have not been included yet.


The neutron-proton mass splitting

31


1 The electromagnetic contribution

As shown by Cottingham (1963), the e.m. self-energy of a hadron can be determined from the knowledge of its electromagnetic structure:

Dispersion relation

ΔMγ Compton amplitude for space-like photons

Cross section for electron scattering

e e

There is one difficulty, however, with this approach: a subtracted dispersion relation is required for the structure function W1 [Harari, 1966]

The phenomenological estimates used still today are those provided by Gasser and Leutwyler 1982, who evaluated the e.m. self-energies of the baryon octet using the Cottingham formula. They acknowledged the need for a subtracted dispersion relation but proceeded to ignore this issue, as the subtraction constant could not be computed.32


Beane, Savage, 2002

The neutron mass is obtained by making the replacement mu ↔ md, so that

For the proton mass, partially quenched HBChPT predicts:

Mn − MP( )QCD = −232α − β( ) mu − md( ) +O mq

3/2( )

MP = M 0 + α + β( ) mu + md( ) + 132α − β( ) mu − md( ) + 2σ mj

sea + mlsea( ) +O mq

3/2( )

known

2 The md-mu contribution

There exist few lattice estimates of (Mn-Mp)QCD. They are based on:

Partially quenched simulations, performed with degenerate sea quark masses and non-degenerate valence quark masses

Partially quenched HBChPT

33


- Predictions in the table are in blue. - Blum et al. 2010 compute the e.m. mass difference (Mn-Mp)e.m. by simulating QCD + quenched QED on the lattice, as discussed before. - (✱) Using a subtracted dispersion relation, one obtains (Mn-Mp)e.m.= -1.39 + cost. [Walker-Loud, Lat 2010], i.e. almost a factor of 2 different from GL 1982

[All masses in MeV]

6.85 (21) 0.86 (30) 5.99 (37) GL 1982

8.08 (8) 0.17 (30) 7.91 (31) GL 1982

1.29 -0.76 (30)(✱) 2.05 (30) GL 1982

1.29 -0.97 (72) 2.26 (72) Beane et al. 2006

2.13 (72) -0.38 (7stat) 2.51 (14stat) Blum et al. 2010

1.29 -2.11 (77stat) 3.40 (77stat) Walker-Loud,Lat 2010

ΔMe.m.

MΞ− − MΞ0

Mn − Mp

ΔMQCDΔM

MΣ− − M Σ+

Mn − Mp

Mn − Mp

Mn − Mp

Estimates of the octet baryon mass splittings

34


Conclusions

At the current level of precision in flavour physics isospin breaking effects, though small, are becoming important

The method looks promising but...

the work is still in progress

35


Isospin Breaking Effects Vittorio from Lattice QCD Lubicz · 2011. 4. 19. · Isospin Breaking...

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