Abstract
The correspondence between theories in anti–de Sitter space and field theories in physical spacetime leads to an analytic, semiclassical model for stronglycoupled QCD which has scale invariance at short distances and color confinement at large distances. These equations, for both mesons and baryons, give a very good representation of the observed hadronic spectrum, including a zero mass pion. Lightfront holography allows hadronic amplitudes in the AdS fifth dimension to be mapped to frameindependent lightfront wavefunctions of hadrons in physical spacetime, thus providing a relativistic description of hadrons at the amplitude level. The meson and baryon wavefunctions derived from lightfront holography and AdS/QCD also have remarkable phenomenological features, including predictions for the electromagnetic form factors and decay constants. The approach can be systematically improved using lightfront Hamiltonian methods. Some novel features of QCD for baryon physics are also discussed.
SLAC–PUB–14381
Applications of AdS/QCD and LightFront Holography
to Baryon Physics
Stanley J. Brodsky, and Guy F. de Téramond
SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA
Universidad de Costa Rica, San José, Costa Rica
1 Introduction
The lightfront wavefunctions (LFWFs) of relativistic bound states in quantum chromodynamics provide a fundamental description of the structure and internal dynamics of hadronic states in terms of their constituent quark and gluons at a fixed lightfront time , the time marked by the front of a light wave, [1, 2] – rather than at instant time , the ordinary time. Unlike instant time quantization, the Hamiltonian equation of motion in the light front (LF) is frame independent. The simple structure of the lightfront vacuum allows an unambiguous definition of the partonic content of a hadron in QCD and of hadronic lightfront wavefunctions which relate its quark and gluon degrees of freedom to their asymptotic hadronic state. For example, the proton’s eigenstate , the lightest , eigenstate of the QCD LF Hamiltonian, can be expanded in terms of the light front Fock components: corresponding to its , etc. projections. Here are the lightfront momentum fractions of the constituents. The plus momentum and transverse momenta are conserved at fixed , and in each parton wavefunction. Remarkably, the are independent of the hadrons 4momentum In lightcone gauge , the gluon quanta have and there are no ghosts. Angular momentum on the LF is simply since there are only independent orbital angular momentum. The angular momentum carried by the gluon in the proton as measured by experiment is simply the mean value of summed over Fock states. The structure functions measured by experiment in deep inelastic proton scattering (DIS) are related to the absolute square of the LFWFs summed over Fock states. The proton’s Dirac and Pauli form factors have an exact representation as the spinconserving and spinflip overlaps of initial and final wavefunctions, respectively. Unlike the ordinary instant form, there are no diagrams where the current couples to vacuum currents, and the boost of the LFWFs is trivial. Given, the LFWFs, one can compute hadronization at the amplitude level from the coalescence of a set of comoving colorsinglet quarks and gluons, [3] in analogy to the formation of moving atoms in quantum electrodynamics. [4]
There are many novel features of baryons which are illuminated using the LFWF representation:

The higher Fock states of baryons contain intrinsic heavy quark [5] such as ; these can arise from gluon splitting, but also from multiconnected amplitudes. The intrinsic heavy quark (IQ) contributions are maximal at minimum offshellness of the LFWF; i.e, when the partons have the same rapidity. This corresponds to , so that the heavy quark have most of the momentum. In a collision the states are materialized and the quarks with the same rapidity such as coalesce to high baryons such as the and the observed at high at the ISR. [6] SELEX [7] has also discovered double charm baryons and at high this way. This suggests using the LHC beam in a fixed target mode to observe very heavy baryons at high such as the . [8] Intrinsic heavy quarks also provide a novel mechanism to produce the Higgs meson and at high . [9, 10]

High hadrons can be created directly from hard subprocesses such as and , rather than from quark of gluon fragmentation. [11] Since the direct hadrons are produced with small transverse size, they are color transparent and can traverse the nuclear medium without absorption. The direct processes have been seen in many experiments. They also account for the remarkable baryon anomaly observed in ionion collisions, [12] as well as the anomalously large powerlaw falloff of the inclusive cross sections for meson and baryon production at fixed and . Since there are no sameside hadrons, the direct processes are energy efficient, requiring the minimum incident parton momentum fractions where the parton distributions are maximal.

The Sivers effect in deep inelastic leptonpolarized proton scattering is most easily computed as an interference of the proton’s and LFWFs. [13] The Sivers spincorrelation is , which reflects the different phases of the and amplitudes due to final state interaction of the scattered quark with the proton’s spectators. The Sivers correlation for each quark is thus seen to be proportional to that quark’s contribution to the proton orbital angular momentum. The Sivers correlation has the opposite sign in DrellYan reactions [14, 15] because it measures initial state scattering of the annihilating quark. There are many other novel factorizationbreaking effects which arise due to initial state or final state scattering of the active quark with the spectators, such as the double BohrMulders correlation which leads to a breakdown of the PQCD LamTung relation in DrellYan reactions. [16]

The quark condensate, normally identified as a vacuum expectation value is actually an “inhadron” condensate associated with the sea quark excitations in the hadron’s higher particle number Fock states. [17, 18] The QCD vacuum is trivial  equal to the vacuum of the free theory in the front form, and there is thus no contribution to the cosmological constant within this framework.
2 AdS/QCD and Light Front Holography
The AdS/CFT correspondence [19] between string states on anti–de Sitter (AdS) spacetime and conformal gauge field theories (CFT) in physical spacetime has brought a new set of tools for studying the dynamics of strongly coupled quantum field theories, and it has led to new analytical insights into the confining dynamics of QCD which is difficult to realize using other methods. Most important, it provides an initial approximation to QCD which is analytically tractable and which can be systematically improved. The original conformal theory can be modified in the far infrared region of AdS space, for example by the introduction of a dilaton background, which yields a confining potential between the colored quarks. The resulting model is usually called AdS/QCD.
One of the most remarkable features of AdS/QCD is the connection between the description of hadronic modes in AdS space and the Hamiltonian formulation of QCD in physical spacetime quantized on the lightfront; i.e., at equal lightfront time . The first step for establishing the correspondence of lightfront QCD in physical 3+1 space with AdS space is to observe that the LF bound state Hamiltonian equation of motion in QCD has an essential dependence in the invariant transverse variable , [20] which measures the separation of the quark and gluonic constituents within the hadron at the same LF time. The variable plays the role of the radial coordinate in atomic systems. The result is a singlevariable lightfront relativistic Schrödinger equation. This first approximation to relativistic QCD boundstate systems is equivalent to the equations of motion that describe the propagation of spin modes in a fixed gravitational background asymptotic to AdS space. [20] The eigenvalues of the LF Schrödinger equation give the hadronic spectrum and its eigenmodes represent the probability amplitudes of the hadronic constituents. By using the correspondence between in the LF theory and in AdS space, one can identify the terms in the dual gravity AdS equations that correspond to the kinetic energy terms of the partons inside a hadron and the interaction terms that build confinement. [20] The identification of orbital angular momentum of the constituents in the lightfront is also a key element in our description of the internal structure of hadrons using holographic principles. This mapping was originally obtained by matching the expression for electromagnetic current matrix elements in AdS space with the corresponding expression for the current matrix element using LF theory in physical space time. [21] More recently we have shown that one obtains the identical holographic mapping using the matrix elements of the energymomentum tensor, [22] thus providing a consistency test and verification of holographic mapping from AdS to physical observables defined on the light front.
3 A Semiclassical LightFront Approximation to Qcd
The eigenmass of hadrons in lightfront theory is determined from the eigenvalue equation
(1) 
where one can expand the initial and final hadronic state in terms of its Fock components. The computation is simplified in the frame where . We find
(2) 
plus similar terms for antiquarks and gluons (. The integrals in (2) are over the internal coordinates of the constituents for each Fock state
(3) 
with phase space normalization .
The LFWF can be expanded in terms of independent position coordinates , , conjugate to the relative coordinates , with . We can also express (2) in terms of the internal impact coordinates with the result
(4) 
The normalization is defined by . To simplify the discussion we will consider a twoparton hadronic bound state. In the limit of zero quark mass
(5) 
The functional dependence for a given Fock state is given in terms of the invariant mass
(6) 
giving the measure of the offenergy shell of the bound state, . Similarly in impact space the relevant variable for a twoparton state is . Thus, to first approximation LF dynamics depend only on the boost invariant variable or , and hadronic properties are encoded in the hadronic mode from the relation
(7) 
thus factoring out the angular dependence and the longitudinal, , and transverse mode with normalization .
We can write the Laplacian operator in (5) in circular cylindrical coordinates and factor out the angular dependence of the modes in terms of the Casimir representation of orbital angular momentum in the transverse plane. Using (7) we find [20]
(8) 
where all the complexity of the interaction terms in the QCD Lagrangian is summed in the effective potential . The LF eigenvalue equation is thus a lightfront wave equation for
(9) 
a relativistic singlevariable LF Schrödinger equation. Its eigenmodes determine the hadronic mass spectrum and represent the probability amplitude to find partons at transverse impact separation , the invariant separation between pointlike constituents within the hadron [21] at equal LF time. Extension of the results to arbitrary follows from the weighted definition of the transverse impact variable of the spectator system [21]: , where is the longitudinal momentum fraction of the active quark. One can also generalize the equations to allow for the kinetic energy of massive quarks using Eqs. (2) or (4). In this case, however, the longitudinal mode does not decouple from the effective LF boundstate equations.
4 A SoftWall AdS/QCD Model for Mesons
The conformal algebraic structure of AdS/CFT can be extended to include a scale . This procedure breaks conformal invariance and provides a solution for the confinement of modes, while maintaining an integrable algebraic structure. It also allows one to determine the stability conditions for the solutions. The resulting model resembles the soft wall model of Ref. [23]. We write the boundstate LF Hamiltonian as a bilinear product of operators plus a constant to be determined:
(10) 
where the LF generator and its adjoint
(11) 
obey the commutation relation
(12) 
For and , the Hamiltonian is positive definite, and . For the Hamiltonian cannot be written as a bilinear product and the Hamiltonian is unbounded from below. The lowest stable solution of the extended LF Hamiltonian corresponds to and and it is massless, . We impose chiral symmetry by choosing and thus identifying the ground state with the pion. With this choice of the constant , the LF Hamiltonian (10) is
(13) 
with eigenfunctions
(14) 
and eigenvalues . This is illustrated in Fig. 1 for the pseudoscalar meson spectra.
The confining model has also an effective classical gravity description corresponding to an AdS geometry modified by a positivesign dilaton background , with sign opposite to that of reference of Ref. [23]. The positive dilaton solution has interesting physical implications, since it leads to a confining potential between heavy quarks [25] and to a convenient framework for describing chiral symmetry breaking. [26] It also leads to the identification of a nonperturbative effective strong coupling and functions which are in agreement with available data and lattice simulations.[27] In the presence of a dilaton profile the wave equation for a spin mode is given by [28]
(15) 
Upon the substitution and , we find the LF wave equation
(16) 
with and . Equation (16) has eigenfunctions given by (14) and eigenvalues . The results for vector mesons is illustrated in Fig. 1, where the spectrum is built by simply adding for a unit change in the radial quantum number, for a change in one unit in the orbital quantum number and for a change of one unit of spin to the ground state value of . Remarkably, the same rule holds for baryons as shown below.
5 Baryons in LightFront Holography
The effective lightfront wave equation which describes baryonic states in holographic QCD is a linear equation determined by the LF transformation properties of spin 1/2 states. We write
(17) 
where is a hermitian operator, , thus . We write as a product , where is the matrix valued (nonhermitian) generator
(18) 
If follows from the square of , , that the matrices and are anticommuting hermitian matrices with unit square. The operator and its adjoint thus satisfy the commutation relation
(19) 
The light front Hamiltonian is
(20) 
The LF equation , has a twocomponent solution
(21) 
where . Thus is the four dimensional chirality operator . In the Weyl representation
(22) 
The effective LF equation for baryons (17) is equivalent to the Dirac equation describing the propagation of spin1/2 hadronic modes, on AdS space
(23) 
where represent the indices of the full space with coordinates and . Upon the transformation , , we recover (17) with and . Higher spin fermionic modes , , with all polarization indices along the 3+1 coordinates follow by shifting dimensions as shown for the case of mesons.
6 A SoftWall LightFront Model for Baryons
An effective LF equation for baryons with a mass gap is constructed by extending the conformal algebraic structure for baryons described above, following the analogy with the mesons. We write the effective LF Dirac equation (17) in terms of the matrixvalued operator and its adjoint
(24) 
with the commutation relation
(25) 
The extended baryonic model also has a geometric interpretation. It corresponds to the Dirac equation in AdS space in presence of a linear potential
(26) 
as can be shown directly by using the transformation , .
As for the case of the mesons Eq. (10) we write the LF Hamiltonian and chose the same value for : , effectively modifying the wave equation (26). With this choice for the LF Hamiltonian is
(27) 
The LF equation , has a twocomponent solution
(28) 
and eigenvalues , identical for both plus and minus eigenfunctions.
The baryon interpolating operator , , is a twist 3, dimension operator with scaling behavior given by its twistdimension . We thus require to match the short distance scaling behavior. Higher spin fermionic modes , , are obtained by shifting dimensions for the fields as in the bosonic case. Thus, as in the meson sector, the increase in the mass squared for higher baryonic states is , and , relative to the lowest ground state, the proton.
The predictions for the positive parity light baryons are shown in Fig. 2. As for the predictions for mesons in Fig. 1, only confirmed PDG [24] states are shown. The Roper state and the are well accounted for in this model as the first and second radial states. Likewise the corresponds to the first radial state of the family. The model is successful in explaining the important parity degeneracy observed in the light baryon spectrum, such as the , pair and the states which are degenerate within error bars. The parity degeneracy of baryons is also a property of the hard wall model, but radial states are not well described by this model. [29] For other recent calculations of the hadronic spectrum based on AdS/QCD, see Refs. [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44].
The proton eigenstate in lightfront holography
(29) 
has and orbital components combined with spin components and respectively. An interesting feature of light front holography for baryons and massless quarks is that the lowest valence Fock states with and have the same probability
(30) 
a manifestation of the chiral invariance of the theory for massless quarks. [45] This implies that the quarks carry zero angular momentum in the proton with and .
There are many other interesting predictions for baryons using AdS/QCD and light front holographic methods such as spacelike and timelike Dirac and Pauli form factors, valence structure functions, etc. The AdS/QCD LF Hamiltonian also creates Fock states with extra quarkantiquark pairs [46] as in QCD(1+1).
7 Conclusions
We have derived a correspondence between a semiclassical first approximation to QCD quantized on the lightfront and hadronic modes propagating on a fixed AdS background. This provides a duality between the bosonic and fermionic wave equations in AdS higher dimensional space and the corresponding LF equations in physical 3 + 1 space. The duality leads to Schrödinger and Diraclike equations for hadronic bound states in physical spacetime when one identifies the AdS fifth dimension coordinate with the LF coordinate . The lightfront equations of motion, which are dual to an effective classical gravity theory, possess remarkable algebraic and integrability properties which follow from the underlying conformal properties of the theory. We also extend the algebraic construction to include a confining potential while preserving the integrability of the mesonic and baryonic boundstate equations.
LightFront Holography is one of the most remarkable features of AdS/CFT. It allows one to project the functional dependence of the wavefunction computed in the AdS fifth dimension to the hadronic frameindependent lightfront wavefunction in physical spacetime. The variable maps to . To confirm this, we have shown that there exists a correspondence between the matrix elements of the energymomentum tensor of the fundamental hadronic constituents in QCD with the transition amplitudes describing the interaction of string modes in antide Sitter space with an external graviton field which propagates in the AdS interior. The agreement of the results for both electromagnetic and gravitational hadronic transition amplitudes provides an important consistency test and verification of holographic mapping from AdS to physical observables defined on the lightfront. As we have discussed, this correspondence is a consequence of the fact that the metric for AdS at fixed lightfront time is invariant under the simultaneous scale change in transverse space and . The transverse coordinate is closely related to the invariant mass squared of the constituents in the LFWF and its offshellness in the lightfront kinetic energy, and it is thus the natural variable to characterize the hadronic wavefunction. In fact is the only variable to appear in the lightfront Schrödinger equations predicted from AdS/QCD. These equations for both meson and baryons give a good representation of the observed hadronic spectrum. The resulting LFWFs also have remarkable phenomenological features, including predictions for the electromagnetic form factors and decay constants. We have also shown that the LF Hamiltonian formulation of quantum field theory provides a natural formalism to compute hadronization at the amplitude level. [3]
The lightfront holographic theory provides successful predictions for the lightquark meson and baryon spectra, as function of hadron spin, quark angular momentum, and radial quantum number. Using the positive dilaton background the pion is massless, corresponding to zero mass quarks, in agreement with chiral invariance arguments. Higher spin lightfront equations can be derived by shifting dimensions in the AdS wave equations. [28] Unlike the topdown string theory approach, one is not limited to hadrons of maximum spin , and one can study baryons with finite color Both the hard and softwall models predict similar multiplicity of states for mesons and baryons as it is observed experimentally. [47] In the hardwall model the dependence has the form: . However, in the softwall model the observed Regge behavior is found: which has the same slope in radial quantum number and orbital angular momentum.
The semiclassical AdS/QCD approximation to lightfront QCD described in this talk breaks down at short distances where hard gluon exchange and quantum corrections become important. However, one can systematically improve the semiclassical approximation by introducing nonzero quark masses and shortrange Coulomb corrections, thus extending the predictions of the model to the dynamics and spectra of heavy and heavylight quark systems. One can also diagonalize the LF Hamiltonian as in DLCQ, but on the orthonormal basis states provided by AdS/QCD. [48] One could also employ LippmannSchwinger perturbation theory, systematically correcting the AdS/QCD eigensolutions.
Acknowledgments
Presented by SJB at the International Conference on the Structure of Baryons
BARYONS’10, December 711, 2010, Osaka, Japan. This research was supported by the Department
of Energy contract DE–AC02–76SF00515. SLACPUB14381
References
 [1] P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949).
 [2] For a review of lightfront quantization, see S. J. Brodsky, H. C. Pauli, S. S. Pinsky, Phys. Rept. 301, 299486 (1998) [hepph/9705477].
 [3] S. J. Brodsky, G. de Teramond, R. Shrock, AIP Conf. Proc. 1056, 314 (2008) [arXiv:0807.2484 [hepph]].
 [4] C. T. Munger, S. J. Brodsky, I. Schmidt, Phys. Rev. D49, 32283235 (1994).
 [5] S. J. Brodsky, P. Hoyer, C. Peterson, N. Sakai, Phys. Lett. B93, 451455 (1980).
 [6] G. Bari, M. Basile, G. Bruni et al., Nuovo Cim. A104, 17871800 (1991).
 [7] M. Mattson et al. [ SELEX Collaboration ], Phys. Rev. Lett. 89, 112001 (2002) [hepex/0208014].
 [8] S. J. Brodsky, .J Fleuret, J. P. Lansberg, C. Lorce (in progress).
 [9] S. J. Brodsky, B. Kopeliovich, I. Schmidt, J.Soffer, Phys. Rev. D73, 113005 (2006) [hepph/0603238].
 [10] S. J. Brodsky, A. S. Goldhaber, B. Z. Kopeliovich,I. Schmidt, Nucl. Phys. B807, 334347 (2009) [arXiv:0707.4658 [hepph]].
 [11] F. Arleo, S. J. Brodsky, D. S. Hwang, A. Sickles, Phys. Rev. Lett. 105, 062002 (2010) [arXiv:0911.4604 [hepph]].
 [12] S. J. Brodsky, A. Sickles, Phys. Lett. B668, 111115 (2008) [arXiv:0804.4608 [hepph]].
 [13] S. J. Brodsky, D. S. Hwang, I. Schmidt, Phys. Lett. B530, 99107 (2002) [hepph/0201296].
 [14] J. C. Collins, Phys. Lett. B536, 4348 (2002) [hepph/0204004].
 [15] S. J. Brodsky, D. S. Hwang, I. Schmidt, Nucl. Phys. B642, 344356 (2002) [hepph/0206259].
 [16] D. Boer, S. J. Brodsky, D. S. Hwang, Phys. Rev. D67, 054003 (2003). [hepph/0211110].
 [17] S. J. Brodsky, R. Shrock, Proc. Nat. Acad. Sci. 108, 4550 (2011) [arXiv:0905.1151 [hepth]].
 [18] S. J. Brodsky, C. D. Roberts, R. Shrock , P. Tandy, Phys. Rev. C82, 022201 (2010) [arXiv:1005.4610 [nuclth]].
 [19] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hepth/9711200].
 [20] G. F. de Teramond and S. J. Brodsky, Phys. Rev. Lett. 102, 081601 (2009) [arXiv:0809.4899 [hepph]].
 [21] S. J. Brodsky and G. F. de Teramond, Phys. Rev. Lett. 96, 201601 (2006) [arXiv:hepph/0602252]; Phys. Rev. D77, 056007 (2008) [arXiv:0707.3859 [hepph]].
 [22] S. J. Brodsky and G. F. de Teramond, Phys. Rev. D 78, 025032 (2008) [arXiv:0804.0452 [hepph]].
 [23] A. Karch, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. D 74, 015005 (2006) [arXiv:hepph/0602229].
 [24] C. Amsler et al. (Particle Data Group), Phys. Lett. B 667 (2008) 1.
 [25] O. Andreev, Phys. Rev. D 73, 107901 (2006) [arXiv:hepth/0603170].
 [26] F. Zuo, Phys. Rev. D 82, 086011 (2010) [arXiv:0909.4240 [hepph]].
 [27] S. J. Brodsky, G. F. de Teramond, A. Deur, Phys. Rev. D81, 096010 (2010) [arXiv:1002.3948 [hepph]].
 [28] G. F. de Teramond, S. J. Brodsky, Nucl. Phys. Proc. Suppl. 199, 8996 (2010) [arXiv:0909.3900 [hepph]].
 [29] G. F. de Teramond and S. J. Brodsky, Phys. Rev. Lett. 94, 201601 (2005) [arXiv:hepth/0501022].
 [30] H. BoschiFilho and N. R. F. Braga, JHEP 0305, 009 (2003) [arXiv:hepth/0212207].
 [31] H. BoschiFilho, N. R. F. Braga and H. L. Carrion, Phys. Rev. D 73, 047901 (2006) [arXiv:hepth/0507063].
 [32] N. Evans and A. Tedder, Phys. Lett. B 642, 546 (2006) [arXiv:hepph/0609112].
 [33] D. K. Hong, T. Inami and H. U. Yee, Phys. Lett. B 646, 165 (2007) [arXiv:hepph/0609270].
 [34] P. Colangelo, F. De Fazio, F. Jugeau and S. Nicotri, Phys. Lett. B 652, 73 (2007) [arXiv:hepph/0703316].
 [35] H. Forkel, Phys. Rev. D 78, 025001 (2008) [arXiv:0711.1179 [hepph]].
 [36] A. Vega and I. Schmidt, Phys. Rev. D 78, 017703 (2008) [arXiv:0806.2267 [hepph]].
 [37] K. Nawa, H. Suganuma and T. Kojo, Mod. Phys. Lett. A 23, 2364 (2008) [arXiv:0806.3040 [hepth]].
 [38] W. de Paula, T. Frederico, H. Forkel and M. Beyer, Phys. Rev. D 79, 075019 (2009) [arXiv:0806.3830 [hepph]].
 [39] P. Colangelo, F. De Fazio, F. Giannuzzi, F. Jugeau and S. Nicotri, Phys. Rev. D 78, 055009 (2008) [arXiv:0807.1054 [hepph]].
 [40] H. Forkel and E. Klempt, Phys. Lett. B 679, 77 (2009) [arXiv:0810.2959 [hepph]].
 [41] H. C. Ahn, D. K. Hong, C. Park and S. Siwach, Phys. Rev. D 80, 054001 (2009) [arXiv:0904.3731 [hepph]].
 [42] Y. Q. Sui, Y. L. Wu, Z. F. Xie and Y. B. Yang, Phys. Rev. D 81, 014024 (2010) [arXiv:0909.3887 [hepph]].
 [43] M. Kirchbach, C. B. Compean, Phys. Rev. D82, 034008 (2010) [arXiv:1003.1747 [hepph]].
 [44] T. Branz, T. Gutsche, V. E. Lyubovitskij et al., Phys. Rev. D82, 074022 (2010) [arXiv:1008.0268 [hepph]].
 [45] S. J. Brodsky, G. F. de Teramond, Acta Phys. Pol. B 41, 2605 (2010) [arXiv:1009.4232 [hepph]].
 [46] G. F. de Teramond, S. J. Brodsky [arXiv:1010.1204 [hepph]].
 [47] E. Klempt and A. Zaitsev, Phys. Rept. 454, 1 (2007) [arXiv:0708.4016 [hepph]].
 [48] J. P. Vary, H. Honkanen, J. Li et al., Phys. Rev. C81, 035205 (2010) [arXiv:0905.1411 [nuclth]].