Determination of the light quark massesFor a more precise determination of the light quark masses it...

Determination of the light quark massesThe mu and md quark masses from η → 3π decay

Martin Zdrahal

Institute of Particle and Nuclear PhysicsFaculty of Mathematics and Physics

Charles University, Czech Republic

Work performed in cooperation with K. Kampf, M. Knecht and J. Novotny

[aka Prague-(Lund)-Marseille Dispersive Treatment].M. Zdrahal (Charles University in Prague) Determination of the light quark masses Mesonnet 2013, June 18 1 / 14

Determination of the light quark masses Methods

Determination of the current masses of u, d and s quarks

quark confinement prohibits their direct determination

determination – from comparison of theoret. prediction for some observabledepending on quark masses with the corresponding experimental value

Methods

effective theories for QCD – chiral perturbation theory

QCD sum-rules – analytic properties of hadronic spectral functions + OPE

lattice QCD

Advantages and disadvantages of SR and lattice

ms and m = mu+md

2determined independently with compatible results

isospin breaking effects on the observables studied by these methods generatedby mu −md difference and by EM interactions of the same order, however,elmag. corrections problematic for both of them

isospin breaking effects in both methods nowadays need additional input

M. Zdrahal (Charles University in Prague) Determination of the light quark masses Mesonnet 2013, June 18 2 / 14

Determination of the light quark masses Use of χPT computation of η → 3π decay rate

Use of χPT

can determine only quark mass ratios and needs some input fixing the physicaldefinition of the masses (Kaplan-Manohar ambiguity)

For a more precise determination of the light quark masses it is therefore usefulto combine isospin symmetric results of lattice QCD and sum rules with someisospin breaking study performed in χPT.

η → 3π in χPT

proceeds via IB effects; moreover direct effect of EM on the amplitude small⇒ proportional to md −mu ∼

1

R; measurement of its decay rate R

two-loop χPT computation exists, but

large corrections in first three successive ordersdiscrepancies between the experimentally measured and χPTpredicted Dalitz parameters (describing energy dependence)poor knowledge of Ci

it inspired studies of the origin of the discrepancy and its effect on R


η → 3π Alternative approaches

η → 3π: Alternative approaches

Origin of the discrepancy?

incorrect determination of NNLO LECs Ci, effect of resonances

higher-order final state rescatterings

influence of slow convergence of ππ scattering or η → 3π amplitude

unexpectedly large electromagnetic contribution, . . .

[B. Kubis, S. Schneider, C. Ditsche; S. Lanz, G. Colangelo, E. Passemar; M. Kolesar,J. Novotny; A. Nehme, S. Zein.]

Its influence on R determinationThe approaches taking different assumptions than ChPT do not fix normalization(there do not appear mq explicitly).⇒ in order to address this question, unavoidable to match to ChPT

⇒ need to find a region where both these approaches compatible.The only approaches addressing this question by employing the η → π+π−π0 so farare the two dispersive approaches.


η → 3π Dispersive approaches

Dispersive approaches – the main differences

isospin breaking parameters (m = mu+md

2, r = ms

m)

Q2 =m2

s−m2

m2

d−m2

u

← Q = 1

2R(r+1)→ R =

ms−m

md−m

u

at NLO

Advantages of using Q

• expressible using only QCD meson masses• reasonably stable w.r.t. Kaplan-Manohar

Advantages of using R

Nothing special (maybe better connectionto baryon physics?)




isospin breaking parameters (m = mu+md

2, r = ms

m)

Q2 =m2

s−m2

m2

d−m2

u

← Q = 1

2R(r+1)→ R =

ms−m

md−m

u

at NLO


• expressible using only QCD meson masses• reasonably stable w.r.t. Kaplan-Manohar


Nothing special (maybe better connectionto baryon physics?)

at higher orders


• both of them lost – the relation obtainsr-dependent corrections• better connection to NLO• historical reasons


• Direct connection to the two-loop χPTcomputation - not explicitly dependent onr (but determination of values of Li is)• Kaplan-Manohar has to be fixed fromelsewhere (Lis from Lattice, large Nc, . . . )




What do we have in common?

construct a parametrization by using dispersive relations and the two-particle unitarity⇒ dispersive

parameters stem from the subtraction polynomials of the dispersive relations

use that parametrization for correcting the chiral results or more-or-less fit theexperimental data in order to determine R

The most obvious differences:

Analytical perturbative dispersiveapproach of KKNZ

proceeds order by order in theconstruction of the amplitude

the original guiding principle was adirect correspondence to χ amplitude

all of the computations are performedanalytically

Numerical fully dispersive approach ofBern

searches a stable point of the dispersiverelations

the original guiding principle was toinclude ππ rescatterings to all orders

the computations are performednumerically




The form of the parametrization

M(s, t, u) = Normalization (Polynomial + Unitary part)

Normalization:

KKNZ

Constant

Bern

Omnes function containing elastic ππ

rescattering in S-channel

Polynomial:

6 parameters – a reasonable way how toestimate the error of neglecting higher

imaginary parts of the parameters – canbe added or neglected (good estimate)

contain many various contributions thatcannot be separated

with a good statistics of data a clean fit(up to the normalization) possible

6 parameters – less number ofparameters was insufficient

imaginary parts of the parameters –Taylor expand the amplitude and seemsits coefficients small or can be added

should be more stable w.r.t. iterations;effect of ππ rescattering separated

do they use the Adler staff really justfor fixing the normalization?






Normalization:

KKNZ

Constant

Bern



Unitary part:

stems from the ππ rescattering in all(crossed) channels

parameters from Polynomial multipliedby polynomials in Mandelstam variablesand by combinations of 5 kinematicalfunctions

depends on ππ scattering parameters

inclusion of higher than 2-loop ordersonly through addition of higher chiralparts of constants and physical ππ p.

stems from the ππ rescattering in T-and U- channels

parameters from Polynomial multipliedby a number of numerical integrals ofparts of the amplitudes from theprevious iterations

depends on ππ phase shifts

one can proceed with further iterationtill finding a stable point






Normalization:

KKNZ

Constant

Bern



Additional extensions:

Would you like to add inelastic contributionsfor the price of having more free parameters?O.K.Would you like to have results that take intoaccount mπ± 6= mπ0? O.K. with additionalfree parameters.


η → 3π Fits to our parametrization

Fits to our parametrization

Fit of a modelThe amplitude can be written in the form

M(s, t, u) = AxfA(s, t, u) +BxfB(s, t, u) + CxfC(s, t, u) +DxfD(s, t, u)

+ Ex(s− s0)3 + F

[

(t− s0)3 + (u− s0)

3]

,

where fi(s, t, u) are ”simple” functions of s, t, u and ππ scattering parameters.

⇒ simple linear fit of the model

Fit of NNLO ChPTWorked well.We have tried to add also higher order ππ rescattering by changing the unitarity part→ the Dalitz plot parameters moved in the right direction, but the difference wasreduced just by 10%. We would need to change also the polynomial part.


η → 3π Fits to our parametrization

Fits to our parametrization

Fit of a modelThe amplitude can be written in the form

M(s, t, u) = AxfA(s, t, u) +BxfB(s, t, u) + CxfC(s, t, u) +DxfD(s, t, u)

+ Ex(s− s0)3 + F

[

(t− s0)3 + (u− s0)

3]

,

where fi(s, t, u) are ”simple” functions of s, t, u and ππ scattering parameters.

⇒ simple linear fit of the model

Fit of an experiment

Data are on |M(s, t, u)|2, which is not linear but quadratic in the parameters.But it has worked so far for:

very optimistic data set constructed from KLOE 2008 (2500 data points)

realistic data set constructed from KLOE 2008 (154 points)

Analysis of real data from WasaCOSY 20 with 50 data points under progress.


η → 3π Correcting the NNLO chiral amplitude

Our analysis I: correcting values of Ci’s

Even though the individual Dalitzparameters differ significantly, there existCi-independent combinations of them andtheir central values are in concordance. ⇒

We study this possibility.

KLOE χPT

a −1.09± 0.02 −1.271 ± 0.075b 0.124 ± 0.012 0.394 ± 0.102d 0.057 ± 0.017 0.055 ± 0.057f 0.14 ± 0.02 0.025 ± 0.160g ∼ 0 0

α −0.030 ± 0.005 0.013 ± 0.016

Ci-independent combinations ofDalitz parameters (mπ± = mπ0)

1 rel1 : 4(b+ d)− a2 − 16α

2 rel2 : a3 − 4ab+ 4ad+ 8f − 8g

3 rel3 : β

KLOE χPT

rel1 0.02± 0.12 −0.03 ± 0.72rel2 0.12± 0.21 −0.13 ± 1.40rel3 ? −0.002 ± 0.025

Change of Ci’s reflects in a shift of the polynomial part of the amplitude ∆P (s, t, u).We fit the “corrected amplitude” to KLOE parametrization. ⇒ Reasonably small ∆P .

Optimistic KLOE distribution

R = 37.7± 2.9.

More conservative estimate

R = 30+14

−12.


η → 3π Direct fit of the KLOE

Analysis II.: Direct fit of the KLOE

imposing nothing about the origin of the Dalitz paramet. discepancy

the parametrization employs just general properties of M(s, t, u) and theobserved hierarchy of various contributions ⇒ Whatever the explanation of thediscrepancies is, we assume the physical amplitude fulfills these assumptions.

Physical normalization of the amplitude

We need at least 1 point from ChPT, where physics is reproduced well.



Where is the Adler zero?in the mu = md = 0 limit the amplitude in the point s = t = 0 (and its s = u

brother) has to be zero

this point can move anywhere in the distance of order O(m2π)

at that point the (real and imaginary parts of) amplitude is of order O(m2π), i.e.

if we use the naturality condition, it should be there significantly smaller than sayat the center of the Dalitz plot

no statement about chiral convergence of the amplitude

0.0

0.1

0.2

0.3

0.4

0.0

0.1

0.2

0.3

0.4



Physical normalization of the amplitude – results

Our requirements for the correct regions

the matching should be dependent on the values of Ci’s as less as possible; X

within this region the chiral expansion should work well;

also the higher corrections should be small there;

the physical amplitude should have the similar behavior as the chiral amplitudeinside the region.

The exact procedure for fixing the normalization

match only the imaginary part of the ChPT under the physical threshold






within this region the chiral expansion should work well; X


the physical amplitude should have the similar behavior as the chiral amplitudeinside the region.










the physical amplitude should have the similar behavior as the chiral amplitudeinside the region.X













fit on t = u cut for s ∈(

0.04GeV2, 4m2π

)







also the higher corrections should be small there;X





0.04GeV2, 4m2π

)

use the interpolation between the ObO and the “resummed” fit to ChPT







also the higher corrections should be small there;X





0.04GeV2, 4m2π

)

use the interpolation between the ObO and the “resummed” fit to ChPT

Result of this analysis: R = 39.6+2.5−5.1



Lesson in the extrapolations

Real part of the amplitude in the s = u cut





0.00 0.05 0.10 0.15 0.20

0

10

20

30

40





0.00 0.05 0.10 0.15 0.20

0

10

20

30

40

Statistical error in FM. Zdrahal (Charles University in Prague) Determination of the light quark masses Mesonnet 2013, June 18 13 / 14




0.00 0.05 0.10 0.15 0.20

0

10

20

30

40

Mild dependence on FM. Zdrahal (Charles University in Prague) Determination of the light quark masses Mesonnet 2013, June 18 13 / 14




0.00 0.05 0.10 0.15 0.20

0

10

20

30

40

0.00 0.05 0.10 0.15 0.20

0

5

10

15

Real part in the s = u cut Imaginary part in the s = u cut


η → 3π Final results

Combined result

These two analyses have differentassumptions and sources of errors,however, lead to compatible results, wecan combine them into

Final result for OPT distribution

R = 37.7 ± 2.2.

Estimating the error connected with thedependence on the “physical”distribution used in the fit, the moreconservative value is

Final result

R2 = 39.6+2.5−5.1.

Combination with the other constraints on quark masses from SR, Lattice, . . .

mu

md

r

0.40 0.45 0.50 0.55 0.6022

24

26

28

30

Our final results for mq characteristics

mu

md

= 0.50+0.02−0.05 [0.48(2)] ,

Q = 23.7+0.8−1.5 [23.2(7)] ,

quark masses in MS at µ = 2GeV

mu = 2.29+0.10−0.17 MeV [2.21(9)MeV] ,

md = 4.57+0.19−0.14 MeV [4.62(12)MeV] .


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