The small axial charge of the N(1535) resonance

There is a natural cancellation between the contributions of the $qqq$ and $qqqq\bar q$ components to the axial charge of the N(1535) resonance. While the probability of the former is larger than that of the latter, its coefficient in the axial charge expression is exceptionally small. The magnitude of two of the corresponding coefficients of the $qqqq\bar q$ components are in contrast large and have the opposite sign. This result provides a phenomenological illustration of the recent unquenched lattice calculation result that the axial charge of the N(1535) resonance is very small, if not vanishing \cite{takah}. The result sets an upper limit on the magnitude of the probability of $qqqq\bar q$ components as well.

A number of phenomenological failures of the constituent quark model for the baryons may be repaired by extending the model space beyond that of the basic three quark configurations qqq [2,3,4]. The question of key interest is then that of the relative magnitude of the sea-quark configurations, and in particular of the most obvious qqqqq configurations. For most electromagnetic and strong decay observables, this is difficult to estimate, because of the very strong contribution from the transition matrix elements between the qqq and qqqqq components [5]. The axial current operator of the baryon resonances is an exception, as for this the transition matrix elements are suppressed -i.e. they involve the small components of the spinors -with respect to the diagonal matrix elements, so that the axial charges, to a good approximation, may be expressed as a sum of the diagonal matrix elements of all possible configurations, which takes the form of numerical coefficients A n times the corresponding probabilities P n : The (diagonal) axial charges of baryon resonances are however not accessible experimentally. It is in this regard that the recent result, obtained numerically by an unquenched QCD lattice calculation, that the axial charge of the N(1535) actually may vanish in the two-flavor case, is so interesting [1]. As that result appears to be insensitive to the quark mass (the magnitude of the value extrapolated to 0 is less than 0.2), it may be taken as a substitute  qqqqq configurations in the N(1535) that have an appropriate symmetry structure and spin and isospin 1/2. These are listed in Table I.
The numbering of these configurations are in order of increasing energy, if the hyperfine interaction between the quarks is assumed to depend either on flavor and spin or on color and spin. In the table the matrix elements of the schematic hyperfine splitting operator  (1) and (2) in Table I have to contain a strange quark-antiquark pair. This is as expected on the basis of the observed large Nη decay branch of the N(1535).
isospinor and the spinor of the antiquark respectively, and ϕ [5] represents the completely symmetrical orbital wave function. The first summation involves The symbols C With the results in Table I Here P 3 is the probability for the conventional qqq configuration, while P (i) 5 represents the probabilities of the qqqqq configurations in Table I. Note that the energetically most favorable qqqqq configuration (1) does not contribute to the axial charge at all.
The fact that the two qqqqq contributions in (4), which are positive, have large coefficients ∼ 1, while the coefficient of the qqq contribution is small and negative (−1/9) immediately suggests the possibility for a considerable cancellation between the qqq valence and the qqqqq sea-quark contributions, as the probability of the latter is likely to be considerably smaller than that of the former. If only the first two terms in the expression (4) are taken into account g A (N(1535)) would vanish if P (2) 5 = 2/15P qqq , which may be a fairly reasonable assumption. The last two remaining qqqqq configurations are in expected to have very small probability, as they are energetically unfavorable (Table I).
In ref. [8] it was in fact found that the quark model prediction for the helicity amplitude  Table I are similar in that both involve an ss pair, but the latter is energetically disfavored by the matrix elements of the hyperfine interaction (2), the helicity amplitude should be similar if the probability P for the configuration (2) in Table I (2) and (3) in Table I (N(1535)) falls in the range -0.02 to -0.05. This shows that the likely range of values for g A (N(1535)) in the extended quark model, which includes explicit qqqqq components -0.05 .. +0.06, brackets 0. This range would bracket 0 also in the case where the relative qqq probability where increased to P 3 = 0.7 and P (2) 5 = 0.3. It does in any case not appear possible to reach the value 0 for g A (N(1535)), with an overall qqqqq probability that is larger than 0.45.
The conclusion is therefore that the very small or possibly vanishing axial charge of the N(1535) already at the present level of accuracy constrains the magnitude of the probability of the sea-quark components in the N(1535) to be less than 45%. A more general observation is that the axial charges of the baryon resonances may be useful for setting limits on the probabilities of their sea-quark configurations.