# Laboratory for Hadronic and Nuclear Physics

Baryons in the NJL model as solutions of the relativistic Faddeev equation.
N.Ishii(Univ. of Tokyo), W.Bentz(Univ. of Tokyo), K.Yazaki(Univ. of Tokyo)
Nuclear Physics A 587 (1995) p617.
The relativistic Faddeev equation for three quarks is solved in the SU(2)fXSU(3)cNJL model to get the nucleon and delta states. We truncate the interacting two-body channels to the scalar and axial vector diquark channels, which are expected to dominant from the non-relativistic analogy. We find that both channels contribute attractively to the nucleon as well as the delta state, and that the principal mechanism for the mass splitting between the nucleon and the delta in this picture is the interaction in the scalar diquark channel, which is not present in the delta state. We show the dependences of the masses of the nucleon and the delta on the specific form of the interaction lagrangian, and derive restrictions on the possible forms of four-fermi interactions.

Nucleon quark distributions in a covariant quark-diquark model.
I.C.Cloet (JLab), W.Bentz (Tokai Univ.), and A.W.Thomas (JLab)
Physics Letters B621 (2005), p246.
Spin-dependent and spin-independent quark light-cone momentum distributions and structure functions are calculated for the nucleon. We utilize a modified Nambu-Jona-Lasinio model in which confinement is simulated by eliminating the unphysical thresholds for nucleon decay into quarks. The nucleon bound state is obtained by solving the Faddeev equation in the quark-diquark approximation, where both scalar and axial vector diquark channels are included. We find excellent agreement between our model results and empirical data.

EMC and polarized EMC effects in nuclei.
I.C.Cloet(JLab), W.Bentz(Tokai Univ.), and A.W.Thomas(JLab)
Physics Letters B 642 (2006) p210.
We determine nuclear structure functions and quark distributions for 7Li, 11B, 15N and 27Al. For the nucleon bound state we solve the covariant quark-diquark equations in a confining Nambu--Jona-Lasinio model, which yields excellent results for the free nucleon structure functions. The nucleus is described using a relativistic shell model, including mean scalar and vector fields that couple to the quarks in the nucleon. The nuclear structure functions are then obtained as a convolution of the structure function of the bound nucleon with the light-cone nucleon distributions. We find that we are readily able to reproduce the EMC effect in finite nuclei and confirm earlier nuclear matter studies that found a large polarized EMC effect.

Medium modifications of nucleon electromagnetic form factors.
T.Horikawa and W.Bentz
Nuclear Physics A762 (2005) p102.
We use the Nambu-Jona-Lasinio model as an effective quark theory to investigate the medium modifications of the nucleon electromagnetic form factors. By using the equation of state of nuclear matter derived in this model, we discuss the results based on the naive quark-scalar diquark picture, the effects of finite diquark size, and the meson cloud around the constituent quarks. We apply this description to the longitudinal response function for quasielastic electron scattering. RPA corrections, based on the nucleon-nucleon interaction derived in the same model, are also taken in to account in the calculation of the response function.

The Stability of Nuclear Matter in the Nambu-Jona-Lasinio Model.
Nuclear Physics A 696 (2001) p 138.
Using the Nambu-Jona-Lasinio model to describe the nucleon as a quark-diquark state, we discuss the stability of nuclear matter in a hybrid model for the ground state at finite nucleon density. It is shown that a simple extension of the model to simulate the effects of confinement leads to a scalar polarizability of the nucleon. This, in turn, leads to a less attractive effective interaction between the nucleons, helping to achieve saturation of the nuclear matter ground state. It is also pointed out that that the same effect naturally leads to a suppression of "Z-graph" contributions with increasing scalar potential.

Phase Transition from nuclear matter to color superconducting quark matter
W.Bentz(Tokai Univ.), T.Horikawa(Tokai Univ.), N.Ishii(RIKEN), A.W.Thomas(Univ. of Adelaide)
We construct the nuclear and quark matter equations of state at zero temperature in an effective quark theory (the Nambu-Jona-Lasinio model), and discuss the phase transition between them. The nuclear matter equation of state is based on the quark-diquark description of the single nucleon, while the quark matter equation of state includes the effects of scalar diquark condensation (color superconductivity). The effect of diquark condensation on the phase transition is discussed in detail.

The phases of isospin asymmetric matter in the two flavor NJL model.
S.Lawley, W.Bentz and A.W.Thomas
Phys.Lett.B632(2006)495.
We investigate the phase diagram of isospin asymmetric matter at T=0 in the two flavor Nambu-Jona-Lasinio model. Our approach describes the single nucleon as a confined quark-diquark state, the saturation properties of nuclear matter at normal densities, and the phase transition to normal or superconducting quark matter at higher densities. The resulting equation of state of charge neutral matter and the structure of compact stars are discussed.

Nucleons, nuclear matter and quark matter: A unified NJL approach
S.Lawley, W.Bentz and A.W.Thomas
Journal of Physics. G32 (2006) 667
We use an effective quark model to describe both hadronic matter and deconfined quark matter. By calculating the equations of state and the corresponding neutron star properties, we show that the internal properties of the nucleon have important implications for the properties of these systems.

The NJL-jet model for quark fragmentation functions.
T.Ito, W.Bentz (Tokai U., Hiratsuka), I.C.Cloet(Washington U., Seattle), A.W.Thomas(Jefferson Lab & William-Mary Coll.), K.Yazaki (Nishina Ctr., RIKEN & Wako, RIKEN)
Physical Review D 80 (2009) 074008.
A description of fragmentation functions which satisfy the momentum and isospin sum rules is presented in an effective quark theory. Concentrating on the pion fragmentation function, we first explain why the elementary (lowest order) fragmentation process q --> q \pi is completely inadequate to describe the empirical data, although the 'crossed' process \pi --> q \bar{q} describes the quark distribution functions in the pion reasonably well. Taking into account cascade-like processes in a generalized jet-model approach, we then show that the momentum and isospin sum rules can be satisfied naturally, without the introduction of ad hoc parameters. We present results for the Nambu--Jona-Lasinio (NJL) model in the invariant mass regularization scheme and compare them with the empirical parametrizations. We argue that the NJL-jet model, developed herein, provides a useful framework with which to calculate the fragmentation functions in an effective chiral quark theory.

Rotational states in deformed nuclei: An analytic approach.
W.Bentz(Tokai U.,Hiratsuka), A.Arima(Musashi Inst. Tech.), J.Enders(Darmstadt, Tech. Hochsch.), A.Richter(Darmstadt, Tech. Hochsch. & ECT, Trento), J.Wambach(Darmstadt, Tech. Hochsch.)
Physical Review C 84 (2011) 014327.
The consequences of the spontaneous breaking of rotational symmetry are investigated in a field theory model for deformed nuclei, based on simple separable interactions. The crucial role of the Ward-Takahashi identities in describing the rotational states is emphasized. We show explicitly how the rotor picture emerges from the isoscalar Goldstone modes and how the two-rotor model emerges from the isovector scissors modes. As an application of the formalism, we discuss the M1 sum rules in deformed nuclei and make the connection to empirical information.

Dressed quark mass dependence of pion and kaon form factors.
Y. Ninomiya (Tokai U., Hiratsuka), W. Bentz (Tokai U., Hiratsuka), I.C. Cloët (Argonne, HEP).
Phys.Rev. C91 (2015) 2, 025202.
The structure of hadrons is described well by the Nambu–Jona-Lasinio (NJL) model, which is a chiral effective quark theory of QCD. In this work we explore the electromagnetic structure of the pion and kaon using the three-flavor NJL model in the proper-time regularization scheme, including effects of the pion cloud at the quark level. In the calculation there is only one free parameter, which we take as the dressed light quark (u and d) mass. In the regime where the dressed light quark mass is approximately 0.25 GeV, we find that the calculated values of the kaon decay constant, current quark masses, and quark condensates are consistent with experiment and QCD based analyses. We also investigate the dressed light quark mass dependence of the pion and kaon electromagnetic form factors, where comparison with empirical data and QCD predictions also favors a dressed light quark mass near 0.25 GeV.

Quark-Jet model for transverse momentum dependent fragmentation functions.
W. Bentz (Tokai U., Hiratsuka & Nishina Ctr., RIKEN), A. Kotzinian (Yerevan Phys. Inst. & INFN, Turin), H.H. Matevosyan (Adelaide U.), Y. Ninomiya (Tokai U., Hiratsuka), A.W. Thomas (Adelaide U.), K. Yazaki (Nishina Ctr., RIKEN).
Phys.Rev. D94 (2016) 3, 034004.
In order to describe the hadronization of polarized quarks, we discuss an extension of the quark-jet model to transverse momentum dependent fragmentation functions. The description is based on a product ansatz, where each factor in the product represents one of the transverse momentum de- pendent splitting functions, which can be calculated by using effective quark theories. The resulting integral equations and sum rules are discussed in detail for the case of inclusive pion production. In particular, we demonstrate that the 3-dimensional momentum sum rules are satisfied naturally in this transverse momentum dependent quark-jet model. Our results are well suited for numerical calculations in effective quark theories, and can be implemented in Monte-Carlo simulations of polarized quark hadronization processes.

Transverse-momentum-dependent quark distribution functions of spin-one targets: Formalism and covariant calculations.
Yu.Ninomiya(Univ. of Tokyo), W.Bentz(Univ. of Tokyo), Ian C. Cloët (Argonne, PHY).
Phys.Rev. C96 (2017) no.4, 045206.
We present a covariant formulation and model calculations of transverse momentum-dependent quark distribution functions (TMDs) for spin-one hadrons. Emphasis is placed on a description of these 3-dimensional distribution functions which is independent of any constraints on the spin quantization axis. We apply our covariant spin description to all nine leading-twist time-reversal even ρ meson TMDs in the framework provided by the Nambu--Jona-Lasinio model, incorporating important aspects of quark confinement via the infrared cut-off in the proper-time regularization scheme. In particular, the behavior of the 3-dimensional TMDs in a tensor polarized spin-one hadron are illustrated. Sum rules and positivity constraints are discussed in detail. Results of particular interest include the finding that the tensor polarized TMDs -- associated with spin-one hadrons -- are very sensitive to quark orbital angular momentum, and that the TMDs associated with the quark operator γ+γTγ5 would vanish were it not for dynamical chiral symmetry breaking. In addition, we find that 44% of the ρ meson's spin is carried by the orbital angular momentum of the quarks, and that the magnitude of the tensor polarized quark distribution function is about 30% of the unpolarized quark distribution. A qualitative comparison between our results for the tensor structure of a quark-antiquark bound state is made to previous experimental and theoretical results for the two-nucleon (deuteron) bound state.

Charge Symmetry Breaking Effects in Pion and Kaon Structure.
Parada T.P. Hutauruk (APCTP, Pohang), Wolfgang Bentz (Nishina Ctr., RIKEN & Tokai U., Hiratsuka), Ian C. Cloët (Argonne, PHY), Anthony W. Thomas (U. Adelaide, Dept. Phys. Math. Phys.).
Phys.Rev. C97 (2018) no.5, 055210.
Charge symmetry breaking (CSB) effects associated with the u and d quark mass difference are investigated in the quark distribution functions and spacelike electromagnetic form factors of the pion and kaon. We use a confining version of the Nambu–Jona-Lasinio model, where CSB effects at the infrared scale associated with the model are driven by the dressed u and d quark mass ratio, which because of dynamical chiral symmetry breaking is much closer to unity than the associated current quark mass ratio. The pion and kaon are given as bound states of a dressed quark and a dressed antiquark governed by the Bethe-Salpeter equation, and exhibit the properties of Goldstone bosons, with a pion mass difference given by mπ+2−mπ02∝(mu−md)2 as demanded by dynamical chiral symmetry breaking. We find significant CSB effects for realistic current quark mass ratios (mu/md∼0.5) in the quark flavor-sector electromagnetic form factors of both the pion and kaon. For example, the difference between the u and d quark contributions to the π+ electromagnetic form factors is about 8% at a momentum transfer of Q2≃10GeV2, while the analogous effect for the light quark sector form factors in the K+ and K0 is about twice as large. For the parton distribution functions we find CSB effects which are considerably smaller than those found in the electromagnetic form factors.

Tokai University, Department of Physics, School of Sience
Professor Wolfgang Bentz