dyson-schwinger equations and quantum chromodynamics · · 2008-08-25first contents back...

First Contents Back Conclusion

Dyson-Schwinger Equationsand Quantum Chromodynamics

Craig D. Roberts

[email protected]

Physics Division

Argonne National Laboratory

http://www.phy.anl.gov/theory/staff/cdr.htmlCraig Roberts: Dyson Schwinger Equations and QCD

25th Students’ Workshop on Electromagnetic Interactions, 31/08 – 05/09, 2008. . . – p. 1/38


Room with a View

Room with a View

Craig Roberts: Dyson Schwinger Equations and QCD



Universal Truths




Universal Truths

Form factors give information about distribution of hadron’s

characterising properties amongst its QCD constituents.




Universal Truths



Calculations at Q2 > 1GeV2 require a Poincaré-covariant

approach.




Universal Truths




approach. Covariance requires existence of quark orbital

angular momentum in hadron’s rest-frame wave function.




Universal Truths






DCSB is most important mass generating mechanism for

matter in the Universe.




Universal Truths







matter in the Universe. Higgs mechanism is irrelevant to

light-quarks.




Universal Truths








light-quarks.

Challenge: understand relationship between parton properties

on the light-front and rest frame structure of hadrons.




Universal Truths








light-quarks.

Challenge: understand relationship between parton properties

on the light-front and rest frame structure of hadrons. Problem

because, e.g., DCSB - an established keystone of low-energy

QCD and the origin of constituent-quark masses - has not

been realised in the light-front formulation.Craig Roberts: Dyson Schwinger Equations and QCD



Form Factors: Why?




Form Factors: Why?

The nucleon and pion hold special places in non-perturbativestudies of QCD.




Form Factors: Why?


An explanation of nucleon and pion structure and interactions iscentral to hadron physics – they are respectively the archetypesfor baryons and mesons.




Form Factors: Why?



Form factors have long been recognized as a basic tool forelucidating bound state properties. They can be studied from verylow momentum transfer, the region of non-perturbative QCD, up toa region where perturbative QCD predictions can be tested.




Form Factors: Why?




Experimental and theoretical studies of nucleon electromagneticform factors have made rapid and significant progress during thelast several years, including new data in the time like region, andmaterial gains have been made in studying the pion form factor.




Form Factors: Why?




Experimental and theoretical studies of nucleon electromagneticform factors have made rapid and significant progress during thelast several years, including new data in the time like region, andmaterial gains have been made in studying the pion form factor.

Despite this, many urgent questions remain unanswered.Craig Roberts: Dyson Schwinger Equations and QCD



Some Questions

What is the role of pion cloud in nucleonelectromagnetic structure?

Can we understand the pion cloud in a morequantitative and, perhaps, model-independentway?




Some Questions

Where is the transition from non-pQCD to pQCD inthe pion and nucleon electromagnetic formfactors?




Some Questions

Do we understand the high Q2 behavior of theproton form factor ratio in the space-like region?

Can we make model-independent statementsabout the role of relativity or orbital angularmomentum in the nucleon?




Some Questions

Can we understand the rich structure of thetime-like proton form factors in terms ofresonances?

What do we expect for the proton form factor ratioin the time-like region?

What is the relation between proton and neutronform factor in the time-like region?

How do we understand the ratio between time-likeand space-like form factors?




Some Questions

What is the role of two-photon exchangecontributions in understanding the discrepancybetween the polarization and Rosenbluthmeasurements of the proton form factor ratio?

What is the impact of these contributions on otherform factor measurements?




Some Questions

How accurately can the pion form factor beextracted from the ep → e′nπ+ reaction?




Status




StatusCurrent status is described in

J. Arrington, C. D. Roberts and J. M. Zanotti“Nucleon electromagnetic form factors,”J. Phys. G 34, S23 (2007); [arXiv:nucl-th/0611050].

C. F. Perdrisat, V. Punjabi and M. Vanderhaeghen,“Nucleon electromagnetic form factors,”Prog. Part. Nucl. Phys. 59, 694 (2007);[arXiv:hep-ph/0612014].




StatusCurrent status is described in

J. Arrington, C. D. Roberts and J. M. Zanotti“Nucleon electromagnetic form factors,”J. Phys. G 34, S23 (2007); [arXiv:nucl-th/0611050].

C. F. Perdrisat, V. Punjabi and M. Vanderhaeghen,“Nucleon electromagnetic form factors,”Prog. Part. Nucl. Phys. 59, 694 (2007);[arXiv:hep-ph/0612014].

Most recently:“ECT∗ Workshop on Hadron Electromagnetic Form Factors”Organisers: Alexandrou, Arrington, Friedrich, Maas, RobertsPresentations, etc., available on-linehttp://ect08.phy.anl.gov/




Answer for the pion




Answer for the pion

Two → Infinitely many . . .




Answer for the pion

Two → Infinitely many . . .Handle thatproperly inquantumfield theory




Answer for the pion

Two → Infinitely many . . .Handle thatproperly inquantumfield theory. . .momentum-dependentdressing




Answer for the pion

Two → Infinitely many . . .Handle thatproperly inquantumfield theory. . .momentum-dependentdressing. . .perceiveddistribution ofmass dependson the resolving scale




Pion Form Factor

Procedure Now Straightforward




Pion Form Factor

Solve Gap Equation⇒ Dressed-Quark Propagator, S(p)

Σ=

D

γΓS




Pion Form Factor

Use that to Complete Bethe Salpeter Kernel, K

Solve Homogeneous Bethe-Salpeter Equation for PionBethe-Salpeter Amplitude, Γπ




Pion Form Factor

Use that to Complete Bethe Salpeter Kernel, K

Solve Homogeneous Bethe-Salpeter Equation for PionBethe-Salpeter Amplitude, Γπ

Solve Inhomogeneous Bethe-Salpeter Equation forDressed-Quark-Photon Vertex, Γµ




Pion Form Factor

Now have all elements for Impulse Approximation toElectromagnetic Pion Form factor

Γπ(k;P )

Γµ(k;P )

S(p)




Pion Form Factor

Now have all elements for Impulse Approximation toElectromagnetic Pion Form factor

Γπ(k;P )

Γµ(k;P )

S(p)

Evaluate this final,three-dimensional integral




Calculated Pion Form Factor

0 1 2 3 4

Q2 [GeV

2]

0

0.1

0.2

0.3

0.4

0.5

Q2 F

π(Q2 )

[GeV

2 ]

Amendolia et al.Ackermann et al.Brauel et al.Tadevosyan et al.

Horn et al.Maris and Tandy, 2005

Calculation first published in 1999; No Parameters VariedNumerical method improved in 2005




Calculated Pion Form Factor

0 1 2 3 4

Q2 [GeV

2]

0

0.1

0.2

0.3

0.4

0.5

Q2 F

π(Q2 )

[GeV

2 ]

Amendolia et al.Ackermann et al.Brauel et al.Tadevosyan et al.

Horn et al.Maris and Tandy, 2005

Calculation first published in 1999; No Parameters VariedNumerical method improved in 2005

Data publishedin 2001.Subsequentlyrevised




Timelike Pion Form Factor




Timelike Pion Form FactorAb initio calculation into timelike regionDeeper than ground-state ρ-meson pole





-1 -0.5 0 0.5 1 1.5 2 2.5 3

Q2 [GeV

2]

10-1

100

101

|Fπ(Q

2 )|Amendolia et al.Ackermann et al.Brauel et al.Tadevosyan et al.

Horn et al.Barkov et al.DSE calculation

Ab initio calculation into timelike regionDeeper than ground-state ρ-meson pole





-1 -0.5 0 0.5 1 1.5 2 2.5 3

Q2 [GeV

2]

10-1

100

101

|Fπ(Q

2 )|Amendolia et al.Ackermann et al.Brauel et al.Tadevosyan et al.

Horn et al.Barkov et al.DSE calculation

Ab initio calculation into timelike regionDeeper than ground-state ρ-meson poleρ-meson not put in “by hand” – generated dynamically as a bound-state of dressed-quark and dressed-antiquark




Dimensionless product: rπ fπ





Improved rainbow-ladder interaction






Repeating Fπ(Q2) calculation







Great strides towards placing nucleon studies on same

footing as mesons







Experimentally: rπfπ = 0.315 ± 0.005








DSE prediction








DSE prediction

Lattice results

– James Zanotti [UK QCD]








DSE prediction

Lattice results

– James Zanotti [UK QCD]

Fascinating result:

DSE and Lattice

– Experimental value

obtains independent of

current-quark mass.Craig Roberts: Dyson Schwinger Equations and QCD







DSE prediction

Fascinating result:

DSE and Lattice

– Experimental value

obtains independent of

current-quark mass.

We have understood this

Implications far-reaching.Craig Roberts: Dyson Schwinger Equations and QCD


First Contents Back ConclusionCraig Roberts: Dyson Schwinger Equations and QCD



New Challenges




New Challenges

Next Steps . . . Applications to excited states and

axial-vector mesons, e.g., will improve understanding of

confinement interaction between light-quarks.




New Challenges

Next Steps . . . Applications to excited states and

axial-vector mesons, e.g., will improve understanding of

confinement interaction between light-quarks.

Move on to the problem of a symmetry preserving treatment

of hybrids and exotics.




New Challenges

Another Direction . . . Also want/need information about

three-quark systems




New Challenges


three-quark systems

With this problem . . . most wide-ranging studies employ

expertise familiar from meson applications circa ∼1995.




New Challenges


three-quark systems



Namely . . . Model-building and Phenomenology,

constrained by the DSE results outlined already.Craig Roberts: Dyson Schwinger Equations and QCD



New Challenges


three-quark systems



However, that is beginning to change . . .




Nucleon . . .Three-body Problem?





What is the picture in quantum field theory?





What is the picture in quantum field theory?

Three →infinitelymany!




Faddeev equation




Faddeev equation

=aΨ

P

pq

pd Γb

Γ−a

pd

pq

bΨP

q




Faddeev equation

=aΨ

P

pq

pd Γb

Γ−a

pd

pq

bΨP

q

Linear, Homogeneous Matrix equation

Yields wave function (Poincaré Covariant Faddeev

Amplitude) that describes quark-diquark relative motion

within the nucleon

Scalar and Axial-Vector Diquarks . . . In Nucleon’s Rest

Frame Amplitude has . . . s−, p− & d−wave correlations




Diquark correlations

QUARK-QUARKCraig Roberts: Dyson Schwinger Equations and QCD



Diquark correlations

QUARK-QUARK

Same interaction that

describes mesons also

generates three coloured

quark-quark correlations:

blue–red, blue–green,

green–red

Confined . . . Does not

escape from within baryon.

Scalar is isosinglet,

Axial-vector is isotriplet

DSE and lattice-QCD

m[ud]0+

= 0.74 − 0.82

m(uu)1+

= m(ud)1+

= m(dd)1+

= 0.95 − 1.02




Nucleon EM Form Factors: A PrécisHoll, et al. : nu-th/0412046 & nu-th/0501033



http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+key+6076386



Nucleon EM Form Factors: A Précis






Nucleon EM Form Factors: A Précis



http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0710.2059




Nucleon EM Form Factors: A PrécisCloet, et al. :arXiv:0710.2059, arXiv:0710.5746 & arXiv:0804.3118








• Interpreting expts. with GeV electromagnetic probes

requires Poincaré covariant treatment of baryons










⇒ Covariant dressed-quark Faddeev Equation











• Excellent mass spectrum (octet and decuplet)

Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%












Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%

(Oettel, Hellstern, Alkofer, Reinhardt: nucl-th/9805054)












Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%

• But is that good?












Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%


• Cloudy Bag: δMπ−loop+ = −300 to −400 MeV!












Easily obtained:(

1

NH

∑

H

[M expH − M calc

H ]2

[M expH ]2

)1/2

= 2%


• Cloudy Bag: δMπ−loop+ = −300 to −400 MeV!

• Critical to anticipate pion cloud effects

Roberts, Tandy, Thomas, et al., nu-th/02010084








Nucleon’s self-energy - pion loop





Σ(P ) = 3

∫

d4k

(2π)4g2PV (P, k)∆π((P − k)2)

× γ · (P − k)γ5 G(k) γ · (P − k)γ5

= iγ · k [A(k2) − 1] + B(k2)





Σ(P ) = 3

∫

d4k

(2π)4g2PV (P, k)∆π((P − k)2)

× γ · (P − k)γ5 G(k) γ · (P − k)γ5

= iγ · k [A(k2) − 1] + B(k2)

• Pseudovector coupling





Σ(P ) = 3

∫

d4k

(2π)4g2PV (P, k)∆π((P − k)2)

× γ · (P − k)γ5 G(k) γ · (P − k)γ5

= iγ · k [A(k2) − 1] + B(k2)

• Pseudovector coupling

• Completely equivalent to pseudoscalar coupling

IF that is treated completely

• Tadpole contribution can’t be neglected

(Hecht, Oettel, Roberts, Schmidt, Tandy, Thomas: nucl-th/0201084)





Σ(P ) = 3

∫

d4k

(2π)4g2PV (P, k) ∆π((P − k)2)

× γ · (P − k)γ5 G(k) γ · (P − k)γ5

= iγ · k [A(k2) − 1] + B(k2)

gPV (P, k), πN vertex function

Calculated using Γπ and ΨN

Always soft: Monopole λ ∼ 0.6 GeV





Σ(P ) = 3

∫

d4k

(2π)4g2PV (P, k) ∆π((P − k)2)

× γ · (P − k)γ5 G(k) γ · (P − k)γ5

= iγ · k [A(k2) − 1] + B(k2)

gPV (P, k), πN vertex function

Calculated using Γπ and ΨN

Always soft: Monopole λ ∼ 0.6 GeV

Corresponds to range rλ ∼ 0.8 fm

. . . pion cloud does not

penetrate deeply within nucleon. 0.0 0.5 1.0 1.5k (GeV)

0.0

0.2

0.4

0.6

0.8

1.0

u2 (k

)Craig Roberts: Dyson Schwinger Equations and QCD




Σ(P ) = 3

∫

d4k

(2π)4g2PV (P, k)∆π((P − k)2)

× γ · (P − k)γ5 G(k) γ · (P − k)γ5

= iγ · k [A(k2) − 1] + B(k2)

G(k) = 1/[iγ · k + M + Σ(P )] Pole Position Not

= −iγ · k σV(k2) + σS(k2) Known a priori

Mass shift calculated via self-consistent solution





Σ(P ) = 3

∫

d4k

(2π)4g2PV (const.)∆π((P − k)2)

× γ · (P − k)γ5 G(k) γ · (P − k)γ5

= iγ · k [A(k2) − 1] + B(k2)

Obtain Integral Equation Kernels∫

dΩk f((P − k)2) =2

π

∫ 1

−1dz√

1 − z2 f(P 2 + k2 − 2Pkz)

E.g.ωB(P 2, k2) =

∫

dΩk(P − k)2

(P − k)2 + m2π

= 1 − 2m2π

a +√

a2 − b2,

a = P 2 + k2 + m2π, b = 2Pk





Σ(P ) = 3

∫

d4k

(2π)4g2PV (P, k)∆π((P − k)2)

× γ · (P − k)γ5 G(k) γ · (P − k)γ5

= iγ · k [A(k2) − 1] + B(k2)

But gPV = gPV (P 2, k2, P · k)

Therefore, In General, Kernel only known Numerically

Complicates analysis . . .

locating, incorporating poles in integrand




Nucleon Self Energy: Chiral Limit

Hecht, et al., nu-th/0201084






Let’s look what happens when mπ → 0

Minkowski Space

Pseudovector Coupling







Minkowski Space


One-loop nucleon self energy

Σ(P) = 3ig2

4M2

∫

d4k

(2π)4∆(k2, m2

π) 6kγ5 G0(P − k) 6kγ5 .

This integral is divergent. Assume a Poincaré covariant regularisation,characterised by a mass-scale λ







Minkowski Space



Σ(P) = 3ig2

4M2

∫

d4k

(2π)4∆(k2, m2

π) 6kγ5 G0(P − k) 6kγ5 .

This integral is divergent. Assume a Poincaré covariant regularisation,characterised by a mass-scale λ

Decompose nucleon propagator into positive and negative energy components

G0(P ) =1

6P − M0

= G+

0(P ) + G−

0(P )

=M

ωN ( ~P )

[

Λ+( ~P )1

P0 − ωN ( ~P ) + iε+ Λ−( ~P )

1

P0 + ωN ( ~P ) − iε

]

(4)

ω2N (~P) = ~P2 + M2, and Λ±(~P) = ( 6 P ± M)/(2M), P = (ω(~P), ~P)







Σ(P) = 3ig2

4M2

∫

d4k

(2π)4∆(k2, m2

π) 6kγ5 G0(P − k) 6kγ5 .







Σ(P) = 3ig2

4M2

∫

d4k

(2π)4∆(k2, m2

π) 6kγ5 G0(P − k) 6kγ5 .

Shift in the mass of a positive energy nucleon nucleon:

δM+ = 12trD

[

Λ+(~P = 0) Σ(P0 = M, ~P = 0)]







Σ(P) = 3ig2

4M2

∫

d4k

(2π)4∆(k2, m2

π) 6kγ5 G0(P − k) 6kγ5 .


δM+ = 12trD

[

Λ+(~P = 0) Σ(P0 = M, ~P = 0)]

Focus on positive energy nucleon’s contribution to the loop integral; i.e.,∆(k) G+(P − k), which we denote: δF M

++







Σ(P) = 3ig2

4M2

∫

d4k

(2π)4∆(k2, m2

π) 6kγ5 G0(P − k) 6kγ5 .


δM+ = 12trD

[

Λ+(~P = 0) Σ(P0 = M, ~P = 0)]

Focus on positive energy nucleon’s contribution to the loop integral; i.e.,∆(k) G+(P − k), which we denote: δF M

++

To evaluate k0 integral, close contour in lower half-plane, thereby encirclingonly the positive-energy pion pole.

δF M++ = −3g2

∫

d3k

(2π)3ωN (~k2) − M0

4 ωN (~k2)

1

ωπ(~k2) [ωπ(~k2) + ωN (~k2) − M0](9)






δF M++ = −3g2

∫

d3k

(2π)3ωN (~k2) − M0

4 ωN (~k2)

1

ωπ(~k2) [ωπ(~k2) + ωN (~k2) − M0](10)






δF M++ = −3g2

∫

d3k

(2π)3ωN (~k2) − M0

4 ωN (~k2)

1

ωπ(~k2) [ωπ(~k2) + ωN (~k2) − M0](14)

On the domain for which the regularised integral has significant support, assumethat M0 is very much greater than all other mass scales.

ωN (~k2) − M ≈~k2

2 M(15)






δF M++ = −3g2

∫

d3k

(2π)3ωN (~k2) − M0

4 ωN (~k2)

1

ωπ(~k2) [ωπ(~k2) + ωN (~k2) − M0](18)


ωN (~k2) − M ≈~k2

2 M(19)

Then δF M++ ≈ −3g2

∫

d3k

(2π)3

~k2

8 M2

1

ω2λi

(~k2)(20)






δF M++ = −3g2

∫

d3k

(2π)3ωN (~k2) − M0

4 ωN (~k2)

1

ωπ(~k2) [ωπ(~k2) + ωN (~k2) − M0](22)


ωN (~k2) − M ≈~k2

2 M(23)


∫

d3k

(2π)3

~k2

8 M2

1

ω2λi

(~k2)(24)

So thatd2 δF M

++

(dm2π)2

≈ −3g2

4M2

∫

d3k

(2π)3

~k2

ω6π(~k2)

= −9

128π

g2

M2

1

mπ. (25)






δF M++ = −3g2

∫

d3k

(2π)3ωN (~k2) − M0

4 ωN (~k2)

1

ωπ(~k2) [ωπ(~k2) + ωN (~k2) − M0](26)


ωN (~k2) − M ≈~k2

2 M(27)


∫

d3k

(2π)3

~k2

8 M2

1

ω2λi

(~k2)(28)

So thatd2 δF M

++

(dm2π)2

≈ −3g2

4M2

∫

d3k

(2π)3

~k2

ω6π(~k2)

= −9

128π

g2

M2

1

mπ. (29)

Namely δF M++ = −

3

32π

g2

M2m3

π + f+(1)

(λ1, λ2) m2π + f

+(0)

(λ1, λ2)

where the last two terms express the necessary contribution from the regulator.






Nucleon’s self energy

δF M++ = −

3

32π

g2

M2m3

π + f+(1)

(λ1, λ2) m2π + f

+(0)

(λ1, λ2)







δF M++ = −

3

32π

g2

M2m3

π + f+(1)

(λ1, λ2) m2π + f

+(0)

(λ1, λ2)

Given that m2π ∝ m in the neighbourhood of the chiral limit, the m3

π isnonanalytic in the current-quark mass on this domain.







δF M++ = −

3

32π

g2

M2m3

π + f+(1)

(λ1, λ2) m2π + f

+(0)

(λ1, λ2)



This is the Leading Nonanalytic Contribution much touted in effective fieldtheory.

Its form is completely fixed by chiral symmetry and the pattern of itsdynamical breaking.

NB. Contribution from negative energy nucleon is ∝1

M3.







δF M++ = −

3

32π

g2

M2m3

π + f+(1)

(λ1, λ2) m2π + f

+(0)

(λ1, λ2)



This is the Leading Nonanalytic Contribution much touted in effective fieldtheory.

Its form is completely fixed by chiral symmetry and the pattern of itsdynamical breaking.

NB. Contribution from negative energy nucleon is ∝1

M3.

The remaining terms are regular in the current-quark mass. Their exact naturedepends on the explicit form of regularisation procedure.







δF M++ = −

3

32π

g2

M2m3

π + f+(1)

(λ1, λ2) m2π + f

+(0)

(λ1, λ2)



The Leading Nonanalytic Contribution is a model-independent result.







δF M++ = −

3

32π

g2

M2m3

π + f+(1)

(λ1, λ2) m2π + f

+(0)

(λ1, λ2)



The Leading Nonanalytic Contribution is a model-independent result.

Unfortunately, it is of limited relevance. In a calculation of the nucleon’s mass, theactual value of the pion loop’s contribution is almost completely determined by theregularisation dependent terms.

It is essential for a framework to veraciously express the leading nonanalyticcontribution . . . it serves as a check that DCSB is truly described.

However, beyond that, one must accept that the world is complex.The pion has a finite size. So does the nucleon.These sizes set the mass-scale which determines the nucleon’s massshift.




Model pion-nucleon coupling





gPV (P, k) =g

2Mexp(−(P − k)2/Λ2)





gPV (P, k) =g

2Mexp(−(P − k)2/Λ2)

• B-Kernel∫

dΩk g2PV ((P − k)2)

[

1 − 2m2π

(P − k)2 + m2π

]

Clearly the sum of two independent terms.





gPV (P, k) =g

2Mexp(−(P − k)2/Λ2)

• B-Kernel∫


[

1 − 2m2π

(P − k)2 + m2π

]

• First term can be evaluated exactly

g2PV (P 2, k2) =

∫


=g2

4M2e−2(P 2+k2)/Λ2 Λ2

2PkI1(4Pk/Λ2) ,





gPV (P, k) =g

2Mexp(−(P − k)2/Λ2)

• B-Kernel∫


[

1 − 2m2π

(P − k)2 + m2π

]

• Second term can be approximated

ωg2(P 2, k2) = 2m2π

∫

dΩkg2PV ((P − k)2)

(P − k)2 + m2π

≈ g2PV (|P − k|2) 2m2

π

a +√

a2 − b2

• Reliable when analytic

structure of gPV is not key to that of solutionCraig Roberts: Dyson Schwinger Equations and QCD




gPV (P, k) =g

2Mexp(−(P − k)2/Λ2)

• B-Kernel∫


[

1 − 2m2π

(P − k)2 + m2π

]

• Total Kernel:

≈ g2PV (P 2, k2) − g2

PV (|P 2 − k2|) 2m2π

a +√

a2 − b2,

=: g2PV (P 2, k2) − g2

PV (P 2, k2))2m2

π

a +√

a2 − b2,

• Analytic structure is transparent




Nucleon’s self energy and mass shift





• Solve DSE Nonperturbatively

M2DA2(−M2

D) = [M + B(−M2D)]2

δM = MD − M






M2DA2(−M2

D) = [M + B(−M2D)]2

δM = MD − M

• Vector self energy






M2DA2(−M2

D) = [M + B(−M2D)]2

δM = MD − M

• Scalar self energy






M2DA2(−M2

D) = [M + B(−M2D)]2

δM = MD − M

(Λ,ΛN ) (Λ,ΛN ) (Λ,ΛN )

(0.9,∞) (0.9, 1.5) (0.9, 2.0)

− δM (MeV) 222 61 99






M2DA2(−M2

D) = [M + B(−M2D)]2

δM = MD − M

(Λ,ΛN ) (Λ,ΛN ) (Λ,ΛN )

(0.9,∞) (0.9, 1.5) (0.9, 2.0)

− δM (MeV) 222 61 99

No suppression for nucleon off-shell in self-energy loop;

i.e, gPV ((P − k2), P 2, k2)

Neglected this dependence






M2DA2(−M2

D) = [M + B(−M2D)]2

δM = MD − M

(Λ,ΛN ) (Λ,ΛN ) (Λ,ΛN )

(0.9,∞) (0.9, 1.5) (0.9, 2.0)

− δM (MeV) 222 61 99

gPV (P 2, k2, P · k) =g

2Me−(P−k)2/Λ2

e−(P 2+M2+k2+M2)/Λ2N

Correct on-shell limit:

gPV (P 2 = −M2, k2 = −M2, (P − k)2 = 0) =g

2MCraig Roberts: Dyson Schwinger Equations and QCD





M2DA2(−M2

D) = [M + B(−M2D)]2

δM = MD − M Range from mesonexchange model phen.

(Λ,ΛN ) (Λ,ΛN ) (Λ,ΛN )

(0.9,∞) (0.9, 1.5) (0.9, 2.0)

− δM (MeV) 222 61 99

gPV (P 2, k2, P · k) =g

2Me−(P−k)2/Λ2

e−(P 2+M2+k2+M2)/Λ2N

ΛN → ∞ ⇒ pointlike nucleon




Pion loop’s effect

Nonpointlike πN -loop

. . . reduces nucleon’s mass by ∼ 100 MeV







There’s also a π∆-loop

. . . reduces nucleon’s mass by not more than 100 MeV









−δMN ∼ 200 MeV









−δMN ∼ 200 MeV

Qualitative effect of this?




Too much of a good thing





• Refit Faddeev model parameters,

allowing for heavier “quark-core” mass





ω0+ ω1+ MN M∆ ωf1ωf2

R

0+ 0.45 - 1.44 - 0.36 0.35 2.32

0+ & 1+ 0.45 1.36 1.14 1.33 0.44 0.36 0.54

0+ 0.64 - 1.59 - 0.39 0.41 1.28

0+ & 1+ 0.64 1.19 0.94 1.23 0.49 0.44 0.25

50% reduction in role of axial-vector diquark






R

0+ 0.45 - 1.44 - 0.36 0.35 2.32

0+ & 1+ 0.45 1.36 1.14 1.33 0.44 0.36 0.54

0+ 0.64 - 1.59 - 0.39 0.41 1.28

0+ & 1+ 0.64 1.19 0.94 1.23 0.49 0.44 0.25

50% reduction in role of axial-vector diquark

10% increase in role of scalar diquark






R

0+ 0.45 - 1.44 - 0.36 0.35 2.32

0+ & 1+ 0.45 1.36 1.14 1.33 0.44 0.36 0.54

0+ 0.64 - 1.59 - 0.39 0.41 1.28

0+ & 1+ 0.64 1.19 0.94 1.23 0.49 0.44 0.25

Unsurprisingly:

Requiring Exact Fit to N , ∆ masses

with only q, (qq)JP Degrees of Freedom

⇒ Forces 1+ to mimic, in part, effect of π




Pseudoscalar mesonsand Form Factors





Light mass of pseudoscalar mesons means they play a very

important role in many aspects of hadron physics.







Indeed, no approach to low-energy hadron physics that

does not explicitly account for pseudoscalar meson degrees

of freedom can be valid.










Another example . . . pseudoscalar mesons also contribute

materially to form factors.










Another example . . . pseudoscalar mesons also contribute

materially to form factors.

Illustrate with γN → ∆ transition form factor. Focus on the

M1 (spin-flip) form factor, G∗M(Q2).




Harry LeePions and Form Factors




http://www.slac.stanford.edu/spires/find/hep/www?key=3365395





Dynamical coupled-channels model . . . Analyzed extensive JLabdata . . . Completed a study of the ∆(1236)

Meson Exchange Model for πN Scattering and γN → πN Reaction, T. Satoand T.-S. H. Lee, Phys. Rev. C 54, 2660 (1996)

Dynamical Study of the ∆ Excitation in N(e, e′π) Reactions, T. Sato andT.-S. H. Lee, Phys. Rev. C 63, 055201/1-13 (2001)












Pion cloud effects are large in the low Q2 region.

0

1

2

3

0 1 2 3 4

Q2(GeV/c)2

DressedBare

Ratio of the M1 form factor in γN → ∆

transition and proton dipole form factor GD .Solid curve is G∗

M(Q2)/GD(Q2) including

pions; Dotted curve is GM (Q2)/GD(Q2)

without pions.












Pion cloud effects are large in the low Q2 region.

0

1

2

3

0 1 2 3 4

Q2(GeV/c)2

DressedBare

Ratio of the M1 form factor in γN → ∆

transition and proton dipole form factor GD .Solid curve is G∗

M(Q2)/GD(Q2) including

pions; Dotted curve is GM (Q2)/GD(Q2)

without pions.

Quark Core

Responsible for only 2/3 ofresult at small Q2

Dominant for Q2 >2 – 3 GeV2








Results: Nucleonand ∆ Masses





Mass-scale parameters (in GeV)

for the scalar and axial-vector

diquark correlations, fixed by

fitting nucleon and ∆ masses

Set A – fit to the actual masses was required; whereas for

Set B – fitted mass was offset to allow for “π-cloud” contributions

set MN M∆ m0+ m1+ ω0+ ω1+

A 0.94 1.23 0.63 0.84 0.44=1/(0.45 fm) 0.59=1/(0.33 fm)

B 1.18 1.33 0.80 0.89 0.56=1/(0.35 fm) 0.63=1/(0.31 fm)

m1+ → ∞: MAN = 1.15 GeV; MB

N = 1.46 GeV












A 0.94 1.23 0.63 0.84 0.44=1/(0.45 fm) 0.59=1/(0.33 fm)

B 1.18 1.33 0.80 0.89 0.56=1/(0.35 fm) 0.63=1/(0.31 fm)

m1+ → ∞: MAN = 1.15 GeV; MB

N = 1.46 GeV

Axial-vector diquark provides significant attractionCraig Roberts: Dyson Schwinger Equations and QCD











A 0.94 1.23 0.63 0.84 0.44=1/(0.45 fm) 0.59=1/(0.33 fm)

B 1.18 1.33 0.80 0.89 0.56=1/(0.35 fm) 0.63=1/(0.31 fm)

m1+ → ∞: MAN = 1.15 GeV; MB

N = 1.46 GeV

Constructive Interference: 1++-diquark + ∂µπCraig Roberts: Dyson Schwinger Equations and QCD



Nucleon-Photon Vertex





M. Oettel, M. Pichowskyand L. von Smekal, nu-th/9909082

6 terms . . .constructed systematically . . . current conserved automatically

for on-shell nucleons described by Faddeev Amplitude





M. Oettel, M. Pichowskyand L. von Smekal, nu-th/9909082

6 terms . . .constructed systematically . . . current conserved automatically

for on-shell nucleons described by Faddeev Amplitude

i

iΨ ΨPf

f

P

Q i

iΨ ΨPf

f

P

Q

i

iΨ ΨPPf

f

Q

Γ−

Γ

scalaraxial vector

i

iΨ ΨPf

f

P

Q

µ

i

i

X

Ψ ΨPf

f

Q

P Γ−

µi

i

X−

Ψ ΨPf

f

P

Q

ΓCraig Roberts: Dyson Schwinger Equations and QCD



Form Factor Ratio:GE/GM

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp

Rosenbluthprecision Rosenbluthpolarization transferpolarization transfer




Form Factor Ratio:GE/GMCombine these elements . . .

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp






Dressed-Quark Core

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp






Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp






Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion

Cloud’s Contribution

0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp






Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion


0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp

covariant Fadeev resultRosenbluthprecision Rosenbluthpolarization transferpolarization transfer





Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion


0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp


All parameters fixed in

other applications . . . Not varied.





Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion


0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp




Agreement with Pol. Trans. data at Q2∼> 2 GeV2





Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion


0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp





Correlations in Faddeev amplitude – quark orbital

angular momentum – essential to that agreement





Dressed-Quark Core

Ward-Takahashi

Identity preserving

current

Anticipate and

Estimate Pion


0 2 4 6 8 10Q

2 [GeV2]

-1

-0.5

0

0.5

1

1.5

2

µ p GEp/ G

Mp





Correlations in Faddeev amplitude – quark orbital

angular momentum – essential to that agreement

Predict Zero at Q2 ≈ 6.5GeV2




Density profileof charge and magnetisation





Proton’s Electromagnetic Form Factor

Appearance of a zero in GE(Q2) – Completely

Unexpected







Unexpected

0 0.2 0.4 0.6 0.8 1r (fm)

0

0.002

0.004

0.006

0.008

0.01

ρ(r)

Peak Position and Height -- as zero moves in toward q*q=0

peak moves out and falls in magnitude-- redistribution of charge ... moves to larger r







Unexpected

0 0.2 0.4 0.6 0.8 1r (fm)

0

0.002

0.004

0.006

0.008

0.01

ρ(r)

Peak Position and Height -- as zero moves in toward q*q=0

peak moves out and falls in magnitude-- redistribution of charge ... moves to larger r

However, Current Density remains peaked at r = 0!Craig Roberts: Dyson Schwinger Equations and QCD






Unexpected

Wave Function is complex and correlated mix of virtual

particles and antiparticles: s−, p− and d−waves







Unexpected

Wave Function is complex and correlated mix of virtual

particles and antiparticles: s−, p− and d−waves

Simple independent-particle three-quark bag-model picture

is profoundly incorrect




Improved current




Improved current

Composite axial-vector diquark correlation

Electromagnetic current can be complicated

Limited constraints on large-Q2 behaviour




Improved current


Electromagnetic current can be complicated

Limited constraints on large-Q2 behaviour

Improved performance of code

Implemented corrections so that large-Q2 behaviour of

form factors could be reliably calculated

Exposed two weaknesses in rudimentary Ansatz




Improved current


Improved performance of code

Implemented corrections so that large-Q2 behaviour of

form factors could be reliably calculated

Exposed two weaknesses in rudimentary Ansatz

Diquark effectively pointlike to hard probe

Didn’t account for diquark being off-shell in recoil




Improved current


Minor but material improvements to current

Introduce form factor: radius 0.8 fm

Increase recoil mass by 10%




Improved current





−2.0

−1.5

−1.0

−0.5

0

0.5

1.0

GE/G

MR

atio

s

0 2 4 6 8 10 12

Q2 (GeV2)

µp × Gp

E

Gp

M

µn × GnE

GnM




Improved current





−2.0

−1.5

−1.0

−0.5

0

0.5

1.0

GE/G

MR

atio

s

0 2 4 6 8 10 12

Q2 (GeV2)

µp × Gp

E

Gp

M

µn × GnE

GnM

Proton – zero shifted:

6.5 → 8.0 GeV2




Improved current





−2.0

−1.5

−1.0

−0.5

0

0.5

1.0

GE/G

MR

atio

s

0 2 4 6 8 10 12

Q2 (GeV2)

µp × Gp

E

Gp

M

µn × GnE

GnM

Proton – zero shifted:

6.5 → 8.0 GeV2

Neutron – peak shifted:

7.5 → 5.0 GeV2

& now predict zero

a little above 11 GeV2




Pion Cloud




Pion CloudF1 – neutron

0 2 4 6 8 10

Q2/MN

2

-0.1

-0.05

0

F1n

Kelly ParametrisationDSE

Comparison

between Faddeev

equation result

and Kelly’s

parametrisation





0 2 4 6 8 10

Q2/MN

2

-0.1

-0.05

0

F1n


Comparison

between Faddeev

equation result

and Kelly’s

parametrisation

Faddeev equation

set-up to describe

dressed-quark

core





0 2 4 6 8 10

Q2/MN

2

-0.1

-0.05

0

F1n


Pion cloud contribution

Comparison

between Faddeev

equation result

and Kelly’s

parametrisation

Faddeev equation

set-up to describe

dressed-quark

core

Pseudoscalar

meson cloud (and

related effects)

significant for

Q2∼< 3 − 4 M2

N




Epilogue




Epilogue

DCSB exists in QCD.




Epilogue

DCSB exists in QCD.

It is manifest in dressed propagators and

vertices




Epilogue

DCSB exists in QCD.


vertices

It predicts, amongst other things, that

light current-quarks become heavy

constituent-quarks

pseudoscalar mesons are unnaturally

light

pseudoscalar mesons couple unnaturally

strongly to light-quarks

pseudscalar mesons couple unnaturally

strongly to the lightest baryons




Epilogue

DCSB exists in QCD.


vertices

It predicts, amongst other things, that

light current-quarks become heavy

constituent-quarks

pseudoscalar mesons are unnaturally

light

pseudoscalar mesons couple unnaturally

strongly to light-quarks

pseudscalar mesons couple unnaturally

strongly to the lightest baryons

It impacts dramatically upon observables.Craig Roberts: Dyson Schwinger Equations and QCD



Epilogue

Form Factors - progress anticipated in near- to medium-term

Quantifying pseudoscalar meson “cloud” effects




Epilogue



Locating and explaining the transition from nonp-QCD to

p-QCD in the pion and nucleon electromagnetic form

factors




Epilogue





factors

Explaining the high Q2 behavior of the proton form factor

ratio in the space-like region




Epilogue





factors



Detailing broadly the role of two-photon exchange

contributions




Epiloguenothing!





factors



Detailing broadly the role of two-photon exchange

contributions

Explaining relationship between parton properties on the

light-front and rest frame structure of hadronsCraig Roberts: Dyson Schwinger Equations and QCD



Vector piece of nucleon’s self energy

−1.0 0.0 1.0 2.0sign(t

2) |t| (GeV)

1.00

1.04

1.08

A(t

2 )

M = 0.94, mπ = 0.14

Λ = 0.9 GeV, gA = 1





−1.0 0.0 1.0 2.0sign(t

2) |t| (GeV)

1.00

1.04

1.08

A(t

2 )

M = 0.94, mπ = 0.14

Λ = 0.9 GeV, gA = 1

Solid line: One-loop;

numerically evaluated,

exact (numerical) kernel





−1.0 0.0 1.0 2.0sign(t

2) |t| (GeV)

1.00

1.04

1.08

A(t

2 )

M = 0.94, mπ = 0.14

Λ = 0.9 GeV, gA = 1




Dotted line: One-loop;

algebraic; code test





−1.0 0.0 1.0 2.0sign(t

2) |t| (GeV)

1.00

1.04

1.08

A(t

2 )

M = 0.94, mπ = 0.14

Λ = 0.9 GeV, gA = 1






Dashed line: Fully

self-consistent numerical

solution; extra pion loops

do little





−1.0 0.0 1.0 2.0sign(t

2) |t| (GeV)

1.00

1.04

1.08

A(t

2 )

M = 0.94, mπ = 0.14

Λ = 0.9 GeV, gA = 1






Dot-dashed line: Fully


solution; continuation to

timelike region return




Scalar piece of nucleon’s self energy

−1.0 0.0 1.0 2.0sign(t

2) |t| (GeV)

0.84

0.88

0.92

M+

B(t

2 ) (G

eV)

M = 0.94, mπ = 0.14

Λ = 0.9 GeV, gA = 1





−1.0 0.0 1.0 2.0sign(t

2) |t| (GeV)

0.84

0.88

0.92

M+

B(t

2 ) (G

eV)

M = 0.94, mπ = 0.14

Λ = 0.9 GeV, gA = 1








−1.0 0.0 1.0 2.0sign(t

2) |t| (GeV)

0.84

0.88

0.92

M+

B(t

2 ) (G

eV)

M = 0.94, mπ = 0.14

Λ = 0.9 GeV, gA = 1






Dashed line: Fully


solution; extra pion loops

do little





−1.0 0.0 1.0 2.0sign(t

2) |t| (GeV)

0.84

0.88

0.92

M+

B(t

2 ) (G

eV)

M = 0.94, mπ = 0.14

Λ = 0.9 GeV, gA = 1






Dot-dashed line: Fully


solution; continuation to

timelike region return



dyson-schwinger equations and quantum chromodynamics · · 2008-08-25first contents back...

Documents