Decay properties of Roper resonance in the holographic QCD

Let us start by briefly summarizing how the baryon states are obtained in the Sakai-Sugimoto model by collectively quantizing the instanton solution. The action of hadron effective theory is composed of the Yang-Mills term $S_{YM}$ and the Chern-Simons term $S_{CS}$,

where

In these equations $N_{c}$ is the number of colors, $\lambda$ the ’t Hooft coupling, and the indices $\mu,\nu=0,1,2,3$ are for the 4-dimensional space-time. The curvatures along the extra dimension $z$ are defined by

The 1-form $\mathcal{A}$ expresses $\mathcal{A}=A_{\alpha}dx^{\alpha}+\hat{A}_{\alpha}dx^{\alpha}$ which consists of the flavor SU(2) part $A_{\alpha}$ and the U(1) part $\hat{A}_{\alpha}$ with $\alpha=0,1,2,3,z$. The Chern-Simons 5-form is given by

In general, it is difficult to analytically solve the equations of motion in the presence of the curvatures $h(z)$ and $k(z)$. However, it can be simplified in the large $\lambda$ limit since the instanton profile is localized around $z\sim 0$ as proportional to $\lambda^{-1/2}$, where we can set $h(z)=k(z)=1$. Therefore, the following instanton solution is available, with $M=1,2,3,z$,

where

with $\left({\bm{X}},Z\right)$ and $\rho$ the location and size of the instanton, respectively. The profile function $f\left(\xi\right)$ is given by

The classical instanton solution needs to be quantized for the physical nucleon and Roper resonances. This can be done by the collective coordinate method, where the relevant time dependent dynamical variables are, ${\bm{X}},Z$, $\rho$ and the rotational variable in the isospin and spin space. As shown in Ref. Hata et al. (2007) the time dependent gauge field is given by

where ${\bm{a}}=a_{4}+ia_{a}\tau^{a}$ is for the spin and isospin rotation. By using this gauge field, we find the collective Hamiltonian as

where $M_{0}=8\pi^{2}\kappa$ is the classical soliton mass Hata et al. (2007), and $y_{I}$ is related to the orientation coordinates by $y_{I}=\rho a_{I}$. The baryon states are labeled by its momentum $\vec{p}$ and quantum numbers $(l,I_{3},s_{3},n_{\rho},n_{z})$, where $l/2$ is the equal isospin and spin values; $I_{3},s_{3}$ are the third components of the isospin and spin; and $n_{\rho},n_{z}$ are the quanta for oscillations along the radial and $z$-directions. For the spin up proton ($I_{3}=1/2,\ s_{3}=1/2$) with a finite momentum $\vec{p}$, the wave functions of ground and Roper resonance are given as Hata et al. (2007); Hashimoto et al. (2008)

where

We note that the wave function for the $z$ oscillation is the lowest ($n_{z}=0$) for both the nucleon and Roper resonance. Thus the only difference between them is in the radial part, $R_{N}(\rho)$ and $R_{N^{*}}(\rho)$.

The BPST instanton that we have summarized in the previous subsection is only an approximate solution in the large $\lambda$ limit where the instanton size is small. This can be used for the computation of baryon masses. However, for the computation of currents which are defined at $|z|\to\infty$ such a solution is not suited. As shown in Ref. Hashimoto et al. (2008) we need to find the solution that is properly extended to the large $|z|$ region to obtain the well-defined currents. In this paper, we simply summarize the final result of such a solution;

where the index $i$ runs $1-3$. In these equations, $G$ and $H$ are given by

The function $\psi_{n}\left(z\right)$ are the solutions of the eigenvalue equation

with the eigenvalue $\lambda_{n}$ Sakai and Sugimoto (2005a), and

Now we are ready to calculate the axial current. Following Ref. Hashimoto et al. (2008), the chiral current is derived from the coupling with the external gauge field $\delta\mathcal{A}_{\alpha}$ which is defined by

They are related to the left and right gauge fields in the four dimensional space at $z\to\pm\infty$,

Substituting this field into the action, the coefficients of the first order in $\mathcal{A}_{L\mu},\ \mathcal{A}_{R\mu}$ is identified with the left and right currents $\mathcal{J}^{\mu}_{L},\ \mathcal{J}^{\mu}_{R}$ with the sign properly taken into account,

where

The vector and axial currents are then obtained by

with $\psi_{0}\left(z\right)=\left(2/\pi\right)\arctan{z}$.

When the instanton oscillates along the $z$ direction in a narrow range in the large $\lambda$ limit, the metrices are approximated as $h\left(Z\right)\simeq k\left(Z\right)\simeq 1$. Then, substituting (13) for (20) gives the following form ($r\equiv|\vec{x}-\vec{X}|$)

where

To go further, it is convenient to present the Fourier transform in the momentum space, (in what follows the dependence on the collective coordinates $\vec{X},Z,\rho,\vec{a}$ are suppressed)

We obtain the following form:

This current is regarded as an operator in terms of the dynamical variable $\vec{X}$, $Z$, $\rho$ and $\vec{a}$, which is used when taking the matrix elements by the corresponding wave functions.

The axial coupling $g^{NN^{*}}_{A}$ for the transition $N^{*}\left(1440\right)\rightarrow N+\pi$ is defined as follows:

The factor 2/3 on the right hand side is needed in the chiral limit Adkins et al. (1983). Using (26) and (9), we obtain

where $\braket{\partial_{Z}\psi_{2n}\left(Z\right)}$ stands for the expectation value using the wave functions of $Z$. The matrix element of $\rho^{2}$ can be computed and the result is

with $\rho_{cl}$ being the classical instanton size given by

We note that the transition matrix element for $N^{*}\left(1440\right)\rightarrow N+\pi$ is related to the nucleon matrix element, an interesting feature of the present model associated with the collective nature of baryons. The axial coupling constant is then defined at $\vec{q}=0$, $g^{NN^{*}}_{A}=g^{NN^{*}}_{A}(\vec{0})$. Using the relation

$g^{NN^{*}}_{a}$ can be expressed in a compact form:

In the above equations, $\langle\cdots\rangle$ stands for the expectation value using the wave functions of $Z$.

There are two parameters of this model, $M_{KK}$ and $\kappa$. Following Adkins et al Adkins et al. (1983) they are determined to reproduce the mass splitting of the nucleon and delta, and the pion decay constant $f_{\pi}=64.5$ MeV,

Then, the prediction of the present model for $g^{NN^{*}}_{A}$ is

The decay width of $N^{*}\left(1440\right)\rightarrow N+\pi$ can be computed by the formula

where the amplitude $t_{fi}$ is given by the Lagrangian

as follows

Here we have expressed the effective $\pi NN^{*}$ coupling $g_{\pi NN^{*}}$ in terms of the axial coupling by using the Goldberger-Treiman relation,

Hence we obtain

Using $M_{N}=940\ {\rm MeV},\ M_{N^{*}}=1370\ {\rm MeV},\ q=342\ {\rm MeV},$ we find

In this computation the value of $g^{NN^{*}}_{A}$ at $\vec{q}=0$ is used. By considering the form factor effect, the $g^{NN^{*}}_{A}$ value at $\vec{q}=342$ MeV becomes about 15 % smaller, and hence $\Gamma_{N^{*}\left(1440\right)\rightarrow N+\pi}\sim 55$ MeV.

These values are smaller than the experimental value (29). This is because the axial coupling $g^{NN^{*}}_{A}$ is small, which is a common feature of the solitonic picture of baryons. In fact, the nucleon $g_{A}^{NN}$ is computed in a similar manner as for $g_{A}^{NN^{*}}$ by using the nucleon wave function $R_{N}(\rho)$. The result is

Though small, it is interesting to observe that the value is somewhat larger than that of Adkins et al. (1983). One possible resolution to recover the experimental value $g_{A}^{NN}=1.25$ is to take into account $1/N_{c}$ corrections (Ref. Hosaka and Toki (1996) and references there). Here, however, we do not discuss this anymore. On the other hand, it is interesting to observe the relation between the axial couplings of the nucleon and that of the Roper-nucleon transition. Inspection of Eq. (32), we find

We emphasize that this relation does not include any model parameters (except for $N_{c}=3$), and so a model independent relation. Experimentally, if we use the partial decay width $\Gamma_{N^{*}\to\pi N}\sim 110$ MeV, we find the ratio

which agrees well with the present model prediction within $\sim$ 20 % accuracy.