fundamental constants and electroweak phenomenology from...

Fundamental constants and electroweak phenomenology from

the lattice

Lecture III: Chiral dynamics and light quark masses

Shoji Hashimoto (KEK)

@ INT summer school 2007,

Seattle, August 2007.

III. Chiral dynamics and light quark masses

Aug 17, 2007S Hashimoto (KEK)2

1. Chiral symmetry breaking and quark massesGMOR relationChiral perturbation theoryQuark mass ratios

2. Lattice calculation of light quark massesBasic strategyPerturbative and non-perturbative matchings

3. Pion loop effectsChiral log effects on chiral extrapolationPion form factor and general strategy

III. Chiral dynamics1. Chiral symmetry breaking and

quark masses


Chiral symmetry breaking


In the QCD vacuum, chiral symmetry is broken.

Flavor SU(3)LxSU(3)R → SU(3)V

Non-zero chiral condensate Nambu-Goldstone bosons (pion, kaon, η) nearly massless; in practice massive due to non-zero mq.

Flavor-singlet axial U(1) is special, due to anomaly. η’ is substantially heavier.

Other hadrons have a mass of O(ΛQCD)

Low energy effective theory for pions (and K, η) can be constructed = chiral perturbation theory (ChPT, χPT).

qq

PCAC relation


Partially Conserved Axial Current (PCAC)

From the QCD Lagrangian,

The axial current may annihilate pion to the vacuum; Lorentz invariance restricts its form.

fπ is called the pion decay constant.Can be measured from the leptonic decay π→μν.

fπ = 131 MeVIts analog for kaon is fK.

fK = 160 MeV

dummA

duA

du 5

5

)(

,

γ

γγμ

μ

μμ

+=∂

=

)()(

;)()0(0

,)()0(0

2

2

xmfxA

mfpA

pfipA

πππμ

μ

ππμ

μ

μπμ

φ

π

π

=∂

=∂

=

φπ(x): operator to create a pion.


Consider two-point functions

Taking derivatives of T-products, we obtain

In the limit of qμ→0, it leads to

( )( )

4 †5

4 †5

( ) 0 T ( ) (0) 0 ,

( ) 0 T ( ) (0) 0

iqx

iqx

q i d xe A x A

q i d xe A x A

μν μ ν

μ νμ ν

Π =

Ψ = ∂ ∂

∫

∫

4 0 0 †5 5

4 0 0 †

( ) ( ) ( ) 0 ( ), (0) 0

( ) 0 ( ), (0) 0

iqx

iqx

q q q q q d xe x A x A

i d xe x A x A

μν ν νμ ν

μμ

δ

δ

⎡ ⎤Π = Ψ − ⎣ ⎦

⎡ ⎤+ ∂⎣ ⎦

∫∫

{ })(1

resonancesother )(

222

022

42

πππ

πππ

mOmf

qmimfidduumm

qdu

+−=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+−−

−=++→

Spontaneous symmetry breaking 005 ≠Q


Gell-Mann-Oakes-Renner (GMOR) relation (1968)

Chiral symmetry is broken = Non-zero chiral condensatePion mass squared proportional to quark mass

Also for kaons,

Quark mass ratios can be predicted up to O(mq2).

{ })(1)( 222πππ mOmfdduumm du +−=++

)()(2

)()(

22

20

2

qdu

qdu

mOmmf

qq

mOmmBm

++−

=

++=

π

π

qq

),()4(31

),()(),()(

20

2

20

220

20

qsdu

qsdKqsuK

mOmmmBm

mOmmBmmOmmBm

+++=

++=++=+

η

Chiral Lagrangian


Low energy effective lagrangian is developed assumingSpontaneous breaking of chiral symmetryPion (and kaon, eta) to be the Nambu-Goldston boson

In the low energy regime, pions are only relevant dynamical degrees of freedom.

Given by a non-linear sigma model.Provides a systematic expansion in terms of mπ

2, p2; the leading order is given above.

( ) ( )2 2

† † †2

0

Tr Tr ,4 4

exp , 2a a

f fL D UD U U U

iU B mf

μμ χ χ

τ π χ

= + +

⎛ ⎞= =⎜ ⎟

⎝ ⎠

For full details, see Bernard’s lectures


Expansion in the pion field gives

Pion mass is obtained as mπ2=2B0m

A chain of interaction terms: 4π, 6π, etc.Loop corrections are calculable.

Pick up a factor of (mπ /4πf)2 or (p/4πf)2

Counter terms must also be added at order (mπ /4πf)2 or (p/4πf)2

introduce the low energy constants (LECs): L1~L10 at the one-loop level

[ ] ...))(())((6

1

)(2422

1

2

22

22

2

+∂∂−∂∂+

+−∂∂=

bbaabbaa

aaaaaa

f

fmmL

ππππππππ

ππππππ

μμ

μμ

ππμμ

For full details, see Bernard’s lectures

One-loop example


Pion self-energy

Log dependence m2ln(m2): called the chiral logarithm.Comes from the infrared end of the integral = long distance effect of (nearly massless) pion loop.Counter terms are necessary in order to renormalize the UV divergence.After subtracting the UV divergences

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−+=

⎥⎦

⎤⎢⎣

⎡+Λ

+Λ=−∫

1ln4ln2)4(

ln)4(

1)2(

2

2

2

2

22

222

2224

4

μπγ

επ

ππ

mm

mmm

mkikd

Cutoff regularization

Dimensional reg

⎥⎦

⎤⎢⎣

⎡+×++= )(

)4(const)(ln

)4(2112 4

2

2

2

2

2

2

02

ππππ

π πμπmO

fmm

fmmBm q

Counter terms


At the order (mπ /4πf)2 or (p/4πf)2, there are 10 possible counter terms

10 new parameters, L1~L10 = low energy constant at NLOc.f. 2 parameters at LO: Σ and f.Depends on how one renormalize the UV divergence, just as in the small coupling perturbation. L1~L10 depends on the renormalization scale μ.Once these parameters are determined (e.g. from pionscattering data), one can predict other quantities.

Lattice QCD may be used to calculate these parameters.

Quark mass ratio


At NLO, the quark mass ratio is given as

Assumes that the isospin breaking mu≠md is negligible.Requires the knowledge of the NLO LEC 2L8-L5.Results in ms/mud=25~30 (PDG 2006); large uncertainty due to the unknown LEC.

Comparison with the exp number gives LECs. But the predictive power is lost.Instead, lattice calculation can be used to fix LECs.

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

−−−+

+= )2()(8ln

)4(21ln

)4(211

2 582

22

2

2

2

2

2

2

2

2

2

2

LLf

mmmf

mmf

mm

mmmm K

ud

udsK πηηππ

π μπμπ

Isospin breaking


In the real world, π± and π0 have different masses; two sources

Small mass difference between up and down quarks.Electromagnetic effect: Qu=+2/3, Qd=-1/3.

Quark mass difference When mu≠md, π0 and η can mix

Then, from 2u d sM m uu m dd m ssπ π π⎡ ⎤= + +⎣ ⎦

0 0 200

0 20 0

3 (( ) )ˆ4

3 (( ) )ˆ4

d ud u

s

d ud u

s

m m O m mm m

m m O m mm m

π π η

η η π

−= + + −

−

−= − + −

−

0

0

0

1 ( )2

1 ( 2 )6

uu dd

uu dd ss

π

η

= −

= + −

0

22 0

0( )( )

ˆ4 ( )d u

u ds

B m mM B m mm mπ

−= + −

−

0

0 0

0

1 12 6

1 12 6

23

K

K

K K

π η π

π π η

η

+ +

−

−

⎛ ⎞+⎜ ⎟

⎜ ⎟⎜ ⎟

− +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠

(phys)

~ 0.2 MeV

c.f. ( ) 4.6 MeV

M

Mπ

π

Δ

Δ =

Electromagnetic effect


Self-energy with a photon propagatorUsing the PCAC relation, related to a current two point function (VV-AA).

Das-Guralnick-Low-Mathur-Young sum rule (1967), a close relative of the Weinberg sum rules (1967)

Sum rule estimate gives ΔMπ~5 MeV, comparative to the exp value 4.59 MeV. Lattice calculation is also possible.At the leading order, the same effect for kaon (Dashen’stheorem)

4 †( ) 0 T ( ) (0) 0iqxJJ q i d x e J x Jμν μ νΠ = ∫

2 2 2 2 22 0

3 ( ) ( )4

emVV AAM dQ Q Q Q

fπαπ

∞⎡ ⎤Δ = Π −Π⎣ ⎦∫

0 02 2 2 2K K

M M M Mπ π+ +− = −

Estimate of mu/md


At the leading order, Weinberg (1977)

Using the GMOR relation and the EM correction,

Combine them to obtain

0 0 0

0

0

0

2 2 2 2 2

2 2 2

2 2 2

2 2 2

( ) ( )0.55,

( )

( )20.1

( )

u K K

d K K

s K K

d K K

M M M M Mmm M M M

M M Mmm M M M

π π π

π

π

π

± ±

± ±

± ±

± ±

+ − − −= =

− −

+ −= =

− −

0

0

20

20

20

20

( )

( )

( )

( )

u d em

u d

u s emK

d sK

M B m m

M B m m

M B m m

M B m m

π

π

±

±

= + + Δ

= +

= + + Δ

= +

Dashen’s theorem

Further estimate of mu/md


At NLO, those simple relations are lost.NLO formula (Gasser-Leutwyler (1985))

A double ratio is free from the NLO correction

which can be written in a simple form

A slight ambiguity comes from a violation of the Dashen’s theorem.

{ }

{ }0

22

2

2 22

2 2

ˆ1 ( )

1 ( )ˆ

sKM

d u

d uK KM

K s

m mM O mM m m

M M m m O mM M m m

π

π

±

+= + Δ +

+

− −= + Δ +

− −

2 2

8 52

8( ) (2 ) logsKM

M M L Lf

π χ−Δ = − +

{ }0

2 2 2 222 2

2 2 2 2 2

ˆ1 ( )s KK

d u K K

m m M MMQ O mm m M M M

π

π ±

− −≡ = +

− −

2 2

2

1 1u s

d d

m mm Q m

⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠from Leutwyler, PLB378 (1996) 313.

NLO constraints


The NLO formula makes an ellipse.

Other constraints:ΔM>0, from large Nc.Another constraint on

from a charmonium decay

2 2

2

1 1u s

d d

m mm Q m

⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

ˆs

d u

m mRm m

−=

−

0

0

( ' )( ' )ψ ψπψ ψη

Γ →Γ →

III. Chiral dynamics2. Lattice calculation of light

quark masses


Inputs


In general, lattice QCD simulation requires inputs forLattice scale 1/a ⇒ determines αs(1/a)

Inputs discussed in Part I.Quark masses for each flavor

up and down quarks mud (often assumed to be degenerate)from pseudo-scalar meson mass mπ, good sensitivity because mπ

2~mud.Strange quark ms

from mK, for the same reason.Charm quark mc

either from D (heavy-light) or J/ψ (heavy-heavy) massBottom quark mb

either from B (heavy-light) or Υ (heavy-heavy) mass

Chiral extrapolation


Lattice simulation is harder for lighter sea quarks.Computational cost grows as mq

-n (n~2).Finite volume effect becomes more important ~exp(-mπL)

Practical calculation involves the chiral extrapolation. At the leading order, it is very simple:1. Fit the pseudo-scalar mass with2. Input the physical pion mass mπ0=135 MeV to get

mud=(mu+md)/2. (Forget about the isospin breaking for the moment.)

3. Renormalize it to the continuum scheme.

Including higher orders is non-trivial…

2 20 ( ) ( )u d qm B m m O mπ = + +

NLO example


Chiral expansion

LO (linearity) looks very good, but if you look more carefully NLO is visible.

mπ2/mq not constant.

Chiral log term has a definite coefficient = curvature fixed.Analytic term has an unknown constant, to be fitted with lattice data = linear slope

2 2 22

0 32 2 2

12 1 ln NNLO2 (4 ) (4 )q

m m mm B m cf fπ π π

ππ ππ μ π

⎡ ⎤= + + +⎢ ⎥

⎣ ⎦

JLQCD (2007)dynamical overlap (Nf=2)

Strange quark


Must consider 2+1-flavor theory.If your simulation contains only 2-flavors (up and down quarks), then a possible choice is to use the Partially Quenched ChPT. Although it is not the correct theory after all, it will provide a consistent description of the lattice data.

NLO effect is less pronounced for strange.

No singularity in the chiral limit. (This may not be the case for other quantities like fK.)Numerical analysis will be more stable.

2 22

0 3 4 2 2

2 22

0 3 4 2 2

1 1ˆ ˆ ˆ2 1 2 (2 ) , ln3 2 (4 )

2 1ˆ ˆ ˆ( ) 1 ( ) (2 ) , ln3 2 (4 )

s

K s s s

m mM mB mK m m Kf

m mM m m B m m K m m K

f

π ππ π η π

η ηη η

μ μ μπ μ

μ μπ μ

⎡ ⎤= + − + + + =⎢ ⎥⎣ ⎦

⎡ ⎤= + + + + + + =⎢ ⎥⎣ ⎦

A case study: MILC+HPQCD 2+1


Bare quark masses taken from the MILC 2+1 asqtad simulations.

Two lattice spacings “coarse” (a=0.125 fm) and “fine” (a=0.090 fm).Complicated fit including the taste-breaking effects of staggered fermion, vanishing at a=0.NNLO analytic terms are included. Non-analytic (chiral log) terms are discarded.

MILC+HPQCD, PRD70, 031504(R) (2004), MILC, PRD70, 114501 (2004).

Renormalization


Once the bare quark mass is fixed on the lattice, it must be converted to the continuum definition, because the pole mass is not adequate.

Just like the conversion of the coupling constant.May use the perturbation theory (Use the renormalized coupling!). But, in most cases, known only at the one-loop level. (Exceptions are HQET, Asqtad, stochastic PT(?).)

Non-perturbative renormalization is desirable.

convert:mMS(μ)=Zm(μa)mlat(a-1)

)(...]05.21[)( 11 −− ++== amam lats

MS αμEx). O(a)-improved Wilson fermion

MILC+HPQCD 2+1


Conversion is done perturbatively.Calculated to two-loop. HPQCD (Mason et al.), PRD73, 114501 (2006).

with q*=1.88/a.With αV(q*)=0.27, this series isZm = 1 + 0.03 + 0.16 + …The uncertainty from higher orders is estimated as 2αV

3(q*)~5%.After the continuum extrapolation with αVa2, they quote

* 2( 1) 1 0.119 ( ) (2.22 0.02 )m V f VZ a q nμ α α= = + + −

(2GeV) 87(0)(4)(4)(0) MeVMSsm =

A case study: QCDSF-UKQCD Nf=2


Another work by QCDSF-UKQCD with the O(a)-improved Wilson fermion, PRD73, 054508 (2006).

One may doubt about the convergence of the perturbativeexpansion. Non-perturbativerenormalization is desirable if possible. Non-perturbative renormalization is done using the RI/MOM scheme. It has its own subtlety: the renormalization constant is not really constant, due to SχSB.NP results are about 20% larger than the one-loop calculation.

21 2 2( ) ( )

( )mZ a c a ca

ψψμ μ

μ+ +

For more details, see the lectures by S. Sint.

QCDSF-UKQCD Nf=2


Lattice data fit to the one-loop PQχPT formula (but the magnitude of the χlog is a free parameter).

Conversion to MSbar is (partially) non-perturbative.

Continuum extrapolation is carried out with 4 data points. Substantial rise in the continuum limit.

(2GeV) 111(6)(4)(6) MeVMSsm =

Present status


A recent compilation by Knechtli, hep-ph/0511033.

Non-perturbativerenormalization yields higher ms?

orNf=3 gives lower ms?

Including other determinations (Plot from Davier, Hocker, Zhang, Rev. Mod. Phys.78,1043 (2006)).

Consistent within large errors.

Up and down quarks


Ratio to strangeBasically obtained by the ChPT formula.At NLO, equivalent to the calculation of the LEC.

MILC obtained ms/mud=27.4(1)(4)(1) from the SχPT fit. Much more delicate.mu/md can also be obtained (up to the EM uncertainty) from the relation

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

−−−+

+= )2()(8ln

)4(21ln

)4(211

2 582

22

2

2

2

2

2

2

2

2

2

2

LLf

mmmf

mmf

mm

mmmm K

ud

udsK πηηππ

π μπμπ

2 2

2

1 1u s

d d

m mm Q m

⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

III. Chiral dynamics3. Pion loop effects


Chiral log effects


We learned that the chiral extrapolation is non-trivial.Especially so, if pions are involved as external states

pion mass, decay constantform factors, ππ-scattering...

Even when pion does not appear as external states, it could be there in the loop. May lead to the chiral log.

Kaon decay constantNucleon (masses, matrix elements)Heavy-light meson (masses, decay constants, form factors)

Thus, could always be a delicate problem when one aims at good precision.

An example


Pion decay constantAt NNLO, it has the form

with ξ=mπ2/(4πfπ)2 .

Leading-log terms have known coefficients.Free parameters in the analytic term (NLO, NNLO) and NLL term (NNLO).Turns out that the chiral log effect is not substantial, but non-negligible.

JLQCD (2007)dynamical overlap (Nf=2)(talk by Noaki at lat07)

~ms/6 ~ms

Pion form factor


The simplest form factor

Momentum transfer qμ by a virtual photon. Space-like (q2<0) in the πe→πe process.Vector form factor FV(q2) normalized as FV(0)=1, because the vector current is conserved.

Vector (or EM) charge radius ⟨r2⟩Vπ is defined through the slope at q2=0.

2( ') ( ) ( ') ( ), 'Vp V p i p p F q q p pμ μ μ μ μ μπ π = + ≡ −

2 2 2 41( ) 1 ( ),6V V

F q r q O qπ

= + +

Lattice calculation of 3pt function


π(p)→π(p’)An interpolating operator for the initial state π(p) at t=t0

Another interpolating operator for the final state π(p’) at t=t1

Current insertion Vμ in the middle t.Spatial momentum inserted at two operators.

(sequential) source methodCalculate a quark propagator starting from a previous quark propagator at t.

t0 t1t

c.f. 2pt func

.2 1 0( ) ( ) ( ) ( )iD m S x e S x x tδ/ + = Γ −q x

Ground state?


Working on the Euclidean latticeOn-shell particle will never appear (except for the massless pion in the chiral limit).Instead, one calculates two-point function

This is a Fourier transform of the two-point function in the space-like regime.

All the information encoded in the space-like two-point function Π(q2).

1 0( )(2)1 0( , ) ~ m t tC t t Z e− −

00

/ /(2) 20 02 2 2/ /

0

( ) ~ ( )2 2

iq ta aiq t

a a

dq dq eC t q em q

π π

π ππ π+ +

− −Π =

+ +∫ ∫ q

2 2

1m q−

lattice

Lattice calculation of 3pt function


At large enough time separations Δt=t-t0, Δt’=t1-t, the ground state pions dominate.

Extra factors can be taken off with 2pt functions.

A recent calculation


Lattice signalLook for a plateau, where the ground state pion dominates. Noisier for larger pionmomentum.

Note:The actual data were obtained using the all-to-all technique, so that the data points at different t0,t,t1 and different momentum combinations can all be averaged.

JLQCD (2007)dynamical overlap (Nf=2)(talk by Kaneko at lat07)

A recent calculation


Many points corresponds to many momentum combinations (p, p’).

(1,0,0) → (0,1,0), … etc.in units of 2π/L.Too large p’s are contaminated by discretization effects (ap)2.

q2 dependence well approximated by a vector meson pole

with the independently calculated mV at the same quark mass.

mq~ms/2

2 212 2

1( ) ...1 / V

F q c qq mπ = + +

−

mq~ms/6

Analyticity


Vector meson dominance is understood using analyticity.

written in terms of the form factor in the time-like region t>0.

In the heavier quark mass region, ρmeson is a nearest isolated pole. ππ is subleading.For the physical quark mass, ππ is nearest. ρ is a part of ππ (broad resonance).

0

22 2

1 ( ) 1 Im ( )( )2 t

F t F tF q dt dti t q t qπ π

∞

= =− −∫ ∫

t

ππ

ρ

( ) ( ') 0p p Vμπ π t

ππ

ρ

Chiral extrapolation


Charge radius has a chiral log contribution.

Must diverge in the chiral limit: pion cloud gets larger.Valid only in the region where 2mπ<mρ.Lattice data actually increases towards the chiral limit. Chiral log further enhance its value.

<r2>Vπ = 0.388(9)(12) fm2

22 2 2

92 2

1 ln 12(4 ) ( )(4 )V

mr L O mf

π ππ

π

ππ μ

⎡ ⎤= − + +⎢ ⎥

⎣ ⎦π π

π π

γ

General problem


Chiral extrapolation is a serious issue in current lattice QCD studies. Questions arise…

How closely one must approach the chiral limit?One-loop enough? Two-loop needed?Finite volume effect might become significant.

How big is the effect of chiral symmetry violation of Wilson, twisted-mass, staggered and domain-wall fermions? Modified χPTs for these lattice actions contain many parameters. Possible to determine all of them to necessary precision?

Answer depends on the process, action, …

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