Quark Structure of the Pion
Hyun-Chul Kim
RCNP, Osaka University &Department of Physics, Inha University
Collaborators: H.D. Son, S.i. Nam

Progress of J-PARC Hadron Physics, Nov. 30-Dec. 01, 2014

Interpretation of the Form factors
Non-Relativistic picture of the EM form factors

Electromagnetic form factors

3-D Fourier Transform
Charge & Magnetisation Densities
Schroedinger Eq. & Wave functions
2

Interpretation of the EMFFs
Traditional interpretation of the nucleon form factors
2

F1 (Q ) =Z

3d xeiQ·x⇢(r)⇢(r) =X†(r) (r)However, the initial and final momenta are different in a relativistic case.
Thus, the initial and final wave functions are different.

Probability interpretation is wrong in a relativistic case!

We need a correct interpretation of the form factors
Belitsky & Radyushkin, Phys.Rept. 418, 1 (2005)
G.A. Miller, PRL 99, 112001 (2007)3

Interpretation of the EMFFs
Modern understanding of the form factors
Transverse Charge densities⇢(b)

Lorentz invariant: independent of any observer.
Infinite momentum framework⇢1 (b) =Xq

e2q

Zdxfq q̄ (x, b)
GPDs

Form factors

F (q 2 ) =Zd2 beiq·b ⇢(b)4

Hadron Tomography
Generalised Parton Distributions
bybyby∼ Q12b⊥xpbx

xMomentum fraction

ρ(b⊥ )

0!

b⊥b⊥dx

xpbxx
R

∼ Q12

∆→0q(x, b⊥ )

xb⊥

bx

x"q(x)

0x

a | Formdensities
factorb | GPD, ξ = 0c | Parton density
Transverse
GPDs
Structure functions
ofFormfactors
igure 2.1 | A simplified sketch of the diﬀerent phenomenological observables and theirnterpretation in the infinite momentum frame: a | The form factor as a charge density in5D.
Brömmel,Dissertation(RegensburgU.)
he perpendicular plane (after a Fourier transform, Sec. 1.2). b | A probabilistic interpre-

Generalised Parton Distributions
Probes are unknown for Tensor form factors
and the Energy-Momentum Tensor form factors!
Pion

Pion

6

Generalised Parton Distributions
Probes are unknown for Tensor form factors
and the Energy-Momentum Tensor form factors!
Pion
Form factors as Mellin moments of the GPDs

Pion

6

What we know about the Pion
Experimentally, we know about the pion
• Pion Mass = 139.57 MeV• Pion Spin = 0

Theoretically• pseudo-Goldstone boson• The lowest-lying meson(1 q + 1anti-q + sea quarks + gluons + …)

The structure of the pion is still not trivial!

The spin structure of the Pion
Vector & Tensor Form factors of the pion

xPion: Spin S=0ubyd¯Longitudinal spin structure is trivial.

xPzh⇡(p0 )| ¯ 3 5 |⇡(p)i = 0

What about the transversely polarized quarks
inside a pion?

Internal spin structure of the pion

8The spin distribution of the quark

Spin probability densities in the transverse plane
An0: Vector densities of the pion,

Bn0: Tensor densities of the pion

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The spin distribution of the quark

Spin probability densities in the transverse plane
An0: Vector densities of the pion,Bn0: Tensor densities of the pion

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The spin distribution of the quark

Spin probability densities in the transverse plane
An0: Vector densities of the pion,

Bn0: Tensor densities of the pion

Vector and Tensor form factors of the pionh⇡(pf )| † µ Q̂ |⇡(pi )i = (pi + pf )A10 (q 2 )9Nonlocal chiral quark model
Gauged Effective Nonlocal Chiral ActionSe↵ =hippNc Tr ln iD/ + im + i M (iD, m)U 5 M (iD, m)
Dµ = @µi µ Vµ

The nonlocal chiral quark model from the instanton vacuum• Fully relativistically field theoretic model.
• “Derived” from QCD via the Instanton vacuum.
• Renormalization scale is naturally given.
•No free parameter
0.3 fm, R

1 fm

Dilute instanton liquid ensembleµ600 MeV

D. Diakonov & V. Petrov Nucl.Phys. B272 (1986) 457
H.-Ch.K, M. Musakhanov, M. Siddikov Phys. Lett. B 608, 95 (2005).
Musakhanov & H.-Ch. K, Phys. Lett. B 572, 181-188 (2003)

Nonlocal chiral quark model
Gauged Effective Nonlocal Chiral ActionSe↵ =hipp
Nc Tr ln iD/ + im + i M (iD, m)U 5 M (iD, m)

U 5 = U (x)1+ 51
i5+ U † (x)=1+22f⇡
22M (iD, m) = M0 F (iD)f (m) = M0 F (iD)F (k) = 2t I0 (t)K1 (t)I1 (t)K0 (t)"ra a)
5 (⇡m21+ 2dmd

1 a a 2(⇡) + ···f⇡2#d ⇡ 0.198 GeV1I1 (t)K1 (t)k ⇢¯tt=2M0 ⇡ 0.35 GeV : It is determined by the gap equation.
D. Diakonov & V. Petrov Nucl.Phys. B272 (1986) 457
H.-Ch.K, M. Musakhanov, M. Siddikov Phys. Lett. B 608, 95 (2005).
Musakhanov & H.-Ch. K, Phys. Lett. B 572, 181-188 (2003)

EM Form factor of the pion
EM form factor (A10 )h⇡(pf )| † µ Q̂ |⇡(pi )i = (pi + pf )A10 (q 2 )S.i. Nam & HChK, Phys. Rev. D77 (2008) 094014

EM Form factor of the pion
EM form factor (A10 )h⇡(pf )| † µ Q̂ |⇡(pi )i = (pi + pf )A10 (q 2 ): Local vertices

S.i. Nam & HChK, Phys. Rev. D77 (2008) 094014

EM Form factor of the pion
EM form factor (A10 )h⇡(pf )| † µ Q̂ |⇡(pi )i = (pi + pf )A10 (q 2 ): Local vertices: Nonlocal vertices

S.i. Nam & HChK, Phys. Rev. D77 (2008) 094014

EM Form factor of the pionr2 ⇥ = 0.675 fm
r2 ⇥ = 0.672 ± 0.008 fm (Exp)

M(Phen.): 0.714 GeV
M(Lattice): 0.727 GeVM(XQM): 0.738 GeV

S.i. Nam & HChK, Phys. Rev. D77 (2008) 094014

Tensor Form factor of the pion⌦ +⇡ (pf )†(0) µ⌫↵µ ⌫µ ⌫
pqqp+(0) ⇡ (pi ) =B10 (Q2 )m⇡p = (pf + pi )/2, q = pf

S.i. Nam & H.-Ch.K, Phys. Lett. B 700, 305 (2011).

pi14

Tensor Form factor of the pion
RG equation for the tensor form factor

Brommel et al. PRL101

B10 (Q2 , µ) = B10 (Q2 , µ0 )↵(µ)↵(µ0 )

33

42Nfp-pole parametrization for the form factor

Q2
B10 (Q ) = B10 (0) 1 +pm2p2

S.i. Nam & H.-Ch.K, Phys. Lett. B 700, 305 (2011).

p

15

Tensor Form factor of the pion

S.i. Nam & H.-Ch.K, Phys. Lett. B 700, 305 (2011).

16

Tensor Form factor of the pion

S.i. Nam & H.-Ch.K, Phys. Lett. B 700, 305 (2011).

17

Spin density of the quark
Unpolarized

Polarized

spin

Polarization

18

Spin density of the quark

Significant distortion appears for the polarized quark!

Results are in a good agreement with the lattice calculation!
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Spin density of the quark

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Stability of the pion
Isoscalar vector GPDs of the pion
12 ab H⇡I=0 (x, ⇠, t) =2

Zd ix (P ·n) a 0 ¯eh⇡ (p )| (2⇡n/2)/n[n/2, n/2] ( n/2)|⇡ b (p)i

The second moment of the GPD
Zdx xH⇡I=0 (x, ⇠, t) = A20 (t) + 4⇠ 2 A22 (t)21Stability of the pion
Isoscalar vector GPDs of the pion
12 ab H⇡I=0 (x, ⇠, t) =2Zd ix (P ·n) a 0 ¯eh⇡ (p )| (2⇡n/2)/n[n/2, n/2] ( n/2)|⇡ b (p)i

The second moment of the GPD
Zdx xH⇡I=0 (x, ⇠, t) = A20 (t) + 4⇠ 2 A22 (t) : Generalized form factors of the pion

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Stability of the pion
Isoscalar vector GPDs of the pion
12 ab H⇡I=0 (x, ⇠, t) =2

Zd ix (P ·n) a 0 ¯eh⇡ (p )| (2⇡n/2)/n[n/2, n/2] ( n/2)|⇡ b (p)i

The second moment of the GPD
Zdx xH⇡I=0 (x, ⇠, t) = A20 (t) + 4⇠ 2 A22 (t) : Generalized form factors of the pion
Energy-momentum Tensor Form factors (Pagels, 1966)a0bh⇡ (p )|Tµ⌫ (0)|⇡ (p)i =Tµ⌫ (x) =ab2[(tgµ⌫qµ qn u)⇥1 (t) + 2Pµ P⌫ ⇥2 (t)]!
1¯(x) { i @ ⌫ } (x) : QCD EMT operator
221

Stability of the pion
Isoscalar vector GPDs of the pion
12 ab H⇡I=0 (x, ⇠, t) =2Zd ix (P ·n) a 0 ¯eh⇡ (p )| (2⇡n/2)/n[n/2, n/2] ( n/2)|⇡ b (p)i

The second moment of the GPD
Zdx xH⇡I=0 (x, ⇠, t) = A20 (t) + 4⇠ 2 A22 (t) : Generalized form factors of the pion
Energy-momentum Tensor Form factors (Pagels, 1966)a0bh⇡ (p )|Tµ⌫ (0)|⇡ (p)i =ab2[(tgµ⌫qµ qn u)⇥1 (t) + 2Pµ P⌫ ⇥2 (t)]EMTFFs (Gravitational FFs)Tµ⌫ (x) =!
1¯(x) { i @ ⌫ } (x) : QCD EMT operator
221

Stability of the pion
Isoscalar vector GPDs of the pion
12 ab H⇡I=0 (x, ⇠, t) =2Z

d ix (P ·n) a 0 ¯eh⇡ (p )| (2⇡n/2)/n[n/2, n/2] ( n/2)|⇡ b (p)i

The second moment of the GPD
Zdx xH⇡I=0 (x, ⇠, t) = A20 (t) + 4⇠ 2 A22 (t) : Generalized form factors of the pion
Energy-momentum Tensor Form factors (Pagels, 1966)a0bh⇡ (p )|Tµ⌫ (0)|⇡ (p)i =ab2[(tgµ⌫qµ qn u)⇥1 (t) + 2Pµ P⌫ ⇥2 (t)]EMTFFs (Gravitational FFs)Tµ⌫ (x) =!
1¯(x) { i @ ⌫ } (x) : QCD EMT operator
221

Stability of the pion
Time component of the EMT matrix element gives the pion mass.

The sum of the spatial component of the EMT matrix element
gives the pressure of the pion, which should vanish!
Zero in the chiral limit(Based on the local model)
H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm.

22

Stability of the pion
Pressure of the pion beyond the chiral limit

H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm.

23

Stability of the pion
Pressure of the pion beyond the chiral limit

Quark condensate

H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm.

23

Stability of the pion
Pressure of the pion beyond the chiral limit

Quark condensate

Pion decay constant

H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm.

23

Stability of the pion
Pressure of the pion beyond the chiral limit

Quark condensate

Pion decay constantby the Gell-Mann-Oakes-Renner relation to linear m order
H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm.

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Energy-momentum Tensor FFs

in the chiral limit

H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm.

24

Energy-momentum Tensor FFs

in the chiral limit

The difference arises from the explicit chiral symmetry breaking.
H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm.

24

Transverse charge density of the pion

H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm.

25

Transverse charge density of the pion

H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm.

26

Transverse charge density of the pion
The transverse charge density is divergent at b=0.

H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm.

26

Effective chiral Lagrangian
Low-Energy Constants

Derivative expansions: pion momentum as an expansion parameter

Weinberg term

Gasser-Leutwyler terms

HChK et al. Prog. Part. Nucl. Phys. Vol.37, 91 (1996)

Effective chiral Lagrangian
Weinberg Lagrangian

Gasser-Leutwyler Lagrangian

H.A. Choi and HChK, PRD 69, 054004 (2004)

Low-energy constants in flat space
Gasser-Leutwyler Lagrangian

H.A. Choi and HChK, PRD 69, 054004 (2004)

Low-energy constants in flat space
Gasser-Leutwyler Lagrangian

H.A. Choi and HChK, PRD 69, 054004 (2004)

Effective chiral Lagrangian in curved space

Gasser-Leutwyler LagrangianEffective chiral Lagrangian in curved space

Gasser-Leutwyler Lagrangian

Low-energy constants in curved spaceEffective chiral Lagrangian in curved space

Gasser-Leutwyler Lagrangian

Low-energy constants in curved space
They can be derived either by expanding the action by
the heat-kernel method or expand the EMTffs with respect to t.

Low-energy constants in curved space
Present Results

XPT Results (Gasser & Leutwyler)

Summary & Conclusion

Summary•We have reviewed recent investigations on the quark

structures of the pion, based on the nonlocal chiral quark modelfrom the instanton vacuum.

• We have derived the EM and tensor form factors of the pion,from which we have obtained the quark transverse spin
densities inside a pion.

•We also have shown the energy-momentum tensor(gravitational or generalised) form factors of the pion.

•The pressure of the pion nontrivially vanishes because of theGell-Mann-Oakes-Renner relation.

•We also have presented the higher-order transverse charge
density of the pion.

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Outlook• The transverse charge and spin densities for the transition
processes can be studied (K-pi transition is done, see the next
talk by Hyeon-Dong Son).

• The excited states for the nucleon and the hyperon can beinvestigated (Generalisation of the XQSM is under way).

•Internal structure of Heavy-light quark systems(Derivation of the Partition function is close to the final result.)• New perspective on hadron tomography
34

Though this be madness,
yet there is method in it.
Hamlet Act 2, Scene 2

Thank you very much!