Structure Functions from Chiral Soliton Models 1

We study nucleon structure functions within the bosonized Nambu{ Jona{Lasinio (NJL) model where the nucleon emerges as a chiral soliton. We discuss the model predictions on the Gottfried sum rule for electron{nucleon scattering. A comparison with a low{scale parametrization shows that the model reproduces the gross features of the empirical structure functions. We also compute the leading twist contributions of the polarized structure functions g1 and g2 in this model. We compare the model predictions on these structure functions with data from the E143 experiment by GLAP evolving them from the scale characteristic for the NJL-model to the scale of the data. The purpose of this investigation is to provide a link between two successful although seemingly unrelated pictures of baryons. On one side we have the quark parton model which successfully describes the scaling behavior of the structure functions in deep inelastic scattering (DIS) processes. The deviations from these scaling laws are computable in the framework of perturbative QCD. On the other side we have the chiral soliton approach which is motivated by the large NC expansion of QCD, NC being the number of color degrees of freedom. For NC !1, QCD is known to be equivalent to an e ective theory of weakly interacting mesons. Although this theory is not explicitly known it can be modeled by assuming that at low energies only the light mesons (pions, kaons, , !) are relevant. When modeling the meson theory one requires the symmetry structure of QCD. In particular besides Pioncar e invariance we require chiral symmetry and its spontaneous breaking. Baryons emerge as non{perturbative (topological) con gurations of the meson elds, the so{called solitons. The link between these two pictures can be established by computing structure functions 1) Talk presented by HW at the 6 Conf. on Intersections between Nuclear and Particle Physics, Big Sky, Mt, May 27th{June 2, 1997. Work supported in part by the Deutsche Forschungsgemeinschaft (DFG) under contract Re 856/2{3 and the US{DOE grant DE{FE{ 02{95ER40923. within a chiral soliton model for the nucleon from the hadronic tensor W ab (q) = 1 4 Z d e hN(P )j h J ( ); J by (0) i jN(P )i ; (1) which describes the strong interaction part of the DIS cross{section. In eq (1) jN(P )i refers to the nucleon state with momentum P and J ( ) to the hadronic current suitable for the process under consideration. In most soliton models { due to the non{perturbative nature of the soliton con guration { the current commutator (1) remains intractable. However, the Nambu and Jona{Lasinio (NJL) model [1] of quark avor dynamics, which can be bosonized by functional integral techniques [2], contains simple current operators. Most importantly, the bosonized version of the NJL{model contains soliton solutions [3]. This paves the way to compute structure functions in the soliton approach. In order to extract the leading twist contributions to the structure function one computes the hadronic tensor in the Bjorken limit q0 = jqj MNx with jqj ! 1 and x = q =2P q xed : (2) Here we con ne ourselves to presenting the key issues of the calculation, details may be traced from refs. [4{6]. THE NUCLEON FROM THE CHIRAL SOLITON IN THE NJL MODEL In this section we brie y summarize the basic features of the chiral soliton in the NJL{model and discuss how states with nucleon quantum numbers are generated. For more details see refs. [3,7] and quotations therein. The NJL{model Lagrangian contains a quartic quark interaction which is chirally symmetric. Derivatives of the quarks elds only appear in form of a free Dirac Lagrangian, hence the current operator is formally free. Upon bosonization the action may be expressed as [2] A = Trln (i@= mU ) + m0m 4G tr U + U y 2 (3) where we have con ned ourselves to the interaction in the pseudoscalar channel. The associated pion elds are contained in the non{linear realization U = exp(i =f ). In eq (3) tr denotes discrete avor trace while Tr also includes the functional trace. The parameters of the model are the coupling constant G, the current quark massm0 and the UV cut{o . The constituent quark massm arises as the solution to the Schwinger{Dyson (gap) equation and characterizes the spontaneous breaking of chiral symmetry. A Bethe{Salpeter equation of the pion eld can be derived from eq (3) which allows one to express the pion mass m = 135MeV and decay constant f = 93MeV in terms of the model parameters. Fixing these quantities leaves one parameter undetermined which maybe expressed in terms of the constituent quark mass m. Subsequently an energy functional for non{perturbative but static eld con gurations U(r) can be extracted from (3). It can be expressed as a regularized sum of single quark energies . For the hedgehog ansatz, UH = exp(i r̂ (r)) the assoicated one{particle Dirac Hamiltonian becomes h = p m exp (i 5 r̂ (r)) ; h = : (4) The distinct level (v), which is bound in the background of UH , is referred to as the valence quark state. Its explicit occupation guarantees unit baryon number. The chiral angle (r) of the soliton is determined by self{consistently minimizing the energy functional. This soliton con guration does not yet carry nucleon quantum numbers. To generate them the (unknown) time dependent eld con guration is approximated by elevating the zero modes to time dependent collective coordinates U(r; t) = A(t)UH(r)A (t); A(t) 2 SU(2): Upon canonical quantization the angular velocities, = 2itr( A _ A), are replaced by the spin operator J via = J= 2 with 2 being the moment of inertia while the nucleon states jNi emerge as Wigner D{functions. To compute nucleon properties the action (3) is expanded in powers of corresponding to an expansion in 1=NC. In particular the valence quark wave{function v(x) acquires a linear correction v(x; t) = e i vA(t) 8 < : v(x) + X 6=v (x) h j jvi 2( v ) 9= ; = e i vA(t) v(x): (5) Here v(x) refers to the spatial part of the body{ xed valence quark wave{ function with the rotational corrections included. STRUCTURE FUNCTIONS IN THE VALENCE QUARK APPROXIMATION The starting point for computing the unpolarized structure functions is the symmetric part of hadronic tensor in a form suitable for localized elds [9], W lm f g(q) = Z dk (2 ) S k sgn (k0) k Z +1 1 dt e00 Z dx1 Z dx2 exp [ i(k + q) (x1 x2)] hN j n ̂ (x1; t)tltm ̂(x2; 0) ̂ (x2; 0)tmtl ̂(x1; t) o jNi: (6) Note that the quark spinors are functionals of the soliton. Here S = g g + g g g g and = 1(2) for the structure functions associated 2) Generalizing this treatment to avor SU(3) indeed shows that the baryons have to be quantized as half{integer objects. For a review on solitons in SU(3) see e.g. [8]. 0.0 0.5 1.0 1.5 x 0.0 1.0 2.0 3.0 Isoscalar Structure Functions