Chiral Dynamics How s and Why s 4 th lecture: a specific example Martin Mojžiš, Comenius University23 rd Students’ Workshop, Bosen, 3-8.IX.2006

Chiral DynamicsChiral DynamicsHowHowss and Why and Whyss

4th lecture: a specific example

Martin Mojžiš, Comenius University23rd Students’ Workshop, Bosen, 3-8.IX.2006

how strange is the nucleon?

• not at all, as to the strangeness SN = 0

• not that clear, as to the strangeness content

NdduuN

NssNy

2

• determination of y is a nice illustration of the use of

the effective theory in symbiosis with other techniques

23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

NssdduuNm

m

N

22

ˆbaryon octet masses

NdduuNm

m

N

2

ˆ)0(

pNdduupNmputpu ,',ˆ)()()'(

22 2,0 MtDF

N scattering (data)

N scattering (CD point)

2)'( ppt

Nmus 4)(

)2( 2 M

the story of 3 sigmas (none of them being the standard deviation)


baryon octet masses

)0(

N scattering (data)


)2( 2 M

26 MeV

64 MeV 64 MeV

64 MeV

Gell-Mann, Okubo Gasser, Leutwyler

Brown, Pardee, Peccei

data

Höhler et al.

simple LET

the story of 3 sigmas


26 MeV

64 MeV

NssdduuNm

m

N

22

ˆ

NdduuNm

m

N

2

ˆ

yNdduuN

NssN

2

ˆ

6.0

OOPS !

big y


NHNm

m QCDN

N 2

1

NssNm

mNdduuN

m

mNHN

m N

s

Nmassless

N 22

ˆ

2

1

2ˆy

m

ms NHNm massless

N2

1

26 0.364 MeV

376 MeV 64 MeV 500 MeV

big y is strange


NssmddmuumNNHNNHN sdumasslessQCD

big why

Why does QCD build up the lightest baryon using so much of such a heavy building block?

statesgluonandquarkiicN i

NHN massless

s d

ssdduumassless mmmNHN ###

sddduu mmm ###

does not work for s with a buddy d with the same quantum numbers

but why should every s have a buddy d with the same quantum numbers?23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

big y

• How reliable is the value of y ?

• What approximations were used to get the values of the three sigmas ?

• Is there a way to calculate corrections to the approximate values ?

• What are the corrections ?

• Are they going in the right directions ?

•Are they large enough to decrease y substantially ?

small y ?

?


N scattering (data)

)2( 2 M

SU(3)

SU(2)L SU(2)R

SU(2)L SU(2)R

analycity & unitarity

group theory

current algebra

current algebra

dispersion relations

the original numbers:


qmmmmmm

qH sdusmass

830

3

ˆ

23

ˆ2

the original numbers:

qmm

qH ssplitmass 8

3

ˆ

ssdduumm s 2

3

ˆ

• controls the mass splitting (PT, 1st order)• is controlled by the transformation properties

• of the sandwiched operator• of the sandwiching vector

hHh splitmass

hsplitmassH

cbNssdduuNmm

ms

N 2

12

3

ˆ

2

1

(GMO)

MeV26

1

41

1 2 IIYcbYam

cbmm

mm

MeVmmb

MeVmmc

s

s

N

21

ˆ ˆ131

26ˆ

19021

3921


the original numbers: 0 22 M

the tool: effective lagrangians (ChPT)chiral symmetry

d

uqext

d

u sm

ms

0

0

n

p

0s Bic 41 dependencetno constt

τs 00ss...4

8 02

0 ππsF

BBs

...55. qavipsDiqL fieldsextQCD

...

21

15. cugmiL A

fieldsexteffective


the original numbers: 22 M

0s)(

4 021 dependencemomentumnos

F

Bic ab

a

b

other contributions to the vertex: • one from , others with c2,c3,c4,c5

• all with specific p-dependence

• they do vanish at the CD point ( t = 2M2 )

for t = 2M2 (and = 0) both (t) and (part of) the N-scattering

are controlled by the same term in the Leff

...

21

15. cugmiL A

fieldsexteffective

...4

8 02

0 ππsF

BBs


the original numbers: MeV856

• a choice of a parametrization of the amplitude

• a choice of constraints imposed on the amplitude

• a choice of experimental points taken into account• a choice of a “penalty function” to be minimized

extrapolation from the physical region to unphysical CD point

• many possible choices, at different level of sophistication

• if one is lucky, the result is not very sensitive to a particular choice

• one is not• early determinations: Cheng-Dashen = 110 MeV, Höhler = 4223 MeV

• the reason: one is fishing out an intrinsically small quantity (vanishing for mu=md=0)

• the consequence: great care is needed to extract from data

• see original papers

• fixed-t dispersion relations

• old database (80-ties)• see original papers

KH analysis

underestimated error


N scattering (data)

)2( 2 M

SU(3)

SU(2)L SU(2)R

SU(2)L SU(2)R

analycity & unitarity

group theory

current algebra

current algebra

dispersion relations

corrections:

ChPT

ChPT

ChPT


corrections:

q

Nq

N

q

m

mmNqqN

m

m

2

Feynman-Hellmann theorem

BorasoyMeißner

• 2nd order Bb,q (2 LECs) GMO reproduced

• 3rd order Cb,q (0 LECs) 26 MeV 335 MeV

• 4th order Db,q (lot of LECs) estimated (resonance saturation)

MeV734

...2,

2/3,,

qqqb

qqqb

qqqbb mDmCmBAm


corrections: 0

3rd order Gasser, Sainio, Svarc

integral loop2

34

2

22

12 tItI

F

MgcMt A

MeVF

MgM A 7

64

302

2

322

4th order Becher, Leutwyler

LECorder 4 one with term1402 th2 MeVM

estimated from a dispersive analysis(Gasser, Leutwyler, Locher, Sainio)

MeVM 4.02.152 2


corrections: 22 M

3rd order Bernard, Kaiser, Meißner

MeVM CDCD 35.02 2

4th order Becher, Leutwyler

LECorder 4 one with term0 th MeVCD

large contributions in both (M

2) and canceling each other

MeV1

estimated


corrections: MeV760





• forward dispersion relations

• old database (80-ties)• see original papers

Gasser, Leutwyler, Sainio

forward disp. relations data = 0, t = 0

linear approximation = 0, t = 0 = 0, t = M2

less restrictive constrains

better control over error propagation23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

)0(

N scattering (data)


)2( 2 M

335 MeV (26 MeV)

447 MeV (64 MeV)

597 MeV (64 MeV)

607 MeV (64 MeV )

data

corrections:

24.025.0 y


new partial wave analysis: MeV90





• much less restrictive -

• up-to-date database +• see original papers

VPI


no conclusions:

• new analysis of the data is clearly called for

• redoing the KH analysis for the new data is quite a nontrivial task

• work in progress (Sainio, Pirjola)

• Roy equations used recently successfully for -scattering

• Roy-like equations proposed also for N-scattering




• Becher-Leutwyler• well under control• up-to-date database• not decided yet

Roy-like equations

• work in progress23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

Documents

Chiral Dynamics How s and Why s 4 th lecture: a specific example Martin Mojžiš, Comenius University23 rd Students’ Workshop, Bosen, 3-8.IX.2006