tensor properties and transverse spin structures of the pion and … · 2015. 3. 25. · the pion...

Hyun-Chul Kim

Department of Physics Inha University

Collaborators: Tim Ledwig (Mainz), S.i. Nam (KIAS)

Tensor properties and transverse Spin Structures

of the pion and Nucleon

Nucleon pion & kaon

11년 11월 2일 수요일

Beihang Univ., 01-03 November, 2011 2

Tensor form factors and the spin structure

of

the Pion

S.i. Nam & HChK, Phys. Lett. B 700, 305 (2011)



The spin structure of the Pion

x

y

zb�

Vector & Tensor Form factors of the pion

u

d̄

�P

Pion: Spin S=0

What is the spin distribution of the quark inside the nucleon?

Transversity of the pion



The spin distribution of the quark

Spin probability density function in impact parameter spaceAn0: Vector form factor of the pion, Bn0: Tensor form factor of the pion

�⇥(pf )|⇤†�µQ̂|⇥(pi)⇥ = (pi + pf )A10(q2)

Vector and Tensor form factors of the pion


Se� [m,�] = �Spln�i/⇤ + im+ i

⇤M(i⇤)U�5(�)

⇤M(i⇤) + ⇥ · T

⇥

µ � 600MeV

� � 0.3 fm, R � 1 fm

Nonlocal chiral quark model

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

The chiral quark model from the instanton vacuum

•Fully relativistically field theoretic model.•Related to QCD via the Instanton vacuum.•Renormalization scale is naturally given.•No free parameter

Effective Chiral Action for the tensor current

Beihang Univ., 01-03 November, 2011



Tensor form factor of the pion



EM Form factor of the pion

EM form factor (A10 ) has been already studied.

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

��r2⇥ = 0.675 fm

��r2⇥ = 0.672± 0.008 fm (Exp)

M(Phen.): 0.714 GeV

M(Lattice): 0.727 GeV

M(This work): 0.738 GeV


Bn0(Q2, µ) = Bn0(Q

2, µ0)

↵(µ)

↵(µ0)

��n/(2�0)

�1 = 8/3, �2 = 8,

�0 = 11Nc/3� 2Nf/3Beihang Univ., 01-03 November, 2011 8

Tensor Form factor of the pion

RG evolution between lattice and model Brommel et al. PRL101

S.i. Nam & HChK, Phys. Lett. B 700, 305 (2011)



Brommel et al. PRL101

Lattice fits

Tensor Form factor of the pion


Beihang Univ., 01-03 November, 2011 10Transverse Polarization

Unpolarized Polarized

Spin density of the quark

spin

Transverse polarization




Results are in a good agreement with lattice!!!

Significant distortion appears for the polarized: a hint for spin structure of the pion!!!




Distorted spin distribution of the quark inside the pion



Tensor form factors and spin structure

of

the Nucleon

T. Ledwig & HChK, arXiv:1107.4952



The spin structure of the Nucleon

x

y

z

�(b�)

b�

x

y

Axial & Tensor Form factors, Axial-vector charges,Tensor charges

�P

x

z

y

b�

f(�)

�

Structure functions

Nucleon Tomography

Spin Structure


The Spin of the nucleon


Quark Model QCD

?

The spin of the proton must be ½!

0.2-0.4 Spin Crisis




Longitudinally polarized quark spin

Polarized DIS



⌦N

�� ¯ �µ�5�� N

↵⇠ Axial-vector charges


S.D. Bass, RMP 77, 1257 (2005)

Singlet Axial vector constant Quark spin content


Transversity: Tensor Charges

⌦N

�� ¯ �µ⌫�� N

↵⇠ Tensor charges


• No explicit probe for the tensor charge! Difficult to be measured.• Chiral odd Parton Distribution Function can get accessed via the SSA of SIDIS (HERMES and COMPASS).

A. Airapetian et al. (HERMES Coll.), PRL 94, 012002 (2005).E.S. Ageev et al. (COMPASS Coll.), NPB 765, 31 (2007).CLAS & CLAS12 Coll. (Talk by H. Avakian)

ppbar Drell-Yan process (PAX Coll.): Technically too difficult for the moment (polarized antiproton: hep-ex/0505054).


Transversity: Tensor Charges


�u = 0.60+0.10�0.24 , �d = �0.26+0.1�0.18 at 0.36GeV

2

Based on SIDIS (HERMES) data: M. Anselmino et al. Nucl. Phys. B, Proc. Suppl. 191, 98 (2009)


hNs0(p0)| (0)i�µ⌫�� (0)|Ns(p)i = us0(p0)H�T (Q

2)i�µ⌫ + E�T (Q2)�µq⌫ � qµ�⌫

2M

+ H̃�T (Q2)(nµq⌫ � qµn⌫)

2M2

�us(p)Z 1

�1dxH�T (x, ⇠, t) = H

�T (q

2),Z 1

�1dxE�T (x, ⇠, t) = E

�T (q

2),Z 1

�1dx H̃�T (x, ⇠, t) = H̃

�T (q

2),

Tensor form factors



H��T (Q2) =

2M

q2

�d�

4⇤�N 1

2(p⇥)|⌅†�kqk⇥�⌅|N 1

2(p)⇥

��T = �H�T (0)�H

��T (0)

Anomalous tensor magnetic form factors

Together with the anomalous magnetic moment, this will allow us to describe the transverse spin quark densities inside the nucleon.



Tensor form factorsTensor charges and anomalous tensor magnetic moments are scale-dependent.

�q(µ2) =

✓↵S(µ2)

↵S(µ2i )

◆4/27 1� 337

486⇡

�↵S(µ

2i )� ↵S(µ2)

��q(µ2i ) ,

↵NLOS (µ2) =

4⇡

9 ln(µ2/⇤2QCD)

"1� 64

81

ln ln(µ2/⇤2QCD)

ln(µ2/⇤2QCD)

#

⇤QCD = 0.248GeV M. Gluck, E. Reya, and A. Vogt, Z.Phys. C 67, 433(1995).



Chiral quark-soliton model

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

Z�QSM =Z

DU exp(�Se↵)

Se↵ = �NcTr lnD(U) D(U) = @4 +H(U) + m̂

H(U) = �i�4�i@i + �4MU�5

Merits of the chiral quark-soliton model• Fully relativistically field theoretic model.• Related to QCD via the Instanton vacuum.• Renormalization scale is naturally given.• All parameters were fixed already.





Nucleon consisting of Nc quarks

HChK et al. Prog. Part. Nucl. Phys. Vol.95, (1995) 11년 11월 2일 수요일



Classical solitons

Hedgehog Ansatz:




Collective quantization


g�T > g�A

Results

T. Ledwig, A. Silva, HChK, Phys. Rev. D 82 (2010) 034022

HChK, M. Polyakov, K. Goeke, PRD 53, 4715R (1996)11년 11월 2일 수요일

Proton This work SU(2) Lattice SIDIS NR|�d/�u| 0.30 0.36 0.25 0.42+0.0003�0.20 0.25

Results


[16] M. Anselmino et al. Nucl. Phys. B, Proc. Suppl. 191, 98 (2009)

[21] M. Goeckeler et al., PLB 627, 113 (2005)


ResultsUp and down tensor form factorscompared with the axial-vector ones



ResultsStrange tensor form factorscompared with the axial-vector ones



Results

Comparison with the lattice results.M. Goeckeler et al., PLB 627, 113 (2005)



Results

Isospin relations

SU(3) relations Effects of SU(3) symmetry breaking are almost negligible!



ResultsFlavor decomposition of the anomaloustensor magnetic form factors.



ResultsUp anomalous tensor magnetic form factorscompared with the lattice one.


M. Goeckeler et al. [QCDSF Coll. and UKQCD Coll.]PRL 98, 222001 (2007)


ResultsDown anomalous tensor magnetic form factorscompared with the lattice one.




Present work SU(3) Present work SU(2) Lattice�uT 3.56 3.72 3.00 (3.70)�dT 1.83 1.83 1.90 (2.35)�sT 0.2 ⇥ �0.2

�uT /�dT 1.95 2.02 1.58

µ2 = 0.36GeV2

Results


The present results are in good agreement with the lattice data!M. Goeckeler et al. [QCDSF Coll. and UKQCD Coll.]PRL 98, 222001 (2007)


Transverse spin density

⌅(b, S, s) =1

2

⇧H(b2)� Si⇥ijbj 1

MN

⇧E(b2)

⇧b2� si⇥ijbj 1

MN

⇧⇤T (b2)

⇧b2

+ siSi⌃HT (b

2)� 14M2N

⇥2H̃T (b2)⌥

+ si�2bibj � b2�ij

⇥Sj

1

M2N

⇤⇧

⇧b2

⌅2H̃T (b

2)

�,

F�(b2) =� �

0

dQ

2�QJ0(bQ)F

�(Q2)

H(b2) = F1(b2), E(b2) = F2(b

2)


]

[S, s] = [(1, 0), (0, 0)], [S, s] = [(0, 0), (1, 0)]


39

Results6

FIG. 4. Transverse up and down quark spin densities of the nucleon from the �QSM with the lowest moment. In the upperleft panel, the density of unpolarized quarks in a transversely polarized nucleon ([S, s] = [(1, 0), (0, 0)]) is drawn and in theupper right panel, that of transversely polarized quarks in a unpolarized nucleon ([S, s] = [(0, 0), (1, 0)]). In the lower panel,we plot the down quark densities.

[S, s] = [(1, 0), (0, 0)], it is shown from Fig. 4 that the down quark density is more distorted in the negative directionof by, compared to the up quark density. The reason can be found in the fact that firstly the down anomalous magneticmoment (�1.80µN ) is negative and its absolute value is larger than the up one (1.35µN ). Secondly, the down Pauliform factor falls o� more slowly than that of the up quark. As for the [S, s] = [(1, 0), (0, 0)], the tensor anomalousmagnetic moments come into play. Since both �uT and �

dT are positive, the both transverse spin densities are deformed

in the direction of the positive by. Moreover, the form factor �dT falls o� rather more slowly than �uT as shown in

Fig. 2, the density for the down quark is more strongly deformed. Note that the Q2 dependence of the form factorsplay a more important role in governing the deformation of the densities. The results for the transverse up and downquark spin densities of the nucleon with the lowest moment are very similar to those from the lattice calculation [17]and as a result we can draw similar conclusions.

Since both the strange Pauli form factor and tensor anomalous magnetic form factor turn out to be rather small,we can expect that the strength of the strange densities will be rather small. Nevertheless, it is of great interest to seehow much the transverse strange densities are distorted. Figure 5 plots the transverse strange quark spin densitieswith the lowest moment. Note that the magnitudes of the densities are smaller than those of the up and down quarksby an order of magnitude.

It is interesting to see that the density of unpolarized strange quarks in a polarized nucleon is negatively shifted.It is due to the fact that the strange Pauli form factor F s2 turns negative from Q

2 ⇥ 0.2GeV2. This negative values

Up quark transverse spin density inside a nucleon

the latter correlation is stronger than the one betweentransverse quark and nucleon spin.

Figure 5 shows the n ! 2 moment of the densities.Obviously, the pattern is very similar to that in Fig. 4,which supports our simple interpretation. The main differ-ence is that the densities for the higher n ! 2 moment aremore peaked around the origin b? ! 0 as already observedin [27] for the vector and axial vector GFFs.

Conclusions.—We have presented first lattice results forthe lowest two moments of transverse spin densities ofquarks in the nucleon. Because of the large and positive

contributions from the tensor GFF !BTn0 for up and fordown quarks, we find strongly distorted spin densities fortransversely polarized quarks in an unpolarized nucleon.According to Burkardt [7], this leads to the prediction of asizable negative Boer-Mulders function [4] for up anddown quarks, which may be confirmed in experiments at,e.g., Jefferson Lab and GSI Facility for Antiproton and IonResearch [28,29].

The numerical calculations have been performed on theHitachi No. SR8000 at LRZ (Munich), the apeNEXT atNIC/DESY (Zeuthen), and the BlueGene/L at NIC/FZJ(Jülich), EPCC (Edinburgh), and KEK (by the Kanazawagroup as part of the DIK research programme). This workwas supported by DFG (Forschergruppe Gitter-Hadronen-Phänomenologie and Emmy-Noether programme), HGF(Contract No. VH-NG-004), and EU I3HP (ContractNo. RII3-CT-2004-506078).

*Electronic address: [email protected][1] V. Barone, A. Drago, and P. G. Ratcliffe, Phys. Rep. 359, 1

(2002).[2] R. L. Jaffe and X. D. Ji, Phys. Rev. Lett. 67, 552 (1991).[3] D. W. Sivers, Phys. Rev. D 41, 83 (1990).[4] D. Boer and P. J. Mulders, Phys. Rev. D 57, 5780 (1998).[5] M. Diehl and Ph. Hägler, Eur. Phys. J. C 44, 87 (2005).[6] M. Burkardt, Nucl. Phys. A735, 185 (2004).[7] M. Burkardt, Phys. Rev. D 72, 094020 (2005).[8] M. Diehl, Phys. Rep. 388, 41 (2003).[9] M. Burkardt, Phys. Rev. D 62, 071503 (2000); 66, 119903

(2002).[10] M. Diehl, Eur. Phys. J. C 19, 485 (2001).[11] Ph. Hägler, Phys. Lett. B 594, 164 (2004); Z. Chen and

X. Ji, Phys. Rev. D 71, 016003 (2005).[12] J. C. Collins and M. Diehl, Phys. Rev. D 61, 114015

(2000); D. Y. Ivanov et al., Phys. Lett. B 550, 65 (2002).[13] A. Ali Khan et al., Phys. Rev. D 74, 094508 (2006).[14] C. Aubin et al., Phys. Rev. D 70, 094505 (2004).[15] M. Göckeler et al., Phys. Lett. B 627, 113 (2005).[16] G. Martinelli et al., Nucl. Phys. B445, 81 (1995);

M. Göckeler et al., Nucl. Phys. B544, 699 (1999).[17] M. Göckeler et al., Phys. Rev. Lett. 92, 042002 (2004).[18] M. Göckeler et al., Nucl. Phys. A755, 537 (2005).[19] Ph. Hägler et al., Phys. Rev. D 68, 034505 (2003).[20] M. Diehl et al., arXiv:hep-ph/0511032.[21] D. Brömmel et al., arXiv:hep-lat/0608021.[22] M. Göckeler et al., arXiv:hep-lat/0610118.[23] A. Ali Khan et al., Nucl. Phys. B689, 175 (2004).[24] S. I. Ando, J. W. Chen, and C. W. Kao, arXiv:hep-ph/

0602200; M. Diehl, A. Manashov, and A. Schäfer,arXiv:hep-ph/0608113.

[25] B. Pasquini, M. Pincetti, and S. Boffi, Phys. Rev. D 72,094029 (2005).

[26] M. Burkardt, arXiv:hep-ph/0611256.[27] Ph. Hägler et al., Phys. Rev. Lett. 93, 112001 (2004).[28] H. Avakian et al., Jefferson Lab Research Proposal

No. PR12-06-112.[29] See baseline technical report, PANDA Collaboration,

http://www.gsi.de/fair/reports/btr.html.

FIG. 5 (color online). Second moment (n ! 2) of transversespin densities. For details, see caption of Fig. 4.

FIG. 4 (color online). Lowest moment (n ! 1) of the densitiesof unpolarized quarks in a transversely polarized nucleon (left)and transversely polarized quarks in an unpolarized nucleon(right) for up (upper plots) and down (lower plots) quarks. Thequark spins (inner arrows) and nucleon spins (outer arrows) areoriented in the transverse plane as indicated.

PRL 98, 222001 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending1 JUNE 2007

222001-4

Lattice results



8

FIG. 6: Transverse up and down quark spin densities with the lowest moment of the nucleon from the �QSM with bx = 0. Inthe upper left panel, the density of unpolarized quarks in a transversely polarized nucleon ([S, s] = [(1, 0), (0, 0)]) is drawn andin the upper right panel, that of transversely polarized quarks in a unpolarized nucleon ([S, s] = [(0, 0), (1, 0)]). In the lowerpanel, we plot the down quark densities.

[10] M. Anselmino, V. Barone, A. Drago and N.N. Nikolaev, Phys. Lett. B 594 (2004) 97.[11] B. Pasquini, M. Pincetti and S. Bo�, Phys. Rev. D 76 (2007) 034020.[12] K. Abe et al. [Belle Colaboration], Phys. Rev. Lett. 96 (2006) 232002.[13] L. Pappalardo for the HERMES collaboration, Apr 2006. 4pp. Prepared for 14th International Workshop on Deep Inelastic

Scattering (DIS 2006), Tsukuba, Japan, 20-24 Apr 2006.[14] A. Airapetian et al., [HERMES Collaboration], Phys. Rev. Lett. 94 (2005) 012002.[15] E.S. Ageev et al., [COMPASS Collaboration], Nucl. Phys. B 765 (2007) 31.[16] M. Anselmino et al., Nucl. Phys. Proc. Suppl. 191 (2009) 98.[17] M. Göckeler et al. [QCDSF Collaboration and UKQCD Collaboration], Phys. Rev. Lett. 98 (2007) 222001.[18] M. Diehl and P. .Hägler, Eur. Phys. J. C44 (2005) 87.[19] M. Burkardt, Phys. Rev. D62 (2000) 071503. [hep-ph/0005108].[20] M. Burkardt, Int. J. Mod. Phys. A18 (2003) 173-208. [arXiv:hep-ph/0207047 [hep-ph]].[21] A. Silva, Ph.D. Dissertation, Ruhr-Universität Bochum, (unpublished).[22] A. Silva, H. -Ch. Kim and K. Goeke, Phys. Rev. D65 (2002) 014016.[23] A. Silva, H. -Ch. Kim and K. Goeke, Eur. Phys. J. A22 (2004) 481.[24] K. Goeke, H.-Ch. Kim, A. Silva and D. Urbano, Eur. Phys. J. A32 (2007) 393.[25] T. Ledwig, A. Silva and H.-Ch. Kim, Phys. Rev. D 82 (2010) 034022.[26] T. Ledwig, A. Silva and H.-Ch. Kim, Phys. Rev. D 82 (2010) 054014.[27] C. C. Carlson and M. Vanderhaeghen, Phys. Rev. Lett. 100 (2008) 032004; Eur. Phys. J. A 41 (2009) 41.[28] L. Tiator and M. Vanderhaeghen, Phys. Lett. B 672 (2009) 344.[29] C. Lorce, B. Pasquini, M. Vanderhaeghen, arXiv:1102.4702[30] C. Lorce, B. Pasquini, arXiv:1106.0139

40

ResultsUp quark transverse spin density inside a nucleon



6

FIG. 4: Transverse up and down quark spin densities with the lowest moment of the nucleon from the �QSM. In the upper leftpanel, the density of unpolarized quarks in a transversely polarized nucleon ([S, s] = [(1, 0), (0, 0)]) is drawn and in the upperright panel, that of transversely polarized quarks in a unpolarized nucleon ([S, s] = [(0, 0), (1, 0)]). In the lower panel, we plotthe down quark densities.

the width of the profile gets narrower and more peaked. On the contrary, the transverse down quark spin density forthe unpolarized quarks in a polarized nucleon is changed to the negative by direction and shows an obvious distortion.That for the polarized quarks in an unpolarized nucleon is shifted to the positive by direction and distorted again.

Figure 7 shows the profiles of the transverse strange quark spin densities with bx = 0. Interestingly, the densityfor the unpolarized strange quarks in a polarized nucleon is only slightly modified. In contrast, the polarized strangequarks are strongly redistributed in an unpolarized nucleon. As a result, the peak position is shifted to the positive bydirection and becomes sharper. Moreover, the density becomes even more negative for the negative value of by thanfor the positive by.

5. We investigated the transverse quark spin densities of the nucleon with the lowest moment, using the results ofthe vector and tensor form factors derived from the SU(3) chiral quark-soliton model. we first recapitulated the flavor-decomposed anomalous vector and tensor magnetic form factors as functions of the momentum transfer. For numericalconvenience, we made parametrizations of the form factors. Having combined these form factors and performed theFourier transformations, we evaluated the transverse quark spin densities of the nucleon with the lowest moment. Weconsidered two di�erent cases, the density of unpolarized quarks in a polarized nucleon and that of polarized quarksin an unpolarized nucleon. The results turned out to be rather similar to the lattice QCD ones. Transversly polarized

41

ResultsDown quark transverse spin density inside a nucleon

the latter correlation is stronger than the one betweentransverse quark and nucleon spin.

Figure 5 shows the n ! 2 moment of the densities.Obviously, the pattern is very similar to that in Fig. 4,which supports our simple interpretation. The main differ-ence is that the densities for the higher n ! 2 moment aremore peaked around the origin b? ! 0 as already observedin [27] for the vector and axial vector GFFs.

Conclusions.—We have presented first lattice results forthe lowest two moments of transverse spin densities ofquarks in the nucleon. Because of the large and positive

contributions from the tensor GFF !BTn0 for up and fordown quarks, we find strongly distorted spin densities fortransversely polarized quarks in an unpolarized nucleon.According to Burkardt [7], this leads to the prediction of asizable negative Boer-Mulders function [4] for up anddown quarks, which may be confirmed in experiments at,e.g., Jefferson Lab and GSI Facility for Antiproton and IonResearch [28,29].

The numerical calculations have been performed on theHitachi No. SR8000 at LRZ (Munich), the apeNEXT atNIC/DESY (Zeuthen), and the BlueGene/L at NIC/FZJ(Jülich), EPCC (Edinburgh), and KEK (by the Kanazawagroup as part of the DIK research programme). This workwas supported by DFG (Forschergruppe Gitter-Hadronen-Phänomenologie and Emmy-Noether programme), HGF(Contract No. VH-NG-004), and EU I3HP (ContractNo. RII3-CT-2004-506078).

*Electronic address: [email protected][1] V. Barone, A. Drago, and P. G. Ratcliffe, Phys. Rep. 359, 1

(2002).[2] R. L. Jaffe and X. D. Ji, Phys. Rev. Lett. 67, 552 (1991).[3] D. W. Sivers, Phys. Rev. D 41, 83 (1990).[4] D. Boer and P. J. Mulders, Phys. Rev. D 57, 5780 (1998).[5] M. Diehl and Ph. Hägler, Eur. Phys. J. C 44, 87 (2005).[6] M. Burkardt, Nucl. Phys. A735, 185 (2004).[7] M. Burkardt, Phys. Rev. D 72, 094020 (2005).[8] M. Diehl, Phys. Rep. 388, 41 (2003).[9] M. Burkardt, Phys. Rev. D 62, 071503 (2000); 66, 119903

(2002).[10] M. Diehl, Eur. Phys. J. C 19, 485 (2001).[11] Ph. Hägler, Phys. Lett. B 594, 164 (2004); Z. Chen and

X. Ji, Phys. Rev. D 71, 016003 (2005).[12] J. C. Collins and M. Diehl, Phys. Rev. D 61, 114015

(2000); D. Y. Ivanov et al., Phys. Lett. B 550, 65 (2002).[13] A. Ali Khan et al., Phys. Rev. D 74, 094508 (2006).[14] C. Aubin et al., Phys. Rev. D 70, 094505 (2004).[15] M. Göckeler et al., Phys. Lett. B 627, 113 (2005).[16] G. Martinelli et al., Nucl. Phys. B445, 81 (1995);

M. Göckeler et al., Nucl. Phys. B544, 699 (1999).[17] M. Göckeler et al., Phys. Rev. Lett. 92, 042002 (2004).[18] M. Göckeler et al., Nucl. Phys. A755, 537 (2005).[19] Ph. Hägler et al., Phys. Rev. D 68, 034505 (2003).[20] M. Diehl et al., arXiv:hep-ph/0511032.[21] D. Brömmel et al., arXiv:hep-lat/0608021.[22] M. Göckeler et al., arXiv:hep-lat/0610118.[23] A. Ali Khan et al., Nucl. Phys. B689, 175 (2004).[24] S. I. Ando, J. W. Chen, and C. W. Kao, arXiv:hep-ph/

0602200; M. Diehl, A. Manashov, and A. Schäfer,arXiv:hep-ph/0608113.

[25] B. Pasquini, M. Pincetti, and S. Boffi, Phys. Rev. D 72,094029 (2005).

[26] M. Burkardt, arXiv:hep-ph/0611256.[27] Ph. Hägler et al., Phys. Rev. Lett. 93, 112001 (2004).[28] H. Avakian et al., Jefferson Lab Research Proposal

No. PR12-06-112.[29] See baseline technical report, PANDA Collaboration,

http://www.gsi.de/fair/reports/btr.html.

FIG. 5 (color online). Second moment (n ! 2) of transversespin densities. For details, see caption of Fig. 4.

FIG. 4 (color online). Lowest moment (n ! 1) of the densitiesof unpolarized quarks in a transversely polarized nucleon (left)and transversely polarized quarks in an unpolarized nucleon(right) for up (upper plots) and down (lower plots) quarks. Thequark spins (inner arrows) and nucleon spins (outer arrows) areoriented in the transverse plane as indicated.

PRL 98, 222001 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending1 JUNE 2007

222001-4

Lattice resultsM. Goeckeler et al. [QCDSF Coll. and UKQCD Coll.]PRL 98, 222001 (2007)


8

FIG. 6: Transverse up and down quark spin densities with the lowest moment of the nucleon from the �QSM with bx = 0. Inthe upper left panel, the density of unpolarized quarks in a transversely polarized nucleon ([S, s] = [(1, 0), (0, 0)]) is drawn andin the upper right panel, that of transversely polarized quarks in a unpolarized nucleon ([S, s] = [(0, 0), (1, 0)]). In the lowerpanel, we plot the down quark densities.

[10] M. Anselmino, V. Barone, A. Drago and N.N. Nikolaev, Phys. Lett. B 594 (2004) 97.[11] B. Pasquini, M. Pincetti and S. Bo�, Phys. Rev. D 76 (2007) 034020.[12] K. Abe et al. [Belle Colaboration], Phys. Rev. Lett. 96 (2006) 232002.[13] L. Pappalardo for the HERMES collaboration, Apr 2006. 4pp. Prepared for 14th International Workshop on Deep Inelastic

Scattering (DIS 2006), Tsukuba, Japan, 20-24 Apr 2006.[14] A. Airapetian et al., [HERMES Collaboration], Phys. Rev. Lett. 94 (2005) 012002.[15] E.S. Ageev et al., [COMPASS Collaboration], Nucl. Phys. B 765 (2007) 31.[16] M. Anselmino et al., Nucl. Phys. Proc. Suppl. 191 (2009) 98.[17] M. Göckeler et al. [QCDSF Collaboration and UKQCD Collaboration], Phys. Rev. Lett. 98 (2007) 222001.[18] M. Diehl and P. .Hägler, Eur. Phys. J. C44 (2005) 87.[19] M. Burkardt, Phys. Rev. D62 (2000) 071503. [hep-ph/0005108].[20] M. Burkardt, Int. J. Mod. Phys. A18 (2003) 173-208. [arXiv:hep-ph/0207047 [hep-ph]].[21] A. Silva, Ph.D. Dissertation, Ruhr-Universität Bochum, (unpublished).[22] A. Silva, H. -Ch. Kim and K. Goeke, Phys. Rev. D65 (2002) 014016.[23] A. Silva, H. -Ch. Kim and K. Goeke, Eur. Phys. J. A22 (2004) 481.[24] K. Goeke, H.-Ch. Kim, A. Silva and D. Urbano, Eur. Phys. J. A32 (2007) 393.[25] T. Ledwig, A. Silva and H.-Ch. Kim, Phys. Rev. D 82 (2010) 034022.[26] T. Ledwig, A. Silva and H.-Ch. Kim, Phys. Rev. D 82 (2010) 054014.[27] C. C. Carlson and M. Vanderhaeghen, Phys. Rev. Lett. 100 (2008) 032004; Eur. Phys. J. A 41 (2009) 41.[28] L. Tiator and M. Vanderhaeghen, Phys. Lett. B 672 (2009) 344.[29] C. Lorce, B. Pasquini, M. Vanderhaeghen, arXiv:1102.4702[30] C. Lorce, B. Pasquini, arXiv:1106.0139

42

ResultsDown quark transverse spin density inside a nucleon



7

FIG. 5: Transverse strange quark spin densities of the nucleon from the �QSM with the lowest moment. In the left panel,the density of unpolarized strange quarks in a transversely polarized nucleon ([S, s] = [(1, 0), (0, 0)]) is drawn and in the rightpanel, that of transversely polarized strange quarks in an unpolarized nucleon ([S, s] = [(0, 0), (1, 0)]).

quarks in an unpolarized proton are both shifted to the positive direction of bx. The shift is more prominent thanthe one occuring for unpolarized quarks in a polarized proton, where the density for the u-quark is shifted to positiveand that of the d-quark to the negative by direction.

We presented in this work the first result of the transverse strange quark spin densities. Since the magnitudes ofstrange Pauli and anomalous tensor magnetic form factors are rather small, the strange densities turn out to be muchsmaller than those for the up and down quarks. However, the density for polarized strange quarks in an unpolarizednucleon is noticeably distorted in the direction of the positive by and becomes more negative for the negative valuesof by.

In the present work, we considered only the transverse spin densities with the lowest moment. The generalized formfactors with higher moments can be calculated in principle within the framework of the chiral quark-soliton model.However, we need to consider them carefully because of the presence of the derivative operators. The correspondingwork is under progress. So far, we did not consider other cases of the transverse polarizations such as [(1, 0), (1, 0)]for which one requires information on the form factor H̃T and its derivative. However, we found that this form factorshowed a numerically sensitivity. A related work addressing that is under way.

Acknowledgments

H.Ch.K is grateful to E. Epelbaum and M.V. Polyakov for hospitalities during his visit to Theoretische PhysikInstitute II in Ruhr-Universität Bochum, where part of the work was carried out. The authors are grateful to A. Metzfor valuable comments and suggestions. The present work was supported by Basic Science Research Program throughthe National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology(grant number: 2010-0016265).

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43

ResultsStrange quark transverse spin density inside a nucleon

This is the first result of the strange quark transverse spin density inside a nucleon



9

FIG. 7: Transverse strange quark spin densities of the nucleon from the �QSM with the lowest moment with bx = 0. In theleft panel, the density of unpolarized strange quarks in a transversely polarized nucleon ([S, s] = [(1, 0), (0, 0)]) is drawn andin the right panel, that of transversely polarized strange quarks in an unpolarized nucleon ([S, s] = [(0, 0), (1, 0)]).

[31] Ph. Hägler et al., Phys. Rev. D 68 (2003) 034505.[32] C.V. Christov et al., Prog. Part. Nucl. Phys. 37 (1996) 91.

44

ResultsStrange quark transverse spin density inside a nucleon

Polarized to the negative direction in the b plane.



45

Summary•We have reviewed recent investigations on the spin structures of the pion and nucleon, based on the chiral quark-(soliton) model.

• The results were compared with those of the lattice QCD and turned out to be in good agreement with them.

•The first strange anomalous tensor magnetic moment was obtained, though it is compatible with zero.

•The transverse quark spin densities inside the proton were presented.

•The strange quark transverse spin density was first announced in this work.



Thank you very much!

Though this be madness,yet there is method in it.

Hamlet Act 2, Scene 2


tensor properties and transverse spin structures of the pion and … · 2015. 3. 25. · the pion...

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